This is to certify that the thesis entitled KINETICS STUDIES OF ION TRANSPORT THROUGH LIPID BILAYER MEMBRANES presented by Thomas Allen Last has been accepted towards fulfillment of the requirements for Ph . D . Chemis try degree in M54 Major professor Date 8/16/77 0-7639 mains-L u.-- 1... 1-. , -- —. LIBRA R Y Michigan State University . \ KINETICS STUDIES OF ION TRANSPORT THROUGH LIPID BILAYER MEMBRANES By Thomas Allen Last A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1977 ABSTRACT KINETICS STUDIES OF ION TRANSPORT THROUGH LIPID BILAYER MEMBRANES By Thomas Allen Last The kinetics of ion transport, through glycerol mono- oleate/n-hexadecane membranes, induced by the actin homologs (monactin and dinactin) were studied using the charge injection technique. Both time dependent and steady state measurements were performed on the transport system. The charge injection technique was found to be par- ticularly valuable for time dependent type measurements, since the forcing function (343;) the charge pulse) is applied quickly (within approximately 100 ns) and thus measurements can be performed on the system almost im- mediately after the perturbation is applied. The method is not as valuable for steady state measurements since the technique causes added complexity in the mathematical expressions describing the transport, and this added complexity tends to offset the advantages which the tech- nique offers. A transient recorder was developed for the purpose Thomas Allen Last of recording the data produced from the charge injection experiments. The maximum recording rate of the transient recorder is 10 MHz. The signal to noise ratio, for an input voltage of 3V, was found to be 378 for a one—shot recording taken at 10 MHz. The recording system was observed to respond to a step voltage change of 3V within 200 ns. The mechanism for the macrotetralide induced ion transport through the membranes was found to be adequately described by the carrier model. Transients recorded at 10 MHz recording rate showed one early relaxation which was due to the rapid (voltage related) increase in the complex molecule translocation rate constant. The rate limiting step for steady state ion transport was found to be the free carrier translocation process. The surface complexation reaction was found to maintain essentially equilibrium conditions at all times except during the early relaxation. ACKNOWLEDGMENTS I would like to express my sincere thanks to Professor Christie Enke for his guidance and friendship during the course of my graduate study. His advice will always be highly valued. Thanks go also to Professor Crouch who served as my second reader, and to the other members of my guidance committee. I wish to thank my fellow members of the Enke research group for their friendship and help. Finally, I wish to express my appreciation to my parents and family for their unfailing support throughout my education. 11 TABLE OF CONTENTS Chapter LIST OF TABLES . . . . . . . . LIST OF FIGURES. . . . . . . . 1. INTRODUCTION . . . . Historical . . . . . . . . Transport Mechanisms . . . The Measurement Problem. . . . Charge Injection Approach. DESCRIPTION OF MEASUREMENT SYSTEM . . . . . . . . . . Pulse Generator. . . . . . . . Electrochemical Cell . Cell Amplifier . . . . . . Analog Storage Register for Fast Transient Recording . . . The Timing Circuit . . The Analog Register. . Performance. . . . . Baseline Correction. . Transient Recording. . . CARRIER TRANSPORT THEORY . . Membrane Energy Barrier. . . . Steady State Approximation Time Dependent Approach. . . . iii Page vi . viii \DU'll-‘H 1h l8 l8 19 21 25 26 26 27 28 29 30 33 37 A3 Chapter A. ROUTINE MEMBRANE EXPERIMENTS . Materials and Methods. Membrane Formation . . Membrane Data Collection . Data Treatment . . . . . . . . MEMBRANE CAPACITANCE MEASUREMENT . Materials, Methods and Results . Discussion and Conclusions . . . STEADY STATE RESULTS.. . . . . . Introduction . . . . . . . . . . Sample Transients. . . . . . . . Discussion of High and Low Field Approximations . . . . . . . . . Data Fitting o . o 0 o o o c o o Equilibrium Assumption . . . . Results and Discussion . . . . Perspective on Steady State Results. TIME DEPENDENT RESULTS Introduction . . . . . . . . . . Data Treatment . . . . . . . . . Results and Conclusions. . . . . Overview of Transport Scheme . . Discussion on Apparent Carrier Buffering Discrepancy. . . . iv Page us as 51 52 53 55 55 60 62 62 53 65 72 7A 79 83 85 85 85 86 102 10“ Chapter 8 O FUTURE WORK 0 O O O O O O O O O O 0 Short Term . . . . . . . . . . . . Long Term. . . . . . . . . . . APPENDIX A - Description of Interface. APPENDIX B - Program Listings. . . . . Program MEMBRN.FTN . . . . . . . . Subroutine COLECT.MAC. . . . . . . Subroutine TH010.MAC . . . . . . . Program KINRLX.FTN . . . . . . . . Subroutine DSTRIP. . . . . . . Subroutine RUNGE . . . . . . . . . Subroutine DIFFEQ. . . . . . . . Subroutine DIFFEQ. . . . . . . . . Program KINPCH.FTN . . . . . . . . Subroutine WRITKF. . . . . . . . . Program EXPFIT.FTN . . . . . . . . Subroutine TLLSQ . . . . . . . . . APPENDIX C - Comparison of Transport REFERENCES Equations Derived in this Work With Those of Other Workers . . . . . . . . . Page 108 108 109 110 128 129 133 135 luO 1U2 1A3 1A4 1AA 1A6 1U? 1A8 1H9 150 151 Table 1-1 6-1 6-2 6-3 6-“ LIST OF TABLES Page Chemical Composition of Membranes . . . . 5 Parameters resulting from curve fitting to the steady state trans- port equation (Equation (3-20)). Duplicate measurements were per- formed on different membranes under identical experimental conditions (monactin transport in GMO/n-hexa- decane membranes) . . . . . . . . . . . . 73 Rate constants for monactin carrier transport in GMO/n-hexadecane membranes. C8 = carrier concentration. I‘t was cal- culated assuming a membrane thickness of 6 nm. (Ionic strength = 3 M) . . . . . . 80 Rate constants for dinactin carrier transport in GMO/n-hexadecane membranes. C8 = carrier concentration. rt was calculated assuming a membrane thickness of 6 nm. (Ionic strength = 3 M.) . . . . 80 Literature rate constants (from Hladky (36)) for nonactin and trinactin carrier transport. (Ionic strength = 1 M) (Note: the units for k1 were changed from those used by the cited author.). . . . . 80 vi Table 7-la 7-lb Results for dinactin. rt = total carrier concentration. Ionic strength = 3M . . . . . . . . . . . Results for monactin. = total Ft carrier concentration. Ionic strength = 392. Theoretical free carrier concentra- tion. . . . . . . . . . . . . . . . . Pin assignments for the experiment end of the Interface card . . . . Unibus Address Assignments. Card Slot assignments . . . . . . . . vii Page 93 93 95 116 117 117 Figure 1-2 1-3 1-5 1-6 LIST OF FIGURES Page Schematic representation of the Davson- Danielli biomolecular lipid membrane model. The polar head groups of the lipids are included in the aqueous phase . . . . . . . . . . . . . . . . . 3 Simplified reaction scheme for car- rier transport. The free carrier is assumed to be membrane bound, and uncharged . . . . . . . . . . . . . . . 8 Electrical representation of electrodes placed in two aqueous compartments separated by a membrane . . . . . . . . ll Simplified representation of experi- ment depicted in Figure l-3 . . . . . . l2 Curves that show how the membrane capacitance affects the voltage step measurement. The ideal cur- rent curve shows a relaxation to steady state conductance. . . . . . . . 13 Membrane capacitance charging curves for the voltage step and high voltage pulse experiments. V0 is the voltage on the membrane capacitance . . . . . . l6 viii Figure Page 2-1 Photograph of electrochemical cell. . . . . . . . . . . . . . . . . . 20 2—2 Cell amplifier schematic. . . . . . . . 22 1 Comparison of transient recorder employing a temporary analog storage register with one which uses all digital storage. . . . . . . . 26 2 Block diagram of a fast transient recorder which employs the SAD 100 (serial analog delay element) . . . . . 26 3 Schematic of the circuit which controls the timing for data col- lection and clocking for the analog storage register. . . . . . . . . . . . 27 A Schematic of circuitry employed for tending the SAD lOO . . . . . . . . . . 27 Response of transient recording \fl systemtxaa 3V at 100 kHz square wave. Data collection rate: 10 MHz . . 28 6 Linearity of the transient recording system over the range of O to U volts . . . . . . . . . . . . . . . . . 28 7 Signal-to-noise ratio of data collected at 10 MHz . . . . . . . . . . 28 8 Recorded decaying exponential transient . . . . . . . . . . . . . . . 28 ix Figure Page 3-1 Mechanistic model for carrier transport . . . . . . . . . . . . . . . 31 3-2 Double Eyring barrier used to represent the membrane barrier to ion transport. . . . . . . . . . . . 35 u—1 Chemical structures of the actin homologs. . . . . . . . . . . . . . . . 50 5-1 Transient obtained from application of a 100 ns pulse to the cell with no membrane present . . . . . . . . . . 57 5-2A Transient resulting from pulse appli- cation to a membrane which contains no carrier (no external resistor) . . . 58 5-28 Transient resulting from pulse applica— tion to a membrane which contains no carrier (no external resistor). . . . . 58 5—3 Transient obtained from pulsing mem- brane (no carrier present) with ex— ternal resistor present. Solid curve results from exponential least squares fit 0 o o o o o o o o o o o o o o o o o 59 7-1 Logarithmic plot of membrane transients. . . . . . . . . . . . . . . 87 7-2 Logarithmic plot of early relaxation. . . . . . . . . . . . . . . 88 Figure 7-3 A-2A A-2B A-7 A-8A A-8B Membrane transient recorded at 1.25 MHz. . . . . . . . . . . . . . Membrane transient recorded at 10 MHz. . . . . . . . . . . . . . Membrane transients with linear fit to middle portion . . . . . . Concentration vs time profiles for various forms of the carrier. . . Block diagram of measurement system. . . . . . . . . . . . . . Schematic showing computer Data Out path. . . . . . . . . . . Schematic showing computer Data In path . . . . . . . . . . . . . . Command Register circuit diagram. Schematic representation of the Address Decoder . . . . . . . . . . Timing section schematic. . . . . Schematic for the Reticon Controller. . . . . . . . Data Driver schematic . . . . . . . Douglas patch card, part A. . . . . Douglas patch card, part B. Modifications to DRll-A . . . . . xi Page 90 91 98 101 111 118 119 120 121 122 123 12“ 125 126 127 CHAPTER 1 INTRODUCTION It is well established that ion transport across membranes is an underlying factor in nerve conduction (1), muscle function (2), conduction of metabolites through cell walls (3,U) and energy assimilation (5). It follows that an investigation of membrane transport phenomena will lead to a better understanding of these important physiological processes. In this research, ion transport through bilayer lipid membranes was studied by kinetics methods and the information received was used to infer the transport mechanism. Historical The first suggestion that cell membranes contained lipids came from Overton (6) in 1899, when he observed that cell membranes were very permeable to these sub- stances. Then in 1917 Langmiur (7) stressed the tendency of lipids to form structured monolayers when placed in contact with water and discussed the forces (Van der Waals) which were responsible for this behavior. Gorter and Grendel (8) suggested in 1925 that the plasma membrane was two lipid molecules thick. Their studies with lipids extracted from mammalian erythrocytes showed that the monolayer formed from the erythrocyte lipids had a surface area 1.8 to 2 times greater than the original erythrocyte surface area. The greatest breakthrough came in 1935 when Danielli and Davson (9) put forth an idealized model for the plasma membrane. This is shown in Figure 1-1. The membrane is depicted as a lamellar bimolecular lipid layer with the polar head groups of the lipid oriented toward the high dielectric aqueous medium which is both inside and outside of the cell. This configuration allows the organic chains of the lipid molecules to be in their own low dielectric environment. Any protein present in the membrane is thought to be adsorbed on the surface of the lipid bilayer. The thickness of membranes of this type would be in the range of 5 to 12 nm, depend- ing on the chain length of the fatty acid used in pre- paring the lipid. This thickness is in good agreement with electron microscopic measurements made on biological membranes. The first successful attempt to prepare a bimolecular lipid membrane (BLM) in vitro was reported by Mueller g§.§l. (10) in 1962, when they formed a BLM by brushing a complex mixture of Brain lipids, n-tetradecane, silicone fluid and mineral oil across a narrow orifice between two aqueous compartments. Although the BLM model has yielded good agreement OVVV/VVIO .‘r/VV/VVb owv/vvo O'r’\/\//\/\n'O ovavvo Figure l-l. Schematic representation of the Davson— Danielli bimolecular lipid membrane model. The polar head groups of the lipids are included in the aqueous phase. with observations of natural membranes, it deviates in two important respects; physical stability and electrical conductivity. Synthetic unmodified BLM's are often ruptured by vibrations and usually last only a few minutes. The electrical resistance of an unmodified BLM is on the order of that of a linear hydrocarbon or about 106 to 108 ohm~cm2. The conductance (or perme- ability to ions) can be increased to biologically realis- tic levels by adding a chemical modifier to enhance the transport properties. This will be discussed in more detail later. To account for the increased strength of biological membranes it has been postulated that protein molecules are somehow incorporated in the BLM as binders. Many configurations of protein-lipid associations have been discussed, leading to the present model (11) of globular proteins situated across the membrane and protruding on either side. It has been suggested (5) that these proteins are responsible for making the membrane asymmet- ric. This asymmetry is a necessary condition for the establishment of an active transport mechanism (iii;’ the transport of a chemical species against its electro- chemical gradient). It is reasonable to expect that proteins would play a significant part in the struc- ture and properties of biological membranes since studies have shown that membranes often contain half protein. The composition of some representative biological membranes is shown in Table 1-1 (12). Table l-l. Chemical Composition of Membranes. Membranes Protein(%) Lipid(%) Carbohydrate(%) Myelin (nerve) 18 79 3 Plasma Membrane Human erythrocyte A9 A3 8 Amoeba 5A A2 A Rat liver cell 58 A2 5-10 Nuclear membrane of rat liver cell 59 35 2.9 Retinal rods, bovine 51 A9 A Mitochondrial outer membrane 52 A8 2-A Chloroplast lamellae, spinach 70 3O 6 Mitochondrial inner membrane 76 2A 1-2 Transport Mechanisms The ion conductance of a BLM can be significantly increased, as was previously mentioned, by adding a chemical modifier. These modifiers have been observed to increase the ion conductance by several orders of magnitude, even though their concentration in the mem- brane phase is in the millimolar range. This enhance- ment of ion transport is thought to occur in one of two ways; pore formation or carrier transport, depending on experimental conditions and the chemical modifier used. Pore formation, as the name implies, results in a pore or hole which extends across the membrane from one inter- face to the other and allows restricted diffusion of ions across the membrane. The environment of the interior of the pore would be similar in nature to the environment in the aqueous solution, except for the restrictions that the pore size might place on the ion. A pore could be formed in the membrane by having long linear (non— lipid) molecules combine to form a helix across the mem- brane, or by having several smaller heterocyclic mole- cules associate in such a way that their centers are aligned giving a path across the membrane. Several investigators (13-15) have observed the formation and dissociation of these channels, and reported the conduc- tance per unit channel. While the pore is stable however, the kinetics of the ion transport would be determined by: a) Electrodiffusion, b) Interaction with charged pore walls, and c) Shedding of the solvent sheath upon entering the pore. The last two parameters would, of course, depend on the pore diameter, and all three would be largely affected by the ion being transported (i¢§;, the ionic radius and the charge). In the carrier transport mechanism model, the ion is envisioned to be shuttled across the membrane by a mobile carrier molecule which is soluble in the lipid phase and capable of forming a fairly stable complex with the ion. This carrier molecule could be a macro- cyclic molecule or a linear molecule capable of chelating with the ion. In either case the carrier must contain hetero—atoms which will complex with the ion, and the like exterior of the complex must appear hydrocarbon-like to the interior of the membrane so that it is still readily soluble in the lipid phase. The mechanism of the carrier— mediated transport has been widely discussed (16-19) and the salient features are shown in schematic form in Figure 1—2. Ions at one interface (1) react with the free carrier (3) to form the complex (is). In the figure, C and I refer to the bulk solution concentration and the surface concentration, respectively. The complex is then transferred to the other interface by electrodiffusion where the ion is again released into the squeous solution. The kinetics of carrier mediated transport are governed by: a) Formation of the ion-carrier complex at the interface, b) Diffusion of the complex across the membrane interior, c) Dissociation of the complex at the other inter- face, and d) Diffusion of the carrier back to the first inter- face. C.+ Figure 1-2. II- ‘1: l l [:‘e —>l:' +Ci Simplified reaction scheme for carrier transport. The free carrier is assumed to be membrane bound, and uncharged. In this case a, b, and d are again dependent on the ion which is being transported. However according to Hladky (16) the step involving the diffusion of the carrier- ion complex across the membrane should be, to a first approximation, independent of the ion being transported (assuming ions of equal charge). This is easily under- standable since the carrier molecules are usually much larger than the ions being transported and the ion is thought to be completely surrounded by the carrier mole- cule. Thus the ion is essentially invisible to the membrane interior. Thus by making kinetics measurements on a given mem- brane and transport system for a family of ions, it should be possible to discriminate between the two transport mechanisms and verify the mechanisms or postulate new ones. In order to evaluate all of the kinetic parameters however it is often necessary to perform measurements on a short time scale and the difficulties with this type of measurement are discussed in the next section. The Measurement Problem In order to measure fast kinetic parameters it is necessary to use some type of perturbation technique. The most widely used method to date has been the voltage step experiment. This technique has not been totally successful in measuring very fast parameters however, 10 due to the membrane capacitance. The experimental set- up for a voltage step study can be electrically repre- sented by Figure 1—3; where: CM — membrane capacitance, G(V) = membrane conductance (function of volt- age). CI = membrane-solution double layer capacitance, :6 II S solution resistance, 0 II E electrode-solution double layer capacitance, and RF = Faradaic resistance at the electrode. This schematic represents two electrodes placed in two aqueous compartments which are separated by the membrane. Since the electrode area is made much larger than the membrane area the capacitance of the electrode solution interface is large and can be neglected. Further, the solution resistance is kept low by adding electrolyte and is much smaller than the resistance due to the membrane transport. Therefore, the effective electrical representation reduces to that shown in Figure l-A; where C is approximately equal to CM. Therefore if we place a voltage step across the experimental cell and monitor the current arising from G(V) we would observe the transient shown in Figure 1-5. Due to the fact that the membrane must be charged, the perturbation is not a clean transition but rather a growing exponential curve whose time constant depends on the resistance and capacitance mentioned above. This slow growth of the forcing function 11 .mcwspEmE w an cmpmanmm mpcmEpsmo IEoo msomsvm 03» CH pmomaa mmGOLpomHm mo coapmpcmmmpamp awofippomam Ae.