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Chloride conductance in xenopus laevis skeletal muscle membrane Loo, Donald Doo Fuey 1978

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CHLORIDE CONDUCTANCE IN XENOPUS LAEVIS SKELETAL MUSCLE MEMBRANE by Donald Doo Fuey|Loo B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1970 M . S c , U n i v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE: FACULTY OF GRADUATE STUDIES in the I n s t i t u t e of Appl ied Mathematics and S t a t i s t i c s and the Department of Physiology We accept t h i s thes i s as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA September, 1978 (_) Donald Doo Fueylioo, 1978 In presenting th is thes is in par t i a l fu l f i lment o f the requirements for an advanced degree at the Univers i ty of B r i t i sh Columbia, I agree that the L ibrary shal l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permiss ion. Department of Mathematics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Oct, 11. 1978 i ABSTRACT The chloride current-voltage characteristics of the membrane of sartorius f i b e r s from Xenopus laevis were studied using a three microelectrode voltage clamp system. In fibers with normal resting potentials (-70 to -90 mV) and in fibers depolarized i n 115 mM KC1 (resting potential -20 mV) the direction and degree of steady state r e c t i f i c a t i o n depended on extra-c e l l u l a r pH. In alk a l i n e solutions (pH 8.4) the current r e c t i f i e d outwards; with large hyperpolarizations the current recorded i n normally polarized fibers was sometimes seen to diminish as the voltage was made extremely negative (the current-voltage r e l a t i o n exhibited a negative slope). In the depolarizing region (in depolarized fibers) the slope of the I-V re l a t i o n became constant ( l i m i t i n g conductance) i n a l k a l i n e solutions. In acid solutions (pH 5.4) the current r e c t i f i e d inwards with hyperpolarization and reached a l i m i t i n g value with depolarization. Chloride currents decay ('inactivate') following changes of membrane potential from the resting potential (for both polarized and depolarized f i b e r s ) . The k i n e t i c s of current relaxation exhibited voltage-dependent time constants depending on the size of the voltage step with a s e n s i t i v i t y of about -1.5 msec/mV but were independent of absolute membrane potential and external pH. i i I nactivation of chloride conductance was studied i n two-pulse (conditioning (v^) a n d test ( V 2 ) , pulses) voltage clamp experiments. In variable experiments the dependence of the i n i t i a l current at the onset of was sigmoidally related to (inactivation r e l a t i o n ) . The slope of the inactivation r e l a t i o n was twice as steep i n acid as in a l k a l i n e solutions, but was independent of the resting potential. In variable experiments, the current-voltage r e l a t i o n was linear over a wide voltage range and for different values of V^, the instan- v.: '/ taneous I-V relations converged i n the outward current region; they also had zero-current potentials that became increasingly negative with respect to the holding potential as was made negative. Instantaneous chloride currents and the kinetics of current relaxation were found to depend on i n i t i a l conditions when the membrane potential was changed under non-stationary conditions. The ina c t i v a t i o n and recovery of i n i t i a l current had similar timecourses as did the prolongation and recovery of the time constants. I n i t i a l currents recovered from conditioning with an exponential or sigmoid timecourse. Relaxation time constants exhibited a similar recovery pattern. The decline of i n i t i a l current was i n i t i a l l y exponentially dependent on the duration of conditioning. The time constant increased sigmoidally, or exponentially as the duration of conditioning increased. Using the data from variable conditioning step and variable test step experiments a manifold (or state space representation) was constructed that enables much of the current-voltage behavior of the chloride permeation system to be predicted. Currents recorded i i i i n voltage clamp experiments can be visualized as time-dependent flows along t r a j e c t o r i e s that are determined by the voltage. The r e c t i f i c a t i o n of the steady state and instantaneous current-voltage relations are related to the dispersion of the traje c t o r i e s . The dependence of time constants of current transients can also be accounted for by the manifold. The results are examined i n l i g h t of models for channel behavior. The instantaneous I-V characteristics exhibit some properties of channels of the ele c t r o d i f f u s i o n type. The steady state current-voltage relations are q u a l i t a t i v e l y s i m i l a r to those of a model incorporating a p a r t i c l e within the chloride channel that either blocks or unblocks i t depending on the ex t r a c e l l u l a r pH. The dependence of relaxation k i n e t i c s on the size of the voltage step and on i n i t i a l conditions suggest the pa r t i c i p a t i o n of a molecule acting i n a c a t a l y t i c role c o n t r o l l i n g the relaxation of current transients. ACKNOWLEDGEMENTS I am deeply indebted to, and i t gives me the greatest pleasure to thank Drs. P. Vaughan and J. McLarnon for their technical assist-ance with the experiments and whose comments and suggestions were most helpful during the writing of this thesis. The intellectual stimuli and encouragement provided by my friend and advisor, Dr. P. Vaughan, are beyond thanks. The financial support from the Muscular Dystrophy Association of Canada and the University of British Columbia i s gratefully acknowledged. V Preface v i In t h i s thesis the mechanism of passive chloride transport across amphibian sk e l e t a l muscle fibers i s examined. At the resting membrane potential chloride i s the most permeant ion i n amphibian skeletal muscle membrane accounting for about two-thirds of the t o t a l ionic conductance (Hodgkin and Horowicz, 1959). I t plays a physiologically important role i n muscle i n the regulation of the resting potential and during e l e c t r i c a l a c t i v i t y , the repolarization of the membrane i s greatly accelerated by the high conductance (Hutter and Noble, 1960; Horowicz, 1964). When the trans-membrane potential i s altered i n voltage clamp experiments, chloride currents exhibit voltage and time dependent relaxations having a complex dependence on the pH of the bathing environment (Warner, 1972). Although the physiological significance of chloride relaxations i s not understood, an increased appreciation of the physical and chemical processes involved i n ion translocation across c e l l membranes can be gained from a study of the voltage and time dependence of chloride currents. The dissertation has been divided into four parts. F i r s t the background and rationale for the investigation i s presented. Experimental re s u l t s are then described i n Chapter 2. In the th i r d chapter, a geometric representation of the data i s presented and f i n a l l y i n Chapter 4, some theoretical models for chloride transport are examined. v i i TABLE OF CONTENTS Page ABSTRACT i ACKNOWLEDGMENTS i v PREFACE v TABLE OF CONTENTS v i i LIST OF TABLES x i i LIST OF FIGURES x i i i Chapter I. Introduction and Methods Background 1 Ion transport mechanisms 2 Experimental aims and objectives 7 Methods Voltage clamp 14 Sign convention 14 Ions contribution to membrane current 19 Solutions 19 Chapter I I . Experimental results Experimental Section I. The behavior of chloride currents i n one pulse voltage clamp experiments. 25 I. Experiments on fibe r s with normal resting potentials 25 A. The instantaneous and steady state current-voltage c h a r a c t e r i s t i c s 25 Fibers i n a l k a l i n e solutions 25 Fibers i n neutral solutions 27 Fibers i n acid solutions 29 Summary of the effect of pH on current-voltage relations and on resting conductance 30 v i i i B. The voltage dependence of chloride current transients 33 Analysis of current transients 35 Kinetics i n acid solutions 38 Kinetics i n neutral and alkalin e solutions 38 Comparison with Rana temporaria Al Dependence of kinetics on the magnitude of voltage step 41 C. Aftercurrents: changes i n zero-current potential with chloride passage across the membrane 43 I I . Experiments on permanently depolarized fibers 45 Experimental rationale 45 Resting potentials of permanently depolarized fi b e r s 47 Value of the inte r n a l r e s i s t i v i t y (R^) i n depolarized fibers 49 Contribution of cation currents 50 A comment on the solutions 52 A. The steady state and instantaneous current-voltage relations i n fibe r s that have been permanently depolarized 55 Fibers i n acid solutions 55 Fibers i n pH range (6.7-7.3) 56 Fibers i n solutions of pH 8.4-8.8 58 Summary of the effect of pH on resting conductance of depolarized f i b e r s 63 Comparison of the current-voltage relations i n polarized and depolarized f i b e r s 64 B. Voltage dependence of current transients i n depolarized f i b e r s . 