HIGH RESOLUTION TERTIARY STRUCTURE OF THE MEMBRANE-ASSOCIATED HIV FUSION PEPTIDES BY SOLID STATE NUCLEAR MAGNETIC RESONANCE By Scott Schmick A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY CHEMISTRY 2012 ABSTRACT HIGH-RESOLUTION TERTIARY STRUCTURE OF THE MEMBRANE-ASSOCIATED HIV FUSION PEPTIDES BY SOLID STATE NUCLEAR MAGNETIC RESONANCE By Scott Schmick HIV gp41 protein catalyzes fusion between viral and host cell membranes, and its apolar N-terminal region or “fusion peptide” binds to host cell membranes and plays a key role in viral and host cell membrane fusion. Gp41 fusion can be dominantly inhibited by dilute amounts of V2E mutant gp41, but a structural basis for this inhibition has not been demonstrated. “HFP” is a construct containing the fusion peptide sequence that induces membrane vesicle fusion, and V2E mutant HFP (V2E-HFP) has reduced membrane vesicle fusion rates. Earlier solid-state NMR (SSNMR) studies showed that when HFP or V2E-HFP are associated with membranes with ~30 mol% cholesterol (mHFP or mV2E-HFP), the apolar N-terminal regions of these constructs have predominant β strand secondary structure. In mHFP, a fraction of the strands form antiparallel β sheet structure with residue 161/116 or 171/117 registries of adjacent strands (i.e t = 16 and t = 17 registries). Other SSNMR and infrared studies have been interpreted to support a large fraction of approximately in-register parallel registry of adjacent strands. However, the samples had many isotopic labels and other structural models were also consistent with the data. The tertiary structure of mHFP was studied using SSNMR with the rotational-echo double resonance (REDOR) pulse sequence to measure a sample’s average couplings. Experimental data were collected for samples with sparser 13 13 CO and 15 CO- N dipolar 15 N labeling and were compared to simulated NMR data. The in-register parallel β sheet fraction was ≤ 0.15, and a much greater fraction of antiparallel registries were identified. The accuracy of the quantitative measurements was enhanced by inclusion of “long range” natural abundance contributions in the data analysis, and the validity of this approach was supported by a negative control sample. Furthermore, mHFP samples were prepared with a single the closest 13 13 CO and a single 15 N label for which 15 CO- N interstrand proximity resulted from a distinct registry. These experimental data were compared to simulated data that incorporated fractional populations, ft, of 17 different 2 registries. These ft, were globally fit using a χ metric which identified a broad distribution of antiparallel β sheet registries (11 < t < 21). Sequential hydrophobic residues in HFP result in intrastrand hydrophobic patches and interstrand overlap of these patches result in interstrand hydrophobic regions. These regions may insert into the vesicle membranes, and a hydrophobicity or insertion energy metric, Gtmin , was developed to quantify each registry’s insertion energy. In general, registries present in our NMR samples had a negative Gtmin while registries that were not present generally had a positive Gtmin . A similar set of experiments were run with mV2E-HFP, and mV2E-HFP had a narrower distribution of registries where the t = 20 registry was significantly more populated in mV2E-HFP than in mHFP. The hydrophobic residues of HFP are located within the first 12 N-terminal amino acids, and the t = 12 registry was more populated in mHFP than mV2E-HFP. The t = 12 registry localizes hydrophobic residues which may result in deeper membrane insertion and increased vesicle fusion rates compared to the t = 20 registry. The t = 20 registry delocalizes the interstrand proximity of N-terminal hydrophobic residues which may result in shallower membrane insertion and reduced membrane fusion rates. These results provide a new, experimentally-based structural model for transdominant inhibition where co-mixing of V2E mutant gp41 and wild type gp41 may energetically favor a non-native registry distribution shifted toward longer registries for the FP region of wild type gp41. TABLE OF CONTENTS TABLE OF CONTENTS ...........................................................................................................iv LIST OF TABLES ................................................................................................................... vii LIST OF FIGURES ................................................................................................................ viii LIST OF ABBREVIATIONS................................................................................................ xviii Chapter I. Introduction ................................................................................................................1 Chapter II. Materials and Methods ............................................................................................14 2.1 HFP Synthesis and Sample Preparation ...........................................................................14 1. General HFP Preparation ...............................................................................................14 2. HFP Solutions ...............................................................................................................17 3. Trimeric HFP (HFPtr) Synthesis ....................................................................................18 2.2 General NMR Background ..............................................................................................22 1. NMR Spectroscopy........................................................................................................22 2. Rotating Frame and Bloch Equations .............................................................................24 2.3 Experimental Setup .........................................................................................................26 1. General Concepts for Setting the Pulses.........................................................................26 2. Setup Compounds..........................................................................................................32 3. Setting the Magic Angle ................................................................................................33 4. Setting Proton π/2 Pulse................................................................................................35 5. Proton Decoupling Field ................................................................................................38 6. Proton-Carbon Cross Polarization..................................................................................41 7. Carbonyl π Pulse ...........................................................................................................42 8. Nitrogen π Pulse ............................................................................................................43 9. Proton TPPM Decoupling to Nitrogen Rabi Frequency Ratio in HCN REDOR .............44 10. Chemical Shift Referencing .........................................................................................46 2.4 MAS Solid-State NMR (REDOR) ...................................................................................46 2.5 SIMPSON Simulations....................................................................................................52 Chapter III. Natural Abundance Calculations for Solid State NMR REDOR Experiments and Quantitative Determination of In-Register Parallel β Sheet Registries in Membrane-Associated HFP ..........................................................................................................................................58 3.1 Background. ....................................................................................................................58 3.2 Materials and Methods. ...................................................................................................63 1. SSNMR samples............................................................................................................63 2. Modeling. ......................................................................................................................65 3.3 Results. ...........................................................................................................................67 1. Qualitative Analysis of the REDOR Data. .....................................................................67 2. Natural Abundance Models............................................................................................70 3. Quantitative Analysis of Registry Populations – Fully Constrained Model.....................77 4. Quantitative Analysis of Registry Populations – Unconstrained Model. .........................85 iv 3.4 Discussion .......................................................................................................................87 CHAPTER IV. Quantitative Identification of the Antiparallel Sheet Registry Distribution in Membrane Associated HFP and V2E-HFP Samples ..................................................................94 4.1 Background. ....................................................................................................................94 4.2 Materials and Methods ....................................................................................................99 4.3 Results. .........................................................................................................................105 1. Fully Constrained Model mHFP Registry Distribution.................................................105 2. Fully Constrained Model mV2E-HFP Registry Distribution ........................................117 4.4 Discussion .....................................................................................................................125 1. Modeled Membrane Insertion Depth mHFP.................................................................125 2. Relevance of Broad Distribution ..................................................................................130 Chapter V. Dissertation Summary and Future Work ................................................................135 5.1 Summary.......................................................................................................................135 5.2 Membrane Location ......................................................................................................139 5.3 Resin Bound Structure...................................................................................................145 APPENDICES ........................................................................................................................149 Appendix I. Files Checklist .....................................................................................................150 Appendix II. Current HIV Inhibitor Drugs...............................................................................155 Appendix III. Simple Number of Strands for Fusion Model.....................................................156 Appendix IV. RP-HPLC Purification, Optimization and Troubleshooting ...............................160 1. Specific Problems............................................................................................................166 1.1 Well separated peak contains “impurities” by MALDI-TOF analysis .........................166 1.2 Forgot to deprotect Fmoc group. ................................................................................166 1.3 Column pressure is increasing over time. ...................................................................167 Appendix V. HFPdm Data and Lyophilized HFP ....................................................................168 Appendix VI. SIMMOL, SIMPSON, and Fortran Files. ..........................................................172 1. Sample SIMMOL Files ...................................................................................................172 13 1.1 SIMMOL input file for Leu-132 CO from the 2IWW.pdb file. ...............................172 13 1.2. SIMMOL output file for the Leu-132 residue CO from the 2IWW.pdb file............172 2. Sample SIMPSON Files ..................................................................................................173 13 2.1. Input file 5 spin (NNCNN) SIMPSON File for Ala-12 CO from 2WII.pdb ............173 2.2 Output file 5 spin (NNCNN) SIMPSON File for Ala-12 2.3 Input file 3 spin (CNN) SIMPSON File for Ala-12 13 13 CO from 2WII.pdb ..........177 CO from 2WII.pdb ..................177 13 2.4 Output file 3 spin (CNN) SIMPSON File for Ala-12 CO from 2WII.pdb................181 2.5 Input file 3 spin (CNC) SIMPSON File from 2WII.pdb .............................................181 2.6 Output file 3 spin (CNC) SIMPSON File from 2WII.pdb...........................................185 2.7 Input file 2 spin (CN) SIMPSON File from 2WII.pdb................................................185 v 2.8 Output file 2 spin (CN) SIMPSON File from 2WII.pdb .............................................189 2 3. Sample Fortran Input Script Files For Global χ Fittings .................................................189 3.1 HFP 3 Registry Fitting qsub Script, “x2_fixed”..........................................................189 3.2 HFP 3 Registry Fitting Main Script, “HFP.f” .............................................................190 3.3 HFP 5 Registry Fitting qsub Script, “x2_HFP” ..........................................................207 3.4 HFP 5 Registry Fitting Main Script, “HFP_5var.f” ....................................................208 3.5 V2E-HFP 3 Registry Fitting qsub Script, “x2_V2E” ..................................................226 3.6 V2E-HFP 3 Registry Fitting Main Script, “V2E.f”.....................................................226 3.7 V2E-HFP 5 Registry Fitting qsub Script, “x2_V2E” ..................................................244 3.8 V2E-HFP 5 Registry Fitting Main Script, “V2E_5var.f” ............................................245 exp lab lab Appendix VII. Chapter III Table of (S/S0) values and 1tuv ( ) or 1t1t2uv ( ) spin geometries with calculated values from SIMPSON. ................................................................264 Appendix VIII. Five Registry Fittings .....................................................................................272 Appendix IX. Chapter IV Unconstrained Fitting......................................................................273 Appendix X. Freed Mutations .................................................................................................276 Appendix XI. Raw Data for mHFP and mV2E-HFP................................................................277 Appendix XII. Boltzmann Fraction Populations. .....................................................................280 Appendix XIII. L9R Mutant Discussion ..................................................................................281 min i , and n Values. ..............................................288 Appendix XIV. HFP, V2E, and L9R Gt t t Appendix XV. Summary of Hessa Biological Hydrophobicity Scale .......................................289 lab lab Appendix XVI. Chapter IV Tables for 1tuv ( ) or 1t1t2uv ( ) spin geometries with calculated values from SIMPSON for the three registry fittings...............................................296 lab Appendix XVII. Chapter IV Tables for the unique 1tuv ( ) spin geometries with calculated values from SIMPSON for the five registry fittings. ................................................................300 REFERENCES .......................................................................................................................301 vi LIST OF TABLES Table 1. HFP construct labeling schemes. .................................................................................17 Table 2. Oligomeric HFP synthesis summary. ...........................................................................20 Table 3. Error analysis for V2E-F8CG13N................................................................................52 Table 4. Error analysis for HFP-L12CA6N. ..............................................................................52 Table 5. Chapter III indices and parameters...............................................................................73 Table 6. Chapter IV indices and parameter ..............................................................................100 Table 7. Three registry fittings for mHFP and mV2E-HFP. .....................................................106 Table 8. Five registry fittings for mHFP and mV2E-HFP. .......................................................107 Table 9. HFP-F8CG13N data sets used in global fittings .........................................................108 Table 10. mHFP and mV2E-HFP 13 CO D avg ..................................................................129 h,it , nt Table 11. Gtmin values for HFP constructs. ..........................................................................139 21 Table 12. Freed fusion activity .............................................................................................158 Table 13. Strands Model 1.......................................................................................................158 Table 14. Strands Model 2.......................................................................................................159 Table 15. HFPdm and L12CA6Nmn Lyophilized (ΔS/S0) Table 16. Chapter III (S/S0) exp exp and σ exp ....................................171 and rms error. .......................................................................264 Table 17. Unconstrained model mHFP. ...................................................................................274 Table 18. mHFP Δ(S/S0) exp .....................................................................................................277 Table 19. mV2E-HFP Δ(S/S0) exp . ...........................................................................................278 Table 20. mHFP fully constrained model and Boltzmann distribution based ft.........................280 Table 21. Energy minimized membrane insertion energy parameters.......................................288 vii LIST OF FIGURES Figure 1. HIV infection model (left) and freeze fracture electron microscopy (right) of (a) binding (b) hemi-fused viral and host cell membranes (c, d) pore formation with HIV infection of 8 the host cell (modified from literature ).......................................................................................2 Figure 2. A summary of the gp41 sequence and regions defined from literature where FP = fusion peptide, FPPR = fusion peptide proximal region, NTH = N-terminal helix, CTH = Cterminal helix, MPER = membrane proximal region, and TMR = transmembrane region. This dissertation primarily focuses on the role of FP in membrane fusion. The ~16 residue FP sequence followed by C-terminal amino acids (HFP) was studied in this dissertation. This figure 15 was adapted from literature . For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation....................................5 Figure 3. Proposed fusion mechanism (Figure provided by Dr. Kelly Sackett). This model for membrane fusion suggests that sequential events occur during viral and host cell membrane fusion: (A  B) The ectodomain of gp41 is fusion inactive prior to contact with the host cell membrane. Contact between the host cell and viral membrane is initiated when gp120 binds to 12 CD4 and a co-receptor . This induces a conformational change in gp41 to the proposed prehairpin intermediate structure and FP binds to the host cell membrane. To my knowledge, there is currently no direct evidence that FP binds to the host cell, but synthetic fusion peptides bind 25,26 and insert into model membranes . Additionally, no structural data have been collected to elucidate the proposed PHI structure; (B  C) The host cell and viral membrane undergo HemiFusion; (C  D) Prior to completion of membrane fusion, a fusion pore is formed within the 26 host cell membrane. HFP constructs have been shown to insert into model membranes which appears to be driven by local hydrophobicity of HFP secondary and tertiary structures (See Chapter IV). Similarly, FP may insert into host cell membranes which may enable FP to be involved in pore formation. The membrane fusion pore is enlarged after or during the formation 27 of hairpin structure . Additionally, peptides with the CTH sequence inhibit membrane fusion by a proposed mechanism where these peptides bind to the NTHs in the Pre-Hairpin Intermediate structure (B and C) which prevents the formation of the 6 helix bundle structure (D). This lead to 22 development of the membrane fusion inhibitor drug Enfuvirtide . .............................................6 Figure 4. (a) Electron micrograph of HIV and a T cell where the width of the HIV membrane is ~100 nm. The increased electron density between the HIV and host cell membranes suggests that multiple proteins from the HIV membrane are in contact with the T cell membrane. (b) Model of HIV contact with a T cell based upon electron micrographs where multiple proteins are believed 46 to be between the HIV and T cell membranes. This figure was modified from literature . .........9 Figure 5. The t = 16, t = 17, and in-register parallel β sheet registries are illustrated. Isotopic labeling schemes to detect these registries are illustrated where red underlined atoms indicate viii 13 CO labeled residues and blue underlined atoms indicate labeling schemes are effective because they have 13 15 N labeled residues. These specific 15 CO- N proximities of ~4 Å......................12 Figure 6. Purification of HFPtr synthesis. The product of each peak was identified by MALDITOF mass spectrometry, Figure 7, and each peak corresponded to the following: (1) HFPWK5CA; (2) HFPdm(Lys) and HFPdm(Cys); (3) HFPdm(Lys) and HFPdm(Cys); (4) HFPtr; and (5) HFPte. Additionally, peaks (4) and (5) were not present in HFPdm(Cys) which further supports that these peaks result from HFPtr and/or HFPte. Peak (5) was not used for any experiments in this dissertation while Peak (4) was collected for the one HFPtr experiment in Chapter IV. The identity of HFPte would have to be confirmed before meaningful experimental data is collected for this peptide. ...............................................................................................20 Figure 7. MALDI-TOF mass spectroscopy from the HFPtr purification displayed in Figure 6. ..21 Figure 8. A dipole moment in the laboratory frame (x, y, and z axes) where the B0 external magnetic field is aligned along the z-axis. A new axis, x’, rotates at the Larmor frequency only if w0 = w1. ....................................................................................................................................26 13 Figure 9. FID’s for cross polarization acquisitions of the CO labeled i4 peptide where the red horizontal line signifies the zero value or baseline for the y-axis. (a) The FID decays to the baseline. (b) The FID decays to a positive y-value as evident by the zero point value of the FID (~2.5 ms) being above the red line.............................................................................................31 Figure 10. A pulse of a single radiofrequency is illustrated in the time domain (a,c) and the frequency domain (b,d). The relationship between pulse length (τp), radiative field (B1), and the energy of the B1 field are illustrated. The excitation frequency is centered about the transmitter frequency, w1. The figure was made using concepts from the literature where the frequency domain is represented as the Fourier transform of the time domain where the Fourier transform 68 of a step function can be represented by a sinc function (b), (d) ..............................................32 Figure 11. The chemical structure of adamantane. .....................................................................33 Figure 12. Magic angle spectrum FID of KBr with 64 acquisitions and a pd = 0.10 s. The 79 transmitter was moved to the Br resonant frequency of KBr, and an exponential decay is observed in the free induction decay (FID). ...............................................................................35 Figure 13. Proton pulse length arrayed using the I4 peptide, a pd = 1.0 sec and 10 acquisitions per spectrum. The H rf ampl parameter was set to 0.2300 and the H 90 pulse parameter was arrayed from 1.0 to 15.0 µs by 1.0 µs increments. Maximum signal was observed at ~5 µs which approximately corresponds to a π/2 pulse. Zero signal should be observed between 10.0 and 11.0 μs which would correspond to a π pulse. The change in signal intensity is the greatest at points 1 surrounding the π pulse so it is more accurate to set the π/2 H pulse length by identifying the π pulse length and dividing pulse length time by 2. To do this for the above data, the H 90 pulse parameter could be arrayed from 10.0 to 11.0 µs by increments of 0.1 µs and increase the number ix of scans per spectrum to enhance the signal to noise ratio. Alternatively, a better approach would be to set the H 90 pulse parameter to 10.0 µs and array the H rf ampl parameter. Upon determining which H rf ampl that yields zero signal, change the 10.0 μs to 5.0 μs to set the π/2 pulse. While the proton amplifier output voltage is approximately linear with respect to the H rf ampl parameter, it is not exactly linear so you need to divide the pulse length by 2 and not the amplifier input parameter. .........................................................................................................38 74 Figure 14. Cross polarization arrayed with the I4 peptide using 5 acquisitions, 1.0 s pulse delay (pd), MAS frequency = 10 kHz, and contact time = 2 ms. Cross polarization is included in the 1 REDOR experiments to transfer magnetization from the highly abundant and polarizable H 13 13 nuclei to the more dilute CO nuclei to increase the CO signal to noise ratio by increasing the signal per acquisition. Additionally, the longitudinal relaxation rate is approximately 4 times 1 13 faster for H than for C nuclei in organic solids which allows for acquisition of ~4 times as many FIDs using cross polarization than for waiting for 13 75 C repolarization . The cp_ramp pulse 1 program was used, and the X cp ampl change parameter was set to 0.04. A 48 kHz H cross polarization and π/2 pulse was used, and the aXcp X cp ampl start parameter was arrayed from 0.00-0.70 by 0.05 increments. Maximum signal intensities were observed between 0.25-0.30 and 0.35-0.40. Either of these regions could be used for CP since the signal intensities are nearly equivalent. Additionally, the contact time also should be arrayed to set up an optimal CP, and it is important to use a setup compound with similar nuclear magnetic relaxation rates. Typical 13 optimal contact times for peptides in REDOR experiments are 1.5-2.0 ms for CO nuclei. 13 Finally, array the X cp ampl change (i.e. the C ramp parameter) parameter to obtain the maximum signal where 0.04 is a typical value, but this value may vary depending upon the amplifier being used. .................................................................................................................41 Figure 15. The cp_zfilter pulse sequence (CP – π/2 – τz – π – acquisition) was used to set the 13 CO π pulse with 10 acquisitions and a 1.0 sec pulse delay using the i4 peptide and a MAS = 10 13 kHz. When the pulse lengths are set correctly, precession of the CO magnetization can be followed using the Bloch Equation, Eq (9) where (1) equilibrium, M = +z; (2) CP, rotates to the xy plane; (3) π/2, rotates to the –z axis; (4) τz - remains along –z axis; (4) π, flips back to +z axis; (5) acquisition with consequent zero signal detection in the rotating frame when pulses are set accurately..................................................................................................................................42 Figure 16. REDOR spectra (32.2 ms dephasing time) of the I4 peptide where each set of black 13 and red spectra have a corresponding CO peak in the S0 and S1 spectra for each arrayed points, respectively. Each S0 and S1 spectrum was the sum of 120 acquisitions with pd = 2.0 s. The aY180 Y 180 ampl parameter was arrayed from 0.07 to 0.21 by 0.01 increments and the pw180Y Y 180 parameter was set to 20.0 sec (wR = 25 kHz as determined by Eq (13)-(16)). Maximum dephasing was observed with the pw 180Y Y 180 ampl = 0.13 where ΔS/S0 = 0.82 where 15 maximum dephasing corresponds to a N  pulse. For further understanding of the REDOR experiment, see 2.4 MAS Solid-State NMR (REDOR). .............................................................43 x 1 15 Figure 17. The ΔS/S0 of the i4 peptide is plotted against the H decoupling wR to N wR ratio 15 for τ = 16.2, 24.2, 32.2, 40.2, and 48.2 ms dephasing times where the N π pulse was 25 μsec (wR = 20 kHz), the 13 C π pulse was 11 μsec (wR = 45 kHz), the pd = 2.0 s, and 350 scans were acquired for S0 and S1 of each data point. These HCN REDOR experiments appear to require a 1 15 H decoupling wR to be at least 3.5 times greater than the N wR to obtain maximum ΔS/S0. 13 1 1 13 The CO nuclei were decoupled from H nuclei at a H decoupling to C wR ratios approximately ≥ 1.5. This is evident since equivalent error bars were obtained for data within a 1 13 dephasing time period for H decoupling fields of ≥ 60 kHz. The CO atoms are not directly 1 1 13 bonded to H atoms which results in weaker H- CO heteronuclear dipolar couplings (~3.8 kHz which was determined using Eq. (19) for a rHC of 2.0 Å between the carbonyl carbon and the 1 15 adjacent residue’s amide proton (See 1K09.pdb) relative to amide heteronuclear H- N dipolar couplings (11.648 kHz corresponds to a rHN = 1.015 Å) which is approximately the width of the dipolar powder pattern for amide 1 15 is unclear whether the H decoupling to 1 76 N in proteins . Based upon the current set of experiments, it 15 N ratio is a causation or correlation relationship. There 15 1 may be a H- N dipolar interaction that results in lower S/S0 values where H decoupling 1 15 fields of >80 kHz may be required to average out effects due to H- N dipolar couplings. ......44 Figure 18. Adamantane 13 C spectrum prior to chemical shift referencing. The transmitter was set 13 13 near the CO Larmor frequencies to increase CO signal intensity in REDOR experiments as described in Figure 10. The chemical shift for the left adamantane peak is 40.5 ppm downfield from the tetramethyl silane (TMS) internal standard reference, but is observed at -113.7 ppm in this figure. Therefore, 154.2 ppm should be added to the chemical shift to correctly reference chemical shifts. Chemical shift referencing is important since referenced chemical shifts of 77 carbon nuclei in peptides provide information about local secondary structure .......................46 13 15 Figure 19. (a) HFPs where red and blue correspond to CO and N labeled residues, respectively. (b) HFP-NC, HFP-P, HFP-A, and HFP-AP were SSNMR samples which each 13 15 contained a mixture of CO and N labeled peptides in 1:2 mol ratio. The HFP-NC sample was a mixture of HFP-F8 and HFP-A6L7 that had been lyophilized separately. The other samples 13 were membrane-associated HFPs that formed  sheet structure with a molecular mixture of CO and 15 N labeled peptides in the sample. (c) Registries probed by the SSNMR REDOR 13 15 experiments and labeled CO/labeled N proximities for the membrane-associated HFPs in these registries. Consideration of residue 116 or 117 registries is based on the fully extended conformation in HFP. For parallel sheets, there is CO (residue h) – HN (residue h+1) hydrogen bonding of adjacent strands. ......................................................................................................61 13 Figure 20. REDOR S0 and S1 C SSNMR spectra at 32.2 ms dephasing time for (a) HFP-NC, (b) HFP-P, (c) HFP-A, or (d) HFP-AP. Each spectrum was processed with 200 Hz line broadening and baseline correction and was the sum of: (a) 38624; (b) 23488; (c) 24914; or (d) xi 13 14240 scans. Relatively narrow CO signals were observed in the HFP-P, HFP-A, and HFP-AP samples because the HFPs were membrane-associated with predominant  sheet conformation at 13 13 the labeled CO site. A broader CO signal was observed in the HFP-NC sample because there was no membrane and there were populations of lyophilized HFP with either  helical or β sheet 13 conformation at the labeled CO site. ......................................................................................68 exp sim Figure 21. (a) Plot of REDOR (ΔS/S0) (filled squares with error bars) and (ΔS/S0) (open sim circles) vs dephasing time for the lyophilized HFP-NC sample. The (ΔS/S0) were calculated using a mixture of nad models with fractional populations:  helical, 0.5; min  sheet, 0.25; max exp  sheet, 0.25. (b) Plots of (ΔS/S0) vs dephasing time for: HFP-NC, open triangles; HFP-P, filled triangles; HFP-A, open circles; HFP-AP, filled circles. The typical  exp is 0.02. exp Variation of 0.02 in (ΔS/S0) was also observed between two different preparations of the same sample type, e.g. HFP-A...................................................................................................69 Figure 22. (a, b) Schematic diagrams of the HFP-F8 region of the HFP-NC sample in antiparallel 13  sheet structure with labeled COs represented as red circles. Panel a shows a model that is fully constrained to a single registry while panel b shows multiple registries. (c, d) β sheet 13 backbone representations of the respective boxed regions of panels a and b with labeled COs in red and possible na 13 15 N sites in blue, i.e. sites for which a na CO. A particular spin geometry will have only one 15 c and each spin geometry will have either one labeled labeled 13 COs and one na 15 15 N is within 7 Å of a labeled N. The min nad model is shown in panel 13 CO and one na 15 N (#1, 2 or 3) or two N (#4, 5, or 6). The max nad model is shown in panel d and each spin geometry will have one labeled 13 CO and one na 15 N. ......................................................72 Figure 23. Schematics of three adjacent HFPs for HFP-A, i.e. u = 2, in (a-d) fully constrained or .................................................................................................................................................80 2 Figure 24. Contour plots of  vs fa parallel and fa antiparallel fractional populations for (a) fully constrained and (b) unconstrained models. In each plot, fa is the sum of populations of 117/117 and 217/116 parallel registries and fb is the sum of populations of 161/116 and 171/117 antiparallel registries. ...............................................................84 Figure 25. Pictorial model of HFP (red lines) binding to membranes followed by antiparallel  sheet formation and membrane insertion and then fusion. Time increases from left-to-right. For reasons of clarity, some lipids are not shown in the right-most picture. Although there are no data yet on fusion peptide structure during HIV/host cell fusion, the antiparallel  sheet structure of the right-most picture is plausible because: (1) the structure is consistent with multiple trimers at the fusion site; and (2) the structure is membrane-inserted with deeper insertion positively correlated with increased membrane perturbation and vesicle fusion rate. .................................90 xii Figure 26. A summary of the gp41 sequence and regions defined from literature where FP = fusion peptide, FPPR = fusion peptide proximal region, NTH = N-terminal helix, CTH = Cterminal helix, MPER = membrane proximal region, and TMR = transmembrane region. This 15 figure was adapted from literature ..........................................................................................96 13 Figure 27. REDOR S0 and S1 C SSNMR spectra at 48.2 ms dephasing time for (a) mHFPA6CG3N, (b) HFP-L12CG5N, (c) HFP-F8CL12N, or (d) HFP-L9CG16N. Each spectrum was processed with 200 Hz line broadening and baseline correction and was the sum of: (a) 23766; exp (b) 21454; (c) 40331; or (d) 39133 scans. (e) (ΔS/S0) (τ = 48.2 ms) for all samples, u. (f) Plots exp of (ΔS/S0) vs dephasing time with the rms error. Isotopic labeling of each mHFP is displayed in the legend that correspond to HFP-A6CG3N (black, square), HFP-L12CG5N (red, circle), HFP-F8CL12N (cyan, triangle), and (d) HFP-L9CG16N (orange, inverted triangle). Variation exp less than 0.02 in (ΔS/S0) was also observed between two different preparations of the same sample type, e.g. HFP-F8CG13N (not displayed here).............................................................109 Figure 28. Sample indices, u, with the corresponding labeling schemes are displayed along with the registries, t, that result in labeled rCN of ~ 4.1 Å and ~ 5.5 Å that respectively correspond to 13 15 hydrogen-bonded and non-hydrogen bonded CO – H N. Membrane inserted regions are highlighted in yellow, and the corresponding n and i values are listed. ....................................115 min Figure 29. Double-y plot where ft populations (black) and ΔGt (red) are plotted for each registry for mHFP (a) and mV2E-HFP (b)...............................................................................116 Figure 30. (a) The (ΔS/S0) exp (τ = 48.2 ms) for mHFP (black) and mV2E-HFP (green). (b) The ft for mHFP (black) and mV2E-HFP (green) for the fully constrained model using the 3 registry fitting method. Both (a) and (b) demonstrate that mV2E-HFP has a smaller population of shorter registries ( t < 16) and that mV2E-HFP has a larger population of longer registries (t > 17).....120 13 Figure 31. REDOR S0 and S1 C SSNMR spectra at 48.2 ms dephasing time for (a) HFPF8CA21N, (b) HFP-F8CG13N, (c) HFPtr-F8CG13N, (d) V2E-F8CG13N, (f) HFP-L12CA6N . (g) V2E-L12CA6N, (h) HFP-L9CG5N, or (i) V2E-L9CG5N. Each spectrum was processed with 200 Hz line broadening and baseline correction and was the sum of: (a) 46816; (b) 36665; (c) exp 19372; (d) 34271; (f) 44931; (g) 46231; (h) 40272; or (i) 46809 scans. (e), (j) Plots of (ΔS/S0) vs dephasing time with the rms error. Isotopic labeling of each mHFP is displayed in the legend that correspond to HFP-F8CA21N, (black, square), HFP-F8CG13N (orange, inverted triangle), HFPtr-F8CG13N (cyan, triangle), (d) V2E-F8CG13N (red, circle), (f) HFP-L12CA6N (dark yellow, square), (g) V2E-L12CA6N (purple, circle), (h) HFP-L9CG5N (green, triangle), or (i) exp V2E-L9CG5N (wine, inverted triangle). Variation less than 0.02 in (ΔS/S0) was also observed between two different preparations of the same sample type, e.g. HFP-F8CG13N. ...124 Figure 32. Membrane insertion depth model as described in the text. Previous work has demonstrated that the Ala-1 carbonyl carbon is ~5 Å from the lipid phosphorus, and this phosphate region is referred to as the water/bilayer interface. ..................................................126 xiii 13 avg , h,it , nt is plotted for HFP (red) and V2E-HFP (blue). .........................................................................128 Figure 33. The calculated average membrane insertion depth of each residue’s Figure 34. Calculated membrane insertion depth of the Ala-6 13 CO, D CO for HFP (red) and V2E-HFP (blue) is plotted for each registry. Additionally, ft for mHFP (black) and mV2E-HFP (green) are plotted for each registry. The mV2E-HFP registry populations are shifted toward longer registries (t > 17) relative to mHFP and the calculated membrane insertion depth of the Ala-6 13 13 CO is  0 for these registries. This is consistent with previous work where the Ala-6 CO is 26 inserted deeper in mHFP relative to mV2E-HFP . .................................................................130 Figure 35. The membrane insertion energies were derived from the Hessa biological hydrophobicity scale for the HFP, V2E-HFP and L9R-HFP by the methods described in Chapter IV. The L9R-HFP has predominantly positive Gtmin whereas both HFP and V2E-HFP have many registries with negative Gtmin which suggests that the distribution of registries should be different between constructs. Additionally, it is not obvious that mL9R-HFP should form membrane inserted β sheets since t < 12 registries were minimally populated in mHFP, and t > 12 registries have positive Gtmin in mL9R-HFP. Unlike mHFP and mV2E-HFP, mL9R does not predominantly form membrane inserted sheets, Appendix XIII. ........................................138 Figure 36. Cholesterol molecules with carbon atoms numbered (a) Cholesterol (b) Cholesterol2,2,3,4,4,6-d6 and (c) Cholesterol-25,26,26,26,27,27,27-d7. ....................................................144 Figure 37. V2E-L9CI4N resin bound (prior to cleavage) with a MAS speed of 10 kHz. ..........147 Figure 38. V2E-L9CI4N resin bound (prior to cleavage) with a MAS speed of 6 kHz. ............148 Figure 39. Chart of commercially available anti HIV drugs. This chart was last updated 12\14\2010 and was taken from www.aidsmeds.com...............................................................155 Figure 40. (a) HFP purification with a “large” C18 column (10-15 µm pore size). (b) HFP purification with a “small” C4 column (10 x 250 mm and 5 µm pore size). Better peak resolution was obtained with the C4 column. (c) Typical MALDI-TOF mass spectroscopy of peak 2 HFP from a purification similar to (b) where the expected mass was 3151 +2 g/mol where the +2 13 15 refers to the mass gain from the C and N isotopes.............................................................161 Figure 41. Preliminary gradient of 15% to 80% solvent B over 40 minutes. In developing purification protocols, small amounts of crude peptide were used to make product peaks narrow. By mass spec, Peak 2 is the confirmed product peak. From Eq (62), the variables have the following values: Ci = 15%; Gs = 1.625 %/min; te ~ 26.5 min; and Pe = 45%. ........................162 xiv Figure 42. A linear 39-48% gradient was run over 18.5 minutes for purifying the product peak. From Eq (62), the variables have the following values: Ci = 39%; Gs = 0.5 %/min; te ~ 15.5 min; and Pe = 46.5%. The initial starting concentration was chosen to make the elution time around 15 minutes which was calculated by Eq (62) and a more gradual gradient was used to better separate peaks 1 and 2 from Figure 41. Also, at the end of the program, the gradient was ramped up to 80% solvent B over 0.5 minutes and the flow rate was increased to 9 mL/min to clean the column after each run. After 5 minutes, the gradient concentration of solvent B was returned to 39% over 0.5 minutes and the column equilibrated at this concentration for 3 minutes to prepare for the next run........................................................................................................................163 Figure 43. The program from Figure 42 was used, but a higher loading volume of the crude peptide was used which resulted in poor separation of our product peak..................................164 Figure 44. A linear 37-47% gradient was run over 20 minutes. The peaks were separated better with minimal peak broadening. ...............................................................................................164 Figure 45. Nonlinear gradients can be used to separate peak 1 from peak 2. The gradient broadened peak 1 using a more gradual slope initially while the gradient was steeper from 15 to 21 minutes to retain the sharpness of peak 2. To optimize the time of the program, it’s best to have your product elute during the period where the ramp is up to 75% solvent B since nothing is achieved during this time in Figure 41-Figure 44. Recall, the elution time of a peak is 8 minutes. Therefore, in this figure, peak 2 was collected during 75% use of solvent B, but peak 2 actually began coming off the column at time te-8 or ~16-18 minutes...................................................165 Figure 46. The ramp was modified to separate peak 2 from peak 3. This program was created because peak 1 also contains peptide with our products molecular weight which was collected for potential future use. Peak 3 should also be collected if the product peak is low relative to other syntheses. Peak 3 can contain HFP with N-terminal or sidechain protecting groups.................165 Figure 47. MALDI-TOF mass spectroscopy of purified HFP-L9G10. In MALDI-TOF experiments, increasing the laser power can increase the signal to noise, but it can also lead to peptide fragmentation where fragmentation can occur C-terminal of amino acids with basic 65,66 sidechain groups . Alternatively, gas phase degredation of the peptide may be unlikely. The peptide degredation may result from hydrolytic cleavage in the matrix or possibly during 64 isolation . Fragmentation of the HFP-L9G19 peptide appeared to occur C-terminal of the Arg22, Lys-29, and Lys-30 where the respective fragments detected were likely AVGIGALFLGFLGAAGSTMGAR (2038 +2 g/mol), AVGIGALFLGFLGAAGSTMGARSWKKKKK (2952 +2 g/mol), and AVGIGALFLGFLGAAGSTMGARSWKKKKKK (3080 +2 g/mol), and the HFP product had an expected mass of 3151 + 2 g/mol. .......................................................................................166 Figure 48. REDOR data for HFP (black boxes) and HFPdm (red circles) are displayed with error bars that are associated with rms deviation and labeling corresponds to (a) L9CG5N and (b) L12CG5N. ..............................................................................................................................169 Figure 49. Chapter III spin geometries and simulated data.......................................................265 xv Figure 50. Flow chart for unconstrained iterative fitting. Each iteration is denoted by the variable 2 κ, and the χ u calculations are found in Chapter IV, Eq (50)....................................................275 min Figure 51. The ΔGt are plotted for registries t = 8-24 for HFP, V2E-HFP, F11G-HFP, and min F11V-HFP. For each registry, the F11G-HFP ΔGt are greater than or equal to the HFP ΔGt min which may contribute toward F11G-HFP’s lower fusion activity. The F11V-HFP ΔGt is approximately equal to the HFP ΔGt min min relative to mHFP for each registry. ........................276 Figure 52. The membrane insertion energies were derived from the Hessa biological hydrophobicity scale for the HFP and L9R-HFP by the methods described in Chapter IV. The L9R-HFP has predominantly positive G min whereas HFP has many registries with negative nt G min which suggests that the distribution of registries should be different between constructs. nt Additionally, it is not obvious that mL9R-HFP should form membrane inserted β sheets since t < 12 registries were minimally populated in mHFP, and t > 12 registries have positive G min in nt mL9R-HFP. ............................................................................................................................284 Figure 53. NMR sample of mV2E-HFP (left) compared to a water standard (right) prior to centrifugation and after mixing overnight. Aggregation of LUV’s is evident in mV2E-HFP under our sample preparation conditions. ..........................................................................................285 Figure 54. NMR samples of mL9R-HFP (L9R), mHFP (WT), mV2E-HFP (V2E) prior to centrifugation and after mixing overnight. The mL9R-HFP sample appeared to be more transparent than both mHFP and mV2E-HFP, but LUV aggregation was evident. ...................286 Figure 55. mL9R-HFP with F8CG13N labeling. The chemical shift of 174.6 ppm and 7.5 ppm line full-width at half maximum height indicate the presence of a distribution of secondary structures since the peak spans chemical shifts of α helical, random coil and β sheet structures. ...............................................................................................................................................287 Figure 56. The model systems were composed of two transmembrane domains (TM1 and TM2), two luminal domains (P1 and P2), and two glycosylation acceptor sites (G1 and G2). A third helical transmembrane domain (H) is illustrated in red. Translocation of the H segment from the membrane allows for glycosylation of both G1 and G2 while membrane insertion of the H 93 segment only allows for glycosylation of G1. This figured was modified from literature ......290 93 Figure 57. The Hessa biological hydrophobicity scale. This figure was taken from literature . ...............................................................................................................................................291 Figure 58. The positional dependence of amino acids. Key points from these figured are 93 discussed below. These figures were taken from literature . ..................................................292 xvi Figure 60. Additional spin geometries and simulated data for Chapter IV five registry fittings. ...............................................................................................................................................300 xvii LIST OF ABBREVIATIONS a: t = 16 or 17 antiparallel registry AIDS: Acquired Immune Deficiency Syndrome CHOL: Cholesterol CTH: C-terminal helix CO: carbonyl CP: Cross polarization DIEA: N, N-diisopropylethylamine DMAP: dimethylaminopyridine DMPC: 1, 2-dimyristoyl-sn-glycerol-3-phosphocholine DTPC: 1, 2-di-O-tetradecyl-sn-glycerol-3phosphocholine DTPG: 1, 2-di-O-tetradecyl-sn-glycerol-3-[phosphor-rac-(1-glycerol)] FID: free induction decay Fmoc: 9-fluorenylmethoxycarbonyl FT: Fourier Transform FTIR: Fourier Transform Infrared FP: Fusion Peptide HBTU: O-benzotriazole-N,N,N’,N’-tetramethyl-uronium-hexafluoro-phosphate HEPES: N-(2-hydroxyethyl)piperazine-N’-2-ethanesulfonic acid HFP: HIV Fusion Peptide HFPdm: HFP dimer HFPmn: HFP monomer HFPtr: HFP trimer xviii HIV: Human Immunodeficiency Virus HOBt: 1-hydroxybenzotriazole HPLC: high-performance liquid chromatography IR: Infrared I4: Ac- AEAAAKEAAAKEAAAKA-NH2 peptide with 13 CO and 15 N labels at A9 and A13, respectively. LUVs: Large Unilamellar Vesicles MALDI: Matrix-Assisted Laser Desorption/Ionization MAS: magic angle spinning max nad: maximum natural abundance dephasing MD: Molecular Dynamics mHFP: HFP associated with membranes containing approximately 30 mol% cholesterol mL9R-HFP: L9R-HFP associated with membranes containing approximately 30 mol% cholesterol mV2E-HFP: V2E-HFP associated with membranes containing approximately 30 mol% cholesterol mHFPdm: HFPdm associated with membranes containing approximately 30 mol% cholesterol mHFPtr: HFPtr associated with membranes containing approximately 30 mol% cholesterol min nad: minimum natural abundance dephasing na: natural abundance nad: natural abundance dephasing NTH: N-terminal helix xix NMR: Nuclear Magnetic Resonance pd: pulse delay PDB: Protein Data Bank PHI: pre-hairpin intermediate POPC: 1, 2-dimyristoyl-sn-glycerol-3-phosphocholine POPG: 1, 2-dimyristoyl-sn-glycerol-3-[phosphor-rac-(1-glycerol)] REDOR: Rotational-echo Double Resonance RMSD: Root-Mean Squared Deviation RP-HPLC: reversed-phase high-performance liquid chromatography rf: Rabi freqency rXY: X-Y internuclear distance SDS: sodium dodecyl sulfate SIMPSON: simulation program for solid-state NMR spectroscopy TFA: trifluoroacetic acid TM: Transmembrane TOF: time-of-flight TPPM: Two-pulse phase modulation WT: wild type xx Chapter I. Introduction Viral replication is initiated by infection of a host cell where infection requires membrane 1 fusion of the viral and host cell membranes , Figure 1. Vaccines have been developed to build resistance to viral infections and minimize the effects of diseases such as measles, mumps, and 2 small pox to name a few . Relative to other viruses, the human immunodeficiency virus (HIV) 3 strains have higher mutation rates , and in the absence of a vaccine, new types of drugs will be needed. HIV infects T helper cells, regulatory T cells, monocytes, macrophages and dendrite 4 cells . Depletion of T cells can result from uncontrolled HIV infection and consequent 5 development of acquired immunodeficiency syndrome (AIDS) which often results in fatality. The consequences of this disease have inspired efforts to develop antiviral therapeutic drugs that 4 target enzymatic activity and protein-protein interactions at various stages of the HIV life cycle . 