ADVANCES IN METAL ION MODELING By Pengfei Li            A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of ChemistryÑDoctor of Philosophy 2016 ABSTRACT ADVANCES IN METAL ION MODELING By Pengfei Li Metal ions play fundamental roles in geochemistry, biochemistry and materials science. With the tremendous increasing power of the computational resources and largely inventions of the computational tools, computational chemistry became a more and more important tool to study various chemical processes. Force field modeling strategy, which is built on physical background, offered a fast way to study chemical systems at atomic level. It could offer considerable accuracy when combined with the Monte Carlo or Molecular Dynamics simulation protocol. However, there are various metal ions and it is still challenging to model them using available force field models. Generally there are several models available for modeling metal ions using the force field approach such as the nonbonded model, the bonded model, the cationic dummy atom model, the combined model, and the polarizable models. Our work concentrated on the nonbonded and bonded models, which are widely used nowadays. Firstly, we focused on filling in the blanks of this field. We proposed a noble gas curve, which was used to describe the relationship between the van der Waals radius and well depth parameters in the 12-6 Lennard-Jones potential. By using the noble gas curve and multiple target values (the hydration free energy, ion-oxygen distance, coordination number values), we have consistently parameterized the 12-6 Lennard-Jones nonbonded model for 63 different ions (including 11 monovalent cations, 4 monovalent anions, 24 divalent cations, 18 trivalent cations, and 6 tetravalent cations) combined with three widely used water models (TIP3P, SPC/E,and TIP4PEW). Secondly, we found there is limited accuracy of the 12-6 model, which makes it hard to simulate different properties simultaneously for ions with formal charge equal or larger than +2. By considering the physical origins of the 12-6 model, we proposed a new nonbonded model, named the 12-6-4 LJ-type nonbonded model. We have systematically parameterized the 12-6-4 model for 55 different ions (including 11 monovalent cations, 4 monovalent anions, 16 divalent cations, 18 trivalent cations, and 6 tetravalent cations) in the three water models. It was shown that the 12-6-4 model could reproduce several properties at the same time, showing remarkable improvement over the 12-6 model. Meanwhile, through the usage of a proposed combining rule, the 12-6-4 model showed excellent transferability to mixed systems. Thirdly, we have developed the MCPB.py program to facilitate building of the bonded model for metal ion containing systems, which can largely reduce human efforts. Finally, an application case of a metallochaperone - CusF was shown, and based on the simulations we hypothesized an ion transfer mechanism.  iv This dissertation is dedicated to my parents, who always love, trust, and support me. v ACKNOWLEDGEMENTS  I thank God for leading my life, loving me, and being my Lord. I thank my family especially my parents for their generous love. My parents have raised me and spared no effort to offer me a good life and excellent education. They have set excellent examples for me about hard work and good behavior. They created a loving home, always love me and are my powerful backing. I would like to thank the brothers and sisters in the Lansing Chinese Christian Church, who have shared the Gospel with me and helped me to know God. I learned a lot about Love and how to practice love in the Church, which really benefits my life. I also thank the Chinese community in the Greater Lansing area, which has enriched my life at Michigan State University (MSU).  I acknowledge my advisor Professor Kenneth M. Merz Jr. for his patient love and kind support of my research. I thank him for being a model theoretical chemist. His positive attitude, broad thoughts, and deep insights educated me greatly. I thank my guidance committee members: Professor Robert I. Cukier, Professor Benjamin G. Levine, and Professor James E. Jackson, and my guidance committee when I was in the University of Florida (UF): Professor Adrian E. Roitberg, Professor Samuel B. Trickey, Professor Benjamin W. Smith, and Professor Clifford R. Bowers, as well as my research advisor during my undergraduate, Professor Zexing Cao in Xiamen University, for their kind help and support during my research career. I thank the members of the Merz research group for their company. Being a member of the Merz research group is a precious memory for me. Among them, I specifically thank Ben for handing me the project of designing van  vi der Waals parameters for the zinc ion, which is the beginning of this research story. I would like to thank Dhruva for his guidance in the early stage of my PhD research, and for his kind help and suggestions during my PhD career. I thank Lin for his hard work concerning parameterization of the nonbonded model for monovalent and highly charged ions. I would like to thank Yipu for his help with computer programming, which helped me greatly in my start-up PhD research. I would also like to thank Mike and David for their help with the language of my manuscripts.  Besides the Merz group members, I thank Shuai Wang (UF) for his help of writing the analysis code for determining the ion-oxygen distance. I would like to thank the AMBER developersÕ team. It is an honor to be a part of the team and I really enjoyed the meetings, which broadened my horizons and offered me precious opportunities to talk with the experts in related fields. Among them I especially thank Professor David A. Case, Dr. Jason M. Swails, and Dr. Hai Nguygen for their generous help with computer programming in the AMBER software package. I thank the financial support from the United States National Institutes of Health (RO1Õs GM044974 and GM066859), and the computing support from the high performance computing centers at UF and MSU. I thank the kind support from the secretary of my advisor, Ms. Carey Byerrum and one of the graduate secretaries in the Department of Chemistry at MSU, Ms. Heidi Wardin, who are always kind and efficient. Finally I thank other teachers, colleagues, administrators, secretaries, and friends who have enriched my life and supported my research.    vii TABLE OF CONTENTS  LIST OF TABLES .............................................................................................................. ix LIST OF FIGURES ........................................................................................................... xii KEY TO ABBREVIATIONS .......................................................................................... xvi CHAPTER 1: INTRODUCTION ........................................................................................ 1 1.1 General Introduction of Models for Metal Ions ........................................... 1 1.2 The 12-6 Nonbonded Model ........................................................................ 3 1.3 The 12-6-4 Nonbonded Model .................................................................... 9 1.4 The Bonded Model and MCPB.py Program ............................................. 11 1.5 The CusF Metallochaperone System ......................................................... 14 CHAPTER 2: PARAMETERIZATION OF THE 12-6 NONBONDED MODEL ........... 15 2.1 Target Values ............................................................................................. 15 2.2 Methods ..................................................................................................... 19 2.3 Validation Tests of Several Available LJ Parameters for M(II) Ions ........ 31 2.4 Parameter Space Scanning ......................................................................... 33 2.5 Parameter Determination ........................................................................... 41 2.6 Parameter Assessment ............................................................................... 47 2.7 Error Analysis ............................................................................................ 53 2.8 Conclusions ............................................................................................... 58 CHAPTER 3: PARAMETERIZATION OF THE 12-6-4 NONBONDED MODEL ....... 62 3.1 Model Origin ............................................................................................. 62 3.2 Methods ..................................................................................................... 64 3.3 Parameter Space Scanning and Parameter Determination ........................ 68 3.4 Discussion .................................................................................................. 72 3.5 Parameter Assessment ............................................................................... 85 3.6 Design Your Own Parameters ................................................................... 91 3.7 Conclusions ............................................................................................... 93 CHAPTER 4: DEVELOPMENT OF THE MCPB.PY PROGRAM ................................ 96 4.1 Code Structure ........................................................................................... 96 4.2 Workflow ................................................................................................... 97 4.3 Usage ....................................................................................................... 100 4.4 Examples ................................................................................................. 103 4.5 Conclusions ............................................................................................. 111 CHAPTER 5: APPLICATION CASE OF THE CUSF METALLOCHAPERONE SYSTEM ......................................................................................................................... 112   viii APPENDICES ................................................................................................................. 116 APPENDIX A: TABLES .................................................................................... 117 APPENDIX B: FIGURES ................................................................................... 164 APPENDIX C: COPYRIGHT NOTICE ............................................................. 208 BIBLIOGRAPHY ........................................................................................................... 209    ix LIST OF TABLES  Table 1. Target values of the HFE, IOD and the CN of the first solvation shell for monovalent ions. .............................................................................................................. 117 Table 2. Target values of the HFE, IOD and the CN of the first solvation shell for divalent metal ions. .......................................................................................................... 118 Table 3. Target values of the HFE, IOD and the CN of the first solvation shell for trivalent and tetravalent metal ions. ................................................................................. 119 Table 4. Parameters of four different water models. ...................................................... 120 Table 5. Absolute HFEs, IODs and CNs of M(II) ions employing parameters available in the AMBER package. ...................................................................................................... 121 Table 6. Absolute HFEs, IODs and CNs of Zn(II) employing parameters developed by Stote and Karplus. ............................................................................................................ 122 Table 7. Absolute HFEs, relative HFEs (relative to the Cd2+ ion), IODs and CNs of M(II) ions with the parameters taken from Babu and Lim. ....................................................... 123 Table 8. Standard deviations of the IOD values for the LJ grids of divalent metal ions. .....   ......................................................................................................................................... 124 Table 9. The experimental LJ parameters for noble gas atoms.a .................................... 125 Table 10. Comparison of results of different scaling methods in the VDW-disappearing and VDW-appearing steps of the HFE calculations for the Be2+ and Ba2+ ions.a ........... 126 Table 11. Final optimized HFE parameter set of divalent metal ions for different water models. ............................................................................................................................. 127 Table 12. The simulated HFE, IOD and CN values of the HFE parameter set for divalent metal ions.a ...................................................................................................................... 128 Table 13. Final optimized IOD parameter set of divalent metal ions.a ........................... 130 Table 14. The simulated HFE, IOD and CN values of the IOD parameter set for divalent metal ions.a ...................................................................................................................... 131 Table 15. Final optimized CM parameter set of divalent metal ions for different water models. ............................................................................................................................. 133 Table 16. The simulated HFE, IOD and CN values of the CM parameter set for divalent metal ions.a ...................................................................................................................... 134  x Table 17. Final optimized HFE parameter set of monovalent ions for different water models. ............................................................................................................................. 136 Table 18. The simulated HFE, IOD and CN values of the HFE parameter set for monovalent metal ions.a ................................................................................................... 137 Table 19. Final optimized IOD parameter set of monovalent ions for all the three water models. ............................................................................................................................. 139 Table 20. The simulated HFE, IOD and CN values of the IOD parameter set for monovalent metal ions.a ................................................................................................... 140 Table 21. Estimated HFE parameter set of highly charged metal ions for different water models. ............................................................................................................................. 142 Table 22. Estimated IOD parameter set of highly charged metal ions for different water models. ............................................................................................................................. 143 Table 23. Comparison between the Rmin/2 parameters from the HFE parameter set for the TIP3P water model and the VDW radii calculated from QMSP method.. ...................... 144 Table 24. The activity derivatives for six different ionic solutions in the SPC/E water model (under ~0.3 M condition). ..................................................................................... 145 Table 25. The average IOD errors for the HFE parameter set of the mono-, di-, tri- and tetravalent cations. ........................................................................................................... 146 Table 26. The average HFE errors for the IOD parameter sets of the mono-, di-, tri- and tetravalent cations. ........................................................................................................... 147 Table 27. Final optimized 12-6-4 parameter set of divalent metal ions for different water models. ............................................................................................................................. 148 Table 28. Computed C4 values for the 12-6-4 parameter set of divalent metal ions for different water models.a ................................................................................................... 149 Table 29. The simulated HFE, IOD and CN values of the 12-6-4 parameter set for divalent metal ions.a ......................................................................................................... 150 Table 30. Final optimized 12-6-4 parameter set of monovalent ions for different water models. ............................................................................................................................. 151 Table 31. The simulated HFE, IOD and CN values of the 12-6-4 parameter set for monovalent ions.a ............................................................................................................ 153 Table 32. Final optimized 12-6-4 parameter set of highly charged metal ions for different water models. ................................................................................................................... 154  xi Table 33. The simulated HFE, IOD and CN values of the 12-6-4 parameter set for highly charged metal ions.a ......................................................................................................... 156 Table 34. Polarizabilities used in the 12-6-4 model for different atom types in the AMBER FF. ..................................................................................................................... 158 Table 35. The IODs of the Mg2+ and Cl- ions at different concentrations.a .................... 159 Table 36. Comparison of the structural properties predicted by the three different Mg2+ parameters used in the simulations of a Mg2+-nucleic acid system. ................................ 160 Table 37. Polarizability values of the AMBER atom types used in the 12-6-4 nonbonded model simulation of a metalloprotein system.a ................................................................ 161 Table 38.  Small model capping schemes for coordinated amino acid residues. ............ 162 Table 39. The RMSD values between the CSD, QM optimized and MM minimized structures.a ....................................................................................................................... 163    xii LIST OF FIGURES  Figure 1. Atomic ions that were parameterized for the 12-6 model in current work. These ions are shown with blue background. The monovalent anions, monovalent cations, divalent cations, trivalent cations, and tetravalent cations are shown in light green, yellow, orange, red, dark red, respectively. The ions that have multiple oxidation states are shown in white, which are Cr2+/Cr3+, Fe2+/Fe3+, Cu+/Cu2+, Ag+/Ag2+, Ce3+/Ce4+, Sm2+/Sm3+, Eu2+/Eu3+, and Tl+/Tl3+. Besides these atomic ions, the H3O+ and NH4+ ions were also parameterized in current work. ........................................................................................ 164 Figure 2. Thermodynamics cycle used to determine the HFE values. ........................... 165 Figure 3. HFE fitting curves for 24 M(II) metal ions for the TIP3P water model. ........ 166 Figure 4. HFE fitting curves for Zn2+ in four different water models.  .......................... 167 Figure 5. IOD fitting curves for Zn2+ in four different water models.  ........................... 168 Figure 6. CN switching during a MD simulation when Rmin/2 =2.2 † and !=0.1 kcal/mol for a M(II) ion in the TIP3P water model. ....................................................................... 169 Figure 7. HFE and IOD fitting curves of six representative M(II) metal ions in the TIP3P water model. .................................................................................................................... 170 Figure 8. Determination of the three parameter sets for Zn2+ in the TIP3P water modelÉ.. ......................................................................................................................................... 171 Figure 9. Radial distribution functions of the three parameter sets for Zn2+ in the TIP3P water model. .................................................................................................................... 172 Figure 10. Fitting curves between the HFE and IOD values for the positive (upper) and negative (lower) monovalent ions in the three water models together with the target values of the ions investigated in the present work. ........................................................ 173 Figure 11. Fitting curves between the HFE and Rmin/2 values for the positive (upper) and negative (lower) monovalent ions in the three water models. ......................................... 174 Figure 12. Fitting curves between the IOD and Rmin/2 values for the positive (upper) and negative (lower) monovalent ions in the three water models. ......................................... 175 Figure 13. Unsigned average errors of the VDW radii of different parameter sets using QMSP calculated values as reference. The parameter sets developed herein are shown in the seven green columns on the right side of the figure. ................................................. 176 Figure 14. PDB entry 4BV1. Waters are not shown in the figure, the ferric ions are shown as sliver spheres. This picture was created using VMD.1  ................................... 177  xiii Figure 15. Chain C in PDB entry 4BV1 (upper) and a close-up figure of the metal site in Chain C (lower). The ferric ion is represented as a silver sphere and it is coordinated by one Cys, four His and one water molecule. The figures were made using VMD.1 ......... 178 Figure 16. RMSDs of the backbone CA, C, N atoms (upper) and the metal site heavy atoms (lower) for the simulations with the IOD and 12-6-4 parameter sets using the initial structure (experimental structure) as reference. .............................................................. 179 Figure 17. RMSFs of the heavy atoms of the protein residues in the simulations using the IOD and 12-6-4 parameter sets. ....................................................................................... 180 Figure 18. HFE percent errors for different parameter sets of monovalent ions in the TIP3P water model. HZ+ and HE+ represent the H+ Zundel and Eigen ions respectively. Since we did not design HFE set of parameters for the HZ+ and HE+ ions, they are not shown in the figure. ......................................................................................................... 181 Figure 19. IOD percent errors for different parameter sets of monovalent ions in the TIP3P water model. HZ+ and HE+ represent the H+ Zundel and Eigen ions respectively. Since we did not design IOD set of parameters for the HZ+ and HE+ ions, they are not shown in the figure. ......................................................................................................... 182 Figure 20. The average errors (upper) and average percentage errors (lower) of the IODs for the HFE parameter set for the mono-, di-, tri- and tetravalent cations in the TIP3P water model. .................................................................................................................... 183 Figure 21. The average errors (upper) and average percentage errors (lower) of the HFEs for the IOD parameter set for the mono-, di-, tri- and tetravalent cations in the TIP3P water model. .................................................................................................................... 184 Figure 22. Fitting curves between the HFE and Rmin/2 values (upper) and fitting curves between the IOD and Rmin/2 values (lower) for the positive and negative monovalent ions in different water models. ................................................................................................ 185 Figure 23. Scheme representing intermolecular interactions: the green double-headed arrow and red double-headed arrow represent the interactions that are included and not included in the 12-6 nonbonded model, respectively. ..................................................... 186 Figure 24. Atomic ions that were parameterized for the 12-6-4 model in current work. These ions are shown with blue background. The monovalent anions, monovalent cations, divalent cations, trivalent cations, and tetravalent cations are shown in light green, yellow, orange, red, dark red, respectively. The ions that have multiple oxidation states are shown in white, which are Cr2+/Cr3+, Fe2+/Fe3+, Cu+/Cu2+, Ce3+/Ce4+, and Tl+/Tl3+. Besides these atomic ions, the H3O+ and NH4+ ions were also parameterized in current work. ............ 187 Figure 25. HFE and IOD fitting curves for the Cu2+, Ni2+, Zn2+, Co2+, and Cr2+ ions in the TIP3P water model. ......................................................................................................... 188  xiv Figure 26. HFE and IOD fitting curves for the Fe2+, Mg2+, V2+, Mn2+, and Hg2+ ions in the TIP3P water model. ................................................................................................... 189 Figure 27. HFE and IOD fitting curves for the Cd2+, Ca2+, Sn2+, Sr2+, and Ba2+ ions in the TIP3P water. .................................................................................................................... 190 Figure 28. Workflow of the MCPB.py program. ............................................................ 191 Figure 29. Examples of capping different metal bound amino acid residues in the small model. Structures on the left are the coordinated amino acids in the protein, and the structures on the right are the capped residues in the small models built by MCPB.py. Panels A and B are for PDB entry 1E67, panels C, D, and E are for PDB entries 1AK0, 1Y79, and 2BZS, respectively. Zinc coordination is shown as dotted lines with the bond lengths given in †. ........................................................................................................... 192 Figure 30. Metal ions currently supported by the MCPB.py program. The metals with a blue background use the VDW parameters from section 2 while the metals with green background use the VDW parameters adapted from UFF.2 ............................................ 193 Figure 31. Structure of chain A from PDB entry 1E67 with the zinc ion represented by a VDW sphere and the metal site residues indicated by sticks. This figure was made using VMD.1  ............................................................................................................................. 194 Figure 32. Content of the MCPB.py input file for the construction of the metal site found in chain A of PDB entry 1E67. ........................................................................................ 195 Figure 33. Commands used for the construction of the metal site found in chain A of PDB entry 1E67 using MCPB.py. The second to the fifth lines are for the QM calculations using Gaussian03, while the last line is to build the topology and coordinate files using tleap. ............................................................................................................... 196 Figure 34. Bond parameters (above) and angle parameters (below) for the metal site in chain A from the 1E67 PDB structure. Here M1 is the atom type of the zinc ion, while Y1, Y2, Y3, Y4 are the atom types for the four atoms bound to the zinc ion, which are the GLY45 backbone oxygen atom, the d nitrogen atom in HIE46, the sidechain sulfur atom in CYM112 and the d nitrogen atom in HIE117, respectively. Here HIE and CYM are the ÒAMBER styleÓ residue names: HIE means a His residue which has the e nitrogen protonated, CYM indicates a Cys residue which is negatively charged (i.e. with a CH2S- sidechain group). ............................................................................................................. 197 Figure 35. RMSD values of the protein backbone N, CA and C atoms (upper) and the heavy atoms in the metal site (lower) over 20 ns of simulation. The RMSD values were calculated using CPPTRAJ.3 ........................................................................................... 198 Figure 36. Structure of residue 1 from CSD entry FAJYAR01. This figure was made using VMD.1  ................................................................................................................... 199  xv Figure 37. Content of the MCPB.py input file for the construction of the metal site found in residue 1 of CSD entry FAJYAR01. ........................................................................... 200 Figure 38. Commands used for the construction of the metal site found in residue 1 of CSD entry FAJYAR01 using MCPB.py. The second to the fifth lines are for the QM calculations using Gaussian03, while the last line is to build the topology and coordinate files using tleap. ............................................................................................................... 201 Figure 39. Bond parameters (above) and angle parameters (below) for the metal site of residue 1 from CSD entry FAJYAR01. Here M1 is the atom type of the Os2+ ion, while Y1 to Y6 are the atom types for the six metal bound nitrogen atoms. ............................ 202 Figure 40. Comparison of the QM and MM calculated normal modes. ......................... 203 Figure 41. Cartoon representation of Cu(I)!CusF (PDB entry 2VB2). Cu+ is shown as a pink sphere, and the metal binding residues are represented as sticks. ........................... 204 Figure 42. Cu+"Trp44 distance and Glu46"Met47 # dihedral coordinates (black lines) for our two-dimensional PMF MD simulations shown in (a) a closed structure and (b) a wide-open structure of Cu(I)!CusF. ................................................................................ 205 Figure 43. Scheme employed to generate the starting conformations for the two-dimensional PMF calculation. (Left) Initial conformations were generated by sampling along the diagonal from the closed to the wide-open conformations. Arrows indicate the direction of sampling along the reaction coordinate. Equilibrated conformation from a window was used as the starting coordinate of the adjacent window as we moved along the diagonal. (Right) Scheme employed to sample conformations starting from windows along the diagonal (solid circles) to the hollow circles. .................................................. 206 Figure 44. (Upper) A two-dimensional free energy landscape calculated for the Cu-Trp44 distance versus the Glu46-Met47 # dihedral from our classical MD simulations of Cu(I)!CusF. Cu(I)!CusF is unable to make the transition from the closed to the open conformations. The closed conformations are represented by smaller Cu+-Trp distances, while the open conformations are represented by larger Cu+-Trp44 distances. (Lower) PMF profile based on the umbrella sampling simulations in which the Cu+-Trp44 distance and Glu46-Met47 # dihedral were gradually modified to force the transition from the closed conformation to the wide-open conformation of Cu(I)!CusF. Error bars were calculated using a bootstrapping algorithm implemented in Alan GrossfieldÕs WHAM code (http://membrane.urmc.rochester.edu/content/wham). Error bars were calculated along each axis of the 2D PMF profile. The error bars along the dihedral dimension are between 0.002 and 0.040 kcal/mol with an average of 0.022 kcal/mol.  The error bars along the distance dimension are in the range of 0.001 to 0.128 kcal/mol with an average of 0.012 kcal/mol. .......................................................................................... 207    xvi KEY TO ABBREVIATIONS  AMBER     Assisted Model Building with Energy Refinement CHA   Charge hydration asymmetry CHARMM  Chemistry at Harvard macromolecular mechanics CM   Compromise CN   Coordination number CPMD  Car-Parrinello molecular dynamics FF   Force field GAFF  General AMBER force field GROMACS Groningen machine for chemical simulations GROMOS  A force field in the GROMACS software package HFE   Hydration free energy IOD   Ion-oxygen distance KB   Kirkwood-Buff LAMMPS  Large-scale atomic/molecular massively parallel simulator LJ   Lennard-Jones MC   Monte Carlo MCPB  Metal center parameter builder MD   Molecular dynamics MM   Molecular mechanics NAMD  Nanoscale molecular dynamics NGC   Noble gas curve  xvii OPLS-AA  Optimized potentials for liquid simulations - all atom PBC   Periodic boundary condition PDB   Protein databank PME   Particle mesh Ewald QM   Quantum mechanics QMCF  quantum mechanical charge $eld QMSP  Quantum scaling principle RDF   Radial distribution function RESP  Restrained electrostatic potential TI   Thermodynamic integration UAE   Unsigned average error VDW  van der Waals ZAFF  Zinc AMBER force field pyMSMT     Python metal site modeling toolbox  1 CHAPTER 1: INTRODUCTION  1.1 General Introduction of Models for Metal Ions  Metal ions play significant roles in chemical disciplines such as geochemistry, biochemistry and materials science. There are 87 metals among the first 112 elements in the periodic table. The elements Al, Fe, Ca, Na, K, Mg and Ti occupy about a quarter of the earthÕs crust. There are about one third of the proteins in the protein databank (PDB) that contain metal ions.4 Metal ions such as calcium, zinc, iron, copper, manganese, nickel, and magnesium ions form complexes with surrounding amino acid residues, and serve significant functional roles including structural, electron transfer and catalytic functions.5-18 There are more than 80% large scale industry processes reply on solid catalysis, which are usually related to the transition metal chemistry.19  There are different ways to simulate these ions using theoretical approaches, for example, the quantum mechanics (QM) method,20-22 the molecular mechanics (MM) method4, 23, 24 and the combined QM/MM method.25-28 Classical force fields (FFs), which use analytical functions to represent the relationship between the energy and configuration of a system, have significant speed advantages over the QM based methods. It is a state-of-the-art tool to study systems at the atomic level when combined with Molecular Dynamics (MD)29, 30 or Monte Carlo (MC) method.31, 32 There are a number of strategies that have been employed in the classical FFs: the bonded model, 4, 33-36 the nonbonded model,37-40 the hybrid model,41 the cationic dummy model,23, 42, 43 the constrained nonbonded model,44  2 the points-on-a-surface model45 and the polarizable model46-48 have been described and parameterized to study a broad range of metal containing complexes.    3 1.2 The 12-6 Nonbonded Model  !!"!#"!$%$!!"!!!!!!!!!!"!!!"!"!!"!"!!!!!"!!"!"!!!!!!!!!!"!!!!"!!"#!!"!!"!"!!!!"#!!"!!"!!!!!!!!!!!"!!!!"!!"!!"!"!!!!"!!"!    (1) The nonbonded model treats the metal ion as a point with an integer charge while the interactions are represented by Columbic and Lennard-Jones (LJ) terms (see equation 1). The first term is a classical Coulomb potential while the second term is a 12-6 LJ potential. In which !!" is the distance between two particles i and j, and the Qi and Qj are the point charges of the two particles while e is the proton charge. For the metal ion, its point charge is usually treated as an integer number according to their oxidation state, hence the parameters (!!"!", !!!"), (Rmin,ij, !ij) or ("ij, !ij) are the only two parameters that need to be determined. In the nonbonded model, the coordination of the metal ion is flexible, which allows coordination number (CN) switching and ligand exchange at the metal center. However, this model oversimplifies the interaction between the ions and their surrounding residues. In addition to Columbic and van der Waals (VDW) interactions, charge transfer, polarization and even covalent interactions could exist between a metal ion and its surrounding ligands.49-51 Furthermore, a single point poorly represents the charge distribution of most ions. The electronic cloud is usually non-symmetrically distributed around the metal ion, which could also further change and redistribute in response to changes in the surrounding environment. Nevertheless, due to the simple form, computational efficiency and excellent transferability characteristics of  4 the nonbonded model,37, 38, 40, 52 it is still extensively used for metal ions in MD simulations even though more sophisticated potential forms exist.  There are numerous systematic studies that have been reported in recent decades regarding the parameterization of the LJ nonbonded model for atomic ions. For example, †qvist pioneered the development of LJ parameters for the alkali and alkaline-earth metal cations.37 Dang and co-workers developed a series of LJ parameters for alkali metal and halide ions from 1992 to 2012 for either non-polarized or polarized water models.53-60 Peng and Hagler parameterized the 9-6 potential for alkali metal cations and halide anions.61 Jensen and Jorgensen have parameterized the LJ potential for the halide ions, alkali metal ions and the ammonium ion using the TIP4P water model.62 Roux and co-workers have parameterized the nonbonded model for the alkali metal and halide ions using the SWM4-DP polarized water model.47, 63 Babu and Lim re-optimized LJ parameters for biologically relevant +2 metal (M(II)) cations based on experimental relative hydration free energy (HFE) values while the nonbonded interactions were truncated by an atom-based force switching function.64 Joung and Cheatham have developed the LJ parameters for alkali metal cations and halide anions for three commonly used water models (TIP3P,65 SPC/E,66  and TIP4PEW67) for use in particle mesh Ewald (PME) simulation.39 Netz and co-workers have designed different parameter sets for alkali metal ions in SPC/E water by treating single-ion and ion-pair properties as targets.68 Hasse and co-workers have developed different parameter sets for alkali metal ions and halide ions in the SPC/E water in order to reproduce different experimental endpoints.69, 70 Reif and Hunenberger have created LJ nonbonded model parameters for  5 alkali metal and halide ions using the SPC or SPC/E water model.71 These parameters have been developed for different combining rules and simulation protocols. Various experimental/theoretical target values were used in the previous parameterization work to simulate ions in various environments (e.g., the gas phase, liquid phase and solid phase, interfacial phase, etc.). For example, QM calculated gas phase ion-water interaction energies for the monohydrates, experimental HFEs, enthalpies and entropies, the ion-oxygen distance (IOD) and CN of the first solvation shell, diffusion coefficients, mean residence times, radial distribution functions (RDFs), electric conductivity, dynamic hydration numbers, ion-pair properties, osmotic coefficients, lattice constants, lattice energies, etc. Overall, it is hard to reproduce all of the available experimental/theoretical results at the same time due to accuracy limitations of the classical nonbonded model.  