66 Priming of the membrane conductance with conditioning pulses i n depolarized f i b e r s i n alka l i n e solutions 67 ix Instantaneous a f t ercurrent s i n depolar ized f i b e r s 72 Discuss ion 74 Steady state current -vo l tage r e l a t i o n s 74 Instantaneous current -vo l tage r e l a t i o n s 75 Relat ions between the instantaneous and steady s tate proper t i e s i n p o l a r i z e d and depolar ized f i b e r s 75 Dependence of c h l o r i d e conductance on concen-t r a t i o n 77 The p H - s e n s i t i v i t y of fast t rans i en t s 78 A f t e r c u r r e n t s and slow t rans i en t s 79 Experimental Sect ion I I . Two pulse vo l tage clamp experiments 80 I . Instantaneous current -vo l tage r e l a t i o n s 80 A. Experiments where the c o n d i t i o n i n g p o t e n t i a l was v a r i e d 80 Normal izat ion procedures: comparison of data from d i f f e r e n t f i b e r s 82 F i b e r s i n a l k a l i n e so lu t ions 82 F i b e r s i n a c i d so lut ions 83 B. Experiments i n which the test p o t e n t i a l was v a r i e d 84 I I . The vol tage dependence of c h l o r i d e current t rans i en t s i n two pulse experiments 89 A. Dependence of time constants on the vo l tage step Iv^vJ. 89 Depolar ized f i b e r s 89 P o l a r i z e d f i b e r s 90 Summary of the vo l tage dependence of current t rans i en t s 93 Appearance of a delayed (non-exponential) f a l l i n g current 93 Discuss ion 95 S h i f t i n g of the S-curves 95 Re la t ion between instantaneous and steady current values 95 The voltage dependence of the k i n e t i c s of current t r a n s i e n t s 98 X Experimental Section I I I . The inactivation and recovery of i n i t i a l currents; dependence of kinetics on i n i t i a l conditions 101 I. Recovery of conductance 101 A. Recovery experiments i n polarized f i b e r s 103 B. Recovery experiments i n depolarized fibers 104 Fibers i n acid solution 105 Fibers i n alkaline solution 111 I I . The rate of inactivation of chloride currents: Experiments i n which the duration of the conditioning step as varied. 112 Comparison of the rates of inactivation and recovery 118 Discussion 121 Dependence of ki n e t i c s on i n i t i a l conditions 121 Slow currents under 'primed' conditions 122 Chapter I I I . The i n i t i a l current-voltage manifold 125 Assumptions 127 Application of the manifold 130 Predictions of the instantaneous currents 130 Dependence of kin e t i c s on i n i t i a l conditions 137 Discussion 138 Relationships between steady state and instantaneous current-voltage relations 138 Chapter IV: Theoretical considerations: Models of membrane permeation process. 143 Introduction 144 Models of membrane permeation process 146 A. Electrodiffusion models 146 Def i n i t i o n of symbols; t y p i c a l dimensions 146 Steady state properties 149 An extension of el e c t r o d i f f u s i o n model 154 x i Transient properties 156 Instantaneous currents and aftercurrents 159 Summary 160 B. Channel or pore models 161 General formulation for a pore model 162 Steady state properties 167 Specifications for a possible chloride channel 170 Variations i n the number of conducting channels 174 Transient'properties 178 Discussion 190 Instantaneous current-voltage relations 191 Effect of pH on current-voltage relations 192 Shi f t s i n zero-current potential 192 Dependence of current transients on voltage and on i n i t i a l conditions 193 F i n a l Discussion 194 Channel versus c a r r i e r transport 195 The nature of the chloride channel 195 Some unsolved problems 197 References 199 Appendix Steady state measurements Transient properties 2 1 2 2 1 6 2 1 7 LIST OF TABLES Solutions Effect of pH on resting conductance Summary of the voltage dependence of the fast time-constants of relaxation of chloride current Resting potentials of depolarized fibers Cable constants i n depolarized fibers Instantaneous and steady state conductances at the holding potential for 5 conducting states i n a depolarized f i b e r at pH 8 . 4 Effect of pH on resting conductance i n depolarized fi b e r s Summary of the voltage dependence of the fast time-constants of relaxation of chloride current'In depolarized fi b e r s Time dependence of the recovery of time constants after priming Dominant fast and slow time constants for channel models. x i i i LIST OF FIGURES Figure Page 1.1 Schematic of the voltage clamp system 15 2.0.1 Waveforms of membrane currents i n polarized 23 fib e r s 2.1.1 Current-voltage relations i n neutral and 26 alkali n e solutions 2.1.2 Current-voltage relations i n neutral solutions 28 2.1.3 Current-voltage relations i n acid solutions 31 2.1.4 Effect of changing pH on current-voltage 34 relations of a single f i b e r 2.1.5 Analysis of current transients 36 2.1.6 Relationship of time constants to voltage 39 step amplitudes i n acid solutions 2.1.7 Relationship between decay time constant and 42 the amplitude of hyperpolarizing voltage steps 2.1.8 Magnitude and d i r e c t i o n of after-currents 44 2.1.9 Potassium current-voltage relations 53 2.1.10 Current-voltage relations i n depolarized 59 f i b e r at pH 5.4 2.1.11 Current-voltage relations i n depolarized 59 ..fibers at pH 6.7 and 7.3 2.1.12 Current-voltage r e l a t i o n s i n depolarized 60 f i b e r at pH 8.4 2.1.13 Dependence of relaxation time-constants on 69 voltage: pH 8.4 x i v LIST OF FIGURES (continued) Figure 2.1.14 2.1.15 2.1.16 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.3.1 2.3.2 2.3.3 Page Current-waveforms: depolarizing fibers 70 at pH 8.4 Currents recorded i n a priming sequence 71 Instantaneous after-currents 76 Protocol of the two-pulse experiment 81 and instantaneous current-voltage relations Normalized data from variable experiments 85 Steady state and instantaneous current-voltage 86 r e l a t i o n i n two-pulse experiments: variable Families of Vj/I (0)' and V^l^Q) relations 88 from a depolarized f i b e r at pH 8.4 Time constants of relaxation of the test currents 91 (I^) i n a two-pulse experiment Time constants of relaxation of test currents as 92 a function of voltage Changes i n the waveform of I 94 Comparison between measured steady state I-V 100 relations and those predicted from instantaneous currents, and between measured instantaneous currents and those predicted from a steady state relations Protocol of recovery experiments 102 a) Recovery of relaxation time constants 106 b) Recovery of 1^(0) Current traces recorded i n recovery experiments 107 XV LIST OF FIGURES (continued) Figure Page 2.3.4 Waveforms seen as t_ i s varied 108 2.3.5 a s a f u n c t i ° n °f t. f° r t w o depolarized 109 fib e r s at pH 5.4 2.3.6 The dependence of the instantaneous test 110 current 1^(0) and the time constant of the relaxation of the test current to i t s steady state on the period of recovery, t, at the holding potential 2.3.7 Current waveforms recorded i n a recovery 113 experiment on a depolarized f i b e r at pH 8.4 2.3.8 Dependence of I 2(0) on _t for the f i b e r of 114 Figure 2.3.7 2.3.9 Time constants for r i s i n g currents i n a f i b e r 115 that showed a rapid t r a n s i t i o n from r i s i n g to f a l l i n g currents as a function of _t 2.3.10 a) Current traces from a depolarized f i b e r 116 at pH 8.4; variable duration V^, and amplitudes fixe d . b) 12(0) as a function of the duration of 2.3.11 Effects of the duration of the conditioning 117 potential (V^) on the k i n e t i c time constant of the test current (I^) i n a f i b e r at pH 8.4 2.3.12 a) Current transients i n a recovery experiment 120 performed on the same f i b e r as Fig.2.3.10 b) Dependence of the relaxation time constants of the test current on t xv i LIST OF FIGURES (continued) Figure Page 2.3.13 Schematic relat i n g postulated states of the 124 conductance mechanism of the 'primed' current to membrane potential 3.1 Steps i n the construction of the manifold 131 3.2 The current-voltage manifold 132 3.3 Use of the manifold to predict the responses 133 to a sequence of voltage steps 3.4 (a,b) The use of the manifold i n predicting 134 currents when voltage steps are made after steady state conditions have been reached (c,d) Simulate a case i n which the voltage step i s made before current at the conditioning voltage has decayed to the steady state 3.5 Representation of a recovery experiment on the 140 current-voltage manifold 3.6 Prediction of instantaneous currents and re- 141 laxation time constants from the current-voltage manifold when steps are made during non-steady conditions 3.7 Relationship between the steady state current- 142 voltage r e l a t i o n and the dispersion of the S-curves 4.1 (a,b) Concentration p r o f i l e s of permeant anion 150 within the membrane for various values of the membrane potential x v i i LIST OF FIGURES (continued) Figure Page c) Current-voltage relations for different zero-current potentials d) The r e c t i f i c a t i o n r a t i o i n constant f i e l d electrodiffusion theory (e,f,g) Current-voltage relations of modified constant f i e l d theory incorporating surface charge 4.2 A potential energy diagram for d i f f u s i o n within 163 an ionic channel 4.3 A schematic diagram i l l u s t r a t i n g a l l states of 168 a channel which permits the entrance of one of two ionic species 4.4 The q u a l i t a t i v e behavior of the current-voltage 179 equation for an ionic channel 4.5 Current-voltage relations predicted from the 182 sigmoid dependence of conductance on pH at different resting potentials 4.6 Current-voltage relations for a channel 189 A.1 Diagrammatic representation of muscle f i b e r 218 and the e l e c t r i c a l parameters defined i n the Appendix 1 Chapter I Introduct ion and Methods 2 Background E a r l y s tudies on ion