>vo L) CC C) .mua magmas 12 G Figure l-A. Simplified representation of equivalent circuit depicted in Figure 1-3. 13 Forcing Function Current o 'G j//f‘\‘~‘I Actual 5 Ideal Current A V 0 TIME —-—> Figure 1—5. Curves that show how the membrane capacitance affects the voltage step measurement. The ideal current curve shows a relaxation to steady state conductance. 1A completely obscures any information in the early part of the relaxation. In a voltage jump experiment we would actually measure the total current passing through the cell, rather than just that current due to G(v,t), so the relaxations would be further obscured by the cur— rent going to charge the membrane capacitance. Charge Injection Approach The membrane capacitance problem has been avoided by applying the charge injection technique (20). This method uses the membrane capacitance to advantage, by Charging the membrane very rapidly and monitoring the voltage decay as the membrane's conductance (ion trans- port) depletes the charge stored on the membrane capaci- tance. The transient produced from this would be a decay- ing exponential if the conductance were under perfect steady state conditions. Relaxations would result in ‘ deviations from the exponential decay, since the early conductance would be larger than for steady state, and it would continually decrease down to the steady state value. The charging of the membrane can be accomplished very quickly by using a pulse generator with a high output voltage compliance and rapid rise time. Since the pulse duration is very short, voltage levels which are in excess, by orders of magnitude, of those that would normally rupture the membrane can be successfully 15 employed. If the pulse generator is capable of opera— tion in the constant current mode, the time necessary to charge the membrane to a given voltage would be given by; t - VC/i, where V is the desired voltage, C is the membrane capacitance, and i is the current during the pulse. If the pulse generator is not capable of current injection, but still has a fairly large output voltage swing,it can be used to advantage in this technique. Even though the membrane capacitance must still be charged through the solution resistance, the time required to charge the mem- brane capacitance to the desired voltage level is sig- nificantly reduced. This is demonstrated in Figure 1-6. When the voltage step is applied in either case, the mem- brane capacitance is charged up through the solution resistance (see Figure l-A). This produces a growing exponential voltage on the capacitor which has a time constant Rng' At the end of one time constant, the veltage on the capacitor is equal to (l-l/e)Vapplied or about 0.63 vapplied‘ If we assume that the desired final voltage in either case is equal to the voltage applied in the voltage step experiment, it is easily seen from Figure 1-6 that the voltage pulse method arrives at the desired voltage excur- sion in much less than one time constant whereas the vol- tage step experiment requires several time constants. In practice this results in an improvement of the charg- ing time from tens of microseconds for the voltage step Inethod, to a hundred nanoseconds for the charge injection 16 Voltage Pulse Voltage Step Figure 1-6. TIME Membrane capacitance charging curves for the voltage step and high voltage pulse experi- ments. VC is the voltage on the membrane capacitance. l7 technique as it was applied in this case. The advantages gained by the current injection method are: l. The fast current injection allows measurements to be made on the system within a few hundred nano- seconds. 2. The complete current xs'voltage curve is obtained from a single experiment. 3. Steady state as well as time dependent data are obtained. A. No current exists in the solution or external circuitry during the measurement. 5. The time course of the decay depends on the ratio of the membrane capacitance to the conductance and therefore is independent of membrane area. 6. The voltage at any time after injection yields the amountcfi'charge which has crossed the membrane since application of the charge pulse. The major disadvantage to the technique is the fact that since the conductance contains both time and voltage dependent factors, the data treatment is mathematically much more difficult than with conventional steady state methods. CHAPTER 2 DESCRIPTION OF MEASUREMENT SYSTEM The charge injection technique, as it was previously described, requires the ability to rapidly inject a charge on the membrane and record the voltage transient produced. The instrument designed to accomplish this measurement is described in this chapter. The chapter is composed of two parts. The first part includes a description of the pulse generator, electrochemical cell and cell amplifier. The second section is an article, published in the Analytical Chemistry journal, which describes the transient recorder. Also related to the measurement system is an appendix (Appendix A) which describes the interface between the measurement system and the PDP ll/AO computer, and is included mainly as a trouble shooting guide. Pulse Generator The Chronetics model PG-33 pulse generator was used for this study. This pulse generator has an output voltage compliance of 112 volts, and a rise time of 6 ns maximum. The device is capable of driving 200 mA into 50 ohms, and can be triggered externally. 18 19 Electrochemical Cell A photograph of the electrochemical cell is shown in Figure 2-1. The cell consists of two rectangular Teflon compartments each having a volume of approximately A5 ml separated by a 0.25 mm Teflon barrier containing a small orifice (dia. 1-2 mm). The barrier was made removable to allow for orifice selection and barrier replacement. Both compartment faces which contact the barrier have face centered 3/8 inch holes (flared toward the inside of the compartment to avoid trapping air bubbles). These holes are surrounded on the outside face by Viton "0" rings (1 in. O.D. x 7/8 in. I.D.) inset in the Teflon with a 0.12 mm protrusion. When the cell is assembled, the barrier is sandwiched by the 0 rings between the two cell compartments which are in turn held together by spring clamps. On the face opposite to the barrier side of each com- partment there is a 1 inch hole with a 3/32 inch lip. A circular glass window (die. 1 inch) is press fit into each compartment. This allows the membrane formed on the ori- face of the barrier to be viewed with a microscope through the window. The membrane is lighted from the rear with polarized light, to give good definition for area measure- ment. Two Ag/AgCl electrodes are pressed into a rectangular piece of Teflon, which, when placed over the solution 2O .Haoo HooHEoQOOLuooHo mo casewOpozm .Hum osswam 21 compartments allows the electrodes to dip into the elec- trolyte solutions. In this fashion the compartments are completely enclosed while the experiment occurs, and yet the cover is easily removed while preparing the membrane on the barrier orifice. The electrodes consist of coiled Ag wires coated with AgCl and have a surface area of about 3.5 cm2. Since the AgCl is depleted by the passage of electrical current, the coating must occasionally be reinforced by anodizing the electrodes in KCl with a Pt counter electrode. It should be noted however that the cell conducts approximately 5 nanocoulombs per transient, and therefore the coating is not depleted rapidly. Cell Amplifier The cell amplifier is shown in schematic form in Figure 2-2. The amplifier is placed as near to the electrodes as possible in order to minimize capacitance and stray signal pickup in the connecting wires. The components are housed in a small aluminum box which rides "piggy back" on the electrochemical cell. The electrodes are attached through a 2 prong Cinch-Jones connector on the bottom of the housing. The input signal in Figure 2-2 comes from the pulse generator via coaxial cable. A toggle switch set for proper pulse polarity, places a set of reverse blocking 22 .m: Ga one moocwpfiowamo .moahmm :H oomom a sfimmzH xm ”a .moflsmm as comma s aamza xm ”m .mwoomq "m .mmoomq ”H .ofipwsonom noHMfiHqu Haoo .mum onswfim AW 1 Foo w, \\W4 .. \\WA xod fins? /._/T a. ixzxzxp. xs yd ihi> 23 diodes (IN5317 and FD300 in series) in the circuit which prevent conduction back through the pulse generator after the pulse is completed. The charge pulse from the pulse generator travels through the solution and charges the membrane which is situated between the two electrodes. The voltage transient produced is amplified by the LH0032 FET input amplifier (gain of X8) to produce a voltage excursion in the range of O-A volts at the output. To avoid saturation of the amplifier during pulse application a set of back-to-back diodes was placed in parallel with the input of the amplifier. These diodes (IN9lA and FD300 in series) prevent the voltage at the input from exceeding approximately 0.7 volts, which is a workable value since voltages in excess of 0.5 volts maintained across the the membrane result in its rupture. The 10 K9 resistor was included to insure that the major current path for the charge pulse remains the membrane pathway. The diodes used for input protection as well as reverse blocking were chosen in combination to yield fast switching characteris- tics as well as low leakage when in their "off" state. The output signal is buffered by a LH0063 cable driver and sent via coaxial cable (terminated with 50 ohms) to the transient recorder which will be discussed in the next section. Power (:15 volts) and common are connected to the cell amplifier by a A prong Cinch-Jones connector located on the side of the housing. The power supply lines are 2A decoupled to common with 33 HF tantalum capacitors at the input, and with smaller ceramic capacitors (see Figure 2-2) near the amplifiers, to allow good transient re- sponse. Reprinted from ANALYTICAL CHEMISTRY, Vol. 49. Page. 19. January 1977 . Copyright 1976 by the American Chemical Society and reprinted by permission of the copyright owner Analog Storage Register for Fast Transient Recording T.A.I.aetandc.G.Eisre' Worm. wsmwmjumm. 48824 Alaettranelentreoerder.mmatemperaryanaleg mafiawaswmypedmm mMaWWWmm-i hmwmmmammmfie maibiommmaummm mbmmmmamemm Mannheim“ Linearity summer betterlian 1% wereebtaliedatreceraigrateeqite 10 Its. There are many measurements (e.g., kinetics studies by perturbation techniques (1)) which require the ability to record the amplitude vs. time function of a short-lived and rapidly varying signal. The most popular solution to this problem has been the “fast digitizer" (2) which converts the incoming analog signal amplitude into a digital data word at regular intervals and then stores the digital words from suc- cessive conversions in a high speed memory. This produces a digital record of the signal amplitude at a number of se- quential points in time. While this digital recording technique can be inexpensively implemented at moderate data rates (up to about 50 KHz), the cost increases rapidly as the speed re- quirements for the analog-to-digital converter (ADC) and digital memory become more demanding. One way to eliminate the need for these high speed com- ponents while maintaining a high recording rate capability is through the useofa temporaryanalogstorage register which canreaddata in atahigh rateand read dataoutatalower rate. A moderately fast data acquisition system is then ade- 25 quate to handle the readout rate from the analog storage register. In fact. the output rate from the analog storage reg- ister can be made slow enough to eliminate the need for any digital storage other than the computer memory in a com- puter-based acquisition system. These two recording schemes are contrasted in Figure l. The transient waveform is clocked very rapidly into the analog memory and then clocked out at a rate compatible with an ordinary ADC. The converted data can then be sent to the computer. at software data transfer rates. where it is stored for later analysis. Since the analog storage register is comparatively inexpensive. the cost of the analog storage transient recorder an be significantly less than a digital storage recorder which has the same recording speed. A transient recorder developed in our laboratory, which utilizes a Reticon SAD 100 serial analog delay element is shown in Figure 2. The timing section of the recorder contains a crystal oscillator which is divided to obtain selectable. ac- curate data clocking rates up to 10 MHz (the analog register’s approximate maximum data rate). This section produces the frequency shifting necessary for changing from the fast clocking rate (while the data are being stored) to the slow rate (while the data are being read out). The Reticon SAD 100 consists of 100 sequentially ad- dressable storage cells contained on a single monolithic inte. grated circuit. Functionally. this is implemented with 2 banks of 50 cells each, which are alternately accessed to increase the storage rate. Each storage cell consists of a read in switch (PET), a read out switch, and a storage capacitor. As the ring counter in each bank is sequenced through its possible states. the nth cell is addressed for read in while the (n + llth cell is ANALYTICAL OVENSTRY. V“. 49. m. 1. JAMMRY 1977 o 19 26 oaaav N '11:... l" T on: "‘ crosses rxsr .’” ADC ’ oeersaav seer ADC sierra; creases I 41 coaeurae converse 'Y00e" crosses I IlIPLAV We tourmalsonottraruontrecordororrployhgamary mmmmmwudrmeaalmw simultaneously addressed for read out. In this manner. data clocked in at any instant will appear at the output 98 clock cycles later. Thus 98 data points can be acquired at a fast rate (up to 10 MHz or 100 as per data point) and then clocked out to the ADC at the appropriate speed while still acquiring data at this slower speed. This is particularly useful for transients of the decaying exponential shape. A primary application of this transient recording system in our laboratory is the study of the transient voltage response produced by pulsing bilipid layer membranes with a brief current pulse. This experiment produces a decaying exponential voltage transient with a time constant ranging from several microseconds to a few milli— wconds. The bandwidth of this type of signal is continuouly decreasing and the early points must be recorded rapidly whereas the later points can be recorded much more slowly. More SAD 100’s could be combined to increase the memory size if desired. Data can be stored in the analog memory for times up to 40 ms with less than 1% retention loss (3). This allows a data acquisition time as long as 400 no per point for the digitisation and storage of the analog-stored data. There are a number of other analog storage devices available (al- though the SAD 100 is at present the fastest). some with up to 1024stmagecells,and otherswithuptob-sretention time. Anexcollentreviswofthetypesofdevicmairrentlyavailable. is given in Reference 4. The Timing Circuit. One important consideration in this schemeoftransientrecordingisthenecessitytobeableto change clocking rates without losing the timing integrity (the accuracy of the time of each data point) of the experiment. Whilerecordingthetramientwaveform itisnecessarytoclock the analog storage register very rapidly. However the recorded data must necessarily be clocked out for digital conversion at some rate slower than the maximum conversion rats. After this clocking frequency shift has been made. the analog reg- ister continues to collect data at the slower rate. In order for the time correlation of the data collected at the two rates to be accurate. the shift from the fast clock rate to the slower clock rate must be made in such a way as to avoid the occur- rence of any spurious clock transitions. A circuit which con- trols the timing of the experiment in this manner is shown in schematic form in Figure 3. The operation of the circuit is as follows: the crystal controlled oscillator frequency (10 MHz) is divided by the synchronous counter to give selectable data in (multiplexer l) and data out (multiplexer 2) clocking rates with the proper phase relationship. These frequencies are pro-selected by the computer. The number of fast data points (up to 98) is pro-set in the 7 bit up/down counter. Prior to the start of the experiment. oscillator gate 1 inhibits clocking the up/down counter, and the fast clocking rate is selected (F172. Q - 0). The analog storage register is in its fast record mode and cycling through its storage cells. This scheme allows a pro-trigger record featuresince, in thewaitingetate, theanalog registeriscontinuouslyrecordingatthefastrate.'l‘hefunction 20 0 AWUIJflflCAL CiIiISTRVK VII” 49.!!1 1..”UIAARY'1977 Al W oodoonoumoccocmouoo. SAD 100 thm“-.bfhl;tw ’ """" m """ m " m on SA!) IOOteeriaiandogdoleyolemem) h- ofthe oscillator gate is to allow only full clock cycles to pass through the gate when it is enabled or inhibited. A detailed description of the operation of the oscillator gate is given in Reference 5. When the transient recorder is armed (FN2 or Manual Start). output Q ofF‘Fngoes to“l" which enables the oscillator gate, and on the next rising clock edge (out of use. gate 1) theexperimentstimulatoristriggered byFF'B,andthe up/down counter begins counting off data points. When the counter reaches 0, the data clock is inhibited (Q of FF2 -r 1) while the synchronous counter is reset. the clocking rate is shifted to the slower rate (FFI, Q - 0) and the sample and hold (S&H) and ADC convert command are enabled. On the clock cycles which follow, data will continue to be collected at the slow rate. while data already collected will be shifted out to the ADC. This triggering approach is useful for those cases where the transient is induced by the recording instru- ment. There are many cases. however, where it is desirable to record an event which occurs spontaneously (e.g., the decay of a nuclear particle). In those cases it is necessary to trigger the transient recorder with the event. This could be imple- mented with the timing circuit by replacing PM 2 (computer start) with a trigger pulse derived from some feature of the transient waveform (e.g.. a Schmidt trigger with a variable voltage level could signal the leading edge of a voltage tran- sient). This trigger pulse would again arm the transient re- corder and it would continue on as described above. The Analog Register. The schematic diagram for the circuit which controls the SAD 100 appears in Figure 4. The clock signal from the timing section is divided by the three Schottlry flip-flops to produce a quadrature clock capable of interleaving the twin banks (Rows AdrB) of storage cells within the SAD 100. The 3-input NAND gate ensures that the phasing of the quadrature clock remains proper. Since the SAD 100 is a MOS device. level shifters must be employed in order to interface to TTL signals. This is ac- complished by the two MH0026 drivers. and also the Zener diode. N -channel MOSFET (SD 210) combinations. The analog signal is amplified by the LH0032 FET input operational amplifier (National Semiconductor). This must be included to avoid loading the signal source, since the input impedanceoftheSAD lOOisonlyafewthousandohms. lnour primary application. the amplifier resides at the experiment. and a buffer amplifier (National Semiconductor LH0063) drives the signal along 504) coaxial cable to the SAD 100. The analog signal is stored in the internal storage cells of the SAD 100 as a quantity of electrical charge. Functionally these cells consist of the capacitance of reverse biased pn junction within the SAD 100. The typical capacitance of a cell is 1.2 pf, which yieldsa maximum stored chargeofabout5 pc. ThereforetheoutputoftheSAD lOOmustbeconnectedtoan 27 mamaummmumuummmunmmw was: err—s. 74876: PFC—O. were: “1-2. 