4 These efforts have decreased the rate of HIV infection within infected patients , and the “death sentence” disease of the late 80’s and early 90’s can be viable to live with as high profile MSU alumni and NBA Hall of Famer Earvin “Magic” Johnson has demonstrated. However, the World Health Organization estimates that there are currently ~33 million people living with HIV and ~2 million deaths per year due to HIV infection world wide (2009 statistics). Thus, development of a HIV vaccine to prevent infection is critical, and continued development of HIV inhibitor drugs is equally important for therapeutic treatment of infected patients. Small molecule candidates for HIV inhibitory drug design have been identified using computational and high throughput 4,6,7 screening methodologies . In general, the success rate of discovery and implementation of 1 small molecule candidates for anti-HIV drugs is enhanced by understanding the HIV life cycle and identifying key interactions within the HIV lifecycle to inhibit. Anti-HIV drugs have targeted different stages of the HIV life cycle (Appendix II), and only two molecules are commercially available that inhibit HIV entry, Enfuvirtide (discussed below, targets gp41) and 4 Maraviroc (binds to chemokine co-receptor CCR5) . Thus, HIV entry inhibitor drugs appear to be an underdeveloped area relative to other stages of the HIV life cycle (Appendix II), and the efficiency of drug design could be enhanced by further knowledge of important protein-protein and protein-membrane interactions that are necessary for HIV entry. Figure 1. HIV infection model (left) and freeze fracture electron microscopy (right) of (a) binding (b) hemi-fused viral and host cell membranes (c, d) pore formation with HIV infection of 8 the host cell (modified from literature ). 2 HIV is an enveloped virus, and its membrane is derived from its infected host cells. HIV entry and infection are initiated by the noncovalently associated glycoproteins gp120 and gp41 where gp120 is located on the exterior of the transmembrane protein gp41 1,9 . Membrane fusion is initiated after gp120 is bound to a CD4 receptor and an additional co-receptor from the chemokine family of a host cell, and binding to these receptors results in removal of gp120 from gp41 10-12 . The previously “covered” gp41 is exposed to aqueous solution and is thought to 13 undergo a series of structural transitions required for infection (see below) . The gp41 protein is composed of ~356 residues 14 and is subdivided into regions from the N-terminus: fusion peptide (FP) (~16 residues), FP proximal region (FPPR) (~13 residues), N-terminal helix (NTH) (~40 residues), loop (~47 residues), C-terminal helix (CTH) (~37 residues), pre-transmembrane region 15 (~18 residues), transmembrane region (~28 residues) , and cytoplasmic endodomain (~160 14 residues) . The ~175-residue N-terminal ectodomain of gp41 lies outside the virus, and X-ray crystal and liquid-state nuclear magnetic resonance (LSNMR) structures have shown organized molecular trimers for constructs that lacked the fusion peptide, transmembrane and endodomain 1,15-19 regions . These ectodomain crystal structures showed protein trimers with three interior parallel α helical NTH segments and three exterior α helical CTH segments packed antiparallel to the NTHs. The overall structure of each monomer was a hairpin, and the trimer formed a sixhelix bundle 15-17,20 . The above domains have been defined by crystal structures of gp41 based constructs with varying interpretations of the number of residues incorporated into each domain. To my knowledge, the largest HIV gp41 crystallized construct to date has shown that the helicity extends beyond the traditionally defined NTH and CTH, Figure 2. The helicity of the NTH and 3 CTH approximately extends from the respective residues Ala-532 to Ile-580 and Asp-627 to Asn-677. Additionally, the NTH and CTH in the SIV gp41 crystal structure span residues Arg-30 18 to Ala-86 and Thr-104 to Lys-146 , respectively, which is analogous to HIV gp41 crystal structure residues Arg-542 to Ser-598 and Ser-616 to Glu-662. One interpretation of these combined results is that under crystallization conditions, the helicity of the NTH and CTH regions are terminated due to the length of the gp41 construct rather than the length of the NTH and CTH of the full length gp41. Of note, these gp41 structures are for gp41 without the presence of a membrane, and these structures also lack the hydrophobic fusion peptide. For HIV gp41, mutations within the FP and FPPR have been shown to inhibit membrane 21 fusion which suggests that both the FP and FPPR are important for membrane fusion . Of special interest, transdominant inhibition of the V2E mutated gp41 (a FP mutation) has demonstrated that more than three gp41 or multiple gp41 trimers are needed to initiate membrane 21 fusion . Whatever the structure/s of the FP and FPPR are, the structure/s must allow for aggregation of FPs between more than 3 gp41 or multiple gp41 trimers. While the structures of the FP region of gp41 are the focal point of this dissertation, it should also be noted that regions that are C-terminal of FP can also be effective targets for fusion inhibitor drugs. The AIDS drug 22 Enfuvirtide is a fusion inhibitor and is a 36-residue peptide containing parts of the C-helix and the pre-transmembrane regions. Enfuvirtide likely binds to exposed N-helical regions in prehairpin intermediate (PHI) gp41 and acts as a competitive inhibitor to the native C-helix regions with consequent prevention of the transition to the final hairpin structure. Cell-cell fusion mediated by gp41 includes in sequence: (1) lipid mixing between the membranes; (2) fusion pore formation; and (3) pore enlargement. After addition of Enfuvirtide, small pores rather than large 4 pores can be closed which indicates that gp41 in the PHI state mediates lipid mixing and initial fusion pore formation while the final hairpin state (or possibly the PHI  hairpin transition) mediates pore stabilization and enlargement 13,22,23 . While Enfuvirtide is generally effective and possesses minimal side effects, Enfuvirtide is not commonly prescribed because it is a peptide 22 that requires two daily injections , and it is not cost-efficient. Continued development of HIV inhibitory drugs is highly valuable to reduce the costs of current HIV treatments 24 and enhance the accessibility to treatment of people in all social classes worldwide. Figure 2. A summary of the gp41 sequence and regions defined from literature where FP = fusion peptide, FPPR = fusion peptide proximal region, NTH = N-terminal helix, CTH = Cterminal helix, MPER = membrane proximal region, and TMR = transmembrane region. This dissertation primarily focuses on the role of FP in membrane fusion. The ~16 residue FP sequence followed by C-terminal amino acids (HFP) was studied in this dissertation. This figure 15 was adapted from literature . For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation. 5 Figure 3. Proposed fusion mechanism (Figure provided by Dr. Kelly Sackett). This model for membrane fusion suggests that sequential events occur during viral and host cell membrane fusion: (A  B) The ectodomain of gp41 is fusion inactive prior to contact with the host cell membrane. Contact between the host cell and viral membrane is initiated when gp120 binds to 12 CD4 and a co-receptor . This induces a conformational change in gp41 to the proposed prehairpin intermediate structure and FP binds to the host cell membrane. To my knowledge, there is currently no direct evidence that FP binds to the host cell, but synthetic fusion peptides bind 25,26 and insert into model membranes . Additionally, no structural data have been collected to elucidate the proposed PHI structure; (B  C) The host cell and viral membrane undergo HemiFusion; (C  D) Prior to completion of membrane fusion, a fusion pore is formed within the 26 host cell membrane. HFP constructs have been shown to insert into model membranes which appears to be driven by local hydrophobicity of HFP secondary and tertiary structures (See Chapter IV). Similarly, FP may insert into host cell membranes which may enable FP to be involved in pore formation. The membrane fusion pore is enlarged after or during the formation 27 of hairpin structure . Additionally, peptides with the CTH sequence inhibit membrane fusion by a proposed mechanism where these peptides bind to the NTHs in the Pre-Hairpin Intermediate structure (B and C) which prevents the formation of the 6 helix bundle structure (D). This lead to 22 development of the membrane fusion inhibitor drug Enfuvirtide . 6 The FP is a potential target for inhibitory drug design because it is a highly conserved region that is essential for HIV infection. While the FP has numerous sequential variations in different strains of HIV, the FP is composed of hydrophobic amino acids mixed with alanine and glycine residues where the AVGIGALFLGFLGAAG sequence was used in this study. The FP is essential for HIV infection which has been demonstrated by in vivo fusion and infection studies where mutations within the FP region of gp41 inhibited fusion and infection 21,28 . The ~16 residue FP sequence followed by C-terminal amino acids are generally referred to as HFP constructs. HFP constructs induce membrane vesicle fusion in the absence of other regions of gp41 or other proteins 25,29,30 . There are good correlations between the effects on gp41-mediated fusion by specific FP mutations and the effects on vesicle fusion by the corresponding mutations in HFP 21,31 . These similar trends support investigation of HFP and elucidation of its membrane- associated structure to better understand the role of FP in fusion and as a fusion inhibitor target. The HFP structure-function literature includes NMR data showing random coil structure 32,33 for HFP in aqueous solution . A fluorescence and infrared (IR) study further elucidated HFP structure and reported the time-resolved courses of HFP structural changes and the intervesicle 34 lipid mixing following addition of a HFP solution to a membrane vesicle solution . The experimental rates (Rs) were ordered RHFP membrane binding > RHFP β sheet formation ~ Rlipid mixing and were consistent with the sequence: (1) random coil HFPs bind to a membrane vesicle; and (2) HFP form oligomeric β sheets and vesicle fusion occurs. Similar structures were observed by solid-state NMR (SSNMR): (1) HFP lyophilized from aqueous solution without vesicles had a distribution of secondary structures as indicated by single site backbone 7 13 CO 35 signals whose ~8 ppm linewidths spanned typical helical and β strand chemical shifts ; and (2) HFP bound to hydrated membranes containing ~30 mol% cholesterol (i.e. the approximate cholesterol content in the membranes oligomers/aggregates with β sheet structure of HIV and its host cells 36-39 ) formed 35,36 . The biological relevance of HFP oligomers is further supported by the molecular trimer structure of soluble regions of the gp41 ectodomain 19 17- . The region approximately between residues Thr-25 and Gly-85 of each molecule was a continuous helix, and the helices of the different molecules formed a parallel coiled-coil. The fusion peptide region was not included in the protein constructs for these structures but would be N-terminal of residue Thr-25. A C-terminally cross-linked HFP trimer (HFPtr) was therefore synthesized to mimic the close proximity of the three Thr-25 residues in the hairpin structure. Relative to HFP monomer, HFPtr induced membrane vesicle fusion at >15 fold faster rate which supported the functional significance of the trimer 25,30 . Although both HFP and HFPtr formed  sheet oligomers in membranes with cholesterol, HFPtr is more deeply inserted which correlates 40 with greater membrane perturbation and reduction of the vesicle fusion activation energy . The in vivo importance of fusion peptide oligomers was also demonstrated by dominant inhibition of fusion and infection in viruses and cells for which a small fraction of the gp41 had the V2E point 21,41 mutation in the fusion peptide region . Analyses of these data supported the involvement of 42 multiple gp41 trimers and fusion peptides in fusion and a model for this dominant inhibition is provided in Appendix III. Electron micrographs of virus-cell contacts have also been interpreted 8 43 to show multiple gp41 trimers at the contact site , Figure 4. Functional importance of fusion peptide trimers has also been demonstrated for fusion peptides of other viruses a) 44,45 . b) Figure 4. (a) Electron micrograph of HIV and a T cell where the width of the HIV membrane is ~100 nm. The increased electron density between the HIV and host cell membranes suggests that multiple proteins from the HIV membrane are in contact with the T cell membrane. (b) Model of HIV contact with a T cell based upon electron micrographs where multiple proteins are believed 46 to be between the HIV and T cell membranes. This figure was modified from literature . Because of the aforementioned functional significance of HIV fusion peptide oligomers, there has been effort to elucidate the membrane-associated HFP structure(s). Efforts to identify the predominant β sheet registry of membrane-associated HFP have lead to conflicting β sheet 47 29 structural models such as in-register parallel , mixed in-register parallel and antiparallel , and 48 β hairpin . However, the data from these studies were not interpreted quantitatively which left room for interpretational ambiguity that, in my opinion, explains the inconsistency of the 9 interpretations of these experiments. The membrane-associated HFP samples (mHFP) that are presented in this dissertation were prepared in a manner similar to that of fusion assays with 37 fusion peptide in aqueous solution added to a membrane vesicle solution . Previous experiments have shown that appendage of a C-terminal lysine tag to HFP greatly reduced HFP aggregation in aqueous solution and allowed separation of pelletted fused vesicles with bound HFP from unbound HFP in the supernatant 30,33,49 . Additionally, SSNMR experiments are capable of detecting structure in membrane bilayers and have greatly contributed to the mHFP literature that has inspired the work of this dissertation. HFP/lipid binding has been supported by SSNMR detection of a HFP Ala-1 13 CO-lipid 31 13 CO(carbonyl)-lipid 31 40 P distance of ~5 Å and the Ala-1 P contact as well as other data suggest that the number of molecules in the 40,50,51 oligomer is small . Intermolecular 13 13 C- C and 13 15 C- N distances of ~4-5 Å have been detected in mHFP and supported  sheet oligomers/aggregates. Additionally, 13 C chemical shifts of residues between Ala-1 and Gly-16 in mHFP were consistent with a fully extended  strand 50 conformation . This dissertation has contributed to the β sheet structure literature by quantitative determination of populations of specific  sheet registries. Prior to my work, the clearest information to-date on this topic has been a SSNMR experiment on membrane-associated HFP with an Ala-14 13 50 CO label and a Gly-3 15 N label whose separation (rCN) was >20 Å along a single  strand . SSNMR can detect labeled 13 3 15 CO- N dipolar coupling (dCN) where dCN = 3080/rCN with d in Hz and r in Å. The minimum detectable dCN ~10 Hz correlates with rCN ~7 10 Å so that detectable dCN in this sample were necessarily ascribed to inter- rather than intramolecular 13 15 CO- N proximity since labeled intrastrand 13 15 CO- N distances were > 7 Å for residues in β sheet structure. SSNMR detection of d > 30 Hz strongly supported a significant fraction of molecules with intermolecular Ala-14-Gly-3 hydrogen bonding and labeled rCN of ~4.1 and ~5.5 Å, i.e. 161/116 antiparallel  sheet registry, Figure 5. In general, antiparallel registries will be described using a registry index t where the t = 16 registry describes the 161/116 antiparallel  sheet registry. Detection of similarly large dCN in an Ala-14 13 CO/Ile-4 15 N HFP sample also supported a fraction of the t = 17 registry, Figure 5. At most half of the membrane-associated HFP molecules were in the t = 16 and t = 17 registries, i.e. a large fraction of the molecules were in registries not detected in either the Ala-14 15 N or Ala-14 13 CO/Ile-4 15 13 CO/Gly-3 N labeled samples. Because of the close proximity of the Thr-25 residues of the three molecules of the gp41 trimer, a reasonable hypothesis for a populated HFP registry is in-register parallel  sheet, i.e. 117/117 in Figure 5. An earlier SSNMR study attempted to test this hypothesis using samples each containing an equimolar mixture of two labeled HFPs, one with three sequential backbone sequential backbone 13 CO/Gly-5-Leu-7 15 15 13 CO labels and the other with three 29 N labels . Detection of an average dCN > 10 Hz for a Gly-5-Leu-7 N sample and a Phe-11-Gly-13 13 CO/Phe-11-Gly-13 15 N sample were consistent with a fraction of in-register parallel HFP molecules. However, because the samples were extensively labeled, the data were also consistent with other parallel or antiparallel registries. In addition, the data reflected an average of many intermolecular dCNs so it was not 11 possible to determine the fraction of molecules with a particular registry. There have also been efforts to detect in-register parallel structure using SSNMR measurement of intermolecular 13 13 C- 3 C dipolar couplings (dCCs) where dCC = 7710/rCC with dCC in Hz and rCC in Å. For HFP with a single dCC ~70 Hz 13 CO label and in-register parallel structure, the labeled interstrand rCC ~5 Å with 52,53 . These parameters will be independent of the residue that is mHFP with Phe-8 13 13 CO labeled. For CO, a best-fit dCC ~70 Hz was detected whereas for mHFPtr, dCC depended on the position of the labeled 13 54,55 CO residue with a range of 10-60 Hz . This residue dependence argued against a major fraction of in-register parallel structure in HFPtr. Figure 5. The t = 16, t = 17, and in-register parallel β sheet registries are illustrated. Isotopic labeling schemes to detect these registries are illustrated where red underlined atoms indicate 13 15 CO labeled residues and blue underlined atoms indicate N labeled residues. These specific labeling schemes are effective because they have 13 15 CO- N proximities of ~4 Å. There was also an IR spectroscopy effort to distinguish between the 117/117 parallel and the t = 16 registries using samples that contained backbone 13 CO labeling at either: (1) Ala-1 47 to Val-3, Gly-5 to Ile-9; (2) Phe-8 to Gly-16; or (3) Aal-1 to Val-3, Gly-5 to Gly-16 . The IR wavenumbers and intensities of different samples were interpreted to support a large fraction of in-register parallel structure and little antiparallel structure. However, in my view, the extensive labeling of the IR samples precluded quantitation of specific registries and greater support for this argument is provided in Chapter III. These in-register parallel studies motivated the work for 12 Chapter III where more accurate quantification of the natural abundance dephasing (nad) in HCN rotational-echo double-resonance (REDOR) experiments was achieved. This established a more quantitative method for detection of the fraction of parallel structure in membraneassociated HFP oligomers. The lack of in-register parallel β sheets motivated the work described in Chapter IV where the complete distribution of antiparallel β sheet registries was quantified in mHFP, and the potential functional significance is discussed. The registry distribution was also quantified in the less active V2E mutant HFP (V2E-HFP). Clear differences were evident in the registry distribution between mHFP and mV2E-HFP. These observed structural differences were the basis for experimentally based structure-function models that are presented in Chapter IV. 13 Chapter II. Materials and Methods 2.1 HFP Synthesis and Sample Preparation 1. General HFP Preparation 9-fluorenylmethoxycarbonyl (Fmoc) protected amino acids and Fmoc-Ala-Wang resin were purchased from Peptides International (Louisville, KY). Isotopically labeled amino acids were purchased from Cambridge Isotopes (Andover, MA) and were Fmoc-protected using 56 57 literature methods . Standard Fmoc chemistry was used to synthesize the gp41 fusion peptide (HFP) with the 23 N-terminal residues (AVGIGALFLGFLGAAGSTMGARS). A WK6A or WK6Aβ tag was added to the native sequence to for the following purposes: (1) non-native W24 was incorporated as an A280 chromophore; (2) non-native lysines were added to reduce HFP aggregation in aqueous solution prior to membrane binding 30,33 . This ensured that membrane- associated  sheet oligomers/aggregates were formed after membrane binding; (3) Wang Resin was preloaded with Ala or β-Ala (~0.27-0.74 mmol/g). Alternatively, Wang Resin preloaded with Lys would have likely been effective for HFP synthesis. HFP was manually synthesized and then cleaved from the resin for three hours in a solution of trifluoroacetic acid (TFA):water:anisole:thioanisole:ethanedithiol in a 90:5:2:2:2 volume ratio. After precipitation with cold diethyl ether, centrifugation, and dissolution of the pellet in water, crude HFP was purified by reversed-phase high-performance liquid chromatography (RP-HPLC) generally with a C4 column and a water-acetonitrile gradient containing 0.1% TFA (See Appendix IV for setup). Acetonitrile, TFA, and residual solvents were removed with nitrogen gas and subsequent lyophilization. HFP purity was >95% as 14 determined by mass spectrometry. HFP amounts were quantified using A280 of aqueous –1 –1 solutions of HFP with  = 5600 M cm . The membrane composition of membrane-associated samples was 1,2-di-O-tetradecyl-snglycero-3-phosphocholine (DTPC) lipid, 1,2-di-O-tetradecyl-sn-glycero-3-[phospho-rac-(1glycerol)] (DTPG) lipid, and cholesterol in a 8:2:5 mol ratio. This composition reflected the large amount of choline lipid and fractions of negatively charged lipid and cholesterol in membranes 58 of host cells of HIV . Ether- rather than more physiologically abundant ester-linked lipids were used because the latter have two COs/molecule that would contribute substantial natural abundance (na) with HFP 59-61 13 CO signal. Bilayer phase is retained for ether-linked lipids with cholesterol and . In addition, membrane-associated HFP has predominant  sheet structure in 54 either ester-linked lipid + cholesterol or ether-linked lipid + cholesterol compositions . Samples were prepared by first dissolving DTPC (20-40 µmol), DTPG (5-10 µmol), and cholesterol (12.5-25 µmol) in chloroform (lipid and cholesterol amounts depended upon the amount of peptide used) and the chloroform was removed with nitrogen gas and vacuum. The lipid film was suspended in 2 mL of 5 mM HEPES buffer at pH 7.0 with 0.01% NaN3 preservative. The suspension was homogenized with ten freeze-thaw cycles and large unilamellar vesicles were formed by extrusion through a polycarbonate filter with 100 nm diameter pores (Avestin, Ottawa, ON). The HFP solution was added drop-wise to the vesicle solution and the combined solution was gently stirred overnight. Ultracentrifugation at ~150000g for four hours 37 pelleted membranes with bound HFP while unbound HFP remained in the supernatant . The pellet was lyophilized, transferred to the SSNMR rotor, and rehydrated with ~30 µL of 5 mM 15 62 HEPES buffer at pH 7.0 for every 50 µmol of lipid . The validity of the lyophilization/rehydration approach was supported by peak within 0.6 ppm of those of samples that were not lyophilized 16 13 50,63 CO chemical shifts that were . Table 1. HFP construct labeling schemes. Peptide Labeled residues 13 Peptide Labeled residues HFP-F8 Phe-8 CO HFP-F8CA21N Phe-8 13CO and Ala-21 15N HFP-L12 Leu-12 13CO V2E-A6CG3N Ala-6 13CO and Gly-3 15N HFP-G5A6 Gly-5 and Ala-6 15N V2E-L7CG3N Leu-7 13CO and Gly-3 15N HFP-A6L7 Ala-6 and Leu-7 15N V2E-F8CG3N Phe-8 13CO and Gly-3 15N HFP-L9G10 Leu-9 and Gly-10 15N V2E-L9CG3N Leu-9 13CO and Gly-3 15N HFP-L12G13 Leu-12 and Gly-13 15N V2E-L9CI4N Leu-9 13CO and Ile-4 15N HFP-G13A14 Gly-13 and Ala-14 15N V2E-L9CG5N Leu-9 13CO and Gly-5 15N HFP-A6CG3N Ala-6 13CO and Gly-3 15N V2E-L12CG3N Leu-12 13CO and Gly-3 15N HFP-L7CG3N Leu-7 13CO and Gly-3 15N V2E-L12CI4N Leu-12 13CO and Ile-4 15N HFP-F8CG3N Phe-8 13CO and Gly-3 15N V2E-L12CG5N Leu-12 13CO and Gly-5 15N HFP-L9CG3N Leu-9 13CO and Gly-3 15N V2E-L12CA6N Leu-12 13CO and Ala-6 15N HFP-L9CI4N Leu-9 13CO and Ile-4 15N V2E-L12CL7N Leu-12 13CO and Leu-7 15N HFP-L9CG5N Leu-9 13CO and Gly-5 15N V2E-F8CL12N Phe-8 13CO and Leu-12 15N HFP-L12CG3N Leu-12 13CO and Gly-3 15N V2E-F8CG13N Phe-8 13CO and Gly-13 15N HFP-L12CI4N Leu-12 13CO and Ile-4 15N V2E-F8CA14N Phe-8 13CO and Ala-14 15N HFP-L12CG5N Leu-12 13CO and Gly-5 15N V2E-F8CA15N Phe-8 13CO and Ala-15 15N HFP-L12CA6N Leu-12 13CO and Ala-6 15N V2E-F8CG16N Phe-8 13CO and Gly-16 15N HFP-L12CL7N Leu-12 13CO and Leu-7 15N V2E-L9CG16N Leu-9 13CO and Gly-16 15N HFP-F8CL12N Phe-8 13CO and Leu-12 15N L9R-F8CG13N Phe-8 13CO and Gly-13 15N HFP-F8CG13N Phe-8 13CO and Gly-13 15N HFPtr-F8CG13N Phe-8 13CO and Gly-13 15N HFP-F8CA14N Phe-8 13CO and Ala-14 15N V2E-L12CA6N Leu-12 13CO and Ala-6 15N HFP-F8CA15N Phe-8 13CO and Ala-15 15N V2E-L12CL7N Leu-12 13CO and Leu-7 15N HFP-F8CG16N Phe-8 13CO and Gly-16 15N V2E-F8CL12N Phe-8 13CO and Leu-12 15N HFP-L9CG16N Leu-9 13CO and Gly-16 15N V2E-F8CG13N Phe-8 13CO and Gly-13 15N 2. HFP Solutions Chapter III. The samples that were prepared to detect in-register parallel and antiparallel β sheets (HFP-P, HFP-A, HFP-AP) used HFP solutions that were prepared with HFP (3.0 mg) and 15 13 CO-labeled N labeled HFP (6.0 mg) in HEPES buffer (32 mL) before it was added drop wise to the ~2 mL lipid solution. 17 Chapter III. The negative control sample (HFP-NC) was a physical mixture of lyophilized HFP-F8 (5.0 mg) and HFP-A6L7 (10.0 mg) without any membrane. Each peptide was lyophilized separately, and the two peptides were then mixed in the solid phase to form a uniform physical mixture. Water and membrane were not added to the physical mixture so that the labeled 13 COs and 15 Ns remained much farther apart than 7 Å which is the approximate REDOR detection limit. All other HFP constructs. The HFP had a single 13 CO label and a single 15 N label. HFP (4.0-6.5 mg) was suspended in HEPES buffer (32 mL) before it was added dropwise to the ~2 mL lipid solution. 3. Trimeric HFP (HFPtr) Synthesis 25 Literature methods were used to synthesize the dimeric and trimeric peptides. Cross linking reactions between two peptides, denoted A and B, are listed as well as the product in Table 2. The HFPdm(Lys) (0.10 mmol) was synthesized by adding Fmoc-Lys(Mtt)-OH, FmocLys-OH with a Mtt sidechain protecting group, at position Lys-30. After addition of the Lys-30 residue, the resin was washed and “capped” with acetic anhydride solution. The Mtt sidechain protecting group was cleaved with an acid solution (5 mL of dichloromethane (DCM) with 1% v/v TFA) using 6 x 6 minute cycles. Fmoc-Cys(Trt)-OH was added to the unprotected Lys sidechain. Standard Fmoc chemistry was used to synthesize the rest of HFPdm(Lys) where HFPdm(Lys) contained HFPWK5KA and HFPWK5C peptides that were cross-linked at the K and C residues indicated in bold font. The HFPdm(Cys) was synthesized by dimerization of the HFPWK5CA peptide. Formation of the disulfide bond between peptides works well at a pH of ~8, and the solutions can 18 be adjusted by addition of acid and base. However, when the pH of the solution becomes too acidic, the peptides become insoluble and precipitate out of solution. Precipitated HFPWK5CA peptides are difficult to dissolve. To avoid this problem, I changed the DMAP concentration 25 from 10 mM (literature values ) to 20 mM. In a 15 mL conical vial, 2.5 mmol of HFPWK5CA was dissolved in 490 µL of 20 mM DMAP solution and gently vortexed in air. After approximately 2 hours, 100 µL of DMAP solution was used to rinse the sides of the 15 mL conical vial and the mixture was then vortexed in air overnight. The sample was additionally sonicated every 30 minutes for the first 3 hours to redissolve any precipitated peptide. The effect of sonication on reaction yield was not measured. The pH was ~8 under these conditions and precipitation of the peptide was not observed. The reaction yielded 5.7 mg of HFPdm(Cys) from 7.7 mg of HFPWK5CA after purification (~74% yield). The same peptide and DMAP concentration was used for the cross-linking reaction of HFPWK5CA and HFPdm(Lys) to yield HFPtr. The ratio of HFPWK5CA:HFPdm(Lys) peptides should affect the molar ratio of reaction products (HFPWK5CA:HFPdm(Lys):HFPdm(Cys):HFPtr:HFPte) 25 where HFPte is tetrameric HFP. A 2:1 mol ratio of HFPWK5CA to HFPdm(Lys) was used and 13.2 mg HFPWK5CA was reacted with 13.2 mg of HFPdm(Lys). This reaction yielded 4.4 mg of HFPdm(Lys)+HFPdm(Cys), 5.0 mg of HFPtr, and 2.9 mg of HFPte after HPLC purification. This reaction was only run once, and a representative HPLC chromatogram is displayed in Figure 6. HFPtr was used in Chapter IV while HFPdm data are displayed in Appendix V. 19 Table 2. Oligomeric HFP synthesis summary. Peptide A Peptide B None HFPWK5CA HFPWK5CA HFPdm(Lys) HFPdm(Lys) HFPdm(Lys) Product HFPWK5CA HFPdm(Cys) HFPWK5CA None HFPWK5CA HFPdm(Lys) HFPdm(Lys) HFPtr HFPte Figure 6. Purification of HFPtr synthesis. The product of each peak was identified by MALDITOF mass spectrometry, Figure 7, and each peak corresponded to the following: (1) HFPWK5CA; (2) HFPdm(Lys) and HFPdm(Cys); (3) HFPdm(Lys) and HFPdm(Cys); (4) HFPtr; and (5) HFPte. Additionally, peaks (4) and (5) were not present in HFPdm(Cys) which further supports that these peaks result from HFPtr and/or HFPte. Peak (5) was not used for any experiments in this dissertation while Peak (4) was collected for the one HFPtr experiment in Chapter IV. The identity of HFPte would have to be confirmed before meaningful experimental data is collected for this peptide. 20 Figure 7. MALDI-TOF mass spectroscopy from the HFPtr purification displayed in Figure 6. 21 Figure 7 (cont’d). The fractions collected from the numbered peaks from HPLC are correlated to the chromatograms and peptides (expected mass) as follows: (1)-(a)-HFPWK5CA (3126 +2 g/mol); (2)-(b)-HPFdm (6250 +4 g/mol) and HFPdm(Cys) (6257 +4 g/mol); (3)-(c)-HPFdm (6250 +4 g/mol) and HFPdm(Cys) (6257 +4 g/mol); (4)-(d)-HFPtr (9381 +6 g/mol); and (5)-(e)15 13 HFPte (12512 +8 g/mol) where an N or CO label adds +1 g/mol each. In (a)-(e), expected product peaks and expected product peaks missing an Ala residue are observed. The missing Ala 64 residue(s) is likely due to laser induced C-terminal fragmentation or hydrolytic cleavage rather than synthetic impurities since acetic anhydride was used to “cap” failed syntheses. The mass of a failed synthesis should correspond to a peak with mass equal to the failed synthesis’s native sequence plus 43 (acyl group). Although not likely here, fragmentation of C-terminal of amino 65,66 acids with basic sidechain groups can occur . Additionally, fragmentation at a disulfide bond 67 may occur , and peptides may contain multiple charges. Either scenario could result in detection of a mass to charge ratio that is a fraction of the expected mass to charge ratio of the dimer, trimer, or tetramer product peaks. 2.2 General NMR Background 1. NMR Spectroscopy NMR experiments have an external magnetic field aligned along the z-axis of the laboratory frame where B0 represents external magnetic field. For this chapter, bold characters are used to represent vector quantities while non-bold characters are used to represent scalars. NMR experiments collect nuclear resolvable data where the vector quantity of the nuclear magnetic moment, µ, for an atom can be described below: µ  I (1) where γ is the gyromagnetic ratio of the nucleus and I is the spin angular momentum of the nucleus. I can be written in terms of scalars and unit vectors by: ˆ ˆ ˆ I = Ix x + I y y  I z z (2) When placed in an external magnetic field, B0, µ has a potential energy, V. V  µ  B0   I  B0 (3) 22 where ˆ ˆ ˆ B0 = Bx x + B y y  Bz z Because external magnetic fields are typically aligned in the z direction, Eq. (3) can be rewritten. V   Iz Bz (4) By replacing Iz with the spin operator ˆz , the Schrodinger equation for a nuclear spin can be I written using the nuclear spin Hamiltonian. ˆ H   Bz ˆ z  E I (5) The selection rules in NMR spectroscopy demonstrate that the wave functions are spin eigenfunctions in that ˆ z  mI where mI = I, I-1,….-I. Therefore, from Eq. (5), energy can I be calculated as: E   m I B z (6) 1 Nuclei have 2I +1 energy levels, and mI = ±½ nuclei, such as H and 13 C, have two energy levels are separated by ΔE. E   B z (7) The two energy levels correspond to the mI = +½ and mI = -½. The equilibrium populations of these spins in a magnetic field are described by a Boltzmann distribution. In NMR experiments, an ensemble of spins are observed and both the mI = +½ and mI = -½ spin states are populated at equilibrium in the presence of B0 where mI = +½ is the lower energy state. 23 2. Rotating Frame and Bloch Equations Simplistically speaking, NMR spectroscopy measurements are made after the magnetization is transferred from a z-axis orientation to the xy-plane. In general, NMR experiments detect coherence of the nuclear magnetic moments as magnetization in the xy-plane. A nuclear magnetic moment can be represented by the vector quantity, µ, and the vector sum of the nuclear magnetic moments is the magnetization, M. n M   µa a 1 (8) where a is an index for nuclear magnetic moments being detected and n represents the number of nuclear magnetic moments detected in a NMR sample. As discussed in 2.4 MAS Solid-State NMR (REDOR), structural information can be obtained by observing decoherence of the nuclear magnetic moments which results in a smaller M. Before discussing coherence and decoherence of the nuclear magnetic moments, it is essential to first understand how nuclear magnetic moments or magnetization interact with the external magnetic field, B0, and radiative magnetic field, B1. First, consider the interaction of M with a magnetic field, B, by the Bloch equation: dM  M  ( B ) dt (9) where B is the vector sum of the external magnetic field, B0, and the radiative magnetic field B1. Before considering the interactions in pulsed experiments, it is important to expand upon Eq. (9). Consider the specific example below: dM = M z   B x  γ M z B x sinφ dt (10) 24 Eq. (10) evaluates the magnitude of a vector quantity where φ is the angle between the z and x axes. In general, directional dependence for evaluating the direction of a cross product can be determined by using the right hand rule where Mz x Bx would have a y direction. Before introducing the concept of a pulsed experiment, it is important to understand the interaction of a dipole moment with the external magnetic field. In this brief discussion, consider a single dipole moment as depicted in Figure 8. Due to the interaction with the external magnetic field, B0, µ precesses about the z axis with an angular frequency of w0, the Larmor frequency. w0   B0 (11) As depicted in Figure 8, a rotating frame with the Cartesian coordinates (cos(wt), sin(wt), z), where z has a constant value, can be thought to precess at the same frequency as the Larmor frequency only if w1 = w0. A rotating frame of reference is created by imagining x’ and y’ axes rotating about the z axis. This is a useful concept since B1 fields are applied in the xy plane at the transmitter frequency w1. The B1 field precesses at the same frequency as the rotating x’ and y’ axes, and the x’ and y’ axes precess at the Larmor frequency only when w0 = w1. The rotating frame (x’ and y’ plane) is defined by w1 because signal detection is performed relative to the transmitter frequency. Thus, the B1 field can arbitrarily be thought to be applied along the x’ o axis. This rotates Mz about the x’ axis in the z-y’ plane. Applying a 90 pulse along the x’ axis is equivalent to the mathematical expression Mz x Bx’ which reorients the magnetization in the y’ direction where the resultant magnetization is denoted My’. These concepts build a foundation 25 for understanding how to set up a pulsed NMR experiment and also establish a foundation for understanding coherence of the nuclear magnetic moments in the rotating frame as discussed in section 2.4 MAS Solid-State NMR (REDOR). Figure 8. A dipole moment in the laboratory frame (x, y, and z axes) where the B0 external magnetic field is aligned along the z-axis. A new axis, x’, rotates at the Larmor frequency only if w0 = w1. 2.3 Experimental Setup 1. General Concepts for Setting the Pulses Many NMR experiments begin by applying a π/2 pulse using an external magnetic field, B1, to change the magnetization from the z-orientation to the xy plane. In a classical picture and for simplicity, consider the x’, y’, and z’ axes to be the axes of the rotating frame where the axes rotate at the transmitter frequency, w1. In a complex system with many nuclear magnetic moments, a sample placed in an external magnetic field has a sum of the nuclear magnetic 26 moments that yield an observable net magnetization oriented along the z-axis, Mz. In NMR spectroscopy, the transverse magnetization (magnetization in the xy plane) is detected in the time domain and is then typically transformed to the frequency domain by a Fourier transformation. In order to detect the magnetization, Mz must be reoriented to the xy plane of the laboratory frame. For simplicity, first consider the scenario where the w0 = w1 (i.e. the precession frequency of B0 = the precession frequency of B1). Because B1 is always stationary in the rotating frame, we can arbitrary assign B1 to the “Bx’” direction. If the pulse duration of B1 is short relative to both T1 and T2 (~ < 50 µs), we can ignore the T1 and T2 effects on the angle that magnetization rotates at the time when B1 is applied. T1 and T2 represent the longitudinal (z-axis) and transverse (xy-plane) magnetic relaxation time constants. Typical relaxation times for membrane-associated peptides studied in this dissertation were on the millisecond timescale for T2’s and seconds time scale for T1’s. Additionally, recall that in an NMR experiment for mz = ½ nuclei, we are inducing a population change in the spin up and spin down states and the energy for this transition can be denoted ΔE, Eq (7). The energy for this transition is supplied by the B1 radiative field and is related to power, P, and the time, τ, for which the power is applied by Eq (12).  E   P   d 0 (12) The power, P(τ), can be related to voltage V(τ) and current I(τ) by Eq (13). P( )  V ( )  I ( ) (13) 27 Therefore, from Eq (12) and Eq (13),  E   V ( )  I   d 0 (14) Experimentally, we can measure the average voltage over a period of time using an oscilloscope. In NMR the Rabi frequency can be thought of as the frequency of oscillation between the spin up and spin down states due to the applied magnetic field, B1, where wR   B1 (15) In the rotating frame, ϑ is the angle that the magnetization rotates about the axis of the radiative field as written below:   wR (16) where ϑ is expressed in radians, ϑ is the pulse length used to achieved a ϑ pulse, and the interaction between the magnetization and the radiative field was previously described by Eq (9). This is a general convention that is used so that a simple expression can be written for calculating the Rabi frequency for a pulse where the Rabi frequency for a 2π pulse can be calculated by Eq 19. 1  2  wR 2 (17) Additionally, the wR/2π can be calculated for a π/2 pulse by Eq (18) assuming that a π/2 pulse takes 1/4 times as long as a 2π pulse. wR 1  2 4   / 2  (18) where τπ/2 is the time required to achieve a π/2 pulse. By varying the strength of the B1 radiative field, there are many variations of wR and τp that can be used to achieve a π/2 pulse, but different 28 combinations of wR and τp used to rotate θ to a fixed angle, such as π/2, affect the bandwidth of excitation where the bandwidth of excitation is approximately proportional to the inverse of the 68 pulse duration , Figure 10. Shorter π/2 pulses have a larger bandwidth of excitation since there is greater uncertainty associated with the approximation of using a square wave to represent a π/2 pulse. In a 13 C NMR experiment where 13 CO nuclei are the primary focus of data analysis, it may be less important to have a large bandwidth of excitation since the range of 69,70 resonant frequencies are narrow (anisotropic chemical shift range of ~140 ppm relative to the spectral width (50 kHz). However, shorter 13 13 CO nuclei or ~14 kHz) C π pulses are still important for REDOR experiments. By Eq (13)-(16), shorter pulses that require more power (i.e. higher voltage measured on the oscilloscope) excite a broader range of resonant frequencies, and the measured voltage required to achieve a π/2 pulse is inversely proportional to the τp. Typical pulse lengths used in the HCN REDOR experiments are 5 μs for π/2 pulses and 10 μs for π 1 pulses for the H and used for 15 13 CO respective nuclei while longer pulses (~20-25 µs) were typically N (see 9. Proton TPPM Decoupling to Nitrogen Rabi Frequency Ratio in HCN REDOR). As displayed in Figure 10 the largest amplitude of excitation is at the transmitter frequency, wm, but it is important not to set the transmitter frequency to the frequency of the observed nuclei, generally 13 CO in HCN REDOR experiments. Detection is performed in the rotating frame which is at the transmitter frequency of the laboratory frame. This commonly leads to detection of the DC voltage at the transmitter frequency which results in a nonzero “baseline” of the free induction decay (FID), Figure 9, and a signal at 0 Hz in a Fourier 29 transformed spectrum. The DC offset correction can be applied prior to Fourier transforming the FID so that the baseline of the FID is correctly offset to zero. However, this processing method is not perfect and often does not entirely eliminate the “blip” at 0 Hz. Thus, it is best to move the transmitter frequency ~15-20 ppm (~1.5-2 kHz) when detecting 13 CO nuclei on a 400 MHz spectrometer) away from the observed nuclei’s resonant frequency so that the “blip” does not overlap and interfere with the 13 CO signal. Additionally, it is important that the observed nuclei have resonant frequencies that are near the maximum excitation energy. This is important since the energy due to the radiative field changes less near transmitter frequency as evident by Figure 10 where the slope of curve is approximately equal to zero at the transmitter frequency, w1. 30 13 Figure 9. FID’s for cross polarization acquisitions of the CO labeled i4 peptide where the red horizontal line signifies the zero value or baseline for the y-axis. (a) The FID decays to the baseline. (b) The FID decays to a positive y-value as evident by the zero point value of the FID (~2.5 ms) being above the red line. 31 Figure 10. A pulse of a single radiofrequency is illustrated in the time domain (a,c) and the frequency domain (b,d). The relationship between pulse length (τp), radiative field (B1), and the energy of the B1 field are illustrated. The excitation frequency is centered about the transmitter frequency, w1. The figure was made using concepts from the literature where the frequency domain is represented as the Fourier transform of the time domain where the Fourier transform 68 of a step function can be represented by a sinc function (b), (d) . 2. Setup Compounds I4 peptide. A 17-residue acetylated and amidated peptide with the sequence Ac13 AEAAAKEAAAKEAAAKA-NH2, and I4 had a CO label at Ala-9 and a 15 N label at Ala-13. The peptide was lyophilized from aqueous solution and is predominantly α helical (83 ± 6%) at 71 Ala-9 . These labeled nuclei should have a 13 15 C- N internuclear distance of 4.1 Å in an α helix. 32 Adamantane. Adamantane was used for chemical shift referencing (see below). The chemical structure of adamantane is much different than the chemical structure of backbone 13 CO in HFP, and adamantane should not be used for optimizing cross polarization (CP) parameters for the REDOR pulse sequence (see below). Figure 11. The chemical structure of adamantane. 3. Setting the Magic Angle In the Fourier transformed spectrum, quadrupolar nuclei have many spinning sidebands where the intensity of the sidebands is sensitive to the magic angle, and the magic angle is 72 54.7° . Thus, quadrupolar nuclei such as 79 Br can be used to set the magic angle. We currently have a rotor packed with adamantane and KBr. Additionally, it is convenient to detect setting the magic angle because (1) Detection of 79 79 Br when 1 Br in KBr does not require H decoupling which reduces the power applied to the probe and allows for shorter pulse delays without damaging the probe; and (2) The 79 Br nuclei have a gyromagnetic ratio that is close to therefore can easily be used without having to change hardware to detect from 13 C to 79 79 13 C and Br signal (i.e. tuning Br can be accomplished only by adjusting tuning rods). In a simple one pulse 33 experiment (“1pulse” on the Infinity Plus spectrometers), a π/2 pulse is applied (say in the –y direction) followed by acquisition. Consequently, the magnetization is transferred to the x axis and magic angle spinning (MAS) during acquisition results in no net evolution of the magnetization over each rotor period due to quadrupolar and chemical shift anisotropy (See 2.4 MAS Solid-State NMR (REDOR) for brief discussion of anisotropy). At the end of each rotor period, the magnetization is realigned on the x-axis which results in a series of “spikes” in the FID that are separated by the spinning frequency. With a 4.0 kHz spinning frequency, the spikes typically extend out to 10-15 ms for ~64-128 scans acquired. It is not critical to set the pulse 68 nutation angle exactly to 90° (see Ernst angle for more detail ). Using KBr, signal averaging of 79 Br requires ~5-10 seconds to resolve spin echoes as observed in Figure 12, and it is sufficient to use the pulse length and amplifier input parameters that would be used for 13 C. Additionally, another helpful pdf file that demonstrates how to setup magic angle spinning from scratch can be found in the folder @..home/mb4b/data/Scott/REDOR/Setup/KBr_Magic_angle.pdf. When preliminarily setting the magic angle, the continuously updated 1-acquisition FID can be used to get near the magic angle whereas more scans should be acquired to finely adjust the magic angle 34 Figure 12. Magic angle spectrum FID of KBr with 64 acquisitions and a pd = 0.10 s. The 79 transmitter was moved to the Br resonant frequency of KBr, and an exponential decay is observed in the free induction decay (FID). 4. Setting Proton π/2 Pulse When setting up experiments, equivalent parameter sets between experiments are essential to obtaining reproducible results. One way that we monitor the consistency between experiments is by measuring the Rabi frequencies, wR, of our pulses by Eq. (16). In general, the pulses are used to induce a radiative transition between the spin up and spin down energy levels. As previously described, the time that the B1 field is applied must be shorter than the relaxation 35 transition between the spin up and spin down states so that rotation of the magnetization is instantaneous relative the longitudinal relaxation time. In Figure 17, it can be observed that the data acquisitions with a lower decoupling field had lower signal to noise (i.e. larger error bars). 1 13 For these points, this suggests that there was a larger observable H- CO coupling. As the decoupling field was increased, the proton decoupling Rabi frequency became greater than the 1 13 1 13 H- CO dipolar coupling frequency, wd, which results in a non-observable H- CO dipolar coupling interaction. A general equation for dipolar coupling frequency between two nuclei is listed below where “a” and “b” identify two nuclei.     wd  0 a b 8 2 r 3 ab (19) Additionally, the dipolar coupling energy is described by Eq. (20).    