permeation showed that membranes i n general behaved as though they have negative f i xed charge and would permit the passage of cat ions and exclude anions (Michae l i s , 1926). Observations on s k e l e t a l muscle were compatible with these views of low anion permeabi l i ty s ince changes i n ex terna l c h l o r i d e concen-t r a t i o n on s u b s t i t u t i o n by var ious l y o t r o p i c anions produced small changes i n the transmembrane p o t e n t i a l (Hober, 1904 ; Fenn, 1936; Horowicz, 1964). However, i n t h e i r study of the d i s t r i b u t i o n of potassium and c h l o r i d e across frog s k e l e t a l muscle, Boyle and Conway (1941) found that the r e s t i n g muscle was permeable to c h l o r i d e and other anions of small i o n i c r a d i u s : the Nernst equation was s a t i s f i e d for both potassium and c h l o r i d e . Since t h i s pioneering work of Boyle and Conway the c h l o r i d e permeation system i n frog twitch muscles has been extens ive ly studied and three i n t e r e s t i n g proper t i e s have emerged. F i r s t , the presence of any one of the anions in the l y o t r o p i c s er i e s (Br~, N0~, I~, C10~ and CNS~) reduces the f luxes of c h l o r i d e ( H a r r i s , 1958; A d r i a n , 1961; Hutter and Warner, 1967c; Moore, 1969) and e l e c t r i c a l conductance (Hutter and Padsha, 1957; Hutter and Warner, 1967a). These anions have been found to have very l i t t l e or no e f f ec t s on N a + and K + movements (Edwards, H a r r i s and N i s h i e , 1957). The r e l a t i v e magnitude of the membrane res i s tance when the c h l o r i d e i n the R inger ' s s o l u t i o n i s replaced by the l y o t r o p i c anions Br , NO^ , and I~ i s (Hutter and Padsha, 1959) : 3 CI : Br : N0 3 : I = 1.0 : 1.5 : 2.0 : 2.3 Second, the chloride permeability i s affected by variations i n ex-ternal pH: in acid solutions, the chloride permeability i s low; in a l k a l i n e solutions i t i s high (Hutter and Warner, 1967a,b; Moore, 1969). The dependence of permeability on pH has been seen to be sigmoid (S-shaped) with the region of sharpest t r a n s i t i o n close to pH 7 (Hutter and Warner, 1967a). Variations in the pH of Ringer's solution from 5.0 to 9.8 caused a three fold increase i n membrane conductance. The anion permeability sequence i s also markedly af-fected by variations i n external pH (Hutter, deMello and Warner, 1968; Hestenes and Woodbury, 1972; Woodbury and Miles, 1973): In a l k a l i n e solutions (pH 9.8) the sequence i s CI > Br > N0.j>I and i s reversed in acid solutions (pH 5.0) I >• N0~>Br >• Cl (Hutter, deMello and Warner, 1968). Third, the chloride conductance has been found to be voltage dependent (Hutter and Warner, 1972). Hodgkin and Horowicz (1959) and subsequently other workers (Adrian, 1961; Harris, 1963; Hutter and Warner, 1972) found that chloride movements in frog twitch f i b e r s i n neutral Ringer's solution (pH 7.2-7.4) could be empirically described by the constant f i e l d theory (Goldman, 1943; Hodgkin and Katz, 1949). Subsequently the constant f i e l d theory has been found to be inadequate to account for chloride permeation at other values of the pH. The dependence of the steady state chloride current on voltage i s strongly dependent on the external pH: i n a l k a l i n e solutions, the current reaches a l i m i t i n g maximum value for large hyperpolarizations whereas in acid solutions, the current continues to increase with hyperpolarizing voltages (Hutter and Warner, 4 1972). Attempts by different workers to study the pH dependence of anionic conductance and anionic interactions have yielded c o n f l i c t i n g r e s u l t s : From a study of the inhibi t o r y effect of the metallic ions copper and zinc on chloride e f f l u x and the near neutral pK of the pH-dependence of chloride conductance, Hutter and Warner (1967c) hypothesized that an imidazole group controls chloride permeability. However, t h i s has been questioned recently by Stephenson and Wood-bury (1976) who found that the enthalpy of the imidazole group of hi s t i d i n e (+8 Kcal/mole) i s much different from the enthalpy values (greater than -20 Kcal/mole) for the reaction: HS ^ ±H + S (S i s the binding s i t e for chloride conductance). Hutter, deMello and Warner (1969) attempted to account for the reversal in the anion s e l e c t i v i t y sequences by the f i e l d strength 1 theory (Eisenman, 1961). They postulated that the cationic f i e l d strength of the si t e s c ontrolling anion permeation increases as the pH of the bathing solution decreases. The s e l e c t i v i t y sequence in a l k a l i n e solutions i s governed by a s i t e with weak f i e l d strength and i n acid solutions, by a s i t e with strong f i e l d strength. For instance, i f a monopolar electrode cationic s i t e i s assumed, then the ionic radius i n acid solutions i s less than .65 A° and the ionic radius i n a l k a l i n e solutions l i e s between .65-.87 A° (Wright and . Diamond, 1977; Fi g . 15). In order to account for the impediment of chloride movement by other anions, the permeability of an ion through the membrane was assumed to decrease with the strength of i t s binding 5 to membrane s i t e s . A prediction of t h i s theory i s that fluoride should be the most permeable anion in alkaline solutions, and chloride should interfere with fluoride movement. However, Hestenes and Woodbury (1972) obtained the s e l e c t i v i t y sequence: CI >Br >I >F at pH 9 and fluoride was found not to interfere with chloride movement. Moreover contrary to Hutter, deMello and Warner(1969), Hestenes and Woodbury reported no changes i n the s e l e c t i v i t y sequence at pH 5.0. Interestingly, the s e l e c t i v i t y sequence of anions (CI >Br >I -^CH^ SO^  ) i n alkal i n e solutions i n mammalian diaphragm fib e r s has been found to be similar to frog twitch fibers (Palade and Barchi, 1977). As i n amphibian f i b e r s , the conductance of a l l permeant anions was found to decrease when the bathing f l u i d i s made acid. However the reduction of each anion conductance was proportional and the sequence remained unchanged, whereas Hutter, deMello and Warner (1969) reported a disproportionately larger reduction of the chloride conductance as compared to the other anions, hence their observation that s e l e c t i v i t y sequence reverses i n acid solutions. Since the work of Hutter, deMello and Warner (1969) i s the only report of a s e l e c t i v i t y sequence reversal with pH, further work i s necessary for c l a r i f i c a t i o n of the different observations of Hutter et a l (1969) and Hestenes and Woodbury (1972). The two recent studies on chloride transport by Woodbury and Miles (1972) arrived at different conclusions concerning the mechanism of chloride transport. Warner (1972) suggested that the pH dependent r e c t i f i c a t i o n of the steady state chloride current-voltage r e l a t i o n s i s analogous to that obtained by Sandblom, Eisenman and Walker (1967) for a model 6 incorporating mobile charged ca r r i e r s . Other s i m i l a r i t i e s between the chloride conductance and the model of mobile charged carriers include linear instantaneous current-voltage relations, the presence of 'after-current' after a conditioning voltage prepulse; and a disagreement between permeability coefficients obtained by e l e c t r i c a l conductance methods and by f l u x measurements (Adrian, 1961; Hutter and Warner, 1967a). Woodbury and co-workers (Hestenes and Woodbury, 1972,1973; Woodbury and Miles, 1973; Parker and Woodbury, 1976; Stephenson and Woodbury, 1976) have presented evidence supporting a channel or pore hypothesis: a family of organic ('benzoate-like') anions whose molecular structure indicates a hydrophobic region and a carboxylate group (for instance, benzoate, valerate, butyrate, proprionate, formate, acetate) was found to have permeability behaviour opposite to that of c h l o r i d e — t h e i r permeability increased as the pH was decreased (Woodbury and Miles, 1973). The permeabilities of these organic anions are lower than any of the lyotropic anions but they did not block chloride movement. The correlation between the sizes of the i r hydrophobic moieties and the i r sequence of conductance led to the postulate that these ions interact with a hydrophobic region of the membrane near the s i t e of a rate l i m i t i n g step. As a r e s u l t of the interaction, the penetration of 'benzoate-like' anions through the membrane i s f a c i l i t a t e d . Moreover, s i m i l a r i t i e s i n the i n h i b i t o r y effect of zinc ions on chloride and 'benzoate-like' ions suggest that they use the same ionic channels (Woodbury and Miles, 1973). 