74121:”1-2. 7061: smc. Germ. 1410i; WM Gasser. 74102; wise. 1475; al we baled "S" «Scrotum ANALOG IN IOOULO 5. YINII. fl is n z >" F 0‘ o r- a n rs U 3 use 3g ssasr &‘ u... cue ; moss MQMNMWMMNW IN \ ANALOG OUT I!) RCA IIIIA Was: #14. 74814; PFC. ureter-assessments mmucmaosocousa. reassure thematic-intense nouroseeummors zmswemeenerdeaee-e11V:m::at1mlehflnhe~rea) extr-ernely high input impedance device Low lead capacitance is also an important consideration. The analog outputs of the SAD 100 are connected to high input impedance (FET) operational amplifiers (RCA 313) A) placedascloeeaspoasibletotheSAD lOOinordsrtokeepthe leads short (This minimises capacitance and stray signal pickup.) The two alternating storage banks are then summed together to reproduce the original input voltage signal. This l-tpartispoesible because theSAD lOOinter-nallyehunteone output line to ground while the other is active. The output signal should besampled and held. because even with the high input impedance (typically 1.5 Til) of the RCA 3130 A, a sig- nificantdroopwouldoccurduringtheresdouttimesttheelow output rate. Acircuitcard(SClw)whichisaimilartothecircuitahown influetissvdlablefromReticonCorpcsatirnL'l‘hecircuit card was found to be inappropriate for this application. howsver.intworespecta.l"irst.itisneceaearytoeccurately controltheclockingrateofdataintotheSAD liltTheon- boardRCoscillatesoftheSClOOisnotsufficientlyaccurste andisnotreadllycomputsrcontrolled.Theeecondproblam arisasintheoutputbuffaringofthechargecontainedinthe SAD Ill). The SC I“) use discrete MOSFET transistor charge amplifiers which are necessarily ac coupled to the output amplifier.'I'heaccoupling camesallaignale whichare notsymmetrical aboutOvoltetobedistorted. Mes-snea'l‘heperformanceofthetransientrecordim system isillustrated inthenextfive figures.Theeefiguree ANALYTBAL WY. V0. 40. no.1. MY 1077 0 21 28 5.0 ' ssssoass (use) 4 A A A A A A A A A . v r v v v v v r ' Hustfiseperweotssnelentracudngsy'atsmotavutoom amwsvafiatacolecttcnrataztom LO! ' var vs V V C b o'”"‘ " 'O.’ ‘ ’ I a eeeeeeee J 0.000. {b { AA A A A A A A A - V v v v v v *v v on ' is n. lbs-t FlgwotLhaarttyottnemreccrdngeystwnovermer-rgeot oresvotts (aviator V..,IV..va. vhataceflectlonrate-sutubnomctadata demonstrate the rise time, linearity, eignal-to-noise ratio, and accuracy of the system. Figure 5 shows the response of the transient recorder to a loo-kHz, 3-V square wave. The data were clocked in at 10 MHz and then clocked out to the ADC atabout39kl-ls. Itcan beseen fromthefigurethatthesystem responds to voltage steps within 2 data points. Furthermore it is apparent that no significant bleeding between adjacent cells occurs while the information is stored in the SAD 100. If bleeding were occurring, it would manifest itself in a round-off at the voltage steps. The linearity of the system is shown in Figure 6. The SAD 100 is specified by the manufacturer (3) to be linear over a range of 0—4 volts in its present configuration. The response of the recording system over this range, is demonstrated by plotting Vow/V5,, vs. V... (Figure Be). It is clear from the figure that an inherent nonlinearity of 2-396 is present. and that it is a systematic nonlinearity. This nonlinearity increases with decreasing readout clock rate and is probably due to the “droop” of the Reticon storage cells. This effect would be ee- sentially constant, however if the readout rate remains con- stant,andthedatatakingrateisgreaterthanthereadoutrate by several-fold. (This is always the case for fast transients.) in the case stated. this nonlinearity would remain constant andcouldbecorrectednumerically.Figure6bshowstlierseult of a computer correction made on the data by performing a as a ANALYTCAL GEMSTRY. V“. 49. no. i. MY 1977 W IJJI ‘l M 3.00 ifl it’lr ' l’ ' M ’ 1.00 n 0.09 0 .a" h. 0 V ' fl - i ' ' T -A'. crocs cvctss Hustsw-b-noleerattootflacolectadd toll-t: u)Mdtwwmmame:Mtoomm (mMMeracuI-IMWetwnmtclneuot WMmumsrm so». A v (van-l w A v an m t : f : - f A f - 0.0 u u 8.0 as u 7.. 0.! are as) see "II ‘psea , mammwm imaumeymemwmeewm Mhpudetfiuwsrecolaculm ton-humbling» Wannesccsuem-(sccoaomnx 10"amm -(s.orseo.ooox t0"a8elelrewwalvsaehswlsadam-se nemmamoummereeaaemmme mm A A A A A 6th order polynomial fit, and then using the coefficients to correctthedata'l‘heeecoeficientscanbestoredandueedto correctsubsequentdataaets.'l‘heecatterinFigme6hbasbeen reduced to about 0.5-1%. Baseline Correction. Early in the development of this system,itwasnoticedthataproblemofglitchesex'ltedinthe output of the SAD 100. Two spurious voltage spikes appeared sttlreoutputaridtheyappearedonceforeech lM'data points collected. This noise is believed to be due to phasing differ- ences which occur during the reset cycle of the ring counters withintheSAD 100.11iisexplanationh-beencorroborated by the manufacturer. Sincethe noisespikesappearad tobeacharacteristicofthe analog storage register, it seemed that the best way to elimi- nate them was to numerically subtract them out in the data processing routine. This was accomplished by collecting a baseline scan and subtracting this set point by point from the data scan. This scheme requires the knowledge of when the noieespihesoccurinthedatastraam,however.andtothisend the cue index circuit (lower part of Figure 4) and a second 29 oscillator gate (upper right hand corner of Figure 3) were addsdPrior‘totheadditicnofcacillatosgatethwuoheerved that upon reset of the transient recorder (i.e., change back to fastrscordingrste),sehortburstofpulaeeoccimed.which wannableofchamingthetimerelatimahipbstweenthetwo noise spikes within the SAD 1006..., changing the relative statssofthstwointernalringcounters).Theoscillatorgate (gateflncwensuraethatnoclockpulaescanleavethetiming ofthemodulo-fiOcounterandfiagcircuitry,gensratesapube once per 100 clock cycles of the SAD 100. Since the noise spikeefromtheSAD 100aregeneratedbyaparticularstate oftheringcountersintemaltothedevice,theirtimerela- tionshiptothecueindexpuleewillremainconstantwhilethe systemisopasm'lh'ntimerelatiomhipisnotreproducible overauccessive power up cycles of thesystem however, so it isneceesarytocollectsbaaelineecaneachtimethesystemis pomssdupmndthenmeth‘nbmelinsecantoca-rectthedata scanswhichfollow. Figure7showstheresultsofabaselinesubtractionmade onadataaetcosfitingoflmdatapointscollsctadstlom'lz from a constant input voltage (~3V). The bottom trace (a) represents lOOsverageeofalowbaselinevoltsge.Themiddle tracs(b)istheeingleehotofdata,andtheuppertrace(c)is thermltofthabmelinecorrection.Thesignal-to-noise(S/N) ratiooftheuncorrecteddataacanis210andthe8/Nratioof thebuelinscorrsctedscanis378.Thiecorrespondstonearly 9hitsofresolution.'l'heS/Nrsticswereobtainedbydividing thesverageofthedatasetbyiteatandarddeviation. Transientleeordlu.Theaccuracyofthetransientre- corderinarsalapplicstionisillustratedinl’igure8.’l‘he tramiurtrmultir'fromanRCdecaywmrecordedatlom-ls. The transient was produced by applying a 200—na current pulse to the parallel combination ofa ace-o. 0.0196 reeistorand a 0.01 -uF, 0.5% capacitor and monitoring the voltage decay. A weighted least squares fit w- performed on the linearized dataandthesbpeofthefitcomparedmtheRCtimemtam forthecomponentementioned. lt'neeenfromthefigurethar thememrredruultisingoodqraamentwiththetheoretical value. ThecostoftheSAD lNisSle).Thatotalpartscostofthe transient recording system as described (not including com- puter and peripherals) was under $1000. The performance of the system with respect to speed and accuracy is certainly competitive with commercial tnmient recorders costing much more. The combination of cost and performance makes this method of transient recording very appealing, and the po- tential applications for this technique should increase still further as MOS monolithic technology expands. LITERATURE CITED (1) J. E. Stew-1. Am. w.. No. 11.01-101(1973). (2) w. P. Cecile and B. J. Moore. Res. Donne. as. 32-35 (1973). (3) DuaareetstchAD tOO.obhhed(onroquast)IromRaticonCorp..910 Busch»... Smnyvale. Call! 04000. , (4) a Hortlck. Anal. am. as. recs—rersnsrs). Elececnlcetor M". w. A. RECEIVED for review June 25, 1976. Accepted October 1, 1976. The authors are grateful to Michigan State University for the Quill Fellowship received by one of us (T.A.L.). This paper was presented at the second annual meeting of the Federation of Analytical Chemistry and Spectroscopy So- cieties. CHAPTER 3 CARRIER TRANSPORT THEORY The molecular model for the carrier transport mechanism is shown in Figure 3-1. Ion transport from left (') to right (") would occur through the following scheme. An ion (concentration Ci) (moles/cm3) in the aqueous solution on the left, migrates to the membrane—solution interface where it reacts with a neutral carrier mole- cule (surface concentration r; (moles/cm2)) to form an ion-carrier complex (surface concentration rig) within the interior of the membrane. Next, the charged complex must cross the energy barrier presented by the low di— electric environment of the membrane interior. Once the complex has crossed to the right side (surface concentra- tion r38) it can dissociate to form an ion (concentration Cg) in solution and a free carrier molecule (surface con- centration Pg). In order to have a sustained net flux of ions, carrier molecules depleted on the left must be replenished from the excess supply on the right. This process is shown in the last step of the mechanism, where a carrier molecule on the right crosses the membrane interior to resupply the carrier concentration at the left interface. Thus the conduction cycle is completed, and a net flow of ions can continue without depleting carrier concentrations within the membrane. The forward and 30 31 [-n' kis A). [—q " Is V I: " kis 1‘1] “-1 k-l F1 C + F' < "S 9 F" C I s ‘ + I ' l Aqueous I I Aqueous Solution I Membrane | Solution ' I Figure 3-1. Mechanistic model for carrier transport. 32 reverse rate constants for the complexation reaction are labeled kl and k-l’ respectively. The rate constant for translocation of the complex is given by k (this rate is constant is equal for transport in either direction, until a potential gradient is placed across the membrane). The rate constant for translocation of free carrier molecules is labeled ks and is voltage independent. The assumptions in the model as written are: 1. The carrier forms a 1:1 complex with the ion. 2. The carrier molecule is membrane bound, and thus no complex forms in the aqueous phase. 3. No dislocation of ion or carrier molecule occurs during complexation, and thus the only voltage dependent step is the one in which complex crosses the membrane interior (i;g;, kis)' u. Ions are not soluble in the low dielectric membrane interior, and therefore the only path for conduc- tion of an ion is via complexation with a carrier molecule. The differential equations resulting from the molecular model (Figure 2-1), are given in Equations (3-1) - (3-3); dF' __§ a .. _. _ ' _ dt kiciré + 181113 k3”; Pg) (3 1) art! ___S_ g _ n n w _ _ _ at klcirs + k-lris ks”: Fé) (3 2) car. amo SUI‘ mem Men! 33 dP' 18 I: v __ v v n n _ 1'5“ klcirs " k—1I‘1s kisris + kisris (3 3) r33 = 1"t ‘ ris ' F; ’ Pg (3—u) Equation (3-h) results from assumption 2 and the additional assumption that the complex and free carrier molecules are concentrated in two planar regions located Just inside the polar head groups of the lipid molecules (26), and it can also be written in differential form. Ft is the total amount of carrier added to the membrane (expressed in surface concentration units; i.e., Carrier Concentration X membrane thickness). Membrane Energy Barrier Since the carrier-ion complex molecule is charged, the low dielectric membrane interior presents a barrier to ion transport through the membrane. The ion starts out in the high dielectric aqueous medium, so energy must be added in order to move the ion into the low dielectric membrane medium. The fact that the carrier molecule tends to disperse the charge over a greater volume, re- duces the energetics greatly ($42,, causes the height of the barrier to decrease). It is this property of the carrier molecule which causes the conductance to be en- hanced by several orders of magnitude when small amounts of carrier are added to the membrane. 3h The simplest barrier to be considered is a single Eyring barrier with the maximum located at the center of the mem- brane (this is necessary to maintain membrane symmetry). This type of barrier (or any smooth curve having a maximum located at the membrane center) results in a current voltage behavior of the following form: I a eu/2 (3-5) where u = VF/RT, V = applied voltage, and F, R, and T have their usual meanings. Experimental evidence (27) shows however that the slopes of log I (steady state current) gs u curves are less than 1/2 for many membranes. If the surface complexation reaction is at equilibrium when the system is under conditions of steady state con- duction (evidence which supports this contention will be presented in Chapter 6), then the rate determining step would be complex translocation across the membrane (1,24, kis)' In this case the most plausible explanation is that the barrier energy maximum must occur at some other point than the membrane center. If membrane symmetry is still to be maintained, this forces consideration of a double Eyring barrier placed symmetrically about the membrane center. An energy barrier of this type is depicted in Figure 3-2. The barrier maxima are located at n and l-n, 35 A j 0 n 1—n 1 Figure 3-2. Double Eyring barrier used to represent the membrane barrier to ion transport. 36 respectively, where n is the decimal fraction represent- ing how far a maximum occurs from the nearest interface relative to the total membrane thickness (g;g;, for a single centrally located Eyring barrier, n = 0.5). The dashed curve represents the energy barrier to ion trans— port in the membrane resting state (i;g;, no applied vol- tage). If a voltage (u - VF/RT) is applied to the mem- brane (the potential gradient within the membrane is assumed to be constant), the barrier is shifted to that represented by the solid curve. It can be seen from the figure that the fraction of the applied potential which goes to shift the height of the barrier maximum (i;g;, change the energy of the transi- tion state) is n. Therefore the form of the voltage de- pendence for each of the maxima in this type of barrier would be given by; I d enu (3-5) where n is that shown in Figure 3-2 (the ions are assumed to be univalent). An identical voltage dependence can be obtained for a smooth "trapezoidal" barrier (28,29). The necessary feature of the barrier is that two maxima occur symmetric- ally about the membrane center. The membrane shape shown in Figure 3-2 was chosen because it is felt that the complex is at a slightly lower energy near the center of the 37 membrane, owing to the relative disorganization which exists near the ends of the lipid tail groups (see Figure 1-1). The voltage dependence of the complex translocation rate constants is given by Equations (3-7) and (3-8); kis = kgs eu/2/2Cosh((0-5-n)U) (3-7) kis = kis e-u/2/2008h((0.5-n)u) (3-8) where kis is the standard rate constant for complex trans- location. Since symmetry has been maintained, the reverse rate constant is decreased by the same factor by which the forward rate constant is increased. The height of the barrier at the membrane center cancels out of the mathe- matical expressions. Steady State Approximation From Figure 3—1, four net flux equations can be written (for transport from left to right) i _ ' J1 ' klcirs “ k-1F1s (3‘9) J3 = k_lrgs - klcirg (3-10) .