2 E 0 a b 1  3cos 2  maI mbI 3 4 r ab   (20) where φ is the angle between B0 and the internuclear vector and µ0 is the magnetic permeability. The process of becoming an NMR spectroscopist is an ongoing learning experience. Because altering Rabi frequencies can potentially alter experimental data (see 9. Proton TPPM Decoupling to Nitrogen Rabi Frequency Ratio in HCN REDOR for example), it is best to try to keep Rabi frequencies consistent and equivalent between experiments. From my experience as a graduate student, the power output/efficiency of the amplifiers/NMR circuitry can vary over time when users are continually altering experimental setup conditions such as probe soldering, 1 H amplifier tuning, and cable or capacitor swapping to name a few examples. Because of this, it is best to set the pulse length by varying the amplifier input parameter (see Carbonyl π pulse 36 below) than by varying the pulse length so that equivalent radiative fields are used for all experimental data sets that will be compared to each other. Variations in the amplifier input parameters between experiments result in no interpretational ambiguity between experimental data sets as long as equivalent Rabi frequencies are used. Alternatively, variation of pulse lengths used between experiments result in different Rabi frequencies for fixed , Eq. (16), that can yield different experimental data for a sample (see 9. Proton TPPM Decoupling to Nitrogen Rabi Frequency Ratio in HCN REDOR). Alternatively, you can set the amplifier amplitude and vary the pulse length (see 4. Setting Proton π/2 Pulse), but I would recommend doing the former since the Rabi frequencies affect simulated and experimental data while variation of the amplifier input parameters between experiments does not. Quite simply put, over time, there may be variation of the amplifier input parameters used to achieve an equivalent output voltage. Since the REDOR experiments are affected by Rabi frequencies, keeping the nutation angles and pulse lengths the same for all REDOR experiments will simplify comparisons between experimental data. Below, I have listed the setup protocols that will be helpful to new users. The protocols are listed in the order that I would generally perform setup experiments before running HCN REDOR experiments. 37 Figure 13. Proton pulse length arrayed using the I4 peptide, a pd = 1.0 sec and 10 acquisitions per spectrum. The H rf ampl parameter was set to 0.2300 and the H 90 pulse parameter was arrayed from 1.0 to 15.0 µs by 1.0 µs increments. Maximum signal was observed at ~5 µs which approximately corresponds to a π/2 pulse. Zero signal should be observed between 10.0 and 11.0 µs which would correspond to a π pulse. The change in signal intensity is the greatest at points 1 surrounding the π pulse so it is more accurate to set the π/2 H pulse length by identifying the π pulse length and dividing pulse length time by 2. To do this for the above data, the H 90 pulse parameter could be arrayed from 10.0 to 11.0 µs by increments of 0.1 µs and increase the number of scans per spectrum to enhance the signal to noise ratio. Alternatively, a better approach would be to set the H 90 pulse parameter to 10.0 µs and array the H rf ampl parameter. Upon determining which H rf ampl that yields zero signal, change the 10.0 µs to 5.0 µs to set the π/2 pulse. While the proton amplifier output voltage is approximately linear with respect to the H rf ampl parameter, it is not exactly linear so you need to divide the pulse length by 2 and not the amplifier input parameter. 5. Proton Decoupling Field In the HCN REDOR experiments, TPPM decoupling 73 was used which is a windowless 1 series of π pulses on the H channel as reflected by the input parameters TPPM aHdec dec. ampl and pw TPPM dcpl pulse. Proton decoupling prevents the loss of 38 13 CO M due 13 1 CO- H dipolar couplings when the proton decoupling Rabi frequency is greater than the frequency of the 1 13 CO- H dipolar couplings. However, in practice this is only an approximation for REDOR experiments. This statement is supported by Figure 17, where decoupling wR’s > 50 kHz did not improve the signal to noise for the 16.2 ms dephasing time points where as decoupling wR’s > 50 kHz provided better signal to noise per acquisition for the 48.2 ms dephasing time points. For the operational range of our probes (input wR‘s < 95 kHz), higher decoupling fields reduce transverse magnetization loss due to 13 1 CO- H dipolar interactions which results in 13 13 CO CO signal enhancement. However, high decoupling fields can lead to sample heating and arcing of the probe. Arcing occurs when current travels across an unintended path (i.e. path of least impedance is not the intended path) which shorts the circuit and often results in excessive heating. Excessive heating or arcing points can generally be identified by spotting black soot in the LC circuit. The REDOR experiments work with the MAS probes with 4 mm diameter rotors with decoupling fields of 80-90 kHz. Currently, the proton amplifier output is approximately linear for the input 1 parameters in Spinsight that yield ~40-100 kHz H decoupling fields (I have experimental data for one of the amplifiers in my 2 nd NMR notebook on page 23 to support this statement while 1 actual input parameter values depend upon the amplifier). The H Rabi frequency of a pulse can 1 be explicitly quantified by Eq (16), and the H decoupling energy can be quantified by 1 measuring the forward voltage of the H π/2 pulse and comparing it to the forward voltage 1 applied for H decoupling by Eq (21). 39 wR1 wR 2  V1 V2 (21) where V represents the measured voltage going into the probe (use oscilloscope). The forward 1 voltage for a H π/2 pulse is easily measured if the H rf ampl is set equal to the aHcp H CP ampl. 1 1 Otherwise, it is difficult to resolve the H π/2 pulse voltage from the H cross polarization voltage. 40 6. Proton-Carbon Cross Polarization 74 Figure 14. Cross polarization arrayed with the I4 peptide using 5 acquisitions, 1.0 s pulse delay (pd), MAS frequency = 10 kHz, and contact time = 2 ms. Cross polarization is included in 1 the REDOR experiments to transfer magnetization from the highly abundant and polarizable H 13 13 nuclei to the more dilute CO nuclei to increase the CO signal to noise ratio by increasing the signal per acquisition. Additionally, the longitudinal relaxation rate is approximately 4 times 1 13 faster for H than for C nuclei in organic solids which allows for acquisition of ~4 times as many FIDs using cross polarization than for waiting for 13 75 C repolarization . The cp_ramp pulse 1 program was used, and the X cp ampl change parameter was set to 0.04. A 48 kHz H cross polarization and π/2 pulse was used, and the aXcp X cp ampl start parameter was arrayed from 0.00-0.70 by 0.05 increments. Maximum signal intensities were observed between 0.25-0.30 and 0.35-0.40. Either of these regions could be used for CP since the signal intensities are nearly equivalent. Additionally, the contact time also should be arrayed to set up an optimal CP, and it is important to use a setup compound with similar nuclear magnetic relaxation rates. Typical 13 optimal contact times for peptides in REDOR experiments are 1.5-2.0 ms for CO nuclei. 13 Finally, array the X cp ampl change (i.e. the C ramp parameter) parameter to obtain the maximum signal where 0.04 is a typical value, but this value may vary depending upon the amplifier being used. 41 7. Carbonyl π Pulse Figure 15. The cp_zfilter pulse sequence (CP – π/2 – τz – π – acquisition) was used to set the 13 CO π pulse with 10 acquisitions and a 1.0 sec pulse delay using the i4 peptide and a MAS = 10 13 kHz. When the pulse lengths are set correctly, precession of the CO magnetization can be followed using the Bloch Equation, Eq (9) where (1) equilibrium, M = +z; (2) CP, rotates to the xy plane; (3) π/2, rotates to the –z axis; (4) τz - remains along –z axis; (4) π, flips back to +z axis; (5) acquisition with consequent zero signal detection in the rotating frame when pulses are set accurately. In Figure 15, the aX X 180 ampl parameter was arrayed from 0.00 – 0.50 by 0.05 increments, the pw90X X 90 pulse parameter was set to 5.50 µs, and the pw180X X 180 pulse set to 11.0 µs. The 13 CO π pulse was set correctly when zero signal intensity was observed. As evident by Figure 15, the correct amplitude parameter for zero signal is between 0.25-0.30. To more accurately set correct 13 CO pulses, array the aX X 180 ampl parameter from 0.25 to 0.30 by 0.01 increments and increase the number of scans to obtain better signal to noise. 42 8. Nitrogen π Pulse 0.07 0.13 Figure 16. REDOR spectra (32.2 ms dephasing time) of the I4 peptide where each set of black 13 and red spectra have a corresponding CO peak in the S0 and S1 spectra for each arrayed points, respectively. Each S0 and S1 spectrum was the sum of 120 acquisitions with pd = 2.0 s. The aY180 Y 180 ampl parameter was arrayed from 0.07 to 0.21 by 0.01 increments and the pw180Y Y 180 parameter was set to 20.0 sec (wR = 25 kHz as determined by Eq (13)-(16)). Maximum dephasing was observed with the pw 180Y Y 180 ampl = 0.13 where ΔS/S0 = 0.82 where 15 maximum dephasing corresponds to a N  pulse. For further understanding of the REDOR experiment, see 2.4 MAS Solid-State NMR (REDOR). 43 9. Proton TPPM Decoupling to Nitrogen Rabi Frequency Ratio in HCN REDOR 1 Figure 17. The ΔS/S0 of the i4 peptide is plotted against the H decoupling wR to for τ = 16.2, 24.2, 32.2, 40.2, and 48.2 ms dephasing times where the (wR = 20 kHz), the 13 15 15 N wR ratio N π pulse was 25 μsec C π pulse was 11 μsec (wR = 45 kHz), the pd = 2.0 s, and 350 scans were acquired for S0 and S1 of each data point. These HCN REDOR experiments appear to require a 1 15 H decoupling wR to be at least 3.5 times greater than the N wR to obtain maximum ΔS/S0. 13 1 1 13 The CO nuclei were decoupled from H nuclei at a H decoupling to C wR ratios approximately ≥ 1.5. This is evident since equivalent error bars were obtained for data within a 1 13 dephasing time period for H decoupling fields of ≥ 60 kHz. The CO atoms are not directly 1 1 13 bonded to H atoms which results in weaker H- CO heteronuclear dipolar couplings (~3.8 kHz which was determined using Eq. (19) for a rHC of 2.0 Å between the carbonyl carbon and the 1 15 adjacent residue’s amide proton (See 1K09.pdb) relative to amide heteronuclear H- N dipolar couplings (11.648 kHz corresponds to a rHN = 1.015 Å) which is approximately the width of the dipolar powder pattern for amide 1 15 is unclear whether the H decoupling to 1 76 N in proteins . Based upon the current set of experiments, it 15 N ratio is a causation or correlation relationship. There 15 1 may be a H- N dipolar interaction that results in lower S/S0 values where H decoupling 1 15 fields of >80 kHz may be required to average out effects due to H- N dipolar couplings. 44 1 Figure 17 (cont’d). However, the causation for the lower S/S0 with lower H decoupling wR to 15 N wR ratios is not understood. The SIMPSON simulations used in this dissertation did not 1 include H atoms so this affect was not simulated. Additionally, as evident by the larger error 1 bars, less signal per scan was acquired for lower H decoupling wR to decoupling varied while the 13 15 1 15 1 N wR ratios because H N wR was held constant. The lower signal per scan implies that 1 C- H dipolar coupling interactions were not averaged to zero. When the H decoupling 1 13 1 frequency is greater than the H- C dipolar coupling frequency, the effective or average H spin 1 13 quantum state is “0”. This results in no observable H- CO dipolar couplings. 45 10. Chemical Shift Referencing Figure 18. Adamantane 13 C spectrum prior to chemical shift referencing. The transmitter was 13 13 set near the CO Larmor frequencies to increase CO signal intensity in REDOR experiments as described in Figure 10. The chemical shift for the left adamantane peak is 40.5 ppm downfield from the tetramethyl silane (TMS) internal standard reference, but is observed at 113.7 ppm in this figure. Therefore, 154.2 ppm should be added to the chemical shift to correctly reference chemical shifts. Chemical shift referencing is important since referenced chemical 77 shifts of carbon nuclei in peptides provide information about local secondary structure . 2.4 MAS Solid-State NMR (REDOR) Data were collected on a 9.4 T spectrometer (Varian Infinity Plus, Palo Alto, CA) using triple and quadruple resonance MAS probes equipped for 4.0 mm rotors and tuned to 46 13 1 C, H, and 15 N nuclei at respective frequencies of 100.8, 400.8, and 40.6 MHz. The 13 C chemical shift was externally referenced to the methylene resonance of adamantane at 40.5 ppm, and the transmitter was set to ~153 ppm. The 13 13 C 15 CO- N dipolar coupling (dCN) was probed with 1 REDOR experiments with typical parameters: (1) 50 kHz H π/2 pulse; (2) 1.6 ms cross1 polarization with 50 kHz H field and 56-63 kHz ramped 13 C field; (3) dephasing period of duration τ for which the “S0” and “S1” acquisitions had 45 kHz 13 C π pulses at the end of each rotor cycle except the last cycle, and the S1 acquisitions additionally had 25 kHz the middle of each rotor cycle; and (4) 13 54,78 C detection 15 N π pulses in . The MAS frequency was 10 kHz, the 1 recycle delay was 2 s, 85 kHz TPPM H decoupling was applied during the dephasing period ( = 2.2, 8.2, 16.2, 24.2, 32.2, 40.2, and 48.2 ms) and detection period. The XY-8 phase cycling (x, y, x, y, y, x, y, x) was used for the heteronuclear train of π pulses during τ except for the last 13 73,79 CO π pulse . Samples were typically cooled by nitrogen gas at –50 °C to enhance 80 signal and reduce motional averaging of dCN . The typical difference between 13 13 CO C shift in cooled and uncooled membrane-associated HFP samples is ≤ 0.5 ppm and indicates little 51 variation in secondary structure with temperature . NMR pulse sequences are designed to distinguish observable interactions between nuclei in an NMR sample. In solid state NMR experiments, this can be challenging because a functional groups, such as a 13 CO, have a unique chemical shift that depends upon the orientation relative to the B0 field. Peptide/protein NMR samples contain many 47 13 13 CO bond CO bonds which results in a broad distribution of 13 CO bond orientations and therefore a broad distribution of chemical shifts (also commonly referred to as a powder pattern). By spinning a sample at a high frequency along an axis, the average orientation of a 13 CO bond over one rotor period lies along the axis of rotation which makes the average orientation of each all 13 13 CO bond equivalent for CO functional groups over a rotor period. When the axis of rotation set to be the magic angle, 54.7°, the dipolar coupling energy has an average value of zero over each rotor period as per Eq (20). This results in much narrower line widths. Peptides and proteins contain many carbonyl and amide functional groups and solid state NMR experiments often detect The 13 13 CO signals to obtain protein and peptide structural information. CO REDOR experiments collect S0 and S1 spectra where individual 13 CO nuclear magnetic moments precess in the xy plane of the rotating frame at different rates due to different shielding from the B0 field and dipolar couplings. These different rates of precession lead to decoherence of the 13 CO nuclear magnetic moments. Because magnetization is the sum of nuclear magnetic moments, decoherence of the 13 CO nuclear magnetic moments in the xy plane leads to a smaller magnitude vector sum and detection of a smaller observable magnetization (i.e. smaller 13 CO signal). Consider the different precession rates of the 13 CO 13 CO nuclear magnetic moments due different shielding from the B0 field. Ignoring longitudinal relaxing contributions that result in the loss of transverse magnetization, coherence of the nuclear magnetic moments can be reestablished by introducing time synchronized 48 13 13 CO CO π pulses 81 as accomplished in the Hartman-Hahn Spin-Echo pulse sequence . Thus, one function of the 13 CO π pulses in a REDOR experiment is to establish coherence of the nuclear magnetic moments in the rotating frame at the time of signal acquisition to enhance the signal acquired per acquisition. Additionally, the REDOR experiments in this dissertation introduced labeled residues making 13 13 CO and 15 N 15 CO- N dipolar couplings (dCNs) an observable property. The dCN can be related to distance between 13 CO and 15 3 N nuclei (rCN) by dCN = 3080/rCN where d and r are expressed in Hz and Å, respectively (See 2.5 SIMPSON Simulations for discussion regarding isolating observable NMR interactions). A dipolar coupling interaction between two nuclei lead to precession of the 13 C magnetization where xy-plane of the rotating frame. The rate of precession of this for 13 13 13 C magnetization is detected in the CO magnetization that results from 15 C - N dipolar coupling interaction can be described by Eq (22) d wd   CN 3cos 2   1 CN   2   (22) where φ is the angle between B0 and the internuclear vector. This observable property can be negated by magic angle spinning (MAS) which makes the average value for φ equal to 54.7°. Experimentally, this is achieved by making a 54.7° angle between the rotor axis and B0. Spinning the rotor at a high frequency makes the average orientation for any internuclear vector equal to 54.7° resulting in an average dipolar coupling equal to zero. Thus, MAS results in no net evolution of 13 C transverse magnetization from a dipolar coupling interaction as observed in the S0 spectra in REDOR experiments. In the S1 spectra, a 49 13 C or 15 N π pulse is applied at the middle and end of each rotor period. The direction of the precession of the magnetic moments in the rotating frame that results from this for interaction can be reversed by either a 13 C or 15 13 82 13 CO nuclear 15 C - N dipolar coupling N π pulse . By applying a the middle and end of each rotor period, the precession of the 13 13 C or 15 N π pulse at CO nuclear magnetic moments in the rotating frame is nonzero. This results in average dipolar coupling interaction making dCN an observable property in the S1 spectra. The precession of the due to 13 13 CO nuclear magnetic moments 15 C- N dipolar coupling interactions results in decoherence of the magnetic moments and consequent loss of 13 CO magnetization (i.e. 13 13 CO nuclear CO signal). This decoherence can be quantified in terms of the average dCN in the S1 spectra relative to the S0 78 spectra . The dCN were quantified by comparing the (no average 13 13 CO evolution due to dCN) to the reduced CO signal intensities of the S0 spectra 13 CO signal intensities of the S1 spectra -3 (evolution from dCN) where dCN α rCN . Greater reduction of the 13 CO signal intensity in the S1 spectrum relative to the S0 spectrum generally occurred for longer . These 13 CO peak intensities were denoted S1 and S0 and were determined from integration over a shift range that encompassed most of the 13 CO signal. The width of the integrated region was different for each sample type, but the normalized dephasing was quantified by: exp exp exp exp  S  S S S 0 1 1     1  exp exp S  S S  0 0 0 (23) 50 and its standard deviation:  exp  S    S   0  2  S   0   2      S       S   1   0 2  S S  S   0  0 1          2 (24) where S0 and S1 were the experimental root-mean-squared deviations of the spectral 83 intensities derived from 12 regions of the spectrum that did not include spectral features . The calculation in Eq (24) is derived from error analysis of (ΔS/S0) and did not consider a covariance term. Alternatively, error analysis could be derived from calculating (1-(S1/S0)).  exp S  1 S  0       S  1  S  1  2    S   0   S   0         2 (25) Eq (24) was used to quantify error in this dissertation and comparisons between these two methods for calculating error are found in Table 3 and Table 4. Typical variation in σ between these error analysis methods appears to be 0.000 to 0.003 for our experiments. 51 exp Table 3. Error analysis for V2E-F8CG13N. τ (ΔS/S0) 48.2 0.400 40.2 0.379 32.2 0.330 24.2 0.262 16.2 0.146 8.2 0.056 2.2 0.009 S0 100.2861 94.9187 106.9589 124.2104 127.5113 70.7462 81.0649 S1 60.4066 58.9761 71.6677 91.6186 108.9537 66.8136 80.3265  S0  S1 0.767 0.980 0.777 0.836 0.637 0.329 0.549 1.435 0.808 0.421 0.878 0.612 0.316 0.303 Error (Eq (24)) 0.017 0.014 0.009 0.010 0.007 0.006 0.008 Error (Eq (25)) 0.015 0.011 0.006 0.009 0.006 0.006 0.008 Error (Eq (24)) 0.021 0.016 0.011 0.013 0.007 0.010 0.009 Error (Eq (25)) 0.019 0.014 0.010 0.012 0.007 0.010 0.009 Table 4. Error analysis for HFP-L12CA6N. τ (ΔS/S0) 48.2 0.275 40.2 0.247 32.2 0.192 24.2 0.155 16.2 0.099 8.2 0.058 2.2 0.004 S0 135.2543 109.808 137.7211 126.5162 194.515 86.3175 74.1142 S1 98.0556 82.7338 111.2584 106.8944 175.3549 81.3029 73.7822  S0  S1 1.658 1.292 1.193 1.231 1.136 0.363 0.323 2.267 1.127 0.988 1.07 0.853 0.758 0.57 2.5 SIMPSON Simulations MAS solid state NMR pulse sequences are typically designed to limit the number of observable spin interactions that affect signal intensities and line shapes in order to simplify experimental data interpretations. REDOR experiments incorporate heteronuclear trains of π pulses that are rotor synchronized and result in time dependent signal attenuation in the S 1 spectra relative to the S0 spectra that results from a loss of transverse magnetization due to dipolar couplings. Quantification of these dipolar couplings can be achieved by comparing the experimental data to simulated data. REDOR experiments can be simulated with the SIMPSON 84 program which was used as a tool to evaluate the evolution of the time-dependent Schrödinger equation. The SIMPSON program is an accessible program that is capable of simulating many 52 different pulse sequences, but I will provide a brief introduction from the literature 84 with key concepts relevant to the REDOR pulse sequence highlighted. The SIMPSON simulations use the Liouville Von Neumann equation d  (t )  i  H (t ),  (t ) dt (26) where specific observables can be quantified using time dependent wavefunction/s (ψ(t,)) timedependent Hamiltonian/s (H(t)) and the density matrix operator (ρ(t)). In general, a time dependent wavefunction, ψ(t), can be used to describe quantum mechanics. While ψ(t) can be a linear combination of many basis states, ψ(t) is defined below for a simple wavefunction with two basis states:  (t )   c p (t ) p p 1,2 (27) where ψp represents two time independent basis states, and the time dependent terms are accumulated in the coefficients, cp(t). Wavefunctions can be incorporated into density matrix theory to follow the time evolution of any operator using the time dependent density matrix operator, ρ(t), where  (t )   (t )  (t ) (28) For a simple two-state quantum mechanics system, this can be represented in matrix form by: 2    c t   c  t  c*  t    c1  t  1 1 2  c*  t  c*  t     (t )    2  c t   1 2   c  t  c*  t  c t   2  1 2  2    When normalized, the trace(ρ(t)) = 1. The expectation value for an operator, such as H(t), can be calculated as follows: 53 (29)  H (t )11 H (t )12   c1  t   *   H (t )   (t ) H (t )  (t )  c1  t  c*  t   2  H (t ) H (t )   c  t    21 22  2  *  c  t  2 H (t )  c  t  c*  t  H (t )  c  t  c1  t  H (t )  c  t  2 H (t ) 2 1 11 1 21 2 12 2 22  c t  2 c  t  c*  t    H (t ) H (t )  2   1 1 11 12   trace  (t ) H (t )   trace      H (t ) 2  H (t )   c  t  c*  t  c t   21 22  1 2  2    (30) A detailed step by step description describing how to use density matrix theory to simulate NMR 85 experiments for a simple spin ½ system is described in literature , and further expansion of 86 density matrix theory to a two spin system is also described in literature . These articles describe the basic principals that were used to build the framework for SIMPSON simulations. Currently, these two aforementioned papers can be obtained through MSU’s interlibrary loan system, and in my graduate school experience, these papers provided the most clear and complete overview of density matrix theory for NMR experiments. Furthermore, the Liouville Von Neumann equation can be rewritten to evaluate the density operator at a given time by:  ( t )  U (t , 0)  (0)U * (t , 0) (31) where ρ(t) is the density operator at a given time, ρ(0) is the density operator at equilibrium, and U(t,0) is a unitary operator used to evaluate the time evolution of spin system for time 0 to t.  t U (t , 0)  T exp i 0 H (t ')dt '  (32) where T is the Dyson time-ordering operator that contains noncommuting terms of the Hamiltonian. This expression can be simplified to express as a time-ordered product. n 1 U (t , 0)   exp  iH ( j t )t  j 0 (33) 54 where n is the number of time increments needed for time 0 to t for which the H(t) can be approximated to be time-independent for each Δt increment. For a simple one spin system, spin I, the components of the Hamiltonian can be described by: (34) H  H RF  H CS  H J  H D  H Q where the Hamiltonians for the resonant frequency pulse (RF), chemical shift (CS), J spin coupling (J), dipole-dipole coupling (D), and quadrupolar coupling (Q) can be incorporated into the unitary operator. The HCN MAS REDOR experiments allow for isolation of the HRF, HCS, HD terms. H RF   wa (t ) ( I ax cos a  I ay sin a ) rf a (35) a H CS   wCS , 0 (t ) I az a (36)    ˆ ˆ H D  0 a b 1  3cos 2  I az Ibz 3 4 r ab   (37) where a and b refer to specific nuclei spins, ϑ is the phase of the applied field,  is the angle between B0 and the internuclear vector, and µ0 is the magnetic permeability. Thus, input parameters that affect these Hamiltonians are incorporated into the SIMPSON simulations and allow for comparisons between experimental data and simulated data. For HCN experiments, the 13 15 CO- N dipolar coupling (dCN) can be related to distance between by 13 CO and 15 N nuclei 3 (rCN) by dCN = 3080/rCN where d and r are expressed in Hz and Å, respectively. NMR samples were modeled as a sum of S0 and a sum of S1 signals of different structure dependent spin geometries of 13 CO and 15 63 N nuclei (see Chapter III). Unless otherwise noted, rCN > 7 Å 55 were not included in simulation and S1 = S0 for these nuclei. The notation (ΔS/S0) sim will be generally used for simulated (ΔS/S0) and can refer to a particular generalized spin geometry or to the population weighted sum using calculations from different spin geometries over the range of experimental dephasing times . For the former case, the (S1/S0) sim   were calculated using the SIMPSON program with input parameters that included dCNs as well as Euler angles in a fixed crystal frame for each 13 84,87 principal axis system SIMMOL program using 15 CO- N vector and for the 13 CO chemical shift anisotropy (CSA) . For 5 spin simulations, these input parameters were calculated by the 13 CO and 15 N coordinates from a region of a high-resolution crystal 88 structure with the appropriate structural motif, e.g. antiparallel  sheet . Similarly, the (α, β, γ) Euler angles of dCN PAS were (0, 0, 0) and (0, ϑ, 0) for 2 and 3 spin simulations, respectively, where ϑ was the angle between two dCN vectors. Specific Protein Data Bank (PDB) files used to model spin geometries associated with a structural motif are noted in the Materials and Methods section of each Chapter that simulations were used. For each spin geometry, (S1/S0) sim was the average of ten different SIMPSON calculations, and each calculation was based on input parameters from a different set of atomic coordinates. The 13 89 CO CSA principal values of 247, 1 176, and 99 ppm were inputs to the SIMPSON calculations , and Hs and relaxation rates were not considered. The same set of chemical shift input parameters were used for all comparisons between simulated and experimental data. Using the same 13 CO CSA principal values as above, the isotropic chemical shift was varied between 167-180 ppm for a 5 spin system (four 56 15 N, one 13 CO) which more than covered the range of typical β sheet isotropic chemical shifts and the (S1/S0) sim at 48 ms was affected by less than 0.01. SIMPSON and SIMMOL script files for each spin geometry type can be found in Appendix VI where spin geometry type refers to the number of 13 CO and 15 N nuclei used in each simulation (not structure type). 57 Chapter III. Natural Abundance Calculations for Solid State NMR REDOR Experiments and Quantitative Determination of In-Register Parallel β Sheet Registries in MembraneAssociated HFP 3.1 Background. The HFP structure-function literature includes NMR data showing random coil structure 32,33 for HFP in aqueous solution . Solid-state nuclear magnetic resonance (SSNMR) has shown predominant  sheet structure for residues 1-16 of membrane-associated HFP (mHFP) where the membranes contained ~30 mol% cholesterol, which is comparable to the mol% cholesterol of membranes of HIV and host cells of HIV 37,50,58,90 . A fluorescence and infrared (IR) study reported the time-resolved courses of HFP structural changes, and the intervesicle lipid mixing 34 function following addition of a HFP solution to a membrane vesicle solution . The experimental rates (Rs) were ordered RHFP membrane binding > RHFP β sheet formation > Rlipid mixing and were consistent with the sequence: (1) random coil HFPs binds to membrane vesicles and HFP structure changes to oligomeric β sheet; (2) vesicle fusion. The biological relevance of HFP oligomers, potentially β sheet conformation, is further supported by the molecular trimer structure of soluble regions of the gp41 ectodomain 17,18 (~175 residues). In these structures, the residue 30-80 region of each molecule forms a continuous helix, and the helices of the different molecules form a parallel coiled-coil. The Thr25 residue places the N-terminal helical regions of three gp41 close to one another. The Thr-25 is C-terminal of the approximate 16 residue N-terminal fusion peptide which motivated the study of a C-terminally cross-linked HFP trimer (HFPtr). Relative to HFP monomer, HFPtr induced membrane vesicle fusion with ~15-40-fold faster rate which supported the functional 58 30 significance of the trimer . Although both mHFP and mHFPtr formed  sheet oligomers, mHFPtr is more deeply inserted which correlates with greater membrane perturbation and a 40 reduction of the vesicle fusion activation energy . The in vivo importance of fusion peptide oligomers was also demonstrated by dominant inhibition of fusion and infection in viruses and cells for which a small fraction of the gp41 had the V2E point mutation in the fusion peptide 21,41 region . Analyses of these data supported the involvement of multiple gp41 trimers and 42 fusion peptides in fusion . Electron micrographs of virus-cell contacts have also been 43 interpreted to show multiple gp41 trimers at the contact site . Functional importance of fusion peptide trimers has also been demonstrated for fusion peptides of other viruses 44,45 . Because of the aforementioned functional significance of HIV fusion peptide oligomers, there has been effort to elucidate the distribution of structures of mHFP oligomers. SSNMR has played a key role in this effort in particular for samples prepared in a manner similar to that of fusion assays with addition of an aqueous fusion peptide solution to a membrane vesicle 37 solution . Appendage of a C-terminal lysine tag to HFP greatly reduced HFP aggregation in aqueous solution and allowed separation of pelleted fused vesicles with bound HFP from unbound HFP in the supernatant of a HFP Ala-1 13 30,33,49 CO(carbonyl)-lipid 31 . HFP/lipid binding was supported by SSNMR detection 40 P distance of ~5 Å . For mHFP, the 13 C chemical shifts derived from an unambiguous assignment were consistent with a fully extended  strand 50 conformation for residues between Ala-1 and Gly-16 . Detection of intermolecular and 13 15 13 13 C- C C- N distances of ~5 Å supported  sheet oligomer/aggregate structure, and the Ala-1 59 13 CO-lipid 31 40,50,51 small P contact and other data suggest that the number of molecules in the oligomer is . This chapter focuses on quantitative determination of populations of in-register parallel  36 sheets relative to the previous identified antiparallel β sheet registries . The clearest information to-date on this topic has been a SSNMR experiment on membrane-associated HFP with an Ala14 13 CO label and a Gly-3 15 N label whose separation (rCN) was >20 Å along a single  50 strand . SSNMR can detect labeled 13 15 3 CO- N dipolar coupling (dCN) where dCN = 3080/rCN with d in Hz and r in Å. The minimum detectable dCN ~10 Hz correlates with rCN ~7 Å so that detectable dCN in this sample were necessarily ascribed to inter- rather than intramolecular 13 15 CO- N proximity. SSNMR detection of dCN > 30 Hz strongly supported a significant fraction of molecules with intermolecular Ala-14-Gly-3 hydrogen bonding and labeled rCN of 4.1 and 5.5 Å, i.e. 161/116 antiparallel  sheet registry (t = 16). Figure 19 displays this registry with isotopic labeling from the present study and not the earlier study. Detection of similarly large dCN in an Ala-14 13 CO/Ile-4 15 N HFP sample supported a fraction of 171/117 antiparallel registry (t = 17). As noted above, multiple gp41 trimers are required for membrane fusion and close proximity of multiple gp41 trimers make both parallel and interleaved antiparallel  sheet oligomers potential models for HFP oligomeric structure in vivo. 60 13 15 Figure 19. (a) HFPs where red and blue correspond to CO and N labeled residues, respectively. (b) HFP-NC, HFP-P, HFP-A, and HFP-AP were SSNMR samples which each 13 15 contained a mixture of CO and N labeled peptides in 1:2 mol ratio. The HFP-NC sample was a mixture of HFP-F8 and HFP-A6L7 that had been lyophilized separately. The other samples 13 were membrane-associated HFPs that formed  sheet structure with a molecular mixture of CO and 15 N labeled peptides in the sample. (c) Registries probed by the SSNMR REDOR 13 15 experiments and labeled CO/labeled N proximities for the membrane-associated HFPs in these registries. Consideration of residue 116 or 117 registries is based on the fully extended conformation in HFP. For parallel sheets, there is CO (residue h) – HN (residue h+1) hydrogen bonding of adjacent strands. 61 At most half of the membrane-associated HFP molecules were in the 161/116 or 171/117 registries, i.e. a large fraction of the molecules were in registries not detected in either the Ala-14 13 CO/Gly-3 15 N or Ala-14 13 CO/Ile-4 15 N labeled samples. Because of the close proximity of the Thr-25 of the three molecules of the gp41 trimer, a reasonable hypothesis for a populated HFP registry is in-register parallel  sheet, i.e. 117/117 in Figure 19c. An earlier SSNMR study attempted to test this hypothesis using samples each containing an equimolar mixture of two labeled HFPs, one with three sequential backbone other with three sequential backbone Gly-5-Leu-7 13 CO/Gly-5-Leu-7 15 15 13 CO labels and the 29 N labels . Detection of an average dCN > 10 Hz for a N sample and a Phe-11-Gly-13 13 CO/Phe-11-Gly-13 15 N sample were consistent with a fraction of in-register parallel HFP molecules. However, because the samples were extensively labeled, the data were also consistent with other parallel or antiparallel registries. In addition, the data reflected an average of many intermolecular dCNs so it was not possible to determine the fraction of molecules with a particular registry. There have also been efforts to detect in-register parallel structure using SSNMR measurement of intermolecular 13 13 3 C- C dipolar couplings (dCCs) where dCC = 7710/rCC with dCC in Hz and rCC in Å. For mHFP with a single 13 CO label and in-register parallel structure, the labeled 52,53 interstrand rCC ~5 Å with dCC ~70 Hz that is 13 . These parameters will be independent of the residue CO labeled. For mHFP with Phe-8 13 CO, a best-fit dCC ~70 Hz was detected whereas for mHFPtr, dCC depended on the position of the labeled 62 13 CO residue with a range of 10-60 54,55 Hz . This residue dependence argued against a major fraction of in-register parallel structure in HFPtr. There was also an IR spectroscopy effort to distinguish between the 117/117 parallel and 161/116 antiparallel registries using samples that contained backbone 13 CO labeling at either: (1) Ala-1 to Val-3, Gly-5 to Ile-9; (2) Phe-8 to Gly-16; or (3) Ala-1 to Val-3, Gly-5 to 47 Gly-16 . The IR wavenumbers and intensities of different samples were interpreted to support a large fraction of in-register parallel structure and little t = 16 antiparallel structure. However, in my view, the extensive labeling of the IR samples precluded quantitation of specific registries and greater support for this argument is provided in the Discussion section. This chapter reports a determination of the fraction of in-register parallel structure in mHFP oligomers. This study was motivated because of: (1) the functional significance of HIV fusion peptide oligomers; and (2) the existing undefinitive and conflicting data and interpretations relevant to this question. As part of this effort, an experimentally validated model was developed to quantify effects of natural abundance 13 C and 15 N nuclei on SSNMR measurements of dCN. 3.2 Materials and Methods. 1. SSNMR samples. As shown in Error! Reference source not found.b, each sample contained a 15 13 CO and a N labeled peptide in a 1:2 mol ratio. Three samples contained mHFP  sheet oligomers/aggregates that were each a statistical mixture of 13 CO and Detection of substantial dCN by SSNMR indicated proximity of the labeled 63 15 13 N labeled HFPs. CO and 15 N nuclei on adjacent strands and was used to estimate the fractional populations of specific registries as detailed below. As shown in Figure 19c, the HFP-P sample was designed to detect parallel 117/117 and 217/116 registries, the HFP-A sample was designed to detect previously observed antiparallel 161/116 and 171/117 registries (t = 16 and t = 17), and the HFPAP sample was designed to detect both parallel 117/117 and 217/116 registries as well as antiparallel 161/116 and 171/117 registries. In addition to the potential proximity of labeled be proximity between labeled proximity between some na 13 13 13 CO and 15 N nuclei, there will always CO and some natural abundance (na) 15 CO and labeled 15 N nuclei as well as N nuclei. These proximities will contribute to the dCN detected in the SSNMR experiment and should be included in the data modeling. Quantitative understanding of these proximities required a negative control (HFP-NC) sample with: (1) the same relative fractions of labeled and HFP-AP samples; and (2) labeled 13 13 CO, CO – labeled 15 N, and na sites as the HFP-P, HFP-A, 15 N rCN that are much greater than the approximate REDOR detection limit of ~7 Å. One possibility was a sample made like HFP-P, HFP-A, and HFP-AP but with labels at sites that do not form intermolecular hydrogen bonds. This possibility was not pursued because the distribution of registries of mHFP was not yet welldefined at the time of these experiments. Instead, the HFP-NC sample was a physical mixture of lyophilized HFP-F8 (5.0 mg) and HFP-A6L7 (10.0 mg) without any membrane. Each peptide was lyophilized separately and the two peptides were then mixed in the solid phase to form a uniform physical mixture. Water and membrane were not added to the physical mixture so that the labeled 13 COs and 15 Ns remained much farther apart than the 7 Å, the approximate REDOR 64 detection limit. Although there were populations of  sheet as well as  helical lyophilized peptides in the HFP-NC sample, each population yielded very similar (S/S0) – see Results section for further details. 2. Modeling. Experimental dephasing of a mHFP sample was modeled as a sum of S0 and a sum of S1 signals of different spin geometries of statistical distributions of na 13 CO and 13 15 CO and 15 N nuclei where the geometries reflected N nuclei as well as geometries of (1) 117/117 and 217/116 parallel adjacent strand registries; (2) 161/116 and 171/117 antiparallel registries; and (3) other “X ” registries where all labeled rCN > 7 Å and S1 = S0. The notation (ΔS/S0) sim will be generally used for simulated (ΔS/S0) and can refer to a particular spin geometry or to the population weighted sum using calculations from different spin geometries. For the former case, the (S1/S0) sim  l were calculated using the SIMPSON program with input parameters that included dCNs as well as Euler angles in a fixed crystal frame for each vector and for the 13 84,87 CO chemical shift anisotropy (CSA) principal axis system parameters were calculated by the SIMMOL program using 13 CO and 15 13 15 CO- N . These input N coordinates from a region of a high-resolution crystal structure with the appropriate structural motif, e.g. parallel  88 sheet . Coordinates were obtained from the following Protein Data Bank (PDB) files: 1JK3, 1IGD, 1NKI, 2E4T, 1CEX, 1MNZ, and 2IWW. For each spin geometry, (S1/S0) sim was the average of ten different SIMPSON calculations and each calculation was based on input 65 parameters from a different set of atomic coordinates. The 13 176, and 99 ppm were inputs to the SIMPSON calculations CO CSA principal values of 247, 89 1 and Hs and relaxation were not considered. Chemical shift and conformational distributions. Figure 20 displays REDOR S0 and S1 13 C SSNMR spectra for  = 32.2 ms. Each S0 spectrum has a ~50% contribution from the labeled 13 CO and ~50% contribution from na 13 COs of the unlabeled residues. The full-width at half- maximum linewidths of the mHFP samples in Figure 20b-d are 3-4 ppm and indicate a distinct secondary structure. For the HFP-AP sample with Phe-8 ppm is the same as was observed for Phe-8 13 13 CO label, the peak 13 CO of mHFP in known  strand conformation and 50,54 is very different from the 178 ppm shift observed in  helical conformation and HFP-A samples with Leu-12 13 CO shift of 173 . For the HFP-P CO label, the 174 ppm peak shift is also the same as  strand HFP and different from the 179 ppm shift of Leu in helical HFP 54,91 . Overall, the shifts and linewidths are consistent with the fully extended conformation that has been observed for the first sixteen residues of HFP associated with membranes with biologically relevant cholesterol 50 content . The linewidth of the lyophilized HFP-NC sample with F8 13 CO label is ~7 ppm and correlates with a broad distribution of secondary structures that is also evidenced by a 176 ppm peak 13 77 CO shift that is midway between typical Phe helical and  strand shifts . S0 and S1 signals were quantified by integration of 8 ppm regions for the HFP-NC spectra and 5 ppm 66 regions were used for HFP-P, HFP-A, and HFP-AP spectra. These regions were chosen to encompass most of the 13 CO signal, and they also resulted in low σ exp . 3.3 Results. 1. Qualitative Analysis of the REDOR Data. Relative to the S0 signals, there is attenuation in the S1 signals of 15 13 COs within ~7 Å of Ns and the associated S/S0 normalized dephasing increased with dephasing time, Figure 20 13 and Figure 21. Because of the physical separation of the NC, the S1 attenuation and (S/S0) Ala-6,Leu-7 15 exp CO and of this sample reflected Phe-8 N proximities but not Phe-8 13 CO-Ala-6,Leu-7 exp Figure 21a. There was similar S1 attenuation and (S/S0) 21b, which demonstrated that there was little Leu-12 13 15 15 13 N labeled HFPs in HFP- CO-na 15 N and na 13 CO- N proximity, Figure 20a and in HFP-P, Figure 20b and Figure CO-Gly-13,Ala-14 15 N proximity in HFP-P and only a small fraction of parallel 117/117 and 217/116 registries. There was much larger S1 attenuation and (S/S0) which indicated significant Leu-12 13 exp for the HFP-A sample, Figure 20c and Figure 21b, CO-Gly-5,Ala-6 15 N proximity and therefore a substantial fraction of antiparallel 161/116 and 1717/117 registries. Comparably large S1 attenuation and (S/S0) exp were observed for the HFP-AP sample, Figure 20d and Figure 21b. The similarity of the HFP-NC and HFP-P data and the similarity of the HFP-A and HFP-AP data were consistent with ascribing Phe-8 13 CO-Leu-9,Gly-10 15 N proximity in HFP-AP to antiparallel 161/116 and 171/117 registries rather than parallel 117/117 and 67 217/116 registries. Detection of a substantial fraction of these antiparallel registries is 50 consistent with earlier SSNMR data for sparsely labeled HFP . Detection of only a small fraction of parallel registries is a new result and disagrees with previous interpretations of SSNMR and IR data for samples with extensive labeling 29,47 . These data highlight the importance of sparse labeling to reduce interpretational ambiguity for systems with a structural distribution like mHFP. 13 Figure 20. REDOR S0 and S1 C SSNMR spectra at 32.2 ms dephasing time for (a) HFP-NC, (b) HFP-P, (c) HFP-A, or (d) HFP-AP. Each spectrum was processed with 200 Hz line broadening and baseline correction and was the sum of: (a) 38624; (b) 23488; (c) 24914; or (d) 13 14240 scans. Relatively narrow CO signals were observed in the HFP-P, HFP-A, and HFP-AP samples because the HFPs were membrane-associated with predominant  sheet conformation at 13 13 the labeled CO site. A broader CO signal was observed in the HFP-NC sample because there was no membrane and there were populations of lyophilized HFP with either  helical or β sheet 13 conformation at the labeled CO site. 68 exp sim Figure 21. (a) Plot of REDOR (ΔS/S0) (filled squares with error bars) and (ΔS/S0) (open sim circles) vs dephasing time for the lyophilized HFP-NC sample. The (ΔS/S0) were calculated using a mixture of nad models with fractional populations:  helical, 0.5; min  sheet, 0.25; max exp  sheet, 0.25. (b) Plots of (ΔS/S0) vs dephasing time for: HFP-NC, open triangles; HFP-P, filled triangles; HFP-A, open circles; HFP-AP, filled circles. The typical  exp Variation of 0.02 in (ΔS/S0) same sample type, e.g. HFP-A. exp is 0.02. was also observed between two different preparations of the 69 2. Natural Abundance Models. exp Quantitative analysis of the (S/S0) to yield the fraction of parallel and antiparallel HFP registries requires an accurate natural abundance dephasing (nad) model, i.e. a model that accounts for effects of labeled 13 CO-na 15 N and na 13 CO-labeled 15 N proximities. Both types of proximities were considered but for conciseness of presentation, the discussion in this chapter focuses on labeled 13 CO-na 15 N. One measure of validity of a nad model was agreement within experimental error between (S/S0) exp of HFP-NC and (S/S0) sim of the model. Consideration was first given to the HFP-F8 regions of HFP-NC including the spin geometries of one or two labeled 13 COs and one na 15 N. Geometries with two or more na because the fractional isotopic abundance of 15 SIMPSON program was used to calculate (S1/S0) 15 Ns were not considered N is only 0.0037. For each geometry, the sim as a function of the dephasing time . Only geometries with rCN < 7 Å were considered because those with rCN > 7 Å do not affect (S1/S0) sim within our experimental signal-to-noise for these experiments. We consider this a “long-range” nad model which is distinguished from a “short-range” model of earlier studies that only considered na nuclei separated by one or two bonds from a labeled nucleus, i.e. rCN < 3 50,54 Å . The broad spectral linewidth of HFP-NC indicated both helical and β strand conformational populations and coordinates of spin geometries for both  helical and β sheet structures were obtained from corresponding regions of high-resolution structure PDB files. For  helical structure, the rCN < 7 Å criterion resulted in geometries with a single labeled 70 13 CO at residue h and a single na 15 N at a residue between h – 3 and h + 5. These nine geometries are one aspect of the  nad model. Figure 22 illustrates relevant labeled 13 COs and na 15 Ns for antiparallel β sheet structure. The strands in panels a and c are “fully constrained” to a single registry with resultant six unique spin geometries. Three geometries had one labeled same strand and three geometries had two labeled 13 13 CO and one na 15 N within the COs on vicinal strands and one na 15 N in the intervening strand. In panels b and d, the strands have different registries so that the labeled 13 CO in the top strand was >7 Å from the nine na 15 N sites of the 13 CO of the third strand. The structure of panels b and d has nine unique spin geometries and is denoted a maximum  sheet nad (max  nad) model while the structure of panels a and c has six geometries and is denoted a minimum  sheet nad (min  nad) model. In either structure there are nine na Å of each labeled 13 “shared” between two na 15 N sites per 13 15 N sites within 7 CO but in the min nad model, some sites (e.g. 4-6 in Figure 22c) are 13 COs, i.e. within 7 Å of two CO and the overall nad. 71 13 COs. This reduces the average number of Figure 22. (a, b) Schematic diagrams of the HFP-F8 region of the HFP-NC sample in 13 antiparallel  sheet structure with labeled COs represented as red circles. Panel a shows a model that is fully constrained to a single registry while panel b shows multiple registries. (c, d) β sheet backbone representations of the respective boxed regions of panels a and b with labeled 13 15 15 COs in red and possible na N sites in blue, i.e. sites for which a na N is within 7 Å of a labeled 13 CO. A particular spin geometry will have only one 15 in panel c and each spin geometry will have either one labeled or two labeled 13 COs and one na 15 N. The min nad model is shown 13 CO and one na 15 N (#1, 2 or 3) N (#4, 5, or 6). The max nad model is shown in panel d and each spin geometry will have one labeled 13 CO and one na 72 15 N. Table 5. Chapter III indices and parameters Index/parameter Description Values ft fractional population of structure t j, k index for n.a. site <7 Å from a labeled 15 13 site: j, n.a. N near labeled CO; k, n.a J, K 13 CO near labeled 15 determined by fitting N number of n.a. sites <7 Å from a 15 labeled site: J, n.a. Ns near labeled 13 CO; K, n.a 13 COs near labeled 15 N  helical structure, J = 9, K = 10; min  sheet structure, J = 6, K = 8; max  sheet structure, J = 9, K = 12 0  no dipolar evolution 1  dipolar evolution l REDOR data type index m datum index Sl, Slj, Slk, Slu REDOR signal intensity t, t1, t2 structural population index; for unconstrained model of membr. assoc. samples, t1 indexes the top/middle 1, 2, 3, 4, 5, 6, or 7 determined by experiment or calculation registry and t2 indexes the middle/bottom registry for membr. assoc. samples: p  parallel registry a  t = 16 or 17 antiparallel registry c  other “X” registry membr. assoc. sample index 1  HFP-P 2  HFP-A 3  HFP-AP v index for arrangement of three adjacent labeled HFPs in membr. assoc. samples – middle HFP has 13 13 CO labeling and the CO is hydrogen bonded to HN of top HFP 1  CO HFP (top), CO HFP (bottom) 15 13 2  N HFP (top), CO HFP (bottom) 13 15 3  CO HFP (top) , N HFP (bottom) 15 15 4  N HFP(top), N HFP(bottom) wv fractional population of arrangement of three adjacent labeled HFPs in membr. assoc. samples u 13 13 fully constrained model: w1 = 1/9, w2 = 2/9, w3 = 2/9, w4 = 4/9; unconstrained model: w1 = 1/81, w2 = 2/81, w3 = 2/81, w4 = 4/81 na lN (), na lC ( ) Sl ( )/S0: lN ( ), labeled 15 15 N pairs; lC ( ), n.a. 13 13 CO-n.a. CO-labeled N pairs 73 0N ( ) = 1; 0C ( ) = 1; 1N ( ) and 1C ( ) determined by calculation Table 5 (cont’d). lab Slu lab ( )/S0 13 lab for arrangement of 15 lab lab lab 0tuv lab ( ) = 1; 0t1t2uv ( ) = 1; ltuv (), lab lt1t2uv ( ) labeled  REDOR dephasing time 2.2, 8.2, 16.2, 24.2, 32.2, 40.2, or 48.2 ms nad 1 = min β nad model 2 = max β nad model 3 = α helical nad model b CO and N nuclei: lab 1tuv ( ) and 1t1t2uv ( ) determined by calculation 1tuv ( ), fully constrained model; lab 1t1t2uv ( ), unconstrained model There are many indices and parameters in the quantitative modeling and descriptions and possible values for them are compiled in Table 5. For the , min , or max  nad models, the average l = (S1/S0) for the relevant spin geometries: J  Slj     na     J 1    lN j  1  S0 j      sim (38) where “na” refers to natural abundance, l = 0 or 1, “N” refers to na 15 N, j is the index of a particular spin geometry, and J is the number of unique spin geometries of the model (J = 9 for  na na or max  nad and J = 6 for min  nad). The 0N ( ) = 1 for all  while 1N ( ) were calculated using [S1j ( )/S0j ( )] sim from the SIMPSON program and generally decreased with increasing . After setting a total labeled relative population affected by na 15 13 CO population of 1.0 for HFP-F8 in HFP-NC, the N is J  0.0037 while the remainder population, [1 – (J  0.0037)], has S1 = S0. There is also a na 13 CO contribution from unlabeled residues in HFP-F8 74 with a relative population 30  0.011 and with S1 = S0. Similar analysis for 15 N labeled HFP- A6L7 in HFP-NC results in: sim na   K 1 K  Slk            lC k 1  S0k    where l = 0 or 1, “C ” refers to na number of unique na 13 (39) 13 CO, k is the index of a particular spin geometry, K is the 15 CO-labeled N spin geometries of the model (K = 10, 8, or 12, na respectively, for , min , or max  nad models), and 0C ( ) = 1. Accounting for the 1:2 ratio of HFP-F8:HFP-A6L7, the total na 13 CO population of HFP-A6L7 is 2  31  0.011 with population 2K  0.011 affected by labeled 15 N and the remainder having S1 = S0. For HFP-NC in total:   S sim     J  0.0037   na      2 K  0.011   na    l , tot lN lC         (40)  2.0   J  0.0037    2 K  0.011 where l = 0 or 1 and the terms in the first braces are- and l-dependent. For each nad model (, min , and max ), (S/S0) sim exp (40) and statistical comparison was then made to (S/S0) sim exp 2    S   S   S   7   0 m  2     S0 m    exp  m 1  m     for each  was calculated with Eq. : (41) 75 where m is the index for an experimental datum, i.e. a particular . The respective  2 for the , min , and max  nad models were 1.2, 3.8, and 2.0 which were all less than the number of degrees of fitting, 7, i.e. the number of data, 7, minus the number of independent fitting parameters, 0. The validity of the approach to nad calculation was supported by good fits for all 83 models . The broad 13 CO linewidth of HFP-NC in Figure 20a was consistent with two HFP populations, one with helical and one with β strand secondary structure. It was therefore reasonable to calculate (ΔS/S0) sim for “mixtures” with contributions from multiple models: 3 S sim      fb  S sim    l , mix l    b 1  (42) where l = 0 or 1, b was the index that referred to the , min , or max  model, fb was fractional sim population with fb = 1, and each Sl distribution of 13 ( ) was calculated using Eq. (40). The HFP-NC CO shifts indicated f  0.5 and fmin + fmax  0.5 but did not provide information about individual fmin or fmax. Fitting using f = 0.5 and fmin = 0.5, fmax = 0.0 2 yielded  = 1.5 while fitting using either f = 0.5, fmin = 0.0, fmax = 0.5 or f = 0.5, fmin = 2 0.25, fmax = 0.25 yielded  = 1.2 and (S/S0) distribution. The (S/S0) sim sim in Figure 21a were calculated with the latter exp from all three conformational distributions fit well to the (S/S0) and these models are statistically similar. Together with previously described good fitting for different secondary structure models show that nad is accurately calculated with these models 76 and only weakly dependent on secondary and tertiary structure. The key feature of all these wellfitting long-range models was consideration of the multiple na sites within 7 Å of a labeled site which led to continually increasing (S/S0) with, Figure 21a. The (S/S0) sim were also calculated using a short-range model that only considered na sites separated by one or two bonds from each labeled site. The (S/S0) sim were systematically less than the (S/S0) exp with resultant 2 poor fit and  = 29. 