7 In order to pursue a more detailed discussion of the mechanism of chloride transport as well as to describe the objective of our experiments, a br i e f review of ion transport mechanisms across mem-brane i s required. Ion transport mechanisms Early quantitative descriptions of ion transport across membranes were based on the work of Nernst (1889) and Planck (1890) on the diff u s i o n of ions i n electrolytes (Cole, 1968). The i n t e r i o r s of membranes were idealized as homogenous, and isotropic with a position independent d i e l e c t r i c constant, and ionic movement within them was viewed as the resultant of two f o r c e s — a concentration dependent di f f u s i o n current and an ionic current driven by the e l e c t r i c f i e l d (Goldman, 1943; Teorell, 1953). With our increasing understanding of the structure of the c e l l mem-brane, and the recognition that i t i s a very poor conductor composed p r i n c i p a l l y of l i p i d s (Singer and Nicholson, 1972) our views on transport mechanisms have changed quite r a d i c a l l y . Ion transport across c e l l membranes i s now postulated to be mediated by special l o c a l i z e d proteinaceous structures i within the l i p i d mosaic. Although no transport mechanism has been worked out i n d e t a i l , there i s much evidence (Armstrong,1975a) supporting the hypothesis that some of the specialized protein subunits form 'pores' across membranes, or alt e r n a t i v e l y , they can act as 'ca r r i e r s ' for the transport of ions. Much of our present understanding of ion transport across c e l l mem-branes has been gained from a study of model systems such as l i p i d bilayers and the transport of ions across them by compounds that form 8 pores across the bilayer, and those that act as c a r r i e r s of ions (Hladky and Haydon, 1972; Eisenman et a l , 1978). Pores and car r i e r s exhibit many common e l e c t r i c a l properties. Studies on compounds that form channels and those that act as carriers in a r t i f i c i a l membranes (Haydon and Hladky, 1972; Eisenman et a l , 1967) have shown that macroscopic conductance (conductance from a c o l l e c t i o n of channels or pores) can be a highly non-linear function of membrane potential (Latorre et a l , 1975; Haydon and Hladky 1972; Bamberg and Lauger, 1973) and that conductances undergo relaxations when the transmembrane potential i s suddenly perturbed (Frehland and Lauger, 1974; Latorre et a l , 1975; Lauger and Stark, 1970; Laprade et a l , 1974; Bamberg and Lauger, 1973; Neher and Stevens, 1977). A l l of these properties have been observed i n natural membrane transport systems. There are, at the present time, two available methods for the determination of whether a particular transport mechanism u t i l i z e s a c a r r i e r or a pore. The f i r s t method, introduced by Krasne, Eisenman and Szabo (1971) studies the temperature dependence of the permeation system and has been successfully applied to a r t i f i c i a l l i p i d bilayer membranes (Krasne et a l , 1971) but not to c e l l membranes. The membrane i s 'frozen' to temperatures lower than the l i p i d t r a n s i t i o n temperature and since c a r r i e r m o b i l i t i e s within l i p i d membranes are very dependent on l i p i d motions, they are d r a s t i c a l l y reduced upon freezing the l i p i d . Transport of ions v i a pores, on the other hand, i s r e l a t i v e l y independent of the t r a n s i t i o n temperature of the l i p i d s (Krasne et a l , 1971). In the second method, the turnover number, that i s , the maximum transport rate of an in d i v i d u a l c a r r i e r or the maximum number of ions crossing a single channel, i s determined. In a r t i f i c i a l membranes, i n which known concentrations of ion carriers have been dissolved, transport 4 numbers of the order of 10 ions/sec have been observed (Lauger, 1972). In other systems where pores are known to form across membranes, the transport numbers are of the order of 10^ ions/sec (Armstrong, 1975a,b). At the present time, the strongest evidence for a particular transport process being v i a a c a r r i e r or a pore i s the determination of the turnover number. In order to determine the turnover number, an estimate of the density of conducting units ( H i l l e , 1970; Keynes, Ritchie and Rojas, 1971; Colquhound et a l , 1974; Aimers and Levinson, 1975) i s required. Alternately, direct measurement of a single unit conductance (Hladky and Haydon, 1972; Krawczyk, 1978) or noise analysis (Anderson and Stevens, 1973; Neher and Stevens, 1977) may be used. However when the turnover number i s unavailable, as i s the case with anions i n amphibian twitch f i b e r s , we have to r e l y on models that are based on analogies between properties of the anion transport system and the properties of transport systems that are known c a r r i e r s or pores. Unfortunately, t h i s does not y i e l d unique answers. For instance, the c a r r i e r hypothesis for chloride was based on the s i m i l a r i t i e s between the r e c t i f i c a t i o n s of the instantaneous and steady-state current voltage relations of chloride and mobile charged c a r r i e r s (Warner, 1972; Sandblom, Eisenman and Walker, 1967). On the other hand, chloride movement i s interfered with by the lyotropic anions (the mechanism of th i s interaction i s not known) and ionic interference and block has been observed i n the potassium (Hodgkin and Keynes, 1955; Chandler and Meves, 1965a;Bezanilla and Armstrong, 1972; French and Adelman. J r . . 1976) 10 sodium (Chandler and Meves, 1965a,b; H i l l e , 1 9 7 1 , 1975a, 1975b; Woodhull, 1973), calcium (Hagiwara and Takahashi, 1967; Hagiwara, Toyama, and Hayashi , 1971) channels, and channels in l i p i d b i l a y e r membranes (Hladky and Haydon, 1972; Bamberg and Lauger, 1977; Eisenman et a l , 1978). Woodbury and M i l e s (1973) have postulated that hydrophobic in t erac t ions play a r o l e i n determining the conductance sequence of the anion channel i n frog twitch muscle membranes s i m i l a r to the apparently acce s s ib l e hydrophobic region of the membrane in the v i c i n i t y of the inner mouth of the potassium channel (Armstrong, 1969, 1971; Armstrong and H i l l e , 1972). The energet ic cost of i o n i c transport across membranes i s represented by a s er i e s of a c t i v a t i o n energy b a r r i e r s with successive e q u i l i b r i u m p o s i t i o n s at p o t e n t i a l energy minima (Glasstone, L a i d l e r and E y r i n g , 1941; Johnson, Eyr ing and P o l i s s a r , 1954 ; P a r l i n and E y r i n g , 1954; C i a n i , 1965; Lauger, 1973; Stark, 1973). When the transport mechanism i s a pore, these minima may be imagined as the places where the ion i s in an e n e r g e t i c a l l y favourable p o s i t i o n with respect to one or severa l coordinat ing l i g a n d s . The a c t i v a t i n g b a r r i e r between successive minima may be e l e c t r o s t a t i c i n o r i g i n such as the i n t e r a c t i o n of the permeating ion with a charged or d i p o l a r s ide group wi th in the pore. In the event that the transport mechanism i s v i a a c a r r i e r , the sequence of a c t i v a t i n g b a r r i e r s could represent success ive ly the a c t i v a t i n g energy of the a s s o c i a t i o n between ion and c a r r i e r at the membrane surface , the d i f f u s i o n and migrat ion of c a r r i e r - i o n complex across the i n t e r i o r of the membrane, and the d i s s o c i a t i o n energy of the i o n - c a r r i e r complex. 1 1 Experimental aims and objectives. The objective of our experiments was to investigate some of the implications of the channel hypothesis for chloride permeation theory i n amphibian twitch f i b e r s . The available experimental evidence on chloride transport does not sel e c t i v e l y support a channel (or a l t e r n a t i v e l y , a carrier) hypothesis and exclude the other. However, a channel hypothesis i s more l i k e l y i n view of the interaction of anions (Harris, 1958; Moore, 1959) and the hypothesized action of the 'benzoate-like' anions (Woodbury and Miles, 1973). The motivating question for our experiments was the following: suppose chloride translocation i s v i a a pore, what are some of the chemical and physical processes that occur within the pore? Even though the question i s formulated in terms of a pore, an analogous formulation can also be applied to c a r r i e r s . P a r t i a l answers to these questions have been provided by Warner (1972), who found,using voltage clamp experiments, that the chloride conductance i s time dependent. In response to hyperpolarizing voltage steps, the chloride currents also depended on external pH: i n neutral and a l k a l i n e solutions, the current decayed from an instantaneous to a steady value whereas in acid solutions, the transients were biphasic, f i r s t exhibiting a f a l l which was then followed by a subsequent r i s e to a steady value. I f the transient properties are interpreted using a channel hypothesis, and since the decaying current transients have comparably large time constants (300 milliseconds, Warner, 1972) as compared with 12 ionic r e d i s t r i b u t i o n times within a channel (less than 1 millisecond), the current transients may be attributed to movement or re-orientation of macromolecules within a channel or alternatively, to the number of conducting channels changing with voltage and time. In the study of e l e c t r i c a l characteristics of b i o l o g i c a l and l i p i d bilayer membranes, r e c t i f i c a t i o n phenomena are commonly encountered (Katz, 1948; Hodgkin and Huxley, 1952; Adrian, 1969; Haydon and Hladky, 1972). Since membrane conductance may depend both on the number of conducting channels and the conductance of single channels, variations i n the number of conducting channels or i n the conductance of a single channel with voltage can result i n non-linear current-voltage relations when current i s measured from a family of channels. The dependence of the conductance of a single channel on voltage also depends on the p r