— fl " - Jis ' kisris - kisris (3 11) 38 JS 3 ks(rg " Pg) (3—12) where Equation (3-9) represents the ion flux across the left interface, Equation (3-10) represents the ion flux across the right interface, Equation (3-11) represents the complex flux across the membrane interior, and Equation (3-12) represents the carrier molecule flux across the membrane. In this case the ion concentrations, in the solutions on either side of the membrane, are equal and the driving force for the ion transport is the electrical potential gradient. If the average free carrier concentration is introduced rs = (r; + r;)/2 (3—13) then from Equations (3-12) and (3—13); JS = kS(2FS - 2ré). (3—1u) At steady state all fluxes written must be equal (i.e., Equations (3—9) — (3-12)), thus from Equation (3-1“); r; = rs - Jig/2kg (3-15) and from Equations (3-13) and (3-15); r; = rs + Jig/2kg. (3—16) 39 Substitution of Equation (3-15) into Equation (3—9) yields; 1'" a (3-17) is k-l and substitution of Equation (3-16) into Equation (3—10) yields; le1(I‘S + Jis/2ks) + Jis F" = (3-18) is k_1 Finally by substituting Equations (3-17) and (3-18) into Equation (3-11), and rearranging, the following is obtained; P (k /k )C (k' - k" ) Jis = s l -l i is is (3_19) k' + k" k C l + ( 18 18)(1 + l 1) k_1 2ks Equation (3-19) describes the flux of complexed ions through the membrane, and therefore, it can be used to determine the conductance in the steady state. By insert- ing the voltage dependence of the complex molecule trans- location rate constants (Equations (3-7) and (3-8)) into Equation (3-19) the following is obtained; k 1 n Sinh(u/2) I‘s E——'Cikis ( 03 -n u ) is (3-20) E-_' Cosh((l/2-n)u3 2k -1 u 1 + kis ( Cosh(u/Z) ) (1 + klci) -1 s H0 and the steady state conductance is given by; G = FJis/V (3-21) where the charge on the carried ion is assumed to be 1. If all forms of the carrier molecule are membrane bound, and if the surface complexation reaction remains at equilibrium during steady state transport, 2FS = Tt/(l + chi) (3-22) where Pt is the total amount of carrier added to the mem- brane solution (expressed in moles/cm2), and K is the l equilibrium constant (kl/k_1) for the surface complexa- tion reaction. This is a particularly useful result since Ft can be easily calculated, and if the above relation— ship holds it can be used to eliminate P8 in Equation (3-20). In a charge injection experiment, the conductance is measured by the rate at which charge is depleted from the membrane capacitance (see Figure l-H). The voltage transient for steady state conductance is given in Equa- tion (3-23); __ = -FJiS/K (3-23) where K is the membrane capacitance (F/cm2). By using Equation (3-20) to substitute for Jis’ N1 Equation (3-23) can be numerically integrated, and used in a non—linear curve fitting routine to solve for the "best" values of the transport parameters. It is interesting to note how the shape of the current- voltage relationship is affected by the high and low electric field approximations. From Equation (3-20) it can be seen that if the voltage gradient across the -u/2 membrane is large (i.e., e can be neglected), the flux equation reduces to n nu rs(kl/k-1)Cikis 8 J = . (3-2“) is F: klci) enu is 1+E:(1+-2-1-(;- From Equations (3-23) and (3-2h); 2 nu du -F ae = (-—-——-——-) (3-25) at KRT l + benu = * — where the assignments a Ps(k1/k_1)C1kis, and b - kis klci (l + 153—) have been made for compactness. Equation -1 8 (3-25) can be integrated to yield e-nu - nbu = (nan/KRT)t + 0 (3-26) where u 3 0.1F/RT, and c is a constant yielding the proper initial conditions. Thus if b is vanishingly small, a nu plot of e' is t would be linear. If on the other hand “2 b is large, a plot of u 1§_t would be linear. If the voltage across the membrane is small, Sinh(x) z x, and Cosh(x) : 1, so Equation (3-20) reduces to Ps(k1/k_1)Cikzs(u/2) Jis = k‘; klc1 ' I (3‘27) 1+:(1+2k31) Thus from Equations (3-23) and (3-27) (73% ‘ F—RT(I‘T"5') “/2 (3-28) where a and b retain the values assigned previously. Equation (3-28) can be integrated to yield F2 in(u) = (1 + ab)—-fif‘t + c (3-29) where u g 0.03 F/RT, and c is a constant which satisfies the initial conditions. In this case a plot of £n(u) 1s t would be linear. Thus from Equation (3-20), three different current- voltage relationships can be obtained under the various circumstances described above. The experimental condi- tions which can bring about these circumstances will be discussed in Chapter 6. “3 Time Dependent Approach Equations (3-1) — (3-“) can be rewritten to include the voltage dependence (i.e., Equations (3-7) and (3-8)) to yield dfé —— = .. ' ' _ I _ fl _ dt klcirs + k‘lris ks(rs PS) (3 3O) dr" __§ = .. H u _ fl _ - dt klcirs + k-l is ks(Ps Pg) (3 31) dF' k! I'- ! ell/2 _ I." e-u/2 is = k C F' - k F' - -l§(-1§ 4§+ ) (3-32) dt 1 i s -l is 2 Cosh((O.S-n)u) " = _. _ _ fl - I‘is rt I.is P; I‘9, (3 33) where the ion concentrations in the aqueous phases on either side of the membrane are equal. If Equations (3-30) and (3-31) are added together and rearranged, Equation (3-3“); d(Pé + F3) dt = k_lrt ' (klci + k_1)(ré + Pg) (3’3“) can be obtained. This has the form; dx/dt + ax + b = 0, and can be integrated directly to rt[1 + e‘(k1C1 + k-1)t] (’5 + rg> = . (3-35) 2(ch1 + 1) MA The initial conditions here are the same as for a voltage step experiment. Since ion concentrations on either side of the membrane are equal, and since at t = 0 no voltage has yet been applied to the membrane, there are no net fluxes, and the system is under equilibrium conditions. At t = 0, Equation (3-35) reduces to V H... _ rs + rs — rt/(chi + 1). (3 36) The equilibrium expression for the surface complexation at either interface is r18 = chirs. (3-37) If Equation (3—37) is used to eliminate the P18 terms in Equation (3-33), an expression equivalent to Equation (3—36) is obtained, which means that Equation (3-35) al- ready satisfies the initial conditions. At this point, several assumptions are made which simplify Equation(3-32). Since the time dependent relaxa- tions occur quickly after pulse application (iLg;, in the first few microseconds), the voltage across the mem- brane throughout the relaxation will be greater than 0.2 V. In this case P"Se'u/2 << r' eu/2 i is tion, and thus Equation (3-32) reduces to is a reasonable assump- dris his nu dt klCirs k--l is 2 rise ' (3 38) "5 Further, if the translocation across the membrane (of the complex molecule) is assumed to be the only voltage dependent step, the voltage relaxation is given by -F2k* d“ . is ' “u (3-39) 6? —2KTR ' 138 ° Equation (3-33) can be used to eliminate Fig from Equation (3732) to yield dris = k C r! + m e-nu g2- + E 92 (3‘u0) dt 1 1 s 2 , dt 2 dt ° F kis F Next, Equation (3-35) can be used to eliminate Pg, Equa- tion (3—39) can be used to eliminate P18, and Equation (3-h0) can be used to eliminate P; from Equation (3-30), resulting in I v drs = _(1 + 2k8 )[dris _ k-lKRT e-nu du - KRT fl] ‘6'? k""lci' _dt__ sz, 6'76 —F2' dt is k r - k KRT + s t [1 + e-(klci + k_1)tJ _ -1 e-nu g3. 2(K 0'"? '1') , dt 1 1 F k is (3-A1) Equation (3-Al) can be integrated to yield us 2k tk F RT 3 t s klai is F2 KlCi + l k r k 2ksk_ KRT s t -l 2 e‘(klci + k-1)t _ 1* 2 e'nu + const 2(klc1 + k_1) nklcikisF (3-u2) where; 1" A k I' t S t 1 const = + F /2 + [1 + 3 W161 + 1) t (1:101 + k__l) 2 K1 1 + 1 + 2ksk_1KRT nk C ka’ l i is One unusual characteristic of Equation (3—u2) is the term (ksrt(t)/(K1C + 1)) which is linear in time. Oh- i viously this equation cannot hold for long times. This term might possibly be a result of dropping the e'u/2 term from the original differential equation (u is a function of time), but at any rate it shouldn't cause any signi- ficant problem, since the time dependent relaxations are finished shortly after charge pulse application. Equation (3-A2) cannot be further used to simplify the differential equations without a prior knowledge of the voltage relaxation function (u(t)). The final two steps of the solution leading to an equation for u(t) could be done numerically, but no solution in closed "7 form has yet been found. Since the entire set of differen- tial equations may be integrated numerically without making any simplifying assumptions, this latter path was chosen for the time dependent data treatment. All of the equations used in this chapter were derived from first principles and the model which was adopted for carrier transport. Some of the equations are, however, similar to equations obtained by other workers, and these similarities are discussed in Appendix C. CHAPTER 4 ROUTINE MEMBRANE EXPERIMENTS Materials and Methods The electrolyte solution used for the charge pulse experiments was 3M LiCl. This was chosen because the actin antibiotics used in the transport experiments are bound more favorably (approximately 200 fold) to potassium ion than to lithium ion. Various volumes of 3M.K01 were added to 20-25 ml of electrolyte solution to bring the K+ concentration up to the desired level. In this way the K+ concentration could be varied, while maintaining constant ionic strength. Reagent grade salts were dis- solved in distilled/deionized water. The salt solutions were filtered with Whatman #2V filter paper to remove any insoluble particulates. The membrane solution was prepared by adding glycerol- monooleate (GMO) (5% by volume) to nahexadecane. The glycerol-monooleate (technical grade) was purchased from Matheson, Coleman & Bell. Thin layer chromatography on silica gel with 50% ethanol/benzene showed two spots (with iodine stain under UV light). The first spot (approxi- mately 5—10% impurity) traveled with the solvent front. The second spot traveled approximately half as far along the TLC plate, and contained the balance of the sample. A8 “9 The 013 NMR spectrum of GMO in 013013 was identical to the literature spectrum for oleic acid, with the addition of three peaks, slightly upfield from CDC13, which were assigned to the three glycerol carbons. The n-hexadecane (99% purity) was purchased from Sigma. Both the GMO and n-hexadecane were used without further purification. The antibiotics used in this study were the actin homologs (nonactin, monactin, and dinactin). The struc- tures of these antibiotics are shown in Figure “-1. The actins were a gift from Drs. Bickel, Waldman, and Scheibli of CIBA-GEIGY LTD. (Basle, Switzerland). Antibiotic solu- tions in the concentration range 10"“ to 10'3 M_were prepared by adding the appropriate amount of drug to the membrane forming solution (GMO/n-hexadecane). It should be noted that 10‘3 M'is a practical upper limit for the solubility of these drugs in the membrane forming solu- tion. For antibiotics which did not readily dissolve, an ultrasonic cleaner was used to agitate the solution. The antibiotics were weighed using a Kahn Electrobalance (Model N100) which is capable of ug resolution. This is necessary since the preciousness of the drugs requires that accurate weighings be made on small samples. 50 H3: H; o w, 3 C "I o o cwacw, cu, "3 CH3 "2 CH3 Figure “-1. Chemical structures of the actin homologs. 51 Membrane Formation The electrochemical cell used for the membrane ion transport study was described in Chapter 2. The membranes were formed on a small (1—2 mm) orifice in a thin Teflon barrier which was sandwiched between the two electrolyte compartments. The membrane formation was accomplished using a Pasteur pipet to apply approximately 0.01 ml of the membrane forming solution to the lip of the orifice, while observing through a microscope. By using the Pasteur pipet to gently create air bubbles along the barrier, Just below the orifice, the membrane solution can be repeatedly swept across the orifice until a membrane forms. Any ex— cess membrane forming solution applied to the orifice will remain around the circumference of the BLM on the lip of the orifice. This region is called the membrane torus, and experience has shown that membranes are most stable when the size of the torus region is minimized. The mem- brane environment should be as vibration free as possible to further increase membrane lifetime. Experiments can be performed on the membrane approximately 30 seconds after formation. (It takes that long for the bi-layer to stabilize.) This time was generally used to measure the membrane area with a microscope and graticule. 52 Membrane Data Collection The program used to control data collection is named "MEMBRN.FTN". A listing of the program is shown in Ap- pendix B. This program can be used to collect, plot, list, and store data in a file on any supported mass storage device. It should be noted that a baseline scan (n averages) must be collected before the program will honor a Store com- mand. This is necessary to allow correction for the transient recorderksbackground noise (see Chapter 2). After the data have been stored, a plot command will plot the baseline corrected data. The program uses the CPU console switch register (SR) as the transient recorder front panel. The recording and readout rates of the transient recorder, as well as the number of data points collected at the fast recording rate, are controlled by various bits in the SR. The bit assign- ments are shown in the program listing under "Data Col- lection Section", and are entitled "Switch Register Options". Note: The transient recorder should be powered up when the program is started, because an initialization routine is automatically run on startup. It should also be noted that it is possible on power up, for the transient recorder to be hung up in a state which cannot be cor- rected by the initialization routine. In this case, the computer will hang up in subroutine "COLECT", waiting for the data flag. Attempts to correct this problem have 53 been unsuccessful; however repeated power ON/OFF: program restart cycles will eventually recover the transient re- corder from this state. Data Treatment The data collected were generally fit to the equations shown in the Theory section using KINFIT (30) on the MSU CDC-6500 computer. The program "KINPCH.FTN" (see Appendix B) was created to facilitate the transfer of data etc. to the CDC-6500. This program decodes the data files, and creates files containing multiple data sets, in the proper format for KINFIT. These files can then be easily appended to subroutine "EQN" (from KINFIT), and shipped to the CDC—6500 via the CEMCOMGRAF facility (MSU Chemistry Department). KINFIT requires that initial estimates for the rate parameters be given by the user. It is often a neces- sary condition for convergence that these initial esti- mates be fairly close to the actual values. The program "KINRLX.FTN" (see Appendix B) was created to allow the user to select proper estimates, through interactive fitting of the rate equations to the data. KINRLX al- lows the user to suggest values for the fitting param- eters, numerically integrates the differential equations using the Runge-Kutta method, and plots the predicted voltage transient along with the actual data. The user 5“ may then revise his estimates until a satisfying fit is achieved. For fast time dependent relaxations, it is often better to use "GEAR.FTN" (31,32) to numerically integrate the differential equations, rather than KINRLX. This method is more suitable for "Stiff Differential Equations" (143;, equations which have rapidly decaying terms early on in the lifetime of the transient and which essentially vanish later on) and gives a better fit for early relaxations. This program is not listed in Appendix B because of its length; however, source listings of all programs useful to this research (including those mentioned above) are contained on a Floppy Diskette which has been placed in the C. G. Enke Group (MSU Chemistry Department) research archive. Only a few of the most important programs are included in Appendix B. "EXPFIT.FTN" is an exponential curve fitting routine used for obtaining the membrane capacitance. The program linearizes the exponential data, calculates the proper weighting coefficients, and then calls a weighted linear least squares fitting subroutine. Under proper experi- mental conditions (see Equation (3-29)) "EXPFIT.FTN" can also be used to obtain conductance estimates. CHAPTER 5 MEMBRANE CAPACITANCE MEASUREMENT Introduction The Charge Injection method (see Chapter 1) of measur— ing membrane ion transport phenomena requires the knowledge of the membrane's capacitance, in order to obtain rate parameters from the recorded data. The method for obtain— ing the membrane capacitance value, and the results of the capacitance measurement,are given in this chapter. Unmodified BLM's have resistances in the range of 106 - 107 ohm-cmz. Therefore, a charge placed on the membrane's capacitance will leak off very slowly if conductance through the membrane is the only discharge path. This property makes it possible to measure the membrane's capacitance by placing an external precision resistor between the electrodes (i;g;, in parallel with the capacitance of the membrane), charging the membrane, and determining the time constant of the RC decay. Materials, Methods and Results Antibiotic—free GMO/n-hexadecane membranes were formed between solutions of pure electrolyte (LiCl with no K+ salt added), and a 10 K0 i 0.005% resistor was placed 55 56 between the electrodes. The results of the charge pulse experiments are shown in Figures 5-1 through 5-3. Figure 5-1 shows the result of pulse application to the cell when neither the membrane nor external precision resistor were present. The pulse width was previously set to 100 ns using an oscilloscope. It can be seen from the figure that the recorded pulse width (at 1/2 height) is approximately 200 ns. Thus, the measurement system can be expected to respond to a step voltage transient within 200 ns. Figures 5-2A and 5-2B show the result of pulse applica— tion to an unmodified GMO membrane with no external bleeder resistor. Figure 5-2A was recorded at the maximum rate of the transient recorder (10 MHz), and Figure 5-2B was recorded at 78.125 KHz. Two important facts can be recognized from these two figures. First, it is clear that discharge through the membrane or external circuitry is minimal, and therefore the membrane capacitance can be determined accurately by this method. Second, it shows that if any dielectric relaxation occurs in the membrane, it must either have a very small amplitude, or it occurs on a time scale which is either very much shorter or very much longer than the time scale of the charge pulse experi- ments. In either case, Figures 5-2A and 5-2B show that the results of the membrane experiments will not be affected by a dielectric relaxation. ‘ Figure 5—3 shows the transient resulting from the 57 .pcomoad ocmpnsms 0: Spas Hamo map 0» mmaso m: 00H m mo coHmefiHoom Bosh oocfimppo pcoamcmpe .Hlm madman 0.? W m: mzHH ofix » L L d1 d a. r H l ,2) 1' C9 0 G) 5 8110A [loom .@ 58 0.60677 U) —— F_. ‘T _J __ ........................................................................................... O _ > qr- 9'099 }}}{%}}%}41 0.0 1 0 TIME,US x10 '1 Figure 5-2A. Transient resulting from pulse application to a membrane which contains no carrier (no ex- ternal resistor). 0 . 6 0 O'T' U) ‘— *— dr- __| "'- ..................................................................................... o -_ > I 0«M» ii—: tt {i '1 q— q;— T ’41 0.0 1.2 TIME,M8 Figure 5-2B. Transient resulting from pulse application to a membrane which contains no carrier (no ex- ternal resistor). 