3. Quantitative Analysis of Registry Populations – Fully Constrained Model. For membrane-associated HFP, there is a single distribution of registries which we model as fractions of: (p) 117/117 and 217/116 parallel registries; (a) 161/116 and 171/117 antiparallel registries; and (c) X registries not detected by any of our labeling schemes, Figure 19c. Fraction p contributed to the (S/S0) to the (S/S0) exp exp of HFP-P, fraction a contributed exp of HFP-A, and fractions p and a contributed to (S/S0) of HFP-AP. The overall goal was best-fit determination of these fractions based on the (S/S0) exp of the three exp samples, Figure 21b, and this analysis required calculation of the nad contribution to (S/S0) Because a 1:2 13 . 15 CO-HFP: N-HFP ratio was used for all samples, this contribution was calculated using models developed for HFP-NC and resulted in a modified Eq. (42) appropriate for HFP-P, HFP-A, and HFP-AP:   S sim     J  0.0037   na      2 K  0.011   na    lu , tot lN lC          (43)  lab  1.0   2 K  0.011  1.0   J  0.0037     ltuv      S na    S lab   l lu (44) 77 na where Sl is the sum of the first two braced terms in Eq. (43) and Slu lab is the third braced term. Each membrane-associated sample is labeled by the index u where u = 1, 2, or 3 respectively refers to HFP-P, HFP-A, or HFP-AP, Table 5 and Figure 19b. The first braced term in Eq. (43) corresponds to labeled and na na 13 13 CO that experience nad, the second braced term corresponds to COs that do not experience nad, and the third braced term corresponds to labeled that do not experience nad but may experience dephasing from labeled 15 13 COs Ns. The secondary structure of membrane-associated HFP was predominantly  sheet, Figure 20b-d, and the best estimates of the nad terms in the first braced term were taken to be the average of the max  and min  calculated values. In the second and third braced terms, K and J were estimated to be their respective average values of 10 and 7.5. S0u S0u lab ( ) = 1.0 – (J x 0.0037) while 1tuv lab lab ( ) was calculated using 0tuv ( ) and therefore S1u lab lab ( ) = 1, and ( ) were first calculated with a “fully constrained” model, Figure 22a,c, in which a  sheet region contained either: (a) 117/117 or 217/116 parallel registries; or (b) 161/161 or 171/117 antiparallel registries; or (c) X registries not directly detected by any of our labeling schemes, Figure 19c. A sample was considered to be a mixture of the three registry types each denoted by index t = a, b, or c and fractional population ft, Table 5. The S1u lab c lab lab lab S1u      ft  1tuv     S0       t a The 1tuv lab ( ) was calculated by modified Eq. (42): (45) ( ) values depended on the labeled dCNs and therefore rCNs which in turn depended on registry type t and sample labeling u, Figure 19c. For some combinations of t and u, all 78 labeled rCN > 7 Å with consequent dCN ~0 and 1tuv lab ( ) = 1. Specific examples are t = p and u = 2, t = a and u = 1, and t = c and u = 1, 2, or 3. For other combinations of t and u, 1tuv lab ( ) were determined from SIMPSON calculations and Figure 23a-d displays schematic examples for t = a, u = 2 with numerical values of 1tuv lab ( ) and more details in the Appendix VII. Column a, b, c, or d corresponds to particular arrangements of 13 CO and 15 N labeled HFPs that are respectively denoted by the index v = 1, 2, 3, or 4. For each v, the typical difference between the calculated 1a2v 1a2v lab ( ) for the 161/116 or 171/117 registry was ≤ 0.01 and the final lab ( ) were the average for the two registries. The antiparallel 1a3v analogously calculated and the parallel 1p1v lab lab ( ) of HFP-AP were ( ) of HFP-P and parallel 1p3v lab ( ) of HFP-AP were calculated using the 117/117 registry and had similar values to s calculated using the 217/116 registry. Fractional weightings wv were based on the 1:2 ratio of 13 15 CO HFP: N HFP with w1 = 1/9, w2 = 2/9, w3 = 2/9, and w4 = 4/9. A more complete version of Eq. (45): S lab    1u 4     lab lab   ft    wv  1tuv      S0      t  p, a, c  v 1   with indices and parameters summarized in Table 5. 79 (46) Figure 23. Schematics of three adjacent HFPs for HFP-A, i.e. u = 2, in (a-d) fully constrained or 80 Figure 23 (cont’d). (e-h) unconstrained models. Red and blue correspond to 13 13 CO and 15 N labeled residues, respectively, and the labeled CO in the middle strand was hydrogen bonded to the HN group of the residue in the top strand. 81 Figure 23 (cont’d). Panels a-d display antiparallel 161/116 (top) or 171/117 (bottom) registries while panels e-h display parallel 117/117, antiparallel 161/116, and X registries where X refers to a registry for which the labeled rCN > 7 Å, i.e. beyond the approximate detection limit of the SSNMR experiment, and which is not 117/117, 217/116, 161/116, or 171/117. Correspondence between columns and the index v are: a and e, v = 1; b and f, v = 2; c and g; v = 3; d and h; v = 4. Both rows of three-strand arrangements in panels a-d correspond to t = b and the row, t1, t2 correspondence in panels e-h is: row 1, t1 = p, t2 = p; row 2, t1 = p, t2 = a; row 3, t1 = p, t2 = c; row 4, t1 = a, t2 = p; row 5, t1 = a, t2 = a; row 6, t1 = a, t2 = c; row 7, t1 = c, t2 = p; row 8, t1 = c, t2 = b; row 9, t1 = c, t2 = c. For each three-strand arrangement enclosed by a box, the 1tuv lab lab ( ) or 1t1t2uv ( ) were calculated by SIMPSON simulation. For arrangements with t, t1, or t2 = b, fitting to experiment used s that were the average of those calculated with 161/116 and 171/117 registries although the latter registry is not displayed in panels e-h. For any arrangement not enclosed by a lab lab box, 1tuv ( ) = 1 or 1t1t2uv ( ) = 1. The values of fp, fa, and fc were the same for the HFP-P, HFP-A, and HFP-AP samples 2 with fc = 1 – fp – fa. Best-fit values of fp and fa were obtained by calculating  (fp, fa) using an expression analogous to Eq. (41): 2 sim    exp    S f p , f a    S    S0  S0   um  3 7    um  2 f p , fa      exp  u 1 m 1   um           (47) 2 2 then selecting the fp and fa values which corresponded to minimum  , i.e.  min. In Eq. (47), m is the index for each , and [S(fp, fa)/S0]um sim was determined using Eqs. (43)-(47). For this 2 fully constrained model, Figure 24a displays a plot of  vs fp and fa with best-fit fp = 0.12 and 2 2 fa = 0.52 and  min = 11. The model was reasonable as evidenced by  min which was smaller 82 than the number of degrees of fitting, 19, i.e. the number of data, 21, minus the number of fitting parameters, 2. The fp fractional parallel population was small which was consistent with qualitative analysis of the data, Figure 21b. The fa antiparallel population was substantially larger and also consistent with Figure 21b. The fc  0.35 indicated a substantial population of X registries not detected by the labeling of the three samples. 83 Figure 24. Contour plots of  2 vs fa parallel and fa antiparallel fractional populations for (a) fully constrained and (b) unconstrained models. In each plot, fa is the sum of populations of 117/117 and 217/116 parallel registries and fb is the sum of populations of 161/116 and 171/117 antiparallel registries. 84 Figure 24 (cont’d). For plot (a), the best-fit values were fa = 0.12 ± 0.03 and fb = 0.52 ± 0.04 2 2 with  min = 11, and for plot (b), fa = 0.11 ± 0.03, fb = 0.46 ± 0.04, and  min = 8. For plot (a), 2 the best-fit values were fa = 0.12 ± 0.03 and fb = 0.52 ± 0.04 with  min = 11, and for plot (b), fa 2 = 0.11 ± 0.03, fb = 0.46 ± 0.04, and  min = 8.Parameter uncertainties were determined by the 2 2 within about three units of  min. In plot (a), the black, red, green, yellow, 2 2 2 2 2 and white regions correspond to  < 14, 14 <  < 17, 17 <  < 20, 20 <  < 23, and  > populations with χ 2 2 2 2 23 and in plot (b), the regions correspond to  < 11, 11 <  < 14, 14 <  < 17, 17 <  < 20, 2 and  > 20. The above fitting was done using a long-range nad model that considered effects of na sites within 7 Å of each labeled nucleus. Fitting displayed in Figure 24a was based on nad calculated from half min  and half max  sheet structure, Figure 22, but the best-fit fp and fa 2 and  min were not sensitive to the structural composition of the long-range nad model. For example, fitting done using nad for half  helical and half max  sheet structure yielded best-fit 2 fp and fa and  min respectively within 0.01, 0.01, and 1 of the corresponding Figure 24a values. For HFP-NC fitting, nad was underestimated by a short-range model that only considered na sites separated by one or two bonds from each labeled site. This effect was also observed when fitting membrane-associated HFP data with the short-range nad model and led to best-fit fp 2 = 0.22 and fa = 0.57 which were significantly higher than the Figure 24a values. The  min = 20 2 using the short-range model was also higher than the  min in Figure 24a. 4. Quantitative Analysis of Registry Populations – Unconstrained Model. In addition to the fully constrained model for strand registries, an alternate “unconstrained” fitting model was also considered for which there was local mixing of: (p) 117/117 parallel registries; (a) 161/116 and 171/117 antiparallel registries; and (c) X registries not directly detected by any of our labeling schemes, Figure 19c. Each pairwise 85 registry type was labeled by t =p, a, or c with fractional population ft. For this unconstrained model, Figure 23e-h displays schematics of three-strand registries with 13 CO labeled HFP in the middle strand. Each e-h row has three-strand registries that were each a combination of two registries labeled by specific t1 and t2 which denote the respective t of the top/middle and middle/bottom strands. As with the fully constrained models, the registries in each e-h column corresponded to a particular or 4. The 13 13 15 CO HFP/ N HFP arrangement which respective label v = 1, 2, 3, 15 CO HFP: N HFP population ratio of 1:2 correlated with a sum weighting of 1/9 for the v = 1 registries with individual registry weighting w1 = 1/(99) = 1/81. The sum weightings for v = 2, 3, or 4 were respectively 2/9, 2/9, or 4/9 with respective individual weightings w2 = 2/81, w3 = 2/81, and w4 = 4/81. Eq. (46) was modified for the unconstrained model: S lab    1u 4       lab lab    ft1  ft2    wv   1t t uv      S0   12  t1  p, a, c t2  p, a, c  v 1    (48) Similar to the fully constrained model, many combinations of t1, t2, u, and v have rCN > 7 Å with consequent dCN  0 and 1t1t2uv lab ( ) = 1. In Figure 23e-g, such registries are not enclosed by a box. Similar to results for the fully constrained model, the 1t1t2uv lab ( ) were similar for the two antiparallel registries and an average value was used. The values of fp, fa, and fc in the unconstrained model were the same for the HFP-P, HFP-A, and HFP-AP samples with fc = 1 – fp – fa. Best-fit values of fp and fa were obtained with 2 Eq. (47) and Figure 24b displays a plot of  vs fp and fa with best-fit fp = 0.11 and fa = 0.46 86 2 and corresponding  min = 8. The unconstrained model was reasonable as evidenced by a bestfit  2 which was smaller than the number of degrees of fitting, 19. The results were similar to the fully constrained model in that the fp fractional parallel population was small, the fa antiparallel population was large, and the fc ~0.4 which suggest that there is a significant population of other structures. This unconstrained model fitting was done with nad calculated with a long-range model and half min  and half max  structure. Similar to the fully constrained model, best-fit fp, fa, and 2  for the unconstrained model were: (1) negligibly affected by the structural distribution of the long-range nad model; and (2) significantly increased by use of a short-range nad model. 3.4 Discussion Chapter III sets an upper limit of ~0.15 on the fraction of mHFP in in-register parallel  sheet structure, and this result is supported by both qualitative analysis of the data, Figure 21b, as well as quantitative analyses with fully constrained and unconstrained models, Figure 24. Both models fit the data well and yielded similar best-fit fractional population of parallel registries and similar populations of antiparallel registries. The small fractional parallel population agrees with some earlier SSNMR studies but differs from interpretations of other SSNMR and IR data which respectively reported ~0.5 and ~1.0 fractions of in-register parallel structure 29,47,54,55 . The Chapter III study used samples with sparse isotopic labeling while the earlier studies interpreted to support a large fraction of parallel structure used samples with extensive labeling. I believe that there was ambiguity of interpretation in the studies with extensively labeled samples and that the data could also be reasonably interpreted in terms of 87 small in-register parallel population. For example, the earlier SSNMR study also used the REDOR technique but with only a single  (24 ms), and with samples that contained equimolar amounts of HFP 13 CO labeled on three sequential residues and HFP sequential residues. The typical (S/S0) exp 15 N labeled on three was ~0.1 and was approximately independent of the positions of the labeled residues that also had some contribution from nad. It was not possible to do unambiguous quantitative analysis of registry distributions because: (1) each sample was extensively labeled so that non-zero (S/S0) was expected for many different registries; (2) exp (S/S0) were only measured for a single  ; and (3) a “HFP-NC”-type sample was not studied and nad was therefore not quantitatively modeled. The samples for the IR study were also extensively labeled with backbone 13 CO labeling at either: (1) Ala-1 to Val-3, Gly-5 to Leu-9; (2) Phe-8 to Gly-16; or (3) Ala-1 to Val-3, Gly-5 to Gly-16. The authors’ interpretation of their spectra to support predominant in-register parallel 13 16 … 13 16 structure was based in part on expected effects of ( C= O electric dipole) ( C= O electric dipole) coupling on 13 C=O vibrational wavenumber and intensity. However, their interpretation appeared to neglect the substantial intramolecular coupling between 13 C=Os on adjacent residues, and is should be noted that this coupling is independent of registry. In addition to these “undiluted” samples, three “diluted” samples were studied that had an equimolar mixture of a labeled and unlabeled peptide. The wavenumber (ν) of a (~5 Å away) C=O vibrations and is higher with neighbors. If there is hydrogen bonding between 13 88 13 12 16 C= O vibration is sensitive to nearby 16 C= O neighbors than with 13 16 C= O CO labeled residues of adjacent strands in an undiluted sample, the corresponding diluted sample should have an increased fraction of 13 12 C=O/ C=O proximities, decreased fraction of 13 13 C=O/ C=O proximities, and ν = νdiluted – νundiluted > 0. If there were a major fraction of parallel 117/117 structure (as claimed by the authors), dilution of (1) Ala-1-Val-3, Gly-5-Leu-9; (2) Phe-8-Gly-16; or (3) Ala-1-Val-3, Gly-5Gly-16 labeled HFPs would have had comparable effect on proximities and resulted in similar ν. However, the experimental νA1-V3,G5-L9  νF8-G16  (νA1-V3,G5-G16)/2 which is inconsistent with a large fraction of in-register parallel structure. Like the earlier SSNMR study on extensively labeled samples, extensive labeling of the IR samples also meant that the IR data were consistent with many registry distributions and precluded more quantitative analysis of the distribution. Overall, the sparse labeling of the present SSNMR study allowed for much more unambiguous and quantitative determination of the populations of specific registries. This general approach can be applied in the future to determine the registry distributions of HFP constructs with very high or low fusogenicity such as HFPtr or V2E mutant, respectively. 89 Figure 25. Pictorial model of HFP (red lines) binding to membranes followed by antiparallel  sheet formation and membrane insertion and then fusion. Time increases from left-to-right. For reasons of clarity, some lipids are not shown in the right-most picture. Although there are no data yet on fusion peptide structure during HIV/host cell fusion, the antiparallel  sheet structure of the right-most picture is plausible because: (1) the structure is consistent with multiple trimers at the fusion site; and (2) the structure is membrane-inserted with deeper insertion positively correlated with increased membrane perturbation and vesicle fusion rate. Figure 25 displays a structure/function model for HFP based on results from this and earlier studies. Prior to membrane binding, HFP is monomeric in aqueous solution and has random coil structure 30,32,33 . HFP sequentially: (1) binds to membranes; (2) forms β sheet oligomers with a significant fraction of 161/116 and 171/117 antiparallel registries; and (3) induces membrane fusion as monitored by intervesicle lipid mixing 30,33,34,37 . It is also known that the Ala-6 and Leu-9 residues of β sheet HFP insert shallowly into the membrane with correlation between membrane insertion depth and both membrane perturbation and fusion 40,61,92 rate . A global structure-function model is non-transmembrane HFP insertion perturbs bilayer structure and moves the membrane on the fusion reaction coordinate towards the highly perturbed transition state with consequent reduction in fusion activation energy and increase in fusion rate. 90 There is functional and electron microscopic evidence that multiple gp41 trimers are required for fusion and Figure 25 (right) shows a  sheet HFP hexamer as would be reasonable for interleaved antiparallel fusion peptides from two gp41 trimers. While there are no data specifically supporting a HFP hexamer, HFP  sheet oligomers likely contain a small number of molecules because: (1) for ~90% of HFP molecules, there is an Ala-1 13 CO-lipid 31 P distance of ~5 Å, i.e. close contact of most HFPs with the membrane; and (2) significant temperature 40,51 dependence of intensities of SSNMR spectra . Interleaved antiparallel fusion peptides from multiple gp41 trimers may also be the fusogenic structure of HIV/host cell fusion. As noted above, this structure can insert into and perturb membranes which are likely requirements for HIV/host cell fusion . The experimentally observed membrane insertion of the central  sheet region (e.g. Ala-6 insertion to Leu-12) of HFP is consistent with G  –6 kJ/mol for the fully constrained 171/117 registry as calculated by summing individual residue insertion energies for Leuinsertion 93 12Ala-6/Ala-6Leu-12, Figure 19c . A similar calculation yielded G insertion for Leu-12Gly-5/Gly-5Leu-12 of the 161/116 registry. The G ~3 kJ/mol ~ –6 kJ/mol for insertion Ala-6Leu-12/Ala-6Leu-12 of the 117/117 parallel registry suggests that G does not underlie the preference for antiparallel over parallel structure. This preference may electrostatic instead be due to G as the HFP N-terminus is located in the high water content lipid + headgroup region and is therefore likely protonated. Closest intermolecular NH3 –NH3 + distance is ~5 Å for the low population 117/117 registry and ~10 Å for the significantly 91 electrostatic populated 171/117 registry. For dielectric = 78 and hexameric HFP, G  +5.1 kJ/mol for the 117/117 registry and +1.5 kJ/mol for the 171/117 registry. It is expected that inclusion of the non-native C-terminal W(K)6A tag does not contribute to the preference for antiparallel over parallel registry because either registry would have similar minimized electrostatic repulsion energy. Such repulsion would be minimized by: (1) extensive solvation of the tag; and (2) large inter-tag distances that are possible because of random coil tag structure. Tag solvation is supported by the previously observed lack of membrane insertion of HFP beyond residue Leu-12 and random coil tag structure is supported by broad NMR linewidths in the C-terminal region of HFP 37,40 . We also note that inclusion of the tag has minor effect on fusion activity and that similar REDOR S/S0 were observed for mixtures of triply 13 CO and 15 N labeled HFPs with or without the tag 29,33 . In contrast to the reasonably large distribution of membrane-associated HFP registries, i.e. significant 161/116, 171/117, and X registries, SSNMR studies of  sheet registries of protein in amyloid fibrils have typically shown a single registry which is usually in-register parallel, e.g. 117/117 52,53 . The width of a fibril is at most a few protein molecules and the length is >200 molecules and along the intermolecular  sheet hydrogen bonding direction. The amyloid fibrils are grown in aqueous solution (without lipid) and their greater registry 94 homogeneity may reflect ordered fibril growth from seeds . One distinctive feature of this chapter is the development of a quantitative nad model that was experimentally validated. Accurate fitting of the HFP-NC data and fitting of the membraneassociated HFP data relied on a long-range nad model which included effects of na nuclei <7 Å 92 from each labeled nucleus. For this model, the nad was approximately independent of secondary and tertiary structure. The nad was systematically underestimated by a short-range model which only considered na nuclei separated by one or two bonds from each labeled nucleus. 93 CHAPTER IV. Quantitative Identification of the Antiparallel Sheet Registry Distribution in Membrane Associated HFP and V2E-HFP Samples 4.1 Background. Viral replication is initiated by infection of a host cell, and vaccines have been developed to build resistance to viral infections which has minimized the effects of diseases such as measles, mumps, and small pox to name a few 2,93 . Relative to other viruses, sequences of HIV 3 have higher mutation rates , and there is not yet an HIV vaccine. Development of an HIV vaccine is critical since vaccines prevent infection, and inhibitory drug design is equally essentially for therapeutic treatment of infected patients. Development of inhibitory drugs would be enhanced by obtaining a deeper understanding of the HIV life cycle. The HIV life cycle’s initial step in infection is membrane fusion, a joining of the HIV and the host cell membrane. The glycoproteins on the ectodomain of the HIV cell membrane are responsible for catalyzing membrane fusion where gp120 binds to a host cell receptor which is shed from gp41 10,12 . Removal of gp120 from gp41 is believed to catalyze structural transitions within gp41 from a unknown structure to a hypothesized extended coiled-coil pre-hairpin intermediate structure 13 (PHI) which eventually folds into a hairpin structure (see below) . The gp41 protein is 14 composed of ~356 residues and is subdivided into regions from the N-terminus: fusion peptide (FP) (~16 residues), FP proximal region (FPPR) (~13 residues), N-terminal helix (NTH) (~40 residues), loop (~47 residues), C-terminal helix (CTH) (~37 residues), pre-transmembrane region 15 (~18 residues), transmembrane region (~28 residues) , and cytoplasmic endodomain (~160 14 residues) . To date, high-resolution crystal structures of gp41 were only for ectodomain 94 constructs that lacked the fusion peptide, transmembrane and endodomain regions. These ectodomain crystal structures showed protein trimers with three interior parallel α helical NTH segments and three exterior α helical CTH segments packed antiparallel to the NTHs. The overall structure of each monomer was a hairpin, and the trimer formed a six-helix bundle 17,20 15- . The above domains have been defined by crystal structures of gp41 based constructs with varying interpretations of the number of residues incorporated into each domain. To my knowledge, the largest HIV gp41 crystallized construct to date has shown that the helicity extends beyond the traditionally defined NTH and CTH, Figure 26. The helicity of the NTH and CTH approximately extends from the respective residues Ala-532 to Ile-580 and Asp-627 to Asn-677. Additionally, the NTH and CTH in the SIV gp41 crystal structure span residues Arg-30 18 to Ala-86 and Thr-104 to Lys-146 , respectively, which is analogous to HIV gp41 crystal structure residues Arg-542 to Ser-598 and Ser-616 to Glu-662. One interpretation of these combined results is that under crystallization conditions, the helicity of the NTH and CTH regions are terminated due to the length of the gp41 construct rather than the length of the NTH and CTH of the full length gp41. Of note, these gp41 structures are for gp41 without the presence of a membrane, and these structures also lack the hydrophobic fusion peptide. For HIV gp41, mutations within the FP and FPPR have been shown to inhibit membrane fusion which 21 suggests that both the FP and FPPR are important for membrane fusion . Of special interest, transdominant inhibition of the V2E mutated gp41 (a FP mutation) has demonstrated that more 21 than three gp41 or multiple gp41 trimers are needed to initiate membrane fusion . Whatever the structure/s of the FP and FPPR are, the structure/s must allow for aggregation of FPs between more than 3 gp41 or multiple gp41 trimers. While the structures of the FP region of gp41 are the 95 focal point of this chapter and dissertation, it should also be noted that regions that are Cterminal of FP can also be effective targets for fusion inhibitor drugs. The AIDS drug Enfuvirtide 22 is a fusion inhibitor and is a 36-residue peptide containing parts of the C-helix and the pre-transmembrane regions. Enfuvirtide likely binds to exposed N-helical regions in PHI gp41 and acts as a competitive inhibitor to the native C-helix regions with consequent prevention of the transition to the final hairpin structure. Cell-cell fusion mediated by gp41 includes in sequence: (1) lipid mixing between the membranes; (2) fusion pore formation; and (3) pore enlargement. After addition of Enfuvirtide, small pores rather than large pores can be closed which indicates that gp41 in the PHI state mediates lipid mixing and initial fusion pore formation while the final hairpin state (or possibly the PHI  hairpin transition) mediates pore stabilization and enlargement 13,22,23 Despite its efficacy, Enfuvirtide is not widely used because it is a 22 peptide and must be administered by injection rather than orally . Figure 26. A summary of the gp41 sequence and regions defined from literature where FP = fusion peptide, FPPR = fusion peptide proximal region, NTH = N-terminal helix, CTH = Cterminal helix, MPER = membrane proximal region, and TMR = transmembrane region. This 15 figure was adapted from literature The FP of gp41 is an alternative fusion inhibitor target and its membrane-associated structure is the subject of the present study. HIV and cellular studies have shown that point 21,28 mutations within the FP lead to inhibition of fusion and infection . Various “HFP” constructs containing the ~16 residue FP have also been made and often catalyze fusion between membrane vesicles. There are good correlations between the effects on gp41-mediated fusion by specific FP 96 31 mutations and the effects on vesicle fusion by the corresponding mutations in HFP . These collective data support FP (HFP) binding to the cell (vesicle) membrane early in the fusion process with fusion catalysis correlated to the membrane perturbation induced by the FP (HFP). The present study of the membrane-associated structures of HFP is important because knowledge of these structures will aid: (1) understanding of the FP as a fusion catalyst; and (2) the potential of the FP as a fusion inhibitor target. Infrared (IR) spectroscopy and liquid-state NMR (LSNMR) have shown that HFP in aqueous solution adopts random coil structure 34,37 . A fluorescence and IR study provided rates of: (1) HFP binding to membranes; (2) the transition to HFP  sheet structure; and (3) vesicle fusion. The study showed that binding to vesicles and β sheet formation occurred before vesicle 34 fusion . Similar structures were observed by solid-state NMR (SSNMR): (1) HFP lyophilized from aqueous solution without vesicles had a distribution of secondary structures as indicated by single site backbone 13 CO signals whose ~8 ppm linewidths spanned typical helical and β strand chemical shifts; and (2) HFP bound to hydrated membranes with ~30 mol% cholesterol was peptide oligomers/aggregates with β sheet structure. The cholesterol content of the membranes approximated that of the membranes of HIV and its host cells 36-39 . These data support β sheet HFP oligomers as an important structure in fusion. Initial SSNMR and IR experiments with extensively labeled HFP respectively suggested that ~50% and ~100% of the membrane-associated oligomers had in-register parallel β sheet structure 47,54 . In some contrast, Chapter III experiments with sparsely labeled HFP showed that 63 in-register parallel structure accounted for ≤15% of the molecules . The REDOR experiments 97 from Chapter III and literature 36 also identified significant populations for at least two distinct and specific antiparallel registries. The “HFPtr” construct has also been studied and contained three C-terminally cross-linked peptides that modeled the close contact of Thr-25 residues in the hairpin structure of the N-helix + C-helix regions of gp41. Relative to non-cross-linked HFP, HFPtr induced vesicle fusion with >15-fold faster rate 25,30 . SSNMR showed that membrane- associated HFPtr had  sheet structure similar to that of HFP including lack of in-register parallel 54 structure . The main structural distinction observed to-date between mHFP and mHFPtr was deeper membrane insertion of mHFPtr 25,54,95 . Other data have also supported small β sheet oligomers as a structural model for FP in gp41: (1) electron micrographs of virus-cell contacts showed multiple nearby gp41 trimers in contact with the cell membrane and (2) fusion and HIV infection were dominantly inhibited in samples containing a small fraction of gp41 with a V2E point mutation and a large fraction of wild-type gp41 21,96,97 . Efforts to identify the predominant β sheet registry of mHFP have led to conflicting β 47 sheet structural models such as in-register parallel , mixed in-register parallel and 29 48 antiparallel , and β hairpin . As described above, Chapter III showed that in-register parallel β sheets are minimally populated while the present Chapter quantitatively describes the distribution of antiparallel registries and accounts for nearly all of the mHFPs. Each registry was considered as a  sheet composed of residues 1 to t of the HFP molecule. The t = 22, 18 and 13 registries are illustrated in Figure 28. In the present Chapter, many HFPs were prepared that differed in the residue numbers of their single 13 CO and single 15 N backbone labels (See Chapter II, Table 1). The sample index u = Ϛ + η -1 and Ϛ and η were respectively the residue 98 numbers of the 13 CO and 15 N labels. The intermolecular labeled 13 15 CO- N distance was denoted rCN and for sample u, the smallest rCN of ~ 4 Å was observed for molecules with registry t = u and an interpeptide 13 … 15 CO H N hydrogen bond. In the solid-state NMR (SSNMR) rotational-echo double-resonance (REDOR) experiments of the present study, this 13 …15 CO N proximity was qualitatively and quantitatively understood by the extent of signal attenuation. The most important and novel result was significant 13 13 CO CO signal attenuation from many different samples; i.e. there was a broad distribution of antiparallel registries. Additionally, similar experiments were run for mV2E-HFP. The mV2E-HFP samples displayed a distribution of antiparallel registries, but the distribution of registries was distinctly different than that of mHFP. These distinct differences in the registry distributions have led to development of new hypotheses to explain the previously observed transdominant inhibition of fusion that was observed when wild type gp41 was co-expressed with dilute amounts of V2E 21 mutant gp41 . 4.2 Materials and Methods 99 Table 6. Chapter IV indices and parameter Index/parameter Description Values ft fractional population of structure t j, k index for n.a. site <7 Å from a labeled 15 13 site: j, n.a. N near labeled CO; k, n.a 13 CO near labeled 15 determined by fitting N J, K number of n.a. sites <7 Å from a labeled 15 13 site: J, n.a. Ns near labeled CO; K, min  sheet structure, J = 6, K = 6; max  sheet structure, J = 9, K = 9 13 15 n.a COs near labeled N l REDOR data type index 0  no dipolar evolution 1  dipolar evolution datum index 1, 2, 3, 4, 5, 6, or 7 correspond to 2.2, 8.2, 16.2, 24.2, 32.2, 40.2 and 48.2 ms dephasing time, respectively. m determined by experiment or calculation Sl , Slj , Slk , Slu REDOR signal intensity membrane associated sample index u = Ϛ+η-1 13 Ϛ = residue number for CO labeled u η = residue number for t, t1, t2 15 N labeled structural population index; for unconstrained model of membr. assoc. samples, t1 indexes the top/middle registry and t2 indexes the middle/bottom registry na lN ( ), na lC ( ) lab  ltu (), lab lt1t2u ( ) Sl ( )/S0: lN ( ), labeled pairs; lC ( ), n.a. Slu 13 lab ( )/S0 CO and lab 15 13 13 CO-labeled N nuclei: 1tu constrained model; 1t1t2u unconstrained model lab ( ), fully ( ), 100 7-25, and X where 8  8 residue antiparallel registry, X  t ≠ (u-1), u, or (u-1) 15 for arrangement of labeled lab 15 η = residue number for N labeled 8  HFP-A6CG3N, 9  HFPL7CG3N, etc. N 0N ( ) = 1; 0C ( ) = 1; 1N ( ) and N pairs 1C ( ) determined by calculation CO-n.a. 15 u = Ϛ+η-1 13 Ϛ = residue number for CO labeled 0tu lab lab ( ) = 1; 0t1t2u lab lab ( ) = 1; 1tu ( ) and 1t1t2u ( ) determined by calculation Table 6 (cont’d).  b 2.2, 8.2, 16.2, 24.2, 32.2, 40.2, or 48.2 ms 1= min n.a.d. model 2= max n.a.d. model REDOR dephasing time Structural population index The data analysis is adapted from literature 63 and similar to Chapter III. Table 6 contains a list of all the indices and parameters. As mentioned in Chapter III, REDOR was used to probe dipolar coupling (dCN) between 13 CO and 15 78 N nuclei average dCN were quantified by comparing the 13 3 3 where dCN = 3080 Hz-Å /rCN . The CO signal intensity of the “S0” spectra, S0, to the generally reduced intensity of the “S1” spectra, S1, as a function of dephasing time, . Signal attenuation was quantified by the normalized dephasing (ΔS/S0) = (S0-S1)/S0 where S0 and S1 were quantified by 3.0 ppm integrations about the 13 CO peak center of the S0 and S1 spectra, respectively. These 3.0 ppm regions about the peak center generally encompassed a majority of the 13 CO signal. Integration of both larger and smaller regions (1.0 ppm and 5.0 ppm) generally yielded similar results (0.01 variation ΔS/S0). In modeling the experimental data, rCN ≤ 7 Å were responsible for most of the dephasing, and r CN > 7 Å were not considered in the analysis. The SIMPSON program was used to simulate γ(τ) for 13 15 CO- N spin geometries. These spin geometries were based upon dCN and orientations between dCN vectors derived from of the crystal structure coordinates of the β barrel outer membrane protein G (OMPG) (PDB file 2IWW). Input parameters included dCNs as well as Euler angles for each dCN vector and for the 13 CO chemical shift anisotropy (CSA) principal axis system (PAS) in a fixed crystal frame 101 84,87 . The (α, β, γ) Euler angles of dCN PAS were (0, 0, 0) and (0, ϑ, 0) for 2 and 3 spin simulations, 13 respectively, where ϑ was the angle between two dCN vectors. The CO CSA principal values 89 na of 247, 176, and 99 ppm were also inputs to the SIMPSON calculations . The γ lC(τ) and na γ lN(τ) were calculated for na 13 CO-labeled 15 N and labeled 13 CO-na 15 N for rCN ≤ 7 Å. Each membrane-associated sample was labeled by an index u where u = Ϛ + η -1 (the residue number minus the 15 13 CO labeled N labeled residue number minus 1). For each sample, u, the t = u-1, u, and u+1 registries resulted in labeled 13 CO-labled 15 N interstrand rCN ≤ 7 Å. These registries resulted in the most significant signal attenuation in S1 relative to S0 and γ1tu lab ≠ 1. For all other registries (in total denoted X), labeled interstrand rCN > 7 Å, and we approximated γ0tu γ1tu lab lab = 98-100 = 1. Intersheet distances were neglected because the typical intersheet rCN > 7 Å . Simulated signal was defined as:   S sim     0.99  J  0.0037   na     0.98  K  0.011   na     lu lN lC          u 1   lab 0.33   K  0.011   0.99   J  0.0037    f X   ft  ltu          t  u 1    The first braced term in Eq. (49) corresponds to labeled and na second braced term corresponds to na term corresponds to labeled from labeled 15 13 13 13 (49) CO that experience nad, the COs that do not experience nad, and the third braced COs that do not experience nad but may experience dephasing Ns where fX =1- ft =u-1- ft = u,- ft= u+1. The K and J were estimated to be 7.5 and were derived from the nad model used in Chapter III (0.5 max nad model and 0.5 min nad). 102 S0u sim ( ) were calculated using 0tu calculated using 1tu lab lab na na ( ) = 0N ( ) =0C ( ) = 1 while S1u sim ( ) were ( ) from the fully constrained model and fractional populations, ft’s. The fractional populations were varied from 0.00 to 1.00 for 7 < t < 25 where ft = 0.00 for t < 8 and t 2 > 24. Similar to Chapter III, ft values were evaluated by using a χ metric, but the larger range of t values considered in this Chapter required minimization of the searching space for each ft to reduce the computational time associated with a global fitting of 17 variables (Eq (51) below). For each t, the maximum ft , ft max , was obtained using the (S /S0) exp u = t data, and the 2 minimum  u = t was determined under the constraints ft = u–1 = ft = u +1 = 0 with variation of ft = u from 0.00 to 1.00 by 0.01 increments by Eq. (50). sim  exp   S  ft  u 1 , ft  u , ft  u 1      S  S0    7   S0 um  um    2u     exp  m 1   um       exp where m is the index for each  and σum 2 (50) is the rms deviation determined by twelve 3 ppm 83 2 integrations of noise regions of both the S0 and S1 spectra . At the minimum  u = t, ft = u was set equal to ft = u for each sample, u. Similarly, each minimum ft, ft 2 searching for the minimum  u = t under the constraints ft = u–1 = ft = u-1 max u+1 min max , was obtained by and ft = u +1 = ft = min with variation of ft = u from 0.00 to 1.00 by 0.01 increments. The ft = u 2 max was the value of ft = u which yielded minimum  u = t. The best-fit f8 – f24 were obtained by fitting all (S 103 /S0) exp min data for u = 8 to 24 by Eq (51). For this fitting, each ft was varied from ft max to ft by 0.01 increments.    sim exp   S  24 7   S  ft  u 1 , ft  u , ft  u 1        2  f8  24       S0   S0   um    um  u  8 m 1  exp    um     2 (51) 2 The best-fit ft’s were those corresponding to the global minimum  (f8–24). This method was min used for determining the best-fit ft’s for both mHFP and mV2E-HFP. Each ft max , ft and best- fit ft are found in Table 7 (see Results). These fittings are generally referred to as 3 registry fittings while the equations were modified for the 5 registry fittings, Appendix VIII, and the results are found in Table 8. To evaluate the reproducibility of these experiments, a second HFP-F8CG13N was synthesized, and incorporated into a new vesicle sample and new (ΔS/S0) The (ΔS/S0) exp exp data were collected. data typically varied by < 0.01 between similar data points, τ, of the two HFP- F8CG13N samples, Table 9. The new HFP-F8CG13N (ΔS/S0) old (ΔS/S0) exp exp data were substituted for the data and a second fitting was performed with resultant f19 = 0.02, f20 = 0.17, f21 = 2 2 0.00 for a fully constrained model at the χ min = 119. The second fitting yielded a χ min that was ~9 higher than the first fitting, but the reproducibility of the (ΔS/S0) exp 21  ft differed by only 0.02 between the two fittings. The t 19 data and best-fit ft’s validate this data analysis whereas variation 104 2 exp in the χ min associated with the second fitting was due in part to the smaller σ20,m s associated with the second data set, Table 9. An unconstrained iterative 3 registry fitting was also considered which yielded similar results, see Appendix IX. 4.3 Results. 1. Fully Constrained Model mHFP Registry Distribution 105 Table 7. Three registry fittings for mHFP and mV2E-HFP. mHFP min max * 2 min ft ft ft Best-fit ft χ ft 0.00 0.00 0.00 1.4 0.00 f8 mV2E-HFP Best-fit ft 0.00 0 max ft * 2 χ 4.0 f9 0.00 0.01 0.00 3.0 0.00 0.00 0 4.0 f10 0.00 0.02 0.01 3.9 0.00 0.00 0 2.7 f11 0.01 0.07 0.04 2.0 0.00 0.00 0 8.1 f12 0.03 0.15 0.07 6.4 0.00 0.03 0.01 6.2 f13 0.08 0.22 0.16 7.8 0.00 0.05 0.05 5.0 f14 0.00 0.19 0.06 9.5 0.00 0.09 0.01 6.4 f15 0.06 0.23 0.15 3.1 0.00 0.17 0.12 4.2 f16 0.04 0.25 0.12 3.4 0.04 0.26 0.13 3.8 f17 0.10 0.28 0.18 5.3 0.05 0.33 0.18 4.8 f18 0.05 0.21 0.13 6.8 0.10 0.36 0.26 3.8 f19 0.00 0.14 0.02 9.4 0.00 0.33 0.03 3.1 f20 0.11 0.18 0.15 8.2 0.28 0.49 0.44 5.0 f21 0.00 0.04 0.00 9.8 0.05 0.24 0.06 10.9 f22 0.00 0.04 0.01 10.6 0.00 0.08 0.01 8.6 f23 0.04 0.05 0.04 6.0 0.03 0.08 0.07 3.3 0.00 0.00 0.00 13.8 0.02 0.04 0.02 3.2 f24 2 * The χ u=t were calculated using Eq 45 where the best-fit ft values were used for ft = u-1, ft = u, and ft = u-1. For mHFP, the global fitting resulted in = 1090. The 24 max  ft  2.08 t 8 24  ft  1.14 t 8 2 24 min  ft  0.52 t 8 2 24 max  ft  2.55 t 8 24  ft  1.39 t 8 2 with χ = 4444. 106 2 with χ min = 87. The 2 and χ χ = 2110. For mV2E-HFP, the global best-fit resulted in with χ = 2071. The 2 with χ min = 110. The 24 min  ft  0.57 t 8 Table 8. Five registry fittings for mHFP and mV2E-HFP. mHFP min max * 2 min ft ft ft Best-fit ft χ ft 0.00 0.00 0.00 1.4 0.00 f8 max ft mV2E-HFP Best-fit ft * 2 χ 0.00 0.00 4.0 f9 0.00 0.01 0.00 3.0 0.00 0.00 0.00 4.0 f10 0.00 0.02 0.01 3.9 0.00 0.00 0.00 2.7 f11 0.00 0.07 0.04 2.0 0.00 0.00 0.00 8.2 f12 0.02 0.15 0.07 6.4 0.00 0.03 0.00 6.5 f13 0.07 0.22 0.16 7.8 0.00 0.05 0.05 5.0 f14 0.00 0.19 0.06 9.5 0.00 0.09 0.01 6.7 f15 0.03 0.23 0.15 3.1 0.00 0.17 0.09 2.9 f16 0.02 0.25 0.12 3.4 0.04 0.26 0.14 4.7 f17 0.07 0.28 0.18 5.3 0.03 0.33 0.14 5.1 f18 0.03 0.21 0.13 6.8 0.05 0.36 0.25 4.7 f19 0.00 0.14 0.02 9.4 0.00 0.33 0.01 4.3 f20 0.09 0.18 0.15 8.2 0.24 0.49 0.44 3.9 f21 0.00 0.04 0.00 9.8 0.00 0.24 0.05 10.0 f22 0.00 0.04 0.01 10.6 0.00 0.08 0.00 4.4 f23 0.03 0.05 0.04 6.0 0.01 0.08 0.07 2.7 0.00 0.00 0.00 13.8 0.00 0.04 f24 0.01 2.9 * The χ were calculated using Eq 60 (Appendix VIII) where the best-fit ft values were used for 2 for ft = u-2, ft = u-1, f t = u, ft = u+1, and for ft = u+2. For mHFP, the global fitting resulted in = 1561. The 24 max  ft  2.08 t 8 24  ft  1.05 t 8 2 24 min  ft  0.36 t 8 2 24 max  ft  2.55 t 8 24  ft  1.26 t 8 2 with χ = 5740. 107 2 with χ min = 83. The 2 and χ χ = 2738. For mV2E-HFP, the global best-fit resulted in with χ = 2929. The 2 with χ min = 111. The 24 min  ft  0.37 t 8 Table 9. HFP-F8CG13N data sets used in global fittings st 1 Fitting τ (ms) (ΔS/S0) exp 2 σ exp (ΔS/S0) nd Fitting exp σ exp 48.2 0.175 0.015 0.206 0.011 40.2 0.177 0.024 0.170 0.012 32.2 0.161 0.012 0.162 0.011 24.2 0.116 0.015 0.108 0.010 16.2 0.068 0.010 0.066 0.010 8.2 0.017 0.012 0.0326 0.010 2.2 0.022 0.012 0.018 0.010 For a sample u, the t = u-1, u and u+1 registries resulted in interstrand labeled labeled 15 13 CO- N rCN < 7 Å while the corresponding intrastrand rCN were always much greater than 7 Å. A small fraction of the labeled (natural abundance) abundance (labeled) 15 13 COs were also close to natural Ns, and this contribution to S1 signal attenuation is detailed below. If all pairs of HFP molecules had a specific antiparallel registry t, then each labeled sample would have a rCN ~ 4.0 Å and a (ΔS/S0) exp 13 CO in a u = t ~ 1 for  = 48 ms. Additionally, the u = t – 1 and u = t + 1 samples would have significant dephasing whereas other samples would have minimal dephasing. If only a fraction of the molecules have t = u-1, u and u+1 registries, the (ΔS/S0) exp at  = 48 ms would be reduced for this sample. A (ΔS/S0) implies that ft ~ 0.00, and comparisons between (ΔS/S0) exp exp ~0.0-0.1 at  = 48 ms can be used as a semi-quantitative measure of the fractional population of t registry, ft. Detection of significant (ΔS/S0) exp for many different samples in the u = 8 – 24 range, Figure 27, demonstrated that there was a broad 108 distribution of antiparallel β sheet registries rather than a single predominant registry. Only  sheet structures were considered because of the observed  sheet 13 CO chemical shift distributions. 13 Figure 27. REDOR S0 and S1 C SSNMR spectra at 48.2 ms dephasing time for (a) mHFPA6CG3N, (b) HFP-L12CG5N, (c) HFP-F8CL12N, or (d) HFP-L9CG16N. Each spectrum was processed with 200 Hz line broadening and baseline correction and was the sum of: (a) 23766; exp (b) 21454; (c) 40331; or (d) 39133 scans. (e) (ΔS/S0) (τ = 48.2 ms) for all samples, u. (f) Plots exp of (ΔS/S0) vs dephasing time with the rms error. Isotopic labeling of each mHFP is displayed in the legend that correspond to HFP-A6CG3N (black, square), HFP-L12CG5N (red, circle), HFP-F8CL12N (cyan, triangle), and (d) HFP-L9CG16N (orange, inverted triangle). Variation exp less than 0.02 in (ΔS/S0) was also observed between two different preparations of the same sample type, e.g. HFP-F8CG13N (not displayed here). Quantitative analysis of this distribution considered that the S0 and S1 signal intensities each contained contributions from labeled and natural abundance (na) 13 CO nuclei. Signal attenuation in S1 resulted from natural abundance dephasing (nad) and registry dependent 109 dephasing (rdd). The nad was due to na 13 CO-labeled 13 proximities. For sample u, the rdd of the labeled 15 N and labeled CO-labeled 15 13 15 CO-na N N interstrand proximities resulted from the t = u-1, t = u, and t = u+1 registries where the corresponding fractional populations of these registries were denoted ft = u-1, ft = u, and ft = u+1. The SIMPSON software was used to calculate (S1/S0) sim = γ sim l for specific β sheet structure spin geometries (see Materials and Methods and Appendix XVI) over the range of experimental dephasing times . Such calculations were done for: (1) different labeled na resultant γ l C’s; and (2) different na 13 CO- labeled Consideration of the fractional populations of na (ΔS/S0) 13 15 13 CO-na 15 N spin geometries with na N spin geometries with resultant γ lN‘s. CO and na 15 N then allowed calculation of nad , i.e. the expected dephasing if there were no rdd. If the calculation of (ΔS/S0) exp accurate, the (ΔS/S0) exp smallest (ΔS/S0) = (ΔS/S0) nad nad were for u samples with ft = u-1 = ft = u = ft = u+1 = 0.00. The for mHFP were derived from the u = 8 and the u = 28 samples and were considered to only be due to nad, i.e. and f7 = f8 = f9 = 0.00 and f27 = f28 = f29 = 0.00. exp Comparison of the (ΔS/S0) to (ΔS/S0) nad 2 2 yielded χ = 1.1 for u = 8 and χ = 1.8 for u = 28 with 7 data points for each sample. Similarly, the mV2E-HFP (ΔS/S0) compared to (ΔS/S0) nad exp from u = 8 sample were 2 , f7 = f8 = f9 = 0.00, which yielded χ = 4.0 for 7 data points for each sample. These combined results and the Chapter III results supported the accuracy and adaptability of the (ΔS/S0) nad calculation for different membrane-associated samples. 