o f i l e of the sequence of activating energy barriers within the pore (Parlin and Eyring, 1954; Woodbury, 1971; H i l l e , 1975b). If the above qu a l i t a t i v e points of view were adopted for the chloride permeating system, then on the application of hyperpolarizing voltage steps from a fixed reference potential (such as the resting potential), the instantaneous current-, voltage r e l a t i o n indicates the dependence of a single channel on voltage. The subsequent current transients may r e f l e c t a decrease either i n the number of conducting channels or the conductance of each channel or erhaps both. The nature of the relaxation processes involved i n the chloride permeating systems may be studied by determining the voltage and time dependence of the current transients. When two voltage pulses are applied in succession, the instantaneous currents corresponding to the second pulse (test pulse) indicate the state 1 3 of the channel at the terminat ion of the previous conditioning pulse. F i n a l l y , the state of the channel during transitions may be studied by applying voltage perturbations under non-stationary conditions such as during a f a l l (or a r i s e ) in current. The dependence of the instantaneous current on voltage at the onset of the perturbation and i t s subsequent dependence on time and transmembrane voltage a l l serve as clues to the molecular nature of the kinetic processes involved i n chloride transport. In the experiments reported here, we have attempted to gain a broad perspective of the behavior of chloride conductance under conditions of different external pH and different chloride concentration gradients across the membrane. Since the variety of experimental conditions i s vast, a detailed study of the chloride conductance under a l l conditions was impractical because of the time required. Rather than studying the conductance i n d e t a i l at one pH and at a fixed concentration gradient, we f e l t a more comparative and extensive over-view i s required before detailed studies be undertaken at one pH. The present experiments have given us a broad over-view of the behavior of chloride conductance. I t w i l l be seen that the experiments provide a comparative basis for a further detailed study of any one aspect of chloride conductance. 1 4 Methods Laboratory raised Xenopus laevis, rather than Rana pipiens were used because the l a t t e r were often infected with parasites. The Xenopus specimens were obtained from the Amphibian F a c i l i t y of the University of Michigan at Ann Arbor, or from Nasco Ltd., Guelph, Ontario. The experiments were done at room temperature (20-22°C) on surface f i b e r s of sartorius. Voltage clamp The experiments were carried out using the three microelectrode voltage clamp technique as f i r s t formulated by Adrian, Chandler and Hodgkin (1970a). The voltage clamp arrangement with the positioning of the three microelectrodes i s shown i n Fig.1.1. The electrodes had resistances between 5 and 10 megaohms and those at A and B, 500 and 1000 microns respectively from the pelvic end, were f i l l e d with 3 M KC1 and had t i p potentials of less than 5 mV. The current in j e c t i n g electrode at C, 1100 microns from the pelvic end, was f i l l e d with 2 M potassium c i t r a t e . The voltage control i n our experiments was different from that used i n other 3 microelectrode experiments. The fi b e r i n t e r i o r at A was held close to v i r t u a l earth and the potential at D, making contact with the bath v i a an Agar-Ringer bridge, was held at minus the membrane potential by a feedback follower arrangement as described by Eisenberg and Gage (1969a). The potential at D was re-inverted and 1 5 Figure 1.1. Schematic of the voltage clamp c i r c u i t . The positioning and function of the electrode i s described i n the text. Amplifier A l i s a Burr-Brown 3307/12C with an open-loop gain of 106 dB: the inside of the f i b e r at A i s n e g l i g i b l y d i f f e r e n t from v i r t u a l earth p o t e n t i a l . Amplifiers A2 and A 5 are Burr-Brown 3400A. Amplifier A3 (the clamp amplifier ) i s a Burr-Brown 3010/25 and the driving amplifier (set at 10 x gain) i s a Burr-Brown 3138/25 (100 v o l t output). - i p h a s e c o m m 16 a p p l i e d , together with a c o n t r o l vo l tage , to a summing a m p l i f i e r (voltage clamp operating i n the summing mode) the output of which i n j e c t e d current at C. The a x i a l vo l tage drop between A and B, denoted in our experiments by AV, was measured d i r e c t l y at B with a capac i ty compensated vol tage fo l lower referenced to earth p o t e n t i a l . This method avoided the need for d i f f e r e n t i a l a m p l i f i c a t i o n to obta in AV. 2 The membrane current densi ty I (Amp/cm ) at A i s r e l a t e d to AV by the r e l a t i o n (see Appendix) d (1) I = — 3 AV 6 R. I2 I i s the i n t e r n a l r e s i s t i v i t y of the sarcoplasm; d i s the f i b e r diameter, and £ i s the distance between e lectrodes A and B. The e r r o r in t h i s method i s l e s s than 5% (Adrian, Chandler and Hodgkin, 1970) i f the separat ion ( £ ) between e lectrodes at A and B i s l e s s than twice the low frequency space constant . The i n t e r n a l r e s i s t i v i t y R^ was estimated i n Xenopus l a e v i s using convent ional small s i g n a l cable a n a l y s i s (Vaughan, 1975). A value of 163-14 ficm was obtained i n Ringer ' s s o l u t i o n . This value i s in c lose agreement with the value obtained by Nakajima and Bast ian (1974) in the i l i o f i b u l a r i s muscle of Xenopus. For the current c a l i b r a t i o n (AV) i n our experimental records , the e lec trode separations were 500 microns (between A and B) . The standard value of 170 ficm for R^ (Hodgkin and Nakajima, 1972) was used. A one m i l l i v o l t d i f f erence i n p o t e n t i a l i n a f i b e r with a diameter of 80 microns corresponds to a current density of 3.14 Amp/cm'1 of membrane area. To bring the f i b e r under voltage control, the electrode closest to the pelvic end (position A, Fig.1.1) was f i r s t inserted into the fi b e r . Small hyperpolarizing constant current pulses were than passed through the c i t r a t e - f i l l e d electrode at C as i t was lowered into the fi b e r . When the current injecting electrode and the electrode A were in the same f i b e r , electrode B i n the solution recorded a potential minus that of electrode A. The electrode at B was then inserted between the current i n j e c t i n g electrode and position A. When a l l three electrodes were in the same f i b e r , the system was switched to voltage control and the clamp gain increased to minimize the rise-time to the command potential at B. The holding potential, E^, was always chosen equal (within 5 mV) to the resting membrane potential unless otherwise indicated. Current and voltage were displayed on a Tektronix R5103N oscilloscope and photographed on 35 mm f i l m for subsequent enlargement and analysis. Sign convention 1. In a l l the experimental records inward current i s shown as downward deflection of the oscilloscope trace unless otherwise noted. 2. A l l graphs are plotted with inward currents below the voltage axis. 3. A l l equations consider the sign of inward current to be positive. 18 Table 1.1 Contents of solutions (mM/Jl). So In Na + K + Rb + C a ^ CI TTX gm/fc sucrose pH buffer A 119 2.5 - 1.8 121.1 - lO" 5 7.2 t r i s B a 129 - 2.5 1.8 12ia_ — - 5.4 phosphate b 119 - 2.5 1.8 121.1 - 6.7 t r i s c 119 - 2.5 1.8 121.1 - — 7.3 t r i s d 119 - 2.5 1.8 121.1 - - 8.4 t r i s C a 154 - 2.5 8.0 - 84.25 - 5.4 phosphate b 150 - 2.5 8.0 - 84.25 - 7.3 t r i s c 150 - 2.5 8.0 - 84.25 - 8.4 t r i s D - 117 - 1.8 120.6 - - 160 7.2 t r i s D* 116 100 - 3.0 218 - - - 7.4 t r i s D a 38 117 - 8.0 - 85.75 - 5.4 phosphate d 34 117 — 8.0 - 85.75 - 8.4 t r i s E a 38 - 117 1.8 120.6 - 160 5.4 phosphate b 34 - 117 1.8 120.6 - 160 6.7 t r i s c 34 - 117 1.8 120.6 - 160 7.3 t r i s d 34 - 117 1.8 120.6 - 160 8.4 t r i s E d* 34 - 117 1.8 120.6 - 400 8.4 t r i s F a 38 - 117 8.0 - 85.75 — 5.4 phosphate d 34 - 117 8.0 - 85.75 - 8.4 t r i s 19 Ions c o n t r i b u t i n g to membrane current Since the aim of the present experiments was to inves t iga te the behavior of c h l o r i d e current s , i t was important that under experimental cond i t ions c o n t r i b u t i o n s of other ions to the membrane current could be obviated or at l eas t accounted f o r . Ch lor ide was the only anion i n the s o l u t i o n except where small amounts of phosphate were introduced by the buffer system. Regenerative sodium currents were blocked by the a d d i t i o n of te trodotox in (10 ~*g/H) to the s o l u t i o n s . An estimate of the rubidium conductance i n rubidium-conta in ing Ringer ' s so lu t ion ( so lut ions B ( a , b , c , d ) , T a b l e 1.1) at a l l pH was made by r e p l a c i n g the c h l o r i d e with sulphate ( so lut ions C ( a , b , c ) ) . The current vol tage r e l a t i o n s of the f i b e r s in so lu t ions C ( a , b , c ) were l i n e a r and -2 were not inf luenced by pH. An averaged value of 20 pmho cm (n=17) was obtained for the r e s t i n g conductance of f i b e r s in so lut ions C. Since sulphate does not permeate the membrane (Hodgkin and Horowicz, 1959; Hutter and Warner, 1972) t h i s conductance i s a t t r i b u t e d to rubidium. Solut ions [Table I.1] So lu t ion A i s R inger ' s s o l u t i o n . So lut ions B ( a , b , c , d ) are standard Ringer ' s so lu t ions with potassium replaced by rubidium and buffered at d i f f e r e n t pH's . Notice the s l i g h t d i f f erences in the sodium concentrat ion depending on the buf fer used. Trizma buf fers (Tris(hydroxymethyl)aminomethane) were used at. n e u t r a l and a l k a l i n e pH's and contr ibuted no sodium. 