59 i .pfim monsoon pmmma rmfipcmcooxm Eopm mpasmmn m>nzo ofiaom .pconmpa nonmanop Hmcmmpxm spas Apcommna pofipnmo ocv ocmmnEmE wcfiwasa Eopm cocfimpno pcmfimsmne .mum opswfim a; Na. m2 3.: m; o.o Tauflunuwnuwuuiuaooé -r 0.90 L] ll. an nu -1 fil I. olrl S if Leone 60 application of a pulse to the parallel combination of the membrane's capacitance and a 10 K9 t 0.005% resistor. The solid curve is the result of a weighted linear least squares fit on the linearized exponential transient decay curve. The average of 12 such experiments, each performed on a new membrane, results in a value of (5.9“ i 0.1“) x 10"7 F/cm2 for the membrane capacitance (this precision is comparable to the estimated error in the membrane area measurement). If the membrane thickness is assumed to be in the range of 5 to 10 nm, this results in a membrane dielectric constant in the range of 3.” to 6.7. Discussion and Conclusions If the membrane's capacitance remains the same upon addition of the carrier molecule to the membrane, the capacitance value listed above can be inserted into the conductance equations (see Equations (3-23) and (3-39)), and the resulting expressions can be used to arrive at the transport rate constants. Since the carrier concentra- tions are generally less than 1.0 millimolar, it seems reasonable to expect no measurable change in the capaci- tance value when the carrier is added to the membrane. In an earlier stage of this research, n-decane was used as the membrane solvent instead of n-hexadecane. It is interesting to note that even though all other experimental conditions remained invariant, the GMO/n- 61 decane membranes consistently gave capacitance values of u.5 x 10‘7 F/cm2. If the basic dielectric structure of the membrane remains the same, this implies that the membrane thickness must have decreased when the solvent was changed from n-decane to n-hexadecane! One possible explanation for this is that the hexadecane forms more compact membranes because its chain length is similar to that of the lipid "tails" (oleate), and therefore the molecule fits neatly into the membrane structure. In the case of n-decane, one molecule is too short, and two mole- cules placed end to end are too long to fit easily within the membrane structure. Thus, even though n-decane is a shorter molecule than n-hexadecane, it produces membranes which are thicker. CHAPTER 6 STEADY STATE RESULTS Introduction When a BLM which contains carrier molecules is placed between two aqueous solutions which have identical (non zero) ion concentrations, the system of reactions respon- sible for ion transport through the membrane will quickly reach equilibrium conditions. If a charge pulse is ap- plied to the membrane's capacitance, the resulting potential gradient will perturb the system away from equilibrium, resulting in a net flow of ions. The flow of ions discharges the potential across the membrane. This response to the electrical transient can be used to obtain the tranSport rate parameters. As the potential across the membrane decreases, the rate of transport (i;g;, the rate of change of potential) also decreases, and thus the system relaxes back to equilibrium conditions. If a voltage step perturbation is applied to a BLM, the conductance initially Jumps to a relatively large value where it remains until the concentration of the transportable ions on either side of the membrane is significantly affected. In a charge inJection experiment, a similar pseudo-steady state (not true steady state 62 63 since the voltage continuously changes) conductance can likewise eventually be obtained. In this case however, there is an additional restriction which must be placed on the system because the voltage across the membrane con- tinuously changes. Since the rate determining transport parameter is voltage dependent (see Equation (3-20)), the voltage must decay to a sufficiently low value such that the rate of change of the rate limiting parameter is small relative to the absolute magnitude of the parameter, before pseudo-steady state conduction can be achieved. It should therefore be possible to perform steady state measurements on the transport system (if the above restric- tions are observed), as well as perturbation type measure- ments, employing the charge inJection method. The results of this type of steady state measurement will be presented in this chapter. Sample Transients Figure 6—1 shows the transients obtained from pulsing a GMO/n-decane membrane which contains 5.” x 10'“ M din- actin. The top trace corresponds to 0.U1U M.K+ in the supporting electrolyte (N3 M LiCl) on both sides of the membrane. Likewise, the middle trace corresponds to 0.79“ M K+, and the bottom trace corresponds to 1.077 M K+ (constant ionic strength). It can be seen from the 6'4 . I .Hox a ems.o mm c>ssc .Hox_m :Hs.o mm c>nsc Hog z mmm.o ”H cease .cacoacac as emm.o antennae "accessesco .puoomcmmp Umpmfivme nofiupmo Eopm wCHpHSmop mpcmfimcmpp campuses .Hnm mhswfim ®._xn NV: .l m: mzae . o.o 1::11: 1 u l l l l .8: . ”...: nlrl . :~: .. . ..... a- .An ..... gs. . any m... an”. 1- 1 ....~.. . H i . : ..mfi“ L- .Qu alomW¢_.o 65 figure, that the 810pe of the decay curve is concentra- tion dependent, as predicted by Equations (3-23) and (3-20). Discussion of High and Low Field Approximations Despite the fact that Equation (3—20) shows a complex current-voltage relationship, workers (20,33) employing the charge pulse technique often assume a linear current voltage relationship in order to simplify the solution to the V(t) transient. This assumption is valid only under the conditions for which Equation (3-27) applies (i;gL, a very small voltage gradient across the membrane interior). Figure 6-2 shows the result of a logarithmic plot (see Equation (3-29)) of a voltage decay curve for K+/dinactin conductance. Although the shape of the entire curve is not particularly linear, the shape of the low voltage end (where the approximation is valid) conforms fairly well to that of a straight line. The negative deviation of the curve, on the high voltage end, may result from the failure of the low voltage approximation. At large vol- tages the data would be more accurately described by Equation (3-26). If the "nbu" term in Equation (3-26), dominates at high potentials, the curve would deviate in the direction observed in the figure. Another possible explanation for the high voltage deviation is that the voltage—dependent rate constant becomes so large that the 66 smm.o "necaeaecco o.m _— - — N m3. .HO& 0 afie.o .ceuccsfie as .ppoamcmpp cfipomcfiv pom oEHp .m> Asvwoa mo uoam .mlm mmswfim 0.Hxn .0sz m.o .L 000.0 (0)907 .....oooJ 67 rate becomes partially controlled by a parameter which is not voltage dependent (Eggg, the surface complexation reaction). In this case the rate would, after a certain potential, increase no further with voltage. The devia- tion probably does not result from non-steady state con- duction since, as will be shown in Chapter 7, this results in positive deviations from the line. Figure 6-3 shows the same set of data plotted (inverse) exponentially (see Equation (3-26)). It can be seen from the figure, that the central portion of the curve is straight, and that both extremes deviate from the line. The high voltage deviation (left side) is probably again either due to the "nbu" term dominating over the "e-nu" term in Equation (3-26), or due to the rate becoming limited by a non-voltage-dependent parameter. Non-steady state conduction would cause deviations in the opposite direc- tion. The low voltage deviation (right side) is probably -1111 due to the failure of the approximation (i.e., e is no longer negligible compared with en“). At any rate, Figures 6-2 and 6-3 show that for the voltage excursions used in this research, the data are in better agreement with the approximations governing Equation (3-2U), than those which apply to Equation (3-27). It is interesting to note that Figure 6—2 is approxi- mately linear out to voltages for which the low voltage approximation does not apply (iggg, ex cannot be accurately represented by (1 + x)). The approximation is accurate 68 .8x .2. 33.0 .5383 as 300.0 "mCOHpHocoo .ppoamGMLp :Huomcfiv you mafia .m> scum mo poam .mlw madman Hx Na: 0 m D m 2 H H o.m m.o _ _ _ _ . _ _ _ _ . a _ E. _ _. _ . _ _ 000 0 Ir! 3 VA [1| Id \/ ll. . r “u . n” (x 0 LI 11000.0 69 to about 25 mV (3“), and the curve appears to be linear to about 80 mV (Note: The vertical scale starts at ap- proximately hO mV). One possible explanation for this is simply that logarithmic plots always tend to make curves more linear to the eye. Another possible explanation is that the voltage drop across the membrane interior really is fairly small, and a maJor portion of the voltage drop occurs across the membrane solution interface. This can come about if the interfacial complexation reaction is the slow step in the transport scheme. Before charge pulse application to the membrane, the reactions responsible for transport through the membrane are all under equilibrium conditions. Therefore, if the equilibrium constant of the surface complexation reaction is not much less than 1, a sizable fraction of the carrier molecule will exist in the form of complex. When a voltage gradient is placed across the membrane the complex trans- location rate increases considerably (see Equation (3-7)), and the charged complex is quickly transported down the voltage gradient to the other interface. This has the effect of charging the membrane—solution double layer capacitance (see Figure 1—3), and thus some of the voltage drop occurs across the membrane-solution interfaces, and the voltage across the membrane interior (where the vol- tage dependent complex translocation step occurs) is de- creased. 70 It is likely,however, that for the actin homologues the conditions described above only exist during time dependent relaxations, where the voltage across the mem- brane interior is large enough to make the complex trans- location rate constant much greater than either the for- ward or backward complexation rate constants. Data will be given in the next section which seem to show that the surface complexation reactions remain under equili- brium conditions when the system is under conditions of steady state transport. Furthermore, a consideration of the total amount of carrier present in the membrane re- veals that probably not more than half of the charge on the membrane (after charge pulse application) could exist on the membrane-solution interface. For antibiotic (carrier) concentrations in the neigh- borhood of 10'6 moles/cm3, the total amount of carrier in the membrane (assuming a membrane thickness of 10'6 cm, and diameter of 0.135 cm) is approximately 1.5 x 10’1” moles. A charge pulse containing 2.8 x 10'1“ Faradays is required to charge the capacitance (see Chapter 5) of a membrane of this size up to 300 my (typical experiments were started at this voltage or higher). Thus, even if all of the carrier molecules were tied up in the form of complex, only slightly more than half of the charge could be placed on the membrane-solution interface. The above discussion is based on the assumption that 71 the only source of carrier molecules is the membrane it- self. It is possible that the torus region (which contains more than 99.99% of the original membrane forming material!) could contribute some additional carrier to the rest of the membrane. However, if the surface complexation reaction remains at equilibrium during the steady state membrane conduction, it is unlikely that the torus could supply carrier molecules to the interface where they are needed as fast as the other interface could resupply them through the dissociation of complex there. This last argument is based primarily on consideration of the relative dimen— sions of the membrane (i;g;, thickness/diameter). The torus could therefore contribute to increasing the total amount of carrier present in the membrane, but if so, it probably does this slowly while the membrane is in its resting state (242;: no potential gradient). Further, the contribution from the torus is probably not sufficiently large to free each carrier molecule from the need to make several passes through the transport mechanism in order to deplete all of the charge placed on the membrane's capacitance. This last prOperty is likely to make the transport system go rapidly to condi- tions of steady state transport. 72 Data Fitting The actual data were fit to Equation (3-23) (Equa- tion (3-20) was used to evaluate Jis)’ using KINRLX.FTN and KINFIT. Values obtained from KINRLX.FTN were often within experimental error of the values eventually ob- tained from KINFIT; however,KINFIT should still be used to remove operator bias. Table 6-1 shows the parameters obtained from fitting membrane transients to the steady state conduction rela- tionship. The parameters listed are combinations of the transport rate constants, etc., given in Equation (3-20), except for the first parameter "n" which is the relative position of the maxima on the complex translocation energy barrier (see Figure 3-2). It can be seen from Equation (3-20) that the coupling between the various rate constants in each of the parameters cannot be resolved without some additional information about the transport system. It should be noted that the parameters n, kis/k-l’ and k1/2kS may vary, from data set to data set, by as much as 110%. The values for these parameters were assigned a constant average value for this set of experiments in order to maintain consistency with the view that the Pskl term is the only concentration dependent parameter. This was done for the purpose of obtaining data which will be used in the next section to test the hypothesis that the surface complexation reaction remains under equilibrium 73 mOH x 0.0 =I0H x m.H 00.0 00.0 mica x 00.0 mloa x ~00.H 00H x 0.0 :IOH x m.H 00.0 00.0 mloa x 00.0 MIOH N ~50.H 00H x 0.0 :IOH x ~.H 00.0 00.0 mloa x 00.0 MIOH x 000.0 moa x 0.0 :IOH N ~.H 00.0 00.0 mloa K m0.0 mloa x 300.0 00H x 0.0 :IOH K 0.0 00.0 00.0 mIOH x 00.0 mloa x :H:.0 moa x 0.0 :IOH x H.0 00.0 00.0 mica x 00.0 mIOH x 0H=.0 00H x 0.0 :IOH N :.0 00.0 00.0 mica K mm.m mloa X 000.0 m0H x 0.0 :IOH x m.0 00.0 00.0 mica x Hm.m mloa x 000.0 00H x 0.0 :IOH K 0.0 00.0 00.0 mica x 00.0 MIOH N mHH.0 00H N 0.0 :IOH N 0.0 00.0 00.0 mloa x 00.0 mloa x mHH.0 mHoE\meo m\Eo II II mcmpmm mEo\oHoe mxmax mep Hlx\mmx c mocmpfiomamo coapmpucmocoo .AmmcmnnEme mcmomcmxonlc\ozw :H ppogmcmnp zapomcoev mCOfipfiocoo Hmpcmsahmgxo Hmofiucmvfi nova: mocwanoE pcmnmmuuc co cmenomnma who: mpGoEmnsmwme mascaaaso .AaomlmV coauwsumv cowpmsuo phoamcwau mumpm homopm map on wcfiupam o>ndo Song wchHSmma mnmpoemnwm .Hlm manna 7H conditions during steady state membrane transport. Equilibrium Assumption The membrane conductance is defined in Equation (3-21). It can be seen from Equation (3-20) that the flux term (J15) in Equation (3-21) contains some voltage dependent terms, and therefore the conductance is not a simple linear function of potential. In order to simplify the con- ductance expression,Equation (3—27) can be used to evaluate the flux term instead of Equation (3-20). As seen from Figure 6-2, some error will result from this simplification. This was done for convenience's sake, and the results were used only to illustrate trends visually. Values for the rate parameters were obtained using Equation (3-20) in all cases. The concentration dependence of the membrane conduc- tance is shown in Figure 6-“. The curve is fairly linear out to K+ concentrations of approximately 0.“ M, where it bends off. This conductance behavior is typical of the actin homologues (35), and is generally attributed to adsorption phenomena. Perhaps a better explanation is that the surface complexation reaction is under equili- brium conditions (during steady state transport), and 0.A M_K+ concentration is approximately the point where P is = T8. In that case, the conductance would appear to increase linearly when Pis << PS and would flatten 75 7r 6r- 5. u- G ‘f" mhos . x10"3 2 1 l J J 08 1.0 1 C1+I\_d Figure 6-H. Plot of conductance versus ion concentration for potassium ion/monactin transport. 76 off when PS << P This type of behavior is certainly is similar to that seen in Figure 6-A. If the surface complexation reaction is under equili- brium conditions during steady state membrane transport, Equation (3-22) can be used to eliminate the coupling between rate constants in the fitting parameters. If both sides of Equation (3-22) are multiplied by k1, the resulting expression can be rearranged to yield 1 = 1 + 2 . (6-1) Thus, if the equilibrium assumption holds, Equation (6-1) states that a plot of l/I‘Skl Kg C1 should be linear, and the slope and intercept can be used to arrive at values for k and k_ An examination of Table 6-1 1 reveals that PBk l. l is conveniently one of the fitting parameters, and also that given values for k and k_ l 1’ all of the rate constants can be uncoupled. Figure 6-5 shows the results of the plot described by Equation (6-1). The curve is satisfyingly linear, and seems to show unmistakably that the surface com- plexation reaction is under equilibrium conditions. The conductance data in Figure 6-A can now be re- plotted, correcting for the rskl term (due to equilibrium). This plot is shown in Figure 6-6. The data seem remark- ably linear between 0.1 _1v; and 1.1 pg K” concentration limits. (Note: This plot has also been corrected for 77 7.5 F 1/Tsk s/cm x 10" 3-0e Figure 6-5. Plot of fitting parameter (l/Fskl) vs. ion concentration for potassium ion/monactin trans- port. 78 113 12 10 001’. mhos 6 X10 O + l3 Figure 6-6. Corrected conductance versus ion concentration for potassium ion/monactin transport. 79 the concentration dependence of the denominator in Equa- tion (3-20)). Results and Discussion The rate constants for monactin and dinactin are shown in Tables 6-2 and 6-3, respectively. Literature values of these same parameters for nonactin and tri- nactin are given in Table 6-h for comparison. Although nonactin and trinactin will not necessarily have rate constants of the same magnitude as dinactin and monactin, the values listed in Table 6-fl are the only literature data available at present for K+ transport through mem- branes of the type studied in this research. It is clear from Tables 6-2, 6-3, and 6-A that the present results are uniformly larger than the reported values for nonactin and trinactin by approximately five-fold. Since the literature values and those reported here differ sig- nificantly, an alternate measurement of the rate constant "kis" was performed. A low amplitude pulse (< 25 mV) experiment was performed on GMO/n-hexadecane membranes which contained monactin. The computations on data received from the low vol- tage experiment, are simplified in two ways. First, the current-voltage relationship becomes linear and thus Equation (3-29) can be used to represent the transient. Second, the voltage across the membrane is low enough Table 6-2. Table 6—3. Table 6-“. 80 Rate constants for monactin carrier transport in GMO/n-hexadecane membranes. CS = carrier concentration. Pt was calculated assuming a membrane thickness of 6 nm. (Ionic strength == '3 El. ) Cs rt k1 k-l kis ks 2 -1 -1 -1 mole/cm3 mole/cm cm3/mole's s s s 2.8a x 10'7 1.7 x 10'13 8.3 x 108 3.9 x 105 1.5 x 105 2.7 x 105 Rate constants for dinactin carrier transport in GMO/n-hexadecane membranes. CS = carrier concentration. Ft was calculated assuming a membrane thickness of 6 nm. (Ionic strength = 3 M. . I C, rt k1 k_1 k k uole/cm3 mole/cm2 cm3/mole-s 3’1 3'1 s'1 9.u1 x 10'7 5.65 x 10'13 1.1 x 109 1.1 x 106 2.3 x 105 2.7 x 105 Literature rate constants (from Hladky (36)) for nonactin and trinactin carrier transport. (Ionic strength = l M.) (Note: the units for kl were changed from those used by the cited author.) Antibiotic k1 k_1 k;' k8 -- cm3/mole'o 0'1 3’1 0'1 Nonactin 2.9 x lo8 5 x 105 2 x lo" 5 x lo" Trinactin 1.85 x 108 1.6 x 10“ 2.3 x 10. 3.7 x lo" 81 to insure that the translocation rate is very much less than the rate of complexation, and thus the surface com- plexation reaction must remain under equilibrium conditions throughout the low voltage transient. Although the absolute magnitude of the rate constants obtained in this work (see Tables 6-2 and 6-3) differ significantly from those in Table 6-“, the relative ratios of the various rate constants (iég;, kl/k-l’ etc.) are quite similar. Therefore Equation (3-29) can be simpli- fied to a relationship having only one parameter, namely a 1 rate constants in either of the Tables (6—2 through 6-A). k s’ and several constants resulting from ratios of the EXPFIT.FTN was used to obtain the slope of the £n(u) is t curve (see Equation (3-29)). The fits obtained were of a quality similar to that seen in Figure 5-3. The values obtained for kis’ by this method, were in the range of (2.0-2.5) x 105 s'l,which is in good agreement with the value given in Table 6-2. Some important features should be noted from the results given in Tables 6-2 and 6-3. First, it can be seen that for ion concentrations of 0.5 M, the term klci/ k_1 is approximately unity (this value agrees well with the K+ concentration dependence observed in Figure 6-h). This means that when the surface complexation reaction is at equilibrium, approximately half of the carrier exists in the complexed form at K+ concentrations of 0.5 M.