110 The γ lab ltu were calculated for the registries that aligned labeled nuclei within 7 Å, t = u-1, t = u, and t = u+1 registries. Because the γ lab 13 CO and labeled 15 N ltu only depended upon generalized spin geometries that were derived from crystal structures (see Materials and Methods), equivalent γ lab ltu values were used for all samples, u, and different γ lab ltu values were used for each generalized spin geometry for the t = u-1, t = u, and t = u+1 registries. Qualitative analysis of the (ΔS/S0) exp at  = 48 ms, Figure 27e, suggests that mHFP forms a distribution of antiparallel registries, and a “fully-constrained” model was used for quantitative fitting of the registry populations for which there was only a single antiparallel registry within a local region of the  sheet. Fully constrained oligomers likely pack with minimal void space that could be energetically favorable if the magnitude of the negative enthalpic contribution to the free energy from local contacts is greater than the positive entropic contribution to the free energy from forming a single rather than multiple registries 101 . It is noted that for mature amyloid fibrils formed in aqueous solution, structural data have been interpreted with a model of a single  sheet 102-105 registry, i.e. the fully constrained model . It is also noted that the number of molecules within a fibril are likely much greater than in mHFP oligomers since most Ala-1 are in close contact with the lipid headgroup of the membrane. This would not occur in larger amyloid fibrils where β sheet stacking would reduce the number of Ala-1 carbonyl in close contact (~ 5 Å) with the phosphorus of the lipid headgroup 106 . For a sample u, the calculated Δ(S/S0) sim depended on ft = u-1, ft = u and ft = u+1 but not on other ft’s, whose fractional population sum is described by fX = 1 - ft = u-1 - ft = u - ft = u+1. As 111 described in the Materials and Methods, sets of distinct (ΔS/S0) na sim for any sample u were na calculated using: (1) γ lC’s, and γ lN‘s from nad spin geometries along with the probabilities of these geometries; (2) γ lab ltu’s from rdd spin geometries; and (3) specific values of ft = u-1, ft = u and ft = u+1. There were 119 Δ(S/S0) exp data from all samples, and these data were globally fit to the ft’s with t in the 8 – 24 range, see Materials and Methods. The best-fit ft’s, Table 7, were 2 correlated with a statistically reasonable χ = 110 which was comparable to the number of 24 degrees of freedom of the fit (102). The sum of best-fit ft’s,  ft  1.14 , was close to 1, and it t 8 is reasonable to approximate the excess to be proportionally distributed among the ft‘s so that ft(true) = ft(best-fit)/1.14. In addition to the aforementioned “fully constrained” model with a single local registry, an alternate “unconstrained” model was also considered for which there 63 2 could be multiple registries within a local oligomeric region . Similar best-fit χ and ft’s were obtained with this model, see Appendix IX. The distribution of registries observed in mHFP appears to be close to the thermodynamic equilibrium state. Evidence to support this view is (1) the reproducibility of the (ΔS/S0) exp data for two different HFP-F8CG13N samples; (2) samples with similar chemical shifts and line shapes were prepared in aqueous solution (this dissertation) and by organic co39 solubilization followed by removal of organic solvent and hydration . The membrane/HFP interactions affect the thermodynamic equilibrium state of mHFP as evidenced by mixed secondary structures for HFP lyophilized without membranes and membrane-inserted β sheet 112 26 structure for mHFP . This Chapter shows that mHFP has a distribution of β sheet registries, and previous work has demonstrated that many factors contribute toward the free energy of any membrane inserted structure 107 . The membrane insertion free energy for each of the twenty naturally occurring amino acids, ΔG ins a, where a is an index referring to each of the twenty naturally occurring amino acids, has been estimated from measurements of the effects of point mutations on the relative quantities of a protein segment in either transmembrane  helical 93 structure or in a location outside of the membrane . The effect of a mutation depended on ins a were based on the effects of mutations residue position within the segment, and the ΔG within the central region of the segment which was located within the membrane interior in the transmembrane population. The secondary, tertiary, and quaternary structures of a sequence result in specific proximities between charged, polar, and hydrophobic residues, and these proximities contribute toward the free energy of the structure and may affect the ΔG ins a values in other systems such as mHFP. In the NMR sample preparation, HFP was initially in aqueous solution with random coil structure and then bound to membranes with a consequent transition to a membrane-inserted β sheet structure. Within the fully constrained model, there was consideration of multiple membrane insertion states for each antiparallel registry, t, where each state was differentiated by the number of amino acids, n, considered to be membrane-inserted and therefore used to calculate the total membrane insertion energy of a HFP strand. Since β sheets typically extend 108 over at least 4 sequential residues , the minimum value allowed for n was 4 and the index i was used to denote the residue number of the first membrane inserted amino acid. The maximum 113 value allowed for n corresponded to the number of residues within a registry, t. The membrane insertion energy of each state, ΔGt,n,i, was the sum of the ΔG ins a for all membrane inserted amino acids and was calculated by: Gt , n,i  i  n 1 ins  Gh h i (52) where h is an index indentifying the residue number of a membrane-inserted residue (i.e. ΔG ins h ins a of amino acid type “a” at residue “h” in the HFP sequence). For each registry, the = ΔG minimum ΔGt,n,i had residue “i” in a strand hydrogen bonded to residue i+n-1 of an adjacent strand, Figure 28. In this model, the first membrane inserted residue, residue i, was the same residue number for adjacent strands. The ΔGt,n,i depend on n and i for a specific t and the minimum ΔGt,n,i for each registry was denoted Gtmin and corresponded to a specific it and nt. The Gtmin for t = 8–24 are displayed in Figure 29 and Appendix XIV. Registries with more negative Gtmin generally had higher best-fit ft’s as determined from the NMR analysis. However, Gtmin is a model that uses approximations and it does not quantitatively predict ft’s accurately (Appendix XII). There are other free energy contributions associated with a peptide/protein’s membrane insertion energy where membrane binding, lipid composition, and peptide/protein secondary and tertiary structures may have non-negligible energetic contributions that contribute to the free energy of a system (for review of this topic, see reference 114 107 ). Figure 28. Sample indices, u, with the corresponding labeling schemes are displayed along with the registries, t, that result in labeled rCN of ~ 4.1 Å and ~ 5.5 Å that respectively correspond to 13 15 hydrogen-bonded and non-hydrogen bonded CO – H N. Membrane inserted regions are highlighted in yellow, and the corresponding n and i values are listed. 115 Figure 29. Double-y plot where ft populations (black) and ΔGt registry for mHFP (a) and mV2E-HFP (b). 116 min (red) are plotted for each 2. Fully Constrained Model mV2E-HFP Registry Distribution A broad distribution of mHFP oligomeric registries were observed, and the importance of a FP oligomeric structure in membrane fusion was suggested by dominant inhibition of fusion and virus infection when a small fraction of V2E mutated gp41 was co-expressed with a major 21 fraction of wild-type gp41 . One interpretation of this result is that V2E mutated gp41 can be incorporated into an oligomeric structure of wild-type gp41 and incorporation of V2E mutated gp41 results in the formation of fusion inactive FP oligomers (Appendix III). To date, this hypothesized fusion inactive FP oligomeric structure has not been identified. Similarly, V2E31 HFP has reduced membrane fusion rates relative to HFP , and structural comparisons between mHFP and mV2E-HFP may help to explain the dominant inhibition of membrane fusion by the gp41 V2E mutant. To investigate potential differences between mHFP and mV2E-HFP, samples were prepared to quantify the registry populations in mV2E-HFP with labeling schemes identical to the mHFP samples. Relative to HFP-WT, larger (ΔS/S0) > 17 while smaller (ΔS/S0) exp exp at τ = 48.2 ms were observed for u at τ = 48.2 ms were observed for u < 16, Figure 30a. These results qualitatively suggest that longer registries (t > 17) are more populated in mV2E-HFP while shorter registries, t < 16, are more populated in mHFP. These qualitative comparisons are in good agreement with the quantitative comparisons between mHFP and mV2E-HFP populations, Figure 30b and Table 7. For the mV2E-HFP registry population fittings, the 2 t  24  ft  1.39 and t 8 2 the χ min = 87 for 102 degrees of freedom. The mV2E-HFP χ min was lower than the mHFP 2 χ min, and one contributing factor is the narrower registry distribution for mV2E-HFP since the 117 2 χ min were generally lower for t  u 1  ft  0.00 than for data sets with populated ft’s, Table 7. t u 1 Another possible contributing factor may be that our calculations only considered the spectral noise contribution to the error and did not account for other errors associated with our data analysis. The largest unaccounted for source of error is likely the error associated with the ft based (ΔS/S0) sim exp data that were fit to the (ΔS/S0) 2 in the χ analyses. As demonstrated by the 2 duplicate analysis of the global χ min with two different F8CG13N mHFP sample preparations, 2 Table 9, smaller rms error lead to similar best fit ft, but larger χ min (+9). This suggests that our experiments were reproducible, but the experimental error may be under approximated when quantified by rms error. Additionally, out of 119 data points, mHFP had smaller spectral noise as indicated by smaller rms error in 64 points while mV2E-HFP had smaller spectral noise in 55 points. If the overall experimental error is generally greater than the spectral noise error, then we 2 2 would expect global χ min(mHFP) > global χ min (mV2E-HFP) since the spectral noise error is smaller for mHFP. There are multiple potential contributing factors that made t  24  ft  1.00 : (1) The rCN t 8 distances may be shorter in mHFP and mV2E-HFP constructs than the modeled rCN that were derived from the 2IWW.pdb β barrel crystal structure. This would result in larger actual dCN than the modeled dCN. Therefore, larger ft = u-1, ft = u, and ft = u+1 registries would be required to model the actual dephasing. Other experiments have demonstrated that larger side chain 118 109 groups can affect the hydrogen bond strength of 310-helices . Since the HFP sequence contains many Gly residues (i.e. the amino acid with the smallest sidechain), the hydrogen bonds in mHFP β sheets may be stronger resulting in shorter rCN than those found in the 2IWW β sheets. However, to my knowledge, there is currently no experimental data for β sheets to support this hypothesis. (2) The dephasing due to the t = u-2 and t = u+2 registries was not accounted for in the 3 registry population fittings for each sample. These registries were 24 accounted for in the 5 parameter (registry) fittings where  ft was reduced by 0.09 and 0.13 t 8 for the mHFP and mV2E-HFP, respectively, Table 7 and Table 8. The modified equations for these fittings are found in Appendix VIII; (3) The β sheets may not be fully constrained β sheets. Other models, such as the unconstrained model (Appendix IX), result in smaller 24  ft than the fully constrained model for the best-fit ft’s. t 8 119 Figure 30. (a) The (ΔS/S0) exp (τ = 48.2 ms) for mHFP (black) and mV2E-HFP (green). (b) The ft for mHFP (black) and mV2E-HFP (green) for the fully constrained model using the 3 registry fitting method. Both (a) and (b) demonstrate that mV2E-HFP has a smaller population of shorter registries ( t < 16) and that mV2E-HFP has a larger population of longer registries (t > 17). 120 This Chapter has shown that mHFP forms a broad distribution of registries that are correlated to membrane insertion energy of each registry, Gtmin , in mHFP. Previous work has shown that a HFP construct with Phe-11 mutated to Gly has significantly reduced membrane fusion activity while the Phe-11 to Val mutation has reduced membrane fusion activity compared 110 to the wild-type HFP construct, but higher activity than the F11G mutation . These results are consistent with the idea that forming structures with more negative Gtmin are important for forming fusion active structures since the F11G mutation increases the calculated Gtmin for most registries with t > 10 (Appendix X). However, mV2E-HFP has a surface membrane 26 location , and it is therefore counterintuitive to expect the negative magnitude of Gtmin to be correlated to the magnitude of the populated registries in mV2E-HFP. The calculated Gtmin for mV2E-HFP are similar to mHFP, and ft’s are very different for mV2E-HFP and mHFP, Figure 30 and Table 7. Therefore, other energy contributions associated with V2E mutation likely affect the distribution of registries and fusion activity (see Discussion). However, it is of note that populated registries in mV2E-HFP generally had negative Gtmin values. Since mV2E-HFP has a shallower membrane insertion depth than mHFP and a larger population of t > 17 registries, these data are consistent with all or a subset of “longer registries”, t > 17, having shallower membrane insertion depth and reduced membrane fusion activity relative to all or a subset of the “shorter registries” t < 16. The t ~ 17 was chosen as a distinguishing registry in this qualitative model because f17 for HFP is approximately equal to the f17 for V2E-HFP, Figure 30. Additionally, for t > 13 in V2E-HFP registries, the charged 121 Glu-2 residue is placed further away from the most hydrophobic region (LFLGFL) of the adjacent strands as t becomes larger. This likely has a favorable contribution to the free energy of these longer registries over the t ≤ 13 registries that contribute to the minimal population of t ≤ 13 registries in V2E-HFP. HFP membrane insertion is likely related to favorable hydrophobic interactions with the membrane relative to water, and the hydrophobic residues in HFP are primarily found in the first 12 N-terminal residues. As previously noted, shorter registries, such as t = 13 registry, cluster hydrophobic residues in HFP oligomers which may be important for achieving deeper membrane insertion and more rapid fusion. The t = 13 registry has it = 2 and nt = 11 at the Gtmin where it and nt are respectively the number of the first membrane-inserted residue and the total number of inserted residues in registry t that correspond to Gtmin . The t = 20 registry has it = 7 and nt = 8. Registries with more residues incorporated into the membrane inserted region may allow for the hydrophobic segments to protrude deeper into the membrane. Additionally, residues that flank the membrane inserted region could affect the membrane insertion depth as well. For mHFP, the t = 13 registry has a membrane inserted region that spans 11 residues (-0.9 kcal/strand) with one flanking residue on each side of the membrane inserted region (+0.9 kcal/strand). For the t = 20 registry, the membrane inserted region spans 8 residues (-0.7 kcal/strand) while the residues flanking this region span six residues on each side (+3.6 kcal/strand). While flanking residues with disordered secondary structure had minimal 93 contributions to the actual Gtmin of helical structure in the Hessa study (Appendix XV) , the large flanking regions of the t = 20 registry may contribute toward the actual Gtmin since the backbone dihedral angles within sheets are more restricted than disordered flanking regions of 122 the α helices in the Hessa experiments. The extension of the β sheet from the membrane inserted region to the flanking region for the t = 20 registry is supported by 13 CO chemical shift data for both mHFP and mV2E-HFP where the Ala-1 and Ile-4 residues have chemical shifts consistent 26 with β sheet structure . In general, this suggests that the β sheet structure extends from the membrane inserted region to the flanking regions in both mHFP and mV2E-HFP. This is different from the Hessa experiments where Pro and Gly residues were inserted adjacent to the membrane inserted region to disrupt secondary structure (Appendix XV). The flanking regions’ strong preference for the bilayer-water interface over the hydrophobic membrane interior may anchor the t = 20 registry to the bilayer-water interface and result in an overall shallower membrane insertion depth than the t = 13 registry. The u = 20 sample, F8CG13N labeling, was also prepared with the highly fusogenic HFPtr. Relative to mHFP, smaller (ΔS/S0) exp were observed for mHFPtr which was opposite to the mV2E-HFP result, Figure 30. These data suggest relative to mHFP and mV2E-HFP, longer registries are less populated in mHFPtr and provide corollary evidence for the hypothesis that the t ~ 20 registries are less fusogenic than shorter registries. These combined data suggest that the FP may provide a target for fusion inhibitory drug design by stabilization of longer (u  20) FP registries. 123 13 Figure 31. REDOR S0 and S1 C SSNMR spectra at 48.2 ms dephasing time for (a) HFPF8CA21N, (b) HFP-F8CG13N, (c) HFPtr-F8CG13N, (d) V2E-F8CG13N, (f) HFP-L12CA6N . (g) V2E-L12CA6N, (h) HFP-L9CG5N, or (i) V2E-L9CG5N. Each spectrum was processed with 200 Hz line broadening and baseline correction and was the sum of: (a) 46816; (b) 36665; (c) exp 19372; (d) 34271; (f) 44931; (g) 46231; (h) 40272; or (i) 46809 scans. (e), (j) Plots of (ΔS/S0) vs dephasing time with the rms error. Isotopic labeling of each mHFP is displayed in the legend that correspond to HFP-F8CA21N, (black, square), HFP-F8CG13N (orange, inverted triangle), HFPtr-F8CG13N (cyan, triangle), (d) V2E-F8CG13N (red, circle), (f) HFP-L12CA6N (dark yellow, square), (g) V2E-L12CA6N (purple, circle), (h) HFP-L9CG5N (green, triangle), or (i) exp V2E-L9CG5N (wine, inverted triangle). Variation less than 0.02 in (ΔS/S0) was also observed between two different preparations of the same sample type, e.g. HFP-F8CG13N. 124 4.4 Discussion 1. Modeled Membrane Insertion Depth mHFP Earlier SSNMR data showed that the deepest membrane insertion depth for mHFP 26 occurred for the Ala-6 and Leu-9 residues . The results of these prior experiments were compared to newly calculated membrane insertion depths of each residue’s 13 CO for Ala-1 through Gly-16. In general, these equations have dependences upon the indices: (1) i, the residue number for the first membrane inserted residue (N-terminal most residue of the membrane inserted region); (2) n, the number of membrane inserted residues; and (3) h, the residue number of the 13 CO for which the membrane insertion depth was calculated. The inserted region of a β sheet was modeled as a semi-circle, and the membrane insertion depth of each residue’s 13 CO, Dh,i,n, was calculated, Eqs (53)-(58). Additionally, Figure 32 provides a visual aid to follow the Dh,i,n calculations. 125 Figure 32. Membrane insertion depth model as described in the text. Previous work has demonstrated that the Ala-1 carbonyl carbon is ~5 Å from the lipid phosphorus, and this phosphate region is referred to as the water/bilayer interface. Dh, i, n  Rn Sin(h,i, n ) (53) where Rn is the radius of the semi-circle for registry t and φh,i,n is the angle between the plane of the water/bilayer interface and Rn when Rn is extended to the carbonyl of residue h. Rn  3.5n  (54) where each amino acid in β sheet conformation spans ~3.5 Å. The angle φh,i,n was calculated by: h,i, n  Ah,i  3.5n (55) where Ah,i, was the arc length along the semi-circle from residue i to h where residue i is Nterminal to the 13 CO of residue h. The arc length, Ah,i was calculated by Eq (56) 126 Ah, i  3.5(h  i  1) (56) where the N-terminus of the semi-circle is the semi-circle is the 13 13 CO of residue i – 1, and the C-terminus of the CO of residue i + n – 1. For h < i, h > i + n -1, or nt = 0 (i.e. the respective conditions where h was N-terminal of the membrane inserted region, h was C-terminal of the membrane inserted region or there was no membrane inserted region) the 13 CO of residue h is not located within the membrane inserted region and Dh,i,n = Rn = φh,i,n = Ah,i = 0. Each t registry could be an ensemble of states which differ in their i and n. In any ensemble, the lowest energy state will be the most populated. In calculating the membrane insertion depth of residue h in registry t, it was approximated that the only populated state was the one with i = it and n = nt corresponding to Gtmin . Under these conditions, Eq (53) can be rewritten as: Dh, i , n  t t  (h  it  1)  3.5nt Sin    nt   (57) Thus, the average membrane insertion depth of each residue’s 13 CO in mHFP for the fully constrained model, Figure 33, was calculated by: 24 ft Dh,it , nt avg D   h,it , nt t  8 24 f  t t 8 (58) 24 where 1.14 and 1.39 correspond to the  ft for mHFP and mV2E-HFP in the fully constrained t 8 model, respecitvely. This calculation is consistent with the membrane insertion depth data in that Ala-6 and Leu-9 were inserted deeper than Ile-4, Leu-12, and Ala-14. For most t < 16 registries, 127 the calculated Ala-6 Dh,i , n > 5 Å while Dh,i , n ~0 Å for t ≥ 16. Relative to mHFP, mV2Et t t t HFP have a larger population of t > 16 registries with consequent smaller D avg for Ala-6 in h,it , nt mV2E-HFP, Figure 34. This is consistent with earlier SSNMR measurements of deeper membrane insertion of the Ala-6 of mHFP relative to the Ala-6 of mV2E-HFP. Figure 33. The calculated average membrane insertion depth of each residue’s is plotted for HFP (red) and V2E-HFP (blue). 128 13 CO, D avg , h,it , nt Table 10. mHFP and mV2E-HFP 13 Residue Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 CO D 13 CO D avg h,it , nt avg h,it , nt mHFP 0.00 -0.0091 -0.99 -2.1 -3.0 -4.6 -6.2 -6.7 -6.2 -5.3 -3.3 -0.90 -0.72 -0.030 -4.3 E-17 0 129 mV2E-HFP 0.0 0.0 0.0 -0.32 -0.72 -1.8 -4.3 -6.1 -6.7 -6.5 -4.9 -2.4 -1.7 -0.17 -1.1 E-16 0.0 Figure 34. Calculated membrane insertion depth of the Ala-6 13 CO for HFP (red) and V2E-HFP (blue) is plotted for each registry. Additionally, ft for mHFP (black) and mV2E-HFP (green) are plotted for each registry. The mV2E-HFP registry populations are shifted toward longer registries (t > 17) relative to mHFP and the calculated membrane insertion depth of the Ala-6 13 13 CO is  0 for these registries. This is consistent with previous work where the Ala-6 CO is 26 inserted deeper in mHFP relative to mV2E-HFP . 2. Relevance of Broad Distribution The quantitative determination of the ft’s is the most significant result of Chapter IV. There is qualitative correlation between the ft’s and the respective negative magnitudes of the Gtmin ’s in mHFP, but there is not quantitative agreement between the ft’s and those from calc thermodynamic calculations, ft , using the Gtmin ’s (Appendix XII). For α helices, one error in the estimation of Gtmin was neglect of the (negative) contribution from local clustering of 130 hydrophobic residues (Support by Appendix XIII, Figure 52). For mHFP, such clustering is most prominent in the first 12 residues in the sequence and may contribute to relatively large f13 since the t = 13 overlaps residues 2-12 (i.e. all of the hydrophobic residues with inclusion of the least possible number of non-hydrophobic residues). In addition, each registry’s energy will 29 include a (negative) enthalpic contribution if void space is minimized . The contribution from such tight packing in β sheets has been evidenced by frequent observation of hydrogen bonding between Phe and Gly residues which have large and small sidechains, respectively 111 . For … mHFP, the Phe Gly hydrogen bonding in the t = 20 and 23 registries may contribute to the results f20 > f19, f21 and f23 > f22, f24. In mV2E-HFP, there is poor qualitative correlation between the negative magnitude of Gtmin and the best-fit ft from the NMR analysis, Figure 29. For example, the G min ~ (1 / 2) G min , but f20 > f18 by 0.18. Electrostatic interactions 20 18 and sidechain packing appear to contribute to the free energy of a particular registry in mV2EHFP. The t = 20 registry may be favored in mV2E-HFP due to the favorable Phe-Gly sidechain packing. Additionally, published crystal structures with charged residues have shown that charged residues are not typically found in β sheet oligomers adjacent to hydrophobic 105 residues . This suggests that it is not energetically favorable to have glutamic acid adjacent to hydrophobic residues which may be due to unfavorable electrostatic interactions between charged and hydrophobic sidechains. This is statistically represented where glutamic acid is more frequently found in β sheets when adjacent to charged residues relative to hydrophobic 105 residues . In mV2E-HFP, longer β sheet registries place Glu-2 near more polar residues, but 131 there is currently no residue specific structural data for the Glu-2 residue in mV2E-HFP. However, the most favorable electrostatic interactions may result from proximity to the Ser-17, Thr-18 or Arg-22 residues (i.e. t ~ 17-23). This may present favorable interstrand electrostatic interactions between sidechains, but having multiple charged residues near the edges of the β sheets could provide more favorable electrostatic interactions for membrane binding and an overall lower free energy than for shorter registries (i.e. t ~ <17). Additionally, mV2E-HFP has 26 shallower membrane insertion relative to mHFP . Relative to mHFP, there may therefore be less correlation between membrane insertion energies and ft’s calculated for mV2E-HFP. Plausible hypotheses to explain the low populations of t < 17 registries and higher populations of t >17 registries in mV2E-HFP relative to mHFP include: (1) Longer β sheet registries place Glu2 near more polar residues, and the most favorable electrostatic interactions may result from Glu2 adjacent to the presumably positively charged Ser-17, Thr-18 or Arg-22 residues (at pH ~7). (2) Our experiments sample the membrane bound structures, and shallow membrane insertion may be required for mHFP and mV2E-HFP to stay bound to the membrane (see Appendix XIII, L9R data, for counter example). The ΔGGlu-2 = +2.68 kcal/mol while the Gtmin of populated registries were typically > -1.7 kcal/strand and < ~ -0.4 kcal/strand, Figure 29. The ΔGGlu-2 = +2.68 kcal/mol reflects Glu’s high preference to stay in a hydrophilic environment over the hydrophobic environment of the interior of a membrane. For mV2E-HFP registries where Glu-2 is located near “membrane inserted” region, it is plausible that Glu’s energetic gain for being located in a hydrophilic environment may out weigh the energetic gain for having the hydrophobic region membrane inserted. As the Glu-2 residue is placed further away from the “membrane inserted” region, is seems possible that the Glu-2 could be placed in a hydrophilic 132 aqueous environment while the “membrane inserted” region could be in a more hydrophobic environment such as the membrane interior or a β sheet aggregate where the hydrophobic residues of adjacent strands are proximate to each other. In the highly populated registries of mHFP-V2E, t > 17, Glu-2 is located far away from the “membrane inserted” region which supports the above hypothesis. This theory is also supported by previous REDOR experiments where residues that were 13 13 31 CO- P mV2E-HFP CO labeled at the Ala-6, Leu-9, and Leu-12 (more hydrophobic region of sequence) dephase less than Ala-1, Ile-4, and Ala-14 (more hydrophilic regions of the sequence) 106 . Additional experiments to further validate this hypothesis have not been designed to date. Relative to the mHFP u = 20 sample, much larger (S/S0) exp were observed for the u = 20 mV2E-HFP sample and the opposite trend was detected for the u = 13 samples. These data demonstrate significant differences in the registry distributions of the two peptides which may correlate to the large differences in their rates of vesicle fusion. Previous experiments have 26 shown that faster vesicle fusion rates are correlated with deeper membrane insertion , and my data and calculations suggest that registries with higher populations (ft > 0.05) have negative free energies for membrane insertion. In general, previous work has supported that membrane insertion of peptides/proteins are governed by thermodynamics 93,107,112,113 . The culmination of the collected data and previous work are consistent with the mHFP and mV2E-HFP registries being governed by thermodynamics. Because thermodynamics appear to govern the membrane inserted structures for proteins/peptides, the following hypotheses are reasonable: (1) The gp41 FP that is membrane bound, has deep membrane insertion depth, and is fusion active may have a 133 distribution of registries similar to mHFP or weighted toward the “shorter” registries (possibly t < 17); and (2) The gp41 FP that is membrane bound, has shallow membrane insertion depth, and is fusion inactive may have a distribution of registries that is similar to mV2E-HFP and weighted toward “longer” registries (possibly t > 17). Alternatively, the FP registry distributions in gp41 may result from interactions with residues C-terminal of FP, but this hypothesis is not supported by the mHFP and mV2E-HFP data. However, this hypothesis has not been proven wrong to date and therefore cannot be entirely ruled out. Previous work has shown that the fusion peptide synthesized with the N-terminal helical region of gp41 (i.e. a longer construct commonly referred to as N70) has a higher vesicle fusion rate relative to HFP, but it has not been shown whether: (1) The faster lipid mixing rate of the N70 construct results from a structural change with the fusion peptide region; or (2) The faster lipid mixing rate of the N70 construct results from an interaction between the N-terminal helix (possibly dehydration of the membrane surface that results from the N-terminal helix binding to the membrane surface). Detection of registry distributions may also be significant for other peptides and proteins. For example, recent data support higher neurotoxicity of small oligomers of amyloid peptides and lower toxicity of large and mature fibrils with well-defined  sheet structure and a single registry (usually in-register parallel) 102-105,114 . There are little data about the registry distribution of the oligomers and the present approach could be applied to determine this distribution. Comparison with the distribution in the fibril will improve understanding of the amyloid structure-function relationship and aid development of new inhibitors of amyloid oligomer formation and amyloid disease. 134 Chapter V. Dissertation Summary and Future Work 5.1 Summary The preliminary focus of this dissertation was to contribute toward the literature of conflicting reports regarding the population of in-register parallel β sheets for membrane29,47 associated HFP constructs . To achieve this, NMR samples were prepared with more sparsely isotopically labeled peptides which provided more specific registry detection and less interpretational ambiguity than for previous studies 29,47 . Additionally, a more quantitative model to account for natural abundance dephasing contributions was developed to improve upon previous models 36,54 which improved the accuracy of quantitative data analysis for REDOR experiments. Very little in-register parallel β sheets were detected (< 15% of the β sheet 63 structure) , and this result argues against previous hypothesizes that in-registry parallel β sheets are required for fusion activity. Later experiments identified the presence of a broad distribution of antiparallel registries in mHFP. The distribution of registries appears to be functionally relevant since the registry distribution of mV2E-HFP is narrower and shifted toward longer registries that delocalize hydrophobic residues. The more fusion active mHFP has a distribution of registries that includes shorter registries that cluster hydrophobic residues. This clustering of hydrophobic residues in mHFP may explain mHFP’s overall deeper membrane insertion relative 26 to mV2E-HFP . For mHFP, there was qualitative correlation between the magnitude of each registry’s ft and the negative magnitude of each registry’s Gtmin which suggests that calculating Gtmin energies for registries may be useful in predicting β sheet registries for any sequence of 135 membrane-bound β sheet structure. Alternatively, the distribution of registries within the less functional mV2E-HFP displayed poor qualitative correlation between the magnitude of each registry’s ft and the negative magnitude of each registry’s Gtmin . While this result may initially be discouraging, it is of note that mV2E-HFP has shallow membrane insertion depth, and therefore, the energy minimized structure should not result from an energetic contribution resulting from the change in free energy in going from aqueous solution to a membrane inserted β sheet. It is also important to keep in mind that the NMR samples contain registries that stay bound to the vesicle aggregates and that the registries observed in mV2E-HFP typically had negative Gtmin . Additionally, the L9R mutant gp41 does not transdominantly inhibit wild type gp41 21 and L9R-HFP registries had predominantly positive Gtmin values, Figure 35, which may inhibit formation of a membrane bound oligomeric structure with the wild type fusion peptide. The β sheet structure was not predominantly formed in mL9R-HFP as observed with mHFP and mV2E-HFP, Appendix XIII. Typical values for Gtmin in populated registries in mHFP ranged from -0.5 to -1.6 kcal/mol. Other literature has shown that sidechain-sidechain hydrogen bonds located at the water-bilayer interfacial region have been shown to contribute ~115 0.4-0.8 kcal/mol . This suggests that hydrogen bonds between polar or charged amino acids contribute to the free energy of a structure in a similar magnitude as that of the Gtmin . These energy contributions that result from charged sidechains may explain why there is poor qualitative agreement between the ft’s in mV2E-HFP and the negative magnitude of the Gtmin ’s. Additionally, sidechain packing interactions may also contribute to the energy 136 minimized structure as previous suggested in the glycine mutant studies 28 and as supported by the higher than expected f20 and f23 in mHFP (Chapter IV). Thus, predicting β sheet registries based upon Gtmin is likely most accurate for hydrophobic sequences lacking or separated from charged residues (i.e. similar to HFP), but other energy contributions should also be considered. Thus, predicting registry populations from each registry’s Gtmin may prove useful for sequences such as the measles fusion peptide and simian immunodeficiency virus (SIV) fusion peptide whereas sequences containing charged residues within or near the hydrophobic sequence (i.e. similar to V2E-HFP and L9R-HFP), will likely have significant energetic contributions from the charged sidechains and poor qualitative agreement between the negative magnitude of Gtmin and ft’s. Alternatively, it seems that charged residues may be incorporated into a sequence with minimally effecting the membrane insertion energies when they are separated by a break in the secondary structure as observed in the Hessa experiments (Appendix XV) observed with the K6 tag in mHFP 36,106 93 and as . Future experiments in other protein-membrane samples should consider quantifying distributions of registries/structures rather than only attempting to qualitatively identify a single registry/structure since the mHFP and mV2E-HFP data have demonstrated that broad distributions of structures can have approximately equal free energies in a membrane environment. In doing this, functional and nonfunctional structures in complex systems may be identified. 137 3 HFP V2E L9R t 1 0 G min (kcal/strand) 2 -1 8 -2 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Registry (t) Figure 35. The membrane insertion energies were derived from the Hessa biological hydrophobicity scale for the HFP, V2E-HFP and L9R-HFP by the methods described in Chapter IV. The L9R-HFP has predominantly positive Gtmin whereas both HFP and V2E-HFP have many registries with negative Gtmin which suggests that the distribution of registries should be different between constructs. Additionally, it is not obvious that mL9R-HFP should form membrane inserted β sheets since t < 12 registries were minimally populated in mHFP, and t > 12 registries have positive Gtmin in mL9R-HFP. Unlike mHFP and mV2E-HFP, mL9R does not predominantly form membrane inserted sheets, Appendix XIII. 138 Table 11. Gtmin values for HFP constructs. Gtmin (kcal/mol) Registry (t) HFP V2E-HFP L9R-HFP 8 -0.08 0.99 -0.08 9 -0.63 0.44 -0.19 10 0.545 -0.3 -0.3 11 0.065 -0.62 -0.62 12 -0.76 -1.17 -0.02 13 -0.87 -0.57 2.26 14 -0.56 -1.31 1.82 15 -1.3 -1.3 1.83 16 -0.7 -0.89 2.24 17 -1.44 -1.44 1.69 18 -1.55 -1.55 1.58 19 -0.81 -1 2.13 20 -0.7 -0.7 2.43 21 -0.59 -0.59 2.54 22 0.15 -0.04 0.61 23 0.99 0.28 0.72 24 1.51 -0.02 -0.02 * Gtmin values were calculated from the methods described in Chapter IV. 5.2 Membrane Location Previous membrane location studies from this group have made lipid distance measurements as well as lipid 19 F – HFP 13 P – HFP 13 CO CO distance measurements to determine the membrane location of mHFP, mV2E-HFP and mHFPtr 26,106 measurements provide a great quantitative tool for measuring 139 31 . The 13 31 13 P- CO distance CO distance from the water- 13 bilayer interface of a vesicle where ~86% of the Ala-1 Additionally, depth. The 19 CO had a 31 13 P- C distance of ~5 Å. 13 F- CO distance measurements were performed to probe membrane insertion 19 13 F- CO setup experiments used a helical peptide with the sequence EQLLKALEFLLKELLEKL where the Phe-9 was substituted with p-fluorophenylalanine from l3 Sigma-Aldrich and the Leu-10 was CO labeled. This setup compound was effective in setting the π pulses, but the fluorinated lipids are not naturally occurring and required H to F substitution at a single site on the lipid acyl chain. Incorporation of fluorinated lipids into lipid vesicles present complications where (1) fluorinated chains tend to cluster together within vesicles, (2) fluorinated chains can form fibrous bands instead of vesicles, (3) fluorinated chains can affect 116,117 membrane permeability and surface tension bilayer phase 118 mol fraction of , and (4) fluorinated chains can disrupt the . These effects were considered in the 19 19 13 F- CO REDOR experiments and the F-DPPC lipids used was varied. A 0.09 mol fraction of 19 F-DPPC lipids exp yielded the largest (ΔS/S0) peptide 13 CO-lipid 19 , and this mol fraction was considered to maximize potential 26 F contacts with minimal disruption to the bilayer . Much like the Chapter exp 2 III and Chapter IV χ analyses, the (ΔS/S0) compared to (ΔS/S0) sim derived from the 19 F experiments were 2 13 sim CO nuclei in contact with one or more the average dipolar coupling (or distance) between these 13 CO-lipid by χ analysis. However, in these experiments, the (ΔS/S0) upon two parameters: (1) the population of 13 13 13 CO and the lipid 19 19 depended F, and (2) F. For all labeled CO in mHFP, mV2E-HFP, and mHFPtr samples, typical best-fit distances between labeled CO and lipid 19 F were ~7-8 Å, and typical best-fit populations ranged from ~0.00 to 0.40. One interpretation of these results is that detection of ~0.4 fractional population may correspond to obtaining maximum (ΔS/S0) exp values for samples with a 0.09 mole fraction of fluorinated lipid. The small molar fraction of fluorinated lipid may make maximal observable (ΔS/S0) exp ~0.4, instead of ~1.0 as could be observed with 1.0 mole fraction of fluorinated lipid. Another interpretation is that for any 19 F experiment using a 0.09 mol fraction of fluorinated lipid, 0.60 140 of the labeled 13 CO’s have a different membrane location than the 13 CO’s close to the 19 Fs. The previous interpretation assumes that the unidentified ~0.6 fraction have the same membrane location as the identified ~0.4 fraction. These two interpretations lie on the extreme ends of how the 19 F data could be interpreted. A positive control experiment could be run to determine the precise meaning of these fractional populations by probing the membrane location using the KALP peptide where KALP peptides form transmembrane α helices in lipid vesicles composed of 1,2-Dilauroyl-sn-glycero-3-phosphocholine (di-C12:0-PC), 1,2-ditridecanoyl-sn-glycero-3phospholcholine (di-C13:0-PC), 1,2-dimyristoyl-sn-glycero-3-phosphocholine (di-C14:0-PC), or 1,2-dioleoyl-sn-glycero-3-phosphocholine (di-C18:1-PC) While membrane location studies using 19 119 . 13 F- CO REDOR experiments provided some insight regarding the membrane location of HFP constructs, future membrane location studies could avoid the problems associated with fluorinated lipids by performing 120,121 REDOR 2 13 H- CO experiments. Deuterated cholesterol is available through Sigma Aldrich in the form of cholesterol-2,2,3,4,4,6-d6 and cholesterol-25,26,26,26,27,27,27-d7, Figure 36. Cholesterol orientation within membranes has been studied by experiments where 13 C spin-lattice relaxation time C spin-lattice relaxation times depended upon the proximity to the paramagnetic agents. These studies used vesicles prepared with compared changes in 13 13 13 C substituted cholesterol and C spin-lattice relaxation times in the presence and absence of a paramagnetic agent to determine the average membrane location of the 122 position 13 C atoms at each . Similar experiments could be run using our NMR sample preparation methods to confirm the average cholesterol orientation in our NMR samples both before and after the addition of the HFP constructs. These studies are important since previous studies have shown that cholesterol position and orientation within a membrane is dependent upon membrane composition. Studies have suggested that cholesterol may lie parallel to the lipid acyl chain with the ringed structure end toward the membrane surface, and it has also been suggested that cholesterol can lie perpendicular to the lipid acyl chain in the center of the membrane 141 123,124 . These differences in membrane location result from different membrane compositions where membranes with higher ratios of poly-unsaturated lipid acyl chains, chains with multiple double bonds, resulted in the cholesterol being located in the center of the membrane. In general, double bonds in unsaturated lipid acyl chains result in “kinked” acyl chains, and membranes containing higher ratios of poly-unsaturated lipids are believed to pack together less favorable than the saturated lipid chains. These less ordered membranes were shown to be correlated to cholesterol being located in the center of the bilayer. While poly-unsaturated lipid chains were not used in the mHFP studies, mHFP constructs have been shown to insert into the membrane bilayer which disorders the membrane bilayer and could alter the cholesterol membrane location. Therefore, defining the cholesterol location before and after the addition of HFP is necessary before using cholesterol to probe mHFP’s membrane location since cholesterol may not be oriented parallel to the lipid acyl chain after the addition of HFP. 2 13 Aside from the membrane location data that could be obtained from D- C REDOR experiments, the presence of cholesterol in membranes favors β sheet secondary structure for HFP, and the cause of the β sheet conformational preference is not understood. Determining the location of HFP relative to cholesterol molecules could demonstrate if or where HFP is in contact with cholesterol which could contribute toward understanding why membranes with cholesterol favor β sheet structure over α helices. Additionally, the mV2E-HFP experiments demonstrated distinct differences in the β sheet conformation where best-fit 31 31 13 P- CO distance for 13 CO labeled Leu-12 residues in 13 P- CO distances and populations were 5.7 ± 0.02 Å, 0.39 ± 0.02 and 8.4 Å ± 0.02, 0.97 ± 0.02 for membranes with and without cholesterol, respectively. The difference in Leu-12 13 CO proximity to 31 P could be due to β sheet stacking in the presence of cholesterol or aggregation of β sheets within a local cholesterol domain that may segregate HFP constructs away from the lipids. Either of these scenarios could additionally result in 13 19 F- CO REDOR experiments that never reach 100% dephasing in the aforementioned samples prepared with fluorinated lipids. Alternatively, the increased 19 13 F- CO and 31 13 P- CO distances in membranes containing cholesterol may be due to the decreased lipid density within vesicles with cholesterol relative to vesicles without cholesterol. 142 Determining the proximity between mHFP constructs and cholesterol has scholarly value and may offer new ideas for inhibitory drug design that target the fusion peptide. Recently, 125 inhibitory drug design has incorporated attachment of peptides to cholesterol . This method is attractive toward inhibiting HIV entry since the host cells that HIV infects contain membranes 37 composed of ~30% cholesterol . Because cholesterol is native to the HIV host cells and the fusion peptide structure is altered by the addition of cholesterol to membranes, future schemes to inhibit HIV fusion may include attaching peptides or even small molecules to cholesterol that inhibit fusion active mHFP structures. In the context of this dissertation, that could involve attaching a small molecule or even peptide, possibly a variation of the V2E sequence, to cholesterol that shifts the distribution of β sheet registries toward longer registries. Attaching molecules to cholesterol has had preliminary success where the inhibitory effects of the C34 125 peptide towards HIV/host-cell fusion are enhanced when attached to cholesterol . If the fusion peptide is found to be proximate to cholesterol, attaching an inhibitor drug that targets the fusion peptide may prove to be an exceptionally efficient method for inhibitory drug design since the HIV host cells contain high concentrations of cholesterol. This may lengthen the time that the inhibitory drug is incorporated into the host cell which could improve its inhibitory effect as observed with the C34 peptide 125 . 143 Figure 36. Cholesterol molecules with carbon atoms numbered (a) Cholesterol (b) Cholesterol2,2,3,4,4,6-d6 and (c) Cholesterol-25,26,26,26,27,27,27-d7. 144 Figure 36 (cont’d) 5.3 Resin Bound Structure Solid phase synthesis using Fmoc chemistry sequentially adds amino acids from the Cterminus to the N-terminus of the synthesized sequence. In general, the coupling step (i.e. attaching a residue) becomes less efficient for residues that are added later in the synthesis (Nterminal residues) than for residues that are added earlier in the synthesis (C-terminal 57 residues) . For HFP, the N-terminal residues are hydrophobic and become more challenging to 26 add than C-terminal residues . One potential explanation for this is that the added amino acids form aggregate structures, such as β sheets, while attached to the resin which may reduce the accessibility of the attachment site making the coupling reactions slower. To test the structural basis for this hypothesis, cross polarization spectra of a resin bound V2E-L9CI4N peptide was acquired, Figure 37 and Figure 38. The Leu-9 carbonyl had a 174.3 ppm chemical shift which is similar to the 174 ppm chemical shift observed for β sheet HFP and different from the 179 ppm 145 chemical shift that is observed for helical HFP 36,77 . Additionally, the Leu-9 has well defined secondary structure as evident by the 4.0 ppm full-width at half maximum height linewidth where as lyophilized HFP had line widths of ~7-8 ppm, Chapter III. These spectra serve as preliminary data to support the aforementioned hypothesis. By sampling more positions on HFP, one would hope to learn (1) where the β sheet structure begins; and (2) whether the β sheet forms as a result of peptide length or hydrophobicity. If the reasons for the formation of β sheet structure were well understood, new approaches for synthesis may be designed to enhance the efficiency of HFP synthesis and possibly the synthesis of other peptides as well. Additionally, sequences greater than ~30 residues are rarely synthesized due to poor yields of these sequences. If the efficiency were improved, synthesis of longer sequences may be more feasible. 146 Figure 37. V2E-L9CI4N resin bound (prior to cleavage) with a MAS speed of 10 kHz. 147 Figure 38. V2E-L9CI4N resin bound (prior to cleavage) with a MAS speed of 6 kHz. 148 APPENDICES 149 Appendix I. Files Checklist Chapter 2 Figure 12 Magic Angle Spinning @ ..\home\sunyan4c\data\Scott\Setup\KBr091809 Magic Angle Spinning Procedure @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\ KBr_Magic_angle.pdf 1 Figure 13 H Pulse @..\home\mb4b\data\Scott\REDOR\Setup\90pulse_array060610 Figure 14 CP @..\home\mb4b\data\Scott\REDOR\Setup\cp_array_012711 Figure 15 Cp_zfilter @..\home\mb4b\data\Scott\REDOR\Setup\cp_zfilter_012711 15 Figure 16 N π pulse @..\home\mb4b\data\Scott\REDOR\Setup\15Narray_012711 Figure 17 @..\home\mb4b\data\Scott\REDOR\Setup\I4\I4_15NH1Rabi_arrays\20kHz_15N\ I4_15N1HRabi_array162 @..\home\mb4b\data\Scott\REDOR\Setup\I4\I4_15NH1Rabi_arrays\20kHz_15N\ I4_15N1HRabi_array242 @..\home\mb4b\data\Scott\REDOR\Setup\I4\I4_15NH1Rabi_arrays\20kHz_15N\ I4_15N1HRabi_array322 @..\home\mb4b\data\Scott\REDOR\Setup\I4\I4_15NH1Rabi_arrays\20kHz_15N\ I4_15N1HRabi_array402 @..\home\mb4b\data\Scott\REDOR\Setup\I4\I4_15NH1Rabi_arrays\20kHz_15N\ I4_15N1HRabi_array482 Figure 18 Adamantane @..\home\mb4b\data\Scott\REDOR\Setup\adam_102010 Chapter 3 SIMPSON SIMULATED Results @..