LJ parameters always have limited transferability between different water models, mixing rules and simulation conditions.72-75 The PME method is now the de facto standard method used to calculate the long-range electrostatic energy in periodic boundary simulation cells.76-79 It calculates the short-range interactions in real space while the long-range interactions are developed in Fourier space. Importantly, it decreases the time complexity of MD simulations from O(N2) to O(NlogN) with reliable precision.76-79 However, there was no systematic parameterization work on various ions, which range from the monovalent to tetravalent, performed specifically for MD simulations employing the PME method. By employing the thermodynamic integration (TI) method80-87 and MD approach, we designed LJ parameters for more than 60 ions (range from monovalent to tetravalent) for three widely used water models (TIP3P, SPC/E and  6 TIP4PEW) respectively. These parameters are specifically designed for the PME method77-79 and different physical properties (e.g. HFE, IOD and CN).  Based on our simulation results, we found that the transferability of existing LJ parameters for divalent ions is limited. Meanwhile, we also found that the simulated HFE, IOD and CN values are highly correlated and there appears to be a one-to-one correspondence between them. Results showed that different water models have different properties and should be treated separately when designing the parameters. Moreover, due to the simplicity of the 12-6 nonbonded model, we could not reproduce all HFE, IOD and CN values simultaneously for divalent metal ions. And the stronger of the coordination interaction between the divalent metal ion and water molecules, in general, the larger of the errors in the nonbonded model. Furthermore, we found that generally the TIP3P, SPC/E, TIP4PEW and TIP4P water models experience a successive increase of error.  Besides the error of the 12-6 model described above, which brings about no ideal LJ parameters for divalent metal ions, there are innumerable combinations of the VDW radius and well depth parameters (Rmin/2 and !) that can generate the same HFE or IOD values,39 which offers a challenge for determining the final parameters. There are different strategies to solve this problem. In some parameter sets, the parameters were designed using fixed ! values for all the negative or positive ions.60, 62, 69, 70 Jensen and Jorgensen designed Òbig and smallÓ ! parameters for negative and positive ions, respectively, in their parameterization work.62 Joung and Cheatham have determined the  7 parameters empirically based on a compromise between different experimental properties, including the HFE, lattice energy and lattice constant.39 For the alkali metal and halide ion series there is an apparent trend in this parameter set where as Rmin/2 increases for the bigger ions the ! parameter also increases.39 In present work, via a consideration of the physical meaning of the VDW interaction, we fitted a curve (herein termed as the Ònoble gas curveÓ Ð NGC) to describe the relationship between the Rmin/2 and ! values based on the experimental results for noble gas atoms. According to NGC we determined the LJ parameters for various ions.  Towards different potential applications, we designed three parameter sets for more than 60 ions (range from monovalent to tetravalent) in TIP3P, SPC/E and TIP4PEW water models based on the NGC: one for reproduction of experimental HFEs, one for experimental IODs, and a final set representing a compromise between the former two properties (we term these the HFE, IOD, and CM parameter sets, respectively). Overall, the HFE parameter set achieved an error range of ~±1.0 kcal/mol for the absolute HFEs and ~±2.0 kcal/mol for the relative HFEs, the CM set could well reproduce the CN values and achieved an error range of ~±2.0 kcal/mol for the relative HFEs, and the IOD set achieved an error range of ~±0.01 † for the IODs.   We found that the derived Rmin/2 parameters coincide well with the VDW radii calculated using the quantum scaling principle (QMSP) method.88 The unsigned average errors (UAEs) of the present parameter sets are smaller than those of other parameter sets with respect to the QMSP calculated VDW radii. We also carried out test simulations on six  8 different ionic solutions. Using the experimental activity derivatives as a basis for comparison, in general, our parameter sets showed better performance than the parameter sets of Joung and Cheatham39 and Horinek et al.68 This supports our assertion that the current parameters have better transferability due to a better balance between the Rmin/2 and ! parameters from NGC. Meanwhile, in test simulations on a Fe(III) containing protein system, stable trajectories were obtained with the metal binding site being well conserved, which further supports the excellent transferability of these parameters. This work marks a systematic investigation and determination of LJ parameters for various ions that can be employed in PME based MD simulations and these parameters are compatible with FFs such as AMBER,89 CHARMM,90 OPLS-AA,91 and GROMOS92 when used with the PME model.  Furthermore, error analysis was carried out and results showed that the 12-6 model could well simulate some monovalent ions with strong ionic characteristics (such as Na+ and K+ ions). However, it yields non-negligible error, making it could not reproduce several experimental properties simultaneously for monovalent ions that have strong polarization interaction with surroundings (such as Tl+ and Ag+) and ions with formal charge equal or larger than +2. Results also showed that the 12-6 model has a much larger underestimation of the ion-water interaction with the increasing of ionÕs formal charge. Hence, we developed a 12-6-4 LJ-type nonbonded model for metal ions and introduced it below.   9 1.3 The 12-6-4 Nonbonded Model  The 12-6 LJ nonbonded model (see equation 1) explicitly includes the Pauli repulsion and induced dipole-induced dipole interactions via the LJ potential, while the charge-charge, charge-dipole, and dipole-dipole interactions are represented by the Coulomb potential. However, it doesnÕt take into account the charge-induced dipole interaction and the dipole-induced dipole interaction explicitly. The former interaction is the dominant one of these two types of interactions in the case of metal ions and its inclusion, in principle, would greatly improve a nonbonded model representing ions. The standard 12-6 LJ nonbonded model is reasonable for neutral systems, while for highly charged systems, the charge-induced dipole interaction, which is proportional to r-4, becomes very significant and needs to be considered. With these considerations, we decided to add an r-4 term into the standard 12-6 LJ nonbonded model in order to include the ion-induced dipole interaction.  We have parameterized the 12-6-4 model for more than 50 ions. The parameterization was also performed separately for TIP3P, SPC/E and TIP4PEW these three widely used water models. Unlike the 12-6 LJ nonbonded model, the new 12-6-4 LJ-type model can reproduce different target values (HFE, IOD and CN) with good accuracy at the same time after selecting appropriate parameters. Afterwards, independent research from Panteva et al. proved the outperformance of the 12-6-4 models over the 12-6 models for simulating the Mg2+-aqueous system.93   10 Meanwhile, we found that the magnitude of final C4 parameters were consistent with the analytical formulation of the charge-induced dipole potential. Furthermore, results also showed that the error of the 12-6 model for cations is approximately proportional to the square of the cationÕs formal charge. This trend is also consistent with the original equation of the charge-induced dipole interaction, which kind of interaction is omitted in the 12-6 nonbonded model. Based on the charge-induced dipole interaction, we proposed a specific combining rule for the C4 terms. By employing the new combining rule it was shown that the 12-6-4 model was readily transferable to mixed systems such as salt solutions, nucleic acid and protein systems. In the salt solution simulations, the 12-6-4 parameter set showed the best performance among the investigated parameter sets, which further illustrated it is a superior model relative to the 12-6 LJ nonbonded model.  Furthermore, the 12-6-4 nonbonded model is compatible with FFs like AMBER,89 CHARMM,90, OPLS-AA,91 and GROMOS92 with almost no additional computing cost when comparing to the 12-6 model. Even so, attention should be paid when the new nonbonded model is employed in systems with strong charge-transfer effects since it may be unphysical to treat the central metal atom as an actual ÒionÓ in such cases.    11 1.4 The Bonded Model and MCPB.py Program  The bonded model is widely used in contemporary FFs like AMBER,89 CHARMM,90 GROMOS92 and OPLS-AA,91 and the generic functional form is shown in equations  1-3  (CHARMM has an additional 1-3 Urey-Bradley nonbonded term, which is not shown in these equations). The total energy of a system is represented by its bonded part and nonbonded part (see equation 2). The bonded energy consists of the bond, angle and torsion terms (see equation 3). The bond and angle terms are described using harmonic equations while the torsion potentials is represented by a Fourier expansion. In equation 3, kr, rij, rij,eq are the bond force constant, bond length and equilibrium bond length; K!, !ij, and !ij,eq are the angle force constant, angle amplitude and equilibrium angle values; Vn, n, ", # are the torsion barrier, periodicity, torsion angle, and phase, respectively. The nonbonded term is described by equation 1 and was introduced in section 1.2. !!"!#$!!!"#$%$!!!"!#"!$%$        (2) !!"#$%$!!!"!!!!!!!"!!!"!!"!!!!!!!!"!!!"!!"!!!""!!"#$%&!""!!"#$%!!!!!""!!"#$%"&$!!!!!!"#!!!"!!!!           (3) There are several ways to obtain force constants parameters: through empirical methods,94 through experiments (e.g., X-ray, NMR, normal mode analysis of spectra etc.),95, 96 or based on theoretically calculated Hessian matrices.97, 98 In general, the empirical method could be applied more broadly but usually offers limited accuracy. Deriving parameters from experimental information is very challenging and time-consuming, thereby, restricting it to specific systems. Parameter determination based on  12 quantum calculations offers considerable accuracy and is applicable to a wide range of molecules.   Seminario proposed a method which uses the Cartesian Hessian matrix to calculate the force constant (referred as the Seminario method) and validated it through a series of small organic molecules.98 Nilsson et al. developed the Hess2FF software to calculate the force constants through the Seminario method. They applied it to 5 different systems with some containing either a Fe or Zn ion.99 Lin and Wang applied the Seminario method and the restrained electrostatic potential (RESP) fitting scheme to zinc complexes with the general AMBER force field (GAFF).100 They showed that the bonded model with RESP fitted charges showed the best performance among the models investigated.100 Peters et al. developed the metal center parameter builder (MCPB) software based on the MTK++ software package using the C++ language.4 It is a semi-automatic tool for the parameterization of metal ion containing molecules. The zinc AMBER force field (ZAFF) has been developed for four-coordinated zinc complexes in protein system using the Seminario/ChgModB combination (shown as the best combination) using MCPB.4 After the availability of MCPB a broad range of metal ion containing systems have been parameterized using this program.34, 36, 101, 102 However, even with the considerable time saving afforded by MCPB, the process is still overly complicated for the non-expert.  Herein we introduce the MCPB.py software, a python based metal center parameter builder, which streamlines much of the functionality of MCPB into a much easier to use program. It was built on the python metal site modeling toolbox (pyMSMT) in  13 AmberTools15.103 It uses a much more optimized workflow, offering a more user-friendly experience with far fewer steps requiring user input. It supports Z-matrix, Seminario and empirical methods for the parameterization. In the current version it supports a variety of AMBER FFs, more than 80 ions, and two widely used QM software packages (Gaussian104 and GAMESS-US105). Together with ParmEd103 or ACEYPE106 the topology and coordinate files of AMBER can be converted to the formats used by CHARMM107 or GROMACS.108 We expect the MCPB.py application to further expand and expedite the modeling of ions in metalloproteins and organometallic compounds using a range of packages and FFs.    14 1.5 The CusF Metallochaperone System  CusF is an Ag+/Cu+ bound metallochaperone in the bacterial pathogen Escherichia coli. It is a member of the CusC(F)BA proteins, which belongs to the resistance-nodulation-cell division superfamily that provide Ag+ and Cu+ resistance for the bacterial.109, 110 It has a specific cation-% coordination in which the metal ion coordinated to the 6-member aromatic ring in a tryptophan residue. In a previous study, the Cu+/Ag+-% interaction in CusF was evaluated to be on the order of ~10 kcal/mol.111 In a prevailing hypothesis of metal ion efflux, CusF metallochaperone transports Cu+/Ag+ ions from the periplasm to the CusCBA metal ion transporter protein-complex, which expels the metal ion from the cell.112, 113 The mechanism of metal ion capture by CusF, and its subsequent transfer to CusB, however, remains poorly understood. Recent experiments showed that CusF interact transiently with the N-terminal disordered region of CusB when either one of them is in the metal bound form, which allows metal ion transfer reversibly.114 Furthermore, metal ion binding quenches conformational dynamics in both proteins. These results indicate that the conformational dynamics of CusF is important for the process of CusF-CusB complex formation and metal ion transferring. Using a bonded/nonbonded hybrid model together with the umbrella sampling technique, we studied the potential of mean force (PMF) changes of the Cu+ bounded CusF from its closed to wide-open conformations. Results showed that the transformation barrier is ~8 kcal/mol, which supports the hypothesis that the open conformation is important for metal ion releasing from the CusF metallochaperone.     15 CHAPTER 2: PARAMETERIZATION OF THE 12-6 NONBONDED MODEL  2.1 Target Values  In this work we designed parameters based on three different target values: HFE, IOD and the CN of the first solvation shell. HFE values represent the thermodynamic properties while the other two represent structural properties. Target HFEs, IODs and CNs for investigated ions are given in Tables 1-3. Overall, we have parameterized the 12-6 LJ nonbonded model for 63 different ions (see Figure 1, including 4 monovalent anions, 11 monovalent cations, 24 divalent cations, 18 trivalent cations, and 6 tetravalent cations) in three widely used water models (TIP3P, SPC/E and TIP4PEW) respectively.  There are different sets of experimental HFE values for ions that exist in the literature.115, 116 Usually these values were obtained based on the protonÕs HFE value but final agreement on this value has not been reached. The general range of it is from -250 to -265 kcal/mol.71, 115, 117-120 For example, in Reif and work on design of the monovalent ion parameters,71 they supported a value of -1100 kJ/mol (~-263 kcal/mol) for the HFE of proton. Herein we used the experimental HFEs of ions from Marcus115 and Schmid et al.116 Marcus obtained the HFEs of cations based on &hydG¡(H+)=-1056 kJ/mol (~ -252 kcal/mol) from &hydH¡(H+)=-1094 kJ/mol (~ -261 kcal/mol) and &hydS¡(H+)=-131 J/(K!mol) or S'[H+(aq)]=-22.2 J/(K!mol). It is one of the most complete databases regarding the thermodynamic properties of ions. Schmid et al. refitted the HFE values of the atomic ions based on &hydH¡(H+)=-1078 kJ/mol (~ -258  16 kcal/mol).116 For monovalent cations, there is a trivial difference (~0.7 kcal/mol on average) between the data sets of Marcus and Schmid et al. For alkaline earth metal ions, the average difference is ~3.9 kcal/mol between the two sets. We have used MarcusÕ HFE values as target values since this set is more complete. The HFE value of the H3O+ ion is from Palascak and Shields121 due to its absence in the data sets of Marcus115 and Schmid et al.116 For the halide ions, Schmid et al.Õs values are ~8 kcal/mol lower than MarcusÕ values on average. Herein, we used Schmid et al.Õs data set as our target since it is consistent with a broader range of experimental values.39, 122   Most of the IOD values of the monovalent ions come from MarcusÕ review123 except those of the Tl+, Cu+, NH4+ and H+ ions (see Table 1). The Tl+ ion is similar to the Sn2+ ion which all have a lone pair electron in the outmost electron shell, resulting in the observation of two IODs in the first solvation shell in the aqueous phase. This effect is hard to reproduce using classical MD simulations due to the assumption of isotropic behavior. Typically, the reported CN value of the Tl+ ion is between 6-8 while Persson et al. determined the CN of Tl+ as 4 for aqueous systems.124 In their work they found a 4-coordinated Tl+ ion with two different IOD values representing two water molecules at 2.73 † and two at 3.18 †. The IOD value of Tl+ (2.96 †) used in the present work is the average of these two values. Shannon calculated the effective ionic radii of 6- and 8-coordinated Tl+ ion as 1.50 † and 1.59 † respectively.125 By adding the effective ionic radius of O2- (1.40 †) from Pauling, we can estimate that the IOD values for Tl+ are 2.90 † and 2.99 † for 6 and 8-coordiante structures, respectively, which is consistent with the IOD value we are using (2.96 †). Vchirawongkwin et al. performed QM/MM MD  17 simulations on the aqueous Tl+ system and observed two different IODs (with 2.79 and 3.16 †) and determined the average CN as 5.9.126 Their theoretical work supports the Òstructure-breakingÓ character of Tl+ in bulk system.126 It is hard to obtain IOD value for the Cu+ ion since it is easily oxidized by water. The IOD value for Cu+ used in present work comes from quantum calculations of Burda et al.127 There is no reliable experimental IOD value for NH4+. We estimated the IOD value (2.85 †) based on the ionic radius of the NH4+ ion (1.45 †) from Detellier and Laszlo 128 and the ionic radius of O2- (1.40 †) from Pauling.129 This value is consistent with classical and Car-Parrinello molecular dynamics (CPMD) simulations.130, 131 H+ in aqueous solution is thought to exist either as the Zundel (H5O2+), Eigen (H9O4+) or hydronium ion (H3O+). The IOD value of H+ in the Zundel form was obtained from the quantum calculations of Meraj and Chaudhari.132 While the IOD value of H+ in the Eigen ion was taken from the calculations of Sobolewski and Domcke done at the MP2/6-31+G** level of theory.133 Blauth et al. investigated aqueous Ag+ by using the quantum mechanical charge field (QMCF) MD method, in which they found the CN of the first solvation shell to be 6. The Ag+ ion also acts as a Òstructure-breakingÓ factor in the bulk system.134 Blumberger et al. investigated the Cu+, Cu2+, Ag+ and Ag2+ ions in the bulk systems using the CPMD simulation method, in which they found the CN of these ions to be 2, 5-6, 4 and 5 respectively.135  Most of the IOD and CN values for M(II) ions were referred from MarcusÕ review,123 while the rest were from some other references (see Table 2). The IOD and CN values for M(III) ions were taken from MarcusÕ review,123 while the IOD and CN values for the  M(IV) ions were taken from a number of sources (see Table 3).136-139 The experimental  18 effective ionic radii of M(III) and M(IV) ions were obtained from Shannon.125 Based on the IOD values and effective ionic radii, we estimated the effective radii of the coordinated water and display the data in Table 3. Some highly charged ions (with charge larger than +2) which readily hydrolyze water such as As3+, Sn4+ and Pb4+ ions26 were not considered in the present work.    19 2.2 Methods  For Rmin,ij (or "ij) in equation 1, there are two combining rules - the Lorentz combining rule (see equation 4) and the Good-Hope combining rule (see equation 5) that are widely used in MM simulations. !!"#!!"!!!!"#!!!!!!"#!! or !!"!!!!!!!            (4) !!"#!!"!!!!"#!!!!!!"#!! or !!"!!!!!!!           (5) For the !ij (potential well depth) parameter in equation 1, the Berthelot combining rule is commonly used (see equation 6).  !!"!!!!!!!!!              (6) The Lorentz-Berthelot combining rules (equations 4 and 6) is used in the AMBER89 and CHARMM90 FFs while the geometric-mean combining rules (equations 5 and 6) is used in the OPLS-AA91 FF. Parameters using one combining rule usually need to be modified when moving to another combining rule. Herein all the simulations use the Lorentz-Berthelot combining rules, but if one wants to use the geometric combining rules, they can adapt the parameters by using equations 4-6. The LJ parameters for the water models (TIP3P,65 SPC/E,66 TIP4P65 and TIP4PEW67) employed in this work are shown in Table 4.  TI method80, 82-87 is a powerful tool to study the free energy difference between two different states of one system. It was used to simulate the HFEs in the present work. It employed a mixed potential of the initial and final states (V0 and V1 respectively in equation 7) during the simulation process. Herein k is an integer number which equals 1  20 when linear mixing is employed, # is a number between 0 and 1, representing the mixing extent of the two states (V(#) is equal to V0 when #=0 while #=1 results in V(#)=V1). !!!!!!!!!!!!!!!!!!!!!!!       (7) To simulate the VDW disappearance or appearance process, the soft-core scaling method with linear mixing (with k=1 in equation 7) was utilized.81 It employs a # dependent modified LJ equation (see equation 8). Herein rij is the distance between the vanishing atom and another atom, $ is a constant set to 0.5 and ( equals Rmin,ij/(21/6). When #=0 it is identical to a normal 12-6 LJ equation while when # approaches 1 it displays a smooth interaction between the Òsoft-coreÓ atom and a surrounding particle, allowing them to approach each other closely with a finite energy penalty. It prevents the Òend-point catastropheÓ when we Òturn onÓ the particle, thereby eliminating related artifacts in an effective way. !!"#$!!"#$!!"#!!!!!!!!!!!!!!!!"!!!!!!"!!!!"!!       (8) Herein we run a mixture of NVT and NPT simulations, so we make the approximation: !"!!!!          (9) The free energy difference of two states is obtained by integration of !"!!" values along the # coordinate (see equation 10) in the NVT and NPT ensembles. The results could be fit to a cubic spline or quadratic curve, while in present work we employed Gaussian quadrature to calculate the integration (see equation 11, where wi are the weights for the different i values). !"!!!!!!!!!!!!!!"!!"!!"!!       (10) !!!!!!!"!!"!           (11)  21 Based on the consideration of balance of accuracy and speed, we used three methods (i.e. Methods 1-3) to simulate the HFE, IOD and CN values of ions. For Methods 1 and 2 all the simulations were carried out using the AMBER 11 suite of programs140 while the modeling and data analyses were performed using the AmberTools suite of programs.140 For Method 3 the AMBER 12 suite of programs140 was used to perform the simulations while the AmberTools suite of  programs140 was utilized to carry out the data analysis. The PME77-79 method and periodic boundary condition (PBC) were used during the MD and TI simulations. The Òtin-foilÓ boundary condition was employed for the systems that are not neutral. The time-step was set as 1 fs and the cut-off was set to 10 † unless specified. The Langevin algorithm was used to control the temperature with a collision frequency of 5 ps-1. For simulations performed in the NPT ensemble, BerendsenÕs barostat with isotropic position scaling was utilized for pressure control with the relaxation time set as 10 ps and 1 ps for the TI and normal MD simulations, respectively. The SHAKE algorithm141 was used to constraint the distances between hydrogen atoms and their attached heavy atoms with a tolerance of 1.0)10-5 † while the Òthree-siteÓ algorithm was used for the water molecules.142 In present work, # values were set to 0.1127, 0.5 and 0.88729 in a 3-window TI simulation. # values were treated as 0.1127, 0.5, 0.88729 and 0.98 in a 4-window TI simulation. # values were chosen as 0, 0.1127, 0.5, 0.88729 and 1 for a 5-window TI simulation. # values were set to 0, 0.04691, 0.23076, 0.5, 0.76923, 0.95308 and 1 for a 7-window TI simulation. While for a 9-window TI simulation # values were treated as 0, 0.2544, 0.12923, 0.29707, 0.5, 0.70292, 0.87076, 0.97455, and 1. Each window began from the final snapshot of the previous window, the windows of #=0, 0.98 (only used in the 4-window TI simulations), and 1  22 served to equilibrate the system and were not considered in the final free energy calculation using equation 11. Herein the thermodynamic cycle shown in Figure 2 was used to simulate HFEs of ions. Generally, in present work, we estimated the uncertainties of the simulated HFE values using the 12-6 model as ~±1 kcal/mol for the monovalent143 and divalent ions,144 and ~±2 kcal/mol for the highly charged ions.145  First, we created a ~(46†!46†)46†) cubic water box surrounding a dummy atom with the closest water molecule at least 1.5 † away from the dummy atom. In total, there were 2439 water molecules in the system for the TIP3P and SPC/E water models while for the TIP4P and TIP4PEW water models this number was 2389. We performed 1000 steps of minimization using the steepest descent algorithm followed by 1000 steps of minimization using the conjugate gradient algorithm. Afterwards a 1 ns heating process was simulated in the NVT ensemble to heat the system from 0 to 300 K. And then a second 1 ns simulation was carried out at 300K in the NVT ensemble to equilibrate the system. To correct the system density, a 1 ns NPT simulation was performed under 1 atmosphere (atm) and 300 K with the final structure was treated as the starting structure for TI simulations using Method 2 (details are shown below). Finally another 1 ns NVT simulation was conducted to prepare the initial structure for TI simulations using Method 1 (details are shown below).  Method 1 was used for two-dimensional scanning of the LJ parameter space of M(II) ions. In this protocol, we performed LJ parameter space scanning for the M(II) ion with a fixed mass of 65.4 g/mol (referred as a Zn2+ ion, the choice of mass has a limited influence on  23 the simulated energetic and structural properties). The range of Rmin/2 was chosen as 0.3-2.5 † with a 0.1 † interval and ! was evenly distributed in the range of 10-6-1 kcal/mol in the logarithmic scale with a interval of 1 for -log(!) (in total there are 23 different Rmin/2 values, 7 different ! values, forming 23)7=161 different combinations of LJ parameters for the M(II) ions). All combinations of the LJ parameters were investigated for each water model in the present work. To balance speed and accuracy, we used a one-step method (turn on the VDW and electrostatic interactions of the metal ion in one step) to obtain &GTotal and -&GTotal (see Figure 2). For each LJ parameter combination, we performed the simulation as described below. First, the ion hydration process was simulated in the NVT ensemble using a 9-window linear TI simulation where each window had a 200 ps simulation time with the dV/d* values in the last 150 ps were collected and averaged. Then the &GTotal value was obtained via Gaussian quadrature using equation 11. Subsequently, we performed a 1 ns MD simulation with snapshots collected every 1000 steps over the last 500 ps simulation. From these snapshots the ion-oxygen RDF was obtained with a resolution of 0.01 † based on the average volume of the trajectory. The IOD value was obtained from two times of quadratic fittings based on the RDF: the first quadratic fitting was done for the data within ±0.1 † of the peak of the first solvation shell (in total 21 points with a 0.01 † interval along the RDF were considered). The second quadratic fitting was performed over the RDF data within ±0.1 † of the point that was closest to the apex of the first fitting. The IOD value was obtained from this final fit and kept with two decimal places. The CN value was determined via integrating the ion-oxygen RDF from the origin to its first minimum. Finally, we carried out the backwards TI simulation to obtain the -&GTotal value using the same method as  24 determining the &GTotal value. Finally the -&GTotal and &GTotal values were used to determine the HFE value.   For our final determination of the LJ parameters for M(II) ions, we employed Method 2, which is a more consistent method to obtain HFE, IOD and CN values. Using TI simulations, we determined the &GVDW, &GEle, -&GEle and Ð&GVDW values (which correspond to the free energy changes for the VDW-appearing, charge-appearing, charge-disappearing and VDW-disappearing steps, respectively Ð see Figure 2) in the NPT ensemble consecutively. For &GVDW and Ð&GVDW, we employed a 3-window soft-core linear scaling method due to the better performance of soft-core method over both the linear and nonlinear scaling methods without using a soft-core potential.81 For the VDW scaling simulations, each window was equilibrated for 100 ps followed by 200 ps of production while for the electrostatic scaling simulations, each window was equilibrated for 50 ps followed by 150 ps of production. Herein the HFE value are computed using HFE = +)(&GVDW+&GEle-(-&GEle-&GVDW)). Finally we modeled the ion-aqueous system again and performed a 2000 step minimization (including 1000 steps of minimization using the steepest descent algorithm followed by 1000 steps of minimization using the conjugate gradient algorithm), a 500 ps NVT heating, a 500 ps NPT equilibration and then a 2 ns NPT production run. Snapshots were stored every 1000 steps during the production run (in total 2000 snapshots were collected). Afterwards the IOD and CN values were determined based on these snapshots and the analysis method described in Method 1.   25 Method 3 was used to simulate the HFE, IOD and CN values of monovalent, trivalent and tetravalent ions. Same as Method 2, we obtained the HFE value based on the free energy changes associated with four processes: &GVDW, &GEle, -&GEle and -&GVDW. At first, a dummy atom was solvated in a 13 † thick water box with the closest water molecule at least 1.5 † away from it (which offers a cubic box with a length of 29 †, and a total VDW size of ~(32† ) 32† ) 32†)). There are 721, 721 and 732 water molecules in the TIP3P, SPC/E and TIP4PEW water boxes, respectively. Then 1000 steps of steepest descent minimization followed by 1000 steps of conjugate gradient minimization were performed to minimize the initial structure. Afterwards a 500 ps simulation in the NVT ensemble was carried out to heat the system from 0 to 300 K. And then a 500 ps simulation in the NPT ensemble was performed under 300 K and 1 atm to correct the system density and further equilibrate the system. The final snapshot was used as the initial structure for the TI simulations in the NPT ensemble. Afterwards, the HFE value was obtained in the same way as described in Method 2 except that we used a 4-window linear soft-core simulations to obtain the &GVDW value (in which the final window was not considered in the free energy calculation but was just used to further equilibrated the system). For determination of the IOD and CN values, we used the same method as described in Method 1, except that the snapshots were stored every 500 fs in the final production run, yielding 4000 snapshots for analysis.   Fitting procedure of the HFE quadratic fitting curves:  26 (1) There are 7 different ! values for each row in Table SI.1 (SI means supporting information), we performed the quadratic fitting and got the equation for HFE versus Ðlog(!) for each row with a certain Rmin/2 value.   (2) Solve the equation to determine the Ðlog(!) value to reproduce the target HFE of a metal ion with the fixed Rmin/2 for each row. Then for each row we obtained a point (Rmin/2, Ðlog(!)) which can reproduce the target HFE. In total there are 23 rows so we can get 23 points, each of which can reproduce the target HFE. (3) Discarded the (Rmin/2, Ðlog(!)) points which were not in the scanned LJ space (with Rmin/2 in 0.3-2.5 † and Ðlog(!) in 0-6) and performed another quadratic fitting of Ðlog(!) versus Rmin/2 for the remaining points to get the final fitting curve for the target HFE.  Fitting procedure of the IOD quadratic fitting curves: (1) There are 7 different ! values for each row in Table SI.2, we performed the quadratic fitting and got the equation for IOD versus Ðlog(!) for each row with a certain Rmin/2 value.   (2) Solve the equation to determine the Ðlog(!) value to reproduce the target IOD of a metal ion with the fixed Rmin/2 for each row. Then for each row we obtained a point (Rmin/2, Ðlog(!)) which can reproduce the target IOD. In total there are 23 rows so we can get 23 points, each of which can reproduce the target IOD. (3) Discarded the (Rmin/2, Ðlog(!)) points which were not in the scanned LJ space (with Rmin/2 in 0.3-2.5 † and Ðlog(!) in 0-6) and performed another quadratic fitting of Ð 27 log(!) versus Rmin/2 for the remaining points to get the final fitting curve for the target IOD.  OpenMM146, 147 (in version 6.1) was used for the ionic solution simulations. 13 Na+ ion and 13 Cl- ions were solvated in a water box with a size of ~(46†!46†!46†), in which there were 2414 SPC/E water molecules (giving a molarity of ~0.3 M). Firstly, 2000 steps of minimization employing the L-BFGS algorithm was performed to optimize the structure, then 1 ns simulation in the NVT ensemble was performed to heat the system with temperature increasing gradually from 0 to 300 K. Afterwards another 1 ns of simulation was performed in the NVT ensemble to equilibrate the system. And then 2 ns of simulation in the NPT ensemble was carried out to equilibrate the system. Finally 10 ns of production run was performed in the NPT ensemble with structures stored for each 0.5 ps, yielding 20000 snapshots for the final analysis. 10 † cut-off was used for the nonbonded interaction. The PME method and PBC were used in these MD simulations and the Langevin algorithm with a 1.0 ps-1 friction coefficient was employed for the temperature control. A MC barostat was utilized for the pressure control with the volume change attempt frequency set as 25 fs-1. The time step was set to 1.0 fs with a Òthree-pointÓ SHAKE algorithm142 was used to constrain the water geometry. The Kirkwood-Buff integrals were calculated based on equation 12, then the activity derivatives acc were obtained from equation 13.  !!"!!!!!!!"!!!!"!!        (12) !!!!!!!!!!!!!!!"!!!!!!!!!!!!!"!!        (13)  28 Herein gij(r) is the RDF between species i and j. The cut-off of the RDF was set to 12 † and dr was treated as 0.01 † for the RDF calculation. %c and %w are the number densities of the ion (here the positive and negative ions are treated as indistinguishable) and water respectively, while Ncc and Ncw are the excess CNs between the ion-ion and ion-water pairs.  For the simulations of a Fe(III) containing protein, the AMBER 12140 and AmberTools140 suites of programs were used for the system building, structure minimizations, MD simulations and data analysis. The structure of Chain C of PDB entry 4BV1 was used for our modeling. The H++ web server148 was utilized to add hydrogen atoms to the protein system with setting the pH, salinity, internal dielectric constant, and external dielectric constant as 7.2, 0.15, 4 and 80, respectively. Different names of the His residues were assigned according to their pronation states. Afterwards the Cys residue that binds to the iron ion was renamed to ÒCYMÓ with its sulfur linked hydrogen atom was deleted. ACE and NME groups were used to cap the protein system. A TIP3P65 water box with thickness of 10 † was employed to solvate the protein system. Afterwards the minimizations and MD simulations were performed as follows: (1) 2000 steps of steepest descent minimization, plus 3000 steps of conjugate gradient minimization were performed with the protein system (excluding the capped residues) and metal ion being held by a force restraint of 500 kcal/mol!