20 The phosphate buf fers used at pH's l e ss than 6 were prepared from sodium s a l t s . Solut ions C ( a , b , c ) are again standard Ringer ' s so lut ions but with sulphate rep lac ing c h l o r i d e . A d d i t i o n a l calcium sa l t was added i n order to maintain the l e v e l of ion ized calcium at i t s normal l e v e l (Hodgkin and Horowicz, 1959). The number given i s for t o t a l ca lc ium. Solut ion D i s R inger ' s s o l u t i o n conta in ing high potassium (sodium replaced by potassium). Th i s s o l u t i o n was used to permanently depolar ize the f i b e r s . So lut ion D i s the d e p o l a r i z i n g so lu t ion used by Hutter and Warner (1972, s o l u t i o n E ) . Solut ions D(a,d) were used to study the potassium conductance in depolar ized f i b e r s . So lut ions E ( a , b , c , d ) were used to measure c h l o r i d e current in depolar ized f i b e r s . The potassium in s o l u t i o n D was replaced by the impermeant ion rubidium and the s o l u t i o n buffered to d i f f e r e n t pH's . Some of the so lu t ions were osmot ica l ly unbuffered. So lu t ion Ed* w a s the hypertonic s o l u t i o n used in some depolar ized f i b e r s to prevent f i b e r c o n t r a c t i o n s . + —2 Solutions F (a .d ) were the solutions used to tes t for Rb and SO, 4 permeation in depolar ized f i b e r s . Further comments on the so lut ions w i l l be provided for i n the text . Chapter I I Experimental Results 22 The membrane current accompanying hyperpolarizing voltage pulses from the resting potential depended on the pH of the solution bathing the fibe r and the size of the voltage pulse. The current transients were q u a l i t a t i v e l y similar to those reported by Warner (1972) for Rana temporaria. In alk a l i n e and neutral rubidium-containing solutions (solutions Be and Bd respectively), the membrane current decreased monotonically from i t s i n i t i a l (instantaneous) maximum, reaching a steady value i n 1.5 sec (Fig.2.0.la). In acid solutions (solutions B(a,b)) the current waveforms varied from fiber to f i b e r and three types of responses could be described. Some fibers showed an i n i t i a l decay i n current from i t s i n i t i a l maximum ( l i k e those i n alkaline and neutral solutions) but the i n i t i a l decrease was followed by an exponential increase to the steady value (Fig.2.0.lb). In other f i b e r s , the i n i t i a l decrease of current became smaller and more rapid and for hyperpolarizations greater than -50 m i l l i v o l t s from the resting potential, only a monotonic increase could be seen. F i n a l l y , a t h i r d groups of fibers had current waveforms that rose monotonically from i n i t i a l to steady state at a l l voltages (Fig.2.0.Ic). In Ringer's solutions containing rubidium sulphate at different pH's (solutions C(a,b,c)), these time dependent current responses were not observed. I f i n the presence of chloride there i s no interaction among the cations, or between cations and chloride such that they contribute a large f r a c t i o n of the membrane current (and there i s no evidence to suggest that there i s i n t e r a c t i o n ) , then the currents described here may be attributed primarily to chloride. 23 Figure 2.0.1 Current waveforms recorded i n response to hyper-p o l a r i z i n g voltage s teps . The numbers beside the traces i n d i c a t e the amplitude of the voltage steps (mV). A l l currents were inward (corresponding to outward c h l o r i d e f l u x ) . Pulse durat ion was three seconds i n a l l cases. a) A l k a l i n e s o l u t i o n (pH 8.4, s o l u t i o n Bd) . The current always f e l l from the i n i t i a l to the steady s t a t e . Holding p o t e n t i a l , -80 mV. b) A c i d s o l u t i o n (pH 5.4, s o l u t i o n Ba) . Current that f e l l and then rose . Traces recorded on a d i g i t a l averager. Holding p o t e n t i a l , -70 mV. c) A c i d s o l u t i o n (pH 5.4, s o l u t i o n Ba) . Waveforms i n which only a r i s i n g current was seen. Recorded on the d i g i t a l averager. Holding p o t e n t i a l , -70 mV. I, . 1 . I . I o I I o I I i I 1 I I c n I 'l u J . I il J In the following section the response of the instantaneous and steady state currents to hyperpolarizing voltage steps and also the influence of the pH of the bathing solution w i l l be described. Instantaneous currents were estimated by analyzing the time course of current relaxations (as w i l l be described in Section I.IB) toward the steady state from.about 20 msec after the onset of the voltage step and extrapolating to zero time as well as by direct reading from the filmed record. The results obtained always agreed within 10%. Since local contractures occurred with depolarizing voltage steps, causing loss of impalement and fiber damage, only very small positive voltage steps (less than 20 millivolts from the resting potential) were applied. The amplitudes of the current transients were too small for analysis, and subsequently w i l l not be discussed. 25 Experimental Section I. The behavior of chloride currents in one pulse voltage clamp experiments I. Experiments on fibers with normal resting potentials A. The instantaneous and steady state current-voltage characteristics Fibers in alkaline solutions (solution Bd, pH 8.4-8.8) The current-voltage relations of 43 fibers bathed in alkaline solutions were studied. Eleven fibers were held at -70 mV, thirty-one were held at -80 mV, and one fiber was held at -90 mV. The resting potentials of the fibers were within 5 mi l l i v o l t s of the holding potential (average resting potential was -78 mV for 43 fibers). The magnitude of the instantaneous current as a function of the voltage was linear for hyperpolarizing potentials of less than 30 mV. For hyperpolarizing voltages that exceeded 50 mV the instan-taneous currents appeared to reach a limit (Fig.2.1.la). This is in contrast to the fibers of Rana temporaria (Warner, 1972) where instantaneous currents were linear as a function of voltage. The steady state current-voltage relations for Xenopus laevis fibers in alkaline solutions (pH 8) always became parallel to the current axis for large hyperpolarizations. This i s shown in Fig.2.1.1b (solid circles) for 17 fibers that had the same holding potential (-80 mV) and had the same steady state current density. For each point, which represents the mean of a number of observations, the sample size was less than 17 because the cells were not a l l studied 26 Figure 2.1.1 a) Instantaneous and steady state current-voltage relations of a f i b e r at pH 8.4 (solution Bd). As the membrane potential was made more negative the membrane currents reached l i m i t i n g values. Open c i r c l e s , instantaneous current; f i l l e d c i r c l e s , steady state current. Holding p o t e n t i a l , -80 _V. The currents include a contribution by rubidium. b) Steady state current-voltage r e l a t i o n s of populations of polarized f i b e r s i n solutions of pH 8.4 ( f i l l e d c i r c l e s ) and 7.3 (open c i r c l e s ) . Bars indicate standard errors of the means (for d e t a i l s , see t e x t ) . 26a 27 at the same voltages. The v e r t i c a l bars depict standard deviations. The dashed l i n e i n the figure indicates the currents recorded' i n rubidium sulphate solutions (solution Cc). Since i n general these were done on different f i b e r s , no attempt was made to subtract the rubidium current from the steady state currents. However, i t can be seen that i f the rubidium currents were subtracted, the chloride current-voltage relations would be seen to have a negative slope region. In several c e l l s , negative slope conductance was observed i n RbCl solutions (solution Bd) without subtraction of the rubidium current. Fibers i n neutral solutions (pH 7.3, solution Be) The current-voltage relations of 20 fibers were studied at pH 7.3. Fifteen c e l l s were held at -80 mV and f i v e at -70 mV. The average resting potential of the fibe r s was -78 mV. Both the instantaneous and steady state current-voltage characteristics were l i n e a r for a greater hyperpolarizing voltage range than the fiber s i n al k a l i n e solutions. For the largest hyperpolarizing voltages applied (70 mV more negative than the holding potential) the conductances (both the instantaneous and steady state) reach l i m i t i n g values (Fig.2.1.2). This i s i n contrast to f i b e r s i n more al k a l i n e solutions where the currents reach l i m i t i n g values. The c i r c l e s of Fig.2.1.1b are averaged data for 15 c e l l s at pH 7.3. Here again, the same size and hence the contribution to the v e r t i c a l bars, was less than 15 (not a l l f i b e r s were studied at the 28 150 V m ( m v j -80 m pA/cm2 l o o o o -»25 Figure 2.1.2 Instantaneous (open c i r c l e s ) and steady state ( f i l l e d c i r c l e s ) current-voltage r e l a t i o n s for a fi b e r at pH 7.3 (solution Be). As the membrane potential was made more negative the instantaneous and steady state conductances reached a l i m i t i n g value. Holding p o t e n t i a l , -80 mV. A contribution by rubidium i s included i n the currents. 