\\poohbah.chemistry.msu.edu\welikyshare$\ SchmickThesis\Chp3Thesis\Par_anti_na.xls Figures Coral Draw File @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\ Chp3Thesis\Figures_mod13.cdr Figure 2a. @..\home\mb4b\data\Scott\REDOR2\neg_control\-50_482_REDOR_050908 Figure 2b. @..\home\mb4b\data\Scott\REDOR2\2ndL12wG13A14\482_REDOR_041008 Figure 2c. @..\home\mb4b\data\Scott\REDOR2\L12wG5A6\482_REDOR_112407 Figure 2d. @..\home\mb4b\data\Scott\REDOR2\F8wL9G10\482REDOR_021708 Figure 3a,b. @..\home\mb4b\data\Scott\REDOR2\neg_control\ -50_482_REDOR_050908 -50_402_REDOR_050908 -50_322_REDOR_050908 -50_242_REDOR_050908 -50_162_REDOR_050908 -50_82_REDOR_050908 -50_22_REDOR_050908 Figure 3b. @..\home\mb4b\data\Scott\REDOR2\2ndL12wG13A14\ 482_REDOR_041008 402_REDOR_041008 322_REDOR_041008 242_REDOR_041008 162_REDOR_041008 82_REDOR_041008 150 22_REDOR_041008 @..\home\mb4b\data\Scott\REDOR2\L12wG5A6\ 482_REDOR_112407 402_REDOR 322_REDOR_112407 242_REDOR_112407 162_REDOR 82_REDOR_112407 22_REDOR @..\home\mb4b\data\Scott\REDOR2\F8wL9G10\ 482REDOR_021708 402REDOR_021708 322REDOR_021708 242REDOR_021708 162REDOR_021708 82REDOR_021708 22REDOR_021708 Chapter IV and all other. mHFP, mV2E-HFP, mHFPdm, and mHFPtr NMR Files. Table 15. File directories for the 48.2, 40.2, 32.2, 24.2, 16.2, 8.2, and 2.2 ms data are shown. @..\home\mb4b\data\Scott\REDOR\happi\3channel\Dimer\F8C-G13N\ @..\home\mb4b\data\Scott\REDOR\dim\L12C-A5N\ @..\home\mb4b\data\Scott\REDOR\HFP\L12C-A6N\dried\ @..\home\mb4b\data\Scott\REDOR\dim\L12C-A6N\ @..\home\mb4b\data\Scott\REDOR\dim\L9C-G5N\ Table 18. File directories for the 48.2, 40.2, 32.2, 24.2, 16.2, 8.2, and 2.2 ms data are shown. @..\home\mb4b\data\Scott\REDOR\HFP\A6C-G3N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-A14N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-A15N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-A21N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-G13N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-G16N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-G3N\ @..\home\mb4b\data\Scott\REDOR\HFP\F8C-L12N\ @..\home\mb4b\data\Scott\REDOR\HFP\L12C-A6N\ @..\home\mb4b\data\Scott\REDOR\HFP\L12C-G3N\ @..\home\mb4b\data\Scott\REDOR\HFP\L12C-G5N\ @..\home\mb4b\data\Scott\REDOR\HFP\L12C-I4N\ @..\home\mb4b\data\Scott\REDOR\HFP\L12C-L7N\ @..\home\mb4b\data\Scott\REDOR\HFP\L7C-G3N\ @..\home\mb4b\data\Scott\REDOR\HFP\L9C-G16N\ @..\home\mb4b\data\Scott\REDOR\HFP\L9C-G3N\ @..\home\mb4b\data\Scott\REDOR\HFP\L9C-G5N\ @..\home\mb4b\data\Scott\REDOR\HFP\L9C-I4N\ Table 19. File directories for the 48.2, 40.2, 32.2, 24.2, 16.2, 8.2, and 2.2 ms data are shown. @..\home\mb4b\data\Scott\REDOR\V2E\A6CG3N\ @..\home\mb4b\data\Scott\REDOR\V2E\F8CA14N\ 151 @..\home\mb4b\data\Scott\REDOR\V2E\F8CA15N\ @..\home\mb4b\data\Scott\happi\3channel\V2E\F8CG13N\ @..\home\mb4b\data\Scott\REDOR\V2E\F8CG16N\ @..\home\mb4b\data\Scott\REDOR\V2E\F8CG3N\ @..\home\mb4b\data\Scott\REDOR\V2E\F8C-L12N\062210\ @..\home\mb4b\data\Scott\REDOR\V2E\L12CA6N\ @..\home\mb4b\data\Scott\REDOR\V2E\L12CG3N\ @..\home\mb4b\data\Scott\REDOR\V2E\L12CG5N\ @..\home\mb4b\data\Scott\REDOR\V2E\L12CI4N\ @..\home\mb4b\data\Scott\REDOR\V2E\L12CL7N\ @..\home\mb4b\data\Scott\REDOR\V2E\L7CG3N\ @..\home\mb4b\data\Scott\REDOR\V2E\L9CG16N\ @..\home\mb4b\data\Scott\REDOR\V2E\L9CG3N\ @..\home\mb4b\data\Scott\REDOR\V2E\L9CG5N\ @..\home\mb4b\data\Scott\REDOR\V2E\L9CI4N\ Chapter IV. Excel Files for Iterative Fittings HFP 3 Registry Fitting Fully Constrained @..\\poohbah.chemistry.msu.edu\welikyshare$\ SchmickThesis\OtherThesis\3Reg\F.C. WT compiled.xls HFP 3 Registry Fitting Unconstrained @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\ OtherThesis\3Reg\2nd_U.C. WT.xls V2E-HFP 3 Registry Fitting Fully Constrained @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\3Reg\ F.C. V2E compiled.xls HFP 5 Registry Fitting Fully Constrained @..\\poohbah.chemistry.msu.edu\welikyshare$\ SchmickThesis\OtherThesis\5Reg\2nd_F.C. WT compiled.xls V2E-HFP 5 Registry Fitting Fully Constrained @..\\poohbah.chemistry.msu.edu\welikyshare$\ SchmickThesis\OtherThesis\5Reg\2nd_F.C. WT compiled.xls ins Chapter IV and Appendices. HFP, V2E-HFP, L9R-HFP Excel Files for ΔG , Fractional ins Populations Calculated from a Boltzmann Distribution using ΔG for Energy, and the Modeled Membrane Insertion Depth. @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\HFP Chi^2_2nd.xls @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\V2E Chi^2_2nd.xls @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\L9R Chi^2_2nd.xls Chapter V. Figure 37. @..\home\mb4b\data\Scott\Thesis\resin_10kHz_V2E_L9CI4N_012011p Figure 38. @..\home\mb4b\data\Scott\Thesis\resin_6kHz_V2E_L9CI4N_012011p Appendix II. Freed Mutant Excel Spreadsheet Calculations. @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\1992_mutant.xls Appendix IV. Clean HPLC Column Protocol. @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\cleancolumn.pdf L9R CP @..\home\mb4b\data\Scott\REDOR\L9R\F8CG13N\cp_ramp_F8CG13N_081210 Resin @..\home\mb4b\data\Scott\cp\resin\cp10kHz_V2E_resinL9CI4N_012011 @..\home\mb4b\data\Scott\cp\resin\cp6kHz_V2E_resinL9CI4N_012011 152 SIMMOL and SIMPSON files @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\ Coordinate\Antiparallel_5_spin\Leu132_change_cord MOL file @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\Coordinate\ Antiparallel_5_spin\Leu132_change_cord SPINSYS file @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\REDOR_files\ Antiparallel_5spin_4left\redor-ALA12_050608.in @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\REDOR_files\ Antiparallel_5spin_4left\redor-ALA12-050608-122.45-138.46-49.79-31.824-82.087-131.98-17.21--71.217-17.1718-44.177-16.768-19.126.fid @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\REDOR_files\ Antiparallel_3spin_bleft\redor-ALA12_050608.in @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\REDOR_files\ Antiparallel_3spin_bleft\redor-ALA12-050608-49.79-31.824--17.21--71.217-16.768-19.126.fid @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\ 3_strand_CNC_REDOR\redor-98.0-050608.in @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\ 3_strand_CNC_REDOR\redor-98.0-050608-0-98.0-0-0-12.35-12.97.fid @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\REDOR_files\ Natural_abundance\b-sheet\redor-2spin-4.0-050608.in @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\Simpson\REDOR\REDOR_files\ Natural_abundance\b-sheet\redor-2spin-4.00-050608-0-0-48.2.fid Fortran Files @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ HFPdata2\x2_fixed @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ HFPdata2\HFP.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\HFPdata2\ output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ HFP2ndF8CG13N\x2_fixed @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ HFP2ndF8CG13N\HFP.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ HFP2ndF8CG13N\output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\data\AppenThesis\Fortran\ 5regHFP\f9=0.00\x2 @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ 5regHFP\f9=0.00\HFP_5var.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\5regHFP\ f9=0.00\output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\5regHFP\ f9=0.01\x2 @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\5regHFP\ f9=0.01\HFP_5var.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\5regHFP\ 153 f9=0.01\output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\V2E_data\ x2_V2E @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\V2E_data\ V2E.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\V2E_data\ output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.00\x2_V2E @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.00\V2E_5var.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.00\output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.01\x2_V2E @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.01\V2E_5var.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.01\output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.02\x2_V2E @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.02\V2E_5var.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.02\output @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.03\x2_V2E @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.03\V2E_5var.f @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\Fortran\ V2E5reg_data\f12=0.03\output 154 Appendix II. Current HIV Inhibitor Drugs Figure 39. Chart of commercially available anti HIV drugs. This chart was last updated 12\14\2010 and was taken from www.aidsmeds.com. 155 Appendix III. Simple Number of Strands for Fusion Model. As mentioned in Chapters I, III, IV, and V, there is literature that supports that β sheet oligomers are a reasonable structure for the fusion active HFP structure. Additionally, syncytia fusion assays have demonstrated that an oligomeric structure is essential for membrane fusion where dilute amounts of V2E mutant gp41 were expressed with WT gp41 and fusion activity of 21 WT gp41 was dominantly inhibited, Table 12 . A simplistic model was created to relate the number of strands within an oligomer required to initiate membrane fusion, s, as a function of experimentally observed fusion activity, A(s). Incorporation of a single V2E gp41 strand into an oligomer was assumed to abrogate the fusion activity of an oligimer. Assuming that oligomerization of V2E gp41 with WT gp41 is random, the fusion activity as a function of s can be described by Eq (59). A( s)  F s (59) where F is the fraction of WT gp41 from Table 12. The calculated activity was compared to the 2 experimental activity by the χ analysis metric. exp 4 { A( s ) j  ( A) j }2 2   exp j 1 ( j )2 (60) 2 where j was an index for each of the experimental activity data points from Table 12. The χ min = 3.1 for s = 7, Table 13, which suggested that 2-3 gp41 trimers were required to form fusion active β sheet oligimers. Additionally, χ 2 analysis was performed where the activity was modeled a fraction of 6 and a fraction of 9 strand β sheet oligimers (2 and 3 trimers) where the 156 2 χ min = 2.6 which corresponded to 50:50 ratio of 6 strand to 9 strand oligomers in this model (See @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\AppenThesis\ 1992_mutant.xls) 157 21 Table 12. Freed fusion activity Fraction of WT 0.50 0.67 0.83 HFP (F) Fusion Activity 0.04 0.06 0.18 exp (.03) (.01) (0.08) (σ ) 0.91 0.39 (0.20) Table 13. Strands Model 1. Number of χ2 Strands (s) 1 3989 2 1573 3 594 4 206 5 61 6 13 7 3 8 6 9 13 10 19 11 25 12 30 An alternative model was also considered where a minimum of two V2E gp41 strands were needed to be incorporated to abrogate the fusion activity of an oligomer. A( s )  F s  F  s 1 (61) 2 The χ min = 14 for s = 10, Table 14, which suggested that 3 or more gp41 trimers were required to form fusion active β sheet oligimers. While these models are not sufficient for determining the number of oligomers required for membrane fusion, both overwhelming suggest that more than 2 one gp41 trimer is required for membrane fusion since s = 3 yields χ values of 594 and 5,000 for the respective models. The excel spreadsheet with complete analysis was uploaded to the ftp. 158 Table 14. Strands Model 2. Number of χ2 Strands (s) 1 29000 2 12000 3 5000 4 2100 5 830 6 320 7 120 8 41 9 17 10 14 11 16 12 21 13 25 14 29 15 32 159 Appendix IV. RP-HPLC Purification, Optimization and Troubleshooting From the currently available columns in our laboratory, the preferred column used for RP-HPLC was a Custom Bioseperation Symmetry 300 C4 steel column with 19 x 300mm dimensions, 5 µm particle size, and 300 Å pore size. The 19 mm diameter provides larger loading volumes than columns with 10 mm diameters, and the 5 µm particle size provided better peak resolution than columns with larger particle sizes, such as 10-15 µm. 160 Figure 40. (a) HFP purification with a “large” C18 column (10-15 µm pore size). (b) HFP purification with a “small” C4 column (10 x 250 mm and 5 µm pore size). Better peak resolution was obtained with the C4 column. (c) Typical MALDI-TOF mass spectroscopy of peak 2 HFP from a purification similar to (b) where the expected mass was 3151 +2 g/mol where the +2 13 15 refers to the mass gain from the C and N isotopes. The RP-HPLC chromatograms below, Figure 41-Figure 46, demonstrate how to develop a purification protocol to purify crude V2E-HFP. In these gradients, solvent A, 100% water with 0.01% TFA, was mixed with solvent B, 90% acetonitrile with 0.01% TFA. The flow rate of 161 solvent A + solvent B was set to 8 mL/min. Products or impurities that “elute” or pass through the column are detected at ~ 8 minutes under these conditions, Figure 42, and the peak at ~ 8 minutes is generally referred to as the elution peak. Therefore, we can approximate the percent elution, Pe, of a peak as a function of the solvent B gradient slope, Gs, the initial concentration of solvent B, Ci, and the elution time, te. Pe  Gs  te  8  Ci (62) The figures below illustrate changes that can be made to the gradient to better resolve peaks. Figure 41. Preliminary gradient of 15% to 80% solvent B over 40 minutes. In developing purification protocols, small amounts of crude peptide were used to make product peaks narrow. By mass spec, Peak 2 is the confirmed product peak. From Eq (62), the variables have the following values: Ci = 15%; Gs = 1.625 %/min; te ~ 26.5 min; and Pe = 45%. 162 Figure 42. A linear 39-48% gradient was run over 18.5 minutes for purifying the product peak. From Eq (62), the variables have the following values: Ci = 39%; Gs = 0.5 %/min; te ~ 15.5 min; and Pe = 46.5%. The initial starting concentration was chosen to make the elution time around 15 minutes which was calculated by Eq (62) and a more gradual gradient was used to better separate peaks 1 and 2 from Figure 41. Also, at the end of the program, the gradient was ramped up to 80% solvent B over 0.5 minutes and the flow rate was increased to 9 mL/min to clean the column after each run. After 5 minutes, the gradient concentration of solvent B was returned to 39% over 0.5 minutes and the column equilibrated at this concentration for 3 minutes to prepare for the next run. 163 Figure 43. The program from Figure 42 was used, but a higher loading volume of the crude peptide was used which resulted in poor separation of our product peak. Figure 44. A linear 37-47% gradient was run over 20 minutes. The peaks were separated better with minimal peak broadening. 164 Figure 45. Nonlinear gradients can be used to separate peak 1 from peak 2. The gradient broadened peak 1 using a more gradual slope initially while the gradient was steeper from 15 to 21 minutes to retain the sharpness of peak 2. To optimize the time of the program, it’s best to have your product elute during the period where the ramp is up to 75% solvent B since nothing is achieved during this time in Figure 41-Figure 44. Recall, the elution time of a peak is 8 minutes. Therefore, in this figure, peak 2 was collected during 75% use of solvent B, but peak 2 actually began coming off the column at time te-8 or ~16-18 minutes. Figure 46. The ramp was modified to separate peak 2 from peak 3. This program was created because peak 1 also contains peptide with our products molecular weight which was collected for potential future use. Peak 3 should also be collected if the product peak is low relative to other syntheses. Peak 3 can contain HFP with N-terminal or sidechain protecting groups. 165 1. Specific Problems 1.1 Well separated peak contains “impurities” by MALDI-TOF analysis Figure 47. MALDI-TOF mass spectroscopy of purified HFP-L9G10. In MALDI-TOF experiments, increasing the laser power can increase the signal to noise, but it can also lead to peptide fragmentation where fragmentation can occur C-terminal of amino acids with basic 65,66 sidechain groups . Alternatively, gas phase degredation of the peptide may be unlikely. The peptide degredation may result from hydrolytic cleavage in the matrix or possibly during 64 isolation . Fragmentation of the HFP-L9G19 peptide appeared to occur C-terminal of the Arg22, Lys-29, and Lys-30 where the respective fragments detected were likely AVGIGALFLGFLGAAGSTMGAR (2038 +2 g/mol), AVGIGALFLGFLGAAGSTMGARSWKKKKK (2952 +2 g/mol), and AVGIGALFLGFLGAAGSTMGARSWKKKKKK (3080 +2 g/mol), and the HFP product had an expected mass of 3151 + 2 g/mol. 1.2 Forgot to deprotect Fmoc group. Dissolve crude peptide in 10 mL of 20% piperidine in DMF solution in a 50 mL conical vial and gently vortex for 20-25 minutes. Cap the conical vial and sonicate every 3 minutes to increase the solubility of the crude peptide. The crude peptide will not fully dissolve in the deprotection solution. Also, try to dissolve peptide that may be stuck to the conical vial. 166 1.3 Column pressure is increasing over time. You may be loading precipitated peptides or dust onto the column in which case you can sonicate and centrifuge your peptide solution. Additionally, you can reverse the column and run a cleaning protocol. Note: when switching solvents, the pressure of the column will change due to swelling of the resin. The recommended flow rates are suggested to keep the column pressure low, but can be increased as long as you pay attention to the pressure. This procedure is effective in reducing the column pressure as it has reduced the “small” C4 column from ~2100-2500 psi to ~1300-1400 psi at a flow rate of 3 mL/min of 40% solvent B. A pdf file with a cleaning protocol can be found in: @..\\poohbah.chemistry.msu.edu\welikyshare$\SchmickThesis\OtherThesis\cleancolumn.pdf 167 Appendix V. HFPdm Data and Lyophilized HFP Chapter IV detected differences in registry distributions between the mV2E-HFP, mHFP, and mHFPtr constructs by using isotopic labeling schemes and the REDOR pulse sequence. Similar labeling schemes were incorporated into mHFPdm, and no clear structural differences were distinguished between mHFPdm and mHFP using this REDOR method. The structure of mHFP and mHFPdm may be identical or the differences are too subtle to detect using the current experimental design. Additionally, the HFP-L12CA6N sample was lyophilized for 24 hours after collection of the 7 data points and the sample was not rehydrated. The HFP-L12CA6N and lyophilized HFP-L12CA6N data were indistinguishable which suggests that mHFP and lyophilized mHFP have the same or similar registries. 168 Figure 48. REDOR data for HFP (black boxes) and HFPdm (red circles) are displayed with error bars that are associated with rms deviation and labeling corresponds to (a) L9CG5N and (b) L12CG5N. 169 Figure 48 (cont’d). REDOR data for HFP (black boxes), lyophilized HFP (orange stars), and HFPdm (red circles) are displayed with error bars that are associated with rms deviation and labeling corresponds to (c) L12CA6N and (d) L12CG5N. 170 Table 15. HFPdm and L12CA6Nmn Lyophilized (ΔS/S0) exp (S/S0) + exp exp and σ exp in parentheses Dephasing time (ms) L9CG5Ndm L12CG5Ndm L12CA6Ndm L12CA6Nmn Lyophilized F8CG13Ndm 2.2 8.2 16.2 24.2 32.2 40.2 48.2 0.012 (.005) 0.039 (.007) 0.086 (.008) 0.125 (.008) 0.170 (.009) 0.212 (.015) 0.236 (.015) 0.016 (.008) 0.049 (.007) 0.085 (.008) 0.138 (.008) 0.178 (.014) 0.232 (.013) 0.277 (.018) 0.005 (.006) 0.047 (.007) 0.097 (.008) 0.149 (.011) 0.211 (.018) 0.216 (.013) 0.253 (.013) 0.012 (.007) 0.034 (.010) 0.107 (.013) 0.161 (.012) 0.205 (.017) 0.234 (.017) 0.272 (.019) 0.024 (.008) 0.040 (.008) 0.066 (.011) 0.113 (.018) 0.151 (.013) 0.157 (.015) 0.201 (.020) 171 Appendix VI. SIMMOL, SIMPSON, and Fortran Files. 1. Sample SIMMOL Files 1.1 SIMMOL input file for Leu-132 13 CO from the 2IWW.pdb file. regsub “.mol” $argv0 {\1.oogl} oogl regsub “.mol” $argv0 {\1.spinsys} spinsys set m [mload “2IWW.pdb”] mloadtensors $m -default #mloadjcouplings $m -default msetspinsysfile $m $spinsys -numbered mselect mselect mselect mselect mselect mselect mselect $m $m $m $m $m $m $m 1 2 3 4 5 6 7 atom atom atom atom atom atom atom 536 357 352 344 727 716 705 #mset $m -solid -ellipsoid shielding -color cpk -nice mdipole $m 1 2 0AA 20AA mdipole $m 1 3 0AA 20AA mdipole $m 1 4 0AA 20AA mdipole $m 1 5 0AA 20AA mdipole $m 1 6 0AA 20AA mdipole $m 1 7 0AA 20AA #mclosespinsysfile $m munload $m puts “Generated: $spinsys” 1.2. SIMMOL output file for the Leu-132 residue spinsys { # 1 # 536N # channels nuclei dipole 1 dipole 2 2 357N 3 352N 4 344C 13 CO from the 2IWW.pdb file 5 727N 15N 13C 15N 15N 15N 13C 15N 15N 15N 4 21.0712 0 101.81 72.787 4 35.747 0 98.531 33.511 172 6 716N 7 705N dipole dipole dipole dipole 3 4 4 4 4 5 6 7 13.194 16.339 19.871 12.326 0 0 0 0 72.5 9.7837 76.984 9.9556 103.81 36.023 129.81 61.397 } 2. Sample SIMPSON Files 2.1. Input file 5 spin (NNCNN) SIMPSON File for Ala-12 13 CO from 2WII.pdb # REDOR simulation for pulse sequence redorxy8xy_pm for a model three spin system # ver. 1.0, last revised on 02/05/07 w/o all real pulses # no cp phase cycling # no output of s0 and s1 spinsys { channels 13C 15N nuclei 13C 15N 15N 15N 15N dipole 1 2 $par(dp12) 0 $par(a) $par(e) dipole 1 3 $par(dp13) 0 $par(b) $par(f) dipole 1 4 $par(dp14) 0 $par© $par(g) dipole 1 5 $par(dp15) 0 $par(d) $par(h) shift 1 174.3p -75.0p 0.97333 0 109.37 -171.38 } par { proton_frequency 400.7797840e6 spin_rate 10000 sw 50000 np 7 crystal_file rep320 dipole_check false gamma_angles 18 start_operator I1x detect_operator verbose variable Crf variable variable variable variable variable variable variable variable variable variable I1p 1101 61728 Nrf 61728 dp12_min 44.177 dp12_max 44.177 dp12_incr 1 dp13_min 12.43 dp13_max 12.43 dp13_incr 1 dp14_min 19.126 dp14_max 19.126 dp14_incr 1 173 variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable dp15_min 13.485 dp15_max 13.485 dp15_incr 1 a_min 138.46 a_max 138.46 a_incr 1 b_min 118.5 b_max 118.5 b_incr 1 c_min 31.824 c_max 31.824 c_incr 1 d_min 40.492 d_max 40.492 d_incr 1 e_min 131.98 e_max 131.98 e_incr 1 f_min 165.09 f_max 165.09 f_incr 1 g_min -71.217 g_max -71.217 g_incr 1 h_min -132.18 h_max -132.18 h_incr 1 bestkaisqr 1e6 c_off 15398 n_off 0 } proc pulseq { } { global par maxdt 1 set Ct180 [expr 0.5e6/$par(Crf)] set Nt180 [expr 0.5e6/$par(Nrf)] set tr [expr 0.5e6/$par(spin_rate)] set tr1 [expr $tr-0.5*$Ct180] set tr2 [expr $tr-0.5*$Ct180-0.5*$Nt180] reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x $par(Nrf) x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 1 #s1 reset offset $par(c_off) $par(n_off) 174 delay pulse delay pulse store #s1 $tr2 $Nt180 0 y $par(Nrf) y $tr2 $Ct180 $par(Crf) y 0 y 2 reset offset $par(c_off) $par(n_off) delay pulse delay pulse delay pulse delay store #s1 $tr2 $Nt180 0 x $par(Nrf) x $tr2 $Ct180 $par(Crf) x 0 x $tr2 $Nt180 0 y $par(Nrf) y $tr1 3 reset foreach i {1 2 1 2 2 1 2 1} { prop $i } store 8 #s1 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x 0 x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 4 #s0 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 y 0 y delay $tr2 pulse $Ct180 $par(Crf) y 0 y store 5 #s0 reset offset $par(c_off) $par(n_off) delay pulse delay pulse delay pulse delay store $tr2 $Nt180 0 x 0 x $tr2 $Ct180 $par(Crf) x 0 x $tr2 $Nt180 0 y 0 y $tr1 6 175 #s0 reset foreach i {4 5 4 5 5 4 5 4} { prop $i } store 9 #s0 foreach j {2 8 16 24 32 40 48} { reset prop $par(swtch1) $j prop $par(swtch2) acq } } proc main { } { global par for {set par(a) $par(a_min)} {$par(a) <= $par(a_max)} {set par(a) $par(a)+$par(a_incr)]} { for {set par(b) $par(b_min)} {$par(b) <= $par(b_max)} {set par(b) $par(b)+$par(b_incr)]} { for {set par© $par(c_min)} {$par© <= $par(c_max)} {set par© [expr $par©+$par(c_incr)]} { for {set par(d) $par(d_min)} {$par(d) <= $par(d_max)} {set par(d) $par(d)+$par(d_incr)]} { for {set par(e) $par(e_min)} {$par(e) <= $par(e_max)} {set par(e) $par(e)+$par(e_incr)]} { for {set par(f) $par(f_min)} {$par(f) <= $par(f_max)} {set par(f) $par(f)+$par(f_incr)]} { for {set par(g) $par(g_min)} {$par(g) <= $par(g_max)} {set par(g) $par(g)+$par(g_incr)]} { for {set par(h) $par(h_min)} {$par(h) <= $par(h_max)} {set par(h) $par(h)+$par(h_incr)]} { for {set par(dp12) $par(dp12_min)} {$par(dp12) <= $par(dp12_max)} par(dp12) [expr $par(dp12)+$par(dp12_incr)]} { for {set par(dp13) $par(dp13_min)} {$par(dp13) <= $par(dp13_max)} par(dp13) [expr $par(dp13)+$par(dp13_incr)]} { for {set par(dp14) $par(dp14_min)} {$par(dp14) <= $par(dp14_max)} par(dp14) [expr $par(dp14)+$par(dp14_incr)]} { for {set par(dp15) $par(dp15_min)} {$par(dp15) <= $par(dp15_max)} par(dp15) [expr $par(dp15)+$par(dp15_incr)]} { foreach p {{0 9 6} {1 8 3}} { set par(swtch1) [lindex $p 1] set par(swtch2) [lindex $p 2] set f [fsimpson] set g[lindex $p 0] [fdup $f] } #fsave $g0 $par(name)-$par(dp12)-s0.fid #fsave $g1 $par(name)-$par(dp12)-s1.fid set gsub [fdup $g0] fsub $gsub $g1 176 [expr [expr [expr [expr [expr [expr [expr {set {set {set {set for {set j 1} {$j <= $par(np)} {incr j} { set a0 [findex $g0 $j -re] set a1 [findex $gsub $j -re] fsetindex $g1 $j [expr $a1/$a0] 0 } #fphase $g1 -scale $par(scl_avg) fsave $g1 $par(name)-$par(a)-$par(b)-$par(c)-$par(d)-$par(e)-$par(f)-$par(g)$par(h)-$par(dp12)-$par(dp13)-$par(dp14)-$par(dp15).fid } } } } } } } } } } } } #set outpt [open $par(name).out a+ 0600] #puts $outpt “$dp12_opt $par(bestkaisqr)” #close $outpt #funload } 2.2 Output file 5 spin (NNCNN) SIMPSON File for Ala-12 13 CO from 2WII.pdb SIMP NP=7 SW=50000 TYPE=FID DATA 0.00890980072 0 0.113440126 0 0.372697496 0 0.63746613 0 0.80436585 0 0.868245862 0 0.890586802 0 END 2.3 Input file 3 spin (CNN) SIMPSON File for Ala-12 13 CO from 2WII.pdb # REDOR simulation for pulse sequence redorxy8xy_pm for a model three spin system # ver. 1.0, last revised on 02/05/07 w/o all real pulses # no cp phase cycling # no output of s0 and s1 177 spinsys { channels 13C 15N nuclei 13C 15N 15N dipole 1 2 $par(dp12) 0 $par(a) $par(c) dipole 1 3 $par(dp13) 0 $par(b) $par(d) shift 1 174.3p -75.0p 0.97333 0 109.37 -171.38 } par { proton_frequency 400.7797840e6 spin_rate 10000 sw 50000 np 7 crystal_file rep320 dipole_check false gamma_angles 18 start_operator I1x detect_operator verbose variable Crf variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable I1p 1101 61728 Nrf 61728 dp12_min 16.768 dp12_max 16.768 dp12_incr 1 dp13_min 19.126 dp13_max 19.126 dp13_incr 1 a_min 49.79 a_max 49.79 a_incr 1 b_min 31.824 b_max 31.824 b_incr 1 c_min -17.21 c_max -17.21 c_incr 1 d_min -71.217 d_max -71.217 d_incr 1 bestkaisqr 1e6 c_off 15398 n_off 0 } proc pulseq { } { global par maxdt 1 178 set set set set set Ct180 [expr 0.5e6/$par(Crf)] Nt180 [expr 0.5e6/$par(Nrf)] tr [expr 0.5e6/$par(spin_rate)] tr1 [expr $tr-0.5*$Ct180] tr2 [expr $tr-0.5*$Ct180-0.5*$Nt180] reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x $par(Nrf) x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 1 #s1 reset offset $par(c_off) $par(n_off) delay pulse delay pulse store #s1 $tr2 $Nt180 0 y $par(Nrf) y $tr2 $Ct180 $par(Crf) y 0 y 2 reset offset $par(c_off) $par(n_off) delay pulse delay pulse delay pulse delay store #s1 $tr2 $Nt180 0 x $par(Nrf) x $tr2 $Ct180 $par(Crf) x 0 x $tr2 $Nt180 0 y $par(Nrf) y $tr1 3 reset foreach i {1 2 1 2 2 1 2 1} { prop $i } store 8 #s1 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x 0 x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 4 #s0 179 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 y 0 y delay $tr2 pulse $Ct180 $par(Crf) y 0 y store 5 #s0 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 delay $tr2 pulse $Ct180 delay $tr2 pulse $Nt180 delay $tr1 store 6 #s0 reset foreach i {4 prop $i } 0 x 0 x $par(Crf) x 0 x 0 y 0 y 5 4 5 5 4 5 4} { store 9 #s0 foreach j {2 8 16 24 32 40 48} { reset prop $par(swtch1) $j prop $par(swtch2) acq } } proc main { } { global par for {set par(a) $par(a_min)} {$par(a) <= $par(a_max)} {set par(a) $par(a)+$par(a_incr)]} { for {set par(b) $par(b_min)} {$par(b) <= $par(b_max)} {set par(b) $par(b)+$par(b_incr)]} { for {set par© $par(c_min)} {$par© <= $par(c_max)} {set par© [expr $par©+$par(c_incr)]} { for {set par(d) $par(d_min)} {$par(d) <= $par(d_max)} {set par(d) $par(d)+$par(d_incr)]} { for {set par(dp13) $par(dp13_min)} {$par(dp13) <= $par(dp13_max)} par(dp13) [expr $par(dp13)+$par(dp13_incr)]} { for {set par(dp12) $par(dp12_min)} {$par(dp12) <= $par(dp12_max)} par(dp12) [expr $par(dp12)+$par(dp12_incr)]} { foreach p {{0 9 6} {1 8 3}} { set par(swtch1) [lindex $p 1] set par(swtch2) [lindex $p 2] set f [fsimpson] 180 [expr [expr [expr {set {set set g[lindex $p 0] [fdup $f] } #fsave $g0 $par(name)-$par(dp12)-s0.fid #fsave $g1 $par(name)-$par(dp12)-s1.fid set gsub [fdup $g0] fsub $gsub $g1 for {set j 1} {$j <= $par(np)} {incr j} { set a0 [findex $g0 $j -re] set a1 [findex $gsub $j -re] fsetindex $g1 $j [expr $a1/$a0] 0 } #fphase $g1 -scale $par(scl_avg) fsave $g1 $par(name)-$par(a)-$par(b)-$par(c)-$par(d)-$par(dp12)$par(dp13).fid } } } } } } #set outpt [open $par(name).out a+ 0600] #puts $outpt “$dp12_opt $par(bestkaisqr)” #close $outpt #funload } 2.4 Output file 3 spin (CNN) SIMPSON File for Ala-12 13 CO from 2WII.pdb SIMP NP=7 SW=50000 TYPE=FID DATA 0.0021804517 0 0.0289860854 0 0.108652919 0 0.228572947 0 0.373536138 0 0.526272979 0 0.670419088 0 END 2.5 Input file 3 spin (CNC) SIMPSON File from 2WII.pdb # REDOR simulation for pulse sequence redorxy8xy_pm for a model three spin system # ver. 1.0, last revised on 02/05/07 w/o all real pulses # no cp phase cycling # no output of s0 and s1 spinsys { 181 channels 13C 15N nuclei 13C 15N 13C dipole 1 2 $par(dp12) 0 $par(a) $par(c) dipole 2 3 $par(dp23) 0 $par(b) $par(d) shift 1 174.3p -75.0p 0.97333 0 109.37 -171.38 } par { proton_frequency 400.7797840e6 spin_rate 10000 sw 50000 np 7 crystal_file rep320 dipole_check false gamma_angles 18 start_operator I1x detect_operator verbose variable Crf variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable variable I1p 1101 61728 Nrf 61728 dp12_min 12.35 dp12_max 12.35 dp12_incr 1 dp23_min 12.97 dp23_max 12.97 dp23_incr 1 a_min 0 a_max 0 a_incr 1 b_min 98.0 b_max 98.0 b_incr 1 c_min 0 c_max 0 c_incr 1 d_min 0 d_max 0 d_incr 1 bestkaisqr 1e6 c_off 15398 n_off 0 } proc pulseq { } { global par maxdt 1 set Ct180 [expr 0.5e6/$par(Crf)] set Nt180 [expr 0.5e6/$par(Nrf)] 182 set tr [expr 0.5e6/$par(spin_rate)] set tr1 [expr $tr-0.5*$Ct180] set tr2 [expr $tr-0.5*$Ct180-0.5*$Nt180] reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x $par(Nrf) x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 1 #s1 reset offset $par(c_off) $par(n_off) delay pulse delay pulse store #s1 $tr2 $Nt180 0 y $par(Nrf) y $tr2 $Ct180 $par(Crf) y 0 y 2 reset offset $par(c_off) $par(n_off) delay pulse delay pulse delay pulse delay store #s1 $tr2 $Nt180 0 x $par(Nrf) x $tr2 $Ct180 $par(Crf) x 0 x $tr2 $Nt180 0 y $par(Nrf) y $tr1 3 reset foreach i {1 2 1 2 2 1 2 1} { prop $i } store 8 #s1 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x 0 x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 4 #s0 reset offset $par(c_off) $par(n_off) delay $tr2 183 pulse delay pulse store #s0 $Nt180 0 y 0 y $tr2 $Ct180 $par(Crf) y 0 y 5 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 delay $tr2 pulse $Ct180 delay $tr2 pulse $Nt180 delay $tr1 store 6 #s0 reset foreach i {4 prop $i } 0 x 0 x $par(Crf) x 0 x 0 y 0 y 5 4 5 5 4 5 4} { store 9 #s0 foreach j {2 8 16 24 32 40 48} { reset prop $par(swtch1) $j prop $par(swtch2) acq } } proc main { } { global par for {set par(a) $par(a_min)} {$par(a) <= $par(a_max)} {set par(a) $par(a)+$par(a_incr)]} { for {set par(b) $par(b_min)} {$par(b) <= $par(b_max)} {set par(b) $par(b)+$par(b_incr)]} { for {set par© $par(c_min)} {$par© <= $par(c_max)} {set par© [expr $par©+$par(c_incr)]} { for {set par(d) $par(d_min)} {$par(d) <= $par(d_max)} {set par(d) $par(d)+$par(d_incr)]} { for {set par(dp23) $par(dp23_min)} {$par(dp23) <= $par(dp23_max)} par(dp23) [expr $par(dp23)+$par(dp23_incr)]} { for {set par(dp12) $par(dp12_min)} {$par(dp12) <= $par(dp12_max)} par(dp12) [expr $par(dp12)+$par(dp12_incr)]} { foreach p {{0 9 6} {1 8 3}} { set par(swtch1) [lindex $p 1] set par(swtch2) [lindex $p 2] set f [fsimpson] set g[lindex $p 0] [fdup $f] } 184 [expr [expr [expr {set {set #fsave $g0 $par(name)-$par(dp12)-s0.fid #fsave $g1 $par(name)-$par(dp12)-s1.fid set gsub [fdup $g0] fsub $gsub $g1 for {set j 1} {$j <= $par(np)} {incr j} { set a0 [findex $g0 $j -re] set a1 [findex $gsub $j -re] fsetindex $g1 $j [expr $a1/$a0] 0 } #fphase $g1 -scale $par(scl_avg) fsave $g1 $par(name)-$par(a)-$par(b)-$par(c)-$par(d)-$par(dp12)$par(dp23).fid } } } } } } #set outpt [open $par(name).out a+ 0600] #puts $outpt “$dp12_opt $par(bestkaisqr)” #close $outpt #funload } 2.6 Output file 3 spin (CNC) SIMPSON File from 2WII.pdb SIMP NP=7 SW=50000 TYPE=FID DATA 0.000517198113 0 0.00693752805 0 0.0267554081 0 0.0589735416 0 0.102783454 0 0.157079165 0 0.220491486 0 END 2.7 Input file 2 spin (CN) SIMPSON File from 2WII.pdb # REDOR simulation for pulse sequence redorxy8xy_pm for a model three spin system # ver. 1.0, last revised on 02/05/07 w/o all real pulses # no cp phase cycling # no output of s0 and s1 spinsys { channels 13C 15N 185 nuclei 13C 15N dipole 1 2 $par(dp12) 0 0 0 shift 1 174.3p -75.0p 0.97333 0 109.37 -171.38 } par { proton_frequency 400.7797840e6 spin_rate 10000 sw 50000 np 13 crystal_file rep320 dipole_check false gamma_angles 18 start_operator I1x detect_operator verbose variable Crf variable variable variable variable variable variable variable variable variable variable variable variable variable I1p 1101 61728 Nrf 61728 dp12_min 48.125 dp12_max 48.125 dp12_incr 1 a_min 0 a_max 0 a_incr 1 e_min 0 e_max 0 e_incr 1 bestkaisqr 1e6 c_off 15398 n_off 0 } proc pulseq { } { global par maxdt 1 set Ct180 [expr 0.5e6/$par(Crf)] set Nt180 [expr 0.5e6/$par(Nrf)] set tr [expr 0.5e6/$par(spin_rate)] set tr1 [expr $tr-0.5*$Ct180] set tr2 [expr $tr-0.5*$Ct180-0.5*$Nt180] reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x $par(Nrf) x delay $tr2 186 pulse $Ct180 $par(Crf) x 0 x store 1 #s1 reset offset $par(c_off) $par(n_off) delay pulse delay pulse store #s1 $tr2 $Nt180 0 y $par(Nrf) y $tr2 $Ct180 $par(Crf) y 0 y 2 reset offset $par(c_off) $par(n_off) delay pulse delay pulse delay pulse delay store #s1 $tr2 $Nt180 0 x $par(Nrf) x $tr2 $Ct180 $par(Crf) x 0 x $tr2 $Nt180 0 y $par(Nrf) y $tr1 3 reset foreach i {1 2 1 2 2 1 2 1} { prop $i } store 8 #s1 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x 0 x delay $tr2 pulse $Ct180 $par(Crf) x 0 x store 4 #s0 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 y 0 y delay $tr2 pulse $Ct180 $par(Crf) y 0 y store 5 #s0 reset offset $par(c_off) $par(n_off) delay $tr2 pulse $Nt180 0 x 0 x 187 delay $tr2 pulse $Ct180 $par(Crf) x 0 x delay $tr2 pulse $Nt180 0 y 0 y delay $tr1 store 6 #s0 reset foreach i {4 5 4 5 5 4 5 4} { prop $i } store 9 #s0 foreach j {2 8 16 24 32 40 48 56 64 72 80 88 96} { reset prop $par(swtch1) $j prop $par(swtch2) acq } } proc main { } { global par for {set par(a) $par(a_min)} {$par(a) <= $par(a_max)} {set par(a) [expr $par(a)+$par(a_incr)]} { for {set par(e) $par(e_min)} {$par(e) <= $par(e_max)} {set par(e) [expr $par(e)+$par(e_incr)]} { for {set par(dp12) $par(dp12_min)} {$par(dp12) <= $par(dp12_max)} {set par(dp12) [expr $par(dp12)+$par(dp12_incr)]} { foreach p {{0 9 6} {1 8 3}} { set par(swtch1) [lindex $p 1] set par(swtch2) [lindex $p 2] set f [fsimpson] set g[lindex $p 0] [fdup $f] } #fsave $g0 $par(name)-$par(dp12)-s0.fid #fsave $g1 $par(name)-$par(dp12)-s1.fid set gsub [fdup $g0] fsub $gsub $g1 for {set j 1} {$j <= $par(np)} {incr j} { set a0 [findex $g0 $j -re] set a1 [findex $gsub $j -re] fsetindex $g1 $j [expr $a1/$a0] 0 } #fphase $g1 -scale $par(scl_avg) fsave $g1 $par(name)-$par(a)-$par(e)-$par(dp12).fid } } 188 } #set outpt [open $par(name).out a+ 0600] #puts $outpt “$dp12_opt $par(bestkaisqr)” #close $outpt #funload } 2.8 Output file 2 spin (CN) SIMPSON File from 2WII.pdb SIMP NP=7 SW=50000 TYPE=FID DATA 0.00785343775 0 0.102004628 0 0.355299988 0 0.661681946 0 0.912493344 0 1.03832576 0 1.03547994 0 END 2 3. Sample Fortran Input Script Files For Global χ Fittings Reducing computational time was essentially for executing the global fittings where up to 144 nodes were simultaneously performing calculations. This work was not possible without using the Michigan State High Performance Computing Center. Additionally, I must thank Dirk Colbry for providing assistance in developing the script files below to reduce computational times. Please see Appendix I for the location of each file. Additionally, each file has an associated compiled output file that can be viewed in excel. Please see Appendix I for the location of the output file for each fitting. For each fitting, 2 input script files were used: (1) A “qsub file” used to split the main job across many nodes; (2) The main script fortran file. The name of each file below is denoted in quotations in the title of each section. 3.1 HFP 3 Registry Fitting qsub Script, “x2_fixed” #!/bin/bash #PBS -l nodes=1:ppn=1,walltime=144:00:00,mem=2gb,feature=gbe #PBS -j oe 189 #PBS -t 0-109 #change to the original working directory cd ${PBS_O_WORKDIR} # Define number of loops for i16 cols=21 # Define number of loops for i22 (not used) rows=4 # Number of jobs = cols*rows (i.e. -t 0-109) #math to figure out the variable values based on the array id i16=`echo "${PBS_ARRAYID} % ( ${cols} + 1 )" | bc` i22=`echo "${PBS_ARRAYID} / ( ${cols} + 1 )" | bc` #display the command we are going to run echo "./x2 ${i16} ${i22} > ${i16}_${i22}.txt" #run the command with the input variables ./x2 ${i16} ${i22} > ${i16}_${i22}.txt # Calculate the runtiem for the job qstat -f ${PBS_JOBID} 3.2 HFP 3 Registry Fitting Main Script, “HFP.f” * This is a comment * This program was written by Scott Schmick 030111 * * t values (1 = a = 482, 2 = b = 402, etc.) real f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 real f17 f18 f19 f20 f21 f22 f23 f24 f25 real sa8 sb8 sc8 sd8 se8 sg8 sh8 real sa9 sb9 sc9 sd9 se9 sg9 sh9 real sa10 sb10 sc10 sd10 se10 sg10 sh10 real sa11 sb11 sc11 sd11 se11 sg11 sh11 real sa12 sb12 sc12 sd12 se12 sg12 sh12 real sa13 sb13 sc13 sd13 se13 sg13 sh13 real sa14 sb14 sc14 sd14 se14 sg14 sh14 real sa15 sb15 sc15 sd15 se15 sg15 sh15 real sa16 sb16 sc16 sd16 se16 sg16 sh16 real sa17 sb17 sc17 sd17 se17 sg17 sh17 real sa18 sb18 sc18 sd18 se18 sg18 sh18 real sa19 sb19 sc19 sd19 se19 sg19 sh19 real sa20 sb20 sc20 sd20 se20 sg20 sh20 real sa21 sb21 sc21 sd21 se21 sg21 sh21 real sa22 sb22 sc22 sd22 se22 sg22 sh22 real sa23 sb23 sc23 sd23 se23 sg23 sh23 real sa24 sb24 sc24 sd24 se24 sg24 sh24 real real real real S0 xa8 xb8 xc8 xd8 xe8 xg8 xh8 xa9 xb9 xc9 xd9 xe9 xg9 xh9 xa10 xb10 xc10 xd10 xe10 xg10 xh10 190 * real real real real real real real real real real real real real real xa11 xa12 xa13 xa14 xa15 xa16 xa17 xa18 xa19 xa20 xa21 xa22 xa23 xa24 xb11 xb12 xb13 xb14 xb15 xb16 xb17 xb18 xb19 xb20 xb21 xb22 xb23 xb24 xc11 xc12 xc13 xc14 xc15 xc16 xc17 xc18 xc19 xc20 xc21 xc22 xc23 xc24 xd11 xd12 xd13 xd14 xd15 xd16 xd17 xd18 xd19 xd20 xd21 xd22 xd23 xd24 xe11 xe12 xe13 xe14 xe15 xe16 xe17 xe18 xe19 xe20 xe21 xe22 xe23 xe24 xg11 xg12 xg13 xg14 xg15 xg16 xg17 xg18 xg19 xg20 xg21 xg22 xg23 xg24 xh11 xh12 xh13 xh14 xh15 xh16 xh17 xh18 xh19 xh20 xh21 xh22 xh23 xh24 real goff1 goff2 goff3 goff4 goff5 goff6 goff7 real gon1 gon2 gon3 gon4 gon5 gon6 gon7 real gnad1 gnad2 gnad3 gnad4 gnad5 gnad6 gnad7 real ft8t1 real ft8t2 real ft8t3 real ft8t4 real ft8t5 real ft8t6 real ft8t7 real ft8t1e real ft8t2e real ft8t3e real ft8t4e real ft8t5e real ft8t6e real ft8t7e real ft9t1 real ft9t2 real ft9t3 real ft9t4 real ft9t5 real ft9t6 real ft9t7 real ft9t1e real ft9t2e real ft9t3e real ft9t4e real ft9t5e real ft9t6e real ft9t7e real ft10t1 real ft10t2 real ft10t3 real ft10t4 real ft10t5 real ft10t6 real ft10t7 real ft10t1e real ft10t2e real ft10t3e real ft10t4e 191 real ft10t5e real ft10t6e real ft10t7e real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft11t1 ft11t2 ft11t3 ft11t4 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e 192 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e 193 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e 194 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft23t1 ft23t2 ft23t3 ft23t4 ft23t5 ft23t6 ft23t7 ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 ft24t2 ft24t3 ft24t4 ft24t5 ft24t6 ft24t7 ft24t1e ft24t2e ft24t3e ft24t4e ft24t5e ft24t6e ft24t7e kaisq bestkaisq bf8 bf9 bf10 bf11 bf12 bf13 bf14 bf15 bf16 bf17 bf18 bf19 bf20 bf21 bf22 bf23 bf24 * DIRK integer i16 i22 character inputarg*128 CALL getarg(1,inputarg) read(inputarg,*) i16 CALL getarg(2,inputarg) read(inputarg,*) i22 195 goff1 goff2 goff3 goff4 goff5 goff6 goff7 gon1 gon2 gon3 gon4 gon5 gon6 gon7 gnad1 gnad2 gnad3 gnad4 gnad5 gnad6 gnad7 = 0.45011191 = 0.581357776 = 0.710329562 = 0.826254485 = 0.918588976 = 0.978470252 = 0.998385086 = 0.093897695 = 0.118554369 = 0.196398598 = 0.378629871 = 0.645290442 = 0.893821899 = 0.991709633 = 0.281406262 = 0.283440096 = 0.298896559 = 0.309714762 = 0.320805841 = 0.325732761 = 0.342143946 ft8t1 ft8t2 ft8t3 ft8t4 ft8t5 ft8t6 ft8t7 ft8t1e ft8t2e ft8t3e ft8t4e ft8t5e ft8t6e ft8t7e ft9t1 ft9t2 ft9t3 ft9t4 ft9t5 ft9t6 ft9t7 ft9t1e ft9t2e ft9t3e ft9t4e ft9t5e ft9t6e ft9t7e ft10t1 ft10t2 ft10t3 ft10t4 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.05708 0.05172 0.03724 0.03827 0.02986 0.01733 0.00602 0.01557 0.01152 0.01107 0.01 0.01 0.01 0.01 0.06324 0.06789 0.04697 0.03341 0.03188 0.00943 0.01236 0.01814 0.01219 0.01063 0.01 0.01 0.01 0.01 0.1108 0.06207 0.03901 0.04643 196 ft10t5 ft10t6 ft10t7 ft10t1e ft10t2e ft10t3e ft10t4e ft10t5e ft10t6e ft10t7e ft11t1 ft11t2 ft11t3 ft11t4 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.03261 0.02239 0.01478 0.02111 0.01698 0.01879 0.01561 0.01157 0.01 0.01 0.1414 0.09741 0.08076 0.06556 0.04604 0.02563 0.01426 0.02165 0.01693 0.01136 0.012 0.01 0.01 0.01 0.21467 0.16973 0.1133 0.09493 0.05973 0.01603 0.01078 0.023 0.0164 0.0113 0.01236 0.01 0.01 0.01 0.25555 0.21818 0.17189 0.10223 0.06738 0.03356 0.01049 0.02456 0.01572 0.01381 0.01466 0.01151 0.01 0.01 0.23479 0.17063 0.13751 0.10895 0.08805 197 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.03327 0.00304 0.02056 0.01094 0.01379 0.01283 0.012 0.01 0.01 0.24363 0.21509 0.17268 0.12293 0.09325 0.04324 0.00788 0.01916 0.01912 0.01382 0.01091 0.01227 0.01 0.01 0.25327 0.23779 0.17889 0.12801 0.09009 0.0444 0.0119 0.0148 0.01449 0.01081 0.01 0.01 0.01 0.01 0.27503 0.24656 0.19215 0.15509 0.0985 0.05809 0.00448 0.02104 0.01588 0.01137 0.01298 0.01 0.01 0.01 0.20074 0.18812 0.17372 0.12562 0.08545 0.05519 198 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.01112 0.0208 0.02035 0.01215 0.01004 0.01077 0.01061 0.01 0.15663 0.14513 0.13131 0.08177 0.06441 0.02196 0.01039 0.01323 0.01279 0.01142 0.01 0.01 0.01 0.01 0.17509 0.17716 0.16072 0.11588 0.06816 0.01721 0.02198 0.01534 0.02447 0.01242 0.0147 0.01 0.01226 0.01231 0.11176 0.07229 0.07415 0.05246 0.02774 0.00517 0.0098 0.01604 0.01604 0.01327 0.01275 0.01133 0.01032 0.01 0.09585 0.08392 0.06998 0.02077 0.041 0.04159 0.01122 199 ft22t1e = 0.01307 ft22t2e = 0.01559 ft22t3e = 0.01072 ft22t4e = 0.01187 ft22t5e = 0.01 ft22t6e = 0.01 ft22t7e = 0.01 ft23t1 = 0.11271 ft23t2 = 0.08928 ft23t3 = 0.05727 ft23t4 = 0.05885 ft23t5 = 0.04872 ft23t6 = 0.0138 ft23t7 = 0.02577 ft23t1e = 0.01934 ft23t2e = 0.01941 ft23t3e = 0.01625 ft23t4e = 0.01762 ft23t5e = 0.01068 ft23t6e = 0.01402 ft23t7e = 0.01024 ft24t1 = 0.0458 ft24t2 = 0.04996 ft24t3 = 0.0152 ft24t4 = 0.02364 ft24t5 = 0.03095 ft24t6 = 0.01628 ft24t7 = 0.00624 ft24t1e = 0.01433 ft24t2e = 0.0138 ft24t3e = 0.01858 ft24t4e = 0.01 ft24t5e = 0.01 ft24t6e = 0.01 ft24t7e = 0.01 bestkaisq = 100000000 S0 = 1.32 Slab = 0.9625275 f7 = 0.00 f25 = 0.00 f8 = 0.00 do while (f8 <= 0.00) f9 = 0.00 do while (f9 <= 0.01) f10 = 0.00 sa8 = (S0-Slab*(gon1*f8+goff1*(f7+f9)+1-f7-f8-f9)-gnad1)/S0 sb8 sc8 sd8 se8 sg8 sh8 = = = = = = (S0-Slab*(gon2*f8+goff2*(f7+f9)+1-f7-f8-f9)-gnad2)/S0 (S0-Slab*(gon3*f8+goff3*(f7+f9)+1-f7-f8-f9)-gnad3)/S0 (S0-Slab*(gon4*f8+goff4*(f7+f9)+1-f7-f8-f9)-gnad4)/S0 (S0-Slab*(gon5*f8+goff5*(f7+f9)+1-f7-f8-f9)-gnad5)/S0 (S0-Slab*(gon6*f8+goff6*(f7+f9)+1-f7-f8-f9)-gnad6)/S0 (S0-Slab*(gon7*f8+goff7*(f7+f9)+1-f7-f8-f9)-gnad7)/S0 200 xa8 = (ft8t1-sa8)**2/ft8t1e**2 xb8 = (ft8t2-sb8)**2/ft8t2e**2 xc8 = (ft8t3-sc8)**2/ft8t3e**2 xd8 = (ft8t4-sd8)**2/ft8t4e**2 xe8 = (ft8t5-se8)**2/ft8t5e**2 xg8 = (ft8t6-sg8)**2/ft8t6e**2 xh8 = (ft8t7-sh8)**2/ft8t7e**2 do while (f10 <= 0.02) sa9 = (S0-Slab*(gon1*f9+goff1*(f8+f10)+1-f8-f9-f10)-gnad1)/S0 sb9 = (S0-Slab*(gon2*f9+goff2*(f8+f10)+1-f8-f9-f10)-gnad2)/S0 sc9 = (S0-Slab*(gon3*f9+goff3*(f8+f10)+1-f8-f9-f10)-gnad3)/S0 sd9 = (S0-Slab*(gon4*f9+goff4*(f8+f10)+1-f8-f9-f10)-gnad4)/S0 se9 = (S0-Slab*(gon5*f9+goff5*(f8+f10)+1-f8-f9-f10)-gnad5)/S0 sg9 = (S0-Slab*(gon6*f9+goff6*(f8+f10)+1-f8-f9-f10)-gnad6)/S0 sh9 = (S0-Slab*(gon7*f9+goff7*(f8+f10)+1-f8-f9-f10)-gnad7)/S0 xa9 = (ft9t1-sa9)**2/ft9t1e**2 xb9 = (ft9t2-sb9)**2/ft9t2e**2 xc9 = (ft9t3-sc9)**2/ft9t3e**2 xd9 = (ft9t4-sd9)**2/ft9t4e**2 xe9 = (ft9t5-se9)**2/ft9t5e**2 xg9 = (ft9t6-sg9)**2/ft9t6e**2 xh9 = (ft9t7-sh9)**2/ft9t7e**2 f11 = 0.01 do while (f11 <= 0.07) sa10 = (S0-Slab*(gon1*f10+goff1*(f9+f11)+1-f9-f10-f11)-gnad1)/S0 sb10 = (S0-Slab*(gon2*f10+goff2*(f9+f11)+1-f9-f10-f11)-gnad2)/S0 sc10 = (S0-Slab*(gon3*f10+goff3*(f9+f11)+1-f9-f10-f11)-gnad3)/S0 sd10 = (S0-Slab*(gon4*f10+goff4*(f9+f11)+1-f9-f10-f11)-gnad4)/S0 se10 = (S0-Slab*(gon5*f10+goff5*(f9+f11)+1-f9-f10-f11)-gnad5)/S0 sg10 = (S0-Slab*(gon6*f10+goff6*(f9+f11)+1-f9-f10-f11)-gnad6)/S0 sh10 = (S0-Slab*(gon7*f10+goff7*(f9+f11)+1-f9-f10-f11)-gnad7)/S0 xa10 = (ft10t1-sa10)**2/ft10t1e**2 xb10 = (ft10t2-sb10)**2/ft10t2e**2 xc10 = (ft10t3-sc10)**2/ft10t3e**2 xd10 = (ft10t4-sd10)**2/ft10t4e**2 xe10 = (ft10t5-se10)**2/ft10t5e**2 xg10 = (ft10t6-sg10)**2/ft10t6e**2 xh10 = (ft10t7-sh10)**2/ft10t7e**2 f12 = 0.03 do while (f12 <= 0.15) sa11 = (S0-Slab*(gon1*f11+goff1*(f10+f12)+1-f10-f11-f12)-gnad1)/S0 sb11 = (S0-Slab*(gon2*f11+goff2*(f10+f12)+1-f10-f11-f12)-gnad2)/S0 sc11 = (S0-Slab*(gon3*f11+goff3*(f10+f12)+1-f10-f11-f12)-gnad3)/S0 sd11 = (S0-Slab*(gon4*f11+goff4*(f10+f12)+1-f10-f11-f12)-gnad4)/S0 se11 = (S0-Slab*(gon5*f11+goff5*(f10+f12)+1-f10-f11-f12)-gnad5)/S0 sg11 = (S0-Slab*(gon6*f11+goff6*(f10+f12)+1-f10-f11-f12)-gnad6)/S0 sh11 = (S0-Slab*(gon7*f11+goff7*(f10+f12)+1-f10-f11-f12)-gnad7)/S0 xa11 = (ft11t1-sa11)**2/ft11t1e**2 xb11 = (ft11t2-sb11)**2/ft11t2e**2 xc11 = (ft11t3-sc11)**2/ft11t3e**2 xd11 = (ft11t4-sd11)**2/ft11t4e**2 xe11 = (ft11t5-se11)**2/ft11t5e**2 xg11 = (ft11t6-sg11)**2/ft11t6e**2 xh11 = (ft11t7-sh11)**2/ft11t7e**2 f13 = 0.08 do while (f13 <= 0.022) sa12 = (S0-Slab*(gon1*f12+goff1*(f11+f13)+1-f11-f12-f13)-gnad1)/S0 201 sb12 sc12 sd12 se12 sg12 sh12 = (S0-Slab*(gon2*f12+goff2*(f11+f13)+1-f11-f12-f13)-gnad2)/S0 = (S0-Slab*(gon3*f12+goff3*(f11+f13)+1-f11-f12-f13)-gnad3)/S0 = (S0-Slab*(gon4*f12+goff4*(f11+f13)+1-f11-f12-f13)-gnad4)/S0 = (S0-Slab*(gon5*f12+goff5*(f11+f13)+1-f11-f12-f13)-gnad5)/S0 = (S0-Slab*(gon6*f12+goff6*(f11+f13)+1-f11-f12-f13)-gnad6)/S0 = (S0-Slab*(gon7*f12+goff7*(f11+f13)+1-f11-f12-f13)-gnad7)/S0 xa12 = (ft12t1-sa12)**2/ft12t1e**2 xb12 = (ft12t2-sb12)**2/ft12t2e**2 xc12 = (ft12t3-sc12)**2/ft12t3e**2 xd12 = (ft12t4-sd12)**2/ft12t4e**2 xe12 = (ft12t5-se12)**2/ft12t5e**2 xg12 = (ft12t6-sg12)**2/ft12t6e**2 xh12 = (ft12t7-sh12)**2/ft12t7e**2 f14 = 0.00 do while (f14 <= 0.19) sa13 = (S0-Slab*(gon1*f13+goff1*(f12+f14)+1-f12-f13-f14)-gnad1)/S0 sb13 = (S0-Slab*(gon2*f13+goff2*(f12+f14)+1-f12-f13-f14)-gnad2)/S0 sc13 = (S0-Slab*(gon3*f13+goff3*(f12+f14)+1-f12-f13-f14)-gnad3)/S0 sd13 = (S0-Slab*(gon4*f13+goff4*(f12+f14)+1-f12-f13-f14)-gnad4)/S0 se13 = (S0-Slab*(gon5*f13+goff5*(f12+f14)+1-f12-f13-f14)-gnad5)/S0 sg13 = (S0-Slab*(gon6*f13+goff6*(f12+f14)+1-f12-f13-f14)-gnad6)/S0 sh13 = (S0-Slab*(gon7*f13+goff7*(f12+f14)+1-f12-f13-f14)-gnad7)/S0 xa13 = (ft13t1-sa13)**2/ft13t1e**2 xb13 = (ft13t2-sb13)**2/ft13t2e**2 xc13 = (ft13t3-sc13)**2/ft13t3e**2 xd13 = (ft13t4-sd13)**2/ft13t4e**2 xe13 = (ft13t5-se13)**2/ft13t5e**2 xg13 = (ft13t6-sg13)**2/ft13t6e**2 xh13 = (ft13t7-sh13)**2/ft13t7e**2 f15 = 0.06 do while (f15 <= 0.23) sa14 = (S0-Slab*(gon1*f14+goff1*(f13+f15)+1-f13-f14-f15)-gnad1)/S0 sb14 = (S0-Slab*(gon2*f14+goff2*(f13+f15)+1-f13-f14-f15)-gnad2)/S0 sc14 = (S0-Slab*(gon3*f14+goff3*(f13+f15)+1-f13-f14-f15)-gnad3)/S0 sd14 = (S0-Slab*(gon4*f14+goff4*(f13+f15)+1-f13-f14-f15)-gnad4)/S0 se14 = (S0-Slab*(gon5*f14+goff5*(f13+f15)+1-f13-f14-f15)-gnad5)/S0 sg14 = (S0-Slab*(gon6*f14+goff6*(f13+f15)+1-f13-f14-f15)-gnad6)/S0 sh14 = (S0-Slab*(gon7*f14+goff7*(f13+f15)+1-f13-f14-f15)-gnad7)/S0 xa14 = (ft14t1-sa14)**2/ft14t1e**2 xb14 = (ft14t2-sb14)**2/ft14t2e**2 xc14 = (ft14t3-sc14)**2/ft14t3e**2 xd14 = (ft14t4-sd14)**2/ft14t4e**2 xe14 = (ft14t5-se14)**2/ft14t5e**2 xg14 = (ft14t6-sg14)**2/ft14t6e**2 xh14 = (ft14t7-sh14)**2/ft14t7e**2 * DIRK f16 = 0.04+0.01*i16 * f16 = 0.00 * do while (f16 <= 0.03) sa15 = (S0-Slab*(gon1*f15+goff1*(f14+f16)+1-f14-f15-f16)-gnad1)/S0 sb15 = (S0-Slab*(gon2*f15+goff2*(f14+f16)+1-f14-f15-f16)-gnad2)/S0 sc15 = (S0-Slab*(gon3*f15+goff3*(f14+f16)+1-f14-f15-f16)-gnad3)/S0 sd15 = (S0-Slab*(gon4*f15+goff4*(f14+f16)+1-f14-f15-f16)-gnad4)/S0 se15 = (S0-Slab*(gon5*f15+goff5*(f14+f16)+1-f14-f15-f16)-gnad5)/S0 sg15 = (S0-Slab*(gon6*f15+goff6*(f14+f16)+1-f14-f15-f16)-gnad6)/S0 sh15 = (S0-Slab*(gon7*f15+goff7*(f14+f16)+1-f14-f15-f16)-gnad7)/S0 xa15 = (ft15t1-sa15)**2/ft15t1e**2 202 xb15 = (ft15t2-sb15)**2/ft15t2e**2 xc15 = (ft15t3-sc15)**2/ft15t3e**2 xd15 = (ft15t4-sd15)**2/ft15t4e**2 xe15 = (ft15t5-se15)**2/ft15t5e**2 xg15 = (ft15t6-sg15)**2/ft15t6e**2 xh15 = (ft15t7-sh15)**2/ft15t7e**2 f17 = 0.10 do while (f17 <= 0.28) sa16 = (S0-Slab*(gon1*f16+goff1*(f15+f17)+1-f15-f16-f17)-gnad1)/S0 sb16 = (S0-Slab*(gon2*f16+goff2*(f15+f17)+1-f15-f16-f17)-gnad2)/S0 sc16 = (S0-Slab*(gon3*f16+goff3*(f15+f17)+1-f15-f16-f17)-gnad3)/S0 sd16 = (S0-Slab*(gon4*f16+goff4*(f15+f17)+1-f15-f16-f17)-gnad4)/S0 se16 = (S0-Slab*(gon5*f16+goff5*(f15+f17)+1-f15-f16-f17)-gnad5)/S0 sg16 = (S0-Slab*(gon6*f16+goff6*(f15+f17)+1-f15-f16-f17)-gnad6)/S0 sh16 = (S0-Slab*(gon7*f16+goff7*(f15+f17)+1-f15-f16-f17)-gnad7)/S0 xa16 = (ft16t1-sa16)**2/ft16t1e**2 xb16 = (ft16t2-sb16)**2/ft16t2e**2 xc16 = (ft16t3-sc16)**2/ft16t3e**2 xd16 = (ft16t4-sd16)**2/ft16t4e**2 xe16 = (ft16t5-se16)**2/ft16t5e**2 xg16 = (ft16t6-sg16)**2/ft16t6e**2 xh16 = (ft16t7-sh16)**2/ft16t7e**2 f18 = 0.05 do while (f18 <= 0.21) sa17 = (S0-Slab*(gon1*f17+goff1*(f16+f18)+1-f16-f17-f18)-gnad1)/S0 sb17 = (S0-Slab*(gon2*f17+goff2*(f16+f18)+1-f16-f17-f18)-gnad2)/S0 sc17 = (S0-Slab*(gon3*f17+goff3*(f16+f18)+1-f16-f17-f18)-gnad3)/S0 sd17 = (S0-Slab*(gon4*f17+goff4*(f16+f18)+1-f16-f17-f18)-gnad4)/S0 se17 = (S0-Slab*(gon5*f17+goff5*(f16+f18)+1-f16-f17-f18)-gnad5)/S0 sg17 = (S0-Slab*(gon6*f17+goff6*(f16+f18)+1-f16-f17-f18)-gnad6)/S0 sh17 = (S0-Slab*(gon7*f17+goff7*(f16+f18)+1-f16-f17-f18)-gnad7)/S0 xa17 = (ft17t1-sa17)**2/ft17t1e**2 xb17 = (ft17t2-sb17)**2/ft17t2e**2 xc17 = (ft17t3-sc17)**2/ft17t3e**2 xd17 = (ft17t4-sd17)**2/ft17t4e**2 xe17 = (ft17t5-se17)**2/ft17t5e**2 xg17 = (ft17t6-sg17)**2/ft17t6e**2 xh17 = (ft17t7-sh17)**2/ft17t7e**2 f19 = 0.00 do while (f19 <= 0.14) sa18 = (S0-Slab*(gon1*f18+goff1*(f17+f19)+1-f17-f18-f19)-gnad1)/S0 sb18 = (S0-Slab*(gon2*f18+goff2*(f17+f19)+1-f17-f18-f19)-gnad2)/S0 sc18 = (S0-Slab*(gon3*f18+goff3*(f17+f19)+1-f17-f18-f19)-gnad3)/S0 sd18 = (S0-Slab*(gon4*f18+goff4*(f17+f19)+1-f17-f18-f19)-gnad4)/S0 se18 = (S0-Slab*(gon5*f18+goff5*(f17+f19)+1-f17-f18-f19)-gnad5)/S0 sg18 = (S0-Slab*(gon6*f18+goff6*(f17+f19)+1-f17-f18-f19)-gnad6)/S0 sh18 = (S0-Slab*(gon7*f18+goff7*(f17+f19)+1-f17-f18-f19)-gnad7)/S0 xa18 = (ft18t1-sa18)**2/ft18t1e**2 xb18 = (ft18t2-sb18)**2/ft18t2e**2 xc18 = (ft18t3-sc18)**2/ft18t3e**2 xd18 = (ft18t4-sd18)**2/ft18t4e**2 xe18 = (ft18t5-se18)**2/ft18t5e**2 xg18 = (ft18t6-sg18)**2/ft18t6e**2 xh18 = (ft18t7-sh18)**2/ft18t7e**2 f20 = 0.11 do while (f20 <= 0.18) sa19 = (S0-Slab*(gon1*f19+goff1*(f18+f20)+1-f18-f19-f20)-gnad1)/S0 203 sb19 sc19 sd19 se19 sg19 sh19 = (S0-Slab*(gon2*f19+goff2*(f18+f20)+1-f18-f19-f20)-gnad2)/S0 = (S0-Slab*(gon3*f19+goff3*(f18+f20)+1-f18-f19-f20)-gnad3)/S0 = (S0-Slab*(gon4*f19+goff4*(f18+f20)+1-f18-f19-f20)-gnad4)/S0 = (S0-Slab*(gon5*f19+goff5*(f18+f20)+1-f18-f19-f20)-gnad5)/S0 = (S0-Slab*(gon6*f19+goff6*(f18+f20)+1-f18-f19-f20)-gnad6)/S0 = (S0-Slab*(gon7*f19+goff7*(f18+f20)+1-f18-f19-f20)-gnad7)/S0 xa19 = (ft19t1-sa19)**2/ft19t1e**2 xb19 = (ft19t2-sb19)**2/ft19t2e**2 xc19 = (ft19t3-sc19)**2/ft19t3e**2 xd19 = (ft19t4-sd19)**2/ft19t4e**2 xe19 = (ft19t5-se19)**2/ft19t5e**2 xg19 = (ft19t6-sg19)**2/ft19t6e**2 xh19 = (ft19t7-sh19)**2/ft19t7e**2 f21 = 0.00 do while (f21 <= 0.04) sa20 = (S0-Slab*(gon1*f20+goff1*(f19+f21)+1-f19-f20-f21)-gnad1)/S0 sb20 = (S0-Slab*(gon2*f20+goff2*(f19+f21)+1-f19-f20-f21)-gnad2)/S0 sc20 = (S0-Slab*(gon3*f20+goff3*(f19+f21)+1-f19-f20-f21)-gnad3)/S0 sd20 = (S0-Slab*(gon4*f20+goff4*(f19+f21)+1-f19-f20-f21)-gnad4)/S0 se20 = (S0-Slab*(gon5*f20+goff5*(f19+f21)+1-f19-f20-f21)-gnad5)/S0 sg20 = (S0-Slab*(gon6*f20+goff6*(f19+f21)+1-f19-f20-f21)-gnad6)/S0 sh20 = (S0-Slab*(gon7*f20+goff7*(f19+f21)+1-f19-f20-f21)-gnad7)/S0 xa20 = (ft20t1-sa20)**2/ft20t1e**2 xb20 = (ft20t2-sb20)**2/ft20t2e**2 xc20 = (ft20t3-sc20)**2/ft20t3e**2 xd20 = (ft20t4-sd20)**2/ft20t4e**2 xe20 = (ft20t5-se20)**2/ft20t5e**2 xg20 = (ft20t6-sg20)**2/ft20t6e**2 xh20 = (ft20t7-sh20)**2/ft20t7e**2 * DIRK f22 = 0.00+0.01*i22 * f22 = 0.00 * do while (f22 <= 0.03) sa21 = (S0-Slab*(gon1*f21+goff1*(f20+f22)+1-f20-f21-f22)-gnad1)/S0 sb21 = (S0-Slab*(gon2*f21+goff2*(f20+f22)+1-f20-f21-f22)-gnad2)/S0 sc21 = (S0-Slab*(gon3*f21+goff3*(f20+f22)+1-f20-f21-f22)-gnad3)/S0 sd21 = (S0-Slab*(gon4*f21+goff4*(f20+f22)+1-f20-f21-f22)-gnad4)/S0 se21 = (S0-Slab*(gon5*f21+goff5*(f20+f22)+1-f20-f21-f22)-gnad5)/S0 sg21 = (S0-Slab*(gon6*f21+goff6*(f20+f22)+1-f20-f21-f22)-gnad6)/S0 sh21 = (S0-Slab*(gon7*f21+goff7*(f20+f22)+1-f20-f21-f22)-gnad7)/S0 xa21 = (ft21t1-sa21)**2/ft21t1e**2 xb21 = (ft21t2-sb21)**2/ft21t2e**2 xc21 = (ft21t3-sc21)**2/ft21t3e**2 xd21 = (ft21t4-sd21)**2/ft21t4e**2 xe21 = (ft21t5-se21)**2/ft21t5e**2 xg21 = (ft21t6-sg21)**2/ft21t6e**2 xh21 = (ft21t7-sh21)**2/ft21t7e**2 f23 = 0.04 do while (f23 <= 0.05) sa22 = (S0-Slab*(gon1*f22+goff1*(f21+f23)+1-f21-f22-f23)-gnad1)/S0 sb22 = (S0-Slab*(gon2*f22+goff2*(f21+f23)+1-f21-f22-f23)-gnad2)/S0 sc22 = (S0-Slab*(gon3*f22+goff3*(f21+f23)+1-f21-f22-f23)-gnad3)/S0 sd22 = (S0-Slab*(gon4*f22+goff4*(f21+f23)+1-f21-f22-f23)-gnad4)/S0 se22 = (S0-Slab*(gon5*f22+goff5*(f21+f23)+1-f21-f22-f23)-gnad5)/S0 sg22 = (S0-Slab*(gon6*f22+goff6*(f21+f23)+1-f21-f22-f23)-gnad6)/S0 sh22 = (S0-Slab*(gon7*f22+goff7*(f21+f23)+1-f21-f22-f23)-gnad7)/S0 xa22 = (ft22t1-sa22)**2/ft22t1e**2 204 xb22 = (ft22t2-sb22)**2/ft22t2e**2 xc22 = (ft22t3-sc22)**2/ft22t3e**2 xd22 = (ft22t4-sd22)**2/ft22t4e**2 xe22 = (ft22t5-se22)**2/ft22t5e**2 xg22 = (ft22t6-sg22)**2/ft22t6e**2 xh22 = (ft22t7-sh22)**2/ft22t7e**2 f24 = 0.00 do while (f24 <= 0.00) sa23 = (S0-Slab*(gon1*f23+goff1*(f22+f24)+1-f22-f23-f24)-gnad1)/S0 sb23 = (S0-Slab*(gon2*f23+goff2*(f22+f24)+1-f22-f23-f24)-gnad2)/S0 sc23 = (S0-Slab*(gon3*f23+goff3*(f22+f24)+1-f22-f23-f24)-gnad3)/S0 sd23 = (S0-Slab*(gon4*f23+goff4*(f22+f24)+1-f22-f23-f24)-gnad4)/S0 se23 = (S0-Slab*(gon5*f23+goff5*(f22+f24)+1-f22-f23-f24)-gnad5)/S0 sg23 = (S0-Slab*(gon6*f23+goff6*(f22+f24)+1-f22-f23-f24)-gnad6)/S0 sh23 = (S0-Slab*(gon7*f23+goff7*(f22+f24)+1-f22-f23-f24)-gnad7)/S0 sa24 sb24 sc24 sd24 se24 sg24 sh24 = = = = = = = (S0-Slab*(gon1*f24+goff1*(f23+f25)+1-f23-f24-f25)-gnad1)/S0 (S0-Slab*(gon2*f24+goff2*(f23+f25)+1-f23-f24-f25)-gnad2)/S0 (S0-Slab*(gon3*f24+goff3*(f23+f25)+1-f23-f24-f25)-gnad3)/S0 (S0-Slab*(gon4*f24+goff4*(f23+f25)+1-f23-f24-f25)-gnad4)/S0 (S0-Slab*(gon5*f24+goff5*(f23+f25)+1-f23-f24-f25)-gnad5)/S0 (S0-Slab*(gon6*f24+goff6*(f23+f25)+1-f23-f24-f25)-gnad6)/S0 (S0-Slab*(gon7*f24+goff7*(f23+f25)+1-f23-f24-f25)-gnad7)/S0 xa23 xb23 xc23 xd23 xe23 xg23 xh23 (ft23t1-sa23)**2/ft23t1e**2 (ft23t2-sb23)**2/ft23t2e**2 (ft23t3-sc23)**2/ft23t3e**2 (ft23t4-sd23)**2/ft23t4e**2 (ft23t5-se23)**2/ft23t5e**2 (ft23t6-sg23)**2/ft23t6e**2 (ft23t7-sh23)**2/ft23t7e**2 xa24 xb24 xc24 xd24 xe24 xg24 xh24 & & & & & & & & & & & & & & & & = = = = = = = = = = = = = = (ft24t1-sa24)**2/ft24t1e**2 (ft24t2-sb24)**2/ft24t2e**2 (ft24t3-sc24)**2/ft24t3e**2 (ft24t4-sd24)**2/ft24t4e**2 (ft24t5-se24)**2/ft24t5e**2 (ft24t6-sg24)**2/ft24t6e**2 (ft24t7-sh24)**2/ft24t7e**2 kaisq = xa8+xb8+xc8+xd8+xe8+xg8+xh8+ xa9+xb9+xc9+xd9+xe9+xg9+xh9+ xa10+xb10+xc10+xd10+xe10+xg10+xh10+ xa11+xb11+xc11+xd11+xe11+xg11+xh11+ xa12+xb12+xc12+xd12+xe12+xg12+xh12+ xa13+xb13+xc13+xd13+xe13+xg13+xh13+ xa14+xb14+xc14+xd14+xe14+xg14+xh14+ xa15+xb15+xc15+xd15+xe15+xg15+xh15+ xa16+xb16+xc16+xd16+xe16+xg16+xh16+ xa17+xb17+xc17+xd17+xe17+xg17+xh17+ xa18+xb18+xc18+xd18+xe18+xg18+xh18+ xa19+xb19+xc19+xd19+xe19+xg19+xh19+ xa20+xb20+xc20+xd20+xe20+xg20+xh20+ xa21+xb21+xc21+xd21+xe21+xg21+xh21+ xa22+xb22+xc22+xd22+xe22+xg22+xh22+ xa23+xb23+xc23+xd23+xe23+xg23+xh23+ xa24+xb24+xc24+xd24+xe24+xg24+xh24 205 IF (kaisq < bestkaisq) THEN bestkaisq = kaisq bf8 = f8 bf9 = f9 bf10 = f10 bf11 = f11 bf12 = f12 bf13 = f13 bf14 = f14 bf15 = f15 bf16 = f16 bf17 = f17 bf18 = f18 bf19 = f19 bf20 = f20 bf21 = f21 bf22 = f22 bf23 = f23 bf24 = f24 endif f24 = f24 + 0.01 enddo f23 = f23 + 0.01 enddo * DIRK * * * DIRK * * f22 = f22 + 0.01 enddo f21 = f21 + 0.01 enddo f20 = f20 + 0.01 enddo f19 = f19 + 0.01 enddo f18 = f18 + 0.01 enddo f17 = f17 + 0.01 enddo f16 = f16 + 0.01 enddo f15 = f15 + 0.01 enddo f14 = f14 + 0.01 enddo f13 = f13 + 0.01 enddo f12 = f12 + 0.01 enddo f11 = f11 + 0.01 enddo f10 = f10 + 0.01 enddo 206 f9 = f9 + 0.01 enddo f8 = f8 + 0.01 enddo * DIRK * * & & & & & OPEN(UNIT = 12, FILE = 'output', STATUS = 'NEW') WRITE(12,*) bestkaisq, WRITE(*,*) bestkaisq, bf8, bf9, bf10, bf11, bf12, bf13, bf14, bf15, bf16, bf17, bf18, bf19, bf20, bf21, bf22, bf23, bf24 end 3.3 HFP 5 Registry Fitting qsub Script, “x2_HFP” For HFP 5 registry fittings, two main script files were created (see Appendix I). In the file below, f9 = 0.00. The second main script file used f9 = 0.01. Splitting the main script file into two separate jobs was required to complete each script file’s computations in less than 168 hours (i.e. the maximum time allowed to occupy a node). This time limit is set by the High Performance Computing Center and jobs that run longer than 168 hours are automatically terminated. #!