†-2. (2) 2000 steps of steepest descent minimization followed by 3000 steps of conjugate gradient minimization were carried out for the system with a 500 kcal/mol!†-2 force  29 constant on the heavy atoms of the protein (including the capped residues) and metal ion. (3) 10000 steps of steepest descent minimization and afterwards 10000 steps of conjugate gradient minimization were performed for the system using a 200 kcal/mol!†-2 restraint on the backbone C, CA and N atoms of the protein (including the capped residues). (4) 5000 steps of steepest descent minimization and then 40000 steps of conjugate gradient minimization were carried out for the entire system.  (5) 400 ps of simulation using the NVT ensemble was performed to heat the system from 0 to 300 K with the protein system (including the capped residues) and metal ion having a force restraint of 10 kcal/mol!†-2. (6) 200 ps of simulation using the NVT ensemble was carried out to equilibrate the system at 300 K. (7) A 2 ns simulation using the NPT ensemble was performed under 300 K and 1 atm to correct the density and further equilibrate the system. (8) Finally, 10 ns of production simulation was performed using the NPT ensemble at 300 K and 1 atm with snapshots being stored every 2 ps. In total there were 5000 frames collected for the data analysis. The Langevin algorithm was used to control the temperature with a collision frequency set at 5.0 ps-1. An isotropic position scaling algorithm was employed to control the pressure with a relaxation time of 2.0 ps. The cut-off value of the nonbonded interaction was set to 10 †. PME was used to handle the long-range electrostatic interactions.77-79 SHAKE141 was employed during the simulation to constrain the bonds involving  30 hydrogen atoms, while for the water molecules a Òthree-pointÓ algorithm142 was employed.   31 2.3 Validation Tests of Several Available LJ Parameters for M(II) Ions  The general philosophy of parameter design is to make the best compromise estimate for different physical properties at the same time. In the first part of our work, we tested the transferability of LJ parameters, using the PME approach and Method 2, for some previously developed parameters for M(II) ions. The data from these simulations are shown in Tables 5-7: Table 5 is for the LJ parameters of Mg2+, Ca2+ and Zn2+ found in the AMBER ff99 which is provided in the AMBER Package.140 In this FF the LJ parameters of Mg2+ and Ca2+ ions were adopted from †qvist37 by utilizing the Lorentz-Berthelot combining rules while the LJ parameters of Zn2+ ion were obtained from Merz.40 Table 6 shows Zn2+ LJ parameters designed by Stote and Karplus38 while Table 7 contains the LJ parameters designed by Babu and Lim.64 These results suggest that the differences between treating the long-range electrostatics with PME method versus other methods cannot be simply overlooked. For example, as shown in Table 5, for the Ca2+ LJ parameters in AMBER ff99, there are large differences of the HFE and IOD values of Ca2+ ion obtained herein, which are based on the PME method, from the HFE and IOD values determined in †qvistÕs earlier work,37 which are based on a different method. The Zn2+ LJ parameters in Table 6 donÕt have big differences (0.02 †) in the simulated IODs but have significant differences in the simulated HFEs (~70 kcal/mol). The values in Table 7 indicate that Babu and LimÕs parameters are shifted by ~45 kcal/mol from the experimental absolute HFEs, while they show good agreement with the PME simulations with respect to reproducing the relative HFEs. However, some metal ions still have notable differences (for example, Be2+ has a 10 kcal/mol difference in the relative HFEs  32 and 0.12 † difference for the IODs between the two different methods). Based on the results shown above, we decided that it was necessary to design a set of parameters for the M(II) metal ions using the state-of-the-art PME based MD simulations.   It is extremely important to note that we are not condemning the earlier efforts all of which were excellent.37, 38, 40 It simply reflects the fact that simulation protocols have evolved to the point where PME is the accepted standard for the treatment of long-range interactions and it is possible to carry out very long MD simulations using this model with, for example, GPU technology.149-156    33 2.4 Parameter Space Scanning  We cannot design a satisfactory unified ion parameter set for all popular water models since the force field parameters of respective water models are different (see Table 4). Hence, we performed simulations for different combinations of the Rmin/2 and ! parameters for the TIP3P, SPC/E, TIP4P and TIP4PEW water models, respectively. Based on the HFE values for TIP3P LJ grids (see Table SI.1a), quadratic fitting (the fitting procedure was shown in section 2.2.) was done for each of the 24 M(II) metal ions in the TIP3P water model and the fitting curves are depicted in Figure 3.   Figure 3 indicates similar trends exist with respect to the fitting curves for monovalent ions from the previous work of Joung and Cheatham:39 HFE increases with a decrease of Rmin/2, and smaller Rmin/2 value with a large ! parameter can yield similar HFE values as a larger Rmin/2 parameter coupled with a smaller ! value. This can be explained by the form of the LJ potential function (see equation 1): since !!"!!!!!"#!" and !!!!!!!!"#!, smaller Rmin/2 parameter with bigger ! value and smaller ! parameter with larger Rmin/2 value yield similar C12 and C6 values.157 Furthermore, ! is directly proportional to C12 and C6 while Rmin /2 is raised to the twelfth and sixth power in the expression for C12 and C6, respectively. This is the reason why the HFE is quasi-linearly dependent on Rmin/2 while its dependency on ! is logarithmic.   Figure 3 shows that generally all of the fitting curves have a similar shape but different Y-intercepts for the TIP3P water model. The HFE values of LJ grids for different water  34 models (see Table SI.1) show that the HFE differences amongst the same LJ parameters within different water models could not be neglected. To clarify the differences of the HFE fitting curves for different water models, we treated the Zn2+ ion as an example and illustrated its HFE fitting curves within four different water models in Figure 4. It can be seen that the two 3-site models (TIP3P and SPC/E) show very similar results and are distinctly different from the two 4-site models (TIP4P and TIP4PEW). Meanwhile, the two 4-site water models also showed a remarkable difference from each other. Therefore, we concluded that it is necessary for us to design different parameters for the same metal ions for use with different water models.  Meanwhile, the IOD and CN values of LJ grids for different water models (see Table SI.2) show that the four water models generated very similar IOD and CN values when using the same LJ parameters for the M(II) metal ion. To elucidate the difference in IOD values for the four studied water models, we carried out a standard deviation analysis of the IOD values between each pair of the four water models and display the data in Table 8. Our results indicate the TIP3P and SPC/E water models have nearly the same IOD values with a 0.00 † systematic difference and 0.01 † standard deviation with each other when using the same LJ parameters for the metal ions while a similar situation exists between the TIP4P and TIP4PEW water models. Generally, there is a 0.02 † systematic difference between the 3-site and 4-site water models for the IOD values when the same LJ parameters are utilized for the metal ions. Meanwhile, IOD fitting curves were obtained from Table SI.2 by following the similar procedure as for the HFE fitting curves (see section 2.2). Again, we treated the Zn2+ ion as an example and depicted the IOD  35 fitting curves in Figure 5. It can be seen from the figure that the two 3-site water models share one curve and the two 4-site water models share the other curve (although in the latter instance there is a slight difference), which is in agreement with the standard deviation analysis (see Table 8).   As discussed in the introduction, the nonbonded model can simulate CN switching processes. There are several non-integer CN values in Table SI.2, which suggests there is CN switching occurring during these simulations. As an example, we show CN switching in the MD simulation of a M(II) metal ion with LJ parameters of Rmin/2 =2.2 † and !=0.1 kcal/mol in a TIP3P water box in Figure 6. We observe the CN switches between 8, 9, 10, and 11 during the simulation.   A detailed examination of the HFE, CN and IOD values of LJ grids for different water models (see Tables SI.1 and SI.2) indicates that, for each water model, for the parameter combinations corresponding to the same HFE, they have almost the same IOD and CN. These results suggest there is likely a relationship between the HFE, IOD and CN values for the metal ion-water systems, hinting at the strong correlation between various solvation properties, which is consistent with earlier work.157 Figure 7 shows the HFE and IOD fitting curves for six representative metal ions with different sizes in the TIP3P water model. From Figure 7 we find that the IOD and HFE fitting curves for each metal ion are almost parallel with each other and do not have any intersection points in the investigated range, implying it is hard to find a parameter to reproduce the experimental HFE and IOD values at the same time for these metal ions. At the same time, the figure  36 also shows that the distance between the HFE and IOD fitting curves of metal ion begins to decrease along with increase of the ion size, which may be due to the simplicity of the nonbonded model. The electrostatic plus LJ potential approximation underestimates the interaction energy of the metal ion and ligating residues at short range, especially when there is a strong charge transfer, polarization or even covalent interaction between them. In this situation, if one wants to reproduce the experimental HFEs, one should have smaller IOD values than the experimental values. Meanwhile, the 12-6 nonbonded model is more appropriate for the monovalent metal ions since polarization and charge transfer effects are likely to decrease (and these ions tend to be mostly ionic in nature), allowing the parameters to be designed to reproduce both the experimental HFE and IOD values simultaneously.39   Therefore, there appears to be no single ÒperfectÓ LJ parameter set for the M(II) ions since none are able to reproduce the experimental HFE and IOD values simultaneously in a simulation. Hence, we concluded that it was necessary to design several sets of parameters for these M(II) metal ions to meet different demands. Since our intention is to design LJ parameters for the M(II) metal ions specifically for PME based MD simulations, we only designed parameters for the TIP3P, SPC/E and TIP4PEW water models. The TIP4P water model was modified to produce the TIP4PEW model, which is designed specifically for PME based simulations.67 First, by treating the experimental HFE values as the target property we designed the HFE parameter set for each of the three water models. Next, we designed the IOD parameter set to reproduce the experimental IOD values (due to the limited experimental data set for IOD values, only  37 16 of 24 divalent ions have IOD parameter sets). In the case of the IOD parameter sets, they ended up being the same for the three water models. In the end, we designed the CM (short for compromise) parameter set for the three water models respectively, which is a compromise between the HFE and IOD parameter sets using the experimental relative HFE and the CN values as targets.  Since numerous points exist on the fitting curves capable of reproducing almost the same HFE, IOD and CN values, it is problematic to pick a single point among them to determine the final LJ parameters. Initially, we wanted to do simulations on the solid-state salts of these M(II) metal ions together with anions for the different combinations of LJ parameters, as employed in the protocol of Joung and Cheatham.39 However, it is difficult to find valid and consistent experimental data for the salts containing the M(II) metal ions we are dealing with. Our second approach was to pick the point that is also capable of reproducing the QM calculated interaction energy between the metal ion and one or several surrounding water molecules. Unfortunately, although for some metal ions (such as Ca2+) we could get reasonable results, we could not obtain valid results for most of the ions, especially for the metal ions capable of strong covalent interactions with the surrounding waters. This likely reflects the simplicity of the nonbonded model. Furthermore, it is also difficult to find a standard QM method protocol, which is largely due to the various possible electronic states for some of the metal ions.   Finally, we selected an alternate way to design the 12-6 LJ parameters. The LJ potential, which was first proposed by Sir John Edward Lennard-Jones to represent the interaction  38 between noble gas atoms in 1924,158 is remarkably accurate for the noble gases and a very good approximation for neutral atoms and molecules. In the 12-6 LJ function the r-12 term represents the interaction caused by Pauli repulsion due to the overlap of the molecular orbitals at close distance. The r-6 term describes the long-range attraction between molecules due to the dispersion force. Generally, the more dispersive electronic cloud one particle has, the bigger Rmin/2 and ! value it should have, as shown in the experimental data.159 Using the experimental data159 and the Lorentz-Berthelot combining rules, we obtained the Rmin/2 and ! parameters for the He, Ne, Ar, Kr, and Xe atoms (see Table 9). For all of the metal ions treated here, they should have smaller Rmin/2 and ! values than those of Xe since the biggest metal ion herein, Ba2+ has a smaller Rmin/2 than Xe because of the same electronic structure but a larger nuclear charge.  Furthermore, if one metal ion has a Rmin/2 value between the Rmin/2 values of the Kr and Xe atoms, it should have a ! value between the ! values of the Kr and Xe atoms as well. Furthermore, if we could get a curve to represent the relationship between Rmin/2 and !, together with the HFE and IOD fitting curves we obtained in the former part, we could determine the LJ parameters for the M(II) metal ions.    To be consistent with the HFE and IOD fitting curves, we produced the curve fitting between the Ðlog(!) and Rmin/2 values shown in Table 9. By treating Rmin/2 as x and Ðlog(!) as f(x), we attempted several fits with different functions. Finally, we found that the Slater function !!!!!!!!!!! with C1=57.36 and C2=2.471 had a better R2 value (0.98265) than the quadratic fitting (R2=0.94509). We named this curve as the noble gas curve (NGC) and determined the final LJ parameters from the points on this curve.   39 It was found that different combinations of parameter pairs could reproduce the same HFE value in both Joung and CheathamÕs work39 and our work described above. In light of this and to balance the accuracy and speed, we constrained one of the two parameters (Rmin/2 and !) in a physically reasonable manner and then vary the other one during the parameter space scannings for the monovalent, trivalent and tetravalent ions. To accomplish this we used NGC described above to select an appropriate ! value followed by a one-dimensional scan over a series of Rmin/2 values.   For the monovalent cations, we performed the parameter scanning using the Na+ ion (which has a mass of 22.99 g/mol) as our reference. We only carried out the scanning at one mass since the atomic mass has only a small effect on the computed thermodynamic and structural properties. We scanned Rmin/2 from 0.8 to 2.3 † with an interval of 0.1 †. For the monovalent anions, we did the parameter scanning using the Cl- ion (which has a mass of 35.45 g/mol) as our reference. The scanning range of Rmin/2 is from 1.7 to 3.2 † with an interval of 0.1 †. A range of larger Rmin/2 values was chosen for the anions because that, in general, they have more dispersed electronic clouds than the cations. The parameters, and the simulated HFE, IOD and CN values of these parameter scannings are shown in Tables SI.3 and SI.4. For trivalent and tetravalent metal ions, we performed the parameter space scanning similar to the monovalent ions. We used the Fe3+ (mass of 55.85 g/mol) and Th4+ (mass of 232.04 g/mol) ions as our references for the parameter scannings of the trivalent and tetravalent metal ions, respectively. We scanned the Rmin/2 from 0.9 to 2.3 † with an interval of 0.1 † while the ! value is obtained for each Rmin/2  40 value based on NGC. Parameter points, and the simulated HFE, IOD and CN values are collected in Tables SI.5 and SI.6.    41 2.5 Parameter Determination  The HFE, CM and IOD fitting curves for the Zn2+ ion and the NGC are shown in Figure 8. It can be seen that the CM fitting curve for the Zn2+ ion is almost in the middle of the HFE and IOD fitting curves. The original LJ parameters can be obtained as the intersection points between the HFE, CM and IOD fitting curves with the NGC. After slightly tuning the parameters, the final LJ parameters can be determined. We employed Method 2 in this part, which is a more accurate way to obtain the HFE, IOD and CN values. In the VDW-disappearing and VDW-appearing steps we employed the soft-core scaling method instead of the linear or nonlinear normal scaling methods due to its better performance over the latter two.81 In the present work, we also conducted tests among the different scaling methods and the data is given in Table 10.  The L, K4, K6 and SC in Table 10 represent the linear scaling, nonlinear scaling with k=4, nonlinear scaling with k=6 and soft-core scaling methods, respectively, while all the windows involved a 300 ps simulation with the last 200 ps used to determine the free energy changes. It can be seen that the soft-core scaling method gives better-converged and consistent results (i.e., the free energies of the VDW-appearing and VDW-disappearing processes have the opposite sign) than the other methods. The linear and nonlinear scaling methods could give consistent results for the small Be2+ ion, but had more difficulty with larger ion like Ba2+ ion.  The HFE parameter set of divalent metal ions is shown in Table 11, and its simulated HFE, IOD and CN values are shown in Table 12. The HFE parameter set achieved a ±1  42 kcal/mol accuracy of reproducing experimental HFEs. The IOD parameter set, and its simulated HFE, IOD and CN values are shown in Tables 13-14, respectively. Generally, the IOD parameter set reproduced the experimental IOD values with an accuracy of ±0.01 †. The CM parameter set, and its simulated HFE, IOD and CN values are listed in Tables 15-16, respectively. During our parameterization efforts we found that it was impossible to simulate all the CN values while simultaneously reproducing the relative HFE values for the CM parameter set so we compromised on the reproduction of the CNs for the Be2+ and Sn2+ ions and tried to best reproduce their relative HFE values. This lead to the CM parameter set having an average error of ~25 kcal/mol in the absolute HFE (while reproducing the relative HFE) for the TIP3P and SPC/E water models, while for the TIP4PEW water model this value increased to ~40 kcal/mol. Generally the CM parameter set reached a ±2 kcal/mol accuracy of the relative HFEs while keeping the CNs of most M(II) metal ions. Here we treated the Zn2+ ion as an example again and showed the RDFs for the different parameter sets in Figure 9. It could be seen that the CM parameter set yields a first solvation peak between those of the other two parameter sets. Meanwhile, the IOD and CN values are 1.67 † with 4.1, 1.93 † with 6.0, and 2.08 † with 6.0 for the HFE, CM, and IOD parameter sets respectively.  We performed curve fits for HFE versus IOD along the scanning range for Rmin/2 for both the monovalent cations and anions (see Figure 10). The target values are also shown in the figure as blue stars with error bars. There are different phenomena that have been observed for the positive and negative ions. For the positive ions, the error of the 12-6 LJ nonbonded model is relatively small. For example, from Figure 10, we can see that it is  43 possible to reproduce the target HFE and IOD values simultaneously for each of the Na+, K+, Rb+, Cs+ and NH4+ ions. Ramaniah et al. analyzed the K+-aqueous system using ab initio MD and also demonstrated that classical potentials can yield good predictions for this system.160 While for the Li+, Tl+, Cu+, Ag+ and H3O+ ions, the error of the 12-6 model is relatively large because of their stronger charge-induced dipole interactions with the surrounding water molecules. The proton is not shown in the figure because it is out of range. From the discussion below, we note a 60-90 kcal/mol difference between the HFE value of the IOD parameter set and the experimental HFE value for the proton. We did not design a CM parameter set for the monovalent ions due to the relatively small error of the 12-6 model that for almost half of the positive ions it is possible to reproduce the IOD values using the HFE parameter set. For the monovalent anions, the TIP3P water model showed the most consistency with the experimental values when employing the 12-6 LJ nonbonded model. Meanwhile, it is intriguing that the 12-6 nonbonded model overestimated the HFEs of the halide ions. It is hard to reproduce both the experimental HFE and IOD values simultaneously and this may be due to the charge hydration asymmetry (CHA) effect, which is further discussed below.  The parameter sets developed herein attempt to reproduce certain target values through a trial and error process. Based on the results of our parameter scans with the 12-6 LJ potential, we have fit curves of HFE versus Rmin/2 for the monovalent positive and negative ions for the three water models (see Figure 11). For the HFE fitting curves of the positive ions, the TIP3P and SPC/E water models are very close with each other with TIP4PEW being a little further away. This is consistent with our work on divalent metal  44 ions. Intriguingly, for the negative ions, the SPC/E and TIP4PEW water models are similar while the TIP3P water model gives more positive HFE values for the ions with the same LJ parameters. This may also come from the CHA effect (discussed below). Since there is a nontrivial difference between the three water models, we performed parameterizations of the monovalent ions for these three water models separately to reproduce the target HFE values. The HFE parameter set is shown in Table 17. Its simulated HFE, IOD and CN values are given in Table 18. These parameters reproduce HFEs within ±1.0 kcal/mol of the target values.  We also analyzed the IOD values versus Rmin/2 values for the monovalent cations and anions with the three water models (see Figure 12). The three water models nearly share the same curve in the figure, for both the cations and anions. This is also consistent with our work on divalent metal ions. The only difference between the cations and anions is that TIP4PEW predicts slightly larger IOD values for the cations but slightly smaller IOD values for the anions than TIP3P and SPC/E do when using the same LJ parameters. Therefore, we designed a united IOD parameter set for all three water models. This parameter set is shown in Table 19 while the simulated HFE, IOD and CN values are shown in Table 20. The final IOD parameter set reproduced the target IOD values within ±0.01 † for most of the investigated ions in all of the three water models.  In general, our parameter sets reproduce the target CN values with reasonable accuracy. The largest errors are found for the Tl+, Cu+, Ag+ and NH4+, H3O+ ions, which are ions that do not share electronic structures with the noble gas atoms. The electronic structure  45 of the outermost shell of Tl+, Cu+ and Ag+ are 6s2, 3d10 and 4d10 respectively. While for the NH4+ and H3O+ ions, it is hard to reproduce their hydrogen bond network because the current model doesnÕt parameterize the hydrogen atoms explicitly. For the Tl+ ion, the two different IODs were not reproduced in the present parameter sets due to the isotropic nature of the nonbonded model. For the Cu+ ion, we could not reproduce the target CN value (which is 2) with any of the solvent models. In all cases the CN of the Cu+ ion is 4. A similar situation has found for the Ag+ ion where the target CN value ranges from 2 to 4 while our parameter sets give CNs from 4.8 to 6.0. Blauth et al. predicted the CN value of Ag+ as 6.0 in the aqueous system using the QMCF MD simulations134 while Blumberger et al. simulated this value as 4 using the CPMD method.135 It is hard to obtain the experimental CN value for the hydronium ion, but it has been proposed that the CNs of water and the hydronium ion are similar to each other.123 The HFE, IOD, and 12-6-4 parameter sets (in which the 12-6-4 parameter set is introduced in section 3) predict the CN value of the H3O+ ion in the range of 4.7-5.2, 7.0-7.1, and 8.2-8.4, respectively. The former one is consistent with the experimental CN value of water (determined as 5.2 from Soper161) while the later two CN values appear somewhat overestimated. The CN value of the NH4+ ion is in the range of 4-11 from the review of Ohtaki and Radnai,162 but a CN value around 4-5 seems more reasonable based on the consideration of its hydrogen bond network. Our parameter sets predicted this CN in the range of 6.8-7.9 while the CPMD simulations done by Brug” et al. predicted the CN as 5.3.131  Based on the quadratic fittings of the data points from the parameter scanning for the M(III) and M(IV) ions, we estimated their HFE and IOD parameter sets. The two  46 parameter sets are shown in Tables 21-22. Due to the large error of the 12-6 LJ nonbonded model for these highly charged ions, we did not design the CM parameter set for them because it would yield huge errors for both the HFE and IOD properties.    47 2.6 Parameter Assessment  In the parameter space, it is possible to reproduce the same properties using different combinations of the Rmin/2 and ! values. This is because different combinations of Rmin/2 and ! can generate similar C12 and C6 values for a certain pair of atom types. However, a poor balance between the Rmin/2 and ! values can cause transferability issues since all of the other C12 and C6 values need to be calculated based on the combining rules for the mixed system. In this way, the balance of the two parameters plays a significant role in parameter transferability.  Our final parameter sets are consistent with each other since they were built in an analogous manner. In particular, they show a clear trend that when an ion has a bigger VDW radius, it has a deeper well depth. In the work of Peng et al., they proposed the following relationships between the VDW parameters of the ions which share the same electronic structure with noble gas atoms: (1) The ion radii and well depth should increase for the ions inside a group; (2) The ion radius of the negative ion should be bigger than the neutral atom then the positive ion, which both have the same electronic configuration; (3) The dispersion term for the negative ion should be larger, comparing that of the neutral atom then that of the positive ion, which both have the same electronic configuration, due to the extent of the electron cloud.61 Our parameters follow these trends precisely. Also, our parameter set follows the trend that when two cations share the same electronic structure, the cation with a larger charge  48 has a smaller VDW radius. For example, the HFE parameter set for the TIP3P water model offers Rmin/2 parameters of the Na+, Mg2+, and Al3+ ions as 1.475, 1.284, and 0.981 †, respectively.  Besides follow the physical trends, our parameters are also quantitatively agreed with experimental results and theoretical calculations based on the QMSP method. For example, experimental results show that the F- ion has smaller ionic radius than K+ ion125 and the IOD parameter set also shows this trend. In Table 23 we compared the Rmin/2 values of 15 ions, which share the same electronic structures with the noble gas atoms, in the HFE parameter set to the VDW radii calculated by Stokes using the QMSP method.88 Generally they are in strong agreement with each other (with the smallest, biggest, and average absolute percentage errors as 0.0%, 9.1%, and 4.0%, respectively). For example, for the three M(III) ions (Al3+, Y3+ and La3+), the estimated Rmin/2 values in the estimated HFE parameter set for the TIP3P water model are 0.981 †, 1.454 †, and 1.628 †, respectively. The calculated VDW radii are 1.046 †, 1.481 † and 1.642 † respectively based on the QMSP method. There are only 6.2%, 1.8%, and 0.9% differences between the two value sets for the three trivalent ions respectively. This further validates the physical consistency of our parameterization work.  Meanwhile, the Rmin/2 values of our parameter sets are more consistent with the VDW radii calculated by the QMSP method than the parameter sets presented elsewhere. In Table SI.7 we compare our parameters of Na+, K+, Rb+, Cs+, F-, Cl-, Br-, and I- ions to the parameter sets developed by other groups (where we adapted their parameter sets for the  49 Lorentz-Berthelot combining rules). The average errors, standard deviations of the errors, and UAEs (using the QMSP VDW radii as reference values) of each parameter set were calculated and are given in Table SI.7. We show the UAEs of the different parameter sets in Figure 13. All our parameter sets (in which the 12-6-4 parameter set is introduced in section 3) show the smallest UAEs (~0.08 †) relative to earlier parameter sets. The metal ion parameters available in AMBER, which were adapted from the pioneering work of †qvist where he fit the ions using a cut-off procedure (this was done prior to the emergence of the PME method), show the largest UAE. In this parameter set there is an imbalance between C12 and C6 values, in which the C6 value is too small while the C12 value is closer to our values (based on the relative ratio). Because of this the transferability of the parameter set will be affected since Rmin,ij is overestimated due to the too small C6 value (recall !!"#!!"!!!!!"!!"!!!!"!). For example, this causes the alkali metal ions to have an inverse trend amongst the ! parameters. The Jensen parameter set has a relatively large UAE due to the choice of the ! value for the positive ions (which is too small). The UAE of the VDW radii of the positive ions is ~1.08 † for this parameter set while the corresponding UAE of the negative ions is only ~0.08 †. This is because the ! value of the positive ions (which is 0.0005 kcal/mol) is too small while the ! value of the negative ions (which is 0.71 kcal/mol) is more reasonable. The large UAE value for the parameter set 5 of Netz and co-workers (2009_HMN-5) is due to the small ! value for the positive ions (~0.0006 kcal/mol).68 The parameter set from Joung and Cheatham has too small ! values for the halide ions, for example, the ! value of Na+ is always bigger than that of F- (~30 to 100 times).39. DangÕs parameter set also shows small UAE value  50 because its ! values are in the range of 0.1-0.2 kcal/mol,53-60 which are reasonable for the monovalent ions.  Moreover, the Rmin/2 values in our 12-6 LJ parameter sets could be used as the VDW radii for the RESP charge fitting procedure. For example, in the work of Kuznetsov et al.,163 they used 1.4 † as the VDW radius for the RESP charge fitting for both the Fe2+ and Fe3+ ions. Herein, the Rmin/2 of Fe2+ was determined as 1.409 † in the IOD parameter set, and the IOD parameter set for the Fe3+ ion estimated the Rmin/2 as 1.386, 1.386, and 1.375 for the TIP3P, SPC/E, and TIP4PEW water models, respectively.  To further validate our parameter sets, we have carried out MD simulations of different ion pair solutions. The Kirkwood-Buff (KB) integrals were calculated to investigate the ionic solution systems. Finally the activity derivatives were computed to evaluate the parameter sets. In the work of Mouka et al.,164 they found that the parameter set from Joung and Cheatham39 (the JC parameter set herein) and the parameter set 5b from Horinek et al.68 (the H2 parameter set herein, the parameters were adapted using the Lorentz-Berthelot combining rules and are shown in Table SI.8) are the best among 13 different parameter sets for simulating the NaCl solutions when using the SPC/E water model. In the work of Fyta and Netz, they re-optimized the mixing rules of the ion pairs to reproduce the activity derivatives of several different kinds of ionic solutions.75 They found it was hard to reproduce the experimental activity derivatives of the NaF, KF, NaI, and CsI salt solutions. When compared to the values obtained using the un-optimized (standard) Lorentz-Berthelot combining rules, they found that a larger eij value for NaF, a  51 larger Rmin,ij value for KF, and a smaller !ij value for NaI and CsI were necessary to reproduce the experimental activity derivatives.   In the present work, we have carried out simulations on NaCl, KCl, NaF, KF, NaI, and CsI salt solutions at a concentration of ~0.30 M. The SPC/E water model was used in these simulations. The simulation procedure is detailed in the section 2.2. Comparison between our parameter sets (HFE, IOD and the 12-6-4 parameter sets for the SPC/E water model, where the 12-6-4 parameter set is introduced in section 3 below) with the JC and H2 parameter sets were made and are shown in Table 24. From Table 24 we can see that the average errors of the parameter sets follow the sequence 12-6-4 < HFE ~ IOD < JC < H2 while the UAEs follows the trend that 12-6-4 < IOD < JC < HFE < H2. In general our three parameter sets showed improved results over the JC and H2 parameter sets, indicating their superior transferability.  Meanwhile, we also performed simulations of a metalloprotein system. PDB entry 4BV1 was used to obtain the starting coordinates for this modeling exercise. It is a superoxide reductase (SOD) found in Nanoarchaeum equitans. It is a protein tetramer with each monomer has a metal site containing a Fe3+ ion. The structure has been determined by using X-ray crystallography to a resolution of 1.90 †. The tetramer structure is shown in Figure 14 while Chain C with its metal site is shown in Figure 15. The metal site contains 4 His groups, 1 Cys group, and 1 water molecule. By treating Chain C as the initial structure, we performed three simulations with different parameter sets (the HFE, IOD, and 12-6-4 parameter sets, in which the 12-6-4 parameter set is introduced in section 3).  52 The TIP3P water model was employed during the simulations. Details of the simulation procedure are given in section 2.2. Totally 10 ns production was performed during the simulation and snapshots were stored every 500 fs. The HFE parameter set prefers a smaller CN (of 4) and the metal ion moves out from the binding pocket, while stable metal complex structures were obtained for the simulations using the IOD and 12-6-4 parameter sets.  A RMSD analysis was performed over the protein backbone CA, C, N atoms and the metal site (metal ion plus the ligating residues and the binding water molecule) heavy atoms for the simulations by treating the initial structure (experimental structure) as reference. The results are depicted in Figure 16. The RMSD of the protein backbone CA, C, N atoms fluctuated around ~1.2 † while the RMSD of the metal site heavy atoms is ~0.5 †. These values illustrate that the metal binding site is stable during the simulations.   We have also performed an RMSF analysis of the heavy atoms for the protein residues together with the oxygen atom in the metal site binding water. The results are shown in Figure 17. From this figure it can be seen that the metal site residues: residue His10 (residue number 11), His 35 (residue number 36), His 41 (residue number 42), Cys 97 (residue number 98) and His 100 (residue number 101) all have relatively small RMSF values (~0.5 †) for their heavy atoms. The oxygen atom of the metal site binding water (residue number 115, is not shown in the figure since the protein ends at residue number 112) has RMSFs of ~0.6 † for the simulations using the IOD and 12-6-4 parameter set. These results further validated that the metal ion site is stable during the simulations.   