29 potentials). The l i n e shows the prediction according to the constant f i e l d equation (Hodgkin and Horowicz, 1959; Hutter and Warner, 1972) with a value of 8.0 X 10 ^ cm sec * for P ., the permeability coefficient. I t can be seen that i n Xenopus l a e v i s , even though the steady state current-voltage relations reach a l i m i t i n g conductance as i n constant f i e l d theory (Goldman, 1943), the r e c t i f i c a t i o n i s greater than predicted from the constant f i e l d theory. Fibers i n acid solutions (pH 5.4-6.7) 31 f i b e r s were studied i n th i s pH range: 15 fibers i n solutions of pH 6.4-6.7, and 16 fibe r s at pH 5.4-5.6. The average resting potential for the fibe r s i n the higher pH range was -75 mV and for the fiber s i n the lower pH range was -70 mV. The pattern of progressive l i n e a r i z a t i o n of the instantaneous current-voltage curves with reduction i n pH as described i n the previous sections continued i n more acid solutions. In the pH range (6.4-6.7) there was a large v a r i a b i l i t y i n the behavior of the f i b e r s . For instance, some f i b e r s i n solutions of pH 6.4 exhibited instantaneous and steady state current-voltage curves that were upwards concave as i n a l k a l i n e and neutral solutions. Other f i b e r s at the same pH displayed l i n e a r i t y both i n instantaneous and steady state current-voltage curves that were upwards concave as i n alk a l i n e and neutral solutions. F i n a l l y , some f i b e r s showed t y p i c a l 'acid' c h a r a c t e r i s t i c s : l i n e a r instantaneous currents and steady state 30 currents that were downwards concave. This wide variety of behavior might be due to a large v a r i a b i l i t y i n the pK of the pH dependent t i t r a b l e s i t e s . In the most acid solutions (pH 5.4) the instantaneous current-voltage relations were always line a r (up to the largest hyperpolar-izations applied, -70 mV from the holding potential). This l i n e a r i t y was independent of the transient behavior of the currents. The steady state current-voltage relations at t h i s pH were always downwards concave for hyperpolarizing pulses as shown i n Fig.2.1.3a. The downwards concavity of the steady state current voltage relations i s due to the r i s i n g component of current transients. This can be observed as follows: The r i s i n g component i s the only con-tri b u t o r to time dependent membrane current for fibers whose transients increased monotonically to the steady state ( f i l l e d c i r c l e s , Fig.2.1.3a). For fi b e r s whose transients f e l l and then rose, since the time constants of the f a l l were approximately 1/6 of the subsequent r i s e , the currents can be approximated as being i n a steady state after the i n i t i a l f a l l , before the r i s i n g component contributes any membrane current. When the current was lea s t , just before the delayed r i s e i n current, the current-voltage relations were l i n e a r , and the subsequent very s l i g h t concavity of the steady state current-voltage relations i n Fig.2.1.3b was a result of t h i s delayed r i s e i n current (cf. Fig.2.1.3a). Summary of the effect of pH on current-voltage r e l a t i o n s and on  resting conductance In a l k a l i n e and neutral solutions, the instantaneous and the 31 Figure 2.1.3 Instantaneous and steady state current -vo l tage r e l a t i o n s for f i b e r s i n the pH range 5.4 to 6.7. a) Instantaneous (open c i r c l e s ) and steady s tate ( f i l l e d c i r c l e s ) r e l a t i o n s i n a f i b e r that demonstrated monotonical ly r i s i n g currents . pH 5.4, holding p o t e n t i a l , -70 mV. b) Re lat ions for a f i b e r that gave a b iphas ic current waveform. Open c i r c l e s , instantaneous current ; f i l l e d t r i a n g l e s , current measured at the l o c a l minimum where f a l l i n g current gave way to r i s i n g current ; f i l l e d c i r c l e s , steady s tate c u r r e n t . The s t ra ight l i n e s were f i t t e d by eye. pH 6.4, holding p o t e n t i a l , -80 mV. A l l currents i n t h i s f i g u r e contain contr ibut ions by rubidium. 3 2 steady state current-voltage curves are concave upwards. The degree of concavity and the slopes of the curves at the holding potential decrease with decreasing pH, u n t i l ultimately the I^ -V relations are linear (around pH 6.4). The magnitudes of the current transients also decrease as the current-voltage curves become linearized. This pattern of l i n e a r i z a t i o n of I-V curves and decreasing magnitudes of current transients depended continuously on pH u n t i l i n acid solutions there i s a divergence of behavior by the introduction of a r i s i n g component of time dependent current. In the simplest behavior the instantaneous and steady state curves from alkaline and neutral solutions coincide and as a result only an instantaneous linear current-voltage r e l a t i o n i s observed and the current r i s e s monotonically to the steady state. In more complex cases, the two curves do not coincide completely and as a result the time dependent current f i r s t decreases and then increases to the steady l e v e l . In our studies on chloride conductance i n Xenopus i t was found that there were large differences i n the current densities and conduc-tances from f i b e r to f i b e r even at the same pH. In spite of these differences, the qu a l i t a t i v e behaviors of the fi b e r s at the same pH, such as the direc t i o n of r e c t i f i c a t i o n , were sim i l a r . In order to minimize t h i s v a r i a b i l i t y when the effects of pH on membrane conductance are studied, experiments would have to be performed at different pH's on the same f i b e r . Most of our information comes from pooled measurements from a population of f i b e r s . Table I I . 1 . 1 shows the resting conductance obtained from the slope of the current-voltage 33 relations at the resting potential. In going from pH 8.4 to pH 6.4, the resting conductance decreased by a factor of 2. Fig.2.1.4 shows the result of an experiment where the solution bathing the fiber was changed from pH 6.4 to pH 8.4 with the three microelectrodes impaling the f i b e r . When the pH was changed from 8.4 to 6.4, the resting conductance decreased three times. These values are s l i g h t l y less than reported by Hutter and Warner (1967a, 1972) i n Rana temporaria. Table II.1.1 Effect of pH on resting conductance pH resting conductance from current-voltage relations 5.4 2.1 X 10" 4 mho -2 cm 6.4 3.75 X l O ' 4 mho -2 cm 7.4 4.0 X lO " 4 mho -2 cm 8.4 7.35 X lO" 4 mho -2 cm B. The voltage dependence of chloride current transients In the preceding section the q u a l i t a t i v e behavior of the current transients i n solutions of different pH was described. In response to hyperpolarizing voltage pulses, current transients are composed es s e n t i a l l y of 3 components: a fast i n i t i a l f a l l with time constants less than 100 msec; a slow 'creeping' f a l l to the steady state that required between 300-500 msec for completion; and a r i s i n g component that i s observed only i n f i b e r s i n acid solutions. 3 A Figure 2.1.4 Steady state and instantaneous current-voltage relations for a f i b e r i n which voltage control was maintained while the pH of the solution was changed from 8.4 to 6.4. The open c i r c l e s indicate the instantaneous current at pH 8.4 and the f i l l e d c i r c l e s the steady state current at the same pH. Instantaneous and steady state currents were indistinguishable at pH 6.4 ( f i l l e d squares). Holding p o t e n t i a l , -70 mV. Contributions of rubidium current have not been subtracted. 34a 35 Analysis of current transients The time constants of current transients were obtained as follows: a) For increasing currents, Fig.2.1.5a. A logarithmic plot of the transient ( I ^ - I ( t ) ) was made againt t , the time from the onset of the voltage pulse (Fig.2.1.5a), I i s the steady current 00 value. A straight l i n e was drawn through the points by eye, as shown i n Fig.2.1.5b. b) For currents that decreased and then increased (at acid pH), Fig.2.1.5c. The time constant of the r i s i n g current transient and the contribution to t o t a l current made by the r i s i n g phase were estimated by method (a). When the r i s i n g current was subtracted from the t o t a l , the time constant of the f a l l i n g phase was estimated from the l i n e of best f i t to the logarithmic plot of (I(t)-I ( j o) against time Fig.2.1.5.d(i) . The p r i n c i p a l time constant of-the f a l l i n g phase was of the order of 50 msec, whereas the time constants of the r i s i n g phase exceeded 250 msec, allowing a simpler analysis to be used. In t h i s method the l o c a l minimum between f a l l i n g and r i s i n g phases was assumed to be the steady state current for the f a l l i n g phase and the beginning (t=0) of the contribution of r i s i n g phase (Fig.2.1.5d(ii)). The differences between the resu l t s of. the two methods were ne g l i g i b l e and the simpler approach was generally used. But i n situations were the two time constants were s i m i l a r i n magnitude, we were unable to resolve the current transients accurately. 