/bin/bash #PBS -l nodes=1:ppn=1,walltime=168:00:00,mem=2gb,feature=gbe #PBS -j oe #PBS -t 0-139 #change to the original working directory cd ${PBS_O_WORKDIR} # Define number of loops for i12 cols=13 # Define number of loops for i20 (not used) rows=9 # Number of jobs = cols*rows (i.e. -t 0-139) #math to figure out the variable values based on the array id i12=`echo "${PBS_ARRAYID} % ( ${cols} + 1 )" | bc` i20=`echo "${PBS_ARRAYID} / ( ${cols} + 1 )" | bc` #display the command we are going to run echo "./x2 ${i12} ${i20} > ${i12}_${i20}.txt" 207 #run the command with the input variables ./x2 ${i12} ${i20} > ${i12}_${i20}.txt # Calculate the runtiem for the job qstat -f ${PBS_JOBID} 3.4 HFP 5 Registry Fitting Main Script, “HFP_5var.f” * This is a comment * This program was written by Scott Schmick 030111 * * t values (1 = a = 482, 2 = b = 402, etc.) real f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 real f17 f18 f19 f20 f21 f22 f23 f24 f25 real sa8 sb8 sc8 sd8 se8 sg8 sh8 real sa9 sb9 sc9 sd9 se9 sg9 sh9 real sa10 sb10 sc10 sd10 se10 sg10 sh10 real sa11 sb11 sc11 sd11 se11 sg11 sh11 real sa12 sb12 sc12 sd12 se12 sg12 sh12 real sa13 sb13 sc13 sd13 se13 sg13 sh13 real sa14 sb14 sc14 sd14 se14 sg14 sh14 real sa15 sb15 sc15 sd15 se15 sg15 sh15 real sa16 sb16 sc16 sd16 se16 sg16 sh16 real sa17 sb17 sc17 sd17 se17 sg17 sh17 real sa18 sb18 sc18 sd18 se18 sg18 sh18 real sa19 sb19 sc19 sd19 se19 sg19 sh19 real sa20 sb20 sc20 sd20 se20 sg20 sh20 real sa21 sb21 sc21 sd21 se21 sg21 sh21 real sa22 sb22 sc22 sd22 se22 sg22 sh22 real sa23 sb23 sc23 sd23 se23 sg23 sh23 real sa24 sb24 sc24 sd24 se24 sg24 sh24 real real real real real real real real real real real real real real real real real real S0 xa8 xb8 xc8 xd8 xe8 xa9 xb9 xc9 xd9 xe9 xa10 xb10 xc10 xd10 xa11 xb11 xc11 xd11 xa12 xb12 xc12 xd12 xa13 xb13 xc13 xd13 xa14 xb14 xc14 xd14 xa15 xb15 xc15 xd15 xa16 xb16 xc16 xd16 xa17 xb17 xc17 xd17 xa18 xb18 xc18 xd18 xa19 xb19 xc19 xd19 xa20 xb20 xc20 xd20 xa21 xb21 xc21 xd21 xa22 xb22 xc22 xd22 xa23 xb23 xc23 xd23 xa24 xb24 xc24 xd24 xg8 xh8 xg9 xh9 xe10 xg10 xe11 xg11 xe12 xg12 xe13 xg13 xe14 xg14 xe15 xg15 xe16 xg16 xe17 xg17 xe18 xg18 xe19 xg19 xe20 xg20 xe21 xg21 xe22 xg22 xe23 xg23 xe24 xg24 xh10 xh11 xh12 xh13 xh14 xh15 xh16 xh17 xh18 xh19 xh20 xh21 xh22 xh23 xh24 real goff1 goff2 goff3 goff4 goff5 goff6 goff7 real gon1 gon2 gon3 gon4 gon5 gon6 gon7 real gnad1 gnad2 gnad3 gnad4 gnad5 gnad6 gnad7 real ft8t1 208 * real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft8t2 ft8t3 ft8t4 ft8t5 ft8t6 ft8t7 ft8t1e ft8t2e ft8t3e ft8t4e ft8t5e ft8t6e ft8t7e ft9t1 ft9t2 ft9t3 ft9t4 ft9t5 ft9t6 ft9t7 ft9t1e ft9t2e ft9t3e ft9t4e ft9t5e ft9t6e ft9t7e ft10t1 ft10t2 ft10t3 ft10t4 ft10t5 ft10t6 ft10t7 ft10t1e ft10t2e ft10t3e ft10t4e ft10t5e ft10t6e ft10t7e ft11t1 ft11t2 ft11t3 ft11t4 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 209 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 210 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 211 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e ft23t1 ft23t2 ft23t3 ft23t4 ft23t5 ft23t6 ft23t7 ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 ft24t2 ft24t3 ft24t4 ft24t5 212 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft24t6 ft24t7 ft24t1e ft24t2e ft24t3e ft24t4e ft24t5e ft24t6e ft24t7e kaisq bestkaisq bf8 bf9 bf10 bf11 bf12 bf13 bf14 bf15 bf16 bf17 bf18 bf19 bf20 bf21 bf22 bf23 bf24 a8 aa8 a9 aa9 a10 aa10 a11 aa11 a12 aa12 a13 aa13 a14 aa14 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 z1 z2 z3 z4 z5 z6 z7 f6 f26 a15 aa15 a16 aa16 a17 aa17 a18 aa18 a19 aa19 a20 aa20 a21 aa21 a22 aa22 a23 aa23 a24 aa24 * DIRK integer i12 i20 character inputarg*128 CALL getarg(1,inputarg) read(inputarg,*) i12 CALL getarg(2,inputarg) read(inputarg,*) i20 goff1 goff2 goff3 goff4 goff5 goff6 goff7 gon1 gon2 gon3 gon4 gon5 = 0.45011191 = 0.581357776 = 0.710329562 = 0.826254485 = 0.918588976 = 0.978470252 = 0.998385086 = 0.093897695 = 0.118554369 = 0.196398598 = 0.378629871 = 0.645290442 213 gon6 = 0.893821899 gon7 = 0.991709633 z1 = 0.9052388602 z2 = 0.9334319587 z3 = 0.95693731265 z4 = 0.97551074985 z5 = 0.98895970015 z6 = 0.99714805251 z7 = 0.9997874594365 gnad1 = 0.281406262 gnad2 = 0.283440096 gnad3 = 0.298896559 gnad4 = 0.309714762 gnad5 = 0.320805841 gnad6 = 0.325732761 gnad7 = 0.342143946 ft8t1 = 0.05708 ft8t2 = 0.05172 ft8t3 = 0.03724 ft8t4 = 0.03827 ft8t5 = 0.02986 ft8t6 = 0.01733 ft8t7 = 0.00602 ft8t1e = 0.01557 ft8t2e = 0.01152 ft8t3e = 0.01107 ft8t4e = 0.01 ft8t5e = 0.01 ft8t6e = 0.01 ft8t7e = 0.01 ft9t1 = 0.06324 ft9t2 = 0.06789 ft9t3 = 0.04697 ft9t4 = 0.03341 ft9t5 = 0.03188 ft9t6 = 0.00943 ft9t7 = 0.01236 ft9t1e = 0.01814 ft9t2e = 0.01219 ft9t3e = 0.01063 ft9t4e = 0.01 ft9t5e = 0.01 ft9t6e = 0.01 ft9t7e = 0.01 ft10t1 = 0.1108 ft10t2 = 0.06207 ft10t3 = 0.03901 ft10t4 = 0.04643 ft10t5 = 0.03261 ft10t6 = 0.02239 ft10t7 = 0.01478 ft10t1e = 0.02111 ft10t2e = 0.01698 ft10t3e = 0.01879 ft10t4e = 0.01561 ft10t5e = 0.01157 ft10t6e = 0.01 214 ft10t7e = ft11t1 = ft11t2 = ft11t3 = ft11t4 = ft11t5 = ft11t6 = ft11t7 = ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 = ft12t2 = ft12t3 = ft12t4 = ft12t5 = ft12t6 = ft12t7 = ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 = ft13t2 = ft13t3 = ft13t4 = ft13t5 = ft13t6 = ft13t7 = ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 = ft14t2 = ft14t3 = ft14t4 = ft14t5 = ft14t6 = ft14t7 = ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e 0.01 0.1414 0.09741 0.08076 0.06556 0.04604 0.02563 0.01426 = = = = = = = 0.21467 0.16973 0.1133 0.09493 0.05973 0.01603 0.01078 = = = = = = = 0.25555 0.21818 0.17189 0.10223 0.06738 0.03356 0.01049 = = = = = = = 0.23479 0.17063 0.13751 0.10895 0.08805 0.03327 0.00304 = = = = = = = 0.02165 0.01693 0.01136 0.012 0.01 0.01 0.01 0.023 0.0164 0.0113 0.01236 0.01 0.01 0.01 0.02456 0.01572 0.01381 0.01466 0.01151 0.01 0.01 0.02056 0.01094 0.01379 0.01283 0.012 0.01 0.01 215 ft15t1 = ft15t2 = ft15t3 = ft15t4 = ft15t5 = ft15t6 = ft15t7 = ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 = ft16t2 = ft16t3 = ft16t4 = ft16t5 = ft16t6 = ft16t7 = ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 = ft17t2 = ft17t3 = ft17t4 = ft17t5 = ft17t6 = ft17t7 = ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 = ft18t2 = ft18t3 = ft18t4 = ft18t5 = ft18t6 = ft18t7 = ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 = 0.24363 0.21509 0.17268 0.12293 0.09325 0.04324 0.00788 = = = = = = = 0.25327 0.23779 0.17889 0.12801 0.09009 0.0444 0.0119 = = = = = = = 0.27503 0.24656 0.19215 0.15509 0.0985 0.05809 0.00448 = = = = = = = 0.20074 0.18812 0.17372 0.12562 0.08545 0.05519 0.01112 = = = = = = = 0.15663 0.01916 0.01912 0.01382 0.01091 0.01227 0.01 0.01 0.0148 0.01449 0.01081 0.01 0.01 0.01 0.01 0.02104 0.01588 0.01137 0.01298 0.01 0.01 0.01 0.0208 0.02035 0.01215 0.01004 0.01077 0.01061 0.01 216 ft19t2 = ft19t3 = ft19t4 = ft19t5 = ft19t6 = ft19t7 = ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 = ft20t2 = ft20t3 = ft20t4 = ft20t5 = ft20t6 = ft20t7 = ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 = ft21t2 = ft21t3 = ft21t4 = ft21t5 = ft21t6 = ft21t7 = ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 = ft22t2 = ft22t3 = ft22t4 = ft22t5 = ft22t6 = ft22t7 = ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e ft23t1 = ft23t2 = 0.14513 0.13131 0.08177 0.06441 0.02196 0.01039 = = = = = = = 0.17509 0.17716 0.16072 0.11588 0.06816 0.01721 0.02198 = = = = = = = 0.11176 0.07229 0.07415 0.05246 0.02774 0.00517 0.0098 = = = = = = = 0.09585 0.08392 0.06998 0.02077 0.041 0.04159 0.01122 = = = = = = = 0.11271 0.08928 0.01323 0.01279 0.01142 0.01 0.01 0.01 0.01 0.01534 0.02447 0.01242 0.0147 0.01 0.01226 0.01231 0.01604 0.01604 0.01327 0.01275 0.01133 0.01032 0.01 0.01307 0.01559 0.01072 0.01187 0.01 0.01 0.01 217 ft23t3 = ft23t4 = ft23t5 = ft23t6 = ft23t7 = ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 = ft24t2 = ft24t3 = ft24t4 = ft24t5 = ft24t6 = ft24t7 = ft24t1e ft24t2e ft24t3e ft24t4e ft24t5e ft24t6e ft24t7e 0.05727 0.05885 0.04872 0.0138 0.02577 = = = = = = = 0.0458 0.04996 0.0152 0.02364 0.03095 0.01628 0.00624 = = = = = = = 0.01934 0.01941 0.01625 0.01762 0.01068 0.01402 0.01024 0.01433 0.0138 0.01858 0.01 0.01 0.01 0.01 bestkaisq = 100000000 S0 = 1.32 Slab = 0.9625275 f6 = 0.00 f7 = 0.00 f25 = 0.00 f26 = 0.00 f8 = 0.00 do while (f8 <= 0.00) f9 = 0.01 do while (f9 <= 0.01) f10 = 0.00 a8 = f7+f9 b8 = f6+f10 aa8 = 1-f6-f7-f8-f9-f10 sa8 = (S0-Slab*(gon1*f8+goff1*a8+z1*b8+aa8)-gnad1)/S0 sb8 = (S0-Slab*(gon2*f8+goff2*a8+z2*b8+aa8)-gnad2)/S0 sc8 = (S0-Slab*(gon3*f8+goff3*a8+z3*b8+aa8)-gnad3)/S0 sd8 = (S0-Slab*(gon4*f8+goff4*a8+z4*b8+aa8)-gnad4)/S0 se8 = (S0-Slab*(gon5*f8+goff5*a8+z5*b8+aa8)-gnad5)/S0 sg8 = (S0-Slab*(gon6*f8+goff6*a8+z6*b8+aa8)-gnad6)/S0 sh8 = (S0-Slab*(gon7*f8+goff7*a8+z7*b8+aa8)-gnad7)/S0 xa8 = (ft8t1-sa8)**2/ft8t1e**2 xb8 = (ft8t2-sb8)**2/ft8t2e**2 xc8 = (ft8t3-sc8)**2/ft8t3e**2 xd8 = (ft8t4-sd8)**2/ft8t4e**2 xe8 = (ft8t5-se8)**2/ft8t5e**2 218 * xg8 = (ft8t6-sg8)**2/ft8t6e**2 xh8 = (ft8t7-sh8)**2/ft8t7e**2 do while (f10 <= 0.02) a9 = f8+f10 b9 = f7+f11 aa9 = 1-f7-f8-f9-f10-f11 sa9 = (S0-Slab*(gon1*f9+goff1*a9+b9*z1+aa9)-gnad1)/S0 sb9 = (S0-Slab*(gon2*f9+goff2*a9+b9*z2+aa9)-gnad2)/S0 sc9 = (S0-Slab*(gon3*f9+goff3*a9+b9*z3+aa9)-gnad3)/S0 sd9 = (S0-Slab*(gon4*f9+goff4*a9+b9*z4+aa9)-gnad4)/S0 se9 = (S0-Slab*(gon5*f9+goff5*a9+b9*z5+aa9)-gnad5)/S0 sg9 = (S0-Slab*(gon6*f9+goff6*a9+b9*z6+aa9)-gnad6)/S0 sh9 = (S0-Slab*(gon7*f9+goff7*a9+b9*z7+aa9)-gnad7)/S0 xa9 = (ft9t1-sa9)**2/ft9t1e**2 xb9 = (ft9t2-sb9)**2/ft9t2e**2 xc9 = (ft9t3-sc9)**2/ft9t3e**2 xd9 = (ft9t4-sd9)**2/ft9t4e**2 xe9 = (ft9t5-se9)**2/ft9t5e**2 xg9 = (ft9t6-sg9)**2/ft9t6e**2 xh9 = (ft9t7-sh9)**2/ft9t7e**2 f11 = 0.00 do while (f11 <= 0.07) a10 = f9+f11 b10 = f8+f12 aa10 = 1-f8-f9-f10-f11-f12 sa10 = (S0-Slab*(gon1*f10+goff1*a10+z1*b10+aa10)-gnad1)/S0 sb10 = (S0-Slab*(gon2*f10+goff2*a10+z2*b10+aa10)-gnad2)/S0 sc10 = (S0-Slab*(gon3*f10+goff3*a10+z3*b10+aa10)-gnad3)/S0 sd10 = (S0-Slab*(gon4*f10+goff4*a10+z4*b10+aa10)-gnad4)/S0 se10 = (S0-Slab*(gon5*f10+goff5*a10+z5*b10+aa10)-gnad5)/S0 sg10 = (S0-Slab*(gon6*f10+goff6*a10+z6*b10+aa10)-gnad6)/S0 sh10 = (S0-Slab*(gon7*f10+goff7*a10+z7*b10+aa10)-gnad7)/S0 xa10 = (ft10t1-sa10)**2/ft10t1e**2 xb10 = (ft10t2-sb10)**2/ft10t2e**2 xc10 = (ft10t3-sc10)**2/ft10t3e**2 xd10 = (ft10t4-sd10)**2/ft10t4e**2 xe10 = (ft10t5-se10)**2/ft10t5e**2 xg10 = (ft10t6-sg10)**2/ft10t6e**2 xh10 = (ft10t7-sh10)**2/ft10t7e**2 f12 = 0.02 *Dirk * f12 = 0.02+0.01*i12 do while (f12 <= 0.15) a11 = f10+f12 b11 = f9+f13 aa11 = 1-f9-f10-f11-f12-f13 sa11 = (S0-Slab*(gon1*f11+goff1*a11+z1*b11+aa11)-gnad1)/S0 sb11 = (S0-Slab*(gon2*f11+goff2*a11+z2*b11+aa11)-gnad2)/S0 sc11 = (S0-Slab*(gon3*f11+goff3*a11+z3*b11+aa11)-gnad3)/S0 sd11 = (S0-Slab*(gon4*f11+goff4*a11+z4*b11+aa11)-gnad4)/S0 se11 = (S0-Slab*(gon5*f11+goff5*a11+z5*b11+aa11)-gnad5)/S0 sg11 = (S0-Slab*(gon6*f11+goff6*a11+z6*b11+aa11)-gnad6)/S0 sh11 = (S0-Slab*(gon7*f11+goff7*a11+z7*b11+aa11)-gnad7)/S0 xa11 = (ft11t1-sa11)**2/ft11t1e**2 xb11 = (ft11t2-sb11)**2/ft11t2e**2 xc11 = (ft11t3-sc11)**2/ft11t3e**2 219 xd11 = (ft11t4-sd11)**2/ft11t4e**2 xe11 = (ft11t5-se11)**2/ft11t5e**2 xg11 = (ft11t6-sg11)**2/ft11t6e**2 xh11 = (ft11t7-sh11)**2/ft11t7e**2 f13 = 0.07 do while (f13 <= 0.22) a12 = f11+f13 b12 = f10+f14 aa12 = 1-f10-f11-f12-f13-f14 sa12 = (S0-Slab*(gon1*f12+goff1*a12+z1*b12+aa12)-gnad1)/S0 sb12 = (S0-Slab*(gon2*f12+goff2*a12+z2*b12+aa12)-gnad2)/S0 sc12 = (S0-Slab*(gon3*f12+goff3*a12+z3*b12+aa12)-gnad3)/S0 sd12 = (S0-Slab*(gon4*f12+goff4*a12+z4*b12+aa12)-gnad4)/S0 se12 = (S0-Slab*(gon5*f12+goff5*a12+z5*b12+aa12)-gnad5)/S0 sg12 = (S0-Slab*(gon6*f12+goff6*a12+z6*b12+aa12)-gnad6)/S0 sh12 = (S0-Slab*(gon7*f12+goff7*a12+z7*b12+aa12)-gnad7)/S0 xa12 = (ft12t1-sa12)**2/ft12t1e**2 xb12 = (ft12t2-sb12)**2/ft12t2e**2 xc12 = (ft12t3-sc12)**2/ft12t3e**2 xd12 = (ft12t4-sd12)**2/ft12t4e**2 xe12 = (ft12t5-se12)**2/ft12t5e**2 xg12 = (ft12t6-sg12)**2/ft12t6e**2 xh12 = (ft12t7-sh12)**2/ft12t7e**2 f14 = 0.00 do while (f14 <= 0.19) a13 = f12+f14 b13 = f11+f15 aa13 = 1-f11-f12-f13-f14-f15 sa13 = (S0-Slab*(gon1*f13+goff1*a13+z1*b13+aa13)-gnad1)/S0 sb13 = (S0-Slab*(gon2*f13+goff2*a13+z2*b13+aa13)-gnad2)/S0 sc13 = (S0-Slab*(gon3*f13+goff3*a13+z3*b13+aa13)-gnad3)/S0 sd13 = (S0-Slab*(gon4*f13+goff4*a13+z4*b13+aa13)-gnad4)/S0 se13 = (S0-Slab*(gon5*f13+goff5*a13+z5*b13+aa13)-gnad5)/S0 sg13 = (S0-Slab*(gon6*f13+goff6*a13+z6*b13+aa13)-gnad6)/S0 sh13 = (S0-Slab*(gon7*f13+goff7*a13+z7*b13+aa13)-gnad7)/S0 xa13 = (ft13t1-sa13)**2/ft13t1e**2 xb13 = (ft13t2-sb13)**2/ft13t2e**2 xc13 = (ft13t3-sc13)**2/ft13t3e**2 xd13 = (ft13t4-sd13)**2/ft13t4e**2 xe13 = (ft13t5-se13)**2/ft13t5e**2 xg13 = (ft13t6-sg13)**2/ft13t6e**2 xh13 = (ft13t7-sh13)**2/ft13t7e**2 f15 = 0.03 do while (f15 <= 0.23) a14 = f13+f15 b14 = f12+f16 aa14 = 1-f12-f13-f14-f15-f16 sa14 = (S0-Slab*(gon1*f14+goff1*a14+z1*b14+aa14)-gnad1)/S0 sb14 = (S0-Slab*(gon2*f14+goff2*a14+z2*b14+aa14)-gnad2)/S0 sc14 = (S0-Slab*(gon3*f14+goff3*a14+z3*b14+aa14)-gnad3)/S0 sd14 = (S0-Slab*(gon4*f14+goff4*a14+z4*b14+aa14)-gnad4)/S0 se14 = (S0-Slab*(gon5*f14+goff5*a14+z5*b14+aa14)-gnad5)/S0 sg14 = (S0-Slab*(gon6*f14+goff6*a14+z6*b14+aa14)-gnad6)/S0 sh14 = (S0-Slab*(gon7*f14+goff7*a14+z7*b14+aa14)-gnad7)/S0 xa14 = (ft14t1-sa14)**2/ft14t1e**2 220 xb14 xc14 xd14 xe14 xg14 xh14 = = = = = = (ft14t2-sb14)**2/ft14t2e**2 (ft14t3-sc14)**2/ft14t3e**2 (ft14t4-sd14)**2/ft14t4e**2 (ft14t5-se14)**2/ft14t5e**2 (ft14t6-sg14)**2/ft14t6e**2 (ft14t7-sh14)**2/ft14t7e**2 f16 = 0.02 do while (f16 <= 0.25) a15 = f14+f16 b15 = f13+f17 aa15 = 1-f13-f14-f15-f16-f17 sa15 = (S0-Slab*(gon1*f15+goff1*a15+z1*b15+aa15)-gnad1)/S0 sb15 = (S0-Slab*(gon2*f15+goff2*a15+z2*b15+aa15)-gnad2)/S0 sc15 = (S0-Slab*(gon3*f15+goff3*a15+z3*b15+aa15)-gnad3)/S0 sd15 = (S0-Slab*(gon4*f15+goff4*a15+z4*b15+aa15)-gnad4)/S0 se15 = (S0-Slab*(gon5*f15+goff5*a15+z5*b15+aa15)-gnad5)/S0 sg15 = (S0-Slab*(gon6*f15+goff6*a15+z6*b15+aa15)-gnad6)/S0 sh15 = (S0-Slab*(gon7*f15+goff7*a15+z7*b15+aa15)-gnad7)/S0 xa15 = (ft15t1-sa15)**2/ft15t1e**2 xb15 = (ft15t2-sb15)**2/ft15t2e**2 xc15 = (ft15t3-sc15)**2/ft15t3e**2 xd15 = (ft15t4-sd15)**2/ft15t4e**2 xe15 = (ft15t5-se15)**2/ft15t5e**2 xg15 = (ft15t6-sg15)**2/ft15t6e**2 xh15 = (ft15t7-sh15)**2/ft15t7e**2 f17 = 0.07 do while (f17 <= 0.28) a16 = f15+f17 b16 = f14+f18 aa16 = 1-f14-f15-f16-f17-f18 sa16 = (S0-Slab*(gon1*f16+goff1*a16+z1*b16+aa16)-gnad1)/S0 sb16 = (S0-Slab*(gon2*f16+goff2*a16+z2*b16+aa16)-gnad2)/S0 sc16 = (S0-Slab*(gon3*f16+goff3*a16+z3*b16+aa16)-gnad3)/S0 sd16 = (S0-Slab*(gon4*f16+goff4*a16+z4*b16+aa16)-gnad4)/S0 se16 = (S0-Slab*(gon5*f16+goff5*a16+z5*b16+aa16)-gnad5)/S0 sg16 = (S0-Slab*(gon6*f16+goff6*a16+z6*b16+aa16)-gnad6)/S0 sh16 = (S0-Slab*(gon7*f16+goff7*a16+z7*b16+aa16)-gnad7)/S0 xa16 = (ft16t1-sa16)**2/ft16t1e**2 xb16 = (ft16t2-sb16)**2/ft16t2e**2 xc16 = (ft16t3-sc16)**2/ft16t3e**2 xd16 = (ft16t4-sd16)**2/ft16t4e**2 xe16 = (ft16t5-se16)**2/ft16t5e**2 xg16 = (ft16t6-sg16)**2/ft16t6e**2 xh16 = (ft16t7-sh16)**2/ft16t7e**2 f18 = 0.03 do while (f18 <= 0.21) a17 = f16+f18 b17 = f15+f19 aa17 = 1-f15-f16-f17-f18-f19 sa17 = (S0-Slab*(gon1*f17+goff1*a17+z1*b17+aa17)-gnad1)/S0 sb17 = (S0-Slab*(gon2*f17+goff2*a17+z2*b17+aa17)-gnad2)/S0 sc17 = (S0-Slab*(gon3*f17+goff3*a17+z3*b17+aa17)-gnad3)/S0 sd17 = (S0-Slab*(gon4*f17+goff4*a17+z4*b17+aa17)-gnad4)/S0 se17 = (S0-Slab*(gon5*f17+goff5*a17+z5*b17+aa17)-gnad5)/S0 sg17 = (S0-Slab*(gon6*f17+goff6*a17+z6*b17+aa17)-gnad6)/S0 221 * sh17 = (S0-Slab*(gon7*f17+goff7*a17+z7*b17+aa17)-gnad7)/S0 xa17 = (ft17t1-sa17)**2/ft17t1e**2 xb17 = (ft17t2-sb17)**2/ft17t2e**2 xc17 = (ft17t3-sc17)**2/ft17t3e**2 xd17 = (ft17t4-sd17)**2/ft17t4e**2 xe17 = (ft17t5-se17)**2/ft17t5e**2 xg17 = (ft17t6-sg17)**2/ft17t6e**2 xh17 = (ft17t7-sh17)**2/ft17t7e**2 f19 = 0.00 do while (f19 <= 0.14) a18 = f17+f19 b18 = f16+f20 aa18 = 1-f16-f17-f18-f19-f20 sa18 = (S0-Slab*(gon1*f18+goff1*a18+z1*b18+aa18)-gnad1)/S0 sb18 = (S0-Slab*(gon2*f18+goff2*a18+z2*b18+aa18)-gnad2)/S0 sc18 = (S0-Slab*(gon3*f18+goff3*a18+z3*b18+aa18)-gnad3)/S0 sd18 = (S0-Slab*(gon4*f18+goff4*a18+z4*b18+aa18)-gnad4)/S0 se18 = (S0-Slab*(gon5*f18+goff5*a18+z5*b18+aa18)-gnad5)/S0 sg18 = (S0-Slab*(gon6*f18+goff6*a18+z6*b18+aa18)-gnad6)/S0 sh18 = (S0-Slab*(gon7*f18+goff7*a18+z7*b18+aa18)-gnad7)/S0 xa18 = (ft18t1-sa18)**2/ft18t1e**2 xb18 = (ft18t2-sb18)**2/ft18t2e**2 xc18 = (ft18t3-sc18)**2/ft18t3e**2 xd18 = (ft18t4-sd18)**2/ft18t4e**2 xe18 = (ft18t5-se18)**2/ft18t5e**2 xg18 = (ft18t6-sg18)**2/ft18t6e**2 xh18 = (ft18t7-sh18)**2/ft18t7e**2 f20 = 0.24 *Dirk * f20 = 0.09+0.01*i20 do while (f20 <= 0.49) a19 = f18+f20 b19 = f17+f21 aa19 = 1-f17-f18-f19-f20-f21 sa19 = (S0-Slab*(gon1*f19+goff1*a19+z1*b19+aa19)-gnad1)/S0 sb19 = (S0-Slab*(gon2*f19+goff2*a19+z2*b19+aa19)-gnad2)/S0 sc19 = (S0-Slab*(gon3*f19+goff3*a19+z3*b19+aa19)-gnad3)/S0 sd19 = (S0-Slab*(gon4*f19+goff4*a19+z4*b19+aa19)-gnad4)/S0 se19 = (S0-Slab*(gon5*f19+goff5*a19+z5*b19+aa19)-gnad5)/S0 sg19 = (S0-Slab*(gon6*f19+goff6*a19+z6*b19+aa19)-gnad6)/S0 sh19 = (S0-Slab*(gon7*f19+goff7*a19+z7*b19+aa19)-gnad7)/S0 xa19 = (ft19t1-sa19)**2/ft19t1e**2 xb19 = (ft19t2-sb19)**2/ft19t2e**2 xc19 = (ft19t3-sc19)**2/ft19t3e**2 xd19 = (ft19t4-sd19)**2/ft19t4e**2 xe19 = (ft19t5-se19)**2/ft19t5e**2 xg19 = (ft19t6-sg19)**2/ft19t6e**2 xh19 = (ft19t7-sh19)**2/ft19t7e**2 f21 = 0.00 do while (f21 <= 0.04) a20 = f19+f21 b20 = f18+f22 aa20 = 1-f18-f19-f20-f21-f22 sa20 = (S0-Slab*(gon1*f20+goff1*a20+z1*b20+aa20)-gnad1)/S0 sb20 = (S0-Slab*(gon2*f20+goff2*a20+z2*b20+aa20)-gnad2)/S0 sc20 = (S0-Slab*(gon3*f20+goff3*a20+z3*b20+aa20)-gnad3)/S0 222 sd20 se20 sg20 sh20 = (S0-Slab*(gon4*f20+goff4*a20+z4*b20+aa20)-gnad4)/S0 = (S0-Slab*(gon5*f20+goff5*a20+z5*b20+aa20)-gnad5)/S0 = (S0-Slab*(gon6*f20+goff6*a20+z6*b20+aa20)-gnad6)/S0 = (S0-Slab*(gon7*f20+goff7*a20+z7*b20+aa20)-gnad7)/S0 xa20 = (ft20t1-sa20)**2/ft20t1e**2 xb20 = (ft20t2-sb20)**2/ft20t2e**2 xc20 = (ft20t3-sc20)**2/ft20t3e**2 xd20 = (ft20t4-sd20)**2/ft20t4e**2 xe20 = (ft20t5-se20)**2/ft20t5e**2 xg20 = (ft20t6-sg20)**2/ft20t6e**2 xh20 = (ft20t7-sh20)**2/ft20t7e**2 f22 = 0.00 do while (f22 <= 0.04) a21 = f20+f22 b21 = f19+f23 aa21 = 1-f19-f20-f21-f22-f23 sa21 = (S0-Slab*(gon1*f21+goff1*a21+z1*b21+aa21)-gnad1)/S0 sb21 = (S0-Slab*(gon2*f21+goff2*a21+z2*b21+aa21)-gnad2)/S0 sc21 = (S0-Slab*(gon3*f21+goff3*a21+z3*b21+aa21)-gnad3)/S0 sd21 = (S0-Slab*(gon4*f21+goff4*a21+z4*b21+aa21)-gnad4)/S0 se21 = (S0-Slab*(gon5*f21+goff5*a21+z5*b21+aa21)-gnad5)/S0 sg21 = (S0-Slab*(gon6*f21+goff6*a21+z6*b21+aa21)-gnad6)/S0 sh21 = (S0-Slab*(gon7*f21+goff7*a21+z7*b21+aa21)-gnad7)/S0 xa21 = (ft21t1-sa21)**2/ft21t1e**2 xb21 = (ft21t2-sb21)**2/ft21t2e**2 xc21 = (ft21t3-sc21)**2/ft21t3e**2 xd21 = (ft21t4-sd21)**2/ft21t4e**2 xe21 = (ft21t5-se21)**2/ft21t5e**2 xg21 = (ft21t6-sg21)**2/ft21t6e**2 xh21 = (ft21t7-sh21)**2/ft21t7e**2 f23 = 0.03 do while (f23 <= 0.05) a22 = f21+f23 b22 = f20+f24 aa22 = 1-f20-f21-f22-f23-f24 sa22 = (S0-Slab*(gon1*f22+goff1*a22+z1*b22+aa22)-gnad1)/S0 sb22 = (S0-Slab*(gon2*f22+goff2*a22+z2*b22+aa22)-gnad2)/S0 sc22 = (S0-Slab*(gon3*f22+goff3*a22+z3*b22+aa22)-gnad3)/S0 sd22 = (S0-Slab*(gon4*f22+goff4*a22+z4*b22+aa22)-gnad4)/S0 se22 = (S0-Slab*(gon5*f22+goff5*a22+z5*b22+aa22)-gnad5)/S0 sg22 = (S0-Slab*(gon6*f22+goff6*a22+z6*b22+aa22)-gnad6)/S0 sh22 = (S0-Slab*(gon7*f22+goff7*a22+z7*b22+aa22)-gnad7)/S0 xa22 = (ft22t1-sa22)**2/ft22t1e**2 xb22 = (ft22t2-sb22)**2/ft22t2e**2 xc22 = (ft22t3-sc22)**2/ft22t3e**2 xd22 = (ft22t4-sd22)**2/ft22t4e**2 xe22 = (ft22t5-se22)**2/ft22t5e**2 xg22 = (ft22t6-sg22)**2/ft22t6e**2 xh22 = (ft22t7-sh22)**2/ft22t7e**2 f24 = 0.00 do while (f24 <= 0.00) a23 = f22+f24 b23 = f21+f25 aa23 = 1-f21-f22-f23-f24-f25 sa23 = (S0-Slab*(gon1*f23+goff1*a23+z1*b23+aa23)-gnad1)/S0 sb23 = (S0-Slab*(gon2*f23+goff2*a23+z2*b23+aa23)-gnad2)/S0 sc23 = (S0-Slab*(gon3*f23+goff3*a23+z3*b23+aa23)-gnad3)/S0 223 sd23 se23 sg23 sh23 sa24 sb24 sc24 sd24 se24 sg24 sh24 = (S0-Slab*(gon4*f23+goff4*a23+z4*b23+aa23)-gnad4)/S0 = (S0-Slab*(gon5*f23+goff5*a23+z5*b23+aa23)-gnad5)/S0 = (S0-Slab*(gon6*f23+goff6*a23+z6*b23+aa23)-gnad6)/S0 = (S0-Slab*(gon7*f23+goff7*a23+z7*b23+aa23)-gnad7)/S0 a24 = f23+f25 b24 = f22+f26 aa24 = 1-f22-f23-f24-f25-f26 = (S0-Slab*(gon1*f24+goff1*a24+z1*b24+aa24)-gnad1)/S0 = (S0-Slab*(gon2*f24+goff2*a24+z2*b24+aa24)-gnad2)/S0 = (S0-Slab*(gon3*f24+goff3*a24+z3*b24+aa24)-gnad3)/S0 = (S0-Slab*(gon4*f24+goff4*a24+z4*b24+aa24)-gnad4)/S0 = (S0-Slab*(gon5*f24+goff5*a24+z5*b24+aa24)-gnad5)/S0 = (S0-Slab*(gon6*f24+goff6*a24+z6*b24+aa24)-gnad6)/S0 = (S0-Slab*(gon7*f24+goff7*a24+z7*b24+aa24)-gnad7)/S0 xa23 xb23 xc23 xd23 xe23 xg23 xh23 (ft23t1-sa23)**2/ft23t1e**2 (ft23t2-sb23)**2/ft23t2e**2 (ft23t3-sc23)**2/ft23t3e**2 (ft23t4-sd23)**2/ft23t4e**2 (ft23t5-se23)**2/ft23t5e**2 (ft23t6-sg23)**2/ft23t6e**2 (ft23t7-sh23)**2/ft23t7e**2 xa24 xb24 xc24 xd24 xe24 xg24 xh24 & & & & & & & & & & & & & & & & = = = = = = = = = = = = = = (ft24t1-sa24)**2/ft24t1e**2 (ft24t2-sb24)**2/ft24t2e**2 (ft24t3-sc24)**2/ft24t3e**2 (ft24t4-sd24)**2/ft24t4e**2 (ft24t5-se24)**2/ft24t5e**2 (ft24t6-sg24)**2/ft24t6e**2 (ft24t7-sh24)**2/ft24t7e**2 kaisq = xa8+xb8+xc8+xd8+xe8+xg8+xh8+ xa9+xb9+xc9+xd9+xe9+xg9+xh9+ xa10+xb10+xc10+xd10+xe10+xg10+xh10+ xa11+xb11+xc11+xd11+xe11+xg11+xh11+ xa12+xb12+xc12+xd12+xe12+xg12+xh12+ xa13+xb13+xc13+xd13+xe13+xg13+xh13+ xa14+xb14+xc14+xd14+xe14+xg14+xh14+ xa15+xb15+xc15+xd15+xe15+xg15+xh15+ xa16+xb16+xc16+xd16+xe16+xg16+xh16+ xa17+xb17+xc17+xd17+xe17+xg17+xh17+ xa18+xb18+xc18+xd18+xe18+xg18+xh18+ xa19+xb19+xc19+xd19+xe19+xg19+xh19+ xa20+xb20+xc20+xd20+xe20+xg20+xh20+ xa21+xb21+xc21+xd21+xe21+xg21+xh21+ xa22+xb22+xc22+xd22+xe22+xg22+xh22+ xa23+xb23+xc23+xd23+xe23+xg23+xh23+ xa24+xb24+xc24+xd24+xe24+xg24+xh24 IF (kaisq < bestkaisq) THEN bestkaisq = kaisq bf8 = f8 bf9 = f9 bf10 = f10 bf11 = f11 224 bf12 bf13 bf14 bf15 bf16 bf17 bf18 bf19 bf20 bf21 bf22 bf23 bf24 endif f24 enddo f23 enddo f22 enddo f21 enddo = = = = = = = = = = = = = f12 f13 f14 f15 f16 f17 f18 f19 f20 f21 f22 f23 f24 = f24 + 0.01 = f23 + 0.01 = f22 + 0.01 = f21 + 0.01 *Dirk * * f20 = f20 + 0.01 enddo f19 = f19 + 0.01 enddo f18 = f18 + 0.01 enddo f17 = f17 + 0.01 enddo f16 = f16 + 0.01 enddo f15 = f15 + 0.01 enddo f14 = f14 + 0.01 enddo f13 = f13 + 0.01 enddo *Dirk * * f12 = f12 + 0.01 enddo f11 = f11 + 0.01 enddo f10 = f10 + 0.01 enddo f9 = f9 + 0.01 enddo f8 = f8 + 0.01 enddo * Dirk * OPEN(UNIT = 12, FILE = 'values5', STATUS = 'NEW') 225 * & & & & & WRITE(12,*) bestkaisq, WRITE(*,*) bestkaisq, bf8, bf9, bf10, bf11, bf12, bf13, bf14, bf15, bf16, bf17, bf18, bf19, bf20, bf21, bf22, bf23, bf24 end 3.5 V2E-HFP 3 Registry Fitting qsub Script, “x2_V2E” #!/bin/bash #PBS -l nodes=1:ppn=1,walltime=144:00:00,mem=2gb,feature=gbe #PBS -j oe #PBS -t 0-137 #change to the original working directory cd ${PBS_O_WORKDIR} # Define number of loops for i16 cols=22 # Define number of loops for i13 (not used) rows=5 # Number of jobs = cols*rows (i.e. -t 0-137) #math to figure out the variable values based on the array id i16=`echo "${PBS_ARRAYID} % ( ${cols} + 1 )" | bc` i13=`echo "${PBS_ARRAYID} / ( ${cols} + 1 )" | bc` #display the command we are going to run echo "./x2 ${i16} ${i13} > ${i16}_${i13}.txt" #run the command with the input variables ./x2 ${i16} ${i13} > ${i16}_${i13}.txt # Calculate the runtiem for the job qstat -f ${PBS_JOBID} 3.6 V2E-HFP 3 Registry Fitting Main Script, “V2E.f” * This is a comment * This program was written by Scott Schmick 030111 * * t values (1 = a = 482, 2 = b = 402, etc.) real f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 real f17 f18 f19 f20 f21 f22 f23 f24 f25 real sa8 sb8 sc8 sd8 se8 sg8 sh8 real sa9 sb9 sc9 sd9 se9 sg9 sh9 real sa10 sb10 sc10 sd10 se10 sg10 sh10 real sa11 sb11 sc11 sd11 se11 sg11 sh11 real sa12 sb12 sc12 sd12 se12 sg12 sh12 real sa13 sb13 sc13 sd13 se13 sg13 sh13 real sa14 sb14 sc14 sd14 se14 sg14 sh14 226 * real real real real real real real real real real sa15 sa16 sa17 sa18 sa19 sa20 sa21 sa22 sa23 sa24 sb15 sb16 sb17 sb18 sb19 sb20 sb21 sb22 sb23 sb24 sc15 sc16 sc17 sc18 sc19 sc20 sc21 sc22 sc23 sc24 sd15 sd16 sd17 sd18 sd19 sd20 sd21 sd22 sd23 sd24 real real real real real real real real real real real real real real real real real real S0 xa8 xb8 xc8 xd8 xe8 xa9 xb9 xc9 xd9 xe9 xa10 xb10 xc10 xd10 xa11 xb11 xc11 xd11 xa12 xb12 xc12 xd12 xa13 xb13 xc13 xd13 xa14 xb14 xc14 xd14 xa15 xb15 xc15 xd15 xa16 xb16 xc16 xd16 xa17 xb17 xc17 xd17 xa18 xb18 xc18 xd18 xa19 xb19 xc19 xd19 xa20 xb20 xc20 xd20 xa21 xb21 xc21 xd21 xa22 xb22 xc22 xd22 xa23 xb23 xc23 xd23 xa24 xb24 xc24 xd24 se15 se16 se17 se18 se19 se20 se21 se22 se23 se24 sg15 sg16 sg17 sg18 sg19 sg20 sg21 sg22 sg23 sg24 sh15 sh16 sh17 sh18 sh19 sh20 sh21 sh22 sh23 sh24 xg8 xh8 xg9 xh9 xe10 xg10 xe11 xg11 xe12 xg12 xe13 xg13 xe14 xg14 xe15 xg15 xe16 xg16 xe17 xg17 xe18 xg18 xe19 xg19 xe20 xg20 xe21 xg21 xe22 xg22 xe23 xg23 xe24 xg24 xh10 xh11 xh12 xh13 xh14 xh15 xh16 xh17 xh18 xh19 xh20 xh21 xh22 xh23 xh24 real goff1 goff2 goff3 goff4 goff5 goff6 goff7 real gon1 gon2 gon3 gon4 gon5 gon6 gon7 real gnad1 gnad2 gnad3 gnad4 gnad5 gnad6 gnad7 real ft8t1 real ft8t2 real ft8t3 real ft8t4 real ft8t5 real ft8t6 real ft8t7 real ft8t1e real ft8t2e real ft8t3e real ft8t4e real ft8t5e real ft8t6e real ft8t7e real ft9t1 real ft9t2 real ft9t3 real ft9t4 real ft9t5 real ft9t6 real ft9t7 real ft9t1e real ft9t2e real ft9t3e 227 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft9t4e ft9t5e ft9t6e ft9t7e ft10t1 ft10t2 ft10t3 ft10t4 ft10t5 ft10t6 ft10t7 ft10t1e ft10t2e ft10t3e ft10t4e ft10t5e ft10t6e ft10t7e ft11t1 ft11t2 ft11t3 ft11t4 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e 228 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e 229 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e 230 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e ft23t1 ft23t2 ft23t3 ft23t4 ft23t5 ft23t6 ft23t7 ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 ft24t2 ft24t3 ft24t4 ft24t5 ft24t6 ft24t7 ft24t1e ft24t2e ft24t3e ft24t4e ft24t5e ft24t6e ft24t7e kaisq bestkaisq bf8 bf9 bf10 bf11 bf12 bf13 bf14 bf15 bf16 bf17 bf18 bf19 231 real real real real real real real real bf20 bf21 bf22 bf23 bf24 a8 aa8 a9 aa9 a10 aa10 a11 aa11 a12 aa12 a13 aa13 a14 aa14 a15 aa15 a16 aa16 a17 aa17 a18 aa18 a19 aa19 a20 aa20 a21 aa21 a22 aa22 a23 aa23 a24 aa24 * DIRK integer i16 i13 character inputarg*128 CALL getarg(1,inputarg) read(inputarg,*) i16 CALL getarg(2,inputarg) read(inputarg,*) i13 goff1 = 0.45011191 goff2 = 0.581357776 goff3 = 0.710329562 goff4 = 0.826254485 goff5 = 0.918588976 goff6 = 0.978470252 goff7 = 0.998385086 gon1 = 0.093897695 gon2 = 0.118554369 gon3 = 0.196398598 gon4 = 0.378629871 gon5 = 0.645290442 gon6 = 0.893821899 gon7 = 0.991709633 gnad1 = 0.281406262 gnad2 = 0.283440096 gnad3 = 0.298896559 gnad4 = 0.309714762 gnad5 = 0.320805841 gnad6 = 0.325732761 gnad7 = 0.342143946 ft8t1 = 0.05455 ft8t2 = 0.05178 ft8t3 = 0.05255 ft8t4 = 0.01784 ft8t5 = 0.02752 ft8t6 = 0.02554 ft8t7 = 0.00864 ft8t1e = 0.01171 ft8t2e = 0.01136 ft8t3e = 0.01377 ft8t4e = 0.01 ft8t5e = 0.01 ft8t6e = 0.01 ft8t7e = 0.01 ft9t1 = 0.0615 ft9t2 = 0.05812 ft9t3 = 0.04719 232 ft9t4 ft9t5 ft9t6 ft9t7 ft9t1e ft9t2e ft9t3e ft9t4e ft9t5e ft9t6e ft9t7e ft10t1 ft10t2 ft10t3 ft10t4 ft10t5 ft10t6 ft10t7 ft10t1e ft10t2e ft10t3e ft10t4e ft10t5e ft10t6e ft10t7e ft11t1 ft11t2 ft11t3 ft11t4 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.03191 0.02722 0.00512 0.01355 0.01671 0.01235 0.01 0.01 0.01 0.01 0.01 0.06781 0.05011 0.04724 0.0408 0.0239 0.01835 0.02391 0.01606 0.01294 0.01184 0.01233 0.01013 0.01 0.0104 0.06249 0.05293 0.04992 0.03957 0.031 0.00575 -0.01026 0.01341 0.01711 0.01107 0.01247 0.01 0.0106 0.01 0.08531 0.08566 0.0591 0.03005 0.02208 0.0117 6.49E-04 0.01586 0.01 0.01 0.01418 0.01 0.01039 0.0106 0.12877 0.07773 0.06296 0.04703 233 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.03825 0.04439 0.0017 0.02489 0.0254 0.02 0.0155 0.01 0.01673 0.01285 0.15352 0.11403 0.07306 0.06083 0.03403 0.04299 0.00683 0.0119 0.01368 0.01192 0.01 0.01019 0.01263 0.01659 0.19465 0.1749 0.14289 0.09534 0.04554 0.03886 0.00352 0.01573 0.01096 0.01 0.01 0.01565 0.01 0.0131 0.28671 0.24526 0.19745 0.12897 0.07304 0.03423 0.01909 0.02292 0.01424 0.01478 0.01189 0.01 0.01226 0.01049 0.31006 0.27143 0.23816 0.17499 0.07795 234 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.0353 0.01897 0.02074 0.01444 0.01598 0.01853 0.01963 0.0198 0.01164 0.30283 0.3016 0.25411 0.19389 0.11342 0.05735 0.02475 0.02213 0.01776 0.01885 0.0122 0.01153 0.01299 0.01682 0.34606 0.27994 0.21297 0.1443 0.06939 0.04606 0.01298 0.01175 0.01249 0.01 0.01044 0.01414 0.01 0.0107 0.39766 0.37867 0.32995 0.26239 0.14554 0.05559 0.00911 0.01651 0.01394 0.01 0.01 0.01 0.01 0.01 0.25706 0.19786 0.17866 0.1297 0.0752 0.04328 235 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e ft23t1 ft23t2 ft23t3 ft23t4 ft23t5 ft23t6 ft23t7 ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 ft24t2 ft24t3 ft24t4 ft24t5 ft24t6 ft24t7 ft24t1e ft24t2e ft24t3e ft24t4e ft24t5e ft24t6e ft24t7e = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.00665 0.01116 0.01489 0.01 0.01 0.01 0.01 0.01 0.15497 0.10269 0.07505 0.05987 0.02889 0.02218 0.0051 0.01455 0.01357 0.01179 0.01182 0.01011 0.01134 0.01 0.11272 0.10263 0.08713 0.07678 0.04061 0.01599 0.01091 0.01183 0.01 0.01055 0.01 0.01 0.01081 0.01 0.10148 0.09025 0.05784 0.05364 0.029 0.01416 4.40E-04 0.01731 0.01521 0.01928 0.01275 0.01 0.01139 0.01256 bestkaisq = 100000000 236 S0 = 1.32 Slab = 0.9625275 f7 = 0.00 f25 = 0.00 f8 = 0.00 do while (f8 <= 0.00) f9 = 0.00 do while (f9 <= 0.00) f10 = 0.00 a8 = f7+f9 aa8 = 1-f7-f8-f9 sa8 = (S0-Slab*(gon1*f8+goff1*(a8)+aa8)-gnad1)/S0 sb8 = (S0-Slab*(gon2*f8+goff2*(a8)+aa8)-gnad2)/S0 sc8 = (S0-Slab*(gon3*f8+goff3*(a8)+aa8)-gnad3)/S0 sd8 = (S0-Slab*(gon4*f8+goff4*(a8)+aa8)-gnad4)/S0 se8 = (S0-Slab*(gon5*f8+goff5*(a8)+aa8)-gnad5)/S0 sg8 = (S0-Slab*(gon6*f8+goff6*(a8)+aa8)-gnad6)/S0 sh8 = (S0-Slab*(gon7*f8+goff7*(a8)+aa8)-gnad7)/S0 xa8 = (ft8t1-sa8)**2/ft8t1e**2 xb8 = (ft8t2-sb8)**2/ft8t2e**2 xc8 = (ft8t3-sc8)**2/ft8t3e**2 xd8 = (ft8t4-sd8)**2/ft8t4e**2 xe8 = (ft8t5-se8)**2/ft8t5e**2 xg8 = (ft8t6-sg8)**2/ft8t6e**2 xh8 = (ft8t7-sh8)**2/ft8t7e**2 do while (f10 <= 0.00) a9 = f8+f10 aa9 = 1-f8-f9-f10 sa9 = (S0-Slab*(gon1*f9+goff1*(a9)+aa9)-gnad1)/S0 sb9 = (S0-Slab*(gon2*f9+goff2*(a9)+aa9)-gnad2)/S0 sc9 = (S0-Slab*(gon3*f9+goff3*(a9)+aa9)-gnad3)/S0 sd9 = (S0-Slab*(gon4*f9+goff4*(a9)+aa9)-gnad4)/S0 se9 = (S0-Slab*(gon5*f9+goff5*(a9)+aa9)-gnad5)/S0 sg9 = (S0-Slab*(gon6*f9+goff6*(a9)+aa9)-gnad6)/S0 sh9 = (S0-Slab*(gon7*f9+goff7*(a9)+aa9)-gnad7)/S0 xa9 = (ft9t1-sa9)**2/ft9t1e**2 xb9 = (ft9t2-sb9)**2/ft9t2e**2 xc9 = (ft9t3-sc9)**2/ft9t3e**2 xd9 = (ft9t4-sd9)**2/ft9t4e**2 xe9 = (ft9t5-se9)**2/ft9t5e**2 xg9 = (ft9t6-sg9)**2/ft9t6e**2 xh9 = (ft9t7-sh9)**2/ft9t7e**2 f11 = 0.00 do while (f11 <= 0.00) a10 = f9+f11 aa10 = 1-f9-f10-f11 sa10 = (S0-Slab*(gon1*f10+goff1*(a10)+aa10)-gnad1)/S0 sb10 = (S0-Slab*(gon2*f10+goff2*(a10)+aa10)-gnad2)/S0 sc10 = (S0-Slab*(gon3*f10+goff3*(a10)+aa10)-gnad3)/S0 sd10 = (S0-Slab*(gon4*f10+goff4*(a10)+aa10)-gnad4)/S0 se10 = (S0-Slab*(gon5*f10+goff5*(a10)+aa10)-gnad5)/S0 sg10 = (S0-Slab*(gon6*f10+goff6*(a10)+aa10)-gnad6)/S0 sh10 = (S0-Slab*(gon7*f10+goff7*(a10)+aa10)-gnad7)/S0 xa10 = (ft10t1-sa10)**2/ft10t1e**2 xb10 = (ft10t2-sb10)**2/ft10t2e**2 237 xc10 = (ft10t3-sc10)**2/ft10t3e**2 xd10 = (ft10t4-sd10)**2/ft10t4e**2 xe10 = (ft10t5-se10)**2/ft10t5e**2 xg10 = (ft10t6-sg10)**2/ft10t6e**2 xh10 = (ft10t7-sh10)**2/ft10t7e**2 f12 = 0.00 do while (f12 <= 0.03) a11 = f10+f12 aa11 = 1-f10-f11-f12 sa11 = (S0-Slab*(gon1*f11+goff1*(a11)+aa11)-gnad1)/S0 sb11 = (S0-Slab*(gon2*f11+goff2*(a11)+aa11)-gnad2)/S0 sc11 = (S0-Slab*(gon3*f11+goff3*(a11)+aa11)-gnad3)/S0 sd11 = (S0-Slab*(gon4*f11+goff4*(a11)+aa11)-gnad4)/S0 se11 = (S0-Slab*(gon5*f11+goff5*(a11)+aa11)-gnad5)/S0 sg11 = (S0-Slab*(gon6*f11+goff6*(a11)+aa11)-gnad6)/S0 sh11 = (S0-Slab*(gon7*f11+goff7*(a11)+aa11)-gnad7)/S0 xa11 = (ft11t1-sa11)**2/ft11t1e**2 xb11 = (ft11t2-sb11)**2/ft11t2e**2 xc11 = (ft11t3-sc11)**2/ft11t3e**2 xd11 = (ft11t4-sd11)**2/ft11t4e**2 xe11 = (ft11t5-se11)**2/ft11t5e**2 xg11 = (ft11t6-sg11)**2/ft11t6e**2 xh11 = (ft11t7-sh11)**2/ft11t7e**2 * f13 = 0.08 * DIRK f13 = 0.00+0.01*i13 * do while (f13 <= 0.22) a12 = f11+f13 aa12 = 1-f11-f12-f13 sa12 = (S0-Slab*(gon1*f12+goff1*(a12)+aa12)-gnad1)/S0 sb12 = (S0-Slab*(gon2*f12+goff2*(a12)+aa12)-gnad2)/S0 sc12 = (S0-Slab*(gon3*f12+goff3*(a12)+aa12)-gnad3)/S0 sd12 = (S0-Slab*(gon4*f12+goff4*(a12)+aa12)-gnad4)/S0 se12 = (S0-Slab*(gon5*f12+goff5*(a12)+aa12)-gnad5)/S0 sg12 = (S0-Slab*(gon6*f12+goff6*(a12)+aa12)-gnad6)/S0 sh12 = (S0-Slab*(gon7*f12+goff7*(a12)+aa12)-gnad7)/S0 xa12 = (ft12t1-sa12)**2/ft12t1e**2 xb12 = (ft12t2-sb12)**2/ft12t2e**2 xc12 = (ft12t3-sc12)**2/ft12t3e**2 xd12 = (ft12t4-sd12)**2/ft12t4e**2 xe12 = (ft12t5-se12)**2/ft12t5e**2 xg12 = (ft12t6-sg12)**2/ft12t6e**2 xh12 = (ft12t7-sh12)**2/ft12t7e**2 f14 = 0.00 do while (f14 <= 0.09) a13 = f12+f14 aa13 = 1-f12-f13-f14 sa13 = (S0-Slab*(gon1*f13+goff1*(a13)+aa13)-gnad1)/S0 sb13 = (S0-Slab*(gon2*f13+goff2*(a13)+aa13)-gnad2)/S0 sc13 = (S0-Slab*(gon3*f13+goff3*(a13)+aa13)-gnad3)/S0 sd13 = (S0-Slab*(gon4*f13+goff4*(a13)+aa13)-gnad4)/S0 se13 = (S0-Slab*(gon5*f13+goff5*(a13)+aa13)-gnad5)/S0 sg13 = (S0-Slab*(gon6*f13+goff6*(a13)+aa13)-gnad6)/S0 sh13 = (S0-Slab*(gon7*f13+goff7*(a13)+aa13)-gnad7)/S0 xa13 = (ft13t1-sa13)**2/ft13t1e**2 xb13 = (ft13t2-sb13)**2/ft13t2e**2 xc13 = (ft13t3-sc13)**2/ft13t3e**2 238 xd13 = (ft13t4-sd13)**2/ft13t4e**2 xe13 = (ft13t5-se13)**2/ft13t5e**2 xg13 = (ft13t6-sg13)**2/ft13t6e**2 xh13 = (ft13t7-sh13)**2/ft13t7e**2 f15 = 0.00 do while (f15 <= 0.17) a14 = f13+f15 aa14 = 1-f13-f14-f15 sa14 = (S0-Slab*(gon1*f14+goff1*(a14)+aa14)-gnad1)/S0 sb14 = (S0-Slab*(gon2*f14+goff2*(a14)+aa14)-gnad2)/S0 sc14 = (S0-Slab*(gon3*f14+goff3*(a14)+aa14)-gnad3)/S0 sd14 = (S0-Slab*(gon4*f14+goff4*(a14)+aa14)-gnad4)/S0 se14 = (S0-Slab*(gon5*f14+goff5*(a14)+aa14)-gnad5)/S0 sg14 = (S0-Slab*(gon6*f14+goff6*(a14)+aa14)-gnad6)/S0 sh14 = (S0-Slab*(gon7*f14+goff7*(a14)+aa14)-gnad7)/S0 xa14 = (ft14t1-sa14)**2/ft14t1e**2 xb14 = (ft14t2-sb14)**2/ft14t2e**2 xc14 = (ft14t3-sc14)**2/ft14t3e**2 xd14 = (ft14t4-sd14)**2/ft14t4e**2 xe14 = (ft14t5-se14)**2/ft14t5e**2 xg14 = (ft14t6-sg14)**2/ft14t6e**2 xh14 = (ft14t7-sh14)**2/ft14t7e**2 * DIRK f16 = 0.04+0.01*i16 * f16 = 0.00 * do while (f16 <= 0.03) a15 = f14+f16 aa15 = 1-f14-f15-f16 sa15 = (S0-Slab*(gon1*f15+goff1*(a15)+aa15)-gnad1)/S0 sb15 = (S0-Slab*(gon2*f15+goff2*(a15)+aa15)-gnad2)/S0 sc15 = (S0-Slab*(gon3*f15+goff3*(a15)+aa15)-gnad3)/S0 sd15 = (S0-Slab*(gon4*f15+goff4*(a15)+aa15)-gnad4)/S0 se15 = (S0-Slab*(gon5*f15+goff5*(a15)+aa15)-gnad5)/S0 sg15 = (S0-Slab*(gon6*f15+goff6*(a15)+aa15)-gnad6)/S0 sh15 = (S0-Slab*(gon7*f15+goff7*(a15)+aa15)-gnad7)/S0 xa15 = (ft15t1-sa15)**2/ft15t1e**2 xb15 = (ft15t2-sb15)**2/ft15t2e**2 xc15 = (ft15t3-sc15)**2/ft15t3e**2 xd15 = (ft15t4-sd15)**2/ft15t4e**2 xe15 = (ft15t5-se15)**2/ft15t5e**2 xg15 = (ft15t6-sg15)**2/ft15t6e**2 xh15 = (ft15t7-sh15)**2/ft15t7e**2 f17 = 0.05 do while (f17 <= 0.33) a16 = f15+f17 aa16 = 1-f15-f16-f17 sa16 = (S0-Slab*(gon1*f16+goff1*(a16)+aa16)-gnad1)/S0 sb16 = (S0-Slab*(gon2*f16+goff2*(a16)+aa16)-gnad2)/S0 sc16 = (S0-Slab*(gon3*f16+goff3*(a16)+aa16)-gnad3)/S0 sd16 = (S0-Slab*(gon4*f16+goff4*(a16)+aa16)-gnad4)/S0 se16 = (S0-Slab*(gon5*f16+goff5*(a16)+aa16)-gnad5)/S0 sg16 = (S0-Slab*(gon6*f16+goff6*(a16)+aa16)-gnad6)/S0 sh16 = (S0-Slab*(gon7*f16+goff7*(a16)+aa16)-gnad7)/S0 xa16 = (ft16t1-sa16)**2/ft16t1e**2 xb16 = (ft16t2-sb16)**2/ft16t2e**2 xc16 = (ft16t3-sc16)**2/ft16t3e**2 xd16 = (ft16t4-sd16)**2/ft16t4e**2 239 xe16 = (ft16t5-se16)**2/ft16t5e**2 xg16 = (ft16t6-sg16)**2/ft16t6e**2 xh16 = (ft16t7-sh16)**2/ft16t7e**2 f18 = 0.10 do while (f18 <= 0.36) a17 = f16+f18 aa17 = 1-f16-f17-f18 sa17 = (S0-Slab*(gon1*f17+goff1*(a17)+aa17)-gnad1)/S0 sb17 = (S0-Slab*(gon2*f17+goff2*(a17)+aa17)-gnad2)/S0 sc17 = (S0-Slab*(gon3*f17+goff3*(a17)+aa17)-gnad3)/S0 sd17 = (S0-Slab*(gon4*f17+goff4*(a17)+aa17)-gnad4)/S0 se17 = (S0-Slab*(gon5*f17+goff5*(a17)+aa17)-gnad5)/S0 sg17 = (S0-Slab*(gon6*f17+goff6*(a17)+aa17)-gnad6)/S0 sh17 = (S0-Slab*(gon7*f17+goff7*(a17)+aa17)-gnad7)/S0 xa17 = (ft17t1-sa17)**2/ft17t1e**2 xb17 = (ft17t2-sb17)**2/ft17t2e**2 xc17 = (ft17t3-sc17)**2/ft17t3e**2 xd17 = (ft17t4-sd17)**2/ft17t4e**2 xe17 = (ft17t5-se17)**2/ft17t5e**2 xg17 = (ft17t6-sg17)**2/ft17t6e**2 xh17 = (ft17t7-sh17)**2/ft17t7e**2 f19 = 0.00 do while (f19 <= 0.33) a18 = f17+f19 aa18 = 1-f17-f18-f19 sa18 = (S0-Slab*(gon1*f18+goff1*(a18)+aa18)-gnad1)/S0 sb18 = (S0-Slab*(gon2*f18+goff2*(a18)+aa18)-gnad2)/S0 sc18 = (S0-Slab*(gon3*f18+goff3*(a18)+aa18)-gnad3)/S0 sd18 = (S0-Slab*(gon4*f18+goff4*(a18)+aa18)-gnad4)/S0 se18 = (S0-Slab*(gon5*f18+goff5*(a18)+aa18)-gnad5)/S0 sg18 = (S0-Slab*(gon6*f18+goff6*(a18)+aa18)-gnad6)/S0 sh18 = (S0-Slab*(gon7*f18+goff7*(a18)+aa18)-gnad7)/S0 xa18 = (ft18t1-sa18)**2/ft18t1e**2 xb18 = (ft18t2-sb18)**2/ft18t2e**2 xc18 = (ft18t3-sc18)**2/ft18t3e**2 xd18 = (ft18t4-sd18)**2/ft18t4e**2 xe18 = (ft18t5-se18)**2/ft18t5e**2 xg18 = (ft18t6-sg18)**2/ft18t6e**2 xh18 = (ft18t7-sh18)**2/ft18t7e**2 f20 = 0.28 do while (f20 <= 0.49) a19 = f18+f20 aa19 = 1-f18-f19-f20 sa19 = (S0-Slab*(gon1*f19+goff1*(a19)+aa19)-gnad1)/S0 sb19 = (S0-Slab*(gon2*f19+goff2*(a19)+aa19)-gnad2)/S0 sc19 = (S0-Slab*(gon3*f19+goff3*(a19)+aa19)-gnad3)/S0 sd19 = (S0-Slab*(gon4*f19+goff4*(a19)+aa19)-gnad4)/S0 se19 = (S0-Slab*(gon5*f19+goff5*(a19)+aa19)-gnad5)/S0 sg19 = (S0-Slab*(gon6*f19+goff6*(a19)+aa19)-gnad6)/S0 sh19 = (S0-Slab*(gon7*f19+goff7*(a19)+aa19)-gnad7)/S0 xa19 = (ft19t1-sa19)**2/ft19t1e**2 xb19 = (ft19t2-sb19)**2/ft19t2e**2 xc19 = (ft19t3-sc19)**2/ft19t3e**2 xd19 = (ft19t4-sd19)**2/ft19t4e**2 xe19 = (ft19t5-se19)**2/ft19t5e**2 xg19 = (ft19t6-sg19)**2/ft19t6e**2 xh19 = (ft19t7-sh19)**2/ft19t7e**2 240 f21 = 0.05 do while (f21 <= 0.24) a20 = f19+f21 aa20 = 1-f19-f20-f21 sa20 = (S0-Slab*(gon1*f20+goff1*(a20)+aa20)-gnad1)/S0 sb20 = (S0-Slab*(gon2*f20+goff2*(a20)+aa20)-gnad2)/S0 sc20 = (S0-Slab*(gon3*f20+goff3*(a20)+aa20)-gnad3)/S0 sd20 = (S0-Slab*(gon4*f20+goff4*(a20)+aa20)-gnad4)/S0 se20 = (S0-Slab*(gon5*f20+goff5*(a20)+aa20)-gnad5)/S0 sg20 = (S0-Slab*(gon6*f20+goff6*(a20)+aa20)-gnad6)/S0 sh20 = (S0-Slab*(gon7*f20+goff7*(a20)+aa20)-gnad7)/S0 xa20 = (ft20t1-sa20)**2/ft20t1e**2 xb20 = (ft20t2-sb20)**2/ft20t2e**2 xc20 = (ft20t3-sc20)**2/ft20t3e**2 xd20 = (ft20t4-sd20)**2/ft20t4e**2 xe20 = (ft20t5-se20)**2/ft20t5e**2 xg20 = (ft20t6-sg20)**2/ft20t6e**2 xh20 = (ft20t7-sh20)**2/ft20t7e**2 f22 = 0.00 do while (f22 <= 0.08) a21 = f20+f22 aa21 = 1-f20-f21-f22 sa21 = (S0-Slab*(gon1*f21+goff1*(a21)+aa21)-gnad1)/S0 sb21 = (S0-Slab*(gon2*f21+goff2*(a21)+aa21)-gnad2)/S0 sc21 = (S0-Slab*(gon3*f21+goff3*(a21)+aa21)-gnad3)/S0 sd21 = (S0-Slab*(gon4*f21+goff4*(a21)+aa21)-gnad4)/S0 se21 = (S0-Slab*(gon5*f21+goff5*(a21)+aa21)-gnad5)/S0 sg21 = (S0-Slab*(gon6*f21+goff6*(a21)+aa21)-gnad6)/S0 sh21 = (S0-Slab*(gon7*f21+goff7*(a21)+aa21)-gnad7)/S0 xa21 = (ft21t1-sa21)**2/ft21t1e**2 xb21 = (ft21t2-sb21)**2/ft21t2e**2 xc21 = (ft21t3-sc21)**2/ft21t3e**2 xd21 = (ft21t4-sd21)**2/ft21t4e**2 xe21 = (ft21t5-se21)**2/ft21t5e**2 xg21 = (ft21t6-sg21)**2/ft21t6e**2 xh21 = (ft21t7-sh21)**2/ft21t7e**2 f23 = 0.03 do while (f23 <= 0.08) a22 = f21+f23 aa22 = 1-f21-f22-f23 sa22 = (S0-Slab*(gon1*f22+goff1*(a22)+aa22)-gnad1)/S0 sb22 = (S0-Slab*(gon2*f22+goff2*(a22)+aa22)-gnad2)/S0 sc22 = (S0-Slab*(gon3*f22+goff3*(a22)+aa22)-gnad3)/S0 sd22 = (S0-Slab*(gon4*f22+goff4*(a22)+aa22)-gnad4)/S0 se22 = (S0-Slab*(gon5*f22+goff5*(a22)+aa22)-gnad5)/S0 sg22 = (S0-Slab*(gon6*f22+goff6*(a22)+aa22)-gnad6)/S0 sh22 = (S0-Slab*(gon7*f22+goff7*(a22)+aa22)-gnad7)/S0 xa22 = (ft22t1-sa22)**2/ft22t1e**2 xb22 = (ft22t2-sb22)**2/ft22t2e**2 xc22 = (ft22t3-sc22)**2/ft22t3e**2 xd22 = (ft22t4-sd22)**2/ft22t4e**2 xe22 = (ft22t5-se22)**2/ft22t5e**2 xg22 = (ft22t6-sg22)**2/ft22t6e**2 xh22 = (ft22t7-sh22)**2/ft22t7e**2 f24 = 0.02 do while (f24 <= 0.04) a23 = f22+f24 241 sa23 sb23 sc23 sd23 se23 sg23 sh23 sa24 sb24 sc24 sd24 se24 sg24 sh24 aa23 = 1-f22-f23-f24 = (S0-Slab*(gon1*f23+goff1*(a23)+aa23)-gnad1)/S0 = (S0-Slab*(gon2*f23+goff2*(a23)+aa23)-gnad2)/S0 = (S0-Slab*(gon3*f23+goff3*(a23)+aa23)-gnad3)/S0 = (S0-Slab*(gon4*f23+goff4*(a23)+aa23)-gnad4)/S0 = (S0-Slab*(gon5*f23+goff5*(a23)+aa23)-gnad5)/S0 = (S0-Slab*(gon6*f23+goff6*(a23)+aa23)-gnad6)/S0 = (S0-Slab*(gon7*f23+goff7*(a23)+aa23)-gnad7)/S0 a24 = f23+f25 aa24 = 1-f23-f24-f25 = (S0-Slab*(gon1*f24+goff1*(a24)+aa24)-gnad1)/S0 = (S0-Slab*(gon2*f24+goff2*(a24)+aa24)-gnad2)/S0 = (S0-Slab*(gon3*f24+goff3*(a24)+aa24)-gnad3)/S0 = (S0-Slab*(gon4*f24+goff4*(a24)+aa24)-gnad4)/S0 = (S0-Slab*(gon5*f24+goff5*(a24)+aa24)-gnad5)/S0 = (S0-Slab*(gon6*f24+goff6*(a24)+aa24)-gnad6)/S0 = (S0-Slab*(gon7*f24+goff7*(a24)+aa24)-gnad7)/S0 xa23 xb23 xc23 xd23 xe23 xg23 xh23 (ft23t1-sa23)**2/ft23t1e**2 (ft23t2-sb23)**2/ft23t2e**2 (ft23t3-sc23)**2/ft23t3e**2 (ft23t4-sd23)**2/ft23t4e**2 (ft23t5-se23)**2/ft23t5e**2 (ft23t6-sg23)**2/ft23t6e**2 (ft23t7-sh23)**2/ft23t7e**2 xa24 xb24 xc24 xd24 xe24 xg24 xh24 & & & & & & & & & & & & & & & & = = = = = = = = = = = = = = (ft24t1-sa24)**2/ft24t1e**2 (ft24t2-sb24)**2/ft24t2e**2 (ft24t3-sc24)**2/ft24t3e**2 (ft24t4-sd24)**2/ft24t4e**2 (ft24t5-se24)**2/ft24t5e**2 (ft24t6-sg24)**2/ft24t6e**2 (ft24t7-sh24)**2/ft24t7e**2 kaisq = xa8+xb8+xc8+xd8+xe8+xg8+xh8+ xa9+xb9+xc9+xd9+xe9+xg9+xh9+ xa10+xb10+xc10+xd10+xe10+xg10+xh10+ xa11+xb11+xc11+xd11+xe11+xg11+xh11+ xa12+xb12+xc12+xd12+xe12+xg12+xh12+ xa13+xb13+xc13+xd13+xe13+xg13+xh13+ xa14+xb14+xc14+xd14+xe14+xg14+xh14+ xa15+xb15+xc15+xd15+xe15+xg15+xh15+ xa16+xb16+xc16+xd16+xe16+xg16+xh16+ xa17+xb17+xc17+xd17+xe17+xg17+xh17+ xa18+xb18+xc18+xd18+xe18+xg18+xh18+ xa19+xb19+xc19+xd19+xe19+xg19+xh19+ xa20+xb20+xc20+xd20+xe20+xg20+xh20+ xa21+xb21+xc21+xd21+xe21+xg21+xh21+ xa22+xb22+xc22+xd22+xe22+xg22+xh22+ xa23+xb23+xc23+xd23+xe23+xg23+xh23+ xa24+xb24+xc24+xd24+xe24+xg24+xh24 IF (kaisq < bestkaisq) THEN bestkaisq = kaisq bf8 = f8 242 bf9 = f9 bf10 = f10 bf11 = f11 bf12 = f12 bf13 = f13 bf14 = f14 bf15 = f15 bf16 = f16 bf17 = f17 bf18 = f18 bf19 = f19 bf20 = f20 bf21 = f21 bf22 = f22 bf23 = f23 bf24 = f24 endif f24 enddo f23 enddo f22 enddo f21 enddo f20 enddo f19 enddo f18 enddo f17 enddo * Dirk * * = f24 + 0.01 = f23 + 0.01 = f22 + 0.01 = f21 + 0.01 = f20 + 0.01 = f19 + 0.01 = f18 + 0.01 = f17 + 0.01 f16 = f16 + 0.01 enddo f15 = f15 + 0.01 enddo f14 = f14 + 0.01 enddo *Dirk * * f13 = f13 + 0.01 enddo f12 = f12 + 0.01 enddo f11 = f11 + 0.01 enddo f10 = f10 + 0.01 enddo f9 = f9 + 0.01 enddo f8 = f8 + 0.01 enddo 243 * Dirk * * & & & & & OPEN(UNIT = 12, FILE = 'values5', STATUS = 'NEW') WRITE(12,*) bestkaisq, WRITE(*,*) bestkaisq, bf8, bf9, bf10, bf11, bf12, bf13, bf14, bf15, bf16, bf17, bf18, bf19, bf20, bf21, bf22, bf23, bf24 end 3.7 V2E-HFP 5 Registry Fitting qsub Script, “x2_V2E” For V2E-HFP 5 registry fittings, four main script files were created (see Appendix I). In the file below, f12 = 0.00. The other main script files used set f12 = 0.01, f12 = 0.02 and f12 = 0.03. Splitting the main script file into four separate jobs was required to complete each script file’s computations in less than 168 hours (i.e. the maximum time allowed to occupy a node). This time limit is set by the High Performance Computing Center and jobs that run longer than 168 hours are automatically terminated. #!