53 2.7 Error Analysis  To further assess the final parameter sets, we performed error analysis. The resultant values of monovalent ions are shown in Table SI.9. Using the TIP3P water model as an example, we depict the error analysis results of monovalent ions in Figures 18-19. For the positive ions, the 12-6 LJ nonbonded model largely reproduces both the experimental HFE and IOD values for Na+, K+, Rb+, Cs+ and NH4+ ions while for the Li+, Tl+, Ag+, H+ and H3O+ ions, the 12-6 LJ nonbonded model yields larger errors. We also analyzed the percentage errors with respect to the experimental HFE and IOD values for the parameter sets of M(II) ions. These results are summarized in Table SI.10. From these data we can estimate the maximum error ranges. The values given in square brackets in Table SI.10a are for the IOD set, while the unbracketed values given in Table SI.10b are for the HFE parameter set, with the former indicating the error in HFEs (e.g. ~18% in TIP3P for Be2+) if we get the IODs correct, while the latter is the error we see in the IOD values (e.g. ~30% in TIP3P for Be2+) if we get HFEs correct. Hence, these values indicate the maximum error range associated with the modeling of M(II) cations using an unpolarized nonbonded model. Moreover we observe the trend that the TIP3P, SPC/E, TIP4PEW and TIP4P water models have increasing errors successively. For all four water models, ions like Be2+, Cu2+ and Zn2+ have larger errors presumably due to their strong coordination interaction with the surrounding waters. The alkaline-earth metal ions, except for Be2+, have the smallest errors likely due to their preference to form ionic bonds. Meanwhile, the estimated absolute and percentage errors of different parameter sets for highly charged ions were shown in Table SI.11. We can see that there are significant errors  54 associated with the 12-6 LJ nonbonded model for highly charged ions, we did not carry out further refinement work on the estimated 12-6 LJ parameters since the resultant parameters would be of limited usefulness.  Furthermore, we have summarized the (estimated) average errors and (estimated) average percent errors for the HFE and IOD parameter sets for the mono-, di-, tri- and tetravalent cations in Tables 25-26. Again using the TIP3P water model as an example we graphically summarize our results in Figures 20-21. From these tables and figures we observe that, not surprisingly, the average error of the monovalent cations are significantly smaller than that of the cations with higher oxidation states. The TIP3P and SPC/E water models show very similar results while the TIP4PEW water model shows the biggest deviation. For example, for the TIP3P water model, the average error of the IOD values for the HFE parameter set increases from -0.14 † for M(I), to -0.27 † for M(II) ions, -0.29 † for M(III) ions and -0.58 † for M(IV) cations. While the average error goes from ~20 kcal/mol for M(I), to ~50 kcal/mol for M(II) ions, ~80 kcal/mol for M(III), and ~240 kcal/mol for M(IV) ions for the IOD parameter set. These results, taken as a whole, shows that the underestimation the 12-6 LJ nonbonded model increases dramatically as the charge on the cation increases.  In present work we used the experimental HFEs of cations from MarcusÕ data set. These data were obtained based on the conventional HFEs of cations, and the HFE of proton, which was treated as -1056 kJ/mol (~ -252 kcal/mol) by Marcus, using the following equation (where Q is the ionÕs formal charge):  55 !!!"!!!!!!"!!"#$!!!!!!!"!!"#$#%!       (14) Other values for the HFE of proton in the general range of -250 to -265 kcal/mol (see section 2.1) can also be used, which would be more likely to decrease the experimental HFEs used herein and then increase the underestimation extent of the 12-6 model for cations. This is because Marcus used a value close to the upper band (-250 kcal/mol).  The error of the 12-6 model for the anions follows the opposite trend from the cations: the 12-6 model overestimated rather than underestimated the ion-water interaction for the halide anions. Which, we proposed, is due to the CHA effect. Rajamani et al. explained that the CHA effect arises from two effects: 1) a positive electric potential is induced by the surrounding water molecules on the surface of a neutral solute, so more energy will be released when changing the neutral solute into a negatively charged one; 2) water has different packing and orientations on the surface of positively and negatively charged solutes.165 Onufriev and co-workers proposed that the CHA effect stems from the asymmetric electric multipole components of water molecules.166 Later on their group developed a charge-asymmetric Born equation based on a statistical point of view, which incorporates the CHA effect efficiently.167 For aqueous systems containing ions, different atoms of the water molecule are coordinated to the counter-ions due to their different electrical properties. It is the oxygen atoms, which have a partial negative charge, greater mass, and most of the electronic cloud coordinated to the positive ion. The opposite situation exists for the negative ions: it is the hydrogen atoms of the water molecules that orient towards the negative ions.    56 The CHA effect can be well reproduced by using the parameters developed herein. Using the K+ and F- ions as an example: they have relatively similar effective ionic radii (1.38† and 1.33† for the 6-coordinated K+ and F- ions, respectively),125 while the HFE difference between them is ~41 or ~49 kcal/mol based on MarcusÕ or Schmid et al.Õs data sets. In the IOD parameter set, they almost have the same Rmin/2 parameters (1.745 and 1.739 † for K+ and F- ions respectively), while the computed HFE differences between them are ~54, ~61 and ~65 kcal/mol for the TIP3P, SPC/E and TIP4Pew water models, respectively. This is consistent with previous work that shows the TIP3P, SPC/E, TIP4P water models have an increasing CHA effect.165, 166 In Table 20 we show the IOD values for the IOD parameter set for the three water models. As we can see, unlike the positive ions, the IOD value sequence is TIP3P > SPC/E > TIP4PEW for the negative ions. This is opposite to the sequence of the hydrogen charge on these water models, which are +0.417e, +0.4238e and +0.52422e for the TIP3P, SPC/E and TIP4PEW water models, respectively. Hence, on one hand, for the positive ions, it seems the dipole moment of a water molecule plays a dominant role for the solvation properties of the ions: the TIP3P and SPC/E water models showed similar performance with each other and their dipole moments are all ~2.35 D, while the TIP4PEW water model showed a bigger underestimation of the ion-water interaction and it has a dipole moment of ~2.32 D. On the other hand, for the negative ions, it appears that the charge of the hydrogen atoms is a dominant factor for the simulated properties of these ions.   To further clarify the CHA effect for the different water models employed herein, we have depicted the CHA effect using the data from our parameter scans and show the  57 results in Figure 22. In this figure we observe that the CHA effect of the negative ions has smaller IOD and more negative HFE values when compared to the positive ions employing the same LJ parameters. It is intriguing that this effect decreases considerably for the HFE values but not the IOD values as the ionÕs VDW radius increases.  58 2.8 Conclusions  First, we tested the transferability of LJ parameters determined in previous work and found that it was necessary to design new parameters for M(II) metal ions in PME simulations. Systematic studies were performed to determine the LJ parameters for divalent cations using different water models with the Lorentz-Berthelot combining rules in PME simulations. HFEs, IODs in the first solvation peaks as well as CNs were determined for various combinations of LJ parameters by employing TI and MD simulations and using the PME summation method to model long-range electrostatics. The results showed there is a correlated relationship for the simulated HFE, IOD and CN values. A series of curves were obtained using the quadratic fitting procedure towards the target values based on the parameter space scanning results. It was observed that different water models give different HFEs but highly similar structural properties when treating the same metal ions with identical parameters.  Generally, it is hard to reproduce all the target properties of the M(II) metal ions in aqueous solution using the 12-6 nonbonded model due to the modelÕs simplicity, which agrees with the former work of Ponomarev et al.50 Overall, the 12-6 nonbonded model underestimates the interaction energy between these metal ions and surrounding water molecules and the underestimation extent is water model dependent. More interaction terms other than the LJ and Coulomb potentials should be considered in the FF in order to perform more accurate modeling of the M(II) metal ions. Polarizable FFs47, 54, 56, 122, 168-171 and the short-long effective functions (SLEF),172 which consider short-range interactions  59 such as polarization and charge transfer effects in more accurate ways, could be promising methods to solve the dilemma of M(II) metal ion parameter design.   Through a consideration of the physical origins of the VDW interaction, we fit a curve from the experimental data of noble gas atoms to represent the relationship between the two parameters in the 12-6 LJ potential. Furthermore, we also investigated the monovalent, trivalent and tetravalent ions. We found that the 12-6 LJ nonbonded model gives rise to relative small errors for the monovalent ions: for almost half of the monovalent cations investigated, it could reproduce the target HFE and IOD values simultaneously. Meanwhile, we found that the 12-6 model yields huge errors for the highly charged ions.  Based on the parameter space scanning results and NGC, we have developed two 12-6 LJ parameter sets (HFE and IOD) for 15 monovalent ions (11 monovalent cations plus 4 monovalent anions) for the TIP3P, SPC/E, and TIP4PEW water models, respectively. Meanwhile, we arrived at three parameter sets (HFE, IOD, and CM parameter sets) for 24 M(II) cations also with each of the TIP3P, SPC/E, and TIP4PEW water models. Furthermore, we also estimated HFE and IOD parameter sets for 24 highly charged metal ions (18 trivalent cations plus 6 tetravalent cations) with the three water models. The HFE parameter set used experimental HFE values as the target; the target property for the IOD parameter set is the experimental IOD values; while the CM parameter set aimed at reproducing the experimental relative HFE and CN values. Generally these parameters accurately reproduced the target properties.  60 We found our final parameters follow the trends of VDW parameters within ion series. Meanwhile, these parameters are consistent with experimentally or theoretically determined results. Results showed that on average the Rmin/2 values in our HFE parameter set for the TIP3P water model are only ~4.0% deviated from the VDW radii determined using the QMSP method. We have also compared our final parameter sets with previous sets from a number of research groups. Using the VDW radii calculated based on the QMSP method as the reference values, we find that our parameter sets show the smallest UAEs among all the parameter sets investigated. This is due to a better balance between the Rmin/2 and ! values in our parameter sets through using NGC. Furthermore, we predicted the activity derivatives for six different ion pair solutions and showed that the parameter sets developed herein (HFE, IOD, and 12-6-4, in which the 12-6-4 parameter set is introduced in section 3) reproduced these quantities better than existing parameter sets, validating their improved transferability.  Finally we performed error analysis for the 12-6 model of ions. For some monovalent cations (e.g. Na+, K+, Rb+, Cs+ and NH4+), it is possible to reproduce both the experimental HFE and IOD values simultaneously by employing the 12-6 LJ nonbonded model while for others like Ag+, Tl+ the 12-6 model has non-negligible errors. We also investigated the underestimation extents of the HFEs by the nonbonded model for different M(II) metal ions and found the errors are larger for the metal ions that could form stronger coordination interactions (i.e., covalent bonds) with surrounding waters. Moreover, we found that generally, with the increasing charge of the cation there is a notable decrease in the accuracy of the 12-6 LJ nonbonded model. Using TIP3P as an  61 example, the average underestimation of the HFE values increases from ~20 kcal/mol for M(I) cations, to ~50 kcal/mol for M(II) cations, ~80 kcal/mol for M(III) cations and ~240 kcal/mol for M(IV) cations when trying to reproduce the target IOD values. The average underestimation of the IOD values increases from 0.14 †, to 0.27 †, 0.29 † and 0.58 † for the M(I), M(II), M(III) and M(IV) cations respectively when trying to reproduce the experimental HFE values. The error tend of the 12-6 model for the halide anions was also discussed. The CHA phenomenon has been observed for the anions in the explicit water models we have investigated. Our results indicate that the CHA effect increases along the series of TIP3P, SPC/E, and TIP4PEW with the TIP3P water model having the smallest error when compared to the experimental results.    62 CHAPTER 3: PARAMETERIZATION OF THE 12-6-4 NONBONDED MODEL  3.1 Model Origin  As discussed in chapter 2, we found that it is impossible to reproduce all the three experimental properties at the same time in simulations using the 12-6 nonbonded model with parameters spanning the typical LJ space when ion has a formal charge >= +2. This is because the 12-6 nonbonded model underestimates the interaction energy between the ion and surrounding water molecules. This underestimation decreases with the increasing of the metal ionÕs size, and increases along with the ion charge.  FFs are designed to accurately describe the interactions in a complex system using potential functions that are relatively easy to compute and consist of terms representing the bonded and nonbonded interactions. The nonbonded model in typical FFs is composed of an electrostatic and a VDW term. Point charges are obtained in a number of ways, but in the AMBER FF RESP charges173 are used and they approximate the charge-charge, charge-dipole and dipole-dipole interactions. The 12-6 LJ potential is used to represent the VDW interaction, which consists of the Pauli repulsion and induced dipole-induced dipole (so called dispersion) interactions. However, as shown in Figure 23, generally there is no term representing the charge-induced dipole and dipole-induced dipole interaction explicitly in the nonbonded model. Furthermore, among these two interactions, the charge-induced dipole interaction is the dominant one in the case of metal ions and it has a potential functional form proportional to r-4. In light of this, we  63 added a new term into the 12-6 nonbonded model between the charged ion and the surrounding particles to represent the charge-induced dipole interaction. The new potential was given in equation 15 and the parameters that need to be determined are Rmin/2, ! and ! (or C4). We determined the final parameters for 55 ions (range from monovalent to tetravalent, see Figure 24) using different water models (TIP3P, SPC/E, and TIP4PEW) by utilizing the experimental HFE, IOD and CN values shown in Tables 1-3 (in terms of the divalent ions, we only designed parameters for the ions with all of the three target values available in Table 2). Details of the process used to design our parameter sets is described in the following section. !!"!!"!!!!"!"!!"!"!!!!!"!!"!"!!!!!"!!"!!!!!!!!!!!"!!!!"!!"#!!"!!"!"!!!!"#!!"!!"!!!!!"!!"!!!!!!!!!!"!!!!"!!"#!!"!!!!"!!!!"#!!"!!"!!!!!!"!!"!!!!!!!!!!"     (15)    64 3.2 Methods  When simulating the HFE values using the 12-6-4 model, the &GEle+Pol and -&GEle+Pol instead of &GEle and -&GEle values were obtained (see Figure 2). For the divalent metal ions, a similar method to Method 3 was used to obtain the HFE values, except that a 3-window TI scaling procedure was used to obtain &GVDW. We also used a similar method as Method 3 to obtain the IOD and CN values except that a bigger box with size of ~(36†)36†)36†) was used for the TIP4PEW water model, in which there were 1085 water molecules. For the monovalent and highly charged ions, Method 3 was used to simulate the HFE, IOD and CN values. The time-step was set to 1 fs for all of the MD simulations except for the simulations performed for the proton-aqueous system using the 12-6-4 model, for which a 0.5 fs time-step was used. We estimated the uncertainties of the simulated HFEs using the 12-6-4 model in present work are about ±1 kcal/mol for the monovalent143 and divalent ions,174 while they are ±2 kcal/mol for the highly charged ions.145  Fitting procedure of the HFE quadratic fitting curves: (1) There are 7 different ! values for each row in Table SI.12. We performed the quadratic fitting and the got the equation for HFE versus ! for each row with a fixed Rmin/2 value.   (2) We solved the equation to determine the ! value to reproduce the target HFE of a metal ion with the fixed Rmin/2. Then for each row we obtained a point (Rmin/2, !)  65 which can reproduce the target HFE. In total there are 16 rows so we got 16 points, with each of which can reproduce the target HFE.  (3) We discarded the points which were not in the parameter space (with Rmin/2 in 0.8-2.3 † and ! in 0-6 †-2) and performed another quadratic fitting of ! versus Rmin/2 for the remaining points to get the final fitting curve for the target HFE.  Fitting procedure of the IOD quadratic fitting curves: (1) There are 7 different ! values for each row in Table SI.13. We performed the quadratic fitting and the got the equation for IOD versus ! for each row with a fixed Rmin/2 value.   (2) We solved the equation to determine the ! value to reproduce the target IOD of a metal ion with the fixed Rmin/2. Then for each row we obtained a point (Rmin/2, !) which can reproduce the target IOD. In total there are 16 rows so we got 16 points, with each of which can reproduce the target IOD.  (3) We discarded the points which were not in range of Rmin/2 in 0.8-2.3 † and ! in 0-10 †-2 and then performed another quadratic fitting of ! versus Rmin/2 for the remaining points to get the final fitting curve for the target IOD.  Mg2+ is one of the most commonly seen divalent cations in cells and play a significant role in the metabolism of organisms. In order to test the performance of our parameters in mixed systems, we carried out the simulations of MgCl2 solutions at different concentrations. For simulating these MgCl2 systems, the Cl- LJ parameters were from the  66 work of Joung and Cheatam,39 a TIP3P water box (including ~2000 water molecules) was used, and three different concentration levels (0.25 M, 0.5 M and 1.0 M) were modeled: (1) For the system with 0.25 M concentration: 10 Mg2+ and 20 Cl- ions were solvated in a cubic water box with 2158 TIP3P water molecules; (2) For the system with 0.5 M concentration: 19 Mg2+ and 38 Cl- ions were solvated in a cubic water box with 2184 TIP3P water molecules; (3) For the system with 1.0 M concentration: 38 Mg2+ and 76 Cl- ions were solvated in a cubic water box with 2102 TIP3P water molecules. The process for obtaining the IOD values was described in Method 3 in section 2.2.  Nucleic acids are the gene carriers. Mg2+ ions play very important roles in nucleic acid functions. We treated a nucleic acid system as an example to assess different parameter sets of the Mg2+ ion. The PDB structure of a DNA fragment (PDB ID: 1D23) was used in the simulation. The water molecules were removed from the PDB file and hydrogen atoms were added using the reduce program175 in AmberTools. Afterwards, 7 Mg2+ ions were added to neutralize the system (plus the two Mg2+ ions already existing in the PDB structure, yielding 9 Mg2+ in the system in total). Then a cubic water box with 3547 TIP4PEW waters was used to solvate the system.  The minimizations and MD simulations were performed in 7 stages as described follows: (1) Firstly, 2000 steps steepest descent minimization plus 3000 steps conjugate gradient minimization were performed to relax the water molecules and ions (with a 500.0 kcal/mol!†-2 force restraint on the other parts of the structure);  67 (2) Afterwards, 2000 steps of steepest descent minimization plus 3000 steps of conjugate gradient minimization were performed to relax the hydrogen atoms, water molecules and ions (again, with a 500.0 kcal/mol!†-2 force restraint on the other parts of the structure); (3) Then 5000 steps of steepest descent minimization plus 40000 steps of conjugate gradient minimization were performed to relax the whole system. (4) 40 ps of simulation in the NVT ensemble was carried out to heat the system from 0 to 300 K. (5) Afterwards, 20 ps of simulation in the NVT ensemble at 300 K was performed to equilibrate the system. (6) Then 2 ns of simulation was performed using the NPT ensemble at 300 K and 1 atm to correct the system density and further equilibrate the system. (7) Finally 10 ns of production simulation in the NPT ensemble was performed at 300K and 1 atm with snapshots being stored every 2 ps for the final data analysis. PBC and PME were employed during all the MD simulations. The time-step was set as 1 fs and the cut-off was set to 10 †. Temperature was controlled by the Langevin dynamics with the collision frequency equals 1.0 ps-1. The pressure relaxation time was set as 2.0 ps in the NPT ensemble. SHAKE141, 142 was used to constrain the bonds involving hydrogen atoms during the MD simulation. The distance between the Mg2+ and backbone phosphate was obtained in a similar way as the IOD value, using the procedure described in Method 3.    68 3.3 Parameter Space Scanning and Parameter Determination  First, we scanned the parameter space for divalent metal ions where Rmin/2 and ! represented the two axes. The range investigated for Rmin/2 was 0.8-2.3 † with an interval of 0.1 † interval, while ! ranged 0-6 †-2 with an interval of 1 †-2. The ! value was fixed for each Rmin/2 value according to NGC, where the Rmin/2 and ! values had a relationship of !!"#!!!!!!!!!!!"#!! with !!!!"!!" and !!!!!!!"#. To be consistent with our work for the 12-6 model, the final parameters represent Rmin/2 with three decimal places, ! with eight, and ! with three. The HFE, IOD and CN values from our parameter space scanning for the TIP3P, SPC/E and TIP4PEW water models are given in Tables SI.12 and SI.13. The HFE and IOD fitting curves were obtained based on these data. The curve fitting procedure is provided section 3.2. It is similar to the method used for the 12-6 model (see section 2.2) with the only difference being for the IOD fitting curve. The intention of this modification was to obtain enough meaningful points to fit the IOD curves.   Using the TIP3P water model as an example, we present the HFE and IOD fitting curves we obtained in Figures 25-27. In contrast to the 12-6 LJ nonbonded model of divalent metal ions, in which the HFE and IOD fitting curves for each metal ion were nearly parallel with each other (see Figure 7), the HFE and IOD fitting curves for each metal ion for the 12-6-4 model have an intersection-point. The fitting curves for the Be2+ ion is not shown in Figures 25-27 because the intersection-point of its HFE and IOD fitting curves is beyond the scanning range examined. Because of the presence of an intersection point  69 there exists a set of parameters that reproduce the experimental HFE and IOD values simultaneously. After obtaining the intersection-point between the two fitting curves and fine-tuning the resultant parameters, we obtained the final parameters for each divalent metal ion. These parameters are shown in Tables 27-28, the simulated HFE, IOD and CN values are given in Table 29. Herein, our final parameters reproduce the target HFEs within ±1 kcal/mol and simultaneously reproduce most of the target IOD and CN values with good accuracy.  The 12-6 LJ nonbonded model still shows some unsatisfactory behavior in that it underestimates (for some positive ions) or overestimates (for the negative ions) ion-water interactions, which leads us to build a 12-6-4 LJ-type nonbonded model for the monovalent ions investigated. Meanwhile, the 12-6 model showed remarkable errors for highly charged ions, for which the 12-6-4 model was also parameterized.  Besides the 12-6 LJ scanning curves of monovalent and highly charged ions, we also scanned the corresponding parallel curves using the 12-6-4 model. These curves share the same Rmin/2 and ! values with the 12-6 scanning curves but have an additional fixed C4 term. This C4 term equals 100 kcal/mol!†4 for monovalent cations. While for the monovalent anions, this C4 term equals -100 kcal/mol!†4. And for the trivalent and tetravalent ions, this term equals 500 kcal/mol!†4. The parameters, and the simulated HFE, IOD and CN values of these parameter scans were shown in Tables SI.3 to SI.6. Via linear interpolations from the 12-6 to the 12-6-4 scanning curves we were able to obtain initial guesses for the final 12-6-4 parameter sets. After initial parameter selection  70 and subsequent fine-tuning, the final 12-6-4 parameters were determined to reproduce the target HFE and IOD values. The final 12-6-4 parameters of monovalent ions are shown in Table 30, while the corresponding computed HFE, IOD and CN values are shown in Table 31. These parameters simultaneously reproduce the target HFEs and IODs with considerable accuracy (within ±1.0 kcal/mol of the target HFEs and within ±0.01 † of the target IODs). The final optimized 12-6-4 parameters for highly charged ions are given in Table 32 while the simulated HFE, IOD and CN values are shown in Table 33. These parameters reproduce the target HFE values within ±1 kcal/mol and target IOD values within ±0.01 † for the M(III) ions while they reproduce the target HFE values within ±2 kcal/mol and the target IOD values within ±0.01 † for the M(IV) ions. It can be seen that in the 12-6-4 parameter set for cations ranged from monovalent to tetravalent, the Rmin/2 terms are similar between the three water models while the C4 term for TIP4PEW water is generally larger than for the same metal ion in the other two water models. This may due to the smaller dipole of the TIP4PEW water model (~2.32 D) relative to the TIP3P (~2.35 D) and SPC/E (~2.35 D) water models.  Comparing to the 12-6 model, we can see there is significant improvement of the accuracy using the 12-6-4 model, which is able to reproduce the experimental HFE and IOD values simultaneously. While for the 12-6 model, if you want to reproduce the experimental HFE value, the error in the simulated IOD value would increase along with the formal charge of the cation (see Table 25 and Figure 20). Vice versa, if you simulate the IOD value using the 12-6 model, the error of the calculated HFE would increase  71 markedly with an increase of the oxidation state of the cation (see Table 26 and Figure 21).    72 3.4 Discussion  The situations are different for the monovalent cations and anions with respect to the obtained parameters. For the positive ions, the C4 terms are always positive, while the C4 terms are negative values for the negative ions. This arises from the fact that the 12-6 LJ nonbonded model ignores the charge-induced dipole interaction for the positive ions, but overestimates the charge transfer effect for the negative ions (see discussion below). Meanwhile, it is interesting that the C4 term decreases as the ionic radii increases for the halide ions when using the TIP3P water model. While for the SPC/E and TIP4PEW water models the corresponding C4 terms are very similar for different halide ions.  The 12-6 LJ nonbonded model represents the interaction between an ion and its environment via a summation of the electrostatic and VDW terms. It doesnÕt consider polarization and charge-transfer effects explicitly, which may be the reason for the negative C4 terms for the halide ions. For the cations, it is the oxygen atom that acts as the coordinated atom while it is the hydrogen atom coordinated to the counter-ion in the anion-water system. Since the electronic cloud focuses on the oxygen atoms, the charge-induced dipole cannot be overlooked for the positive ion-water systems. However, the polarization effect in the halide ion-water interaction should be relatively weak given the nature of the electron cloud on the hydrogen atoms in a water molecule.   Meanwhile, the charge-transfer effect behaves differently in ion-water systems for the positive and negative ions. The charge transfer direction is from the water molecules to  73 the ions for the former systems but in the opposite direction for the later systems. Thompson and Hynes found that, for anion-water systems, there is charge transfer from the ion to the water, and the strength of this effect decreases as the halide ionÕs size increases.176 This is counter-intuitive to what would be expected based solely on the electron affinities of the halogen atoms. They also found that when the distance between the halide ion and water is in the range of 0.7-1.0 †, there is a strong polarization effect. However, as the distance continues to increase the charge transfer effect becomes the dominant part where the charge is mainly transferred between the halide ion and the oxygen atom but not the hydrogen atoms of the water molecule.176 Moreover, the closer the halide ion is to a water molecule the stronger of the charge transfer effect. The red shift caused by the charge transfer effect is hard to simulate using the 12-6 model while it is demonstrated that a two-state valence bond model is capable of doing that.176 This is consistent with our observation that as the halide ion size increases, the HFE error of the IOD parameter set decreases (true for the SPC/E and TIP4PEW water models while it is a almost a constant for the TIP3P water model). Taking the charge transfer effect into account, the charge of the halide ions are less than -1e in aqueous solution,177 so the 12-6 LJ nonbonded model overestimates the halide ion-water interactions, making a negative C4 term necessary in the 12-6-4 LJ-type nonbonded model.  For the alkaline earth metal ions the C4 term decreases monotonically with an increase of the metal ionÕs radius. This may due to the shielding of the metal ionÕs ability to polarize the surrounding water molecules when the ionÕs radius increases. Another interesting observation is that although Be2+ is the smallest ion studied in the present work, it doesnÕt  74 have the largest C4 value among all the metal ions. For example, in the TIP3P or SPC/E water model, the C4 value of the Be2+ ion is around 190 †4.kcal/mol, which is ~20% less than the value obtained for Zn2+. This may arise from a significant charge-transfer effect between Be2+ and its surrounding water molecules. Pavlov et al. showed that there is about -1.28e transferred from the surrounding water molecules to the Be2+ ion in the [Be(H2O)4]2+ complex according to a Mulliken population analysis at the B3LYP/6-311+G(2d,2p) level of theory.178   For the divalent metal ions in the first transition row, their C4 terms are also consistent with the Irving-William series179 where the magnitudes of the C4 terms varies as Mn2+< Fe2+< Co2+< Ni2+< Cu2+ > Zn2+, which arises due to the interplay between the covalent and ionic interactions.180 Through density functional theory, it was found that the charge transfer effect follows the Irving-William series sequence.180 There is also a larger bond order between the Cu2+ ion and itÕs ligating ligands than for the other M(II) ions in the Irving-William series. Moreover, the ionic and covalent interactions can compensate one another. When there is a stronger charge transfer, there is usually a stronger covalent interaction but a weaker ionic interaction between the metal ion and its ligands.180 Another thing to bear in mind is that the d9 electronic structure of the Cu(H2O)62+ complex results in the Jahn-Teller effect where the axial bond lengths are about 0.44 † longer than the equatorial ones.123 This effect could not be modeled using the present 12-6-4 nonbonded potential due to the isotropic approximation employed.   75 Cd2+ has a smaller C4 value than that of Zn2+, which is due to the longer distance between the Cd2+ ion and the surrounding water molecules. However, it is interesting that Hg2+, the ion that has the highest atomic number among these +2 metal ions, has a C4 value bigger than that of Zn2+. Tai and Lim found that Hg2+ is a much better charge acceptor than Zn2+ according to their calculations,181 which they ascribed to the relativistic effect. Hg2+ can accept more negative charge from its surrounding ligands than the Zn2+ ion. This phenomenon not only happened when the ligating atom is the ÒsoftÓ sulfur atom but also when the ligating atom is a ÒharderÓ atom like nitrogen or oxygen.181   The Sn2+ cation also has an unusual C4 value among these +2 metal ions. This may be because its two outermost electrons occupy a 5s orbital and Sn also has a +4 oxidation state that is slightly more stable. Experiment has also revealed two different bond lengths with an asymmetric structure in the first hydration shell of Sn2+, which can not be well reproduced with our model.162   The main group and transition trivalent metal ions have much stronger ion-water interactions than the Ln3+ ions. They form a stable octahedral structure with water molecules in the first solvation shell. Data in Table 3 indicates that the average effective radius of the first solvation shell water is ~1.35 † for the first several metal ions, which is consistent with strong interactions between the coordinated water molecules and these metal ions. Meanwhile, these values are close to previously proposed coordinated water radius (~1.34 †) based on experimental data of Cesium alums.182 The corresponding average value is ~1.49 † and ~1.46 † for the Ln3+ and the M(IV) metal ions respectively,  76 which implies a smaller electronic cloud overlap between the metal ion center and each of the coordinated water molecules.    Al3+, In3+, Tl3+ are group IV ions. For the C4 parameters derived herein we obtained a sequence of Tl3+ > Al3+ > In3+. The Al3+ ion is the smallest M(III) ion, resulting in a relatively large C4 term due to its strong covalent interaction with the coordinated water molecules. Tl has two oxidation states: +1 and +3 and the HFE values of the Tl+ and K+ ions are almost the same in MarcusÕ HFE set.115 Tl3+ could have very strong covalent interactions with the surrounding residues. The reduced electric potentials (electric potentials of reaction M(III) + 3e- = M) are -1.67 eV,  -0.3382 eV and +0.72 eV for Al3+, In3+ and Tl3+, respectively.183 The positive reduction potential of Tl3+ makes it a very reactive species. It readily obtains electrons from its surroundings, which may be the reason for a strong charge-transfer effect between the Tl3+ ion and the surrounding water molecules. The 12-6-4 parameters of In3+ and Tl3+ ions gave an excellent prediction for the HFE and IOD values but overestimated the CNs (8 instead of 6). This may be due to the lack of a correction term for the water-water interactions in the first solvation shell during the simulations. The water-water interactions were parameterized to reproduce the pure liquid water properties in the original parameter designs. However, the first solvation shell waters of the highly charged metal ions should more strongly repel one another due to their bigger charge separations. This effect is smaller for the M(I) and M(II) metal ions, but it dramatically increases for the highly charged ions. Meanwhile, this kind of effect may decrease in protein systems due to the pre-organizations of the metal ion binding sites.  77 Fe3+ has a larger C4 term than Al3+, the smallest M(III) ion, which suggests that Fe3+ has a stronger interaction with its surrounding water molecules. This is consistent with QMCF MD simulations, which showed the force constant between the ion and the oxygen of the first solvation shell waters (kion-O) is 198N/m for Fe3+, compared to 185 N/m for Al3+.26 This is a consequence of both the electrostatic and covalent interactions. The Fe3+ ion has an average charge of +1.85e (from a Mulliken analysis) in the QMCF simulation184 while the Al3+ ion has a corresponding value of +2.5e,185 which implies there is a stronger charge-transfer effect between the Fe3+ ion and the coordinated water molecules than for the Al3+ ion. There is a slight overestimation of the CN for the Fe3+ ion. As discussed above, this may be due to the underestimation of the interactions between the first solvation water molecules. While this effect is operative in aqueous solution, it could be less of an issue in protein systems (see section 3.5).  The +3 oxidation state is the typical oxidation state of the Ln elements, with the exception that Eu2+ and Ce4+ could also be observed. This is because Eu2+ has a half-filled 4f orbital while Ce4+ has the same electronic configuration as Xe. The interaction of the Ln3+ ions with surrounding water molecules would be expected to have more ionic character than the M(III) ions discussed above. For example, the C4 terms between the Ln3+ ions and water molecules are in the range of 152-282 kcal/mol!†4, which is smaller than the range of 258-456 kcal/mol!