- 82 Another feature that can be seen from Tables 6-2 and 6-3, is that kis is not very much less than klci’ or k_1 (note that kl must be multiplied by C1 in order to maintain consistent units). In fact, Equation (3—20) implies that kis becomes larger than klc1 (or k_1) at potentials above 150 mV. Therefore, the equilibrium approximation (for the surface complexation reaction) would be invalid above 150 mV, and some question exists whether or not the approxi- mation is valid at any potential. The fact that each carrier molecule must make several passes through the membrane in order to deplete all of the charge placed on the membrane by a charge pulse will tend to make the trans- port go to steady state conditions very rapidly if any difference exists between rate constants. (This also allows for ks to be the rate determining step.) Further, Figure 6-5 seems to show incontrovertibly that the sur- face complexation reaction is at equilibrium conditions during steady state conduction. It is interesting to note also that the ratio of kS/k;8 for both monactin and dinactin is less than two. This does not seem unusual until one realizes that the complex carries a charge, whereas the free carrier is neutral. Except for this difference, both molecules are probably almost indistinguishable within the membrane. Therefore, the carrier molecule must do a remarkable Job of diffusing and/or screening this electric charge 83 from the interior of the membrane. Perspective on Steady State Results The validity of steady state rate constants obtained by this and other methods, depends on a whole range of assumptions and approximations. For instance, the model proposed must be valid or the values obtained for the rate constants will be meaningless. Also, the experi- mental conditions must be carefully controlled to insure that conditions implicit in the assumptions are main- tained. The assumptions in the carrier model used in this work are covered in Chapter 3. In order to extract the rate constants from steady state measurements, two addi- tional assumptions were necessary. First, it was assumed that for transmembrane voltages below 200 mV, the system was under conditions of steady state. Although this seems like a reasonable assumption, no data have been presented to substantiate it. The second assumption is that the surface complexation reaction remains under equilibrium conditions during steady state transport. This assumption was necessary to uncouple the rate constants within each curve fitting parameter. Although Figure 6-5 supports this assumption, the values obtained for the transport rate constants seem to conflict with this view. Both of the above assumptions tend to become 8“ more valid as the voltage across the membrane decreases. Unfortunately, the low voltage limit is the region of worst signal-to-noise ratio for this method, since the desired information is contained in the rate at which the voltage changes. Thus the technique is better suited for time dependent measurements than for steady state determinations. It was mentioned previously in the Chapter that the rate constants within the fitting parameters were coupled and that this was eliminated through the use of the equili- brium assumption (see Figure 6-5). It should be mentioned here that a fairly strong coupling also exists between the curve fitting parameters themselves. As a result, the error estimates received from KINFIT can be fairly large. This coupling results from the complexity and nature of the model, and probably cannot be resolved from steady state data alone. Thus, time dependent data are necessary to provide an alternate measure of the rate constants. CHAPTER 7 TIME DEPENDENT RESULTS Introduction In the previous chapter, the need for perturbation type kinetics measurements on the membrane ion transport system was demonstrated. In this chapter, the results of the time dependent measurements are given, and the con- cepts of the torus contribution to the carrier molecule concentration and the rate limiting step in the transport mechanism are further developed. Data Treatment Values for the rate constants shown in Figure 3-1 were obtained by curve fitting the differential equations (Equations (3-30)-(3-33) and Equation (3-39)) to membrane transients. The differential equations were numerically integrated using the "Gear" algorithm (31,32). This method is particularly well suited for transients of the type obtained from charge pulse experiments (i;g;, steep transients occur quickly after pulse application, and slower more gently sloping ones dominate later). The curve fitting was accomplished using a Modified Simplex (37,38) optimization routine. All of the curve fitting 85 86 and computations were performed on a PDP 11/40 computer using double precision word lengths (1:3;J 56 bit mantissa). The simplex was allowed to start at several different locations in the parameter space to see if it would always converge on the same final results. In some cases local minima (minimum residual between theoretical and actual curves) were encountered. These local minima could cause as much as 50% error in the final parameters if they were not recognized as false minima, but the magnitude of the residual could easily be used to discriminate between false minima and the best value obtained for the parameters. This behavior demonstrates the need to start the simplex in several different parameter space locations for each set of data. Results and Conclusions Logarithmic plots of the membrane voltage 3; time response are shown in Figures 7-1 and 7-2. The first point in each figure represents the first reliable data point to be recorded after completion of the applied charge pulse. This point was typically recorded 500 nano- seconds after the start of the pulse. The varying datum point density observed in Figure 7-1 is a result of super- imposing several transients collected at different record- ing rates. This was necessary in order to achieve an optimum data rate, and still cover a fairly large time 87 0.0 1» -b . — p . — p — . a q — a .. Q; r m3 mZHH -_ .mpcmfimcwpp mcmppEoE mo poaa oHEcqumwoq uh- .Hue enemas (A)90"| 000.0: 88 0.0 .COHpmmemp mfipmo 00 poaa Qfienpfipmwoq m3.mzae e _ d dbl — - q— 4- o. ..... 00o . a t _ . . . a _ .mus madman o.o ,000.0u an. Aug Aug \./ .An (\ 0¢.0n 89 interval. Good reproducibility of the measurement system is demonstrated by the degree with which the data from consecutive transients coincide. It can be seen from Figure 7-1 that an early relaxa- tion occurs within the first few microseconds after pulse application. An expansion of the early time domain is shown in Figure 7-2 (likewise a logarithmic plot). After the early relaxation, the log plot (Figure 7-1) becomes linear out to times greater than 50 microseconds after pulse application, where the curve seems again to gently bend off. This second bend in the curve may be due to another relaxation. However, the signal-to-noise ratio of the data has become too low to allow an accurate apprais- al. It is clear from the figure, however, that at least one relaxation (besides the normal voltage decay always encount- ered in charge pulse experiments) has been recorded. Linear plots of the same data used to generate the previous two figures are shown in Figures 7-3 and 7-0. The transient shown in Figure 7-3 was recorded at 1.25 M. The membrane forming solution contained 9.u5 x lO-LI MHz recording rate. The membrane forming solution con- tained 9.h5 x 10'” M dinactin, and the aqueous phases on both sides of the membrane contained 1 M_KC1 (ionic strength was maintained at 3 M with LiCl). The first 9 microseconds of the transient shown in Figure 7-3 are expanded in Figure 7—A. The same membrane was used for both figures. The first 90 .Nmz m0.H um vovhoomp pconCMLp ocmanoz .mln opswfim To; m:.mzie 0.0 0.0 “101“1“ 1 "i“i“ 1““ some A 0 HI. II.— S 00.0 91 up qL. .Nm: 0H 00 Umupoomp pcmfimswpp wasanmz mods: n )— — — p b q — u d a q 4r ‘— .alu onswfim Si-IOA 92 two points on the plot were recorded prior to charge pulse application. The next five points on the transient are due to the charge pulse. This was established using a "dummy" cell composed of resistors and capacitors having comparable value to those of the solution resistance and membrane capacitance. The eighth point on the transient was taken to be the first valid point for the membrane relaxation. The solid curve represents the result of the fit obtained from the simplex optimization routine. It can be seen from Figure 7-h that an early relaxa- tion takes place within the first few microseconds of the transient, followed by a region of nearly constant slope. It can also be seen from the figure that good fits of the data are obtained from the simplex curve fitting routine. The results obtained for a variety of experimental conditions are shown in Tables 7-la and 7-lb, for dinactin and monactin, respectively. The various results are in good agreement, considering the number of parameters which must be fit. It was necessary to assume a higher Ft (approximately 1.5 x larger) than would be arrived at from C8 X membrane thickness, in order to obtain satisfactory fits between the theoretical relationship and the data. The extra carrier molecules probably come from the torus region of the membrane. Values for It were chosen which yielded the best fit for each set of data. It can be seen from Tables 7-la and 7-1b that rt 93 mOH K 3.0 30H x 0.0 00H x m.H moa K 0.: MHIOH K m.0 N|OH N H:.m MIOH K 0.H mOH K 0.0 :0H N o.m 00H N m.H 00H N 0.: MHIOH N 0.0 NIOH K H:.@ MIOH N m.H mOH K 0.0 30H x 0.: 00H K 0.H 00H K 5.: MHIOH x m.m NIOH K H:.m MIOH K m.H Hum Him Hum n.HoE\mEo 050\moHoE mEo\mmHoE mEo\moHoE ox _on flux as be no o ..amu npwcohpm oHCOH .COHpmapcoocoo pmfipnmo HmpOp u up .capomsos pom muasmom .naln oHnt mOH K N.m 30H N 0.0 mOH N 0.: moa K 0.0 MHIOH K 0.0 NIOH N 00.0 MIOH K O.H moa K 3.: 30H x 0.0 mOH N 0.0 00H x 0.m MHIOH K m.b NIOH N m=.m MIOH x 0.0 mOH K N.m 30H M m.m mOH K 0.0 @0H K m.m MHIOH N 0.0 NIOH x m=.m mlOH N O.H mOH K m.m 30H N 0.0 00H N 0.H 00H N m.z MHIOH K 0.0 NIOH K m=.m MIOH K m.H mOH K m.m 20H K h.m 00H w 0.H 00H N =.= MHIOH N 0.0 NIOH N m=.m MIOH K m.H Hum Hum Hum m.HoE\mEo NEo\mmHoE mEo\onoE mEo\moHoE as awe a-u as pa mo o .mmu summonum oHQOH .COHumanoocoo nmfihpwo HmpOp u up .c«pomcao pom mpHSmom .malw canoe 9“ decreases along with decreasing ion concentration. This trend is consistent with the notion that the free carrier, rather than total carrier, is being buffered by the torus. If the surface complexation reactions maintain equilibrium conditions throughout the transient (this fact will be demonstrated later), the free carrier concentration can be calculated from Equation (7-1): rs = rt/2(1 + K1C1) (7-1) where K1 is the equilibrium constant for the surface complexation reaction (i;g;, kl/k-l)' The calculated values of P8 are shown in Table 7-2. It can be seen from the table that the value of PS remains fairly constant over the range of conditions. Some definite trends can be observed from the data in Tables 7-la and 7-1b. First, it is apparent that the value obtained for k_1 decreases along with decreasing ion concentration. This trend is probably due to coupling between the equilibrium constant (K1) and the value chosen for Pt. It can be seen from Equation (7-1) that if Pt increases while k_1 decreases, the value for rs and there- fore r18) can be maintained constant. Another trend obvious from Tables 7-1a and 7-lb, is that the value obtained for kis increases with decreasing ion concentration. This effect is probably due to the time uncertainty in the beginning of the transient. 95 Table 7-2. Theoretical free carrier concentration. Dinactin Monactin Ci I‘s Ci I's moles/cm3 moles/cm2 moles/cm3 moles/cm2 1.5 x 10.3 6.9 x lOull4 1.5 x 10..3 8.8 x lO-ll‘l 1.0 x 10"3 7.2 x 10'1“ 1.3 x 10‘3 9.8 x 10'1“ 0.6 x 10'3 6.8 x 10'1” 1.0 x 10‘3 9.1 x 10‘“l * 1.0 x 10"3 7.3 x 10'1“ a This value was scaled to give the same It as for the other entries in the table (i.e., X 9.H5/2.88). 96 Since the charge pulse has a finite width, reliable data cannot be obtained until several hundred nanoseconds after the start of pulse application (this is much faster than for a voltage step experiment). Therefore the perturba- tion will have been present for approximately half of the charging interval before valid data can be acquired. This time domain uncertainty in the first datum point causes a variant uncertainty in the initial conditions used for the numerical integration routine, since a greater relaxa- tion would occur during the "dead time" when greater ion and carrier concentrations were present. Thus, for the higher ion concentrations listed in Tables 7-la and 7—lb, a greater portion of the early relaxation is obscured by the charging pulse so a smaller value of kis is ob- tained. Several conclusions about the plots in Figures 7-1 and 7—A can be drawn from the values for the rate constants shown in Table 7-1. First, the early relaxation is a result of kis becoming very large due to the voltage gradient across the membrane. (The voltage gradient is assumed to go from left to right.) When kis becomes much larger than any of theother rate constants, the complex which was formed prior to charge pulse application (;;g;, during the equilibrium state) is immediately transported through the membrane, and this redistribution of charge causes the voltage across the membrane to decrease. After the early relaxation most of the complex at the left 97 interface will have been transported to the right inter- face so the rate becomes limited by another of the rate constants. Thus, the next more gently sloping portion on the curve is produced. Finally, when the voltage gradient across the membrane becomes small (l;2;: <100 mV), kis becomes smaller than the other rate constants and another relaxation should occur. This may be the cause of the slight departure of the low voltage portion of the curve in Figure 7-1. According to the above discussion the part of the transient which occurs immediately after the early relaxa- tion should be controlled by a non-voltage dependent parameter. To test this idea, an experiment was performed in which pulses of differing charge content were added to the membrane. The results of this experiment are shown in Figure 7-5. Both transients in the figure were recorded at 5 MHz. If the middle portion of the transient is controlled by a non-voltage dependent parameter, the curve should be essentially linear in that region. The solid line in the figure is the result of a linear least squares fit to the central portion of the transient. It can be seen from the figure that a reasonably good fit is achieved. In addition, the slopes of the curves agree within A% (within experimental error) whereas, a slope ratio of 1.3 would be expected if the reaction rate were limited by the voltage-dependent step. Therefore, Figure 7-5 shows that the middle portion of the transient is 98 .COproa oHUUHE 0p 0H0 nmocfia spas mpcmfimnmnp ocmnnEmz .mlu opswfim H-o;. \ . 03 mzHH 0 p _ 0.0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . Wye . . q _ _ _ _ _ _ _ . _ . _ _ a .000.0 8170A 000.0 99 indeed voltage independent. From the results given in Table 7-1, it seems likely that kS is ultimately the rate limiting parameter in the transport mechanism for continuous steady state transport. Even though the step governed by k8 ($024; free carrier translocation) is the final step in the mechanism, this step is probably responsible for the middle portion of the charge pulse transient (see Figure 7-3b). This is possible since the mechanism is cyclic, and because an examination of the amount of total carrier present in the membrane shows that each carrier molecule must complete the transport cycle several times in order to deplete all of the charge placed on the membrane's capacitance during a charge pulse experiment. The amount of charge necessary to bring the membrane's capacitance (5.9“ x 10'7 F/cm2) to 300 mV is approximately 1.8 x 10"7 coulombs. The largest carrier concentration (in the membrane forming solution) used in this work was 9.A5 x 10'!4 M. This results in a total carrier concentra- 13 tion of approximately 5.7 x 10' moles/cm2 (assuming a membrane thickness of 6 nm) in the membrane. Therefore, even if the torus region of the membrane supplies enough additional carrier to double the amount of total carrier in the membrane, there is still approximately a twofold excess of ions to be transported over total carrier in tfluamembrane. Further, the carrier is distributed between two chemical forms (i.e., free carrier and complex) at 100 two interfaces. If ks is the rate limiting step for the central portion of the membrane transient (see Figure 7-0), then another relaxation (caused by the surface reaction) should theo- retically occur after the initial fast relaxation due to the voltage related increase in kis' This relaxation is not observed in Figure 7-2,either because the amplitude of the second relaxation is very small, or because it is obscured by the curved portion at the end of the first relaxation in Figure 7-2. The theoretical carrier concentration XE time profiles are shown in Figure 7-6. These curves were generated by numerically integrating the transport differential equations (Equations (3-30)-(3-33), and (3-39)) using the parameters obtained from the curve fitting routine. Curve 1 represents P; (1:52; the free carrier concentration at the left inter- face), curve 2 represents Pg, curve 3 represents Pig, and curve A is for P38. The potential gradient goes from left to right. Before charge pulse application the transport system is under equilibrium conditions. Therefore, at time t=0, r; = I; and P' = r" is is“ is applied to the membrane, the complex at the left inter- When the potential gradient face (rig) is rapidly transported to the right interface (P38). Next, the surface complexation reactions readJust causing P; to decrease, and P; to increase until a pseudo- steady state value is achieved for each form of the 101 .nmfihpmo onu mo meaom msofipm> you mmafimopa oEHp .m> coaumpucoocoo .010 mpsmfim CD 01 O O — p b p — n — — — F b p — p b n — p . ._ . a. _ . _1_. _ i_ . _ ..m00~.01 ............. a 1+1 In“ .. “H 3 i... H mm. W H X 0 H I .1. 1......10 /. H To 1.”. 