/bin/bash #PBS -l nodes=1:ppn=1,walltime=168:00:00,mem=2gb,feature=gbe #PBS -j oe #PBS -t 0-143 #change to the original working directory cd ${PBS_O_WORKDIR} # Define number of loops for i16 cols=17 # Define number of loops for i13 (not used) rows=7 # Number of jobs = cols*rows (i.e. -t 0-137) #math to figure out the variable values based on the array id i15=`echo "${PBS_ARRAYID} % ( ${cols} + 1 )" | bc` i23=`echo "${PBS_ARRAYID} / ( ${cols} + 1 )" | bc` #display the command we are going to run echo "./x2 ${i15} ${i23} > ${i15}_${i23}.txt" #run the command with the input variables ./x2 ${i15} ${i23} > ${i15}_${i23}.txt # Calculate the runtiem for the job qstat -f ${PBS_JOBID} 244 3.8 V2E-HFP 5 Registry Fitting Main Script, “V2E_5var.f” * This is a comment * This program was written by Scott Schmick 030111 * * t values (1 = a = 482, 2 = b = 402, etc.) real f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 real f17 f18 f19 f20 f21 f22 f23 f24 f25 real sa8 sb8 sc8 sd8 se8 sg8 sh8 real sa9 sb9 sc9 sd9 se9 sg9 sh9 real sa10 sb10 sc10 sd10 se10 sg10 sh10 real sa11 sb11 sc11 sd11 se11 sg11 sh11 real sa12 sb12 sc12 sd12 se12 sg12 sh12 real sa13 sb13 sc13 sd13 se13 sg13 sh13 real sa14 sb14 sc14 sd14 se14 sg14 sh14 real sa15 sb15 sc15 sd15 se15 sg15 sh15 real sa16 sb16 sc16 sd16 se16 sg16 sh16 real sa17 sb17 sc17 sd17 se17 sg17 sh17 real sa18 sb18 sc18 sd18 se18 sg18 sh18 real sa19 sb19 sc19 sd19 se19 sg19 sh19 real sa20 sb20 sc20 sd20 se20 sg20 sh20 real sa21 sb21 sc21 sd21 se21 sg21 sh21 real sa22 sb22 sc22 sd22 se22 sg22 sh22 real sa23 sb23 sc23 sd23 se23 sg23 sh23 real sa24 sb24 sc24 sd24 se24 sg24 sh24 real real real real real real real real real real real real real real real real real real S0 xa8 xb8 xc8 xd8 xe8 xa9 xb9 xc9 xd9 xe9 xa10 xb10 xc10 xd10 xa11 xb11 xc11 xd11 xa12 xb12 xc12 xd12 xa13 xb13 xc13 xd13 xa14 xb14 xc14 xd14 xa15 xb15 xc15 xd15 xa16 xb16 xc16 xd16 xa17 xb17 xc17 xd17 xa18 xb18 xc18 xd18 xa19 xb19 xc19 xd19 xa20 xb20 xc20 xd20 xa21 xb21 xc21 xd21 xa22 xb22 xc22 xd22 xa23 xb23 xc23 xd23 xa24 xb24 xc24 xd24 xg8 xh8 xg9 xh9 xe10 xg10 xe11 xg11 xe12 xg12 xe13 xg13 xe14 xg14 xe15 xg15 xe16 xg16 xe17 xg17 xe18 xg18 xe19 xg19 xe20 xg20 xe21 xg21 xe22 xg22 xe23 xg23 xe24 xg24 xh10 xh11 xh12 xh13 xh14 xh15 xh16 xh17 xh18 xh19 xh20 xh21 xh22 xh23 xh24 real goff1 goff2 goff3 goff4 goff5 goff6 goff7 real gon1 gon2 gon3 gon4 gon5 gon6 gon7 real gnad1 gnad2 gnad3 gnad4 gnad5 gnad6 gnad7 real ft8t1 real ft8t2 real ft8t3 real ft8t4 real ft8t5 real ft8t6 real ft8t7 245 * real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft8t1e ft8t2e ft8t3e ft8t4e ft8t5e ft8t6e ft8t7e ft9t1 ft9t2 ft9t3 ft9t4 ft9t5 ft9t6 ft9t7 ft9t1e ft9t2e ft9t3e ft9t4e ft9t5e ft9t6e ft9t7e ft10t1 ft10t2 ft10t3 ft10t4 ft10t5 ft10t6 ft10t7 ft10t1e ft10t2e ft10t3e ft10t4e ft10t5e ft10t6e ft10t7e ft11t1 ft11t2 ft11t3 ft11t4 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e 246 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e 247 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e 248 real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e ft23t1 ft23t2 ft23t3 ft23t4 ft23t5 ft23t6 ft23t7 ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 ft24t2 ft24t3 ft24t4 ft24t5 ft24t6 ft24t7 ft24t1e ft24t2e ft24t3e ft24t4e 249 real real real real real real real real real real real real real real real real real real real real real real real real real real real real ft24t5e ft24t6e ft24t7e kaisq bestkaisq bf8 bf9 bf10 bf11 bf12 bf13 bf14 bf15 bf16 bf17 bf18 bf19 bf20 bf21 bf22 bf23 bf24 a8 aa8 a9 aa9 a10 aa10 a11 aa11 a12 aa12 a13 aa13 a14 aa14 b8 b9 b10 b11 b12 b13 b14 b15 b16 b17 b18 b19 b20 b21 b22 b23 b24 z1 z2 z3 z4 z5 z6 z7 f6 f26 a15 aa15 a16 aa16 a17 aa17 a18 aa18 a19 aa19 a20 aa20 a21 aa21 a22 aa22 a23 aa23 a24 aa24 * DIRK integer i15 i23 character inputarg*128 CALL getarg(1,inputarg) read(inputarg,*) i15 CALL getarg(2,inputarg) read(inputarg,*) i23 goff1 = 0.45011191 goff2 = 0.581357776 goff3 = 0.710329562 goff4 = 0.826254485 goff5 = 0.918588976 goff6 = 0.978470252 goff7 = 0.998385086 gon1 = 0.093897695 gon2 = 0.118554369 gon3 = 0.196398598 gon4 = 0.378629871 gon5 = 0.645290442 gon6 = 0.893821899 gon7 = 0.991709633 z1 = 0.9052388602 z2 = 0.9334319587 z3 = 0.95693731265 z4 = 0.97551074985 250 z5 = 0.98895970015 z6 = 0.99714805251 z7 = 0.9997874594365 gnad1 gnad2 gnad3 gnad4 gnad5 gnad6 gnad7 ft8t1 ft8t2 ft8t3 ft8t4 ft8t5 ft8t6 ft8t7 ft8t1e ft8t2e ft8t3e ft8t4e ft8t5e ft8t6e ft8t7e ft9t1 ft9t2 ft9t3 ft9t4 ft9t5 ft9t6 ft9t7 ft9t1e ft9t2e ft9t3e ft9t4e ft9t5e ft9t6e ft9t7e ft10t1 ft10t2 ft10t3 ft10t4 ft10t5 ft10t6 ft10t7 ft10t1e ft10t2e ft10t3e ft10t4e ft10t5e ft10t6e ft10t7e ft11t1 ft11t2 ft11t3 ft11t4 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.281406262 0.283440096 0.298896559 0.309714762 0.320805841 0.325732761 0.342143946 0.05455 0.05178 0.05255 0.01784 0.02752 0.02554 0.00864 0.01171 0.01136 0.01377 0.01 0.01 0.01 0.01 0.0615 0.05812 0.04719 0.03191 0.02722 0.00512 0.01355 0.01671 0.01235 0.01 0.01 0.01 0.01 0.01 0.06781 0.05011 0.04724 0.0408 0.0239 0.01835 0.02391 0.01606 0.01294 0.01184 0.01233 0.01013 0.01 0.0104 0.06249 0.05293 0.04992 0.03957 251 ft11t5 ft11t6 ft11t7 ft11t1e ft11t2e ft11t3e ft11t4e ft11t5e ft11t6e ft11t7e ft12t1 ft12t2 ft12t3 ft12t4 ft12t5 ft12t6 ft12t7 ft12t1e ft12t2e ft12t3e ft12t4e ft12t5e ft12t6e ft12t7e ft13t1 ft13t2 ft13t3 ft13t4 ft13t5 ft13t6 ft13t7 ft13t1e ft13t2e ft13t3e ft13t4e ft13t5e ft13t6e ft13t7e ft14t1 ft14t2 ft14t3 ft14t4 ft14t5 ft14t6 ft14t7 ft14t1e ft14t2e ft14t3e ft14t4e ft14t5e ft14t6e ft14t7e ft15t1 ft15t2 ft15t3 ft15t4 ft15t5 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.031 0.00575 -0.01026 0.01341 0.01711 0.01107 0.01247 0.01 0.0106 0.01 0.08531 0.08566 0.0591 0.03005 0.02208 0.0117 6.49E-04 0.01586 0.01 0.01 0.01418 0.01 0.01039 0.0106 0.12877 0.07773 0.06296 0.04703 0.03825 0.04439 0.0017 0.02489 0.0254 0.02 0.0155 0.01 0.01673 0.01285 0.15352 0.11403 0.07306 0.06083 0.03403 0.04299 0.00683 0.0119 0.01368 0.01192 0.01 0.01019 0.01263 0.01659 0.19465 0.1749 0.14289 0.09534 0.04554 252 ft15t6 ft15t7 ft15t1e ft15t2e ft15t3e ft15t4e ft15t5e ft15t6e ft15t7e ft16t1 ft16t2 ft16t3 ft16t4 ft16t5 ft16t6 ft16t7 ft16t1e ft16t2e ft16t3e ft16t4e ft16t5e ft16t6e ft16t7e ft17t1 ft17t2 ft17t3 ft17t4 ft17t5 ft17t6 ft17t7 ft17t1e ft17t2e ft17t3e ft17t4e ft17t5e ft17t6e ft17t7e ft18t1 ft18t2 ft18t3 ft18t4 ft18t5 ft18t6 ft18t7 ft18t1e ft18t2e ft18t3e ft18t4e ft18t5e ft18t6e ft18t7e ft19t1 ft19t2 ft19t3 ft19t4 ft19t5 ft19t6 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.03886 0.00352 0.01573 0.01096 0.01 0.01 0.01565 0.01 0.0131 0.28671 0.24526 0.19745 0.12897 0.07304 0.03423 0.01909 0.02292 0.01424 0.01478 0.01189 0.01 0.01226 0.01049 0.31006 0.27143 0.23816 0.17499 0.07795 0.0353 0.01897 0.02074 0.01444 0.01598 0.01853 0.01963 0.0198 0.01164 0.30283 0.3016 0.25411 0.19389 0.11342 0.05735 0.02475 0.02213 0.01776 0.01885 0.0122 0.01153 0.01299 0.01682 0.34606 0.27994 0.21297 0.1443 0.06939 0.04606 253 ft19t7 ft19t1e ft19t2e ft19t3e ft19t4e ft19t5e ft19t6e ft19t7e ft20t1 ft20t2 ft20t3 ft20t4 ft20t5 ft20t6 ft20t7 ft20t1e ft20t2e ft20t3e ft20t4e ft20t5e ft20t6e ft20t7e ft21t1 ft21t2 ft21t3 ft21t4 ft21t5 ft21t6 ft21t7 ft21t1e ft21t2e ft21t3e ft21t4e ft21t5e ft21t6e ft21t7e ft22t1 ft22t2 ft22t3 ft22t4 ft22t5 ft22t6 ft22t7 ft22t1e ft22t2e ft22t3e ft22t4e ft22t5e ft22t6e ft22t7e ft23t1 ft23t2 ft23t3 ft23t4 ft23t5 ft23t6 ft23t7 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 0.01298 0.01175 0.01249 0.01 0.01044 0.01414 0.01 0.0107 0.39766 0.37867 0.32995 0.26239 0.14554 0.05559 0.00911 0.01651 0.01394 0.01 0.01 0.01 0.01 0.01 0.25706 0.19786 0.17866 0.1297 0.0752 0.04328 0.00665 0.01116 0.01489 0.01 0.01 0.01 0.01 0.01 0.15497 0.10269 0.07505 0.05987 0.02889 0.02218 0.0051 0.01455 0.01357 0.01179 0.01182 0.01011 0.01134 0.01 0.11272 0.10263 0.08713 0.07678 0.04061 0.01599 0.01091 254 ft23t1e ft23t2e ft23t3e ft23t4e ft23t5e ft23t6e ft23t7e ft24t1 ft24t2 ft24t3 ft24t4 ft24t5 ft24t6 ft24t7 ft24t1e ft24t2e ft24t3e ft24t4e ft24t5e ft24t6e ft24t7e = = = = = = = = = = = = = = = = = = = = = 0.01183 0.01 0.01055 0.01 0.01 0.01081 0.01 0.10148 0.09025 0.05784 0.05364 0.029 0.01416 4.40E-04 0.01731 0.01521 0.01928 0.01275 0.01 0.01139 0.01256 bestkaisq = 100000000 S0 = 1.32 Slab = 0.9625275 f6 = 0.00 f7 = 0.00 f25 = 0.00 f26 = 0.00 f8 = 0.00 do while (f8 <= 0.00) f9 = 0.00 do while (f9 <= 0.00) f10 = 0.00 a8 = f7+f9 b8 = f6+f10 aa8 = 1-f6-f7-f8-f9-f10 sa8 = (S0-Slab*(gon1*f8+goff1*a8+z1*b8+aa8)-gnad1)/S0 sb8 = (S0-Slab*(gon2*f8+goff2*a8+z2*b8+aa8)-gnad2)/S0 sc8 = (S0-Slab*(gon3*f8+goff3*a8+z3*b8+aa8)-gnad3)/S0 sd8 = (S0-Slab*(gon4*f8+goff4*a8+z4*b8+aa8)-gnad4)/S0 se8 = (S0-Slab*(gon5*f8+goff5*a8+z5*b8+aa8)-gnad5)/S0 sg8 = (S0-Slab*(gon6*f8+goff6*a8+z6*b8+aa8)-gnad6)/S0 sh8 = (S0-Slab*(gon7*f8+goff7*a8+z7*b8+aa8)-gnad7)/S0 xa8 = (ft8t1-sa8)**2/ft8t1e**2 xb8 = (ft8t2-sb8)**2/ft8t2e**2 xc8 = (ft8t3-sc8)**2/ft8t3e**2 xd8 = (ft8t4-sd8)**2/ft8t4e**2 xe8 = (ft8t5-se8)**2/ft8t5e**2 xg8 = (ft8t6-sg8)**2/ft8t6e**2 xh8 = (ft8t7-sh8)**2/ft8t7e**2 255 do while (f10 <= 0.00) a9 = f8+f10 b9 = f7+f11 aa9 = 1-f7-f8-f9-f10-f11 sa9 = (S0-Slab*(gon1*f9+goff1*a9+b9*z1+aa9)-gnad1)/S0 sb9 = (S0-Slab*(gon2*f9+goff2*a9+b9*z2+aa9)-gnad2)/S0 sc9 = (S0-Slab*(gon3*f9+goff3*a9+b9*z3+aa9)-gnad3)/S0 sd9 = (S0-Slab*(gon4*f9+goff4*a9+b9*z4+aa9)-gnad4)/S0 se9 = (S0-Slab*(gon5*f9+goff5*a9+b9*z5+aa9)-gnad5)/S0 sg9 = (S0-Slab*(gon6*f9+goff6*a9+b9*z6+aa9)-gnad6)/S0 sh9 = (S0-Slab*(gon7*f9+goff7*a9+b9*z7+aa9)-gnad7)/S0 xa9 = (ft9t1-sa9)**2/ft9t1e**2 xb9 = (ft9t2-sb9)**2/ft9t2e**2 xc9 = (ft9t3-sc9)**2/ft9t3e**2 xd9 = (ft9t4-sd9)**2/ft9t4e**2 xe9 = (ft9t5-se9)**2/ft9t5e**2 xg9 = (ft9t6-sg9)**2/ft9t6e**2 xh9 = (ft9t7-sh9)**2/ft9t7e**2 f11 = 0.00 do while (f11 <= 0.00) a10 = f9+f11 b10 = f8+f12 aa10 = 1-f8-f9-f10-f11-f12 sa10 = (S0-Slab*(gon1*f10+goff1*a10+z1*b10+aa10)-gnad1)/S0 sb10 = (S0-Slab*(gon2*f10+goff2*a10+z2*b10+aa10)-gnad2)/S0 sc10 = (S0-Slab*(gon3*f10+goff3*a10+z3*b10+aa10)-gnad3)/S0 sd10 = (S0-Slab*(gon4*f10+goff4*a10+z4*b10+aa10)-gnad4)/S0 se10 = (S0-Slab*(gon5*f10+goff5*a10+z5*b10+aa10)-gnad5)/S0 sg10 = (S0-Slab*(gon6*f10+goff6*a10+z6*b10+aa10)-gnad6)/S0 sh10 = (S0-Slab*(gon7*f10+goff7*a10+z7*b10+aa10)-gnad7)/S0 xa10 = (ft10t1-sa10)**2/ft10t1e**2 xb10 = (ft10t2-sb10)**2/ft10t2e**2 xc10 = (ft10t3-sc10)**2/ft10t3e**2 xd10 = (ft10t4-sd10)**2/ft10t4e**2 xe10 = (ft10t5-se10)**2/ft10t5e**2 xg10 = (ft10t6-sg10)**2/ft10t6e**2 xh10 = (ft10t7-sh10)**2/ft10t7e**2 f12 = 0.00 do while (f12 <= 0.00) a11 = f10+f12 b11 = f9+f13 aa11 = 1-f9-f10-f11-f12-f13 sa11 = (S0-Slab*(gon1*f11+goff1*a11+z1*b11+aa11)-gnad1)/S0 sb11 = (S0-Slab*(gon2*f11+goff2*a11+z2*b11+aa11)-gnad2)/S0 sc11 = (S0-Slab*(gon3*f11+goff3*a11+z3*b11+aa11)-gnad3)/S0 sd11 = (S0-Slab*(gon4*f11+goff4*a11+z4*b11+aa11)-gnad4)/S0 se11 = (S0-Slab*(gon5*f11+goff5*a11+z5*b11+aa11)-gnad5)/S0 sg11 = (S0-Slab*(gon6*f11+goff6*a11+z6*b11+aa11)-gnad6)/S0 sh11 = (S0-Slab*(gon7*f11+goff7*a11+z7*b11+aa11)-gnad7)/S0 xa11 = (ft11t1-sa11)**2/ft11t1e**2 xb11 = (ft11t2-sb11)**2/ft11t2e**2 xc11 = (ft11t3-sc11)**2/ft11t3e**2 xd11 = (ft11t4-sd11)**2/ft11t4e**2 xe11 = (ft11t5-se11)**2/ft11t5e**2 xg11 = (ft11t6-sg11)**2/ft11t6e**2 xh11 = (ft11t7-sh11)**2/ft11t7e**2 f13 = 0.00 256 do while (f13 <= 0.05) a12 = f11+f13 b12 = f10+f14 aa12 = 1-f10-f11-f12-f13-f14 sa12 = (S0-Slab*(gon1*f12+goff1*a12+z1*b12+aa12)-gnad1)/S0 sb12 = (S0-Slab*(gon2*f12+goff2*a12+z2*b12+aa12)-gnad2)/S0 sc12 = (S0-Slab*(gon3*f12+goff3*a12+z3*b12+aa12)-gnad3)/S0 sd12 = (S0-Slab*(gon4*f12+goff4*a12+z4*b12+aa12)-gnad4)/S0 se12 = (S0-Slab*(gon5*f12+goff5*a12+z5*b12+aa12)-gnad5)/S0 sg12 = (S0-Slab*(gon6*f12+goff6*a12+z6*b12+aa12)-gnad6)/S0 sh12 = (S0-Slab*(gon7*f12+goff7*a12+z7*b12+aa12)-gnad7)/S0 xa12 = (ft12t1-sa12)**2/ft12t1e**2 xb12 = (ft12t2-sb12)**2/ft12t2e**2 xc12 = (ft12t3-sc12)**2/ft12t3e**2 xd12 = (ft12t4-sd12)**2/ft12t4e**2 xe12 = (ft12t5-se12)**2/ft12t5e**2 xg12 = (ft12t6-sg12)**2/ft12t6e**2 xh12 = (ft12t7-sh12)**2/ft12t7e**2 f14 = 0.00 do while (f14 <= 0.09) a13 = f12+f14 b13 = f11+f15 aa13 = 1-f11-f12-f13-f14-f15 sa13 = (S0-Slab*(gon1*f13+goff1*a13+z1*b13+aa13)-gnad1)/S0 sb13 = (S0-Slab*(gon2*f13+goff2*a13+z2*b13+aa13)-gnad2)/S0 sc13 = (S0-Slab*(gon3*f13+goff3*a13+z3*b13+aa13)-gnad3)/S0 sd13 = (S0-Slab*(gon4*f13+goff4*a13+z4*b13+aa13)-gnad4)/S0 se13 = (S0-Slab*(gon5*f13+goff5*a13+z5*b13+aa13)-gnad5)/S0 sg13 = (S0-Slab*(gon6*f13+goff6*a13+z6*b13+aa13)-gnad6)/S0 sh13 = (S0-Slab*(gon7*f13+goff7*a13+z7*b13+aa13)-gnad7)/S0 xa13 = (ft13t1-sa13)**2/ft13t1e**2 xb13 = (ft13t2-sb13)**2/ft13t2e**2 xc13 = (ft13t3-sc13)**2/ft13t3e**2 xd13 = (ft13t4-sd13)**2/ft13t4e**2 xe13 = (ft13t5-se13)**2/ft13t5e**2 xg13 = (ft13t6-sg13)**2/ft13t6e**2 xh13 = (ft13t7-sh13)**2/ft13t7e**2 * f15 = 0.00 * Dirk f15 = 0.00+0.01*i15 * do while (f15 <= 0.17) a14 = f13+f15 b14 = f12+f16 aa14 = 1-f12-f13-f14-f15-f16 sa14 = (S0-Slab*(gon1*f14+goff1*a14+z1*b14+aa14)-gnad1)/S0 sb14 = (S0-Slab*(gon2*f14+goff2*a14+z2*b14+aa14)-gnad2)/S0 sc14 = (S0-Slab*(gon3*f14+goff3*a14+z3*b14+aa14)-gnad3)/S0 sd14 = (S0-Slab*(gon4*f14+goff4*a14+z4*b14+aa14)-gnad4)/S0 se14 = (S0-Slab*(gon5*f14+goff5*a14+z5*b14+aa14)-gnad5)/S0 sg14 = (S0-Slab*(gon6*f14+goff6*a14+z6*b14+aa14)-gnad6)/S0 sh14 = (S0-Slab*(gon7*f14+goff7*a14+z7*b14+aa14)-gnad7)/S0 xa14 = (ft14t1-sa14)**2/ft14t1e**2 xb14 = (ft14t2-sb14)**2/ft14t2e**2 xc14 = (ft14t3-sc14)**2/ft14t3e**2 xd14 = (ft14t4-sd14)**2/ft14t4e**2 xe14 = (ft14t5-se14)**2/ft14t5e**2 257 xg14 = (ft14t6-sg14)**2/ft14t6e**2 xh14 = (ft14t7-sh14)**2/ft14t7e**2 f16 = 0.04 do while (f16 <= 0.26) a15 = f14+f16 b15 = f13+f17 aa15 = 1-f13-f14-f15-f16-f17 sa15 = (S0-Slab*(gon1*f15+goff1*a15+z1*b15+aa15)-gnad1)/S0 sb15 = (S0-Slab*(gon2*f15+goff2*a15+z2*b15+aa15)-gnad2)/S0 sc15 = (S0-Slab*(gon3*f15+goff3*a15+z3*b15+aa15)-gnad3)/S0 sd15 = (S0-Slab*(gon4*f15+goff4*a15+z4*b15+aa15)-gnad4)/S0 se15 = (S0-Slab*(gon5*f15+goff5*a15+z5*b15+aa15)-gnad5)/S0 sg15 = (S0-Slab*(gon6*f15+goff6*a15+z6*b15+aa15)-gnad6)/S0 sh15 = (S0-Slab*(gon7*f15+goff7*a15+z7*b15+aa15)-gnad7)/S0 xa15 = (ft15t1-sa15)**2/ft15t1e**2 xb15 = (ft15t2-sb15)**2/ft15t2e**2 xc15 = (ft15t3-sc15)**2/ft15t3e**2 xd15 = (ft15t4-sd15)**2/ft15t4e**2 xe15 = (ft15t5-se15)**2/ft15t5e**2 xg15 = (ft15t6-sg15)**2/ft15t6e**2 xh15 = (ft15t7-sh15)**2/ft15t7e**2 f17 = 0.03 do while (f17 <= 0.33) a16 = f15+f17 b16 = f14+f18 aa16 = 1-f14-f15-f16-f17-f18 sa16 = (S0-Slab*(gon1*f16+goff1*a16+z1*b16+aa16)-gnad1)/S0 sb16 = (S0-Slab*(gon2*f16+goff2*a16+z2*b16+aa16)-gnad2)/S0 sc16 = (S0-Slab*(gon3*f16+goff3*a16+z3*b16+aa16)-gnad3)/S0 sd16 = (S0-Slab*(gon4*f16+goff4*a16+z4*b16+aa16)-gnad4)/S0 se16 = (S0-Slab*(gon5*f16+goff5*a16+z5*b16+aa16)-gnad5)/S0 sg16 = (S0-Slab*(gon6*f16+goff6*a16+z6*b16+aa16)-gnad6)/S0 sh16 = (S0-Slab*(gon7*f16+goff7*a16+z7*b16+aa16)-gnad7)/S0 xa16 = (ft16t1-sa16)**2/ft16t1e**2 xb16 = (ft16t2-sb16)**2/ft16t2e**2 xc16 = (ft16t3-sc16)**2/ft16t3e**2 xd16 = (ft16t4-sd16)**2/ft16t4e**2 xe16 = (ft16t5-se16)**2/ft16t5e**2 xg16 = (ft16t6-sg16)**2/ft16t6e**2 xh16 = (ft16t7-sh16)**2/ft16t7e**2 f18 = 0.05 do while (f18 <= 0.36) a17 = f16+f18 b17 = f15+f19 aa17 = 1-f15-f16-f17-f18-f19 sa17 = (S0-Slab*(gon1*f17+goff1*a17+z1*b17+aa17)-gnad1)/S0 sb17 = (S0-Slab*(gon2*f17+goff2*a17+z2*b17+aa17)-gnad2)/S0 sc17 = (S0-Slab*(gon3*f17+goff3*a17+z3*b17+aa17)-gnad3)/S0 sd17 = (S0-Slab*(gon4*f17+goff4*a17+z4*b17+aa17)-gnad4)/S0 se17 = (S0-Slab*(gon5*f17+goff5*a17+z5*b17+aa17)-gnad5)/S0 sg17 = (S0-Slab*(gon6*f17+goff6*a17+z6*b17+aa17)-gnad6)/S0 sh17 = (S0-Slab*(gon7*f17+goff7*a17+z7*b17+aa17)-gnad7)/S0 xa17 = (ft17t1-sa17)**2/ft17t1e**2 xb17 = (ft17t2-sb17)**2/ft17t2e**2 xc17 = (ft17t3-sc17)**2/ft17t3e**2 258 xd17 = (ft17t4-sd17)**2/ft17t4e**2 xe17 = (ft17t5-se17)**2/ft17t5e**2 xg17 = (ft17t6-sg17)**2/ft17t6e**2 xh17 = (ft17t7-sh17)**2/ft17t7e**2 f19 = 0.00 do while (f19 <= 0.33) a18 = f17+f19 b18 = f16+f20 aa18 = 1-f16-f17-f18-f19-f20 sa18 = (S0-Slab*(gon1*f18+goff1*a18+z1*b18+aa18)-gnad1)/S0 sb18 = (S0-Slab*(gon2*f18+goff2*a18+z2*b18+aa18)-gnad2)/S0 sc18 = (S0-Slab*(gon3*f18+goff3*a18+z3*b18+aa18)-gnad3)/S0 sd18 = (S0-Slab*(gon4*f18+goff4*a18+z4*b18+aa18)-gnad4)/S0 se18 = (S0-Slab*(gon5*f18+goff5*a18+z5*b18+aa18)-gnad5)/S0 sg18 = (S0-Slab*(gon6*f18+goff6*a18+z6*b18+aa18)-gnad6)/S0 sh18 = (S0-Slab*(gon7*f18+goff7*a18+z7*b18+aa18)-gnad7)/S0 xa18 = (ft18t1-sa18)**2/ft18t1e**2 xb18 = (ft18t2-sb18)**2/ft18t2e**2 xc18 = (ft18t3-sc18)**2/ft18t3e**2 xd18 = (ft18t4-sd18)**2/ft18t4e**2 xe18 = (ft18t5-se18)**2/ft18t5e**2 xg18 = (ft18t6-sg18)**2/ft18t6e**2 xh18 = (ft18t7-sh18)**2/ft18t7e**2 f20 = 0.24 do while (f20 <= 0.49) a19 = f18+f20 b19 = f17+f21 aa19 = 1-f17-f18-f19-f20-f21 sa19 = (S0-Slab*(gon1*f19+goff1*a19+z1*b19+aa19)-gnad1)/S0 sb19 = (S0-Slab*(gon2*f19+goff2*a19+z2*b19+aa19)-gnad2)/S0 sc19 = (S0-Slab*(gon3*f19+goff3*a19+z3*b19+aa19)-gnad3)/S0 sd19 = (S0-Slab*(gon4*f19+goff4*a19+z4*b19+aa19)-gnad4)/S0 se19 = (S0-Slab*(gon5*f19+goff5*a19+z5*b19+aa19)-gnad5)/S0 sg19 = (S0-Slab*(gon6*f19+goff6*a19+z6*b19+aa19)-gnad6)/S0 sh19 = (S0-Slab*(gon7*f19+goff7*a19+z7*b19+aa19)-gnad7)/S0 xa19 = (ft19t1-sa19)**2/ft19t1e**2 xb19 = (ft19t2-sb19)**2/ft19t2e**2 xc19 = (ft19t3-sc19)**2/ft19t3e**2 xd19 = (ft19t4-sd19)**2/ft19t4e**2 xe19 = (ft19t5-se19)**2/ft19t5e**2 xg19 = (ft19t6-sg19)**2/ft19t6e**2 xh19 = (ft19t7-sh19)**2/ft19t7e**2 f21 = 0.00 do while (f21 <= 0.24) a20 = f19+f21 b20 = f18+f22 aa20 = 1-f18-f19-f20-f21-f22 sa20 = (S0-Slab*(gon1*f20+goff1*a20+z1*b20+aa20)-gnad1)/S0 sb20 = (S0-Slab*(gon2*f20+goff2*a20+z2*b20+aa20)-gnad2)/S0 sc20 = (S0-Slab*(gon3*f20+goff3*a20+z3*b20+aa20)-gnad3)/S0 sd20 = (S0-Slab*(gon4*f20+goff4*a20+z4*b20+aa20)-gnad4)/S0 se20 = (S0-Slab*(gon5*f20+goff5*a20+z5*b20+aa20)-gnad5)/S0 sg20 = (S0-Slab*(gon6*f20+goff6*a20+z6*b20+aa20)-gnad6)/S0 sh20 = (S0-Slab*(gon7*f20+goff7*a20+z7*b20+aa20)-gnad7)/S0 xa20 = (ft20t1-sa20)**2/ft20t1e**2 xb20 = (ft20t2-sb20)**2/ft20t2e**2 xc20 = (ft20t3-sc20)**2/ft20t3e**2 259 * xd20 = (ft20t4-sd20)**2/ft20t4e**2 xe20 = (ft20t5-se20)**2/ft20t5e**2 xg20 = (ft20t6-sg20)**2/ft20t6e**2 xh20 = (ft20t7-sh20)**2/ft20t7e**2 f22 = 0.00 do while (f22 <= 0.08) a21 = f20+f22 b21 = f19+f23 aa21 = 1-f19-f20-f21-f22-f23 sa21 = (S0-Slab*(gon1*f21+goff1*a21+z1*b21+aa21)-gnad1)/S0 sb21 = (S0-Slab*(gon2*f21+goff2*a21+z2*b21+aa21)-gnad2)/S0 sc21 = (S0-Slab*(gon3*f21+goff3*a21+z3*b21+aa21)-gnad3)/S0 sd21 = (S0-Slab*(gon4*f21+goff4*a21+z4*b21+aa21)-gnad4)/S0 se21 = (S0-Slab*(gon5*f21+goff5*a21+z5*b21+aa21)-gnad5)/S0 sg21 = (S0-Slab*(gon6*f21+goff6*a21+z6*b21+aa21)-gnad6)/S0 sh21 = (S0-Slab*(gon7*f21+goff7*a21+z7*b21+aa21)-gnad7)/S0 xa21 = (ft21t1-sa21)**2/ft21t1e**2 xb21 = (ft21t2-sb21)**2/ft21t2e**2 xc21 = (ft21t3-sc21)**2/ft21t3e**2 xd21 = (ft21t4-sd21)**2/ft21t4e**2 xe21 = (ft21t5-se21)**2/ft21t5e**2 xg21 = (ft21t6-sg21)**2/ft21t6e**2 xh21 = (ft21t7-sh21)**2/ft21t7e**2 f23 = 0.01 *Dirk * f23 = 0.00+0.01*i23 do while (f23 <= 0.08) a22 = f21+f23 b22 = f20+f24 aa22 = 1-f20-f21-f22-f23-f24 sa22 = (S0-Slab*(gon1*f22+goff1*a22+z1*b22+aa22)-gnad1)/S0 sb22 = (S0-Slab*(gon2*f22+goff2*a22+z2*b22+aa22)-gnad2)/S0 sc22 = (S0-Slab*(gon3*f22+goff3*a22+z3*b22+aa22)-gnad3)/S0 sd22 = (S0-Slab*(gon4*f22+goff4*a22+z4*b22+aa22)-gnad4)/S0 se22 = (S0-Slab*(gon5*f22+goff5*a22+z5*b22+aa22)-gnad5)/S0 sg22 = (S0-Slab*(gon6*f22+goff6*a22+z6*b22+aa22)-gnad6)/S0 sh22 = (S0-Slab*(gon7*f22+goff7*a22+z7*b22+aa22)-gnad7)/S0 xa22 = (ft22t1-sa22)**2/ft22t1e**2 xb22 = (ft22t2-sb22)**2/ft22t2e**2 xc22 = (ft22t3-sc22)**2/ft22t3e**2 xd22 = (ft22t4-sd22)**2/ft22t4e**2 xe22 = (ft22t5-se22)**2/ft22t5e**2 xg22 = (ft22t6-sg22)**2/ft22t6e**2 xh22 = (ft22t7-sh22)**2/ft22t7e**2 f24 = 0.00 do while (f24 <= 0.04) a23 = f22+f24 b23 = f21+f25 aa23 = 1-f21-f22-f23-f24-f25 sa23 = (S0-Slab*(gon1*f23+goff1*a23+z1*b23+aa23)-gnad1)/S0 sb23 = (S0-Slab*(gon2*f23+goff2*a23+z2*b23+aa23)-gnad2)/S0 sc23 = (S0-Slab*(gon3*f23+goff3*a23+z3*b23+aa23)-gnad3)/S0 sd23 = (S0-Slab*(gon4*f23+goff4*a23+z4*b23+aa23)-gnad4)/S0 se23 = (S0-Slab*(gon5*f23+goff5*a23+z5*b23+aa23)-gnad5)/S0 sg23 = (S0-Slab*(gon6*f23+goff6*a23+z6*b23+aa23)-gnad6)/S0 sh23 = (S0-Slab*(gon7*f23+goff7*a23+z7*b23+aa23)-gnad7)/S0 260 sa24 sb24 sc24 sd24 se24 sg24 sh24 a24 = f23+f25 b24 = f22+f26 aa24 = 1-f22-f23-f24-f25-f26 = (S0-Slab*(gon1*f24+goff1*a24+z1*b24+aa24)-gnad1)/S0 = (S0-Slab*(gon2*f24+goff2*a24+z2*b24+aa24)-gnad2)/S0 = (S0-Slab*(gon3*f24+goff3*a24+z3*b24+aa24)-gnad3)/S0 = (S0-Slab*(gon4*f24+goff4*a24+z4*b24+aa24)-gnad4)/S0 = (S0-Slab*(gon5*f24+goff5*a24+z5*b24+aa24)-gnad5)/S0 = (S0-Slab*(gon6*f24+goff6*a24+z6*b24+aa24)-gnad6)/S0 = (S0-Slab*(gon7*f24+goff7*a24+z7*b24+aa24)-gnad7)/S0 xa23 xb23 xc23 xd23 xe23 xg23 xh23 (ft23t1-sa23)**2/ft23t1e**2 (ft23t2-sb23)**2/ft23t2e**2 (ft23t3-sc23)**2/ft23t3e**2 (ft23t4-sd23)**2/ft23t4e**2 (ft23t5-se23)**2/ft23t5e**2 (ft23t6-sg23)**2/ft23t6e**2 (ft23t7-sh23)**2/ft23t7e**2 xa24 xb24 xc24 xd24 xe24 xg24 xh24 & & & & & & & & & & & & & & & & = = = = = = = = = = = = = = (ft24t1-sa24)**2/ft24t1e**2 (ft24t2-sb24)**2/ft24t2e**2 (ft24t3-sc24)**2/ft24t3e**2 (ft24t4-sd24)**2/ft24t4e**2 (ft24t5-se24)**2/ft24t5e**2 (ft24t6-sg24)**2/ft24t6e**2 (ft24t7-sh24)**2/ft24t7e**2 kaisq = xa8+xb8+xc8+xd8+xe8+xg8+xh8+ xa9+xb9+xc9+xd9+xe9+xg9+xh9+ xa10+xb10+xc10+xd10+xe10+xg10+xh10+ xa11+xb11+xc11+xd11+xe11+xg11+xh11+ xa12+xb12+xc12+xd12+xe12+xg12+xh12+ xa13+xb13+xc13+xd13+xe13+xg13+xh13+ xa14+xb14+xc14+xd14+xe14+xg14+xh14+ xa15+xb15+xc15+xd15+xe15+xg15+xh15+ xa16+xb16+xc16+xd16+xe16+xg16+xh16+ xa17+xb17+xc17+xd17+xe17+xg17+xh17+ xa18+xb18+xc18+xd18+xe18+xg18+xh18+ xa19+xb19+xc19+xd19+xe19+xg19+xh19+ xa20+xb20+xc20+xd20+xe20+xg20+xh20+ xa21+xb21+xc21+xd21+xe21+xg21+xh21+ xa22+xb22+xc22+xd22+xe22+xg22+xh22+ xa23+xb23+xc23+xd23+xe23+xg23+xh23+ xa24+xb24+xc24+xd24+xe24+xg24+xh24 IF (kaisq < bestkaisq) THEN bestkaisq = kaisq bf8 = f8 bf9 = f9 bf10 = f10 bf11 = f11 bf12 = f12 bf13 = f13 bf14 = f14 bf15 = f15 261 bf16 bf17 bf18 bf19 bf20 bf21 bf22 bf23 bf24 = = = = = = = = = f16 f17 f18 f19 f20 f21 f22 f23 f24 endif f24 = f24 + 0.01 enddo *Dirk * * f23 = f23 + 0.01 enddo f22 = f22 + 0.01 enddo f21 = f21 + 0.01 enddo f20 = f20 + 0.01 enddo f19 = f19 + 0.01 enddo f18 = f18 + 0.01 enddo f17 = f17 + 0.01 enddo f16 = f16 + 0.01 enddo *Dirk * * f15 = f15 + 0.01 enddo f14 = f14 + 0.01 enddo f13 = f13 enddo f12 = f12 enddo f11 = f11 enddo f10 = f10 enddo f9 = f9 + enddo f8 = f8 + + 0.01 + 0.01 + 0.01 + 0.01 0.01 0.01 enddo * Dirk * * & OPEN(UNIT = 12, FILE = 'values5', STATUS = 'NEW') WRITE(12,*) bestkaisq, WRITE(*,*) bestkaisq, bf8, 262 & & & & bf9, bf10, bf11, bf12, bf13, bf14, bf15, bf16, bf17, bf18, bf19, bf20, bf21, bf22, bf23, bf24 end 263 exp Appendix VII. Chapter III Table of (S/S0) values and 1tuv geometries with calculated values from SIMPSON. exp Table 16. Chapter III (S/S0) (ΔS/S0) Dephasing time (ms) 2.2 8.2 16.2 24.2 32.2 40.2 48.2 (HFP-NC) exp 0.019 (.013) 0.056 (.020) 0.071 (.020) 0.077 (.023) 0.086 (.017) 0.093 (.018) 0.089 (.028) exp exp +σ (HFP-P) lab ( ) or 1t1t2uv lab ( ) spin and rms error. in parentheses exp 0.027 (.013) 0.053 (.013) 0.061 (.015) 0.108 (.013) 0.087 (.018) 0.126 (.023) 0.162 (.025) (HFP-A) exp 0.012 (.023) 0.077 (.036) 0.107 (.021) 0.202 (.026) 0.244 (.025) 0.306 (.031) 0.325 (.060) (HFP-AP) 0.017 (.028) 0.055 (.026) 0.128 (.022) 0.219 (.016) 0.265 (.020) 0.294 (.023) 0.344 (.020) Figure 49 illustrates the spin geometries and associated calculated 1tuv t, u, and v in the fully constrained model or calculated 1t1t2uv exp lab ( ) for specific lab ( ) for specific t1, t2, u, and v in the unconstrained model. In each spin geometry schematic, N or C respectively denote a nucleus or a 13 CO nucleus included in the simulation, and Y denotes either X registry or a HFP. Each arrow denotes 13 15 13 N CO 15 CO- N dipolar coupling considered in the simulation. For unconstrained model geometries, the top/middle strand registry is at the top of the schematic and the middle/bottom strand registry is at the bottom of the schematic. When more than one spin geometries are shown, each  ( ) is the average for the displayed geometries. 264 N N N N C N C N 116/161 N N N 117/171 N N N C C Y Y 116/161 Y C 116/161 C N 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 t lab  (ms) 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 u v 2 2 0.9889 0.8599 0.5566 0.2791 0.1326 0.0885 0.0725 ( ) 2 3 4 4 t u v 2 2 0.9907 0.8805 0.5972 0.2859 0.0698 -0.0082 0.0112 ( ) 2 3 2 2 117/171 Y N lab  (ms) N N  (ms) 2.2 8.2 16.2 24.2 32.2 40.2 48.2 1tuv lab ( ) 0.9982 0.9754 0.9075 0.8044 0.6779 0.5417 0.4092 117/171 Figure 49. Chapter III spin geometries and simulated data. 265 t u v 2 2 2 3 3 3 Figure 49 (cont’d). N N C N N lab  (ms) 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 t u v 1 1 0.9903 0.8770 0.6043 0.3394 0.1782 0.1094 0.0782 ( ) 1 3 4 4 117/117 N N lab Y 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 C  (ms) 0.9933 0.9134 0.6961 0.4263 0.1902 0.0436 -0.0055 117/117 266 ( ) t u v 1 1 1 3 2 2 Figure 49 (cont’d).  (ms) Y C N N 2.2 8.2 16.2 24.2 32.2 40.2 48.2 117/117 267 1tuv lab ( ) 0.9979 0.9717 0.8941 0.7773 0.6364 0.4883 0.3488 t u v 1 1 1 3 3 3 Figure 49 (cont’d). 116/161 N N 117/171 N N 116/161 N N 161/116  (ms) 2.2 8.2 16.2 24.2 32.2 40.2 48.2 N N C C C N C N 117/171 N N 171/117 1tuv lab ( ) 0.9889 0.8605 0.5582 0.2806 0.1318 0.0847 0.0681 N N N 161/116 t1 2 2 268 t2 2 2 171/117 u v 2 3 4 4 N Figure 49 (cont’d). 116/161 N 117/171 N N N C C Y Y 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 0.9907 0.8805 0.5972 0.2859 0.0698 -0.0082 0.0112  (ms) 1tuv ( ) t1 2 2 2 2 2 2 2 2 u v 2 3 2 2 2 2 3 3 t2 2 2 1 1 3 3 3 3 2 2 2 4 2 4 2 4 u v 2 3 2 2 2 2 3 3 3 3 3 4 3 4 3 4 Y Y lab  (ms) C N C N N N 161/116 2.2 8.2 16.2 24.2 32.2 40.2 48.2 lab ( ) 0.9982 0.9754 0.9075 0.8044 0.6779 0.5417 0.4092 171/117 117/117 N N C N N lab  (ms) 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 0.9903 0.8770 0.6043 0.3394 0.1782 0.1094 0.0782 117/117 269 ( ) t1 2 2 1 1 3 3 3 3 t2 2 2 2 2 2 2 2 2 t1 1 1 t2 1 1 u v 1 3 4 4 Figure 49 (cont’d). 117/117 N  (ms) N C 2.2 8.2 16.2 24.2 32.2 40.2 48.2 1tuv lab ( ) 0.9933 0.9134 0.6961 0.4263 0.1902 0.0436 -0.0055 Y  (ms) Y C N N 2.2 8.2 16.2 24.2 32.2 40.2 48.2 117/117 270 1tuv lab ( ) 0.9979 0.9717 0.8941 0.7773 0.6364 0.4883 0.3488 t1 1 1 1 1 1 1 1 1 t2 1 1 2 2 3 3 3 3 u v 1 3 1 1 1 1 3 3 2 2 2 4 2 4 2 4 t1 1 1 2 2 3 3 3 3 t2 1 1 1 1 1 1 1 1 u v 1 3 1 1 1 1 3 3 3 3 3 4 3 4 3 4 Figure 49 (cont’d). 117/171 N 116/161 N C C N N 171/171 117/117 N N C 116/161 2.2 8.2 16.2 24.2 32.2 40.2 48.2 0.9903 0.8766 0.5998 0.3263 0.1591 0.0937 0.0712 1tuv 2.2 8.2 16.2 24.2 32.2 40.2 48.2 0.9904 0.8782 0.6061 0.3389 0.1753 0.1079 0.0805 ( ) t1 2 t2 1 u v 3 4 t1 1 t2 2 u v 3 4 171/171 N N N 117/117 C N N 1tuv  (ms) N N lab  (ms) N N N 171/117 271 lab ( ) Appendix VIII. Five Registry Fittings The 5 registry fittings were performed identical to the 3 registry fittings (Chapter IV). The natural abundance calculations were identical, but the number of antiparallel registries that were considered to contribute to the rdd in a given sample was increased from 3 to 5.   S sim     0.99  J  0.0037   na    0.98  K  0.011   na     lu lN lC         (63)  u2   lab 0.33   K  0.011  0.99   J  0.0037    f X   ft  ltu          t u  2    2 The χ u analysis was modified from below.    S ft  u  2 , ft  u 1, ft  u , ft  u 1, ft  u  2  S0   2  7   u   exp m 1   um       sim     um exp   S       S0 um         2 (64) The global fitting was modified below.  2  f8  24      sim exp  24 7   S  ft  u  2 , ft  u 1 , ft  u , ft  u 1 , ft  u  2        S  S0   S0     um    um   u  8 m 1  exp    um     272 2 (65) Appendix IX. Chapter IV Unconstrained Fitting Similar to Chapter III, an alternate “unconstrained” fitting model was also considered for the mHFP data in Chapter IV. In Chapter IV, the unconstrained model had local mixing of registries t = 8-24 where different registries could be locally mixed in a single HFP oligomer. exp The overall goal was to approximate the best-fit ft’s based on the (S/S0) of many mHFP samples that were labeled differently using an unbiased data analysis method. Previous work has shown that “fully constrained” and “unconstrained” models generally yield similar fractional 36,63 populations for REDOR experiments and η were the residue numbers of the 13 . Each sample was indexed by u = Ϛ + η – 1 where Ϛ CO and 15 N labeled residues. For a particular “middle” HFP, the indices t1 and t2 describe the registries with the two adjacent strands. The adjacent strand of the t1 registry is hydrogen bonded to the labeled 13 CO of the middle HFP while t2 describes the adjacent strand that is not hydrogen bonded to the labeled 13 CO of the middle HFP. The simulated signals for the unconstrained model are described by Eq (66).   S sim     0.99  J  0.0037   na    0.98  K  0.011   na     0.33   K  0.011 lu lN lC         (66)   24 24   lab       0.99   J  0.0037       ft1  ft 2      lt1t 2u  t  8 t  8  1 2    Similar to the fully constrained model, the 1t1t2u lab ( ) = 1 except when t1 = u – 1, t1 = u, t1 = u + 1, t2 = u – 1, t2 = u, and/or t2 = u + 1. In these latter cases, the 1t1t2u lab ( ) were determined by SIMPSON calculations. Additionally, the spin geometry of an adjacent strand is denoted X 273 for t1 when t1 ≠ u – 1, t1 ≠ u, or t1 ≠ u + 1 and is denoted X for t2 when t2 ≠ u – 1, t2 ≠ u, or t2 ≠ u + 1. The iterative fitting for the unconstrained model is illustrated in Figure 50 and the χ analysis for each sample u data set is described by Eq (50). Table 17. Unconstrained model mHFP. 2 ft Best-fit ft χ 0.00 1.4 f8 f9 0.00 3.0 f10 0.01 3.8 f11 0.03 1.8 f12 0.06 5.3 f13 0.13 2.4 f14 0.05 11.0 f15 0.13 3.9 f16 0.10 6.1 f17 0.14 10.7 f18 0.11 12.5 f19 0.01 12.2 f20 0.13 6.2 f21 0.00 8.4 f22 0.01 10.1 f23 0.03 5.6 f24 0.00 12.1 The 24  ft  0.94 t 8 2 and the χ min = 116. 274 2 Figure 50. Flow chart for unconstrained iterative fitting. Each iteration is denoted by the variable 2 κ, and the χ u calculations are found in Chapter IV, Eq (50). 275 Appendix X. Freed Mutations min Figure 51. The ΔGt are plotted for registries t = 8-24 for HFP, V2E-HFP, F11G-HFP, and min F11V-HFP. For each registry, the F11G-HFP ΔGt are greater than or equal to the HFP ΔGt min which may contribute toward F11G-HFP’s lower fusion activity. The F11V-HFP ΔGt is approximately equal to the HFP ΔGt min relative to mHFP for each registry. 276 min Appendix XI. Raw Data for mHFP and mV2E-HFP. Table 18. mHFP Δ(S/S0) exp . Δ(S/S0) exp exp u HFP 48.2 ms 40.2 ms 32.2 ms σ 24.2 ms 8 A6CG3N 0.057 0.016 0.052 0.012 0.037 0.011 0.038 0.009 0.030 0.006 0.017 0.006 0.006 0.007 9 L7CG3N 0.063 0.018 0.068 0.012 0.047 0.011 0.033 0.009 0.032 0.009 0.009 0.006 0.012 0.006 10 F8CG3N 0.111 0.021 0.062 0.017 0.039 0.019 0.046 0.016 0.033 0.012 0.022 0.007 0.015 0.010 11 L9CG3N 0.141 0.022 0.097 0.017 0.081 0.011 0.066 0.012 0.046 0.009 0.026 0.006 0.014 0.005 12 L9CI4N 0.215 0.023 0.170 0.016 0.113 0.011 0.095 0.012 0.060 0.009 0.016 0.009 0.011 0.009 13 L9CG5N 0.256 0.025 0.218 0.016 0.172 0.014 0.102 0.015 0.067 0.012 0.034 0.008 0.010 0.005 14 L12CG3N 0.235 0.021 0.171 0.011 0.138 0.014 0.109 0.013 0.088 0.012 0.033 0.009 0.003 0.006 15 L12CI4N 0.244 0.019 0.215 0.019 0.173 0.014 0.123 0.011 0.093 0.012 0.043 0.008 0.008 0.008 16 L12CG5N 0.253 0.015 0.238 0.015 0.179 0.011 0.128 0.008 0.090 0.010 0.044 0.009 0.012 0.007 17 L12CA6N 0.275 0.021 0.247 0.016 0.192 0.011 0.155 0.013 0.099 0.007 0.058 0.010 0.004 0.009 18 L12CL7N 0.201 0.021 0.188 0.020 0.174 0.012 0.126 0.010 0.085 0.011 0.055 0.011 0.011 0.007 19 F8CL12N 0.157 0.013 0.145 0.013 0.131 0.011 0.082 0.008 0.064 0.006 0.022 0.005 0.010 0.007 20 F8CG13N 0.175 0.015 0.177 0.024 0.161 0.012 0.116 0.015 0.068 0.009 0.017 0.012 0.022 0.012 21 F8CA14N 0.112 0.016 0.072 0.016 0.074 0.013 0.052 0.013 0.028 0.011 0.005 0.010 0.010 0.009 22 F8CA15N 0.096 0.013 0.084 0.016 0.070 0.011 0.021 0.012 0.041 0.009 0.042 0.009 0.011 0.009 277 16.2 ms 8.2 ms 2.2 ms Table 18 (cont’d). 23 F8CG16N 0.113 0.019 0.089 0.019 0.057 0.016 0.059 0.018 0.049 0.011 0.014 0.014 0.026 0.010 24 L9CG16N 0.046 0.014 0.050 0.014 0.015 0.019 0.024 0.008 0.031 0.006 0.016 0.007 0.006 0.005 28 F8CA21N 0.044 0.021 0.043 0.017 0.045 0.013 0.032 0.013 0.021 0.011 0.017 0.010 0.016 0.009 Table 19. mV2E-HFP Δ(S/S0) exp . Δ(S/S0) exp exp u mV2E-HFP 48.2 ms 40.2 ms 32.2 ms σ 24.2 ms 8 A6CG3N 0.055 0.012 0.052 0.010 0.053 0.014 0.018 0.010 0.028 0.010 0.026 0.010 0.009 0.010 9 L7CG3N 0.062 0.017 0.058 0.012 0.047 0.010 0.032 0.010 0.027 0.010 0.005 0.010 0.014 0.010 10 F8CG3N 0.068 0.016 0.050 0.013 0.047 0.012 0.041 0.012 0.024 0.010 0.018 0.010 0.024 0.010 11 L9CG3N 0.062 0.013 0.053 0.017 0.050 0.011 0.040 0.012 0.031 0.010 0.006 0.011 -0.010 0.010 12 L9CI4N 0.085 0.016 0.086 0.010 0.059 0.010 0.030 0.014 0.022 0.010 0.012 0.010 0.001 0.010 13 L9CG5N 0.129 0.025 0.078 0.025 0.063 0.020 0.047 0.016 0.038 0.010 0.044 0.017 0.002 0.013 14 L12CG3N 0.154 0.012 0.114 0.014 0.073 0.012 0.061 0.010 0.034 0.010 0.043 0.013 0.007 0.017 15 L12CI4N 0.195 0.010 0.175 0.011 0.143 0.010 0.095 0.010 0.046 0.016 0.039 0.010 0.004 0.013 16 L12CG5N 0.287 0.023 0.245 0.014 0.197 0.015 0.129 0.012 0.073 0.010 0.034 0.012 0.019 0.010 17 L12CA6N 0.310 0.020 0.271 0.014 0.238 0.016 0.175 0.019 0.078 0.020 0.035 0.020 0.019 0.012 18 L12CL7N 0.303 0.022 0.302 0.018 0.254 0.019 0.194 0.012 0.113 0.012 0.057 0.013 0.025 0.017 278 16.2 ms 8.2 ms 2.2 ms Table 19 (cont’d). 19 F8CL12N 0.346 0.012 0.280 0.012 0.213 0.010 0.144 0.010 0.069 0.014 0.046 0.010 0.013 0.011 20 F8CG13N 0.398 0.017 0.379 0.014 0.330 0.010 0.262 0.010 0.146 0.010 0.056 0.010 0.009 0.010 21 F8CA14N 0.257 0.011 0.198 0.015 0.179 0.010 0.130 0.010 0.075 0.010 0.043 0.010 0.007 0.010 22 F8CA15N 0.155 0.015 0.103 0.014 0.075 0.012 0.060 0.012 0.029 0.010 0.022 0.011 0.005 0.010 23 F8CG16N 0.113 0.011 0.103 0.010 0.087 0.011 0.077 0.010 0.041 0.010 0.016 0.011 0.011 0.010 24 L9CG16N 0.101 0.018 0.090 0.015 0.058 0.019 0.054 0.013 0.029 0.010 0.014 0.011 0.000 0.013 279 Appendix XII. Boltzmann Fraction Populations. A Botlzmann distribution was used to calculate fractional populations of each registry by Eq. (67). ft  Gtmin  kT exp (67) 24 G min  t  kt t  8 exp Table 20. mHFP fully constrained model and Boltzmann distribution based ft. Registry (t) 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 mHFP ft (fully constrained) 0.00 0.00 0.01 0.04 0.07 0.16 0.06 0.15 0.12 0.18 0.13 0.02 0.15 0.00 0.01 0.04 0.00 280 calc ft 0.00 0.01 0.02 0.01 0.08 0.03 0.13 0.12 0.03 0.20 0.29 0.04 0.02 0.01 0.00 0.00 0.00 Appendix XIII. L9R Mutant Discussion Previous experiments have demonstrated that V2E mutated gp41 expressed with wild type gp41 can dominantly inhibit membrane fusion, Appendix III, while L9R mutated gp41 21 expressed with wild type gp41 only stoichiometrically inhibits fusion . Chapter IV of this dissertation demonstrated that mV2E-HFP forms a narrower distribution of registries that were shifted toward longer registries relative to mHFP. To date, there has not been any registry specific membrane insertion data collected for any HFP construct, but mV2E-HFP has a 26 membrane surface location and a predominant population of longer registries which is consistent with longer registries having a surface membrane location. The mV2E-HFP forms 31 stable β sheet oligomers (Chapter IV) that are less fusion active than mHFP β sheet oligomers . Since both mHFP and mV2E-HFP favorably fold into t ~20 registries, a reasonable hypothesis is that the transdominant inhibition may occur when V2E mutant gp41 forms β sheet oligomers with wild type gp41 that preferentially fold into t ~20 registries, and these registries have a membrane surface location that result in fusion inactive oligomers. Similar to the V2E-HFP, the L9R-HFP has one hydrophobic residue mutated to a charged residue, but L9R-HFP did not dominantly inhibit membrane fusion. While this effect may have been due to amino acid type, I hypothesize that it’s due to the sequential placement of the charged residue. For mV2E-HFP, the magnitude of the each Gtmin value was not correlated to the ft’s (i.e. Gtmin does not accurately approximate each registry’s free energy and other 107 energetic contributions must be considered), but registries with ft > 0.00 generally had negative Gtmin values. It would seem that registries with negative Gtmin can stay bound to 281 the lipids while registries with positive Gtmin do not since they are typically not observed in membrane-associated HFP constructs. While mV2E-HFP has a partially inserted or membrane surface location, it stays bound to membranes and has a predominantly β sheet structure whereas 38 lyophilized HFP without membranes forms a broad distribution of secondary structures . Both HFP and V2E-HFP had registries with negative and similar Gtmin values for most registries. Using similar analysis, L9R-HFP has predominantly positive Gtmin values and a minimum Gtmin = -0.62 kcal/strand which is approximately equal to the t = 11 registry in HFP. In HFP, the G min = -0.62 kcal/strand and the G min = -0.70 kcal/strand. The magnitudes of these 11 20 registries’ insertion energies are similar which suggests that other factors, such as side chain packing, are more favorable for the t = 20 registry than the t = 11 registry. Since mL9R-HFP only has a couple registries with negative Gtmin and those registries are not formed in mHFP or mV2E-HFP, there are four hypotheses for mL9R-HFP structure: (1) mL9R-HFP will form longer registries like mV2E-HFP; (2) mL9R-HFP will form predominant t = 11 β sheet registries since this registry has the most negative Gtmin ; (3) mL9R-HFP will not preferentially fold into β sheets since the registries with favorable Gtmin may have unfavorable energetic contributions from other factors, such as sidechain packing; or (4) mL9R-HFP does not bind to the membrane. In mHFP, the most hydrophobic intrastrand sequence, L7-L12, likely plays a key role in stabilizing the membrane inserted structures. In mL9R-HFP, this region has a net positive contribution to G min nt because 282 G Arg 2.58 kcal/mol while 2GLeu  2GPhe  GGly   1.00 kcal/mol. This may not allow for stable binding/partial membrane insertion of β sheet oligomers. A pilot experiment was run to test whether mL9R-HFP forms longer registries like mV2E-HFP using mL9R-HFP with F8CG13N labeling. This sample was prepared identical to mHFP samples and the L9R-HFP induced aggregation among LUV’s, but the LUV’s stuck to the conical vial and precipitated out of solution making a clearer overall solution when compared to mHFP or mV2E-HFP, Figure 54. Due to difficulties acquiring signal above the noise level for REDOR experiments at dephasing times greater than 8 ms, a cross polarization spectrum was taken to observe the chemical shift distribution of the Phe-8 13 CO to obtain secondary structure information, Figure 55. The reduced signal intensity was likely due to L9R-HFP forming a distribution of membrane bound and unbound structures where more L9RHFP is likely unbound than observed for mHFP and mV2E-HFP samples. The NMR sample does not contain the unbound peptide. No data were collected to quantify the amount of unbound L9R-HFP. The mL9R-HFP with F8CG13N labeling had 174.6 ppm chemical shift and a 7.5 ppm line full-width at half maximum height that suggested the presence of a broad distribution of secondary structures since the chemical shifts spanned that of both α helical and β sheet 77 secondary structures . This result was similar to the HFP-NC sample (Chapter III) where 175.9 ppm chemical shift was observed with lyphophilized F8C and A6NL7N labeled HFPs with a 7.0 ppm line full-width at half maximum height (Chapter III). Conversely, the HFP-F8CG13N sample had a 173.1 ppm chemical shift and a 3.5 ppm line width at half maximum height which strongly indicated β sheet secondary structure. These combined results suggest that mL9R-HFP does not favorably fold into membrane bound/partially inserted β sheet oligomers to the extent of mHFP and mV2E-HFP. These data are also consistent with the idea that L9R mutated gp41 does not form an oligomeric structure with wild type gp41 since mL9R-HFP does not preferentially 283 form β sheet oligomers. These results combined with the mV2E-HFP results may explain why V2E mutated gp41 transdominantly inhibits wild type gp41 while L9R mutated gp41 only stoichiometrically inhibits wild type gp41. 3 HFP L9R 1 0 G min t (kcal/strand) 2 -1 8 -2 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Registry (t) Figure 52. The membrane insertion energies were derived from the Hessa biological hydrophobicity scale for the HFP and L9R-HFP by the methods described in Chapter IV. The L9R-HFP has predominantly positive G min whereas HFP has many registries with negative nt G min which suggests that the distribution of registries should be different between constructs. nt Additionally, it is not obvious that mL9R-HFP should form membrane inserted β sheets since t < 12 registries were minimally populated in mHFP, and t > 12 registries have positive G min in nt mL9R-HFP. 284 Figure 53. NMR sample of mV2E-HFP (left) compared to a water standard (right) prior to centrifugation and after mixing overnight. Aggregation of LUV’s is evident in mV2E-HFP under our sample preparation conditions. 285 Figure 54. NMR samples of mL9R-HFP (L9R), mHFP (WT), mV2E-HFP (V2E) prior to centrifugation and after mixing overnight. The mL9R-HFP sample appeared to be more transparent than both mHFP and mV2E-HFP, but LUV aggregation was evident. 286 Figure 55. mL9R-HFP with F8CG13N labeling. The chemical shift of 174.6 ppm and 7.5 ppm line full-width at half maximum height indicate the presence of a distribution of secondary structures since the peak spans chemical shifts of α helical, random coil and β sheet structures. 287 min i , and n Values. Appendix XIV. HFP, V2E, and L9R Gt t t Table 21. Energy minimized membrane insertion energy parameters. HFP V2E L9R Gtmin Gtmin Gtmin t it nt it nt it (kcal/strand) (kcal/strand) (kcal/strand) nt 8 -0.08 1 8 0.99 3 4 -0.08 1 8 9 -0.63 1 9 0.44 3 5 -0.19 2 7 10 -0.74 2 8 -0.3 4 4 -0.3 4 4 11 -0.62 4 5 -0.62 4 5 -0.62 4 5 12 -1.17 4 6 -1.17 4 6 -0.02 5 4 13 -0.87 2 11 -0.57 5 5 2.26 2 11 14 -1.31 6 4 -1.31 6 4 1.82 6 4 15 -1.3 4 9 -1.3 4 9 1.83 4 9 16 -0.89 6 6 -0.89 6 6 2.24 6 6 17 -1.44 6 7 -1.44 6 7 1.69 6 7 18 -1.55 7 6 -1.55 7 6 1.58 7 6 19 -1 8 5 -1 8 5 2.13 8 5 20 -0.7 7 8 -0.7 7 8 2.43 7 8 21 -0.59 7 9 -0.59 7 9 2.54 7 9 22 -0.04 8 8 -0.04 8 8 0.61 10 4 23 0.28 9 7 0.28 9 7 0.72 10 5 24 -0.02 11 4 -0.02 11 4 -0.02 11 4 288 Appendix XV. Summary of Hessa Biological Hydrophobicity Scale Amino acid properties, such as hydrophobicity, are thought to play a key role in determining membrane protein structure. Amino acid solvent partitioning experiments have been used to generate hydrophobicity scales for the twenty naturally occurring amino acids 112,126 . These scales were developed in part to help with predicting membrane protein structures. Solvent partitioning experiments model hydrophobicity well, but the free energy associated with proteinmembrane interactions, such as binding and folding to name a few, are not accounted for in these 107 experiments . These concerns have lead to the development of a thermodynamically based biological hydrophobicity scale where α helices with unique amino acid sequences were inserted 93 into membranes (more details below). I have provided a basic overview of the experiments and highlighted key points that are relevant to structure of membrane-associated HFP constructs in the remainder of this appendix. Figure 56 illustrates the experimental setup where an α helical segment was mutated into the integral membrane protein leader peptidase. Membrane insertion of this H-segment buries the glycosylation site 2 (G2) into the cytoplasm and exposes glycosylation site 1 (G1) for glycosylation of G1 while translocation of this H-segment lead to glycosylation of both G1 and G2. SDS-PAGE gels were used to quantify the fraction of protein with only G1 glycosylated, fg1, and the fraction of protein with both G1 and G2 glycosylated, fg2. These fractions were expressed as the observed or “apparent” equilibrium constant for membrane insertion of the H segment where Kapp = fg1/fg2. The apparent change in free energy between the membrane inserted and translocated H-segments was calculated by ΔG 289 ins,seg app = -RTlnKapp. Therefore, determination of the change in free energy for membrane insertion was determined by mutating a single amino acid within a sequence and comparing the ΔG by ΔG ins,seg app between the two sequences ins ins,mutant ins,seg a = ΔG app – ΔG app. Figure 56. The model systems were composed of two transmembrane domains (TM1 and TM2), two luminal domains (P1 and P2), and two glycosylation acceptor sites (G1 and G2). A third helical transmembrane domain (H) is illustrated in red. Translocation of the H segment from the membrane allows for glycosylation of both G1 and G2 while membrane insertion of the H 93 segment only allows for glycosylation of G1. This figured was modified from literature . 290 93 Figure 57. The Hessa biological hydrophobicity scale. This figure was taken from literature . 291 Figure 58. The positional dependence of amino acids. Key points from these figured are 93 discussed below. These figures were taken from literature . 292 Figure 58 (cont’d). The H-segments were generally composed of GGPG-X19-GPGG where X is the Hsegment primarily composed of Ala and Leu residues, and the Pro and Gly residues were added adjacent to the α helical segment so that flanking residues had a low probability of secondary structure formation. Sequences composed of three or four Leu and sixteen or fifteen Ala, 293 respectively, were used to evaluated the flanking residues contribution to ΔG ins,seg app using the sequence GzPG-X19-GPGz where z was varied from 2 to 6. The resultant ΔG ins,seg app values varied by ±0.2 kcal/mol from the z = 2 construct which suggested that residues flanking the Hsegments had minimal contributions to the ΔG ins,seg app. Additionally, different sequences of the H-segment were developed with 3 and 4 Leu residues where the six flanking Gly residues were mutated to Asn residues, NNPN-X19-NPNN, and the ΔG ins,seg app increased by an average of only +0.5 kcal/mol whereas six Gln residues placed in the central region of the H-segment should increase ΔG ins,seg app by ~+14.2 kcal/mol. These combined results suggest that flanking residues have a minimal contribution to the ΔG ins,seg app values, and support that the K6 tag in our HFP sequence minimally affects the insertion energy of specific β sheet registries since the K6 tag is C-terminal of the membrane inserted region. Additionally, there are residues without regular secondary structure between the membrane inserted region and the K6 tag. In Chapter IV, the insertion of energy of a structure was determined by summing ΔG values, but the “apparent” ΔG ins a ins a , where the subscript “a” denotes amino acid type, are only approximately additive since the insertion energies have different positional dependences within a structure. For example, the hydrophobic residues Leu and Phe demonstrated a small positional dependence, Figure 58a,b, while more polar residues, such as Tyr, Trp, Asn, Gln, Lys, and Ser, ins,seg app were lowered as these residues demonstrated a large positional dependence where ΔG were placed away from the central region and toward the edges of the H-segment, Figure 58a,c,d. Additionally, Pro residues are known to disrupt secondary structure. All secondary 294 structure types have hydrogen bonding between amide and carbonyl groups that result from dipole-dipole interactions. Thus, a sequence with secondary structure is generally considered to be less polar and more energetically favorable for membrane insertion than its unordered counterpart. In this study, residues within the H-segment were mutated into Pro residues at different positions, Figure 58e,f. Pro mutations in the central region of the H-segment drastically increased ΔG ins,seg app. As Pro was moved away from the central region of the H-segments, the ins,seg app became more negative which was consistent with the observed position dependence ΔG of the more polar residues. Polar residues are frequently found in secondary structures and do not inherently disrupt secondary structure; however, relative to an aqueous environment, it is energetically unfavorable to insert polar residues into the hydrophobic environment of the membrane interior. 295 lab lab Appendix XVI. Chapter IV Tables for 1tuv ( ) or 1t1t2uv ( ) spin geometries with calculated values from SIMPSON for the three registry fittings. Similar to Appendix VII, the spin geometries are illustrated for the associated calculated 1tuv lab ( ) for specific t, u, and v in the fully constrained model or calculated 1t1t2uv lab ( ) for specific t1, t2, u, and v in the unconstrained model. In each spin geometry schematic, N or C respectively denote a 15 N nucleus or a registry. Each arrow denotes 13 13 CO nucleus included in the simulation, and X denoes X 15 CO- N dipolar coupling considered in the simulation. For unconstrained model geometries, the top/middle strand registry is at the top of the schematic and the middle/bottom strand registry is at the bottom of the schematic. When more than one spin geometries are shown, each  ( ) is the average for the displayed geometries. N C N lab N Time ( ) 1tuv ( ) 48.2 0.4501 40.2 0.5814 32.2 0.7103 24.2 0.8263 16.2 0.9186 8.2 0.9785 2.2 0.9984 C t u-1 & u+1 N Figure 59. Spin geometries and simulated data for Chapter IV three registry fittings. 296 Figure 59 (cont’d). N Time ( ) 48.2 40.2 32.2 24.2 16.2 8.2 2.2 C lab 1tuv ( ) 0.0939 0.1186 0.1964 0.3786 0.6453 0.8938 0.9917 t u N X C N X C N 297 Time ( ) 48.2 40.2 32.2 24.2 16.2 8.2 2.2  lab   lt1t2 uv 0.6917 0.7769 0.8521 0.9143 0.9608 0.9898 0.9992 t1 x t2 u-1 & u+1 Figure 59 (cont’d). N N Time ( )  lab   lt1t2 uv 48.2 C X 40.2 32.2 24.2 16.2 8.2 2.2 C X 0.6592 0.7520 0.8348 0.9039 0.9560 0.9885 0.9991 N Time ( ) 48.2 40.2 32.2 24.2 16.2 8.2 2.2 C t1 u-1 & u+1  lab   lt1t2 uv -0.0470 -0.0230 0.1236 0.3778 0.6710 0.9064 0.9928 t2 x t1 u t2 x X X Time ( ) 48.2 40.2 32.2 24.2 16.2 8.2 2.2 C N 298  lab   lt1t2 uv 0.5675 0.6824 0.7870 0.8754 0.9427 0.9850 0.9989 t1 x t2 u Figure 59 (cont’d). N N Time ( ) 48.2 C C 40.2 32.2 24.2 16.2 8.2 2.2 N N N N Time ( ) 48.2 C C N N N C N 40.2 32.2 24.2 16.2 8.2 2.2 N Time ( ) 48.2 C N 299 40.2 32.2 24.2 16.2 8.2 2.2  lab   lt1t2 uv 0.0011 0.0156 0.1288 0.3565 0.6476 0.8974 0.9921  lab   lt1t2 uv 0.3739 0.5132 0.6571 0.7914 0.9012 0.9737 0.9980  lab   lt1t2 uv 0.4926 0.6037 0.7200 0.8293 0.9191 0.9785 0.9984 t1 u t2 u-1 & u+1 t1 u-1 & u+1 t2 u t1 u+1 or u-1 t2 u-1 or u+1 lab Appendix XVII. Chapter IV Tables for the unique 1tuv ( ) spin geometries with calculated values from SIMPSON for the five registry fittings. Similar to Appendix VII, the spin geometries are illustrated for the associated calculated 1tuv lab ( ) for specific t, u, and v in the fully constrained model. In each spin geometry schematic, N or C respectively denote a simulation. Each arrow denotes 13 15 N nucleus or a 13 CO nucleus included in the 15 CO- N dipolar coupling considered in the simulation. 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