†4 for the other +3 metal ions discussed above. Previous simulations found that the kion-O values are much smaller for the Ln3+ ions. For example, La3+, Ce3+, Lu3+ and Er3+ have kion-O values ~110 N/m, while the values for the Al3+ and Fe3+ ions are 185 and 203 N/m, respectively.186 The Ln3+ ions have effective  78 ionic radii in the range of 0.86-1.03 † and IOD values in the range of 2.34-2.55†. These values are similar to those of the Ca2+ ion (whose effective ionic radius and IOD value are 1.00 †125 and 2.46 †,187 respectively). Therefore, they have been used as probes to investigate the roles of the Ca2+ ions in biological systems.188  From Table 32 we observe that the La3+ and Gd3+ ions have the smallest C4 terms among the Ln3+ ions. This may be because they have either totally empty or half-filled 4f orbitals, making them more likely to form isolated ions, which reduces the covalent character of their bonds with the coordinated water molecules. It is easy to see the Òlanthanide contractionÓ effect from Table 3. The effective ion radius decreases monotonically with an increase of the metal ionsÕ atomic number due to the poor shielding of the 4f electrons towards the 5s and 5p orbitals.189 A similar tendency can also be seen for the HFE and IOD value along the series. Our final Rmin/2 parameters are consistent with this pattern as well. Meanwhile, the CN also decreases along the Ln3+ ion series. Previous work reached the conclusion that the lighter Ln3+ ions (La3+ to Nd3+) prefer a CN of ~9 and the heavier ions (Gd3+ to Tb3+) prefer a CN of ~8 while the middle ions such as Sm3+ and Eu3+, have CNs between these two values.190-195 It was proposed that the former Ln3+ ions form tricapped trigonal prism structures with their first solvation shells, while the structures shift to distorted bicapped trigonal prism structures for the heavier elements as one of the two capping water leaves the first solvation shell.196 Generally speaking, the 12-6-4 parameter set could well reproduce the target HFE, IOD and CN values with the exception that some of the CN values were slightly overestimated. The final parameters gave CNs in the range of 9-10 for the Ln3+ ions, rather than the range of 8-9 reported in  79 the literature. As discussed above, this may be due to the fact that there is no water-water interaction correction term in the present parameterization process. Moreover, the CN values of the Ln3+ ions given by Marcus115 (as shown in Table 3) likely also vary under different experimental conditions (counter-ions used, solute concentration, etc.). Among all the trivalent metal ions investigated herein, Al3+, Y3+, and La3+ have the same electron configurations as the noble gas atoms Ne, Kr, and Xe respectively. Using the TIP3P water model as an example, we can see their C4 values decrease monotonically from 399, to 216, and 152 kcal/mol!†4.  There are only a few M(IV) ions that exist in aqueous solution while the others are readily hydrolyzed into polynuclear complexes in water.136-139 Table 3 shows the M(IV) ions examined herein. These ions exist at least in highly acidic solutions. The CN values of these metal ions are greater than 8 with some of them being ~10 according to experiment.136-139 Earlier research found that Pu(IV), Th(IV) and U(IV) could strongly bind to transferrin, a iron-transport protein.197 Hence, the parameters developed herein might facilitate theoretical research on the bio-toxicity of these M(IV) ions.   Zr4+ and Hf4+ are in the IVB group. Even though Hf4+ has a larger atomic number than Zr4+, due to the Òlanthanide contractionÓ effect, it has a smaller effective radius, a smaller IOD, a more negative HFE, and a bigger C4 term than Zr4+. These observations reflect its stronger interaction with the surrounding water molecules. In contrast, Ce4+ and Th4+ are in the same group where the one with larger atomic number (Th4+) has a bigger ionic radius, a less negative HFE, and a smaller C4 term. This may be because that Ce4+ and  80 Th4+ share the same electronic structures of Xe and Rn, respectively. Besides these ions, Zr4+ is another M(IV) ion which shares the same electronic structure with a noble gas atom (Kr). There is also a single trend in the C4 terms of the Zr4+, Ce4+, Th4+ ions for each specific water model. For instance, their C4 terms in the TIP3P water model are 761, 706, and 512 kcal/mol!†4 , respectively.  Th, U and Pu elements are in the An series and are the largest elements investigated in the present work. Their tetravalent metal ions only exist in highly acidic solutions. Canaval et al. investigated Th4+ in aqueous solution using the QMCF MD simulation method. They found a stable 9-coordinate complex, and even third solvation shell water molecules had a bigger mean residence time than that of pure water, implying they are stabilized by the highly charged Th4+ ion.198 U4+ fluoresces due to the electron transition between the 5f16d1 and 5f2 electronic configurations.199 The U4+ ion has the largest C4 term in all the M(IV) metal ions investigated. Frick et al. investigated the U4+ ion in aqueous solution using the QMCF MD method and the CN value was characterized as 9 while the average charge of U4+ was predicted to be +2.68e from the Mulliken population analysis.200 Odoh et al. simulated the Pu3+, Pu4+, PuO2+ and PuO22+ ions in waters using the CPMD method.201 They predicted the pKa values of the first hydrolysis step for the Pu3+, Pu4+, PuO2+ and PuO22+ ions are 6.65, 0.17, 9.51, and 5.70 respectively, showing a general tendency that the larger of the charge of the metal center, the lower the pKa value of the first hydrolysis reaction.   81 Hf4+, Zr4+ and Pu4+ have relatively smaller IOD values among the tetravalent ions where they all have experimental CNs of ~8.139 Ce4+ was determined to have an experimental CN of ~9136 while U4+ and Th4+ have CNs between 9-11.139 Soderholm et al. proposed that counter-ions play a key role in the first solvation shell structure while the 9, 10 or 11-coordinated Th4+ have very small energy differences and are in a dynamic equilibration.202 The simulated HFE and IOD values of the 12-6-4 parameter set are in excellent agreement with experiment. The simulated CN values of most of the M(IV) ions are ~10 with Hf4+ having a CN ~8 for the TIP4PEW and SPC/E water models. Herein, the TIP3P model always predicted a larger CN value than the other two water models, which may be because it has a smaller C12 term (~582.0)105 kcal.†12/mol) than those of the SPC/E (~629.4)105 kcal.†12/mol) and TIP4PEW(~656.1)105 kcal.†12/mol) models.   There are several redox pairs available in our 12-6-4 parameter set. We analyzed them below to explore the consistency of the C4 parameters with respect to the behavior of these pairs in aqueous solution. We also calculated the relative HFEs inside some redox pairs for the TIP3P water model. A 9-window TI simulation (50 ps of equilibration and 150 ps of production for each window) was performed forward and backwards to obtain the final results. The simulated relative HFEs of Cr2+/Cr3+, Fe2+/Fe3+ and Ce3+/Ce4+ ion pairs were 516.9, 580.2, and 698.2 kcal/mol while the experimental values are 516.2, 579.6, and 697.6 kcal/mol respectively.115 These results further validated the method employed in the present work.   82 The Cr2+ and Cr3+ ions have the [Ar]3d4 and [Ar]3d5 electronic structures, respectively, where the Cr(H2O)62+ complex has a strong Jahn-Teller effect while Cr(H2O)63+ molecule has a standard octahedral configuration. The IOD values decreases from 2.08 to 1.96 † for the Cr2+ and Cr3+ ions. Using the TIP3P water model as a representative example we observe that the Rmin/2 parameter decreases from 1.431 to 1.415 † while the C4 term increases from ~137 to ~258 kcal/mol!†4 for the Cr2+ and Cr3+ ions, which trends well follow the intrinsic physics.  Fe2+/Fe3+ redox pair exists broadly in biologically related systems such as the Fe-S proteins and heme structures,203, 204 and it plays fundamental roles in many electron transfer processes. The experimental IOD value shrinks about ~0.08 † from 2.11 † of Fe2+ to 2.03 † of Fe3+ ion. Moin et al. investigated the ferrous and ferric ions in water using the QMCF MD simulation method. They obtained the kion-O force constant of 193 N/m for Fe3+, which is almost twice as strong as that of Fe2+ (93 N/m). While the effective charges (from a Mulliken population analysis) are in the range of +1.25e to +1.45e (with an average of +1.36e) for the Fe2+ ion during the simulation and for the Fe3+ ion the effective charges are in the range of +1.70e to +1.95e (with an average of +1.85e).184 By comparing the 12-6-4 parameters for the Fe2+ and Fe3+ ions in the TIP3P water model, we find that the Rmin/2 decreases slightly as the outer shell electron number decreases while the C4 term increases by about 2.5 times. This is consistent with the ratio between the C4 terms of a trivalent and a divalent ion ([3/2]2 = 2.25) that is derived from the original ion-induced dipole equation (see section 3.5).   83 The ratio between the experimental HFEs of the Cu+ and Cu2+ ions is ~3.8 (using -480.4/-125.5), which is close to 4 and consistent with the Born equation. By treating the TIP3P water model as an example, we can see that the Cu2+ ion has a much bigger C4 term than that of the Cu+ ion (290.9 versus 7 †4.kcal/mol). However, the Rmin/2 parameter of the Cu+ ion is less than that of the Cu2+ ion (1.217 versus 1.476 †), because it has a smaller IOD value (1.87 † of Cu+ versus 2.11 † of Cu2+). Meanwhile, the 12-6-4 model could not reproduce the CN of the Cu+ ion (giving it as 4 instead of 2) or the Jahn-Teller effect of the Cu2+ ion. These effects are related to the atomic orbitals and cannot be simulated using the 12-6-4 model which treats the ion as a symmetry particle.   The ratio between the experimental HFEs of the Tl3+ and Tl+ ions is ~13.2 (using -948.9/-71.7), which is deviated from 9, the ratio between the squares of their oxidation states. Again, using the TIP3P water model as an example, we can see that the Rmin/2 parameter decreases by 0.322 † when the Tl+ ion loses two electrons. Meanwhile, the ratio between the C4 terms of the Tl3+ and Tl+ ions is ~9.1 (using 456/20), which strongly agreed with the original ion-induced dipole equation (see section 3.5). These trends are consistent with the intrinsic physics and further validate the 12-6-4 model.  Ce has both the +3 and +4 oxidation states. Ce4+ is the most stable state because it shares the same electronic configuration with Xe. Just like the Cr2+/Cr3+, Fe2+/Fe3+, and Tl+/Tl3+ redox pairs discussed above we find that the Rmin/2 value decreases while the C4 term increases significantly with the increasing of the ionÕ formal charge. For example, for the 12-6-4 parameters determined for the TIP3P water model, the Rmin/2 parameter decreases  84 by ~0.03 † while the C4 term increases by ~480 kcal/mol!†4 when Ce3+ loses one electron.    85 3.5 Parameter Assessment  !!!!!!!!!!!!!!!!!"#$!!!(16) !!!!!!!!!!!!!!!!!!!!!!!!!!"#!!!!!(17)!To further assess our final parameters we employed equation 16, which is based on the equation for the charge-induced dipole interaction (equation 17), to approximate the C4 term between the M(II) ion and the surrounding water molecules. In these two equations, q is the charge of the metal ion, ,0 is the polarizability of the particle interacting with the metal ion. & is the angle between the induced dipole and electronic field generated by the metal ion. To calculate the C4 term between an M(II) ion and a water molecule, we assumed $0=1.444 †3 (obtained from the book of Eisenberg and Kauzmann205), q=+2e, &=0¡ and !r=1 (as in the AMBER FF89). The calculation yields a value of ~960.0 †4.kcal/mol for the C4 term, which is on the same order of magnitude (except for some large ions in the TIP3P or SPC/E water model) as but bigger than our final C4 parameters for divalent metal ions. This may arise because the fixed-charge water models are over-polarized in their original designs. For example, the TIP3P and SPC/E water molecules have dipoles of ~2.35 D while the TIP4PEW water molecule has a dipole of ~2.32 D. These values are greater than the experimentally determined permanent dipole (1.855 D)206 for the gas-phase water molecule. These fixed-charge water models included the polarization effect to some extent by over-fitting the permanent dipole moment while omitting the induced dipole. This approximation may also give some insights into to the nonbonded modelÕs tendency to underestimate the HFEs for +2 metal ions: in particular,  86 the three water models (TIP3P, SPC/E, and TIP4PEW) all have smaller dipole moments than the dipole moment of a liquid phase water molecule determined from experiment (2.95 ± 0.2 D)207 or ab intio MD simulation (~3.0 D)208. In the work of Wu et al., they found that the charge transfer effect between Zn2+ ion and water molecule could be incorporated into the polarization energy term when representing the charge and polarizability of Zn2+ as +2e and 0.260 †3 respectively.209 This may come from the fact that the AMOEBA water model has a total dipole of 2.54 D,210 which is bigger than the dipole moments of the three non-polarizable water models discussed here.  Meanwhile, as shown in Table 28, the C4 term generally decreases when the metal ionÕs radius increases. This may arise from the cos& term in equation 16 because the nearby water molecules would be more readily polarized than the remote ones. When there is a greater distance between the metal ion and its surrounding water molecules, the latter would be more randomly oriented due to a reduction in the influence from the metal ion. Table 28 also shows that the TIP3P and SPC/E water models give very similar C4 values while the TIP4PEW model gives slightly larger values. This may arise from the fact that the negative charge of the TIP4PEW water model is on the dummy atom, which is placed on the bisector of the hydrogen-oxygen-hydrogen angle, yielding a slightly smaller dipole, which is then responsible for the observed larger underestimation of the ion-water interaction. This representation results in a greater distance between the positive charge on the metal ion and the negative charge on water than those of the two 3-site water models and, thereby, increases the C4 values.    87 Furthermore, from Table 26 and Figure 21 we observe that the underestimation extent of the absolute HFEs for the IOD parameter set increases roughly proportional to the charge square of the cation. This is consistent with the equation for the charge-induced dipole interaction (see equation 17), which interaction was not considered in the 12-6 LJ equation (see section 3.1). For example, the ratio of the underestimations of the absolute HFEs for the IOD parameter set in the TIP3P water model is roughly 1.0 : 3.0 : 4.8 : 14.2 between the mono-, di-, tri-, and tetravalent cations (calculated from 17.2 : 51.1 : 82.7 : 244.3 kcal/mol). This trend is close to the ratio: 12 : 22 : 32 : 42, which is predicted from the original equation of charge-induced dipole interaction (see equation 16). This further validates the proposed C4 term in the 12-6-4 model developed herein.  Herein we showed some test cases regarding the 12-6-4 model. These simulations proved the outperformance of the 12-6-4 model over the 12-6 model, and also the excellent transferability of the 12-6-4 model.  During the simulations of mixed systems using the 12-6-4 model, the C4 parameter between the metal ion and each atom type (except the oxygen atom in water because the C4 term between the metal ion and it has been parameterized) is evaluated using the following equation: !!!"#$!!"#$!!!!!!!!!!!!!!!!!!!!"#$!!"#$!   (18) We set the polarizability of a water molecule to 1.444 †3, which was taken from Eisenberg and Kauzmann.205 Since the electron cloud almost centers on the oxygen atom  88 in water, here we treated the oxygen atom as having all of the polarizability of the water molecule while the hydrogen atoms have a polarizability equals zero.  Panteva et al. have benchmarked 17 different nonbonded models for simulating the Mg2+-aqueous system in predicting the thermodynamic, structural, kinetic, and mass transport properties.93 Results showed that none of the 12-6 models could reproduce all the properties at the same time. They also found that the 12-6-4 models offered improvement over the 12-6 models, and the 12-6-4 model combined with the SPC/E water model could accurately reproduce all the experimental properties simultaneously.  To further test the 12-6-4 nonbonded model proposed here, we performed simulations on aqueous MgCl2 systems at different concentrations and on a Mg2+-nucleic acid system. In total three parameter sets were tested in the simulations, which included the original AMBER FF parameters for Mg2+, the CM and 12-6-4 parameter sets for Mg2+ developed herein. For the 12-6-4 parameter set, the Rmin/2 and ! parameters of Mg2+ were from Table 27 based on the water model used and the C4 parameter between Mg2+ and water oxygen atom is from Table 28 based on the water model used. Polarizabilities of different atom types, which were used to obtain the C4 terms between the metal ion and different atom types in the 12-6-4 model, were shown in Table 34.  The IOD values of the Mg2+ and Cl- ions in the MgCl2 systems are shown in Table 35. It can be seen that both the 12-6-4 parameter set (which gives the IOD of ~2.11 † for Mg2+) and the CM parameter set (which gives the IOD of ~2.08 † for Mg2+) outperformed the  89 original AMBER FF parameters (which gives the IOD of ~2.00 † for Mg2+) with respect to the experimental IOD value of 2.09±0.04 †.123 For the Mg2+-nucleic acid system, Table 36 shows the IOD values and the Mg2+-backbone phosphate distances. We can see that both the 12-6-4 and CM parameter sets offered remarkable improvements over the original Mg2+ parameter set in the AMBER FF while the 12-6-4 model gives the best results.  We also performed test simulations for several ionic solutions. The method was the same as the method used in the simulations of ionic solutions described in section 2.2 except a 12-6-4 model was used. We used equation 18 (for the C4 terms between the cation and cation, between the anion and anion, and between the ion and the hydrogen atoms in water) and equation 19 (for the C4 term between the cation and anion) to estimate the C4 terms between the particle pairs (except for the C4 term between cation and water oxygen atom, and the C4 term between anion and water oxygen atom, both of which have been parameterized and were shown in Table 30). The polarizability values of various monovalent ions were taken from Sangster and Atwood.211 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!     (19) As described in section 2.6, we compared the 12-6-4, HFE and IOD parameter sets to two of the best parameter sets developed previously regarding their abilities to reproduce the activity derivatives of different ionic solutions. The results are shown in Table 24. We can see that, indeed, the 12-6-4 parameter set showed the best performance among these parameter sets. Besides its better performance in simulating the single-ion properties, it  90 also showed improved results in reproducing the ion pair properties, further validating its advantage over the 12-6 model in simulating ions.  Moreover, a metalloprotein system was also tested. The method was the same as the method of simulating a Fe(III) containing protein described in section 2.2 except a 12-6-4 model was used. The C4 term between the Fe3+ ion and water oxygen atom is from Table 32 and equation 18 was used to obtain the C4 terms between the Fe3+ ion and other atom types. Polarizabilities of different atom types were shown in Table 37. As described in section 2.6, we examined the performance of the HFE, IOD and 12-6-4 parameter sets for simulating the Fe(III) containing protein system (see Figures 14-15). Results showed that the IOD and 12-6-4 parameter sets yielded stable metal complex structures (see Figures 16-17), while the HFE parameter set failed to do that.    91 3.6 Design Your Own Parameters  It is important to note that the HFEs of ions may vary among different experimental analyses and the subsequent theoretical treatments of the experimental data. The experimental HFEs of ions can be determined in different ways. In some works, the HFE values of ions were obtained based on the National Bureau of Standards (NBS) compilation of the conventional HFE values. In this situation, the HFEs of ions would change if the HFE of a proton changes. The present work relied on the approach of Marcus,115 in which he obtained the experimental HFEs of cations as described above (based on the protonÕs HFE and the conventional HFE values obtained from the NBS compilation,212 see equation 14). Marcus treated &hydG0[H+]=-1056 kJ/mol ±6 kJ/mol in his literature,115 which comes from &hydH0[H+]=-1094 kJ/mol, &hydS0[H+]=-131 J/(K!mol) or S'[H+(aq)]=-22.2 J/(K!mol). However, there are different HFE values that have been determined for the proton.117, 118, 120, 213, 214 For example, Tissandier et al. estimated the &hydG0[H+] value by employing the cluster-pair-approximation,118 while several other works concerning the computational estimation of the HFE of a proton have been published.117, 213, 214 Meanwhile, there were also experimental and theoretical efforts that predicted the HFE values for some of the investigated cations which were different from the target values used in this work.116, 215 For example, Schmid et al. proposed a new set of HFE values for several monovalent and divalent ions116 while Asthagiri et al. calculated the HFEs for the first transition row metal ions using a quasi-chemical theory of solutions.215 Regardless of the choices made, the data in SI from our parameter space  92 scans for different water models would facilitate the parameter design targeting data sets other than we used herein.115   93 3.7 Conclusions  We showed that the electrostatic plus 12-6 LJ potential nonbonded model underestimates the interactions between metal ions and the surrounding water molecules in PME simulations (see section 2). Via a consideration of the physical origins, we hypothesized that the charge-induced dipole interaction is responsible for the majority of this underestimation. In light of this we proposed and parameterized a 12-6-4 LJ-type nonbonded model in order to take into account the charge-induced dipole interaction.  And we found that unlike the 12-6 LJ nonbonded model, the new 12-6-4 model could reproduce the experimental HFE, IOD and CN values simultaneously after systematic parameterization. It improves the accuracy of the 12-6 model remarkably with just a slight increase of the computational cost. In the present work we have parameterized the 12-6-4 LJ-type nonbonded model for 55 ions (including 11 monovalent cations, 4 monovalent anions, 16 divalent cations, 18 trivalent cations, and 6 tetravalent cations) for three extensively used water models (the TIP3P, SPC/E and TIP4PEW water models). The 12-6-4 parameters have been parameterized to reproduce both the experimental HFE and IOD values simultaneously. Generally they reproduce the target HFEs within ±1kcal/mol for the ions that have formal charges equal or less than +3 and within ±2 kcal/mol for the ions that have formal charges of +4, with reproducing the target IOD values within ±0.01 †.   94 Moreover, excellent quantitative and qualitative agreement with previous experimental and computational works supports the validity of the 12-6-4 LJ-type nonbonded model. For example, both the three parameters (Rmin/2, !, and C4) are in physical range. An independent study from Panteva et al. showed that the proposed 12-6-4 model offered improved performance over the 12-6 model for simulating the Mg2+-aqueous system.93 A new combining rule was proposed for the C4 term, and tests in mixed systems further validated the new model: the 12-6-4 parameter set showed the best performance over all the parameter sets in the ionic solution tests, further supporting its advantage over the 12-6 LJ model, while simulations of aqueous MgCl2, Mg2+--nucleic acid, and metalloprotein systems consistently revealed good performance and transferability of the parameters determined herein.  In summary, the 12-6-4 LJ-type nonbonded model provides a significant improvement over the former 12-6 LJ nonbonded model. It reproduces several different kinds of experimental data at the same time, which eliminates the need to develop compromise parameters as was done for the former nonbonded model in PME simulations. It is easy to incorporate the present model into typical biomolecular FFs with minimal additional computational cost. We believe that the parameter sets developed herein will improve our ability to model ions in biological systems. However, further parameterization efforts may be needed to increase the performance of the current model. One caveat is that the new model doesnÕt consider the charge-transfer effect explicitly in the potential form, which may influence its ability to simulate systems with strong charge-transfer effects. Hence, care should be taken for systems like Be2+ and Hg2+ in aqueous solution due to the  95 existence of strong charge-transfer effects.178, 181 Finally, experimental values for the same metal ions may be variable due to different assumptions and standards employed. However, our data presented in SI allows for the straightforward design of LJ parameters for these ions, which could be helpful for the researchers who want to target different experimental HFE, IOD and CN values than used in our work.    96 CHAPTER 4: DEVELOPMENT OF THE MCPB.PY PROGRAM  4.1 Code Structure  MCPB.py is built using the molecule and atom classes in the pyMSMT package. There are different parsers (e.g. PDB parser, mol2 parser) available that read and write different format files. Users can build their own programs based on these data structures as well. The installed main program of MCPB.py is found in the $AMBERHOME/bin directory. The source code for the MCPB.py program is freely available in the AmberTools suite of programs for those interested in delving into the details. The original version of the code is available as a part of the AmberTools15 package103 and updated versions can be obtained from GitHub (https://github.com/Amber-MD/pymsmt). We suggest users that begin with the two examples in the SI to familiarize themselves with the program and then check the source code for particulars.    97 4.2 Workflow  Figure 28 illustrates the workflow of the MCPB.py program. It facilitates metal site modeling in classical FFs with parameters derived from QM calculations. It is designed as a bridge between several QM software packages (including Gaussian104 and GAMESS-US105) and MD software employing FFs (including AMBER,140 CHARMM,107 GROMACS,108 NAMD,216 OpenMM,146 LAMMPS,217 etc.) for the modeling of metal-containing systems. The version 1.0 of MCPB.py supports the following AMBER FFs: ff94, ff99, ff99SB, ff03, ff03.r1, ff10, ff12SB, ff14SB and GAFF. The default is ff14SB for the protein while GAFF is used for the small organic ligands. The program should be used with AmberTools software package because it uses code, data files and the resp program from it.  Similar to MCPB,4 MCPB.py uses two models to do the parameterization in order to strike a balance between accuracy and speed (as shown in Figure 28): a smaller one to obtain the metal associated bond and angle parameters and a larger one to parameterize the partial charges. Besides supporting the parameterization of metal centers which only have amino acid sidechains and ligands coordinating to the metal ion, MCPB.py further supports modeling for complexes, which have backbone, terminal oxygen or nitrogen atoms coordinated to the metal ions, or a system with a mixed sidechain/backbone/terminal/ligand binding mode. To differentiate it from the earlier MCPB program, we use Òsmall modelÓ instead of Òsidechain modelÓ because MCPB.py is more versatile in dealing with metal centers of different binding modes.  98 MCPB.py builds the small model based on the schemes shown in Table 38. The non-amino-acid residues are kept during the modeling process. Generally, MCPB.py uses three approaches to build the small model: (1) uses the CH3R (where R represents a sidechain group) groups to mimic residues which have sidechain atoms bound to the metal ion; (2) uses a ACE (CO-CH3) or NME (NH-CH3) residue (together with a capped ACE or NME group, if necessary) to mimic the backbone when there is a backbone atom bound to the metal ion; or (3) builds a CH2R-CO or NH-CH2R group (with a capped ACE or NME group, if necessary) if both backbone and sidechain atoms of one residue are bound to the central metal ion. These approaches will mimic the chemical environment at an affordable computational cost with several examples shown in Figure 29. Similar to MCPB, MCPB.py supports modeling for those metal sites with multiple metal centers as well.   For the large model, MCPB.py uses an approach adapted from MCPB, in which the metal site amino acid residues are capped by ACE and NME groups, and if there are two residues that are both bound to the metal ion with less than 5 residues between them, the intermediate residues will be retained and simplified to GLY residues. There is a variable (large_opt) in MCPB.py to control whether a full geometry optimization, hydrogen-only optimization or no optimization is specified in the Gaussian input file of the large model. It is not suggested to carry out full geometry optimization for the large model due to the computational expense and the chance that the structure might undergo large structural changes in the absence of the protein environment constraints. In general, we recommend a hydrogen-only geometry optimization if some structural relaxation is necessary, but a  99 full optimization can be done if the computational resources allow and if it is confirmed that the structure doesnÕt dramatically change upon optimization.   In order to conserve the total charge of the metal site, RESP charge fits173 for the large model are performed with the capped ACE, NME and intermediate GLY residues in their neutral forms. Similar to antechamber,218 there are two stages of charge fitting performed in MCPB.py: the first stage is carried out with small restraints on the heavy atoms (0.0005 for each heavy atom) and no restraints on the hydrogen atoms. The second stage refits the charges on the CH2 and CH3 group (while the charges on the other groups are kept at the values from the first stage), including symmetrizing the charges on the hydrogen atoms bound to the same carbon atom, with a 0.001 restraint on each of the carbon atoms and no restraints on the hydrogen atoms.  Figure 30 shows the 81 metal ions presently supported by the MCPB.py program. Most of the VDW parameters (VDW radii for the RESP fits and LJ parameters) of the related metal ions are from section 2, while the remaining are adapted from UFF.2    100 4.3 Usage  The MCPB.py code was designed with an optimized structure and offers a more user-friendly experience than the MCPB program. The command line input for MCPB.py is: MCPB.py Ði input file Ðs step number                 [--fchk Gaussian fchk file] [--logf Gaussian/GAMESS-US log file] (4) Note: The --fchk and --logf options are not necessary, in the case where the fchk file or log file do not use the default name (users can consult the manual for further details).  There are a series of input variables available to meet different usersÕ demands. In the examples below we kept the input files simple. For more advanced cases users can consult the software manual for full details. The modeling process of MCPB.py involves 4 steps (as shown in Figure 28) which are indicated as steps 1, 2, 3 and 4, respectively. The first step is preparing the input files for the small and large models (input files for the standard model will also be generated for the subsequent steps). The second step is generating the frcmod file for the system, which contains the bond, angle, torsion and VDW FF parameters. The bond and angle parameters can be generated based on the Seminario method (default, also using step 2s), Z-matrix method (using step 2z), and an empirical method (using step 2e, only supports zinc currently). As in MCPB,4 MCPB.py assigns zero torsion barriers for dihedral angles.  The third step involves the RESP charge fitting steps and building the mol2 files for each metal site residue based on the fitted RESP charges. The program will rename these  101 residues automatically to differentiate them from the amino acid residues in the standard AMBER FF library. There are several options (with steps labeled as 3a, 3b, 3c(default) and 3d) available for performing the RESP charge fits. They correspond to charge fitting approaches involving: (1) all charges treated flexibly (ChgModA); (2) charges of the backbone heavy atoms (CA, N, C and O) are fixed (ChgModB); (3) charges of all the backbone atoms (N, H, C, O, CA, HA) are fixed (ChgModC); and (4) charges of all the backbone atoms and the CB atom are fixed (ChgModD). These fixed charges are assigned values from the AMBER FF employed. If a residue has a backbone atom bound to the metal ion, all atomic charges in it will be flexible no matter which algorithm is chosen. This is because metal binding may have a strong influence on the charge distribution of the atoms in the backbone.   The fourth step is generating the new PDB file with the renamed metal site residues and creating the leap input file to build the AMBER topology and coordinate files. If there are any missing parameters needed by leap, users can use the CartHess2FC.py program (available in https://github.com/Amber-MD/pymsmt) to calculate all the bond and angle parameters of the small model and manually add the related parameters into the frcmod file. If there are any metal related dihedral parameters missing, users can give them a zero barrier in the frcmod file.    After completion of these steps users can transfer the AMBER topology and coordinate files to another format if they want to use alternate software to run the minimization and MD simulation. There are a few programs available to do this conversion: for example,  102 ParmEd103 converts the AMBER format into CHARMMÕs format; ACEYPE106 converts the AMBER format into that of GROMACS; finally, amber2lmp converts the AMBER format into LAMMPSÕs.    103 4.4 Examples  In what follows we delineate two sample examples: one is a metalloprotein and the other is an organometallic compound. The related modeling files could be found in the SI of ref 219.  Our first example is from PDB entry: 1E67, which is a zinc containing azurin protein from Pseudomonas aeruginosa and the structure has a resolution of 2.14 †.220 The protein is a tetramer in which each monomer contains 128 amino acid residues and has a metal site with four residues bound to the central zinc ion (two HIS residues and one CYS residue which have sidechain atoms coordinated to the zinc ion, along with a GLY residue which has its backbone oxygen bound to the zinc ion). Here we only treat chain A (see Figure 31) as an example Ð the generated parameters are also applicable to the other three chains. First we prepare the system by using the H++ web server148 to add hydrogen atoms (note: users need to manually add the zinc ion into the final PDB file and modify the residues which bind to the zinc ion because the H++ web server will delete the metal ions and water molecules, and ignore them when adding the hydrogen atoms). Followed this we use the MCPB.py input file to generate the necessary files for the small, standard, and large models.  In order to keep the example straightforward there are only a few variables in the MCPB.py input file (see Figure 32). For more complicated cases users can consult the manual for the details concerning all of the available options. Here we specify the PDB  104 file name we are using (after the original_pdb variable name), the group name of the system (the default group name is ÒMOLÓ while here it is 1E67), the bond cut-off value between the metal ion and ligands (here it is 2.7 † and the default is 2.8 - sometimes adjustment of this value is needed to specify which residue is bound to the metal ion), the atom IDs of the central ions in the metal site of the PDB file (here we only have one number because it is a single center metal site), and the mol2 file of the zinc ion (in which we specify its atom type as ÒZNÓ and its charge as +2, and to keep consistent with the zinc ion in the PDB file, we treat the zinc ion with a residue name of ÒZNÓ and an atom name of ÒZNÓ in the mol2 file as well).  Figure 33 lists the commands used during the MCPB.py modeling procedure. First we use the command: ÒMCPB.py Ði 1E67.in Ðs 1Ó, which generates the small, standard and large models of the metal ion coordination sphere. In the fingerprint file for the standard model (see the files supplied in SI of ref 219), there is atom type specification for each atom of the residues in the metal site (i.e., the central metal ion plus its ligating residues), and the linkage information between the metal ion and its ligating atoms. The third and fifth columns of atom information show the original and new atom types, respectively. The Ò-s 1Ó flag tells the program to assign the atom types for the central metal ion and the metal bound atoms automatically (each one is assigned differently). Users can change the atom types and/or linkage information manually based on their own preferences (e.g. by treating some of the bound atoms identically).   105 With all the files in hand we then use the small model to calculate the Hessian matrix and the large model to perform the Merz-Kollman population analysis221 by using Gaussian03,104 Gaussian09222 or GAMESS-US.105 By default MCPB.py treats the output files as being from Gaussian03. If Gaussian09 or GAMESS-US is being used, users need to add Òsoftware_version g09Ó or Òsoftware_version gmsÓ as a line in the input file before performing the second and third steps. Users can also modify the QM input files based on their own needs, i.e. changing the number of CPUs, memory usage, level of theory, etc. Here we used the B3LYP/6-31G* level of theory to perform the Gaussian calculations of example 1.  