83 H W 11”. * U. .x. am 8 . ..... H ...................... t lHu nmoomé 102 carrier molecule. It can be seen from the figure that, except during the initial fast relaxation, the surface complexation reactions remain essentially under condi- tions of equilibrium. Overview of Transport Scheme From the results presented in this chapter and Chapter 6, an overview of the transport mechanism for the actin homologs can now be given. When a membrane which contains carrier molecules is formed between two aqueous phases which contain trans- portable ions, an equilibrium between the complexed and uncomplexed form of the carrier molecule is quickly es- tablished. Simultaneously a slower exchange of free car- rier molecules takes place between the torus region and the membrane. This exchange establishes a partition which tends to maintain a constant free carrier concentration within the membrane. It is important to note that the free carrier, rather than total carrier, is being buffered by the torus region. This means that besides the electro- static difference, there must also be a chemical differ- ence between the free carrier molecule and the ion-carrier complex. This is consistent with the finding of Dobler g2 al. (39,“0) that the actins undergo a conformational change upon complexation with a K+ ion. If a voltage is quickly placed across the membrane, 103 the voltage-dependent rate constant (kis) becomes very large which results in a net flow of ion-carrier complex molecules in the direction of the electric field. This relocation of charge within the membrane is responsible for the early relaxation observed in the transients shown previously in the chapter. After a short time the concentration of the ion-carrier complex becomes depleted at one interface due to the charge redistribution mentioned above. At this point the rate becomes controlled by the diffusion of free carrier mole- cules through the membrane, which is not a voltage depen- dent process. Finally as the voltage across the membrane becomes diminished by the transport of ions through the membrane, the voltage-dependent parameter (kis) becomes small enough to be rate limiting. This leads to three distinct regions which can be observed in the transients obtained from charge pulse experiments. This also means that either ks or kis can be observed to be the rate limiting parameter for conventional steady state transport experiments, de- pending on the transmembrane voltage used in the deter- mination. The exchange rate in the surface complexation reaction is fast enough to maintain virtually equilibrium conditions throughout even the early portions of the membrane tran- sients observed in this study. Very rapid measurements 100 and large relaxation amplitudes will be necessary in order to obtain any information about the surface complexation reaction other than the equilibrium constant. Discussion on Apparent Carrier Buffering Discrepancy In Chapter 6, the conductance Kg ion concentration behavior (see Figure 6-A) was interpreted to be due to the equilibrium of the surface complexation reaction. It was postulated that F (and therefore the conductance) is increased linearly with ion concentration until an ap- preciable amount of Ts had been consumed, where the con- ductance XE ion concentration curve flattened off. An implicit (and necessary!) assumption in this argument is that the total supply of carrier (all chemical forms) in the membrane remains constant. I In this chapter, the results of curve-fitting to the membrane transients indicated that PS remained constant due to buffering by the membrane torus region. A dis- crepancy between the results in Chapters 6 and 7 exists at both interfaces) in that PS and Ft (i.e., F + P S is cannot both remain constant, while the ion concentration is varied, if equilibrium is to be maintained. The results from both chapters overwhelmingly show that the surface complexation reaction maintains equilib- rium conditions essentially at all times during the re— corded transient. Therefore, the discrepancy must exist 105 in the conclusions regarding carrier buffering by the torus. There are several possible explanations which could account for the discrepancy observed. First, it is pos- sible that the conductance Kg ion concentration behavior is a result of equilibrium; however, the limiting factor which causes the curve to flatten off is some other factor rather than a limited supply of PS. It has been postulated (M3) that the ion-carrier complex molecule is adsorbed to the membrane surface. In this case the limiting factor, in the conductance X§_ion concentration curve could be the number of surface sites available for the complex. In this case a similar behavior to that seen in Figure 6-u would be observed. Another possible explanation is that rs does not remain constant, and the change observed in P in Tables 7-1a t and 7-lb results from some other phenomenon. One cause for the observed change in r could be the coupling between t Pt and K1 which has been previously mentioned. Since k_l was observed to decrease with ion concentration, the change in Ft would be offset by the shift in the equi- librium. It should be pointed out, however, that the relative change in k_ was larger than that obtained 1 for Pt. Another possible cause for the observed change in Ft could be that carrier molecules can exist inside the 106 membrane, rather than Just at the interface. In this case, the surface reaction would still be sensitive only to carrier (or complex) concentrations at the interface, and the interfacial carrier concentration could be buf- fered by the carrier in the membrane interior. This could cause Pt to become apparently larger with increasing ion concentration, Just as it would in the case of torus buffering. The maJor difference in this case is that the buffering of rs would be much more limited than in the case of torus buffering. Therefore, the conductance Kg ion concentration behavior seen in Figure 6-A could still be explained in terms of a limited supply of carrier molecules. Experiments designed to test the proposed surface adsorption site limit are presently underway. No de- finitive results have yet been obtained. If the surface adsorption hypothesis can be discounted, the evidence in support of the notion that r remains constant will be t substantial. In that case the observed trend in the Ft values seen in Tables 7-la and 7-lb will have to be con- sidered an experimental artifact. It should also be pointed out that since the surface complexation reaction maintains equilibrium conditions during the transient, the theoretical expressions could be simplified by replacing kl/k_1 by the equilibrium constant (Kl)° In fact, the ratio of kl/k_1 is probably 107 all that is attainable from the measurements performed in this research, and thus, separate values for kl and k_l should not be listed in Tables 7—1a and 7-lb. CHAPTER 8 FUTURE WORK Short Term The only change planned for the measurement system is to make the cell amplifier gain selectable over a X5 to X100 range. This would permit low amplitude charge pulse experiments to be easily carried out. These experi- ments are useful since they produce low voltage transi- ents. The computations on these data are easier than for large amplitude charge pulse experiments because low voltage approximations can be used to simplify the theo— retical expressions. One of the maJor sources of uncertainty with either of the kinetics methods used in this study is the de- termination of the total carrier concentration in the membrane. For the time dependent analysis, values for rt were found which gave the best fit between the theo- retical relationship and the data. In the case of the steady state analysis the value for It had to be esti- mated. An independent measurement of Pt, or some form of the carrier molecule, is necessary in order to improve the uncertainty in the determinations. One approach which could be attempted is to label the carrier molecule with, for example, a fluorescent group. Then spectrometric 108 109 measurements could be used to measure the total carrier concentrations in the membrane at any time. Membranes which contain pores or pore forming agents should be tested, using the charge pulse technique, in order to see ifaxw*early relaxations can be observed which are similar to that seen in Figure 7-h. It seems likely that, for pore transport, no early relaxations will occur, and thus the charge pulse experiments could be used to discriminate between the two transport mechanisms. Long Term The charge pulse technique has been shown to be a valuable tool for measuring membrane ion transport phenom- ena. It is hoped that the method can eventually be used to study reconstituted or natural membranes. These systems are much too complicated to be studied at present, but as the method becomes perfected, it should prove to be useful in the study of these systems. APPENDICES APPENDIX A DESCRIPTION OF INTERFACE The purpose of this section is to describe the instru— ment in detail. Schematics are included for all circuit cards, and edge connector assignments are shown. On the schematics which follow, all NAND gates shown are 7&00 and all monostables are 7u12l unless otherwise specified. Edge connector contacts are labeled according to the con- nector positioned in slot #1 of the instrument. A block diagram of the measurement system which demonstrates the interconnection of the circuit modules, and the direction of signal flow appears in Figure A-l. The instrument is interfaced to the PDP ll/AO through a DEC DRll-A General Device Register (M786). This device provides data transfer both into and out from the PDP ll Unibus, and two request or interrupt channels. Schematics for this device along with a physical description can be found in the DEC 1970 Peripherals Handbook or 1970 Unibus Manual. The DRll-A was modified to produce two different output addresses in order that both a command register and data register can exist in the instrument. This allows for indirect addressing at the experiment end of the interface, whereby certain bits in the command register 110 .Empwmm pcmsopsmmoe mo Emmwmflv xooam .H1¢ madman 111 wooumo wmwmoo< :ao < «0.305200 < .. a fi ZOOnhmm Tull OZHZHH |w0A RIBBON CABLE > O) V O 3 0.) (O m 11.. s :8 * O (D > 0') XMIT 2 < O H N Iv it «I- to N O 4- Vm Y4: 1 0 N O «8 Wm V; ¢ ii i- *- N no N «B -< D 2 x O D I D 3) H O --1 "N08 0000 0001 m N g. N 0002 -n it 0 9 VJ V w V O * N D D D Q N 0003 a: * up «>17 A; NV *% c: c: G) u: 0000 :: fin. * a 17 ...1 .c * '0 c: c: 0» .B 0005 N a 011/ H ‘D V * O c: c> UI 0006 R H a O 0"! D a, V * ~ 07 U D 0: CD 0007 O O 0.0 O R 2 N N N a * * i ii {1’ ... ‘yN N N 011p: 1n» Va:\/ 0. 1w to he ¢ 0') v9 v0! v0) it * 'I- i C < 8 ... D O D O O O O O .... Q 0 O 0 (.0 CD \1 0009 0011 '— N 'I» VG?\I 5.. ‘9 V .1. q C 0 v N 0012 M ii. 1m 5 v 'I' U) D D N 0013 N * O VG) .. 1 CD * 3 O O 0) 0014 0: co n: i 1* 01 \No 0» V“,‘V * 'I' z ‘o c» c: c> (3 III. 0.. (n s 0015 Figure A-2A. Schematic showing computer Data Out path. 119 30'-3§ CONDUCTOR RIBBON CABLE BB Figure A-ZB. Schematic showing computer Data In path. 120 mzu bzu mzm nzu tzm mzu Nzu «Zn mzu M in non M m. :2. m M MW“ e am” "so a N can IUh<4 .I .- I .I .mua mmswflm 121 homo memo mama 30mm momo Nome dawn oomo 2— 0.2 Z>Dhm .nmcoomo mmmnoo¢ mnp mo coapmpcwmmpamp ofipwamnom .:I< maswam zoo . >m ,Pum mu xuoquATl‘ - - _ m m mh————r§ -—-— + -——-10 L_‘E’_ .24, 19 IE:§ I xn17>——————L_____q. fins FN1>————-—J“_"*-—- [-—-u a ea>o14 cue eua>“ TIe7>£————1fi’ L__, FN4>F FN1>£~ CONVERT>~fl-———4 o ”s 1 a ‘ s12 convem j1>saH A0c22a 015 ADC Y 76 FN3 Figure A-8A. Douglas patch card, part A. 126 s #4 PULSE TR] ‘ GEN. MANUA START +1sv>R 7805 11:>+5v . a v ~15v>-. 7995 >- v L____, 5 Figure A-8B. Douglas patch card, part B. 127 OUT STROBE DATA HIGH: "—:1iD> ‘ ’”’——"—1 ‘thjr-r—4:>-{:»———>Non OUT LON: STROBE DATA F’ (LON BYTE) SELB >-T——' *5 , 38 ~>As 59- I * . I 479% +5 Figure A-9. Modifications to DRll-A. APPENDIX B PROGRAM LISTINGS 128 OOOOOOOOOOOOOOOOOOOOOOO HOOOOO O 200 210 20 300 3 10 320 325 129 MEMBRN. FT'N VERSION 04 MANAGER PROGRAM FOR MEMBRANE ION TRANSPORT STUDY AUTHOR: '1‘. A. LAST DATE: 16-Nov-75 THE OPTIONS AVAILABLE TO THIS PROGRAM ARE GIVEN IN THE OPTIONS LIST (IE. FORMAT 200) . AVERAGE: TAKES N SCANS. ALLIGNS THE NOISE GLIT‘C- AND AVERAGES THE SCANS (USED TO OBTAIN BASELINE). NOTE: BASELINE MUST BE TAKEN BEFORE DATA CAN BE STORED SEPARATE DATA AND BASELINE ARRAYS ARE MAINTAINED. AND UPON STORAGE THE DATA ARE AUTOMATICALLY BASELINE SUBTRACTED AND LINEARITY CORRECTED. BY ISSUING A PLOT COMMAND AFTER DATA STORAGE, THE CORRECTED DATA CAN BE VIEWED. FILES ARE CREATED IN THE STANDARD ENKE FORMAT ON THE DEVICE SPECIFIED. <<<< WARNING--MUST COMPILE WITH '/ON’ SWITCH >>>> BYTE FILNAM( 24) ,DATI9) .TIM( B) DIMENSION A( 500) .B(100).C(IO) A'DATA ARRAY. B=BASELINE ARRAY. C=LINEARITY CORRECTION COEFF. INTEGER POINTS( 500) .OPTION. AUTO *2: D I SPATCHER *8: IQUE=2OO IQUE=200 MEANS DATA COLLECTION ROUTINE SHOULD INITIALIZE. CALL COLECTI NPTS. IPUSH. POINTS. IQUE. ISROPT) WRITE(6.200) FORMAT( IX. ’ OPTIONS: COLLECT DATA. AVERAGE. LIST DATA 1. PLOT DATA’/’ STORE DATA. BASELINE 2 PLOT. EXIT’/’I?') READ(6.2IO)OPTION FORMAT(AI) IF (OPTION.E0. IHC) GOTO 20 IF (OPTION.EQ.IHL) GOTO 40 IF (OPTION.EQ. IHB) GOTO 59 IF (OPTION.EQ.IHP) GOTO 60 IF (OPTION.EQ.1HS) GOTO 80 IF (OPT‘ION.EQ.IHA) GOTO [00 IF (OPTION.NE.1HE) GOTO 10 STOP *8 DATA COLLECTION SECTION *1: WRITE( 6 . 300) FORMAT( 'OENTER # OF DATA POINTS (DEFAULT3 128) : ') READ( 6 .310) NPTS FORMATI I4) IF (NPTS.LE.O) NPTS=128 WRITE( 6 .320) FORMATI ' IAUTO OR MANUAL?') READIG. ZIOJAUTO IPUSH=O IF (AUTO.EQ. IHA) IPUSII=I WRITE(6.325) FORMAT( IX.’ SET UP SWITCH REGISTER OPTIONS. TYPE CO') SWITCH REGISTER OPTIONS : MEMBRN.FTN 00000000000000000 0 30 40 330 50 340 0000 60 61 0000000 BITS 0-6 8-10 11-13 130 PAGE FUNCTION SET # FAST DATA POINTS COLLECTED AFTER TRIG. MAX.= 9B. (PRE-TRIG. RECORD 8. 98.- # SET) SET FAST DATA RATE RATE 3 10 MHZ/(233* #SET) SET SLOW DATA RATE (A/D CONVERT RATE) RATE 3 39.0625 KHZ/(23H: #SET) NOTE: 39.0625 KHZ IS FROM 10 MHZ/(2388) FOR#SET (BITS 11-13) > 4 THE RETICON DROOP BECOMES SIGNIFICANT PAUSE CALL COLECT( NPTS . IPUSB. POINTS . IQUE. JSROPT) 00 GET DATA DO 30 I=I.NPTS A( I)=FLOAT( POINTS( 1)) CONTINUE GOTO 10 8* DATA LISTING SECTION ** WRITEI6o380) FORMATI ’SENTER DESTINATION (LPJCB) : ') READI6.210)AUTO IVAR=6 IF (AUTO.EQ. IHL) IVAR=5 WRITEI IVAR.340) (AI I) . [=1 .N'PTS) FORMATI5I4X.F7.2)) WRITE( IVAR.350) [QUE FORMAT( ' NOISE QUE 3 ’ . I4/2X/21V' 1 ’) GOTO 10 an: DATA PLOT'I'ING SECTION *8! LB=1 KPTS= 100 GOTO 61 LB'O KPTS=NPTS YSCALES 700 . 0/2048 IBCALE= 1000.0/KPTS CALL T4010(0. I. I) CALL T4010I5.I.1) DO 70 I=I.KPTS 1Y8 INT( AI I)*YSCALE) IF (LB.EQ.1) IYa INT( BI I)*YSCALE) IX: INT( HEIBCALE) CALL T4010I6. IX. IY) CONTINUE CALL T4010I1.0.780) CALL T4010I3.1.1) GOTO 10 8* DATA STORAGE SECTION an: ASCII FILES - FIELD WIDTH (DATA) 3 15 FILE STRUCTURE: (NU) . # DATA POINTS 2 131 MEMBRN.FTN 800000000000000 6) a a “O Ou- O 00 q 0 375 82 838 0G *3 (RD) . DATA BLOCK < ED) . END OF DATA BLOCK (BA) . BASELINE AVERAGE (GU) . NOISE QUE (SR) . SR OPTIONS (Fl). FILE NAME.EXT I DA) . DATE TAKEN (TI) . TIME TAKEN SAMPLE RECORD: CODE. FORMAT. FORMAT FOR REAL DATA: RD. 115. F15.2 IF (BASAVG.NE.0) GOTO 81 WRITEI6.355) FORMATI ' NO BASELINE--NO ACTION TAKEN’) GOTO 10 WRITEI6.360) FORMATI 'SENTER FILNAME: ') READI6.370)FILNAM FORMAT( 24A1) CALL DATEI DAT) CALL TIME(TIM) IFRR=0 CALL FORPIP(FILNAM. IFRR) IF (IFRR.EQ.0) GOTO 82 WRITEI6,375) FORMATI ' FILE ALREADY EXISTS--NO ACTION TAKEN") GOTO 10 CALL ASSIGNI4.FILNAM.24. IERR) WRIT'EI4.380)NPTS FORMATI ' NU’ . IN) BASELINE SUBTRACTION (ALLIGNS NOISE QUE FIRST) K8 IQUE JBJBQUE IOFF=MODINPTS. 100) DO 84 I'1.NPTS AI K) 8A( IO -B( J) +BASAVG K=K+1 J=J+I IF (K.GT.NPTS) GOTO 85 IF (J.GT. 100) J=J-100 CONTINUE GOTO 87 K'K-NPTS J'J+100-IOFF GOTO 83 NTERIB=6 LINEARITY COEFFICIENTS: I FROM POLFIT ) CI 1)=-0.00035 CI 2) =0 . 96329 C(3)8-0.01033 C(4)=0.01107 C(5)=-0.00271 C(6)=0.00024 DO 90 I'I.NPTS Y=000 TERM=1.0 YI=(U.0/2068) *A( I) LINEARITY CORRECTION DO 88 N=1.NTERIB Y=Y+CI N)*TERM TERM=TERMIKYI CONTINUE PAGE 132 MEMBRN.FTN 90 390 392 395 400 405 410 420 100 500 105 505 110 120 I80 185 000 WRITEI4,390) I.Y CONTINUE FORMAT( ’RD' , I15.F15.5) BASAVE= BASAVG*5 . 0/2048 WRITEI4.392) FORMAT( ’ ED’) WRITEI4,395)BASAVE. IQUE WRITEI 4.400)JSROPT FORMAT( 'BA' .F10.6/'O.U‘ . I4) FORMAT( 'SR’ .06) WRITEI4.405)FILNAM FORMAT( 'FI ' . 1X.24A1) WRITEI4.410)DAT FORMAT( ’DA' . IX.9AI) WRITEI4.420)TIM FORMAT( 'TI ’ , IX.8AI) END FILE 4 GOTO 10 an: BASE LINE AVERAGING SECTION 8* WRITE(6.500) FORMAT( ’SHOW MANY SCANS?: ') READI 6 . 505) NSCANS FORMAT( 16) IF (NSCANS.LE.0) NSCANS= 10 DO 110 I' I . 100 BI I) =0.0 CONTINUE DO 180 J=I.NSCANS CALL COLECT( 100,1.POINTS.JBQUE.ISROPT) K'JBQUE DO 120 II= 1 . 100 BI I) =BI I)+FLOAT( POINTSIIO ) I'M-1'1 IF (ILGT. 100) EUR-100 CONTINUE CONTINUE BASAVG=0.0 DO 185 I= I . 100 BI I) =BI I)/NSCANS BASAVG= BASAVG+BI I) CONTINUE BASAVG= BASAVG/ 100 . 0 JBQUE=I GOTO 10 END THE LINK nsscmr'mn muons: MEMBRM PIEHBRN.FORPIP.COLECT.T4010.FTNLIB/E PAGE 4 .TITLE COLECT.MAC . IDENT /V0.04/ cAUTHOR: T.A. LAST 8 DATE: n-nov-Ta .GIJOBL COLECT .NLIST BEX.T'I'M . PSECT COLECT. RW. REL I ROSSO R1881 R2382 R3833 R4=Z4 R5'X5 SP=X6 PC'X? P83177776 SRFI77570 TPS=177564 TPB8177566 CRFI5 LF=12 FLAG=164000 DATO=164002 DATI=164004 CMRSI64006 : POINTR: .WORD 0 11sz .1108!) 0 IPUSH: .WORD 0 SROPT: .1108!) 0 POL: .WOBD o I .MCALL .BIN2D I COLECT: nov ozcnmma mv mans nov «(35). [PUSH nov «mum nov a4,Pom'm cm can our 610(R5).#100. BLOS PA882 nov #30.DATO nov “woman CLR 910mm uov # 1000mm 31's as PAsszz cm POL WAIT ‘ mv PS.-(SP) nov #340,1’8 nov 911.80 mv R0.SROPT DEC no mv nomxro nov #400,011}! mv #10.Rl nov 911.110 swan no mv 110.112 BIC #177770.RO ASH £0.11: 133 ““00““ ..."... 00“..“ “00“-..“ : MEMBRANE DATA COLLECTION SUB. REGISTER DEFINITIONS PROCESSOR STATUS WORD SWITCH REGISTER TELEPRINTER STATUS TELEPRINTER BUFFER DATA READY FLAG DATA OUT REGISTER DATA IN REGISTER EXPERIMENT COMMAND REGISTER POINTS TO FORTRAN DATA ARRAY CONTAINS 1) OF DATA POINTS '1 FOR AUTO PULSE TRIGGER SWITCH REGISTER OPTIONS 8 1 FOR NEGATIVE DATA BINARY TO DECIMAL CONVERSION ROUTINE R5 CALLING CONVENTION SEE ABOVE FOR NPTS. IPUSH. POINTR POINT TO EXPERIMENT REG. 0 CHECK FOR FIRST PASS NO - NOT FIRST PASS FIRST PASS INITIALIZATION WAIT FOR LINE FREQ. INTERRUPT SAVE PROCISOR STATTB MAKE CPU UNINTERRUPTABLE SET UP EXP. CONDITIONS SAVE SWITCH REGISTER OPTIONS (EXP. COUNTS DOWN TO UNDERFLOW) POINT TO A/D CONVERTER. INITIALIZE SET UP WAIT LOOP FOR PRE-TRIGGER RECORD _ ,- 01* COLECT. MAC CHECK: STORE: CUE: OVER: : EEO l : DATA: ASH #75.R2 BIC #177770,” HOV #20.” ASE R2.” ADD 110.111 ADD #20.R1 808 R10 o-l mV #2000.CMR TS'I' IPUSH BEQ CHECK mV # IOOO.CPIR BIT # 100000. FLAG BEQ CHECK IDV DATI.(R4)+ MOV # 2000.CMR SOB R3.CHECK INJV (SP)+.PS CLR R2 MOV POINTR.R4 INC R2 mv (R4) .R1 COM R1 BMI ERR CMP R2. #100. RBI OVER BIT # 40000.R1 BEQ OVER MOV R2.010(R5) BIT# 4000.R1 BEQ POS INJV #I.POL BIC #174000.R1 INN R1.(R4)+ CMP R2.NPTS BLO STORE MV SROPT.R1 BIC #140200.R1 MOV R1.012(R5) RTS R5 . PAGE . BIN2D #DATA. R2 MOV #ER01.R3 TSTB (R3) BEQ CUE TST'B TPS BPL PUTRO MOVB (R3)+.TPB BR PTIBRB 1311 PAGE SET UP WAIT LOOP FOR FLAG THIS STATEMENT CAUSES ’A’ ERROR -- OK CLEAR FLAG IIggECK IF AUTO TRIG. TRIGGER SEE IF DATA READY NO - GO BACK GET DATA CLEAR FLAG MORE DATA? RESTORE OLD PROCESSOR STATUS ARRAY STORAGE ROUTINE INVERT TO POSITIVE LOGIC TEST FOR A/D CONV. ERROR TFST FOR NOISE CUE NO LOG THE CUE NEGATIVE VOLTAGE? NO NOTE THAT IT IS NEG. CLEAR GARBAGE - 12 BIT CONVERTER STORE DATA POINT IN FORTRAN ARRAY MORE DATA? YFS - GO BACK GET SWITCH REG. OPTIONS CLEAR GARBAGE SEND TO FORTRAN mum: ROUTINE RETURN TO CALLER TAG BAD DATA POINT (DECIMAL) GET ERROR MESSAGE PRINT MESSAGE END OF MESSAGE. GO BACK TELETYPE COMMUNICATION .ASCII SA/D CONVERSION ERROR. CELL #S .BLICB 5 .ASCIZ .END COLECT 135 .TITLE T4010 BT'TL PRELIMINARIES DENT /T40 10/ .PSECT T4010.REL.RW. I .GLOBL T4010 E T4010 WRITTEN BY TOM ATKINSON AND JIM HORNSHU'H FOR THE C.G.ENKE RESEARCH GROUP 18-JUN-75 T4010 IS A SUBROUTINE FOR GRAPHIC [/0 WITH A TEKTRONICS 4010 TERMINAL AND CURRENTLY HAS SEVENI7) IDDES OF OPERATION 0 - ERASE SCREEN.RETURN TO ALPHA MODE.AND GO HOMEIO.780) 1 - GO TO GRAPHIC MODE.CONSTRUCT DARK VECTOR - CONSTRUCT LIGHT VECTOR - RETURN TO ALPHA MOE GRAPHIC INPUT.RETURN CROSS-HAIR LOCATON IN DUMMY - PLOT AXIS AND TICK MARKS - GO TO GRAPHIC nonzmom'r PLOT - GRAPHIC INPUT WITHOUT SCALE SHIFT FOR USE WHEN NOT READING DATA PLOT'TED WITH THIS SUBROUTINE “10‘“ 00010 1 FOR MODE 5 OPERATION CALL T4010I IMODE. IXDIV. IYDIV) I MODE 3 MODE OF OPERATION IXDIV 8 NUMBER OF X DIVISIONS DESIRED IYDIV 8 NUMBER OF Y DIVISIONS DESIRED FOR ALL OTHER MODES CALL T4010I IMODE. IXCORD. IYCORD) IXCORD = INTEGER VALUE 01" X COORDINATE (0-1000) IYCORD ‘ INTEGER VALUE OF Y COORDINATE (0-600) THIS SUBROUTINE SHIFTS THE SCREEN ORIGIN TO (10.10) FOR ALL PLO’I'I‘ING AND GRAPHIC INPUT EXCEPT FOR mm: 7. THIS SHIFTING IS DONE INTERNALLY AND NEED NOT CONCERN THE PROGRAMMER EXCEPT TO TAKE NOTE THEREOF. . PAGE . NLIST T'TM R0=%0 : REGISTER DEFINITION R1 =81 R2= X2 R3= X3 R4= X4 R5335 SP=X6 PC= X7 MASKR5= 177740 MASI-5= 176037 TXCSR= 177564 TXBUF= 177566 RVCSR= 177560 RVBUF= 177562 MASKR5 8 .NOT. 000037 MASKSR 8 .NOT.001740 T4010 TRANSMITTER COMMAND. STATUS REG T4010 TRANSMITTER DATA REGISTER T4010: RECEIVER COMMAND. STATUS REG T4010: RECEIVER DATA REGISTER CSK=33 T4010: ERASE. GO TO ALPHA MODE. AND CL=14 T4010: RETURN TO HOME (0.780) CSO= 37 T4010: GO TO ALPHA MODE CSM=35 T4010: GO TO GRAPHICS MDE C2332 T4010: STARTS CROSS-HAIR WHEN PRECEEDED BY CSK 136 T40 10 . MAC PAGE 2 . PAGE . SBT'I'L MACROS . MCALL . F4DEF .F4DEF 0 ; ASSIGN R5 CONVENTION TO MACROS I .MACRO INPUT.DUMMI.DUMMY DUMMI: TSTB RVCSR :T'EST BIT7. IS DATA READY BPL DUMMI :BRANCH IF NOT READY MOVB RVBUF.DUMMY :DATA TO CPU .ENDM I .MACRO COORD.HI.LO.VALUE CLR R0 MOVB HI.R0 :HI BYTE->R0 BIC #MASKR5.R0 :MASK OFF RIGHT 5 BITS ROL R0 TROTAT'E R0 5 TIMES TO LEFT ROL R0 ROL R0 ROL R0 ROL R0 BIC #MASKS5.R0 :MASK OFF BITS5-9 CLR R1 MOVB LO.R1 3L0 BYTE->R1 . BIC #MASKR5.R1 :MASK OFF BITS 0-4 ADD R1.R0 :ADD HI AND LO BYTES-)COORDINATE MOV R0. VALUE .ENDM I .NLIST ME I .PAGE .SBT'TL MAIN T4010: MOV R0.