For the force constant determination there are two substeps: the first substep is QM geometry optimization (line 2 in Figure 33) and the second substep is the calculation of the Cartesian Hessian matrix (line 3 in Figure 33). The Cartesian Hessian matrix, which will be used by the Seminario method to obtain force constants, was stored in the Gaussian binary chk file for Gaussian03 and Gaussian 09 (the chk file should be converted to a fchk file using the formchk program). Alternatively, this matrix could be in a GAMESS-US log file (if using GAMESS-US) after the second QM calculation substep. The internal force constants (based on the Z-matrix method) are stored in the Gaussian log file, while Z-matrix method is not supported using the GAMESS-US software in the current version of MCPB.py. For the Merz-Kollman population analysis of the large model, there is only one calculation performed (line 5 in Figure 33), in which the VDW radius for the zinc ion was taken from the CM parameter set of the Zn2+ ion for the TIP3P water model.  106 After performing the QM calculations we can use ÒMCPB.py Ði 1E67.in Ðs 2Ó to generate the final force constant parameters using the Seminario method. The resultant metal ion related bond and angle parameters have the ÒCreated by Seminario method using MCPB.pyÓ comment specified in the end of such lines (see Figure 34). We observe see that the generated parameters are in a physically meaningful range:  the bond and angle force constants are all lower than 100 (AMBER frcmod files use the unit of kcal/mol!†-2 for the bond force constants and kcal/mol!Rad-2 for the angle force constants). Following this step we use the ÒMCPB.py Ði 1E67.in Ðs 3Ó command to perform a RESP charge fit for the large model and generate the mol2 files for the metal site residues. From the RESP fit we obtain the zinc ion has a partial charge of ~+0.5e, which represents a strong electron transfer effect from the ligating residues to the zinc ion. This is consistent with a previous work from Merz et al.,223 which showed that refitting the charges of the metal site, other than simply assigning a +2 charge to the zinc ion, is needed to accurately mimic the metal site charge distribution. Finally, we use the ÒMCPB.py ÐI 1E67.in Ðs 4Ó command to generate the PDB file with the renamed metal site residues and the leap source file for the system. Afterwards, the topology and coordinate files can be created using the Òtleap Ðs Ðf 1E67_tleap.in > 1E67_tleap.outÓ command. With all files in hand we can perform minimization and MD simulations using AMBER, or transfer the topology and coordinate files to another format if desired (see Figure 28). In this example, we used pmemd.MPI from AMBER 14224 to perform the energy minimization, MD heating simulation and subsequent MD equilibration simulation. Afterwards, pmemd.cuda225, 226 from AMBER 14224 was used for the production MD simulation.  107 Except the parameters obtained from MCPB.py, we used the AMBER ff14SB FF to model the protein system. The protein structure was solvated in a rectangular TIP3P65 water box (with size of ~(67†)57†)65†) and 5549 water molecules) with water molecules at least 1.5 † away from the protein surface. No counter-ions were added because the system was neutral. Afterwards four minimization steps were performed. 1000 steps of steepest descent minimization and 1000 steps of conjugated gradient minimization were performed for each of the first three stages. The first stage involved a 200 kcal/mol!†-2 restraint on the protein, and the second stage used a 200 kcal/mol!†-2 restraint on the heavy atoms in the protein, while the third stage used a 200 kcal/mol!†-2 restraint on the backbone N, CA, C atoms of the protein. In the fourth stage of minimization, 2000 steps of steepest descent minimization followed by 3000 steps of conjugated gradient minimization were carried out without any restraints. Afterwards 1 ns of simulation was performed in the NVT ensemble to heat the system from 0 to 298.15 K, followed by another 1 ns of simulation in the NVT ensemble to equilibrate the system at 298.15 K. Then a 1 ns of simulation was carried at 298.15 K and 1 atm in the NPT ensemble to correct the system density. Finally, 20 ns of production simulation was performed at 298.15 K in NVT ensemble with snapshots stored every 10 ps. In total, 2000 frames were collected for the final analysis. The Langevin algorithm was used to control the temperature with a collision frequency of 1.0 ps-1 while the Berendsen barostat227 was employed to control the pressure with a relaxation time of 1.0 ps. SHAKE141 was used to constrain the bonds between the hydrogen atoms and their connected heavy atoms while a specific Òthree-pointÓ algorithm142 was used for the water molecules.  108 Figure 35 shows the RMSD values of the protein backbone N, CA, C atoms and the metal site heavy atoms over 2000 snapshots across 20 ns of sampling. The RMSD values of the metal site heavy atoms are fluctuating around 0.3 † while the RMSD values of the whole protein backbone N, CA, C atoms are ~1.0 †. These results indicate that the metal site structure was well conserved during the simulation, thereby, further validating the parameterization accomplished by MCPB.py.  Os2+ complexes are attracting interests due to their special electronic properties.228 Here we treat the Os[(phen)3]2+ complex as the second example to prove the ability of MCPB.py of parameterizing the organometallic compounds. In the present work we use the Os[(phen)3]2+ structure of residue 1 of the Cambridge Structural Database (CSD) entry: FAJYAR01 (see Figure 36) from the work of Demadis et al.228 The content of the MCPB.py input file is shown in Figure 37. The commands used for processing the MCPB.py construction are shown in Figure 38.  We employed the B3LYP/SDD level of theory to do the QM calculations. The VDW radius of Os2+ during the Merz-Kollman population analysis was treated as 1.56 †, which was adapted from the VDW parameters of the ÒOs6+6Ó atom type in UFF.2 The finally obtained bond and angle parameters are shown in Figure 39.  The MM minimization and normal mode analysis were carried out using the nucleic acid builder (NAB) module in AmberTools15.103 The minimization was performed with two steps in gas-phase and a cut-off value of 100 †. The first step used the conjugate gradient  109 minimization algorithm with the convergence criterion of the energy gradient RMS set to 5)10-5 kcal/mol over a maximum of 20000 steps. The second step used the Newton-Raphson algorithm with the convergence criterion of energy gradient RMS set as 2)10-12 kcal/mol over a maximum of 200 steps. Afterwards the normal mode analysis was performed based on the minimized structure.  We have calculated the RMSD values between each pair of the CSD, QM optimized and MM minimized structures (see Table 39). It is noted that the CSD structure is from a crystal, with two PF6- anions and 1/2 water molecule for each Os[(phen)3]2+ complex while the QM optimization and MM minimization were all performed in the gas phase for the independent Os complex. We can see that in general the RMSD values are relative small (less than 0.4 †).   We performed a linear fit of the QM and MM calculated normal modes. The R2 value of the fitting is ~0.99, which is reasonable. Meanwhile, we also compared the QM and MM calculated normal modes in Figure 40, with the normal mode numbers along the X axis, while the frequencies are on the Y axis. Generally the agreement is good with the exception of some modes occupy the medium (around 1500 cm-1) and high (around 3200 cm-1) frequencies. This may due the underestimation of the force constants of the hydrogen involving bonds in GAFF, as shown in the work regarding parameterization of zinc complexes by Lin and Wang.100 However, even improvement could be obtained after tuning these parameters, Lin and Wang noted that hydrogen containing bonds are usually  110 constrained using SHAKE.141 Hence there would only be a small impact whether these parameters were fully optimized or not.    111 4.5 Conclusions  Metal ion modeling in FFs remains a challenging issue due to the range of coordination modes available to transition metal ions. MCPB.py is an efficient tool that facilitates the construction of reliable FF parameters for metal ion containing systems utilizing the bonded model. It supports two widely used QM software packages - Gaussian and GAMESS-US for the parameterization process. It has an optimized code structure and has more options than the original C++ based MCPB program. Far fewer steps overall and fewer steps requiring user interventions are capable of affording a reliable metal ion FF. The ability of MCPB.py to handle FF parameterizations for different metal centers has been shown by two examples outlined. It has a GNU_GPL_v3 license and is free to download and distribute. It lowers the barrier of the molecular modeling for metal ion containing systems and offers a clean interface for non-expert users interested in doing related simulations.    112 CHAPTER 5: APPLICATION CASE OF THE CUSF METALLOCHAPERONE SYSTEM  The CusF metallochaperone system has a special cation-% interaction in its metal site (see Figure 41), which attracts peopleÕs interests.229 In a former research theoretical calculations showed this cation-% interaction is bigger than 10 kcal/mol, which stabilizes the metal site structure considerably.102 Based on MD simulations we observed that the apo-CusF system has an open-close cycle at the microsecond time scale.230 In a putative mechanism of metal ion transfer between Cu+¥CusF and apo-CusB, it is likely that the interactions between the proteins drive Cu+¥CusF to sample the open conformation. Hence the conformational change of CusF may play an important role in the metal ion transfer process.  In an attempt to estimate the free energy difference between the closed and open conformations of Cu+¥CusF, we performed a two-dimensional PMF calculation. Here the hybrid model from a former research was used to describe the metal binding site: the chemical bonds between the Cu+ ion and the three ligating atoms in the His36, Met47, and Met49 residues were represented by the bonded model, while the cation-% interaction between the Cu+ ion and the Trp44 residue was represented by a 12-6 nonbonded model. The 12-6 LJ parameters of the CE3 and CZ3 atoms in the Trp44 aromatic ring were adjusted to reproduce the QM calculated interaction strength between the Cu+ ion and Trp44.102 Our simulations of the apo and metal-bound CusF systems suggest that the distance between the Cu+ ion and the CZ3 atom in Trp44 and the protein backbone # dihedral angle between Glu36 and Met47 provided a useful metric to distinguish between  113 the closed and open states of this metallochaperone (see Figure 42). In a closed conformation, the Cu+-CZ3(Trp44) distance is 2.33 † and the dihedral angle is -147.53¡ (see Figure 42). While in a wide-open conformation observed in our simulation, the corresponding Cu+-CZ3(Trp44) distance increases to 17.42 † and the dihedral angle changes to 47.61¡ (see Figure 42).  In order to efficiently generate conformations for a detailed sampling study, PMF simulations were first performed to transition from state 1 (closed state) to state 2 (wide-open state) along the diagonal described in Figure 43. The total reaction pathway was divided into a number of windows such that values were incremented in each successive window. In this process, the final snapshot of the starting window was treated as the initial structure of the subsequent window.  Conformations were equilibrated for 1 ns of MD simulation for each window. Conformations from the diagonal scan were used as starting structures for a detailed scan along both coordinates while maintaining the same overlap between adjacent windows. For the initial scan, the total reaction coordinate pathway was divided into 39 windows with an increment of 0.5 † and 6.48¡ in each successive window along the distance and dihedral coordinates respectively (see the left panel in Figure 43). The starting and final conformations from these calculations along the diagonal have coordinates of (2.33 †, -147.52¡) and (21.33 †, 98.72¡) respectively, along the Cu+-Trp44 distance and the Glu46-Met47 dihedral angle. After the initial scan, the scan directions went in two directions (see the right panel in Figure 43). Similar to the initial scan, each window was equilibrated for 1 ns of MD simulation, and the equilibrated structure from a window was used as a starting conformation for the  114 neighboring window(s). For the final scan, we sampled an area covering 39 windows along the Cu+-Trp44 distance, and 44 windows across the Glu46-Met47 dihedral angle, adding up to 1716 windows. 5 ns of production PMF MD simulation was performed after the 1 ns of equilibration for each of the 1716 windows. Step size was set to 1 fs and data points were stored after every 10 fs of sampling time for all these simulations. A harmonic potential with a force constant of 10 kcal/mol!†-2 was applied to the Cu+-Trp44(CZ3) distance while a harmonic potential using a force constant of 500 kcal/mol!Rad-2 was applied to the dihedral angle in these calculations. In order to sample adequately around minima, an additional 125 ns of biased MD was sampled for each of the four windows with coordinates of (15.83 †, -56.80¡), (16.83 †, 46.88¡), (16.33†, -56.80¡), and (17.33†, 46.88¡) which characterize two minima, yielding over 9 µs of PMF-MD sampling in the end. The data from these biased simulations was collected and unbiased using the weighted histogram analysis method (WHAM) to calculate a two-dimensional free energy profile.  While these PMF calculations are limited in terms of the conformational changes owing to the choice of coordinates, they provide us a lower estimate of the energetic penalty (~8 kcal/mol) for the protein to undergo this transition (see Figure 44). Afterwards, a nonbonded model of Cu+, employing the HFE parameter set of Cu+ for the TIP4PEW water model, was used to study the Cu+¥CusF system and it was found that the Cu+¥CusF rapidly transitions to an open or askance conformation in the process of metal ion release and then returns to its closed conformation.230 While these simulations are qualitative and cannot be relied upon at this time to give time-scale information, they do provide support  115 for the general hypothesis that the open conformation plays an important role in metal ion release in CusF function.  116 APPENDICES 117 APPENDIX A: TABLES  Table 1. Target values of the HFE, IOD and the CN of the first solvation shell for monovalent ions. Ions Mass (g/mol) Electronic structures HFE (kcal/mol) IOD (†)d CNd Li+ 6.94 [He] -113.5a 2.08±0.06 4-6 Na+ 22.99 [Ne] -87.2a 2.35±0.06 4-8 K+ 39.10 [Ar] -70.5a 2.79±0.08 6-8 Rb+ 85.47 [Kr] -65.7a 2.89±0.10 --- Cs+ 132.91 [Xe] -59.8a 3.13±0.07 7-8 Tl+ 204.38 [Xe]4f145d106s2 -71.7a 2.96e 4e Cu+ 63.55 [Ar]3d10 -125.5a 1.87f 2f Ag+ 107.87 [Kr]4d10 -102.8a 2.41±0.02 2-4 NH4+ 18.04 -- -68.1a 2.85g 4-11j H+ (Zundel cation) 1.007 -- -251.0a 1.24h 2h H+ (Eigen cation) 1.007 -- -251.0a 1.01i 1i H3O+ 19.02 -- -103.4b 2.75±0.01 4 F- 19.00 [Ne] -119.7c 2.63±0.02 4.1-6.8 Cl- 35.45 [Ar] -89.1c 3.18±0.06 6-8.5 Br- 79.90 [Kr] -82.7c 3.37±0.05 6 I- 126.9 [Xe] -74.3c 3.64±0.03 6-8.7 a. From Marcus.115 b. From Palascak and Shields.121 c. From Schmid et al.116 d. From Marcus unless specified otherwise. e. Weighted average value of four bonds (two at 2.73 † and two at 3.18 †) from Persson et al.124 f. From Burda et al.127 g. Obtained by addition of the ionic radius of NH4+ from Detellier and Laszlo128 and the effective ionic radius of O2- from Pauling.129 h. From quantum calculations done by Meraj and Chaudhari.132 i. From Sobolewski and Domcke at the MP2/6-31+G** level of theory.133 j. From Ohtaki and Radnai.162    118 Table 2. Target values of the HFE, IOD and the CN of the first solvation shell for divalent metal ions. Ions Electron configuration HFE (kcal/mol)a Relative HFE  (M2+-Cd2+) (kcal/mol) CN IOD (†) Be2+ [He] -572.4 -152.9 4b 1.67b Cu2+ [Ar]3d9 -480.4 -60.9 6b Eq: 1.96±0.04 Ax: 2.40±0.10b Weighted mean distance: 2.11f  Ni2+ [Ar]3d8 -473.2 -53.7 6b 2.06±0.01b Pt2+ [Xe]4f145d8 -468.5 -49.0 n n Zn2+ [Ar]3d10 -467.3 -47.8 6b 2.09±0.06b Co2+ [Ar]3d7 -457.7 -38.2 6b 2.10±0.02b Pd2+ [Kr]4d8 -456.5 -37.0 n n Ag2+ [Kr]4d9 -445.7 -26.2 n n Cr2+ [Ar]3d4 -442.2 -22.7 6d Eq:2.08d Fe2+ [Ar]3d6 -439.8 -20.3 6b 2.11±0.01b Mg2+ [Ne] -437.4 -17.9 6b 2.09±0.04b V2+ [Ar]3d3 -436.2 -16.7 6c 2.21c Mn2+ [Ar]3d5 -420.7 -1.2 6b 2.19±0.01b Hg2+ [Xe]4f145d10 -420.7 -1.2 6b 2.41b Cd2+ [Kr]4d10 -419.5 0.0 6b 2.30±0.02b Yb2+ [Xe]4f14 -360.9 58.6 n n Ca2+ [Ar] -359.7 59.8 8e 2.46e Sn2+ [Kr]4d105s2 -356.1 63.4 6d Eq: 2.33-2.34;d Ax: 2.38-2.90d Weighted mean distance: 2.62b Pb2+ [Xe]4f145d106s2 -340.6 78.9 n n Eu2+ [Xe]4f7 -331 88.5 n n Sr2+ [Kr] -329.8 89.7 8-15d 2.64±0.04b Sm2+ [Xe]4f6 -328.6 90.9 n n Ba2+ [Xe] -298.8 120.7 9g 2.83g Ra2+ [Rn] -298.8 120.7 n n a. From Marcus.115 b. From Marcus.123 c. From Miyanaga et al.231 d. From Ohtaki and Radnai.162 e. From Jalilehvand et al.187 f. Calculated by the authors from the experimental data. g. From Smirnov and Trostin.232 n. Either no experimental data were available or the data were deemed unreliable by Ohtaki and Radnai.162   119 Table 3. Target values of the HFE, IOD and the CN of the first solvation shell for trivalent and tetravalent metal ions. Metal ion Electron configuration HFE (kcal/mol)a IOD (†)b CNb Effective Ion Radii (†) First Shell Water Radii (†) Al3+ [Ne] -1081.5 1.88 6 0.54 1.34 Fe3+ [Ar]3d5 -1019.4 2.03 6 0.65 1.38 Cr3+ [Ar]3d3 -958.4 1.96 6 0.62 1.34 In3+ [Kr]4d10 -951.2 2.15 6 0.80 1.35 Tl3+ [Xe]4f145d10 -948.9 2.23 4-6 0.89 1.34 Y3+ [Kr] -824.6 2.36 8 0.90 1.46 La3+ [Xe] -751.7 2.52 8.0-9.1 1.03 1.49 Ce3+ [Xe]4f1 -764.8 2.55 7.5 1.01 1.54 Pr3+ [Xe]4f2 -775.6 2.54 9.2 0.99 1.55 Nd3+ [Xe]4f3 -783.9 2.47 8.0-8.9 0.98 1.49 Sm3+ [Xe]4f5 -794.7 2.44 8.0-9.9 0.96 1.48 Eu3+ [Xe]4f6 -803.1 2.45 8.3 0.95 1.50 Gd3+ [Xe]4f7 -806.6 2.39 8.0-9.9 0.94 1.45 Tb3+ [Xe]4f8 -812.6 2.40 8.0-8.2 0.92 1.48 Dy3+ [Xe]4f9 -818.6 2.37 7.4-7.9 0.91 1.46 Er3+ [Xe]4f11 -835.3 2.36 6.3-8.2 0.89 1.47 Tm3+ [Xe]4f12 -840.1 2.36 8.1 0.88 1.48 Lu3+ [Xe]4f14 -840.1 2.34 8 0.86 1.48  Hf4+ [Xe]4f14 -1664.7 2.16c 8c 0.85 1.31 Zr4+ [Kr] -1622.8 2.19c 8c 0.86 1.33 Ce4+ [Xe] -1462.7 2.42d 9d 0.87 1.55 U4+ [Rn]5f16d1 -1567.9 2.42e 9-11e 0.89 1.53 Pu4+ [Rn]5f4 -1520.1 2.39f 8f 0.86 1.53 Th4+ [Rn] -1389.8 2.45e 9-11e 0.94 1.51 a. From Marcus unless specified.115 b. Referenced or calculated from data from Marcus.123 c. From Hagfeldt et al.139 d. From Sham.136 e. From Moll et al.138 f. From Ankudinov et al.137     120 Table 4. Parameters of four different water models. Water model Q(O) Q(H) or Q(M)a r(O-H) (†) H-O-H angle (degree) r(O-M)b (†) Rmin/2 for O atom (†) ! for O atom (kcal/mol) TIP3P -0.834 +0.417 0.9572 104.52 ------- 1.7683 0.1520 SPC/E -0.8476 +0.4238 1.0 109.47 ------- 1.7767 0.1553 TIP4P -1.04 +0.52 0.9572 104.52 0.15 1.7699 0.1550 TIP4PEW -1.04844 +0.52422 0.9572 104.52 0.125 1.775931 0.16275 a. Q(M) is for TIP4P and TIP4PEW which are 4-site water models while Q(O) is for TIP3P and SPC/E which are 3-site water models. M represents the dummy atom in a 4-site water model. b. Distance between the dummy atom and oxygen atom in the water model, only valid for 4-site water models.      121 Table 5. Absolute HFEs, IODs and CNs of M(II) ions employing parameters available in the AMBER package.  LJ parameters Results from Method 2 Results from †qvist37  Rmin/2 (†) ! (kcal/mol) HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) Zn2+ 1.10 0.0125 -443.8 1.93 6.0 -------- ----------- Mg2+ 0.7926 0.8947 -432.6 1.99 6.0 -455.9±2.6 2.00 Ca2+ 1.7131 0.459789 -307.0 2.70 8.9 -380.6±1.3 2.40     122 Table 6. Absolute HFEs, IODs and CNs of Zn(II) employing parameters developed by Stote and Karplus.  LJ parameters Results from Method 2 Results from Stote and Karplus38  Rmin/2 (†) ! (kcal/mol) HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN Zn2+ 1.094 0.250 -399.9  2.10 6.0 -472.7 2.12 6.0     123 Table 7. Absolute HFEs, relative HFEs (relative to the Cd2+ ion), IODs and CNs of M(II) ions with the parameters taken from Babu and Lim.  LJ parameters Results from Method 2 Results from Babu and Lim64 Ions Rmin/2 (†) ! (kcal/mol) HFE (kcal/mol) Relative HFE (kcal/mol) IOD (†) CN Relative HFE (kcal/mol) IOD (†) CN Be2+ 0.5637 0.0032 -521.4 -145.7 1.45 3.3 -156.3 1.57 4 Cu2+ 1.033 0.0427 -436.5 -60.8 1.96 6.0 -59.9 1.94 6 Ni2+ 1.0941 0.0366 -430.3 -54.6 1.98 6.0 -53.2 1.97 6 Pt2+ 1.1376 0.0332 -425.6 -49.9 1.99 6.0 -48.3 1.97 6 Zn2+ 1.1489 0.0325 -423.7 -48.0 2.00 6.0 -47.5 2.00 6 Co2+ 1.2267 0.0286 -414.6 -38.9 2.03 6.0 -38.0 2.02 6 Pd2+ 1.236 0.0282 -413.3 -37.6 2.03 6.0 -37.3 2.02 6 Ag2+ 1.3107 0.0266 -403.1 -27.4 2.07 6.0 -26.6 2.06 6 Cr2+ 1.3344 0.0264 -399.2 -23.5 2.08 6.0 -22.4 2.07 6 Fe2+ 1.3488 0.0264 -397.0 -21.3 2.09 6.0 -20.5 2.08 6 Mg2+ 1.3636 0.0266 -394.4 -18.7 2.10 6.0 -17.5 2.08 6 V2+ 1.3706 0.0266 -393.5 -17.8 2.10 6.0 -16.2 2.11 6 Mn2+ 1.4544 0.03 -377.2 -1.5 2.17 6.1 -1.5 2.16 6 Cd2+ 1.46 0.0304 -375.7 0.0 2.18 6.1 0.0 2.17 6 Yb2+ 1.9298 0.0309 -317.2 58.5 2.57 8.3 57.7 2.47 8 Ca2+ 1.9364 0.0318 -316.7 59.0 2.58 8.3 58.1 2.58 8 Sn2+ 1.954 0.0346 -313.2 62.5 2.61 8.5 62.6 2.58 8 Pb2+ 2.0195 0.0557 -298.2 77.5 2.71 8.9 78.2 2.68 8.5 Eu2+ 2.0846 0.0647 -288.7 87.0 2.78 9.0 88.8 2.74 9 Sr2+ 2.0923 0.0664 -287.8 87.9 2.79 9.0 89.3 2.75 9 Sm2+ 2.0997 0.068 -286.5 89.2 2.79 9.0 90.1 2.75 9 Ba2+ 2.2451 0.1993 -257.8 117.9 3.04 9.8 120.4 3.01 9.5      124 Table 8. Standard deviations of the IOD values for the LJ grids of divalent metal ions.  TIP3P SPC/E TIP4P TIP4PEW TIP3P 0.00† 0.00†±0.01† 0.02†±0.01† 0.02†±0.01† SPC/E ------ 0.00† 0.02†±0.02† 0.02†±0.01† TIP4P ------ ------ 0.00† 0.00†±0.01† TIP4PEW ------ ------ ------ 0.00†     125 Table 9. The experimental LJ parameters for noble gas atoms.a  Rmin (†) ! (meV) Rmin/2 (†) ! (kcal/mol)b -log(!) He 2.97 0.92 1.485  0.02121603 1.67333588 Ne 3.10 3.6 1.55  0.08301924 1.08082125 Ar 3.76 12.2 1.88  0.28134298 0.55076392 Ke 4.00 17.2 2.00  0.39664748 0.40159530 Xe 4.40 24 2.20  0.55346160 0.25691251 a. Adapted from page 408 of the book of Rad‚tfisig et al.159 according to the Lorentz-Berthelot combining rules. b. Using the conversion factor 1 eV=23.0609 kcal/mol.     126 Table 10. Comparison of results of different scaling methods in the VDW-disappearing and VDW-appearing steps of the HFE calculations for the Be2+ and Ba2+ ions.a  VDW-appearing VDW-disappearing L K4 K6 SC L K4 K6 SC Be2+ 3 Windows ----- ----- ----- 0.44 ----- ----- ----- -0.45 5 Windows 0.25 0.19 0.17 ----- -0.26 -0.38 -0.38 ----- 7 Windows 0.28 0.24 0.23 ----- -0.29 -0.38 -0.39 ----- 9 Windows 0.30 0.27 0.25 ----- -0.29 -0.39 -0.37 ----- Ba2+ 3 Windows ----- ----- ----- 1.56 ----- ----- ----- -1.41 5 Windows -0.40 -1.34 -1.74 ----- 0.54 -2.08 -1.76 ----- 7 Windows 0.12 -0.74 -1.10 ----- 0.05 -1.83 -1.90 ----- 9 Windows 0.31 -0.45 -0.59 ----- -0.18 -1.74 -1.95 ----- a. In kcal/mol.    127 Table 11. Final optimized HFE parameter set of divalent metal ions for different water models.  TIP3P SPC/E TIP4PEW Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) !     (kcal/mol) Be2+ 0.907 0.00000080 0.915 0.00000105 0.815 0.0000000221 Cu2+ 1.144 0.00040203 1.149 0.00044254 1.078 0.00010063 Ni2+ 1.162 0.00056491 1.166 0.00060803 1.101 0.00016733 Pt2+ 1.173 0.00069036 1.176 0.00072849 1.114 0.00022027 Zn2+ 1.175 0.00071558 1.178 0.00075490 1.115 0.00022490 Co2+ 1.211 0.00132548 1.217 0.00146124 1.141 0.00037931 Pd2+ 1.215 0.00141473 1.217 0.00146124 1.145 0.00040986 Ag2+ 1.263 0.00294683 1.265 0.00303271 1.171 0.00066591 Cr2+ 1.273 0.00339720 1.276 0.00354287 1.181 0.00079606 Fe2+ 1.277 0.00359255 1.284 0.00395662 1.194 0.00099751 Mg2+ 1.284 0.00395662 1.288 0.00417787 1.208 0.00126172 V2+ 1.290 0.00429223 1.293 0.00446856 1.210 0.00130393 Mn2+/Hg2+ 1.339 0.00799176 1.338 0.00789684 1.276 0.00354287 Cd2+ 1.339 0.00799176 1.344 0.00848000 1.279 0.00369364 Yb2+ 1.526 0.04772212 1.518 0.04490976 1.464 0.02883819 Ca2+ 1.528 0.04844326 1.520 0.04560206 1.467 0.02960343 Sn2+ 1.543 0.05408454 1.532 0.04990735 1.479 0.03280986 Pb2+ 1.620 0.08965674 1.609 0.08389240 1.551 0.05726270 Eu2+ 1.666 0.11617738 1.656 0.11008622 1.596 0.07737276 Sr2+ 1.672 0.11991675 1.659 0.11189491 1.606 0.08235966 Sm2+ 1.680 0.12499993 1.667 0.11679623 1.606 0.08235966 Ba2+/ Ra2+ 1.842 0.24821230 1.825 0.23380842 1.768 0.18767274     128 Table 12. The simulated HFE, IOD and CN values of the HFE parameter set for divalent metal ions.a  TIP3P SPC/E   HFE (kcal/mol) Relative HFE  (M2+-Cd2+) (kcal/mol) IOD (†) CN HFE (kcal/mol) Relative HFE  (M2+-Cd2+) (kcal/mol) IOD (†) CN Be2+ -572.3 -152.5 1.14 2.0 -571.4 -152.4 1.15 2.0 Cu2+ -481.2 -61.4 1.63 4.0 -481.2 -62.2 1.64 4.0 Ni2+ -473.0 -53.2 1.65 4.0 -473.2 -54.2 1.66 4.0 Pt2+ -467.8 -48.0 1.67 4.1 -468.4 -49.4 1.67 4.1 Zn2+ -467.4 -47.6 1.67 4.1 -467.3 -48.3 1.68 4.3 Co2+ -457.0 -37.2 1.87 6.0 -456.9 -37.9 1.89 6.0 Pd2+ -457.0 -37.2 1.88 6.0 -456.8 -37.8 1.89 6.0 Ag2+ -445.0 -25.2 1.93 6.0 -446.0 -27.0 1.94 6.0 Cr2+ -441.6 -21.8 1.94 6.0 -441.9 -22.9 1.95 6.0 Fe2+ -439.5 -19.7 1.94 6.0 -439.4 -20.4 1.96 6.0 Mg2+ -437.7 -17.9 1.95 6.0 -437.6 -18.6 1.96 6.0 V2+ -435.9 -16.1 1.95 6.0 -435.8 -16.8 1.96 6.0 Mn2+ -419.9 -0.1 2.01 6.0 -420.6 -1.6 2.01 6.0 Hg2+ -419.9 -0.1 2.01 6.0 -420.6 -1.6 2.01 6.0 Cd2+ -419.8 0.0 2.01 6.0 -419.0 0.0 2.02 6.0 Yb2+ -360.2 59.6 2.33 7.4 -360.2 58.8 2.30 7.0 Ca2+ -360.2 59.6 2.33 7.4 -360.6 58.4 2.31 7.0 Sn2+ -356.5 63.3 2.36 7.7 -356.4 62.6 2.33 7.2 Pb2+ -340.9 78.9 2.46 8.0 -340.4 78.6 2.45 7.9 Eu2+ -331.2 88.6 2.51 8.0 -330.9 88.1 2.51 8.0 Sr2+ -329.7 90.1 2.52 8.1 -330.6 88.4 2.51 8.0 Sm2+ -328.4 91.4 2.53 8.1 -328.6 90.4 2.52 8.0 Ba2+ -299.4 120.4 2.74 9.0 -299.2 119.8 2.72 8.8 Ra2+ -299.4 120.4 2.74 9.0 -299.2 119.8 2.72 8.8 Average Error 0.1 0.4 -0.27 -0.4 0.0 -0.5 -0.26 -0.5 Standard Deviation 0.5 0.5 0.14 1.0 0.5 0.5 0.14 1.0 UAE 0.4 0.5 0.27 0.6 0.4 0.6 0.26 0.6 a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 2). The target IOD values in Table 2 (without considering the error bars) were treated as the reference for the simulated IOD values.  129 Table 12 (contÕd)  TIP4PEW   HFE (kcal/mol) Relative HFE (M2+-Cd2+) (kcal/mol) IOD (†) CN Be2+ -572.8 -153.2 0.87 1.0 Cu2+ -481.1 -61.5 1.57 4.0 Ni2+ -472.4 -52.8 1.60 4.0 Pt2+ -468.0 -48.4 1.61 4.0 Zn2+ -467.4 -47.8 1.61 4.0 Co2+ -458.6 -39.0 1.64 4.0 Pd2+ -456.1 -36.5 1.65 4.0 Ag2+ -445.2 -25.6 1.69 4.1 Cr2+ -442.3 -22.7 1.84 5.6 Fe2+ -440.4 -20.8 1.88 6.0 Mg2+ -436.5 -16.9 1.89 6.0 V2+ -435.6 -16.0 1.90 6.0 Mn2+ -419.9 -0.3 1.96 6.0 Hg2+ -419.9 -0.3 1.96 6.0 Cd2+ -419.6 0.0 1.96 6.0 Yb2+ -361.3 58.3 2.19 6.1 Ca2+ -359.8 59.8 2.20 6.2 Sn2+ -356.2 63.4 2.23 6.5 Pb2+ -339.7 79.9 2.39 7.7 Eu2+ -331.0 88.6 2.45 7.9 Sr2+ -329.5 90.1 2.46 8.0 Sm2+ -328.8 90.8 2.46 8.0 Ba2+ -299.1 120.5 2.67 8.8 Ra2+ -299.1 120.5 2.67 8.8 Average Error 0.1 0.2 -0.36 -0.8 Standard Deviation 0.5 0.5 0.17 1.1 UAE 0.5 0.4 0.36 0.9     130 Table 13. Final optimized IOD parameter set of divalent metal ions.a  Rmin/2 (†) ! (kcal/mol) Be2+ 1.168 0.00063064 Cu2+ 1.409 0.01721000 Ni2+ 1.373 0.01179373 Zn2+ 1.395 0.01491700 Co2+ 1.404 0.01636246 Cr2+ 1.388 0.01386171 Fe2+ 1.409 0.01721000 Mg2+ 1.395 0.01491700 V2+ 1.476 0.03198620 Mn2+ 1.467 0.02960343 Hg2+ 1.575 0.06751391 Cd2+ 1.506 0.04090549 Ca2+ 1.608 0.08337961 Sn2+ 1.738 0.16500296 Sr2+ 1.753 0.17618319 Ba2+ 1.913 0.31060194 a. The parameters are same for the four different water models.    131 Table 14. The simulated HFE, IOD and CN values of the IOD parameter set for divalent metal ions.a  TIP3P SPC/E  HFE (kcal/mol) Relative HFE   (M2+-Cd2+) (kcal/mol) IOD (†) CN HFE (kcal/mol) Relative HFE   (M2+-Cd2+) (kcal/mol) IOD (†) CN Be2+ -469.8 -103.9 1.66 4.0 -472.0 -108.1 1.66 4.0 Cu2+ -395.2 -29.3 2.10 6.0 -395.4 -31.5 2.10 6.0 Ni2+ -407.6 -41.7 2.05 6.0 -408.2 -44.3 2.05 6.0 Zn2+ -400.0 -34.1 2.08 6.0 -400.0 -36.1 2.08 6.0 Co2+ -397.0 -31.1 2.09 6.0 -397.1 -33.2 2.09 6.0 Cr2+ -402.8 -36.9 2.07 6.0 -403.2 -39.3 2.07 6.0 Fe2+ -394.8 -28.9 2.10 6.0 -395.3 -31.4 2.10 6.0 Mg2+ -400.3 -34.4 2.08 6.0 -400.5 -36.6 2.08 6.0 V2+ -372.8 -6.9 2.21 6.6 -372.2 -8.3 2.19 6.1 Mn2+ -375.6 -9.7 2.18 6.3 -375.4 -11.5 2.18 6.0 Hg2+ -350.1 15.8 2.41 7.9 -346.4 17.5 2.40 7.6 Cd2+ -365.9 0.0 2.29 7.1 -363.9 0.0 2.28 6.8 Ca2+ -342.3 23.6 2.45 8.0 -340.6 23.3 2.45 7.9 Sn2+ -317.6 48.3 2.61 8.6 -314.8 49.1 2.61 8.2 Sr2+ -314.2 51.7 2.63 8.6 -311.8 52.1 2.62 8.3 Ba2+ -288.2 77.7 2.82 9.2 -285.1 78.8 2.82 9.0 Average Error 51.1 -2.5 -0.01 0.4 51.9 -3.7 -0.01 0.3 Standard Deviation 25.2 25.2 0.00 0.8 24.3 24.3 0.00 0.7 UAE 51.1 20.0 0.01 0.4 51.9 19.7 0.01 0.3 a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 2). The target IOD values in Table 2 (without considering the error bars) were treated as the reference for the simulated IOD values.    132 Table 14 (contÕd)  TIP4P TIP4PEW  HFE (kcal/mol) Relative HFE   (M2+-Cd2+) (kcal/mol) IOD (†) CN HFE (kcal/mol) Relative HFE   (M2+-Cd2+) (kcal/mol) IOD (†) CN Be2+ -435.0 -91.9 1.68 4.4 -446.8 -97.3 1.68 4.0 Cu2+ -368.7 -25.6 2.11 6.0 -378.6 -29.1 2.12 6.0 Ni2+ -380.9 -37.8 2.07 6.0 -390.4 -40.9 2.07 6.0 Zn2+ -373.5 -30.4 2.10 6.0 -382.0 -32.5 2.10 6.0 Co2+ -370.7 -27.6 2.11 6.0 -379.6 -30.1 2.11 6.0 Cr2+ -376.0 -32.9 2.09 6.0 -385.0 -35.5 2.09 6.0 Fe2+ -369.3 -26.2 2.11 6.0 -378.4 -28.9 2.12 6.0 Mg2+ -373.9 -30.8 2.09 6.0 -382.9 -33.4 2.10 6.0 V2+ -349.5 -6.4 2.26 7.0 -356.2 -6.7 2.22 6.3 Mn2+ -351.6 -8.5 2.22 6.6 -359.9 -10.4 2.20 6.1 Hg2+ -329.4 13.7 2.42 8.0 -335.1 14.4 2.42 7.9 Cd2+ -343.1 0.0 2.31 7.4 -349.5 0.0 2.31 7.1 Ca2+ -323.3 19.8 2.46 8.0 -329.2 20.3 2.47 8.0 Sn2+ -299.8 43.3 2.63 8.8 -305.2 44.3 2.63 8.6 Sr2+ -297.7 45.4 2.65 8.8 -301.7 47.8 2.65 8.7 Ba2+ -273.3 69.8 2.84 9.4 -276.9 72.6 2.84 9.2 Average Error 74.8 -1.6 0.01 0.5 67.2 -2.8 0.01 0.4 Standard Deviation 29.3 29.3 0.01 0.9 27.5 27.5 0.00 0.8 UAE 74.8 22.3 0.01 0.5 67.2 21.5 0.01 0.4       133 Table 15. Final optimized CM parameter set of divalent metal ions for different water models.  TIP3P SPC/E TIP4PEW Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Be2+ 0.956 0.00000395 0.961 0.00000460 0.918 0.00000116 Cu2+ 1.218 0.00148497 1.223 0.00160860 1.195 0.00101467 Ni2+ 1.255 0.00262320 1.253 0.00254709 1.221 0.00155814 Pt2+ 1.266 0.00307642 1.272 0.00334975 1.251 0.00247282 Zn2+ 1.271 0.00330286 1.276 0.00354287 1.252 0.00250973 Co2+ 1.299 0.00483892 1.305 0.00523385 1.288 0.00417787 Pd2+ 1.303 0.00509941 1.305 0.00523385 1.288 0.00417787 Ag2+ 1.336 0.00770969 1.337 0.00780282 1.323 0.00657749 Cr2+ 1.346 0.00868178 1.348 0.00888732 1.333 0.00743559 Fe2+ 1.353 0.00941798 1.354 0.00952704 1.343 0.00838052 Mg2+ 1.360 0.01020237 1.360 0.01020237 1.353 0.00941798 V2+ 1.364 0.01067299 1.365 0.01079325 1.353 0.00941798 Mn2+/Hg2+ 1.407 0.01686710 1.406 0.01669760 1.401 0.01586934 Cd2+ 1.412 0.01773416 1.412 0.01773416 1.406 0.01669760 Yb2+ 1.642 0.10185975 1.634 0.09731901 1.654 0.10888937 Ca2+ 1.649 0.10592870 1.635 0.09788018 1.657 0.11068733 Sn2+ 1.666 0.11617738 1.651 0.10710756 1.670 0.11866330 Pb2+ 1.745 0.17018074 1.731 0.15989650 1.758 0.17997960 Eu2+ 1.802 0.21475916 1.786 0.20184160 1.823 0.23213110 Sr2+ 1.810 0.22132374 1.794 0.20826406 1.827 0.23548950 Sm2+ 1.819 0.22878796 1.800 0.21312875 1.838 0.24480038 Ba2+/ Ra2+ 2.019 0.40664608 1.980 0.37126402 2.050 0.43454345     134 Table 16. The simulated HFE, IOD and CN values of the CM parameter set for divalent metal ions.a  TIP3P SPC/E  HFE (kcal/mol) Relative HFE   (M2+-Cd2+) (kcal/mol) IOD (†) CN HFE (kcal/mol) Relative HFE  (M2+-Cd2+) (kcal/mol) IOD (†) CN Be2+ -547.3 -153.7 1.21 2.0 -547.5 -153.0 1.22 2.0 Cu2+ -455.0 -61.4 1.88 6.0 -454.6 -60.1 1.90 6.0 Ni2+ -447.9 -54.3 1.92 6.0 -448.4 -53.9 1.92 6.0 Pt2+ -443.7 -50.1 1.93 6.0 -443.8 -49.3 1.94 6.0 Zn2+ -442.0 -48.4 1.93 6.0 -441.6 -47.1 1.95 6.0 Co2+ -433.0 -39.4 1.96 6.0 -431.9 -37.4 1.98 6.0 Pd2+ -431.5 -37.9 1.97 6.0 -432.2 -37.7 1.98 6.0 Ag2+ -420.8 -27.2 2.00 6.0 -420.4 -25.9 2.01 6.0 Cr2+ -417.3 -23.7 2.02 6.0 -417.0 -22.5 2.03 6.0 Fe2+ -414.6 -21.0 2.02 6.0 -415.3 -20.8 2.03 6.0 Mg2+ -412.1 -18.5 2.03 6.0 -412.8 -18.3 2.04 6.0 V2+ -410.7 -17.1 2.04 6.0 -411.4 -16.9 2.05 6.0 Mn2+ -396.1 -2.5 2.09 6.0 -396.5 -2.0 2.10 6.0 Hg2+ -396.1 -2.5 2.09 6.0 -396.5 -2.0 2.10 6.0 Cd2+ -393.6 0.0 2.10 6.0 -394.5 0.0 2.10 6.0 Yb2+ -335.7 57.9 2.48 8.0 -335.3 59.2 2.48 8.0 Ca2+ -334.6 59.0 2.49 8.0 -334.5 60.0 2.48 8.0 Sn2+ -331.2 62.4 2.51 8.1 -331.5 63.0 2.50 8.0 Pb2+ -316.2 77.4 2.62 8.7 -315.7 78.8 2.60 8.2 Eu2+ -305.8 87.8 2.69 8.9 -305.2 89.3 2.67 8.5 Sr2+ -304.9 88.7 2.70 8.9 -304.5 90.0 2.68 8.6 Sm2+ -303.0 90.6 2.71 8.9 -302.7 91.8 2.69 8.6 Ba2+ -273.1 120.5 2.94 9.7 -274.7 119.8 2.90 9.2 Ra2+ -273.1 120.5 2.94 9.7 -274.7 119.8 2.90 9.2 Average Error 25.1 -0.8 -0.13 0.0 25.0 0.0 -0.12 0.0 Standard Deviation 0.4 0.4 0.14 0.8 0.6 0.6 0.13 0.7 UAE 25.1 0.8 0.15 0.3 25.0 0.5 0.14 0.3 a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 2). The target IOD values in Table 2 (without considering the error bars) were treated as the reference for the simulated IOD values.    135 Table 16 (contÕd)  TIP4PEW  HFE (kcal/mol) Relative HFE (M2+-Cd2+) (kcal/mol) IOD (†) CN Be2+ -532.2 -153.6 1.17 2.0 Cu2+ -439.9 -61.3 1.88 6.0 Ni2+ -432.4 -53.8 1.91 6.0 Pt2+ -427.8 -49.2 1.94 6.0 Zn2+ -427.8 -49.2 1.94 6.0 Co2+ -416.8 -38.2 1.97 6.0 Pd2+ -416.8 -38.2 1.97 6.0 Ag2+ -405.1 -26.5 2.01 6.0 Cr2+ -403.1 -24.5 2.02 6.0 Fe2+ -400.0 -21.4 2.03 6.0 Mg2+ -396.4 -17.8 2.05 6.0 V2+ -395.6 -17.0 2.05 6.0 Mn2+ -380.7 -2.1 2.11 6.0 Hg2+ -380.7 -2.1 2.11 6.0 Cd2+ -378.6 0.0 2.11 6.0 Yb2+ -320.1 58.5 2.52 8.0 Ca2+ -319.0 59.6 2.53 8.0 Sn2+ -316.4 62.2 2.54 8.1 Pb2+ -301.3 77.3 2.66 8.7 Eu2+ -290.6 88.0 2.74 8.9 Sr2+ -289.6 89.0 2.74 8.9 Sm2+ -288.1 90.5 2.75 9.0 Ba2+ -258.4 120.2 3.00 9.9 Ra2+ -258.4 120.2 3.00 9.9 Average Error 40.3 -0.6 -0.11 0.1 Standard Deviation 0.5 0.5 0.16 0.8 UAE 40.3 0.6 0.16 0.3    136  Table 17. Final optimized HFE parameter set of monovalent ions for different water models.  TIP3P SPC/E TIP4PEW  Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Li+ 1.267 0.00312065 1.258 0.00274091 1.226 0.00168686 Na+ 1.475 0.03171494 1.454 0.02639002 1.432 0.02154025 K+ 1.719 0.15131351 1.683 0.12693448 1.669 0.11803919 Rb+ 1.834 0.24140216 1.792 0.20665151 1.767 0.18689752 Cs+ 1.988 0.37853483 1.953 0.34673208 1.936 0.33132862 Tl+ 1.703 0.14021803 1.668 0.11741683 1.646 0.10417397 Cu+ 1.201 0.00112300 1.192 0.00096394 1.156 0.00050520 Ag+ 1.341 0.00818431 1.330 0.00716930 1.294 0.00452863 NH4+ 1.779 0.19628399 1.743 0.16869420 1.707 0.14295367 H3O+ 1.337 0.00780282 1.327 0.00691068 1.292 0.00440914 F- 1.783 0.19945255 1.819 0.22878796 1.842 0.24821230 Cl- 2.252 0.60293097 2.308 0.64367011 2.321 0.65269755 Br- 2.428 0.72070940 2.470 0.74435812 2.520 0.77034233 I- 2.724 0.85418187 2.770 0.86877007 2.819 0.88281946     137 Table 18. The simulated HFE, IOD and CN values of the HFE parameter set for monovalent metal ions.a  TIP3P SPC/E  HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN Li+ -113.1 1.95 4.1 -113.5 1.94 4.0 Na+ -88.2 2.36 5.8 -87.9 2.33 5.6 K+ -71.2 2.73 7.0 -70.7 2.69 6.7 Rb+ -65.9 2.90 8.0 -66.1 2.85 7.5 Cs+ -60.5 3.10 9.2 -60.2 3.09 8.9 Tl+ -72.0 2.72 6.6 -71.5 2.67 6.6 Cu+ -125.4 1.84 4.0 -126.1 1.83 4.0 Ag+ -102.7 2.14 5.3 -102.6 2.10 4.9 NH4+ -68.4 2.82 7.7 -68.0 2.77 7.1 H3O+ -103.3 2.13 5.2 -102.8 2.10 4.9 F- -119.1 2.71 6.9 -119.7 2.74 6.5 Cl- -89.5 3.30 8.9 -89.1 3.34 7.9 Br- -83.1 3.48 8.2 -83.0 3.52 7.5 I- -74.9 3.79 8.8 -74.5 3.81 9.5 Average Error -0.2 -0.05 -- -0.1 -0.06 -- Standard Deviation 0.5 0.20 -- 0.3 0.22 -- UAE 0.4 0.13 -- 0.3 0.15 -- a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 1). The target IOD values in Table 1 (without considering the error bars) were treated as the reference for the simulated IOD values.     138 Table 18 (contÕd)  TIP4PEW  HFE (kcal/mol) IOD (†) CN Li+ -114.0 1.90 4.0 Na+ -88.0 2.32 5.8 K+ -70.8 2.68 6.6 Rb+ -65.8 2.83 7.4 Cs+ -59.8 3.05 8.5 Tl+ -71.7 2.65 6.4 Cu+ -125.5 1.79 4.0 Ag+ -103.2 2.05 4.8 NH4+ -68.8 2.74 6.8 H3O+ -103.6 2.05 4.7 F- -120.2 2.77 6.3 Cl- -89.5 3.35 7.4 Br- -82.4 3.56 7.6 I- -74.3 3.87 9.2 Average Error -0.3 -0.11 -- Standard Deviation 0.3 0.28 -- UAE 0.3 0.22 --         139 Table 19. Final optimized IOD parameter set of monovalent ions for all the three water models. Ions Rmin/2 (†) ! (kcal/mol) Li+ 1.315 0.00594975 Na+ 1.465 0.02909167 K+ 1.745 0.17018074 Rb+ 1.820 0.22962229 Cs+ 2.000 0.38943250 Tl+ 1.870 0.27244486 Cu+ 1.214 0.00139196 Ag+ 1.500 0.03899838 NH4+ 1.790 0.20504355 H+ (Zundel cation) 0.925 0.00000147 H+ (Eigen cation) 0.841 0.0000000661 H3O+ 1.720 0.15202035 F- 1.739 0.16573832 Cl- 2.162 0.53154665 Br- 2.331 0.65952968 I- 2.590 0.80293907     140 Table 20. The simulated HFE, IOD and CN values of the IOD parameter set for monovalent metal ions.a  TIP3P SPC/E  HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN Li+ -113.1 1.95 4.1 -113.5 1.94 4.0 Na+ -88.2 2.36 5.8 -87.9 2.33 5.6 K+ -71.2 2.73 7.0 -70.7 2.69 6.7 Rb+ -65.9 2.90 8.0 -66.1 2.85 7.5 Cs+ -60.5 3.10 9.2 -60.2 3.09 8.9 Tl+ -72.0 2.72 6.6 -71.5 2.67 6.6 Cu+ -125.4 1.84 4.0 -126.1 1.83 4.0 Ag+ -102.7 2.14 5.3 -102.6 2.10 4.9 NH4+ -68.4 2.82 7.7 -68.0 2.77 7.1 H3O+ -103.3 2.13 5.2 -102.8 2.10 4.9 F- -119.1 2.71 6.9 -119.7 2.74 6.5 Cl- -89.5 3.30 8.9 -89.1 3.34 7.9 Br- -83.1 3.48 8.2 -83.0 3.52 7.5 I- -74.9 3.79 8.8 -74.5 3.81 9.5 Average Error -0.2 -0.05 -- -0.1 -0.06 -- Standard Deviation 0.5 0.20 -- 0.3 0.22 -- UAE 0.4 0.13 -- 0.3 0.15 -- a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 1). The target IOD values in Table 1 (without considering the error bars) were treated as the reference for the simulated IOD values.      141 Table 20 (contÕd)  TIP4PEW  HFE (kcal/mol) IOD (†) CN Li+ -114.0 1.90 4.0 Na+ -88.0 2.32 5.8 K+ -70.8 2.68 6.6 Rb+ -65.8 2.83 7.4 Cs+ -59.8 3.05 8.5 Tl+ -71.7 2.65 6.4 Cu+ -125.5 1.79 4.0 Ag+ -103.2 2.05 4.8 NH4+ -68.8 2.74 6.8 H3O+ -103.6 2.05 4.7 F- -120.2 2.77 6.3 Cl- -89.5 3.35 7.4 Br- -82.4 3.56 7.6 I- -74.3 3.87 9.2 Average Error -0.3 -0.11 -- Standard Deviation 0.3 0.28 -- UAE 0.3 0.22 --          142 Table 21. Estimated HFE parameter set of highly charged metal ions for different water models.  TIP3P SPC/E TIP4PEW  Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Al3+ 0.981 0.00000832 0.991 0.00001107 0.876 0.00000026 Fe3+ 1.082 0.00011017 1.091 0.00013462 0.984 0.00000907 Cr3+ 1.188 0.00089969 1.196 0.00103208 1.096 0.00015019 In3+ 1.202 0.00114198 1.209 0.00128267 1.110 0.00020260 Tl3+ 1.206 0.00122067 1.213 0.00136949 1.114 0.00022027 Y3+ 1.454 0.02639002 1.459 0.02759452 1.375 0.01205473 La3+ 1.628 0.09399072 1.629 0.09454081 1.553 0.05807581 Ce3+ 1.595 0.07688443 1.597 0.07786298 1.519 0.04525501 Pr3+ 1.568 0.06441235 1.571 0.06573030 1.492 0.03655251 Nd3+ 1.548 0.05605698 1.551 0.05726270 1.471 0.03064622 Sm3+ 1.522 0.04630154 1.526 0.04772212 1.445 0.02431873 Eu3+ 1.503 0.03994409 1.507 0.04122946 1.425 0.02014513 Gd3+ 1.495 0.03745682 1.499 0.03868661 1.417 0.01863432 Tb3+ 1.481 0.03336723 1.485 0.03450196 1.403 0.01619682 Dy3+ 1.468 0.02986171 1.472 0.03091095 1.389 0.01400886 Er3+ 1.431 0.02133669 1.436 0.02236885 1.350 0.00909668 Tm3+ 1.421 0.01937874 1.426 0.02034021 1.340 0.00808758 Lu3+ 1.421 0.01937874 1.426 0.02034021 1.340 0.00808758  Hf4+ 1.087 0.00012321 1.098 0.00015685 0.977 0.00000741 Zr4+ 1.139 0.00036479 1.149 0.00044254 1.031 0.00003240 Ce4+ 1.353 0.00941798 1.360 0.01020237 1.257 0.00270120 U4+ 1.209 0.00128267 1.218 0.00148497 1.105 0.00018227 Pu4+ 1.273 0.00339720 1.281 0.00379705 1.172 0.00067804 Th4+ 1.463 0.02858630 1.468 0.02986171 1.370 0.01141046     143 Table 22. Estimated IOD parameter set of highly charged metal ions for different water models.  TIP3P SPC/E TIP4PEW  Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Rmin/2 (†) ! (kcal/mol) Al3+ 1.297 0.00471279 1.296 0.00465074 1.285 0.00401101 Fe3+ 1.386 0.01357097 1.386 0.01357097 1.375 0.01205473 Cr3+ 1.344 0.00848000 1.343 0.00838052 1.333 0.00743559 In3+ 1.461 0.02808726 1.461 0.02808726 1.450 0.02545423 Tl3+ 1.513 0.04321029 1.513 0.04321029 1.502 0.03962711 Y3+ 1.602 0.08034231 1.602 0.08034231 1.590 0.07447106 La3+ 1.718 0.15060822 1.718 0.15060822 1.707 0.14295367 Ce3+ 1.741 0.16721338 1.741 0.16721338 1.729 0.15845086 Pr3+ 1.733 0.16134811 1.734 0.16207614 1.722 0.15343866 Nd3+ 1.681 0.12564307 1.681 0.12564307 1.669 0.11803919 Sm3+ 1.659 0.11189491 1.659 0.11189491 1.647 0.10475707 Eu3+ 1.666 0.11617738 1.666 0.11617738 1.655 0.10948690 Gd3+ 1.623 0.09126804 1.623 0.09126804 1.612 0.08544204 Tb3+ 1.630 0.09509276 1.630 0.09509276 1.619 0.08912336 Dy3+ 1.609 0.08389240 1.609 0.08389240 1.597 0.07786298 Er3+ 1.602 0.08034231 1.602 0.08034231 1.590 0.07447106 Tm3+ 1.602 0.08034231 1.602 0.08034231 1.590 0.07447106 Lu3+ 1.588 0.07351892 1.588 0.07351892 1.577 0.06841702  Hf4+ 1.499 0.03868661 1.501 0.03931188 1.483 0.03393126 Zr4+ 1.519 0.04525501 1.521 0.04595090 1.503 0.03994409 Ce4+ 1.684 0.12758274 1.689 0.13084945 1.667 0.11679623 U4+ 1.684 0.12758274 1.689 0.13084945 1.667 0.11679623 Pu4+ 1.662 0.11371963 1.666 0.11617738 1.645 0.10359269 Th4+ 1.708 0.14364160 1.713 0.14710519 1.690 0.13150785     144 Table 23. Comparison between the Rmin/2 parameters from the HFE parameter set for the TIP3P water model and the VDW radii calculated from QMSP method. Ions Rmin/2 of HFE parameter set determined for the TIP3P water model (†) VDW radius determined by QMSP methoda (†) Absolute Percentage Errorb Na+ 1.475 1.352 9.1% K+ 1.719 1.671 2.9% Rb+ 1.834 1.801 1.8% Cs+ 1.988 1.997 0.5% Mg2+ 1.284 1.180 8.8% Ca2+ 1.528 1.480 3.2% Sr2+ 1.672 1.625 2.9% Ba2+ 1.842 1.802 2.2% Al3+ 0.981 1.046 6.2% Y3+ 1.454 1.481 1.8% La3+ 1.628 1.642 0.9% F- 1.783 1.909 6.6% Cl- 2.252 2.252 0.0% Br- 2.428 2.298 5.7% I- 2.724 2.548 6.9% a. From Stokes.88 b. Yielding an average absolute percentage error of 4.0%.    145 Table 24. The activity derivatives for six different ionic solutions in the SPC/E water model (under ~0.3 M condition).  JC H2 HFE IOD 12-6-4 Experimental valuesa NaCl 1.08 1.00 1.00 1.03 1.01 0.93 KCl 1.11 1.13 1.03 1.07 0.91 0.90 NaBr 1.07 1.16 1.06 1.00 1.06 0.94 KF 0.87 0.76 0.84 0.87 0.86 0.92 NaI 0.93 0.98 0.87 0.90 0.93 0.97 CsI 0.87 1.09 1.03 0.97 0.92 0.86 Average Error 0.07 0.10 0.05 0.05 0.03 -- Standard Deviation 0.08 0.09 0.04 0.04 0.04 -- UAE 0.10 0.15 0.11 0.09 0.06 -- a. From the book of Robinson and Stokes.233    146 Table 25. The average IOD errors for the HFE parameter set of the mono-, di-, tri- and tetravalent cations.   M(I) M(II) M(III) M(IV) TIP3P Average IOD Error -0.14 (-5.3%) -0.27 (-12.4%) -0.29 (-12.8%) -0.58 (-25.0%) IOD Error Standard Deviation 0.20 (7.2%) 0.14 (7.8%) 0.14 (7.4%) 0.15 (7.0%) SPC/E Average IOD Error -0.17 (-6.5%) -0.26 (-12.3%) -0.28 (-12.4%) -0.57 (-24.5%) IOD Error Standard Deviation 0.20 (7.2%) 0.14 (7.6%) 0.13 (7.2%) 0.14 (6.6%) TIP4PEW Average IOD Error -0.25 (-10.1%) -0.36 (-16.8%) -0.41 (-18.1%) -0.74 (-32.0%) IOD Error Standard Deviation 0.22 (9.7%) 0.17 (10.2%) 0.16 (9.2%) 0.17 (8.2%)    147 Table 26. The average HFE errors for the IOD parameter sets of the mono-, di-, tri- and tetravalent cations.   M(I) M(II) M(III) M(IV) TIP3P Average HFE Error 17.2 (-10.0%) 51.1 (-11.5%) 82.7 (-9.3%) 244.3 (-15.7%) HFE Error Standard Deviation 26.9 (12.8%) 25.2 (4.5%) 42.7 (3.6%) 62.7 (3.3%) SPC/E Average HFE Error 18.4 (-11.6%) 51.9 (-11.7%) 81.8 (-9.2%) 244.9 (-15.8%) HFE Error Standard Deviation 26.1 (12.1%) 24.3 (4.3%) 41.6 (3.5%) 61.8 (3.2%) TIP4PEW Average HFE Error 22.3 (-14.3%) 67.2 (-15.2%) 108.0 (-12.3%) 283.0 (-18.2%) HFE Error Standard Deviation 30.1 (12.6%) 27.5 (4.5%) 46.4 (3.7%) 65.1 (3.3%)    148 Table 27. Final optimized 12-6-4 parameter set of divalent metal ions for different water models.  TIP3P SPC/E TIP4PEW  Rmin/2   (†) ! (kcal/mol) - (†-2) Rmin/2   (†) ! (kcal/mol) - (†-2) Rmin/2   (†) ! (kcal/mol) - (†-2) Be2+ 1.203 0.00116124 10.200  1.205 0.00120058 9.800 1.205 0.00120058 11.650 Cu2+ 1.476 0.03198620 1.789 1.482 0.03364841 1.758 1.475 0.03171494 2.000 Ni2+ 1.431 0.02133669 1.742 1.424 0.01995146 1.714 1.430 0.02113456 2.035 Zn2+ 1.455 0.02662782 1.623 1.454 0.02639002 1.588 1.450 0.02545423 1.877 Co2+ 1.458 0.02735051 1.442 1.457 0.02710805 1.410 1.455 0.02662782 1.688 Cr2+ 1.431 0.02133669 1.120 1.424 0.01995146 1.096 1.425 0.02014513 1.440 Fe2+ 1.457 0.02710805 1.128 1.450 0.02545423 1.095 1.450 0.02545423 1.386 Mg2+ 1.437 0.02257962 1.046 1.429 0.02093385 0.987 1.436 0.02236885 1.362 V2+ 1.494 0.03715368 1.080 1.502 0.03962711 1.060 1.495 0.03745682 1.280 Mn2+ 1.485 0.03450196 0.851 1.495 0.03745682 0.828 1.485 0.03450196 1.067 Hg2+ 1.641 0.10128575 0.741 1.641 0.10128575 0.751 1.632 0.09620220 0.855 Cd2+ 1.535 0.05102457 0.811 1.541 0.05330850 0.819 1.531 0.04953859 0.995 Ca2+ 1.642 0.10185975 0.223 1.634 0.09731901 0.230 1.633 0.09675968 0.325 Sn2+ 1.777 0.19470705 0.275 1.778 0.19549490 0.286 1.765 0.18535099 0.338 Sr2+ 1.777 0.19470705 0.121 1.778 0.19549490 0.137 1.763 0.18380968 0.175 Ba2+ 1.936 0.33132862 0.062 1.937 0.33223312 0.072 1.924 0.32049456 0.096 !    149 Table 28. Computed C4 values for the 12-6-4 parameter set of divalent metal ions for different water models.a  TIP3P SPC/E TIP4PEW Be2+ 186.5  188.1 228.5 Cu2+ 290.9 304.4 339.2 Ni2+ 212.8 205.2 259.2 Zn2+ 231.6 231.2 272.3 Co2+ 209.7 209.2 252.8 Cr2+ 136.8 131.2 177.4 Fe2+ 163.0 155.4 201.1 Mg2+ 132.9 122.2 180.5 V2+ 195.7 206.6 244.8 Mn2+ 146.1  154.9 192.3 Hg2+ 288.8 300.2 335.2 Cd2+ 185.6 198.8 233.7 Ca2+ 87.3 89.0 128.0 Sn2+ 187.9 201.1 231.4 Sr2+ 82.7 96.3 118.9 Ba2+ 71.9 85.8 112.5 a. In †4.kcal/mol.         150 Table 29. The simulated HFE, IOD and CN values of the 12-6-4 parameter set for divalent metal ions.a  TIP3P SPC/E TIP4PEW  HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN Be2+ -572.8 1.64 4.4 -572.6 1.65 4.3 -572.4 1.64 4.0 Cu2+ -479.8 2.11 6.2 -481.4 2.11 6.0 -480.0 2.11 6.1 Ni2+ -473.1 2.06 6.0 -473.1 2.06 6.0 -472.7 2.06 6.0 Zn2+ -467.0 2.09 6.0 -468.2 2.09 6.0 -467.4 2.09 6.0 Co2+ -457.5 2.10 6.0 -456.8 2.10 6.0 -456.7 2.10 6.0 Cr2+ -441.4 2.08 6.0 -442.0 2.08 6.0 -442.1 2.08 6.0 Fe2+ -439.5 2.11 6.0 -439.8 2.11 6.0 -439.7 2.11 6.0 Mg2+ -436.6 2.09 6.0 -436.7 2.09 6.0 -437.9 2.09 6.0 V2+ -435.9 2.21 7.0 -435.5 2.21 6.8 -436.1 2.22 7.0 Mn2+ -421.2 2.19 6.8 -420.3 2.20 6.5 -420.9 2.20 6.7 Hg2+ -420.0 2.41 8.0 -420.6 2.41 8.0 -421.6 2.41 8.0 Cd2+ -419.4 2.30 7.8 -419.4 2.30 7.5 -420.1 2.30 7.7 Ca2+ -360.6 2.46 8.0 -360.6 2.46 8.0 -359.6 2.46 8.0 Sn2+ -356.1 2.62 9.0 -356.1 2.62 8.8 -356.8 2.62 8.9 Sr2+ -330.4 2.64 8.9 -330.3 2.64 8.7 -329.4  2.64 8.9 Ba2+ -299.4 2.83 9.3 -298.6 2.83 9.2 -298.2 2.83 9.5 Average Error  0.1 0.00 0.6 0.0 0.00 0.5 0.0 0.00 0.6 Standard Deviation 0.5 0.01 0.9 0.6 0.01 0.9 0.5 0.01 0.9 UAE  0.4 0.00 0.6 0.4 0.00 0.5 0.4 0.00 0.6 a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 2). The target IOD values in Table 2 (without considering the error bars) were treated as the reference for the simulated IOD values.     151 Table 30. Final optimized 12-6-4 parameter set of monovalent ions for different water models.  TIP3P SPC/E  Rmin/2 (†) ! (kcal/mol) C4 (kcal/mol!†4) Rmin/2 (†) ! (kcal/mol) C4 (kcal/mol!†4) Li+ 1.325 0.00674244 27 1.327 0.00691068 33 Na+ 1.473 0.03117732 0 1.472 0.03091095 6 K+ 1.758 0.17997960 8 1.760 0.18150763 19 Rb+ 1.831 0.23886274 0 1.826 0.23464849 7 Cs+ 2.008 0.39668797 2 2.004 0.39306142 12 Tl+ 1.893 0.29273756 50 1.889 0.28918714 61 Cu+ 1.217 0.00146124 7 1.218 0.00148497 9 Ag+ 1.533 0.05027793 83 1.536 0.05140063 92 NH4+ 1.802 0.21475916 4 1.797 0.21069138 13 H+ (Zundel cation) 0.992 0.00001138 108 0.987 0.00000988 106 H+ (Eigen cation) 0.871 0.00000022 51 0.870 0.00000021 51 H3O+ 1.774 0.19235093 190 1.773 0.19156806 205 F- 1.725 0.15557763 -27 1.726 0.15629366 -53 Cl- 2.150 0.52153239 -38 2.153 0.52404590 -55 Br- 2.314 0.64785703 -39 2.324 0.65475744 -51 I- 2.567 0.79269938 -45 2.579 0.79809803 -51    152 Table 30 (contÕd)  TIP4PEW  Rmin/2 (†) ! (kcal/mol) C4 (kcal/mol!†4) Li+ 1.313 0.00580060 36 Na+ 1.459 0.02759452 9 K+ 1.751 0.17467422 24 Rb+ 1.817 0.22712223 13 Cs+ 1.997 0.38670945 16 Tl+ 1.883 0.28387745 65 Cu+ 1.214 0.00139196 21 Ag+ 1.522 0.04630154 94 NH4+ 1.791 0.20584696 20 H+ (Zundel cation) 0.997 0.00001309 126 H+ (Eigen cation) 0.876 0.00000026 64 H3O+ 1.770 0.18922704 209 F- 1.728 0.15773029 -67 Cl- 2.154 0.52488228 -66 Br- 2.326 0.65612582 -68 I- 2.585 0.80075128 -62       153 Table 31. The simulated HFE, IOD and CN values of the 12-6-4 parameter set for monovalent ions.a  TIP3P SPC/E TIP4PEW  HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN Li+ -113.4 2.09 5.3 -113.2 2.09 5.1 -112.9 2.09 5.4 Na+ -88.2 2.36 5.7 -88.2 2.35 5.7 -87.9 2.35 5.9 K+ -70.4 2.78 7.8 -70.5 2.79 7.2 -70.5 2.79 7.5 Rb+ -66.0 2.90 7.6 -65.9 2.89 7.8 -65.7 2.89 7.5 Cs+ -60.2 3.13 9.4 -60.4 3.14 8.9 -60.3 3.13 9.3 Tl+ -71.4 2.96 8.5 -71.4 2.97 8.9 -71.6 2.96 8.5 Cu+ -125.0 1.86 4.0 -125.0 1.86 4.0 -124.5 1.86 4.0 Ag+ -102.0 2.41 6.0 -102.8 2.41 6.0 -102.2 2.41 6.0 NH4+ -67.8 2.86 7.4 -67.9 2.85 7.4 -68.4 2.86 7.5 H+ (Zundel cation) -250.9 1.24 2.0 -251.7 1.23 2.0 -250.7 1.25 2.0 H+ (Eigen cation) -251.2 1.01 1.0 -251.3 1.01 1.0 -250.6 1.02 1.0 H3O+ -102.8 2.75 8.4 -104.2 2.75 8.4 -103.3 2.75 8.2 F- -119.3 2.63 6.4 -119.8 2.63 6.2 -119.5 2.63 6.0 Cl- -89.1 3.18 7.6 -88.5 3.18 7.3 -89.4 3.18 6.9 Br- -82.6 3.37 7.7 -82.1 3.37 7.3 -81.8 3.37 7.0 I- -74.5 3.64 8.5 -74.7 3.64 7.7 -74.1 3.64 8.5 Average Error 0.1 0.00 -- -0.1 0.00 -- 0.2 0.00 -- Standard Deviation 0.4 0.01 -- 0.5 0.01 -- 0.5 0.01 -- UAE 0.3 0.00 -- 0.4 0.00 -- 0.4 0.00 -- a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 1). The target IOD values in Table 1 (without considering the error bars) were treated as the reference for the simulated IOD values.      154 Table 32. Final optimized 12-6-4 parameter set of highly charged metal ions for different water models.  TIP3P SPC/E  Rmin/2 (†) ! (kcal/mol) C4 (kcal/mol!†4) Rmin/2 (†) !  (kcal/mol) C4 (kcal/mol!†4) Al3+ 1.369 0.01128487 399 1.375 0.01205473 406 Fe3+ 1.443 0.02387506 428 1.450 0.02545423 442 Cr3+ 1.415 0.01827024 258 1.414 0.01809021 254 In3+ 1.491 0.03625449 347 1.487 0.03507938 349 Tl3+ 1.571 0.06573030 456 1.569 0.06484979 455 Y3+ 1.630 0.09509276 216 1.624 0.09180886 209 La3+ 1.758 0.17997960 152 1.763 0.18380968 165 Ce3+ 1.782 0.19865859 230 1.786 0.20184160 242 Pr3+ 1.780 0.19707431 264 1.782 0.19865859 272 Nd3+ 1.724 0.15486311 213 1.735 0.16280564 235 Sm3+ 1.711 0.14571499 230 1.703 0.14021803 224 Eu3+ 1.716 0.14920231 259 1.721 0.15272873 273 Gd3+ 1.658 0.11129023 198 1.646 0.10417397 186 Tb3+ 1.671 0.11928915 235 1.666 0.11617738 227 Dy3+ 1.637 0.09900804 207 1.637 0.09900804 206 Er3+ 1.635 0.09788018 251 1.629 0.09454081 247 Tm3+ 1.647 0.10475707 282 1.633 0.09675968 262 Lu3+ 1.625 0.09235154 249 1.620 0.08965674 247 Hf4+ 1.600 0.07934493 827 1.592 0.07543075 810 Zr4+ 1.609 0.08389240 761 1.609 0.08389240 760 Ce4+ 1.766 0.18612361 706 1.761 0.18227365 694 U4+ 1.792 0.20665151 1034 1.791 0.20584696 1043 Pu4+ 1.752 0.17542802 828 1.750 0.17392181 828 Th4+ 1.770 0.18922704 512 1.773 0.19156806 513     155 Table 32 (contÕd)  TIP4PEW  Rmin/2 (†) !  (kcal/mol) C4 (kcal/mol!†4) Al3+ 1.377 0.01232018 488 Fe3+ 1.448 0.02499549 519 Cr3+ 1.408 0.01703790 322 In3+ 1.486 0.03478983 425 Tl3+ 1.564 0.06268139 535 Y3+ 1.624 0.09180886 294 La3+ 1.755 0.17769767 243 Ce3+ 1.776 0.19392043 315 Pr3+ 1.774 0.19235093 348 Nd3+ 1.720 0.15202035 297 Sm3+ 1.706 0.14226734 314 Eu3+ 1.711 0.14571499 345 Gd3+ 1.652 0.10769970 280 Tb3+ 1.665 0.11556030 313 Dy3+ 1.639 0.10014323 298 Er3+ 1.628 0.09399072 328 Tm3+ 1.638 0.09957472 356 Lu3+ 1.617 0.08806221 331   Hf4+ 1.599 0.07884906 956 Zr4+ 1.610 0.08440707 895 Ce4+ 1.761 0.18227365 835 U4+ 1.791 0.20584696 1183 Pu4+ 1.753 0.17618319 972 Th4+ 1.758 0.17997960 625     156 Table 33. The simulated HFE, IOD and CN values of the 12-6-4 parameter set for highly charged metal ions.a  TIP3P SPC/E  HFE (kcal/mol) IOD (†) CN HFE (kcal/mol) IOD (†) CN Al3+ -1082.0 1.87 6.0 -1081.3 1.88 6.0 Fe3+ -1019.4 2.02 6.9 -1019.2 2.02 6.8 Cr3+ -957.8 1.95 6.0 -957.8 1.96 6.0 In3+ -951.5 2.15 8.0 -950.4 2.15 8.0 Tl3+ -948.1 2.22 8.0 -948.5 2.23 8.0 Y3+ -824.9 2.36 9.0 -824.6 2.36 9.0 La3+ -750.7 2.53 9.7 -752.1 2.52 9.2 Ce3+ -765.2 2.55 9.9 -765.1 2.55 9.7 Pr3+ -775.4 2.54 9.9 -776.6 2.54 9.7 Nd3+ -783.6 2.46 9.0 -784.3 2.47 9.0 Sm3+ -795.3 2.44 9.0 -794.8 2.44 9.0 Eu3+ -802.1 2.44 9.0 -803.4 2.45 9.0 Gd3+ -806.5 2.39 9.0 -807.2 2.39 9.0 Tb3+ -813.5 2.40 9.0 -812.2 2.40 9.0 Dy3+ -818.4 2.37 9.0 -819.0 2.37 9.0 Er3+ -834.8 2.36 9.0 -834.9 2.36 9.0 Tm3+ -840.7 2.37 9.0 -840.2 2.36 9.0 Lu3+ -839.3 2.34 9.0 -840.4 2.34 9.0 Average Error 0.1 0.00 -- 0.0 0.00 -- Standard Deviation 0.6 0.01 -- 0.5 0.00 -- UAE 0.5 0.00 -- 0.4 0.00 -- Hf4+ -1663.9 2.16 10.0 -1663.9 2.16 8.0 Zr4+ -1622.7 2.19 9.9 -1622.9 2.19 9.8 Ce4+ -1462.2 2.42 10.0 -1462.1 2.42 10.0 U4+ -1566.6 2.41 10.0 -1566.0 2.41 10.0 Pu4+ -1519.4 2.39 10.0 -1520.3 2.39 10.0 Th4+ -1389.3 2.44 10.0 -1388.3 2.45 10.0 Average Error 0.6 0.00 -- 0.8 0.00 -- Standard Deviation 0.4 0.01 -- 0.8 0.00 -- UAE 0.6 0.00 -- 0.9 0.00 -- a. All the average errors and standard deviations were obtained by treating the corresponding target values as the reference (see Table 3).     157 Table 33 (contÕd)  TIP4PEW  HFE (kcal/mol) IOD (†) CN Al3+ -1080.8 1.88 6.0 Fe3+ -1020.2 2.03 6.8 Cr3+ -957.5 1.95 6.0 In3+ -952.2 2.15 7.9 Tl3+ -949.7 2.22 8.0 Y3+ -824.9 2.36 9.0 La3+ -752.4 2.52 9.4 Ce3+ -764.6 2.55 9.8 Pr3+ -775.9 2.54 9.8 Nd3+ -783.4 2.46 9.0 Sm3+ -795.4 2.44 9.0 Eu3+ -802.8 2.45 9.0 Gd3+ -807.6 2.39 9.0 Tb3+ -812.9 2.40 9.0 Dy3+ -819.3 2.38 9.0 Er3+ -836.2 2.36 9.0 Tm3+ -840.6 2.36 9.0 Lu3+ -840.9 2.34 9.0 Average Error -0.3 0.00 -- Standard Deviation 0.6 0.00 -- UAE 0.6 0.00 --   Hf4+ -1663.3 2.16 8.0 Zr4+ -1622.6 2.19 9.9 Ce4+ -1462.0 2.42 10.0 U4+ -1569.3 2.42 10.0 Pu4+ -1520.4 2.40 10.0 Th4+ -1388.0 2.44 10.0 Average Error 0.4 0.00 -- Standard Deviation 1.2 0.01 -- UAE 1.0 0.00 --       158 Table 34. Polarizabilities used in the 12-6-4 model for different atom types in the AMBER FF. Atom type in       AMBER FF Polarziability  (†3) Atom type in AMBER FF Polarziability  (†3) HO/ H1/ H2/ H4/ HA/ H/ HC/ H5 0.387a P 1.538a CM/CA/C/CK/CB/CQ 1.352a MG 0.048b CT 1.061a Cl- 1.910c N*/N2/NC/NB/NA 1.090a OW 1.444d O/O2 0.569a EP/HW 0.000 OS 0.637a   OH 0.637a   a. Referenced or adopted from Miller.234 b. Calculated at the B3LYP/6-311++G(2d,2p) level of theory. c. From Applequist et al.235 d. From Eisenberg and Kauzmann.205     159 Table 35. The IODs of the Mg2+ and Cl- ions at different concentrations.a  0.25 M 0.5 M 1.0 M Original AMBER FF parameter 2.01/3.16 2.00/3.16 2.00/3.16 12-6 CM set of parameter 2.07/3.17 2.08/3.17 2.08/3.17 12-6-4 model 2.11/3.21 2.11/3.21 2.11/3.21 a. The units are † while the experimental results are 2.09±0.04 † for Mg2+ and 3.18±0.06† for Cl- (from Marcus123).         160 Table 36. Comparison of the structural properties predicted by the three different Mg2+ parameters used in the simulations of a Mg2+-nucleic acid system.  IOD (†) Mg2+-backbone phosphate distance  (†) Original AMBER FF parameter 2.02 3.30 12-6 CM set of parameter 2.05 3.35 12-6-4 model in present work 2.09 3.43 Experimental values 2.09±0.04a 3.6b a. From Marcus.123 b. From Caminiti.236               161 Table 37. Polarizability values of the AMBER atom types used in the 12-6-4 nonbonded model simulation of a metalloprotein system.a Atom type Polarizability Atom type Polarizability Atom type Polarizability HW 0.000 O2 0.569 CB 1.352 FE 0.264 OH 0.637 CC 1.352 H 0.387 CT 1.061 CN 1.352 H1 0.387 N 1.090 CR 1.352 H4 0.387 N2 1.090 CV 1.352 H5 0.387 N3 1.090 CW 1.352 HA 0.387 NA 1.090 OW 1.444 HC 0.387 NB 1.090 S 3.000 HO 0.387 C 1.352 SH 3.000 HP 0.387 C* 1.352   O 0.569 CA 1.352   a. Polarizability of Fe3+ (with atom type FE) was calculated at the B3LYP/6-311++G(2d,2p) level of theory using Gaussian 09 Revision C.01,237 polarizability of the water oxygen (OW) was taken from Eisenberg and Kauzmann,205 while polarizabilities of the other atom types were adopted from Miller.234        162 Table 38.  Small model capping schemes for coordinated amino acid residues. Atom bound to central metal ion Original residue Small model Sidechain atom in an amino acid NH-CHR-CO CH3R Backbone O atom in a non-terminal residue NH-CHR-CO CH3-CO or CH2R-CO (if this residue also has a sidechain atom bound to the metal ion) *C-terminal converted to NH-CH3 (if it doesnÕt bond to the metal ion), or to NH-CH2R (if it has a sidechain atom bound to the metal ion) Backbone N atom in a non-terminal residue NH-CHR-CO NH-CH3 or NH-CH2R (if this residue also has a sidechain atom bound to the metal ion) *N-terminal converted to CH3-CO (if it doesnÕt bonds to the metal ion), or to CH2R-CO (if it has a sidechain atom bound to the metal ion) Backbone CO2- in the C-terminal residue NH-CHR-CO2- CH3-CO2- Backbone NH2 in the N-terminal residue NH2-CHR-CO NH2-CH3 Backbone N and O in one residue NH-CHR-CO NH-CH2-CO *N-terminal modeled as CH3-CO (or to CH2R-CO if it has a sidechain atom bound to the metal ion) and the C-terminal to NH-CH3 (or to NH-CH2R if it has a sidechain atom bound to the metal ion)     163 Table 39. The RMSD values between the CSD, QM optimized and MM minimized structures.a  CSD structure QM optimized MM minimized CSD structure -- 0.280(0.219) 0.377(0.327) QM optimized 0.280(0.219) -- 0.201(0.193) MM minimized 0.377(0.327) 0.201(0.193) -- a. Units are †. The values outside brackets are for all atoms while the values inside brackets are only for heavy atoms. The CSD structure is from residue 1 in CSD entry FAJYAR01.    164 APPENDIX B: FIGURES   Figure 1. Atomic ions that were parameterized for the 12-6 model in current work. These ions are shown with blue background. The monovalent anions, monovalent cations, divalent cations, trivalent cations, and tetravalent cations are shown in light green, yellow, orange, red, dark red, respectively. The ions that have multiple oxidation states are shown in white, which are Cr2+/Cr3+, Fe2+/Fe3+, Cu+/Cu2+, Ag+/Ag2+, Ce3+/Ce4+, Sm2+/Sm3+, Eu2+/Eu3+, and Tl+/Tl3+. Besides these atomic ions, the H3O+ and NH4+ ions were also parameterized in current work.   165   Figure 2. Thermodynamics cycle used to determine the HFE values.    166    Figure 3. HFE fitting curves for 24 M(II) metal ions for the TIP3P water model.  !"#!"$!"%&&"'&"#&"$&"%''"''"#!&'(#)$*+,-.'/01-23456+786207/09:;8.+687  <='>?@'>A,'>B4'>C-'>?6'>BD'>12'>?5'>E='>F2'>G'>F-'>.H2'>?D'>IJ'>?;'>K-'>BJ'>L@'>K5'>K+'><;'>.*;'> 167  Figure 4. HFE fitting curves for Zn2+ in four different water models.   !"#!"$!"%&&"'&"#&"$&"%''"''"#!&'(#)$*+,-.'/01-23456+786207/09:;8.+687  <=>(>?>@.A<=>#><=>#>AB 168  Figure 5. IOD fitting curves for Zn2+ in four different water models.     !"#$$"%$"&$"'$"#%%"%%"&!$%(&)'*+,-.%/0-12345+6751/68/9:;7.+576  <=>(>?>@.A<=>&><=>&>AB 169 Figure 6. CN switching during a MD simulation when Rmin/2 =2.2 † and !=0.1 kcal/mol for a M(II) ion in the TIP3P water model.    170  Figure 7. HFE and IOD fitting curves of six representative M(II) metal ions in the TIP3P water model.  !"#!"$!"%&&"'&"#&"$&"%''"''"#!&'(#)$*+,-.'/01-23456+786207/09:;8.+687  <='>?@A<='>BCDE,'>?@AE,'>BCDF2'>?@AF2'>BCDG;'>?@AG;'>BCDH5'>?@AH5'>BCD<;'>?@A<;'>BCD 171  Figure 8. Determination of the three parameter sets for Zn2+ in the TIP3P water model.    !"#!"$!"%&&"'&"#&"$&"%''"''"#!&'(#)$*+,-.'/01-23456+786207/09:;8.+687  <-'=>?@<-'=AB<-'=CDEFGA 172  Figure 9. Radial distribution functions of the three parameter sets for Zn2+ in the TIP3P water model.    173   Figure 10. Fitting curves between the HFE and IOD values for the positive (upper) and negative (lower) monovalent ions in the three water models together with the target values of the ions investigated in the present work.    174   Figure 11. Fitting curves between the HFE and Rmin/2 values for the positive (upper) and negative (lower) monovalent ions in the three water models.    175   Figure 12. Fitting curves between the IOD and Rmin/2 values for the positive (upper) and negative (lower) monovalent ions in the three water models.  176  Figure 13. Unsigned average errors of the VDW radii of different parameter sets using QMSP calculated values as reference. The parameter sets developed herein are shown in the seven green columns on the right side of the figure.   !"!!!#!"$!!#!"%!!#!"&!!#!"'!!#!"(!!#!")!!#!"*!!#!"+!!#!",!!#$"!!!#$"$!!#Unsigned average error of VDW radii (†) 177  Figure 14. PDB entry 4BV1. Waters are not shown in the figure, the ferric ions are shown as sliver spheres. This picture was created using VMD.1    178  Figure 15. Chain C in PDB entry 4BV1 (upper) and a close-up figure of the metal site in Chain C (lower). The ferric ion is represented as a silver sphere and it is coordinated by one Cys, four His and one water molecule. The figures were made using VMD.1    179    Figure 16. RMSDs of the backbone CA, C, N atoms (upper) and the metal site heavy atoms (lower) for the simulations with the IOD and 12-6-4 parameter sets using the initial structure (experimental structure) as reference.    180  Figure 17. RMSFs of the heavy atoms of the protein residues in the simulations using the IOD and 12-6-4 parameter sets.    181  Figure 18. HFE percent errors for different parameter sets of monovalent ions in the TIP3P water model. HZ+ and HE+ represent the H+ Zundel and Eigen ions respectively. Since we did not design HFE set of parameters for the HZ+ and HE+ ions, they are not shown in the figure.    182   Figure 19. IOD percent errors for different parameter sets of monovalent ions in the TIP3P water model. HZ+ and HE+ represent the H+ Zundel and Eigen ions respectively. Since we did not design IOD set of parameters for the HZ+ and HE+ ions, they are not shown in the figure.    183  Figure 20. The average errors (upper) and average percentage errors (lower) of the IODs for the HFE parameter set for the mono-, di-, tri- and tetravalent cations in the TIP3P water model.    184  Figure 21. The average errors (upper) and average percentage errors (lower) of the HFEs for the IOD parameter set for the mono-, di-, tri- and tetravalent cations in the TIP3P water model.    185  Figure 22. Fitting curves between the HFE and Rmin/2 values (upper) and fitting curves between the IOD and Rmin/2 values (lower) for the positive and negative monovalent ions in different water models.    186 !Figure 23. Scheme representing intermolecular interactions: the green double-headed arrow and red double-headed arrow represent the interactions that are included and not included in the 12-6 nonbonded model, respectively.         187    Figure 24. Atomic ions that were parameterized for the 12-6-4 model in current work. These ions are shown with blue background. The monovalent anions, monovalent cations, divalent cations, trivalent cations, and tetravalent cations are shown in light green, yellow, orange, red, dark red, respectively. The ions that have multiple oxidation states are shown in white, which are Cr2+/Cr3+, Fe2+/Fe3+, Cu+/Cu2+, Ce3+/Ce4+, and Tl+/Tl3+. Besides these atomic ions, the H3O+ and NH4+ ions were also parameterized in current work.    188  Figure 25. HFE and IOD fitting curves for the Cu2+, Ni2+, Zn2+, Co2+, and Cr2+ ions in the TIP3P water model.     189  Figure 26. HFE and IOD fitting curves for the Fe2+, Mg2+, V2+, Mn2+, and Hg2+ ions in the TIP3P water model.     190  Figure 27. HFE and IOD fitting curves for the Cd2+, Ca2+, Sn2+, Sr2+, and Ba2+ ions in the TIP3P water.     191  Figure 28. Workflow of the MCPB.py program.    192  Figure 29. Examples of capping different metal bound amino acid residues in the small model. Structures on the left are the coordinated amino acids in the protein, and the structures on the right are the capped residues in the small models built by MCPB.py. Panels A and B are for PDB entry 1E67, panels C, D, and E are for PDB entries 1AK0, 1Y79, and 2BZS, respectively. Zinc coordination is shown as dotted lines with the bond lengths given in †.   A)B)C)D)E) 193  Figure 30. Metal ions currently supported by the MCPB.py program. The metals with a blue background use the VDW parameters from section 2 while the metals with green background use the VDW parameters adapted from UFF.2    194  Figure 31. Structure of chain A from PDB entry 1E67 with the zinc ion represented by a VDW sphere and the metal site residues indicated by sticks. This figure was made using VMD.1     195  Figure 32. Content of the MCPB.py input file for the construction of the metal site found in chain A of PDB entry 1E67.   original_pdb 1E67_fixed_H.pdbgroup_name 1E67cut_off 2.7ion_ids 1931ion_mol2files ZN.mol2 196  Figure 33. Commands used for the construction of the metal site found in chain A of PDB entry 1E67 using MCPB.py. The second to the fifth lines are for the QM calculations using Gaussian03, while the last line is to build the topology and coordinate files using tleap.   MCPB.py -i 1E67.in -s 1g03 < 1E67_small_opt.com > 1E67_small_opt.logg03 < 1E67_small_fc.com > 1E67_small_fc.logformchk 1E67_small_opt.chkg03 < 1E67_large_mk.com > 1E67_large_mk.logMCPB.py -i 1E67.in -s 2MCPB.py -i 1E67.in -s 3MCPB.py -i 1E67.in -s 4tleap -s -f 1E67_tleap.in > 1E67_tleap.out 197  Figure 34. Bond parameters (above) and angle parameters (below) for the metal site in chain A from the 1E67 PDB structure. Here M1 is the atom type of the zinc ion, while Y1, Y2, Y3, Y4 are the atom types for the four atoms bound to the zinc ion, which are the GLY45 backbone oxygen atom, the d nitrogen atom in HIE46, the sidechain sulfur atom in CYM112 and the d nitrogen atom in HIE117, respectively. Here HIE and CYM are the ÒAMBER styleÓ residue names: HIE means a His residue which has the e nitrogen protonated, CYM indicates a Cys residue which is negatively charged (i.e. with a CH2S- sidechain group).    198   Figure 35. RMSD values of the protein backbone N, CA and C atoms (upper) and the heavy atoms in the metal site (lower) over 20 ns of simulation. The RMSD values were calculated using CPPTRAJ.3    199  Figure 36. Structure of residue 1 from CSD entry FAJYAR01. This figure was made using VMD.1     200  Figure 37. Content of the MCPB.py input file for the construction of the metal site found in residue 1 of CSD entry FAJYAR01.   original_pdb FAJYAR01.pdbgroup_name FAJYAR01ion_ids 1ion_mol2files OS.mol2naa_mol2files RES.mol2frcmod_files RES.frcmod 201  Figure 38. Commands used for the construction of the metal site found in residue 1 of CSD entry FAJYAR01 using MCPB.py. The second to the fifth lines are for the QM calculations using Gaussian03, while the last line is to build the topology and coordinate files using tleap.   MCPB.py -i FAJYAR01.in -s 1g03 < FAJYAR01_small_opt.com > FAJYAR01_small_opt.logg03 < FAJYAR01_small_fc.com > FAJYAR01_small_fc.logformchk FAJYAR01_small_opt.chkg03 < FAJYAR01_large_mk.com > FAJYAR01_large_mk.logMCPB.py -i FAJYAR01.in -s 2MCPB.py -i FAJYAR01.in -s 3MCPB.py -i FAJYAR01.in -s 4tleap -s -f FAJYAR01_tleap.in > FAJYAR01_tleap.out 202  Figure 39. Bond parameters (above) and angle parameters (below) for the metal site of residue 1 from CSD entry FAJYAR01. Here M1 is the atom type of the Os2+ ion, while Y1 to Y6 are the atom types for the six metal bound nitrogen atoms.  203  Figure 40. Comparison of the QM and MM calculated normal modes.    204    Figure 41. Cartoon representation of Cu(I)!CusF (PDB entry 2VB2). Cu+ is shown as a pink sphere, and the metal binding residues are represented as sticks.     205  Figure 42. Cu+"Trp44 distance and Glu46"Met47 # dihedral coordinates (black lines) for our two-dimensional PMF MD simulations shown in (a) a closed structure and (b) a wide-open structure of Cu(I)!CusF.     206   Figure 43. Scheme employed to generate the starting conformations for the two-dimensional PMF calculation. (Left) Initial conformations were generated by sampling along the diagonal from the closed to the wide-open conformations. Arrows indicate the direction of sampling along the reaction coordinate. Equilibrated conformation from a window was used as the starting coordinate of the adjacent window as we moved along the diagonal. (Right) Scheme employed to sample conformations starting from windows along the diagonal (solid circles) to the hollow circles.        207   Figure 44. (Upper) A two-dimensional free energy landscape calculated for the Cu-Trp44 distance versus the Glu46-Met47 # dihedral from our classical MD simulations of Cu(I)!CusF. Cu(I)!CusF is unable to make the transition from the closed to the open conformations. The closed conformations are represented by smaller Cu+-Trp distances, while the open conformations are represented by larger Cu+-Trp44 distances. (Lower) PMF profile based on the umbrella sampling simulations in which the Cu+-Trp44 distance and Glu46-Met47 # dihedral were gradually modified to force the transition from the closed conformation to the wide-open conformation of Cu(I)!CusF. Error bars were calculated using a bootstrapping algorithm implemented in Alan GrossfieldÕs WHAM code (http://membrane.urmc.rochester.edu/content/wham). Error bars were calculated along each axis of the 2D PMF profile. The error bars along the dihedral dimension are between 0.002 and 0.040 kcal/mol with an average of 0.022 kcal/mol.  The error bars along the distance dimension are in the range of 0.001 to 0.128 kcal/mol with an average of 0.012 kcal/mol.    208 APPENDIX C: COPYRIGHT NOTICE  Figure 1 and Figure 24 of this dissertation are copyrighted by PENGFEI LI 2016. Figure 41 is reprinted with permission from ref 111. Copyright 2011 American Chemical Society. The remaining parts of this dissertation (including its supporting information) are adapted with permissions from several publications listed below: (1) Adapted with permission from ref 143. Copyright 2015 American Chemical Society. (2) Adapted with permission from ref 144. Copyright 2013 American Chemical Society. (3) Adapted with permission from ref 145. Copyright 2014 American Chemical Society. The paper can be accessed from webpage: http://pubs.acs.org/doi/abs/10.1021/jp505875v. This is an unofficial adaptation of an article that appeared in an ACS publication. ACS has not endorsed the content of this adaptation or the context of its use. (4) Adapted with permission from ref 174. Copyright 2014 American Chemical Society. (5) Adapted with permission from ref 219. Copyright 2016 American Chemical Society. (6) Adapted with permission from ref 230. Copyright 2016 American Chemical Society.   209 BIBLIOGRAPHY 210 BIBLIOGRAPHY  1. Humphrey, W.; Dalke, A.; Schulten, K., Vmd: Visual Molecular Dynamics. J. Mol. Graphics 1996, 14, 33-38. 2. Rapp”, A. K.; Casewit, C. J.; Colwell, K. S.; Goddard III, W. A.; Skiff, W. M., Uff, a Full Periodic Table Force Field for Molecular Mechanics and Molecular Dynamics Simulations. J. Am. Chem. Soc. 1992, 114, 10024-10035. 3. Roe, D. R.; Cheatham III, T. E., Ptraj and Cpptraj: Software for Processing and Analysis of Molecular Dynamics Trajectory Data. J. Chem. Theory Comput. 2013, 9, 3084-3095. 4. Peters, M. B.; Yang, Y.; Wang, B.; Fti-Moln⁄r, L.; Weaver, M. N.; Merz Jr, K. M., Structural Survey of Zinc-Containing Proteins and Development of the Zinc Amber Force Field (Zaff). J. Chem. Theory Comput. 2010, 6, 2935-2947. 5. Tus, A.; Rakipovi., A.; Peretin, G.; Tomi., S.; /iki., M., Biome: Biologically Relevant Metals. Nucleic Acids Res. 2012, 40, W352-W357. 6. Waldron, K. J.; Robinson, N. J., How Do Bacterial Cells Ensure That Metalloproteins Get the Correct Metal? Nat. Rev. Microbiol. 2009, 7, 25-35. 7. Dupureur, C. M., Roles of Metal Ions in Nucleases. Curr. Opin. Chem. Biol. 2008, 12, 250-255. 8. Andreini, C.; Bertini, I.; Cavallaro, G.; Holliday, G.; Thornton, J., Metal Ions in Biological Catalysis: From Enzyme Databases to General Principles. J Biol Inorg Chem 2008, 13, 1205-1218. 9. Chaturvedi, U. C.; Shrivastava, R., Interaction of Viral Proteins with Metal Ions: Role in Maintaining the Structure and Functions of Viruses. FEMS Immunol. Med. Microbiol. 2005, 43, 105-114. 10. Rosenzweig, A. C., Metallochaperones: Bind and Deliver. Chem. Biol. 2002, 9, 673-677. 11. Thomson, A. J.; Gray, H. B., Bio-Inorganic Chemistry. Curr. Opin. Chem. Biol. 1998, 2, 155-158. 12. Berridge, M. J.; Bootman, M. D.; Lipp, P., Calcium - a Life and Death Signal. Nature 1998, 395, 645-648. 13. Christianson, D. W., Structural Chemistry and Biology of Manganese Metalloenzymes. Prog. Biophys. Mol. Biol. 1997, 67, 217-252.  211 14. Lipscomb, W. N.; Str−ter, N., Recent Advances in Zinc Enzymology. Chem. Rev. 1996, 96, 2375-2434. 15. Linder, M. C.; Hazegh-Azam, M., Copper Biochemistry and Molecular Biology. Am. J. Clin. Nutr. 1996, 63, 797S-811S. 16. Holm, R. H.; Kennepohl, P.; Solomon, E. I., Structural and Functional Aspects of Metal Sites in Biology. Chem. Rev. 1996, 96, 2239-2314. 17. Coleman, J. E., Zinc Proteins: Enzymes, Storage Proteins, Transcription Factors, and Replication Proteins. Annu. Rev. Biochem. 1992, 61, 897-946. 18. Berg, J. M., Zinc Finger Domains: Hypotheses and Current Knowledge. Annu. Rev. Biophys. Biophys. Chem. 1990, 19, 405-421. 19. Jacoby, M., How Computers Are Helping Chemists Discover Materials That DonÕt yet Exist. Chem. Eng. News 2015, 93, 8-11. 20. Dudev, T.; Lim, C., Principles Governing Mg, Ca, and Zn Binding and Selectivity in Proteins. Chem. Rev. 2003, 103, 773-788. 21. Dudev, T.; Lim, C., Competition among Metal Ions for Protein Binding Sites: Determinants of Metal Ion Selectivity in Proteins. Chem. Rev. 2014, 114, 538-556. 22. Valenzano, L.; Civalleri, B.; Sillar, K.; Sauer, J., Heats of Adsorption of Co and Co2 in MetalÐOrganic Frameworks: Quantum Mechanical Study of Cpo-27-M (M= Mg, Ni, Zn). J. Phys. Chem. C 2011, 115, 21777-21784. 23. Pang, Y.-P., Successful Molecular Dynamics Simulation of Two Zinc Complexes Bridged by a Hydroxide in Phosphotriesterase Using the Cationic Dummy Atom Method. Proteins: Struct., Funct., Bioinf. 2001, 45, 183-189. 24. Yang, Q.; Zhong, C., Molecular Simulation of Carbon Dioxide/Methane/Hydrogen Mixture Adsorption in Metal-Organic Frameworks. J. Phys. Chem. B 2006, 110, 17776-17783. 25. Wu, R.; Hu, P.; Wang, S.; Cao, Z.; Zhang, Y., Flexibility of Catalytic Zinc Coordination in Thermolysin and Hdac8: A Born-Oppenheimer Ab Initio Qm/Mm Molecular Dynamics Study. J. Chem. Theory Comput. 2009, 6, 337-343. 26. Hofer, T. S.; Weiss, A. K.; Randolf, B. R.; Rode, B. M., Hydration of Highly Charged Ions. Chem. Phys. Lett. 2011, 512, 139-145. 27. Sokol, A. A.; Bromley, S. T.; French, S. A.; Catlow, C. R. A.; Sherwood, P., Hybrid Qm/Mm Embedding Approach for the Treatment of Localized Surface States in Ionic Materials. Int. J. Quantum Chem. 2004, 99, 695-712.  212 28. Yu, D.; Yazaydin, A. O.; Lane, J. R.; Dietzel, P. D.; Snurr, R. Q., A Combined Experimental and Quantum Chemical Study of Co2 Adsorption in the MetalÐOrganic Framework Cpo-27 with Different Metals. Chem. Sci. 2013, 4, 3544-3556. 29. Duan, Y.; Kollman, P. A., Pathways to a Protein Folding Intermediate Observed in a 1-Microsecond Simulation in Aqueous Solution. Science 1998, 282, 740-744. 30. Shaw, D. E.; Maragakis, P.; Lindorff-Larsen, K.; Piana, S.; Dror, R. O.; Eastwood, M. P.; Bank, J. A.; Jumper, J. M.; Salmon, J. K.; Shan, Y., Atomic-Level Characterization of the Structural Dynamics of Proteins. Science 2010, 330, 341-346. 31. Jorgensen, W. L.; Ravimohan, C., Monte Carlo Simulation of Differences in Free Energies of Hydration. J. Chem. Phys. 1985, 83, 3050-3054. 32. Jorgensen, W. L.; Duffy, E. M., Prediction of Drug Solubility from Monte Carlo Simulations. Bioorg. Med. Chem. Lett. 2000, 10, 1155-1158. 33. Hoops, S. C.; Anderson, K. W.; Merz, K. M., Force Field Design for Metalloproteins. J. Am. Chem. Soc. 1991, 113, 8262-8270. 34. Roberts, B. P.; Miller III, B. R.; Roitberg, A. E.; Merz Jr, K. M., Wide-Open Flaps Are Key to Urease Activity. J. Am. Chem. Soc. 2012, 134, 9934-9937. 35. Chakravorty, D.; Wang, B.; Lee, C.; Guerra, A.; Giedroc, D.; Merz, K., Jr., Solution Nmr Refinement of a Metal Ion Bound Protein Using Metal Ion Inclusive Restrained Molecular Dynamics Methods. J. Biomol. NMR 2013, 56, 125-137. 