-(SP) : SAVE REGISTERS mv R1.-(SP) MOV R2.-(SP) MOV R3.’(SP) MOV R4.‘(SP) MOV 2(R5) .R4 : IMODE ADDRESS -) R4 MOV (R4).R0 : [MODE VALUE -> R0 TST R0 : TEST IF )0 BGE L0 : YES. CONTINUE JMP ERR : NO.RETURN L0: CMP #7.R0 : IMODE .GT. 7 IS ILLEGAL BEQ A002 3 ADD 2 TO GET OVER JUMP IN TABLE CMP #6.R0 BEQ ADDI BGE L1 JMP ERR A002: ADD #2.R0 BR LI ADDI: INC R0 L1: ROL R0 : MULTIPLY BY TWO ADD AJT. R0 : CONSTRUCT JMP ADDR JMP (R0) JTABLE: BR ERASE BR GRAPHl BR VECTR2 BR ALPHA BR GIN JMP AXIS JMP PTPLT JMP GIN ERASE : MOV #CSK. R0 T4010.MAC GRAPHI : ALPHA: GIN: GINNS: AXIS: BEQ JSR JSR CLR R5.0UTPUT #CL.R0 R5.0UTPUT EXTT 4(R5).R4 (R4).XNUM #12.XNUM 6(R5).R4 (R4).YNUM #12,YNUM R5.GRAPH EXIT 4(R5).R4 IRA).XNUM #12.XNUM 6(R5).R4 (RA).YNUM #12.YNUM R5.VECTR EXTT #CSO.R0 R5.0UTPUT EXIT #100.RVCSR #CSK.R0 R5.0UTPUT #CZ.RO R5.0UTPUT AGAIN.R0 AGN1.HIX AGN2.LOX AGN3.HIY AGN4.LOY HIX. LOX. XCORD HI Y. LOY. YCORD 2015),!!!» (no) .111 #7.!11 stuns # 12.xconn #12.Yconn «95) .m xconmuw) saw) ,no mommnm #Too.nvcsn EXIT 1762.111“)?! 12.1mm 35.0mm #12.xmm R5.VECTR #uaznmtm ”.mm no #1762.RI 4015).!12 (112) .na 137 PAGE :SHIFT ORIGIN TO (10.10) 3 SET TO ALPHA MODE :NON-SCALED GRAPHIC INPUT :SHIFT DATA BACK TO ORIGIN (0.0) :RESET INT ENBL BIT mm x DIVISIONS REQUIRED :YES. CONTINUE 3110. SET XOIV=0 T4010JIAC 110v 111142.111 110v 6(R5).R2 NOV 1112).” TST 113 311E m 01.11 YDIV TST xmv BNE XTICK .111? EXIT A112: 111v 113.110 110v R0.Yn1v mv #12.XNUII XTICK: TST xmv BEQ YTICK XNEXI‘: ADD xnIv.xmm 110v 1112.111111]! CMP #1762.XNUH 13111 LASTX JSR 115.cm11 mv 1120.11111111 JSR R5.VECTR .111? XNEXT LASTX: 110v 111762.le .1811 35.011011 110v 1120.1111111 JSR R5.VECTR 1'1ch: 110v #12.YIIUII 110v 41123111111 TST YDIV BM: 00 JMP EXIT 00: ADD YDIV.YNUM cur 1111423111111 BGE CONT LASTY: 110v 1111423111111 .1811 85.01111” 110v 4120.1011111 .1811 115.VEOT11 110v 1112.11111111 .1111: EXIT CONT: JSR 115.cmn 110v #20.xmm .1811 R5.VECTR 110v #12.xmm JMP co PTPLT: 110v 411151.114 110v (1111.1011111 ADD “2.1111011 110v 6(R5).R4 110v (114).?111111 ADD “2.1111011 .1811 115.cwn .1811 R5.VECTR .1111- EXIT 111111: EXIT: 110v (8P)+.R4 110v (SP)+.R3 110v (SP)+.R2 110v (SN-1,111 110v (SP)+.RO 1118 1111 .PAGE . SBT'TL SUBROUTINES 138 PAGE 4 :ARE Y DIVISIONS REQUIRED :YES. CONTINUE :ARE X DIVISIONS REQUIRED :YES DRAW TICKS 1N0. EXIT :DRAW FINAL TICK MARK AT 1000 :DRAW FINAL Y TICK MARK AT 600 : ADD OFFSET BIAS 8 RESTORE REGISTERS 8ANDRETURN SUBROUTINE GRAPH: DRAW DARK OR LIGHT VECTOR ON TERMINAL SCREEN T6010. MAC 8 GRAPH: OUT1 1 HIX: LOX: HIY: LOYS XCORD: YCORD: XDIV: YDIV: X31111: YNUM: AJT: MOV JSR MOV JSR BIC BIS JSR PDV BIC BIS JSR MOV JSR BIC BIS JSR WV BIC BIS JSR RTS 139 PAGE :DOUPPERY I 8 I I ROTATE 5 RIGHT MASK OFF RIGHT 5 BITS INSERT UPPER Y TAG SEND UPPER Y BYTE 1DOLOWERY MASK OFF RIGHT 5 BITS INSERT LOWER Y TAG DO UPPER X ROTATE 5 RIGHT MASK OFF RIGHT 5 BITS INSERT UPPER X TAG DO LOWER X MASK OFF RIGHT 5 BITS INSERT LOWER X TAG SUBROUTINE ROT5: ROTA'I'E REGISTER O FIVE TIMES TO THE RIGHT ROR ROR ROR ROR ROR RTS SUBROUTINE OUTPUT: TXCSR OUTPUT TXCSR. -( SP) TSTB BPL MOV CLR B I 8 MOVE TSTB BPL MOV RTS . BYTE . BYTE . BYTE . BYTE . EVEN . WORD . WORD . WORD . WORD . WORD . WORD . WORD . END (SP)+.TXCSR JTABLE T40 10 RETURN OUTPUT ONE BYTE TO THE T4010 IS TERMINAL BUSY? STATUS REGISTER POS MEANS BUSY SAVE CURRENT TERM TRANS STATUS‘ BE SURE INTERRUPT IS OFF SET BIT 8 OF BYTE SEND IT OUT TERMINAL BUSY? NO RESTORE STATUS RETURN 1ADDR OF JTABLE 5 00000000000000 100 200 210 220 10 225 230 235 240 I5 245 250 20 1‘40 KINRLX.FTN PROGRAM TO TEST-KINFIT RESULTS CAN ALSO BE USED TO OBTAIN PARAMETER ESTIMATES ESTIMATES FOR THE PARAMETERS ARE PLUGGED INTO THE CONDUCTANCE DIFFERENTIAL EQUATIONS (UP TO 10) AND THE D.EQ. ARE NUMERICALLY INTEGRATED (RUNGE-KUTTA). THE SOLID PLOT IS THE RESULT OF THE FIT. NOTE: WHEN USING KINRLX FOR TIME DEPENDENT FITTING. TRY TO PICK.THE FIRST FITTING POINT AS CLOSE TO T=O (PULSE APPLICATION) AS POSSIBLE. ELSE. THE INITIAL CONDITIONS WILL NOT BE VALID.. COMMON P(10).C(10) DOUBLE PRECESION Y.DY DIMENSION A1500).Y(10),DY(10) BYTE FILNAM(21).DAT(9).TIM(B) WRITE(6.IOO) FORMAT(!X.’ ENTER INPUT FILENAME:'/'O ’) READ(6,200)FILNAM FORMAT(24AI) CALL DSTRIP(FILNAM.A.NPTS.ICUE.ISROPT,DAT.TIM) WRITE(6.210) FORMAT(‘OENTER BASELINE AVERAGE:') READ(6.220)AVE FORMAT(E) AVE=AVE/7.B DO 10 I=I.NPTS A(I)=A(I)/7.B CONTINUE WRITE(6.225) FORMAT(‘OENTER.FIRST.LAST POINTS TO FIT:‘) READ(6.230)NPTI.NLAST FORMAT(2I5) WRITE(6.235) FORMAT(IX’ ENTER.3 CONSTS; ION CONC.PERIOD.CARR. CONC:'/) READ(6.240)C(I).C(3).C(4) FORMAT(QE) MEMBRANE CAPACITANCE FOLLOWS C(2)=5.94E-7 CALL SETERR(3.0) CALL SETERR( 4.0) AVAL1=.1 WRITE(6.245) FORMAT(IX’ ENTER 5 PARAMETER ESTIMATE81KINFIT’ORDER'. 1' [INTEG. IVAL]:’/) READ(6.250)UI.U2.U3.U4.U5.U6 FORMAT(GE) IF (U1.NE.0.0) P(l)8Ul IF (U2.NE.0.0) P(2)=U2 IF (U3.NE.0.0) P(3)=U3 IF (U4.NE.0.0) P(4)=U4 IF (U5.NE.0.0) P(5)IU5 ' IF (U6.NE.0.0) AVALI=U6 YSCALE=700.0/0.64 XSCALE=I000.0/NPTS CALL T4010(0.l.l) CALL T4010(5.l.l) DO 20 I'l.NPTS IXPINT((I+I)*XSCALE) IYilNT(A(I)*YSCALE) CALL T4010(6.IX.IY) CONTINUE IX=INT(NPTI*XSCALE) 1111 KINRLX. FTN 24 25 30 32 33 40 45 50 300 C C C IYIII INT( A( NPTI)*YSCALE) CALL T4010( 1. IX. IY) Y( I)=DBLE( A( NPTl)-AVE) NIB-1 CALL DIFFEQ(N.Y.DY) KFLAG=0 DO 40 I=NPTI .NLAST AJa I . AVALBAVALI IF (KFLAG.EQ. I) AVALS-AVAL CALL RUNGE( Y. DY. AVAL) IF ((KFLAG.EQ. I) .AND. (AVAL.NE.-AVALI)) AVALa-AVAL DO 30 J=1.N YIJ)=Y(J)+DY(J)*AVAL CONTINUE YY=SNGL( Y( N)) IF (ICFLAG.EQ.0) GOTO 32 AJ'AJ'I-AVAL GOTO 33 AJ'AJ-AVAL CALL TSTERR(3. IER) IF (IER.EQ.I) GOTO 50 CALL 'ISTERR(4. IER) IF (IER.EQ.1) GOTO 50 IF (AJ.GT.0.0) GOTO 25 IF (KFLAG.EQ. l) IflK-l IX= INTH 1+1)*XSCALE) IY= INT( (YY+AVE) *YSCALE) CALL T4010(2. IX. IY) IF (KFLAG.EQ.I) 00'“) 45 CONTINUE KFLAG=I IX' INT( NPT1*)BCALE) IY' INT( A(NPTI)*YSCALE) CALL T4010( 1 . IX. IY) KaNPTI Y( I)=DBLE( A( NPTl)-AVE) N3-I CALL DIFFEQ(N. Y. DY) KSK-l IF (K.GT.0) GOTO 24 CALL T4010(1 .0.780) CALL T4010“). 1 . l) WRITE(6.300)P(1).P(2).P(3) .P(4) .P(5) FORMAT( 'OPARAMETERS: '5E) GOTO 15 ‘END LINK DESCRIPTOR FOLLOWS: KINRLX< KINRLX. DSTRIP . T4010 . RUNGE. DIFFEQ. HYPER/CC. FTNLIB/E P AGE 0000000000 3 200 l 10 120 30 102 I30 40 140 50 150 60 I60 70 170 1112 SUBROUTINE DSTRIP(FILNAM. D. NPTS. ICUE. ISROPT. DAT,TIM) BYTE LINE(BO). FILNAM(24). DAT(9). TIM(B) DIMENSION D(500) EQUIVALENCE (JFLAG.LINE( 1)) FLAGS: NU 3 DATA POINTS RD 3 REAL DATA. FIELD WIDTH 8 I5 QU =3 NOISE INDEX PULSE SR = SWITCH REGISTER OPTIONS DA 8 DATE TI 3 TIME ED 8 END OF DATA BLOCK K80 CALL ASSIGN(4.FILNAM.24. IERR) READ(4.1!0.END=200)LINE IF (JFLAG.EQ.2HRD) GOTO 30 IF (JFLAG.EQ.2HNU) GOTO 20 IF (JFLAG.EQ.2HQU) GOTO 40 IF (JFLAG.EQ.2HDA) GOTO 50 IF (JFLAG.EQ.2HTI) GOTO 60 IF (JFLAG.EQ.2-R) GOTO 70 GOTO 10 END FILE 4 RETURN FORMAT(BOAI) DECODE(7. 120. LINE) NPTS FORMAT( 2X. I5) GOTO 10 K=K+I FORMAT(F15.5) DECODE(32.130.LINE)D(IO FORMAT( 2X. 15X.Fl5.5) 001!) 10 DECODE(6. l40.LINE) ICUE FORMAT( 2X. I4) GOTO 10 DECODE( I2. I50.LINE) DAT FORMAT( 3X. 9A1) GOTO IO DECODE(II.160. LINE) TIM FORMAT( 3X. BA! ) GOTO 10 DECODE( 8. I70 . LINE) ISROPT‘ FORMAT( 2X. 06) GOTO [0 END Tm 00000000000000 10 20 30 35 40 50 55 60 65 70 80 1113 SUBROUTINE RUNGE(Y.DY.AVAL) AUTO SCALING RUNGE-KUTTA NUMERICAL INTEGRATOR. CALLING PROGRAM SUGGESTS STEP SIZE IF A VARIABLE TRIES TO GO NEGATIVE. RUNGE SENSES THIS AND SOUEEZES THE STEP INTERVAL DOWN SUCH THAT THE VARIABLE MAY ONLY ASYMPTOTICALLY APPROACH 0. Y=DEPENDANT VARIABLE (COMES IN WITH INITIAL CONDITIONS) DY(J)=DERIVITIVE OF THE JTH VARIABLE AVAL=STEP SIZE (COMES IN AS SUGGESTED. RETURNS ACTUAL USED) RUNGE CALLS DIFFEQ(N.Y.DY) WHICH HOLDS THE USER.DIFF. EQN’S N8 OF DIFF EQ. Y(J)8CURRENT VALUE OF DEP VAR. DYIJ)8DERIV RET. DOUBLE PRECISION Y.DY.YY.EM COMMON P(10).C(10) DIMENSION Y(I).DY(I).YY(10).EM(4.10) K80 CALL DIFFEQ(N.Y.DY) DO 10 J81.N YY(J)=Y(J)+DY(J)*AVAL IF (YY(J).LT.0.0) K8J EM( I.J)=DY(J) CONTINUE IF (K.EQ.0) GOTO 30 K80 DO 20 J8I.N IF (AVAL*DY(J).GE.0.0) GOTO 20 AINT=-SNGL(Y(J)/(2.*DY(J))) IF (ABS(AINT).LT.ABS(AVAL)) AVAL8AINT CONTINUE GOTO 5 DO 40 J=2.4 CALL DIFFEQ(N.YY.DY) DO 35 I=I.N EM(J.I)=DY(I) CONTINUE GOTO (50.50.60.70) J CONTINUE DO 55 I8I.N YY(I)8Y(I)+EM(I.I)*AVAL/2 CONTINUE GOTO 40 DO 65 I81.N YY(I)=Y(I)+EM(2,I)*AVAL CONTINUE GOTO 40 DO 80 I8I.N DY(I)8(EM(I.I)+4*EM(3.I)+EM(4.I))/6 CONTINUE RETURN END 000 N000 11H: SUBROUTINE DIFFEQ(N.Y.DY) SUBROUTINE FOR STEADY STATE DATA COMMON P( 10) .C( 10) DOUBLE PRECISION Y. DY.YYY.R.S DIMENSION Y( I) .DY( I) IF (N.LT.0) GOTO 20 DIFFERENTIAL EQUATIONS GO HERE N8 I YYY=9.649E+4*C(3)/C( 2) UU8COSH( SNGL( (0. 5-P( 1)) *.3895E2*Y( 1))) R=C( I)*P(3) *P( 2)*SINH( SNGL( . 19471211111 I) ) )IUU S8 I . ODO+P( 2) *( COSH( SNGL( . I947E2*Y( I) ) )IUU)*( I .0D0+C( I)*P( 4) ) DY( I)=-YYY*R/S RETURN INITIAL CONDITIONS GO HERE N8I Y( I)8Y( I) RETURN END 000 N000 1115 SUBROUTINE DIFFEQ(N. Y. DY) SUBROUTINE FOR TIME DEPENDANT DATA COMMON P( IO) .C( 10) DOUBLE PRECISION Y.DY.UU.YYY.R. DIMENSION Y( I) .DY( I) IF (N.LT.0) GOTO 20 DIFFERENTIAL EQUATIONS GO HERE N85 UU=Y(5)*3B.95 YYY=P( 4) *C( 3) *( Y( I)-Y( 2)) R8 ( Y( 3) *DEXP( UU/2)-Y( 4) *DEXP( -UU/2) )IDCOSH( ( . 5-P( 5) )*UU) DY( I)=P( 2)*C( 3) *Y( 3)-P( I)*C(3)*C( I)*Y( I)-YYY DY( 2) =P( 2) *C( 3) *Y( 4) -P( l)*C( 3) *C( I)*Y( 2) +YYY DY( 3) =P( I)*C(3)*C( I)*Y( I)-P(2) *C(3)*Y(3)-R*C( 3) *P(3)/2 DY( 5) =l--RPRC( 3) *9 . 649E+4*P( 3) /( 2*C( 2) ) DY(4) =-DY( I)-DY( 2) -DY(3) RETURN INITIAL CONDITIONS GO HERE N85 Y(5)=Y( I) Y( I)=DBLE( C(4)/2/( I+P( I)/P(2)*C( 1))) Y(2)=Y( I) Y( 3) =DBLE( C(4)/2-Y( 1)) Y(4)=DBLE(C(4)-Y(3)-Y(2)-Y(1)) RETURN END 0000000 N 400 420 100 200 2 IO 220 225 250 260 270 IO 20 000 1116 KINPCHJ’TN PROGRAM READS A STANDARD ENKE FILE AND PRODUCES A DATA FILE APPROPRIATE FOR KINFIT DATA. (IE. X.XDEV.Y. YDEV). SEVERAL INPUT FILES CAN BE WRITTEN INTO ONE MULTIPLE OUTPUT FILE. WHICH CAN BE MERGED WITH KINFIT.FT'N (USING PIP) AND FED INTO THE CDC 6500 FROM THE DEPT. PDP II/40. DIMENSION X( 500) .Y( 500) BYTE SRCFIL(24). OUTFIL(24) . DAT(9).TIM(B) WRITE(6.300) FORMAT( IX.’ ENTER OUTPUT FILENAME: ’/'S ') READ(6.400)OUTFIL FORMAT(24AI) IFRR=O CALL FORPIP(OUTFIL. IFRR) IF (IFRR.m.0) GUIO 4 WRITE(6.420) FORMAT(’ FILE ALREADY EXISTS -- NO ACTION TAKEN') GOTO 2 CALL ASSIGN(3.0UTFIL.24. IFRR) WRIT‘E(6. IOO) FORMAT( IX.’ ENTER INPUT FILENAME:'/'O ') READ(6.200)SRCFIL FORMAT( 24A!) WRITE(6.210) FORMAT( ‘OLAST INPUT FILE? [Y/NI : ') READ(6.220) NY FORMAT(AI) CALL DSTRIP(SRCFIL.Y. NPTS. ICUE. ISROPT.DAT.TIM) WRITE(6.225) FORMAT( 'SENTER FIIBT BASELINE POINT:’) IEAD( 6 . 250) NAV FORMAT( I5) WRITE(6.260) FORMAT( ‘OENTER THE FIRST. LAST POINTS (DEFAULT83.92):’) READ(6.270) IFIRST.LAST FORMAT(2I5) IF (IFIRST.EQ.0) IFIRST83 IF (LAST.EQ.0) LAST-92 AVE=0.0 DO IO I=NAV.NPTS AVE=AVE+Y( I) CONTINUE AVE= AVE/( NPTS-NAV+ I) NPTS=LAST-IFIRST+I K81 DO 20 I8 IFIRST, LAST X(K)8I.O*K Y(K)8(Y( I)-AVE)/7.B 7.B=GAIN OF CELL AMP K=K+I CONTINUE CALL WRITKF( SRCFIL.X.Y.NPTS. ICUE. ISROPT. DAT. TIM) IF (NY.EQ.IHY) GOTO 30 GOTO 5 END FILE 3 STOP END LINK DESCR I PTOR FOLLOWS : KINPCH< KINPCH. DSTRIP. WRITKF . FORPIP. FTNLIB/E 00000000000 220 240 260 270 273 275 280 300 20 30 1117 SUBROUTINE WRITKF( SRCF IL. X. Y.NPTS. ICUE. ISROPT. DAT. TIM) BYTE SRCFIL(24) . DAT(9) . TIM(B) DIMENSION X( 500) .Y( 500) DEFINITION OF TERPB: C( I) 8 ION CONCENTRATION C( 2) 8 MEMBRANE CAPACITANCE CG!) 8 PERIOD BETWEEN DATA POINTS (SEC) U( I) 8 N ' U(2) 8 KIS/K-I U(3) 8 GAMMA*KI 0(4) 8 Kl/2KS NOTE: THESE DEFINITIONS SHOULD ALSO BE USED IN KINFIT ( IE. EQN).. JODD8 MOD( NPT‘S . 2) IF (JODD.NE.O) NPTS8NPTSFI WRITE(6.200) FORMAT(‘OENTER 3 KINFIT CONSTINTS:') READ(6.220)CI.C2.C3 FORMAT(3E) WRITE(6.240) FORMAT(’0ENTER 4 KINFIT ESTIMATES:') READ(6.260)UI.U2.U3.U4 FORMAT(4E) WRITE(3.270)NPTS THE FOLLOWING NUMBERS HAVE RELAVENCE TO KINFIT SEE KINFIT (SUBROUTINE EQN) FOR AN EXPLANATION FORMAT( I5. ’ O 50 I I IO -I 3 0.004') WRITE(3.273)SRCFIL FORMAT( 24A!) WRITE(3.275)CI.C2.C3 FORMAT( 3EIO.4) WRITE(3.280)UI.U2.U3.U4 FORMAT( 4EIO.4) DO 20 I8I.NPT‘S.2 J8 I+I XVAR8I.OE-6 YVAR8 1.76E-4 WRITE(3.300)X( I) .XVAR.Y( I) .YVAR.X(J) .XVAR.Y(J) .YVAR FORMAT( BEIO.4) CONTINUE RETURN END 000000000000 IOO I01 102 103 20 IO4 105 30 106 107 C C C 1118 EXPFIT.FTN PROGRAM TO LINEARIZE DECAYING EXPONENTTAL DATA CALCULATE THE WEIGHTING COEFFICIENTS AND CALL THE WEIGHTED LINEAR LEAST SQUARES SUBROUTINE AUTHOR: T.A. LAST DATE: 22-MARr76 READS A STANDARD ENKE FILE RETURNS THE SLOPE. INTERCEPT. AND THEIR STD. DEV. CREATES A FILE REPRESENTING THE FIT TO EXP(-X) DIMENSION X(200) BYTE FILNAM(24). DAT(9). TIM(B) WRITE(6.IOO) FORMAT(‘OENTER FILNAME:'/'S ') READ(6.IOI)FILNAM FORMAT(24AI) WRITE(6.IO2) FORMAT(‘OENTER.INITIAL AND FINAL POINTS AND BASELINE:‘) READ(6.IO3)J.K.BASE FORMAT(2I5,FIO.6) CALL DSTRIP(FILNAM.X.NPTS.ICUE.ISROPT.DAT.TIM) CALL TLLSQ(J.Y.XDEV.YDEV.I) DO 20 I8J.K Y8 ALOG( X( I) -BASE) Z8FLOAT(I-J) YDEV=3.2/(350*X(I)) CALL TLLSQIZ,Y.XDEV.YDEV.2) CONTINUE CALL TLLSQ(Z.Y.XDEV.YDEV.3) WRITE(6.IO4)Z.Y.XDEV.YDEV FORMAT(' SLOPE 8’.FIO.7.’ INTERCEPT 8’.FIO.7.' SDEV 8’.FIO.7 I' IDEV 8’.FIO.7) CALL ASSIGN(4.'FL:FIT.TMP’.IO.IERR) A8 EXP( Y) DO 30 I8J.K V8A*EXP(Z*(I-J))+BASE WRITE(4,IO5)I.V FORMAT(‘RD’.II5.FI5.7) CONTINUE WRITE(4.IOG)Z.Y.XDEV.YDEV FORMAT(‘ED'/‘1',4FIO.7) WRITE(4.IO7)FILNAM FORMAT(24AI) STOP END LINK DESCRIPTOR FOLLOWS: EXPFITKEXPFIT.DSTRIP.TLLSQ.FORPIP.FTNLIB/E 00000000000000000000 fl 1119 SUBROUTINE TLLSQIX.Y.XDEV.YDEV.MODE) LINEAR LEAST SQUARES FOR Y8MX§B WEIGHTED WITH ERROR ESTIMATES AUTHOR: T. A. LAST DATE: 21-MARr76 MODES: I8ZERO SUMS 28ACCUMULATE SUMS 38CALCULATE VALUES VALUES GIVEN: X, Y. YDEV (8 ERROR.ESTIMNTE), MODE VALUES RETURNED: X?SLOPE Y8INTERCEPT XDEV8SLOPE STANDARD DEVIATION YDEV8INTERCEPT STANDARD DEVIATTON GOTO (1.2.3) .MODE AI8O.O A28O.0 BI8O.O B28O.O CI8O.O C28O.O RETURN SIG8YDEV AI'AI+X**2/SIG**2 A28A2+X/SIG**2 BI8BI+X/SIG**2 B28BZ+IISIG**2 CI8CI+(Y*X)/SIG**2 C28C2+YVSIG**2 RETURN D8A2*Bl-B2*AI X?(BI*C2-B2*CI)/D XDEV8SQRT(-B2/(A28t2-A1882)) Y8(A2*Cl-AI¥C2)/D YDEV8SQRT(-AI/(A2**2-A1382)) RETURN END APPENDIX C COMPARISON OF TRANSPORT EQUATIONS DERIVED IN THIS WORK WITH THOSE OF OTHER WORKERS The model used in this study is similar to that used by Lafiger and Stark (19) and Ciani,gt'§l. (Al). The only difference is that, in this work all forms of the carrier molecule are assumed to be membrane bound, whereas the other workers assumed that ion-carrier complex could also form in the aqueous solution. The assumption that the carrier is membrane bound is thought to be a valid one since the carrier molecule favors the membrane phase over the aqueous phase by a factor of approximately 10“. Equations (3-1) through (3-h) result directly from the model, and are essentially the same as Equations A-7 used by Benz and Lafiger (33) with new notation. Equation (3-19) is virtually identical to that obtained by Hladky (A2) (Equation 70) with new notation. The only difference is that Hladky makes a distinction between the surface complexation reactions which occur at the right inter- face and those which occur at the left. This is unneces- sary for symmetrical membranes. The voltage dependence observed in Equations (3-7) and (3-8), which results from the double Eyring barrier, is the same as that obtained by Feldberg and Kissel (20) for the same type of barrier. 150 REFERENCES 100 11. 120 13. 1A. 15. 16. 17. REFERENCES Hodgkin, "The Conduction of the Nervous Impulse", Liverpool University Press, Liverpool (196“). Ashley in "Membrane and Ion Transport", Vol. 2, Ed. Bittar, pp. 1-32, Wiley, NY (1970). Lehninger, "Biochemistry", Ch. 28, Worth, NY (1975). F. Alverado, and A. Mahmood, Biochem. , V. 13(11), 2882- 2890 (197“). Jain, "The Bimolecular Lipid Membrane", Ch. 7, Van Nostrand, NY (1972). E. Overton, VJsehr. Naturf. Ges. Zurich, 33, 88-135 (1899). I. Langmuir, J. Amer. Chem. Soc., 3 , 18fl8-1906 (1917). E. Gorter, and F. Grendel, J. Exptl. Med., Ll, A39- 113 (1925) J. Danielli, and H. Davson, J. Cell Phys., Vol. i (#4). “95-508 (1935). P. Mueller, D. Rudin, H. Tien, and W. Wescott, Nature, I24. 979 (1962)- %. Singer, and G. Nicholson, Science, 17 75, 720-731 1972 G. Guidotti, Ann. Rev. of Biochem. Vol. L1(#80A), 731-752 (1972) E. Bamberg, and P. Lauger, J. Membrane Biol., ll, 177-19“ (1973). M. Eisenberg, J. Hall, and C. Mead, J. Membrane Biol., 15. 113-175 (1973). E. Bamberg et al., Biochem. et Biophys. Acta , ”19, 223-228 (1976). D. Haydon, and S. Hladky, Quart. Rev. Biophys., 5 (#2), 187- 282 (1972). G. Eisenman gt al., Bioenergetics, 3, 93-128 (1973). 151 18. 19. 20. 21. 22. 23. 2M. 25. 26. 27. 28. 29. 30. 31. 32. 33. 3H. 35. 152 G. Stark, and R. Benz, J. Membrane Biol., i, 133-153 (1971). P. Lafiger and G. Stark, Biochem. et Biophys. Acta, 211, 958-566 (1970). S. Feldberg, and G. Kissel, J. Memb. Biol., 22, 269- 300 (1975). J. E. Stuart, Am. Lab., 11, 91-101 (1973). W. P. Cargile, and B. J. Moore, Res. Develop., 23, 32-35 (1973). Data Sheets for SAD 100, obtained (on request) from Reticon Corp., 910 Benicia Ave., Sunnyvale, CA 9&086. G. Horlick, Anal. Chem., 38, 783A-787A (1976). H. Malmstadt, and C. Enke, "Digital Electronics for Scientists", w. A. Benjamin, NY, 1969, p. 226. M. Hsu, and S. Chan, Biochem., 12, 3872 (1973). G. Szabo, G. Eisenman, and S. Ciani, J. Memb. Biol., A: 3146-382 (1969) o J. Hall, C. Mead, and G. Szabo, J. Memb. Biol., 11, 75-97 (1973). B. geumke, and P. Lauger, Biophys. J., 9, 1160-1170 (19 9). V. A. Nicely, and J. L. Dye, J. Chem. Ed., 38, ““3 C. W. Gear, Information Processing 68, North Holland Publishing 00., Amsterdam (1969). C. W.)Gear, Communications of the ACM, 11(3), 185 1971 . R. Benz, and P. Lauger, J. Memb. Biol., 21, 171-191 (1975). J. M. Kudirka, and C. G. Enke, Anal. Chem., 33, 619-615 (1972). Laprade g£_al., in "Membranes, A Series of Advances", Vol. 3, Ed. G. Eisenman, Marcel Dekker, NY (1975). 36. 37. 38. 39. no. Al. A2. 93. 153 S. B. Hladky, Biochim. et. Biophys. Acta, 3_5, 327- 399 (1975) J. Nelder, and R. Mead, Computer J., l, 308 (1965). S. Deming, and S. Morgan, Anal. Chem., 32, 276A-282A 1973 . M. Dobler, Helv. Chim. Acta, 22, 1371 (1972). M. Dobler, J. Dunitz, and B. Kilbourn, Helv. Chim. Acta, 52,, 2573 (1969). s. Ciani 33.31., J. Memb. Biol., £1, 255-292 (1973). S. B. Hladky, J. Memb. Biol., 1Q, 67—91 (1972). W. Knoll, and G. Stark, J. Memb. Biol., 25, 2H9 (1975).