36. Ucisik, M. N.; Chakravorty, D. K.; Merz Jr, K. M., Structure and Dynamics of the N-Terminal Domain of the Cu (I) Binding Protein Cusb. Biochemistry 2013, 52, 6911-6923. 37. †qvist, J., Ion-Water Interaction Potentials Derived from Free Energy Perturbation Simulations. J. Phys. Chem. 1990, 94, 8021-8024. 38. Stote, R. H.; Karplus, M., Zinc Binding in Proteins and Solution: A Simple but Accurate Nonbonded Representation. Proteins: Struct., Funct., Bioinf. 1995, 23, 12-31. 39. Joung, I. S.; Cheatham, T. E., Determination of Alkali and Halide Monovalent Ion Parameters for Use in Explicitly Solvated Biomolecular Simulations. J. Phys. Chem. B 2008, 112, 9020-9041. 40. Merz, K. M., Carbon Dioxide Binding to Human Carbonic Anhydrase Ii. J. Am. Chem. Soc. 1991, 113, 406-411. 41. Chakravorty, D. K.; Wang, B.; Ucisik, M. N.; Merz, K. M., Insight into the Cation" Interaction at the Metal Binding Site of the Copper Metallochaperone Cusf. J. Am. Chem. Soc. 2011, 133, 19330-19333.  213 42. Saxena, A.; Sept, D., Multisite Ion Models That Improve Coordination and Free Energy Calculations in Molecular Dynamics Simulations. J. Chem. Theory Comput. 2013, 9, 3538-3542. 43. Duarte, F.; Bauer, P.; Barrozo, A.; Amrein, B. A.; Purg, M.; †qvist, J.; Kamerlin, S. C. L., Force-Field Independent Metal Parameters Using a Non-Bonded Dummy Model. J. Phys. Chem. B 2014, 118, 4351-4362. 44. Hu, L.; Ryde, U., Comparison of Methods to Obtain Force-Field Parameters for Metal Sites. J. Chem. Theory Comput. 2011, 7, 2452-2463. 45. Hay, B. P., Methods for Molecular Mechanics Modeling of Coordination Compounds. Coord. Chem. Rev. 1993, 126, 177-236. 46. Sakharov, D. V.; Lim, C., Force Fields Including Charge Transfer and Local Polarization Effects: Application to Proteins Containing Multi/Heavy Metal Ions. J. Comput. Chem. 2009, 30, 191-202. 47. Yu, H.; Whitfield, T. W.; Harder, E.; Lamoureux, G.; Vorobyov, I.; Anisimov, V. M.; MacKerell, A. D.; Roux, B. t., Simulating Monovalent and Divalent Ions in Aqueous Solution Using a Drude Polarizable Force Field. J. Chem. Theory Comput. 2010, 6, 774-786. 48. Zhang, J.; Yang, W.; Piquemal, J.-P.; Ren, P., Modeling Structural Coordination and Ligand Binding in Zinc Proteins with a Polarizable Potential. J. Chem. Theory Comput. 2012, 8, 1314-1324. 49. Soniat, M.; Rick, S. W., The Effects of Charge Transfer on the Aqueous Solvation of Ions. J. Chem. Phys. 2012, 137, 044511. 50. Ponomarev, S. Y.; Click, T. H.; Kaminski, G. A., Electrostatic Polarization Is Crucial in Reproducing Cu(I) Interaction Energies and Hydration. J. Phys. Chem. B 2011, 115, 10079-10085. 51. Molina, J. J.; Lectez, S.; Tazi, S.; Salanne, M.; Dufreche, J.-F.; Roques, J.; Simoni, E.; Madden, P. A.; Turq, P., Ions in Solutions: Determining Their Polarizabilities from First-Principles. J. Chem. Phys. 2011, 134, 014511. 52. Jorgensen, W. L., Quantum and Statistical Mechanical Studies of Liquids. 10. Transferable Intermolecular Potential Functions for Water, Alcohols, and Ethers. Application to Liquid Water. J. Am. Chem. Soc. 1981, 103, 335-340. 53. Dang, L. X., Development of Nonadditive Intermolecular Potentials Using Molecular Dynamics: Solvation of Li+ and F- Ions in Polarizable Water. J. Chem. Phys. 1992, 96, 6970-6977. 54. Dang, L. X.; Garrett, B. C., Photoelectron Spectra of the Hydrated Iodine Anion from Molecular Dynamics Simulations. J. Chem. Phys. 1993, 99, 2972-2977.  214 55. Dang, L. X., Free Energies for Association of Cs+ to 18-Crown-6 in Water. A Molecular Dynamics Study Including Counter Ions. Chem. Phys. Lett. 1994, 227, 211-214. 56. Smith, D. E.; Dang, L. X., Computer Simulations of Nacl Association in Polarizable Water. J. Chem. Phys. 1994, 100, 3757-3766. 57. Dang, L. X., Mechanism and Thermodynamics of Ion Selectivity in Aqueous Solutions of 18-Crown-6 Ether: A Molecular Dynamics Study. J. Am. Chem. Soc. 1995, 117, 6954-6960. 58. Chang, T.-M.; Dang, L. X., Detailed Study of Potassium Solvation Using Molecular Dynamics Techniques. J. Phys. Chem. B 1999, 103, 4714-4720. 59. Dang, L. X., Computational Study of Ion Binding to the Liquid Interface of Water. J. Phys. Chem. B 2002, 106, 10388-10394. 60. Peng, T.; Chang, T.-M.; Sun, X.; Nguyen, A. V.; Dang, L. X., Development of Ions-Tip4p-Ew Force Fields for Molecular Processes in Bulk and at the Aqueous Interface Using Molecular Simulations. J. Mol. Liq. 2012, 173, 47-54. 61. Peng, Z.; Ewig, C. S.; Hwang, M.-J.; Waldman, M.; Hagler, A. T., Derivation of Class Ii Force Fields. 4. Van Der Waals Parameters of Alkali Metal Cations and Halide Anions. J. Phys. Chem. A 1997, 101, 7243-7252. 62. Jensen, K. P.; Jorgensen, W. L., Halide, Ammonium, and Alkali Metal Ion Parameters for Modeling Aqueous Solutions. J. Chem. Theory Comput. 2006, 2, 1499-1509. 63. Lamoureux, G.; Roux, B., Absolute Hydration Free Energy Scale for Alkali and Halide Ions Established from Simulations with a Polarizable Force Field. J. Phys. Chem. B 2006, 110, 3308-3322. 64. Babu, C. S.; Lim, C., Empirical Force Fields for Biologically Active Divalent Metal Cations in Water€. J. Phys. Chem. A 2005, 110, 691-699. 65. Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L., Comparison of Simple Potential Functions for Simulating Liquid Water. J. Chem. Phys. 1983, 79, 926-935. 66. Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P., The Missing Term in Effective Pair Potentials. J. Phys. Chem. 1987, 91, 6269-6271. 67. Horn, H. W.; Swope, W. C.; Pitera, J. W.; Madura, J. D.; Dick, T. J.; Hura, G. L.; Head-Gordon, T., Development of an Improved Four-Site Water Model for Biomolecular Simulations: Tip4p-Ew. J. Chem. Phys. 2004, 120, 9665-9678.  215 68. Horinek, D.; Mamatkulov, S. I.; Netz, R. R., Rational Design of Ion Force Fields Based on Thermodynamic Solvation Properties. J. Chem. Phys. 2009, 130, 124507. 69. Deublein, S.; Vrabec, J.; Hasse, H., A Set of Molecular Models for Alkali and Halide Ions in Aqueous Solution. J. Chem. Phys. 2012, 136, 084501. 70. Reiser, S.; Deublein, S.; Vrabec, J.; Hasse, H., Molecular Dispersion Energy Parameters for Alkali and Halide Ions in Aqueous Solution. J. Chem. Phys. 2014, 140, 044504. 71. Reif, M. M.; Hunenberger, P. H., Computation of Methodology-Independent Single-Ion Solvation Properties from Molecular Simulations. Iv. Optimized Lennard-Jones Interaction Parameter Sets for the Alkali and Halide Ions in Water. J. Chem. Phys. 2011, 134, 144104. 72. †qvist, J., Comment on "Transferability of Ion Models". J. Phys. Chem. 1994, 98, 8253-8255. 73. Marrone, T. J.; Merz, K. M., Transferability of Ion Models. J. Phys. Chem. 1993, 97, 6524-6529. 74. Marrone, T. J.; Merz, K. M., Reply to Comment on "Transferability of Ion Models". J. Phys. Chem. 1994, 98, 8256-8257. 75. Fyta, M.; Netz, R. R., Ionic Force Field Optimization Based on Single-Ion and Ion-Pair Solvation Properties: Going Beyond Standard Mixing Rules. J. Chem. Phys. 2012, 136, 124103. 76. Toukmaji, A.; Sagui, C.; Board, J.; Darden, T., Efficient Particle-Mesh Ewald Based Approach to Fixed and Induced Dipolar Interactions. J. Chem. Phys. 2000, 113, 10913-10927. 77. Petersen, H. G., Accuracy and Efficiency of the Particle Mesh Ewald Method. J. Chem. Phys. 1995, 103, 3668-3679. 78. Cheatham, T. E., III; Miller, J. L.; Fox, T.; Darden, T. A.; Kollman, P. A., Molecular Dynamics Simulations on Solvated Biomolecular Systems: The Particle Mesh Ewald Method Leads to Stable Trajectories of DNA, Rna, and Proteins. J. Am. Chem. Soc. 1995, 117, 4193-4194. 79. Darden, T.; York, D.; Pedersen, L., Particle Mesh Ewald: An Nlog(N) Method for Ewald Sums in Large Systems. J. Chem. Phys. 1993, 98, 10089-10092. 80. Beutler, T. C.; Mark, A. E.; van Schaik, R. C.; Gerber, P. R.; van Gunsteren, W. F., Avoiding Singularities and Numerical Instabilities in Free Energy Calculations Based on Molecular Simulations. Chem. Phys. Lett. 1994, 222, 529-539.  216 81. Steinbrecher, T.; Mobley, D. L.; Case, D. A., Nonlinear Scaling Schemes for Lennard-Jones Interactions in Free Energy Calculations. J. Chem. Phys. 2007, 127, 214108. 82. Kollman, P., Free Energy Calculations: Applications to Chemical and Biochemical Phenomena. Chem. Rev. 1993, 93, 2395-2417. 83. Hummer, G.; Szabo, A., Calculation of Free-Energy Differences from Computer Simulations of Initial and Final States. J. Chem. Phys. 1996, 105, 2004-2010. 84. Mitchell, M. J.; McCammon, J. A., Free Energy Difference Calculations by Thermodynamic Integration: Difficulties in Obtaining a Precise Value. J. Comput. Chem. 1991, 12, 271-275. 85. Simonson, T.; Carlsson, J.; Case, D. A., Proton Binding to Proteins:0 Pka Calculations with Explicit and Implicit Solvent Models. J. Am. Chem. Soc. 2004, 126, 4167-4180. 86. Mezei, M., The Finite Difference Thermodynamic Integration, Tested on Calculating the Hydration Free Energy Difference between Acetone and Dimethylamine in Water. J. Chem. Phys. 1987, 86, 7084-7088. 87. Straatsma, T. P.; Berendsen, H. J. C., Free Energy of Ionic Hydration: Analysis of a Thermodynamic Integration Technique to Evaluate Free Energy Differences by Molecular Dynamics Simulations. J. Chem. Phys. 1988, 89, 5876-5886. 88. Stokes, R., The Van Der Waals Radii of Gaseous Ions of the Noble Gas Structure in Relation to Hydration Energies. J. Am. Chem. Soc. 1964, 86, 979-982. 89. Cornell, W. D.; Cieplak, P.; Bayly, C. I.; Gould, I. R.; Merz, K. M.; Ferguson, D. M.; Spellmeyer, D. C.; Fox, T.; Caldwell, J. W.; Kollman, P. A., A Second Generation Force Field for the Simulation of Proteins, Nucleic Acids, and Organic Molecules. J. Am. Chem. Soc. 1995, 117, 5179-5197. 90. MacKerell, A. D.; Bashford, D.; Bellott, M.; Dunbrack, R.; Evanseck, J.; Field, M. J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S. a., All-Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins. J. Phys. Chem. B 1998, 102, 3586-3616. 91. Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J., Development and Testing of the Opls All-Atom Force Field on Conformational Energetics and Properties of Organic Liquids. J. Am. Chem. Soc. 1996, 118, 11225-11236. 92. van Gunsteren, W. F.; Daura, X.; Mark, A. E. Gromos Force Field. In Encyclopedia of Computational Chemistry; John Wiley & Sons, Ltd: 2002. 93. Panteva, M. T.; Giamba1u, G. M.; York, D. M., Comparison of Structural, Thermodynamic, Kinetic and Mass Transport Properties of Mg2+ Ion Models Commonly Used in Biomolecular Simulations. J. Comput. Chem. 2015, 36, 970-982.  217 94. Vedani, A.; Dobler, M.; Dunitz, J. D., An Empirical Potential Function for Metal Centers: Application to Molecular Mechanics Calculations on Metalloproteins. J. Comput. Chem. 1986, 7, 701-710. 95. Nakagawa, I.; Shimanouchi, T., Infrared Absorption Spectra of Aquo Complexes and the Nature of Co-Ordination Bonds. Spectrochimica Acta 1964, 20, 429-439. 96. Sacconi, L.; Sabatini, A.; Gans, P., Infrared Spectra from 80 to 2000 Cm-1 of Some Metal-Ammine Complexes. Inorg. Chem. 1964, 3, 1772-1774. 97. Pulay, P.; Fogarasi, G.; Pang, F.; Boggs, J. E., Systematic Ab Initio Gradient Calculation of Molecular Geometries, Force Constants, and Dipole Moment Derivatives. J. Am. Chem. Soc. 1979, 101, 2550-2560. 98. Seminario, J. M., Calculation of Intramolecular Force Fields from SecondDerivative Tensors. Int. J. Quantum Chem. 1996, 60, 1271-1277. 99. Nilsson, K.; Lecerof, D.; Sigfridsson, E.; Ryde, U., An Automatic Method to Generate Force-Field Parameters for Hetero-Compounds. Acta Crystallographica Section D: Biological Crystallography 2003, 59, 274-289. 100. Lin, F.; Wang, R., Systematic Derivation of Amber Force Field Parameters Applicable to Zinc-Containing Systems. J. Chem. Theory Comput. 2010, 6, 1852-1870. 101. Chakravorty, D. K.; Wang, B.; Lee, C. W.; Giedroc, D. P.; Merz Jr, K. M., Simulations of Allosteric Motions in the Zinc Sensor Czra. J. Am. Chem. Soc. 2011, 134, 3367-3376. 102. Chakravorty, D. K.; Wang, B.; Ucisik, M. N.; Merz Jr, K. M., Insight into the Cation- Interaction at the Metal Binding Site of the Copper Metallochaperone Cusf. J. Am. Chem. Soc. 2011, 133, 19330-19333. 103. Case, D. A.; Berryman, J. T.; Betz, R. M.; Cerutti, D. S.; Cheatham III, T. E.; Darden, T. A.; Duke, R. E.; Giese, T. J.; Gohlke, H.; Goetz, A. W.; Homeyer, N.; Izadi, S.; Janowski, P.; Kaus, J.; Kovalenko, A.; Lee, T. S.; LeGrand, S.; Li, P.; Luchko, T.; Luo, R.; Madej, B.; Merz Jr, K. M.; Monard, G.; Needham, P.; Nguyen, H.; Nguyen, H. T.; Omelyan, I.; Onufriev, A.; Roe, D. R.; Roitberg, A.; Salomon-Ferrer, R.; Simmerling, C. L.; Smith, W.; Swails, J.; Walker, R. C.; Wang, J.; Wolf, R. M.; Wu, X.; York, D. M.; Kollman, P. A. Amber 2015, University of California, San Francisco: 2015. 104. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; J. A. Montgomery, J.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J.  218 J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03, Revision E. 01, Gaussian Inc.: Wallingford, CT, 2004. 105. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S., General Atomic and Molecular Electronic Structure System. J. Comput. Chem. 1993, 14, 1347-1363. 106. Sousa da Silva, A. W.; Vranken, W. F., Acpype - Antechamber Python Parser Interface. BMC Res. Notes 2012, 5, 1-8. 107. Brooks, B. R.; Brooks, C. L.; MacKerell, A. D.; Nilsson, L.; Petrella, R. J.; Roux, B.; Won, Y.; Archontis, G.; Bartels, C.; Boresch, S., Charmm: The Biomolecular Simulation Program. J. Comput. Chem. 2009, 30, 1545-1614. 108. van der Spoel, D.; Lindahl, E.; Hess, B.; Groenhof, G.; Mark, A. E.; Berendsen, H. J., Gromacs: Fast, Flexible, and Free. J. Comput. Chem. 2005, 26, 1701-1718. 109. Ma, Z.; Jacobsen, F. E.; Giedroc, D. P., Coordination Chemistry of Bacterial Metal Transport and Sensing. Chem. Rev. 2009, 109, 4644-4681. 110. Su, C.-C.; Long, F.; Zimmermann, M. T.; Rajashankar, K. R.; Jernigan, R. L.; Edward, W. Y., Crystal Structure of the Cusba Heavy-Metal Efflux Complex of Escherichia Coli. Nature 2011, 470, 558-562. 111. Chakravorty, D. K.; Wang, B.; Ucisik, M. N.; Merz Jr, K. M., Insight into the Cation"  Interaction at the Metal Binding Site of the Copper Metallochaperone Cusf. J. Am. Chem. Soc. 2011, 133, 19330-19333. 112. Kim, E.-H.; Nies, D. H.; McEvoy, M. M.; Rensing, C., Switch or Funnel: How Rnd-Type Transport Systems Control Periplasmic Metal Homeostasis. J. Bacteriol. 2011, 193, 2381-2387. 113. ChacŠn, K. N.; Mealman, T. D.; McEvoy, M. M.; Blackburn, N. J., Tracking Metal Ions through a Cu/Ag Efflux Pump Assigns the Functional Roles of the Periplasmic Proteins. Proceedings of the National Academy of Sciences 2014, 111, 15373-15378. 114. Mealman, T. D.; Zhou, M.; Affandi, T.; ChacŠn, K. N.; Aranguren, M. E.; Blackburn, N. J.; Wysocki, V. H.; McEvoy, M. M., N-Terminal Region of Cusb Is Sufficient for Metal Binding and Metal Transfer with the Metallochaperone Cusf. Biochemistry 2012, 51, 6767-6775.  219 115. Marcus, Y., Thermodynamics of Solvation of Ions. Part 5.-Gibbs Free Energy of Hydration at 298.15 K. J. Chem. Soc., Faraday Trans. 1991, 87, 2995-2999. 116. Schmid, R.; Miah, A. M.; Sapunov, V. N., A New Table of the Thermodynamic Quantities of Ionic Hydration: Values and Some Applications (Enthalpy-Entropy Compensation and Born Radii). Phys. Chem. Chem. Phys. 2000, 2, 97-102. 117. Tawa, G. J.; Topol, I. A.; Burt, S. K.; Caldwell, R. A.; Rashin, A. A., Calculation of the Aqueous Solvation Free Energy of the Proton. J. Chem. Phys. 1998, 109, 4852-4863. 118. Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.; Earhart, A. D.; Coe, J. V.; Tuttle, T. R., The Proton's Absolute Aqueous Enthalpy and Gibbs Free Energy of Solvation from Cluster-Ion Solvation Data. J. Phys. Chem. A 1998, 102, 7787-7794. 119. Fawcett, W. R., Thermodynamic Parameters for the Solvation of Monatomic Ions in Water. J. Phys. Chem. B 1999, 103, 11181-11185. 120. Zhan, C.-G.; Dixon, D. A., Absolute Hydration Free Energy of the Proton from First-Principles Electronic Structure Calculations. J. Phys. Chem. A 2001, 105, 11534-11540. 121. Palascak, M. W.; Shields, G. C., Accurate Experimental Values for the Free Energies of Hydration of H+, Oh-, and H3o+. J. Phys. Chem. A 2004, 108, 3692-3694. 122. Grossfield, A.; Ren, P.; Ponder, J. W., Ion Solvation Thermodynamics from Simulation with a Polarizable Force Field. J. Am. Chem. Soc. 2003, 125, 15671-15682. 123. Marcus, Y., Ionic Radii in Aqueous Solutions. Chem. Rev. 1988, 88, 1475-1498. 124. Persson, I.; Jalilehvand, F.; Sandstrım, M., Structure of the Solvated Thallium (I) Ion in Aqueous, Dimethyl Sulfoxide, N, N'-Dimethylpropyleneurea, and N, N-Dimethylthioformamide Solution. Inorg. Chem. 2002, 41, 192-197. 125. Shannon, R., Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 1976, 32, 751-767. 126. Vchirawongkwin, V.; Hofer, T. S.; Randolf, B. R.; Rode, B. M., Tl (I)the Strongest StructureBreaking Metal Ion in Water? A Quantum Mechanical/Molecular Mechanical Simulation Study. J. Comput. Chem. 2007, 28, 1006-1016. 127. Burda, J. V.; Pavelka, M.; /im⁄nek, M., Theoretical Model of Copper Cu (I)/Cu (Ii) Hydration. Dft and Ab Initio Quantum Chemical Study. J. Mol. Struct.: THEOCHEM 2004, 683, 183-193.  220 128. Detellier, C.; Laszlo, P., Role of Alkali Metal and Ammonium Cations in the Self-Assembly of the 5'-Guanosine Monophosphate Dianion. J. Am. Chem. Soc. 1980, 102, 1135-1141. 129. Pauling, L., The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry. Cornell University Press: 1960; Vol. 18. 130. Jorgensen, W. L.; Gao, J., Monte Carlo Simulations of the Hydration of Ammonium and Carboxylate Ions. J. Phys. Chem. 1986, 90, 2174-2182. 131. Brug”, F.; Bernasconi, M.; Parrinello, M., Ab Initio Simulation of Rotational Dynamics of Solvated Ammonium Ion in Water. J. Am. Chem. Soc. 1999, 121, 10883-10888. 132. Meraj, G.; Chaudhari, A., Hydrogen Bonded Hydrated Hydronium and Zl Ion Complexes. J. Mol. Liq. 2014, 190, 1-5. 133. Sobolewski, A. L.; Domcke, W., Ab Initio Investigation of the Structure and Spectroscopy of Hydronium-Water Clusters. J. Phys. Chem. A 2002, 106, 4158-4167. 134. Markus Blauth, C.; Pribil, A. B.; Randolf, B. R.; Rode, B. M.; Hofer, T. S., Structure and Dynamics of Hydrated Ag+: An Ab Initio Quantum Mechanical/Charge Field Simulation. Chem. Phys. Lett. 2010, 500, 251-255. 135. Blumberger, J.; Bernasconi, L.; Tavernelli, I.; Vuilleumier, R.; Sprik, M., Electronic Structure and Solvation of Copper and Silver Ions: A Theoretical Picture of a Model Aqueous Redox Reaction. J. Am. Chem. Soc. 2004, 126, 3928-3938. 136. Sham, T., Electronic Structure of Hydrated Ce4+ Ions in Solution: An X-Ray-Absorption Study. Phys. Rev. B 1989, 40, 6045-6051. 137. Ankudinov, A.; Conradson, S.; de Leon, J. M.; Rehr, J., Relativistic Xanes Calculations of Pu Hydrates. Phys. Rev. B 1998, 57, 7518-7525. 138. Moll, H.; Denecke, M.; Jalilehvand, F.; Sandstrım, M.; Grenthe, I., Structure of the Aqua Ions and Fluoride Complexes of Uranium (Iv) and Thorium (Iv) in Aqueous Solution an Exafs Study. Inorg. Chem. 1999, 38, 1795-1799. 139. Hagfeldt, C.; Kessler, V.; Persson, I., Structure of the Hydrated, Hydrolysed and Solvated Zirconium (Iv) and Hafnium (Iv) Ions in Water and Aprotic Oxygen Donor Solvents. A Crystallographic, Exafs Spectroscopic and Large Angle X-Ray Scattering Study. Dalton Trans. 2004, 2142-2151. 140. Case, D. A.; Cheatham, T. E.; Darden, T.; Gohlke, H.; Luo, R.; Merz, K. M.; Onufriev, A.; Simmerling, C.; Wang, B.; Woods, R. J., The Amber Biomolecular Simulation Programs. J. Comput. Chem. 2005, 26, 1668-1688.  221 141. Ryckaert, J.-P.; Ciccotti, G.; Berendsen, H. J., Numerical Integration of the Cartesian Equations of Motion of a System with Constraints: Molecular Dynamics of N-Alkanes. J. Comput. Phys. 1977, 23, 327-341. 142. Miyamoto, S.; Kollman, P. A., Settle: An Analytical Version of the Shake and Rattle Algorithm for Rigid Water Models. J. Comput. Chem. 1992, 13, 952-962. 143. Li, P.; Song, L. F.; Merz Jr, K. M., Systematic Parameterization of Monovalent Ions Employing the Nonbonded Model. J. Chem. Theory Comput. 2015, 11, 1645-1657. 144. Li, P.; Roberts, B. P.; Chakravorty, D. K.; Merz Jr, K. M., Rational Design of Particle Mesh Ewald Compatible Lennard-Jones Parameters for +2 Metal Cations in Explicit Solvent. J. Chem. Theory Comput. 2013, 9, 2733-2748. 145. Li, P.; Song, L. F.; Merz Jr, K. M., Parameterization of Highly Charged Metal Ions Using the 12-6-4 Lj-Type Nonbonded Model in Explicit Water. J. Phys. Chem. B 2015, 119, 883-895. 146. Eastman, P.; Pande, V., Openmm: A Hardware-Independent Framework for Molecular Simulations. Comput. Sci. Eng. 2010, 12, 34-39. 147. Eastman, P.; Friedrichs, M. S.; Chodera, J. D.; Radmer, R. J.; Bruns, C. M.; Ku, J. P.; Beauchamp, K. A.; Lane, T. J.; Wang, L.-P.; Shukla, D., Openmm 4: A Reusable, Extensible, Hardware Independent Library for High Performance Molecular Simulation. J. Chem. Theory Comput. 2012, 9, 461-469. 148. Gordon, J. C.; Myers, J. B.; Folta, T.; Shoja, V.; Heath, L. S.; Onufriev, A., H++: A Server for Estimating Pkas and Adding Missing Hydrogens to Macromolecules. Nucleic Acids Res. 2005, 33, W368-W371. 149. Stone, J. E.; Hardy, D. J.; Ufimtsev, I. S.; Schulten, K., Gpu-Accelerated Molecular Modeling Coming of Age. J. Mol. Graphics Model. 2010, 29, 116-125. 150. Yang, J.; Wang, Y.; Chen, Y., Gpu Accelerated Molecular Dynamics Simulation of Thermal Conductivities. J. Comput. Phys. 2007, 221, 799-804. 151. Stone, J. E.; Phillips, J. C.; Freddolino, P. L.; Hardy, D. J.; Trabuco, L. G.; Schulten, K., Accelerating Molecular Modeling Applications with Graphics Processors. J. Comput. Chem. 2007, 28, 2618-2640. 152. Rodrigues, C. I.; Hardy, D. J.; Stone, J. E.; Schulten, K.; Hwu, W.-M. W., In Proceedings of the 5th conference on Computing frontiers; ACM: Ischia, Italy, 2008, pp 273-282. 153. Preis, T.; Virnau, P.; Paul, W.; Schneider, J. J., Gpu Accelerated Monte Carlo Simulation of the 2d and 3d Ising Model. J. Comput. Phys. 2009, 228, 4468-4477.  222 154. Liu, W.; Schmidt, B.; Voss, G.; Mler-Wittig, W., Accelerating Molecular Dynamics Simulations Using Graphics Processing Units with Cuda. Comput. Phys. Commun. 2008, 179, 634-641. 155. Liu, W.; Schmidt, B.; Voss, G.; Mler-Wittig, W. Molecular Dynamics Simulations on Commodity Gpus with Cuda High Performance Computing Ð Hipc 2007. In Aluru, S.; Parashar, M.; Badrinath, R.; Prasanna, V., Eds.; Springer Berlin / Heidelberg: 2007; Vol. 4873, pp 185-196. 156. Friedrichs, M. S.; Eastman, P.; Vaidyanathan, V.; Houston, M.; Legrand, S.; Beberg, A. L.; Ensign, D. L.; Bruns, C. M.; Pande, V. S., Accelerating Molecular Dynamic Simulation on Graphics Processing Units. J. Comput. Chem. 2009, 30, 864-872. 157. Netz, R. R.; Horinek, D., Progress in Modeling of Ion Effects at the Vapor/Water Interface. Annu. Rev. Phys. Chem. 2012, 63, 401-418. 158. Jones, J. E., On the Determination of Molecular Fields. Ii. From the Equation of State of a Gas. Proc. R. Soc. London, A 1924, 106, 463-477. 159. Rad‚tfisig, A.; Smirnov, B., Reference Data on Atoms, Molecules, and Ions. Springer-Verlag (Berlin and New York): 1985. 160. Ramaniah, L. M.; Bernasconi, M.; Parrinello, M., Ab Initio Molecular-Dynamics Simulation of K+ Solvation in Water. J. Chem. Phys. 1999, 111, 1587-1591. 161. Soper, A., Orientational Correlation Function for Molecular Liquids: The Case of Liquid Water. J. Chem. Phys. 1994, 101, 6888-6901. 162. Ohtaki, H.; Radnai, T., Structure and Dynamics of Hydrated Ions. Chem. Rev. 1993, 93, 1157-1204. 163. Kuznetsov, A.; Zueva, E.; Masliy, A.; Krishtalik, L., Redox Potential of the Rieske Iron-Sulfur Protein Quantum-Chemical and Electrostatic Study. Biochim. Biophys. Acta 2010, 1797, 347-359. 164. Mou2ka, F.; Nezbeda, I.; Smith, W. R., Molecular Simulation of Aqueous Electrolytes: Water Chemical Potential Results and Gibbs-Duhem Equation Consistency Tests. J. Chem. Phys. 2013, 139, 124505. 165. Rajamani, S.; Ghosh, T.; Garde, S., Size Dependent Ion Hydration, Its Asymmetry, and Convergence to Macroscopic Behavior. J. Chem. Phys. 2004, 120, 4457-4466. 166. Mukhopadhyay, A.; Fenley, A. T.; Tolokh, I. S.; Onufriev, A. V., Charge Hydration Asymmetry: The Basic Principle and How to Use It to Test and Improve Water Models. J. Phys. Chem. B 2012, 116, 9776-9783.  223 167. Mukhopadhyay, A.; Aguilar, B. H.; Tolokh, I. S.; Onufriev, A. V., Introducing Charge Hydration Asymmetry into the Generalized Born Model. J. Chem. Theory Comput. 2014, 10, 1788-1794. 168. Piquemal, J.-P.; Perera, L.; Cisneros, G. A.; Ren, P.; Pedersen, L. G.; Darden, T. A., Towards Accurate Solvation Dynamics of Divalent Cations in Water Using the Polarizable Amoeba Force Field: From Energetics to Structure. J. Chem. Phys. 2006, 125, 054511. 169. Jiao, D.; King, C.; Grossfield, A.; Darden, T. A.; Ren, P., Simulation of Ca2+ and Mg2+ Solvation Using Polarizable Atomic Multipole Potential. J. Phys. Chem. B 2006, 110, 18553-18559. 170. Dang, L. X., Development of Nonadditive Intermolecular Potentials Using Molecular Dynamics: Solvation of Li[Sup + ] and F[Sup - ] Ions in Polarizable Water. J. Chem. Phys. 1992, 96, 6970-6977. 171. Sakharov, D. V.; Lim, C., Zn Protein Simulations Including Charge Transfer and Local Polarization Effects. J. Am. Chem. Soc. 2005, 127, 4921-4929. 172. Wu, R.; Lu, Z.; Cao, Z.; Zhang, Y., A Transferable Nonbonded Pairwise Force Field to Model Zinc Interactions in Metalloproteins. J. Chem. Theory Comput 2011, 7, 433-443. 173. Bayly, C. I.; Cieplak, P.; Cornell, W.; Kollman, P. A., A Well-Behaved Electrostatic Potential Based Method Using Charge Restraints for Deriving Atomic Charges: The Resp Model. J. Phys. Chem. 1993, 97, 10269-10280. 174. Li, P.; Merz Jr, K. M., Taking into Account the Ion-Induced Dipole Interaction in the Nonbonded Model of Ions. J. Chem. Theory Comput. 2014, 10, 289-297. 175. Word, J. M.; Lovell, S. C.; Richardson, J. S.; Richardson, D. C., Asparagine and Glutamine: Using Hydrogen Atom Contacts in the Choice of Side-Chain Amide Orientation. J. Mol. Biol. 1999, 285, 1735-1747. 176. Thompson, W. H.; Hynes, J. T., Frequency Shifts in the Hydrogen-Bonded Oh Stretch in Halide-Water Clusters. The Importance of Charge Transfer. J. Am. Chem. Soc. 2000, 122, 6278-6286. 177. Lee, H. M.; Kim, D.; Kim, K. S., Structures, Spectra, and Electronic Properties of Halide-Water Pentamers and Hexamers, X"(H2o) 5, 6 (X= F, Cl, Br, I): Ab Initio Study. J. Chem. Phys. 2002, 116, 5509-5520. 178. Pavlov, M.; Siegbahn, P. E.; Sandstrım, M., Hydration of Beryllium, Magnesium, Calcium, and Zinc Ions Using Density Functional Theory. J. Phys. Chem. A 1998, 102, 219-228.  224 179. Irving, H.; Williams, R., 637. The Stability of Transition-Metal Complexes. J. Chem. Soc. (Resumed) 1953, 3192-3210. 180. Gorelsky, S. I.; Basumallick, L.; Vura-Weis, J.; Sarangi, R.; Hodgson, K. O.; Hedman, B.; Fujisawa, K.; Solomon, E. I., Spectroscopic and Dft Investigation of [M{Hb(3,5-Ipr2pz)3}(Sc6f5)](M=Mn, Fe, Co, Ni, Cu, and Zn) Model Complexes: Periodic Trends in Metal-Thiolate Bonding. Inorg. Chem. 2005, 44, 4947-4960. 181. Tai, H.-C.; Lim, C., Computational Studies of the Coordination Stereochemistry, Bonding, and Metal Selectivity of Mercury. J. Phys. Chem. A 2006, 110, 452-462. 182. Beattie, J. K.; Best, S. P.; Skelton, B. W.; White, A. H., Structural Studies on the Caesium Alums, Csmiii[So4]2312h2o. J. Chem. Soc., Dalton Trans. 1981, 2105-2111. 183. Bard, A. J.; Parsons, R.; Jordan, J., Standard Potentials in Aqueous Solution. CRC press: 1985; Vol. 6. 184. Moin, S. T.; Hofer, T. S.; Pribil, A. B.; Randolf, B. R.; Rode, B. M., A Quantum Mechanical Charge Field Molecular Dynamics Study of Fe2+ and Fe3+ Ions in Aqueous Solutions. Inorg. Chem. 2010, 49, 5101-5106. 185. Hofer, T. S.; Randolf, B. R.; Rode, B. M., Al (Iii) Hydration Revisited. An Ab Initio Quantum Mechanical Charge Field Molecular Dynamics Study. J. Phys. Chem. B 2008, 112, 11726-11733. 186. Canaval, L. R.; Sakwarathorn, T.; Rode, B. M.; Messner, C. B.; Lutz, O. M.; Bonn, G. n. K., Erbium (Iii) in Aqueous Solution: An Ab Initio Molecular Dynamics Study. J. Phys. Chem. B 2013, 117, 15151-15156. 187. Jalilehvand, F.; Sp„ngberg, D.; Lindqvist-Reis, P.; Hermansson, K.; Persson, I.; Sandstrım, M., Hydration of the Calcium Ion. An Exafs, Large-Angle X-Ray Scattering, and Molecular Dynamics Simulation Study. J. Am. Chem. Soc. 2000, 123, 431-441. 188. Martin, B.; Richardson, F. S., Lanthanides as Probes for Calcium in Biological Systems. Q. Rev. Biophys. 1979, 12, 181-209. 189. Cotton, S., Lanthanide and Actinide Chemistry. John Wiley & Sons: 2007; Vol. 27. 190. Habenschuss, A.; Spedding, F. H., The Coordination (Hydration) of Rare Earth Ions in Aqueous Chloride Solutions from X Ray Diffraction. I.Tbcl3,Dycl3, Ercl3,Tmcl3,and Lucl3. J. Chem. Phys. 1979, 70, 2797-2806. 191. Habenschuss, A.; Spedding, F. H., The Coordination (Hydration) of Rare Earth Ions in Aqueous Chloride Solutions from XRay Diffraction. Iii. Smcl3, Eucl3, and Series Behavior. J. Chem. Phys. 1980, 73, 442-450.  225 192. Habenschuss, A.; Spedding, F. H., The Coordination (Hydration) of Rare Earth Ions in Aqueous Chloride Solutions from XRay Diffraction. Ii. Lacl3, Prcl3, and Ndcl3a). J. Chem. Phys. 1979, 70, 3758-3763. 193. Yamaguchi, T.; Nomura, M.; Wakita, H.; Ohtaki, H., An Extended XRay Absorption Fine Structure Study of Aqueous Rare Earth Perchlorate Solutions in Liquid and Glassy States. J. Chem. Phys. 1988, 89, 5153-5159. 194. Kanno, H.; Hiraishi, J., Raman Spectroscopic Evidence for a Discrete Change in Coordination Number of Rare Earth Aquo-Ions in the Middle of the Series. Chem. Phys. Lett. 1980, 75, 553-556. 195. DÕAngelo, P.; Zitolo, A.; Migliorati, V.; Chillemi, G.; Duvail, M.; Vitorge, P.; Abadie, S.; Spezia, R., Revised Ionic Radii of Lanthanoid (Iii) Ions in Aqueous Solution. Inorg. Chem. 2011, 50, 4572-4579. 196. D'Angelo, P.; Zitolo, A.; Migliorati, V.; Persson, I., Analysis of the Detailed Configuration of Hydrated Lanthanoid (Iii) Ions in Aqueous Solution and Crystalline Salts by Using K- and L3-Edge Xanes Spectroscopy. Chem. Eur. J. 2010, 16, 684-692. 197. Taylor, D. M., The Bioinorganic Chemistry of Actinides in Blood. J. Alloys Compd. 1998, 271, 6-10. 198. Canaval, L. R.; Weiss, A. K.; Rode, B. M., Structure and Dynamics of the Th4+-Ion in Aqueous SolutionÐan Ab Initio Qmcf-Md Study. Comp. Theor. Chem. 2013, 1022, 94-102. 199. Kirm, M.; Krupa, J.; Makhov, V.; Negodin, E.; Zimmerer, G.; Gesland, J., 6d5f and 5f2 Configurations of U4+ Doped into Liyf4 and Yf3 Crystals. J. Lumin. 2003, 104, 85-92. 200. Frick, R. J.; Pribil, A. B.; Hofer, T. S.; Randolf, B. R.; Bhattacharjee, A.; Rode, B. M., Structure and Dynamics of the U4+ Ion in Aqueous Solution: An Ab Initio Quantum Mechanical Charge Field Molecular Dynamics Study. Inorg. Chem. 2009, 48, 3993-4002. 201. Odoh, S. O.; Bylaska, E. J.; de Jong, W. A., Coordination and Hydrolysis of Plutonium Ions in Aqueous Solution Using CarÐParrinello Molecular Dynamics Free Energy Simulations. J. Phys. Chem. A 2013, 117, 12256-12267. 202. Wilson, R. E.; Skanthakumar, S.; Burns, P. C.; Soderholm, L., Structure of the Homoleptic Thorium (Iv) Aqua Ion [Th(H2o)10]Br4. Angew. Chem. 2007, 119, 8189-8191. 203. Stephens, P.; Jollie, D.; Warshel, A., Protein Control of Redox Potentials of Iron-Sulfur Proteins. Chem. Rev. 1996, 96, 2491-2514. 204. Sono, M.; Roach, M. P.; Coulter, E. D.; Dawson, J. H., Heme-Containing Oxygenases. Chem. Rev. 1996, 96, 2841-2888.  226 205. Eisenberg, D. S.; Kauzmann, W., The Structure and Properties of Water. Oxford University Press: 1969. 206. Dyke, T. R.; Muenter, J. S., Electric Dipole Moments of Low J States of H2o and D2o. J. Chem. Phys. 1973, 59, 3125-3127. 207. Gubskaya, A. V.; Kusalik, P. G., The Total Molecular Dipole Moment for Liquid Water. J. Chem. Phys. 2002, 117, 5290-5302. 208. Silvestrelli, P. L.; Parrinello, M., Structural, Electronic, and Bonding Properties of Liquid Water from First Principles. J. Chem. Phys. 1999, 111, 3572-3580. 209. Wu, J. C.; Piquemal, J.-P.; Chaudret, R.; Reinhardt, P.; Ren, P., Polarizable Molecular Dynamics Simulation of Zn (Ii) in Water Using the Amoeba Force Field. J. Chem. Theory Comput. 2010, 6, 2059-2070. 210. Ren, P.; Ponder, J. W., Polarizable Atomic Multipole Water Model for Molecular Mechanics Simulation. J. Phys. Chem. B 2003, 107, 5933-5947. 211. Sangster, M.; Atwood, R., Interionic Potentials for Alkali Halides. Ii. Completely Crystal Independent Specification of Born-Mayer Potentials. J. Phys. C 1978, 11, 1541-1555. 212. Wagman, D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, I. The Nbs Tables of Chemical Thermodynamic Properties. Selected Values for Inorganic and C1 and C2 Organic Substances in Si Units; DTIC Document: 1982. 213. Topol, I. A.; Tawa, G. J.; Burt, S. K.; Rashin, A. A., On the Structure and Thermodynamics of Solvated Monoatomic Ions Using a Hybrid Solvation Model. J. Chem. Phys. 1999, 111, 10998-11014. 214. Mejias, J. A.; Lago, S., Calculation of the Absolute Hydration Enthalpy and Free Energy of H+ and Oh-. J. Chem. Phys. 2000, 113, 7306-7316. 215. Asthagiri, D.; Pratt, L. R.; Paulaitis, M. E.; Rempe, S. B., Hydration Structure and Free Energy of Biomolecularly Specific Aqueous Dications, Including Zn2+ and First Transition Row Metals. J. Am. Chem. Soc. 2004, 126, 1285-1289. 216. Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, R. D.; Kale, L.; Schulten, K., Scalable Molecular Dynamics with Namd. J. Comput. Chem. 2005, 26, 1781-1802. 217. Plimpton, S., Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1-19. 218. Wang, J.; Wang, W.; Kollman, P. A.; Case, D. A., Automatic Atom Type and Bond Type Perception in Molecular Mechanical Calculations. J. Mol. Graphics Model. 2006, 25, 247-260.  227 219. Li, P.; Merz Jr, K. M., Mcpb.Py: A Python Based Metal Center Parameter Builder. J. Chem. Inf. Model. 2016, 56, 599-604. 220. Nar, H.; Huber, R.; Messerschmidt, A.; Filippou, A. C.; Barth, M.; Jaquinod, M.; van de Kamp, M.; Canters, G. W., Characterization and Crystal Structure of Zinc Azurin, a by-Product of Heterologous Expression in Escherichia Coli of Pseudomonas Aeruginosa Copper Azurin. Eur. J. Biochem. 1992, 205, 1123-1129. 221. Besler, B. H.; Merz Jr, K. M.; Kollman, P. A., Atomic Charges Derived from Semiempirical Methods. J. Comput. Chem. 1990, 11, 431-439. 222. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Montgomery Jr, J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J. Gaussian 09, Revision C. 01, Gaussian Inc.: Wallingford, CT, 2009. 223. Merz Jr, K. M., Carbon Dioxide Binding to Human Carbonic Anhydrase Ii. J. Am. Chem. Soc. 1991, 113, 406-411. 224. Case, D. A.; Babin, V.; Berryman, J. T.; Betz, R. M.; Cai, Q.; Cerutti, D. S.; Cheatham III, T. E.; Darden, T. A.; Duke, R. E.; Gohlke, H.; Goetz, A. W.; Gusarov, S.; Homeyer, N.; Janowski, P.; Kaus, J.; Kolossv⁄ry, I.; Kovalenko, A.; Lee, T. S.; LeGrand, S.; Luchko, T.; Luo, R.; Madej, B.; Merz Jr, K. M.; Paesani, F.; Roe, D. R.; Roitberg, A.; Sagui, C.; Salomon-Ferrer, R.; Seabra, G.; Simmerling, C. L.; Smith, W.; Swails, J.; Walker, R. C.; Wang, J.; Wolf, R. M.; Wu, X.; Kollman, P. A. Amber 14, University of California, San Francisco: 2014. 225. Gıtz, A. W.; Williamson, M. J.; Xu, D.; Poole, D.; Le Grand, S.; Walker, R. C., Routine Microsecond Molecular Dynamics Simulations with Amber on Gpus. 1. Generalized Born. J. Chem. Theory Comput. 2012, 8, 1542-1555. 226. Salomon-Ferrer, R.; Gıtz, A. W.; Poole, D.; Le Grand, S.; Walker, R. C., Routine Microsecond Molecular Dynamics Simulations with Amber on Gpus. 2. Explicit Solvent Particle Mesh Ewald. J. Chem. Theory Comput. 2013, 9, 3878-3888. 227. Berendsen, H. J.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J., Molecular Dynamics with Coupling to an External Bath. J. Chem. Phys. 1984, 81, 3684-3690.  228 228. Demadis, K. D.; Dattelbaum, D. M.; Kober, E. M.; Concepcion, J. J.; Paul, J. J.; Meyer, T. J.; White, P. S., Vibrational and Structural Mapping of [Os(Bpy)3]3+/2+ and [Os(Phen)3]3+/2+. Inorg. Chim. Acta 2007, 360, 1143-1153. 229. Xue, Y.; Davis, A. V.; Balakrishnan, G.; Stasser, J. P.; Staehlin, B. M.; Focia, P.; Spiro, T. G.; Penner-Hahn, J. E.; O'Halloran, T. V., Cu (I) Recognition Via Cation- and Methionine Interactions in Cusf. Nat. Chem. Biol. 2008, 4, 107-109. 230. Chakravorty, D. K.; Li, P.; Tran, T. T.; Bayse, C. A.; Merz Jr, K. M., Metal Ion Capture Mechanism of a Copper Metallochaperone. Biochemistry 2016, 55, 501-509. 231. Miyanaga, T.; Watanabe, I.; Ikeda, S., Amplitude in Exafs and Ligand Exchange Reaction of Hydrated 3d Transition Metal Complexes. Chem. Lett. 1988, 17, 1073-1076. 232. Smirnov, P. R.; Trostin, V. N., Structural Parameters of Close Surroundings of Sr2+ and Ba2+ Ions in Aqueous Solutions of Their Salts. Russ. J. Gen. Chem. 2011, 81, 282-289. 233. Robinson, R. A.; Stokes, R. H., Electrolyte Solutions. Courier Dover Publications: 2002. 234. Miller, K. J., Additivity Methods in Molecular Polarizability. J. Am. Chem. Soc. 1990, 112, 8533-8542. 235. Applequist, J.; Carl, J. R.; Fung, K.-K., Atom Dipole Interaction Model for Molecular Polarizability. Application to Polyatomic Molecules and Determination of Atom Polarizabilities. J. Am. Chem. Soc. 1972, 94, 2952-2960. 236. Caminiti, R., Complex Formation and Phosphate-H2o Interactions in a Concentrated Aqueous Mg(H2po4)2 Solution. J. Mol. Liq. 1984, 28, 191-204. 237. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.; Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.; Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven, T.; Jr., J. A. M.; Peralta, J. E.; Ogliaro, F.; Bearpark, M.; Heyd, J. J.; Brothers, E.; Kudin, K. N.; Staroverov, V. N.; Keith, T.; Kobayashi, R.; Normand, J.; Raghavachari, K.; Rendell, A.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi, M.; Rega, N.; Millam, J. M.; Klene, M.; Knox, J. E.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Martin, R. L.; Morokuma, K.; Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.; Daniels, A. D.; Farkas, O.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J., Gaussian 09, Revision C.01 Gaussian, Inc., Wallingford CT 2010.