36 Figure 2.1.5 Current waveforms and the analysis of transients. a) Waveform of current recorded i n response to a hyperpolarizing voltage step of 70 mV from a hold-ing potential of -70 mV at pH 5.A. The trace was recorded from a d i g i t a l averager and has been inverted by comparison with out usual records. b) The r i s i n g transient of (a) i s f i t t e d by a single exponential. c) Biphasic response from a different f i b e r at pH 5.4. The hyperpolarizing voltage step was 45 mV from a holding potential of -70 mV. The double arrow indicates the l o c a l minimum current. d) The delayed r i s e i n (c) i s f i t t e d by a single exponential (open c i r c l e s ) . When th i s i s projected to zero time and the resulting current subtracted from the t o t a l , the remaining f a l l i n g current i s f i t t e d by a single exponential ( f i l l e d c i r c l e s ) . I f the f a l l i n g phase i s analysed assuming a steady current at the l e v e l of the l o c a l minimum a s l i g h t l y different time constant i s obtained (open diamonds). e) A current waveform that f e l l to a steady value. Holding potential, -80 mV, pH 7.4. f) Currents that only decline were usually resolved into two time constants. Current c a l i b r a t i o n (bar), 5 mV. The corresponding —6 2 current density i s 3.14 x 10 Amp/cm per mV. In (b), (d) and (f) the abscissa i s i n milliseconds. 3 6 a 37 c) Currents that decreased to a steady value, Fig.2.1.5e. The logarithm of the transient component of the current was plotted against t , the time from the onset of the voltage step. The l i n e of best f i t was made by eye to the points between 200 and 400 msec. This l i n e had a time constant of about 500 msec. After subtracting t h i s slow process from the t o t a l current transient, the difference was straight l i n e with a time constant varying from 10 to 100 msec (Fig,2.1.5f). The time course of the creeping decay of current appeared to consist of a family of exponentials, possibly indicating the relaxation spectrum of a d i s t r i b u t i v e system. In the approximate analysis of Fig.2.1.5f, the time period (200-400 milliseconds) was chosen because the exponential approximation f i t t e d the data for the longest time i n t e r v a l and the current waveforms could consistently be resolved into two components. The results that we describe were obtained by these approximate methods. The slow creeping f a l l i n current to the steady state was found not to depend on the membrane potential nor on the pH of the bathing medium. Only the more rapidly evolving current transients were sensitive to voltage. In the following section we w i l l describe the voltage dependence of the i n i t i a l f a l l i n g current i n a l k a l i n e and neutral solutions and the f a l l and subsequent r i s e i n the current transients in acid solutions. 38 K i n e t i c s i n a c i d so lut ions (pH 5.4-6.7) The i n i t i a l f a l l i n g currents and the r i s i n g currents i n ac id so lu t ions appeared to act independently. The voltage dependence of the rate of i n i t i a l f a l l of current was independent of the subsequent k i n e t i c behavior . That i s , i t d id not depend on the existence of a subsequent r i s e . Likewise the time constants of the r i s i n g currents were independent of the presence of any preceding f a l l . The time constants for the decrease i n i n i t i a l current depended l i n e a r l y on the amplitude of the hyperpo lar i z ing pulse as shown i n F i g . 2 . 1 . 6 a . The slope of the l i n e (obtained by regress ion) i s -1.17 msec/mV. The absc i s sa of F i g . 2 . 1 . 6 a . represents the s i ze of the h y p e r p o l a r i z i n g vo l tage . The time constants of the increas ing component of current t rans i en t s for f i b e r s . i n a c i d so lu t ions depended n o n - l i n e a r l y on the h y p e r p o l a r i z i n g vo l tage . For h y p e r p o l a r i z i n g vo l tage steps greater than 50 m i l l i v o l t s , the time constant approached a l i m i t i n g va lue of 250 m i l l i s e c o n d s ( F i g . 2 . 1 . 6 b ) . K i n e t i c s i n n e u t r a l and a l k a l i n e s o l u t i o n s (pH 7.3-8.8) A summary of the vo l tage dependence of the time constants of the i n i t i a l decay of current i s presented i n Table I I J.. 2. I t can be seen that i n n e u t r a l and a l k a l i n e s o l u t i o n s , as i n a c i d s o l u t i o n s , the time constants of the f a l l of i n i t i a l current depended l i n e a r l y on the h y p e r p o l a r i z i n g vo l tage s tep . L i n e a r regress ion on data i n n e u t r a l s o l u t i o n s (pH 7.3) gave slopes 39 « Figure 2.1.6 The dependence of relaxation time constants on the size of the membrane potential step i n fib e r s bathed in acid solutions (pH 5.4, solution Ba). The holding potentials of the f i b e r s studied were -70 mV. a) The voltage dependence of the time constant of the i n i t i a l f a l l i n g current transient. b) The dependence of the time constant of the sub-sequent r i s i n g phase. Summary of the voltage dependence of the fast time-constants of relaxation of chloride current. pH Voltage dependence (msec/mV) 5.4 -1 .17* 7.3 -1.55 ± 0.44 * * (n=8) 8.4 -1.56 ± 0.38 * * (n=4) * Refer to F i g . 2 . 1 . 6 a . * * Standard d e v i a t i o n . In pH's 7.3 and 8.4, l i n e a r regress ion was performed on data from i n d i v i d u a l f i b e r s and the slopes (voltage dependence of time constants) were then averaged to o b t a i n the mean and standard d e v i a t i o n . Al of -1.55 ± O.AA (S.D. n=8) msec/mV and i n alkaline solutions solutions (pH 8.A) gave slopes of -1.56 ± 0.38 (S.D. n=4) msec/mV (Table II.1.2). Comparison with Rana temporaria Even though the quali t a t i v e behavior of the fibers i n solutions of different pH's was similar between fibers of Xenopus laevis and Rana temporaria, there are quantitative differences between our results and those reported by Warner (1972) for Rana. The decay of current was resolved into a single time constant between 100 and 200 msec (Warner, 1972) with a voltage dependence of -1.5A msec/mV in pH 9.8 and -1.33 msec/mV, for fibers in pH 7.A, whereas i n our experiments, current decays were resolved into two time constants, one pH and voltage dependent (-1.55 msec/mV) but independent of pH. Dependence of ki n e t i c s on the magnitude of voltage steps In the previous section i t was found that the time constants of currents depended l i n e a r l y on the hyperpolarizing voltage. However, since the comparisons were made on different f i b e r s with the same holding potential, we can not distinguish whether t h i s l i n e a r de-pendence was on the absolute potential or on the size of the negative-going voltage step. To d i f f e r e n t i a t e between the two alternatives, several experiments were conducted with the same fib e r held at different potentials. Fig.2.1.7 shows the resu l t s from one f i b e r i n solution of pH 7.3, held successively at -60, -70 and 4 2 Figure 2.1.7 Relationship between decay time constant and the amplitude of hyperpolarizing voltage steps from the holding potential at pH 7.3 (solution Be), for three holding potentials. F i l l e d c i r c l e s , open c i r c l e s and triangles are for holding potentials of -60 mV, -70 mV and -80 _V, respectively. The l i n e s were f i t t e d by least squares line a r regression. The abscissa i s the size of the hyperpolarizing voltage step and the ordinate i s the fast decay time constant, i n m i l l i -seconds. The slow time constant found to be independent of voltage. 42a 43 -80 mV. The absc i s sa i s the s i ze of the vol tage step. The slopes of the l i n e s of best f i t f or data from each of the three holding p o t e n t i a l s for pulses between 10 and 50 mV amplitude are not s i g n i f i c a n t l y d i f f e r e n t . That i s the ra te of the fast decay depends only on the magnitude of the hyperpo lar i z ing pulse and not on the holding p o t e n t i a l or the absolute membrane p o t e n t i a l . This conclus ion w i l l be supported by two pulse experiments and experiments on f i b e r s with low r e s t i n g p o t e n t i a l s . A f t e r c u r r e n t s : changes i n zero current p o t e n t i a l with c h l o r i d e passage  across the membrane On subsequent re turn to the holding p o t e n t i a l a f t er a hyperpo lar i z ing test pu l se , the membrane current , instead of re turn ing to the ho ld ing current , was t r a n s i e n t l y outward. The magnitudes of the a f t ercurrent s were u s u a l l y very smal l , making accurate ana lys i s of t h e i r amplitudes and time courses very d i f f i c u l t . However, i t d id appear that the magnitudes of the instantaneous a f t e r c u r r e n t s i n f i b e r s i n n e u t r a l and a l k a l i n e so lut ions (pH greater than 7) reached a l i m i t i n g value for large h y p e r p o l a r i z a t i o n s . This can be seen i n F ig .2 .1 .8a< where a sequence of vo l tage steps and assoc iated a f t e r c u r r e n t s from a f i b e r at pH 7.3 i s shown; the magnitudes of the instantaneous a f t e r c u r r e n t s from the same f i b e r but at two d i f f e r e n t ho ld ing p o t e n t i a l s (open and f i l l e d c i r c l e s ) i s shown i n F i g . 2 . 1 . 8 b . Warner (1972) observed s i m i l a r a f t e r c u r r e n t s with h y p e r p o l a r i z i n g vo l tage steps i n Rana temporaria and termed the phenomenon ' s tored 44 Figure 2.1.8 Magnitude and direction of after-currents. a) Outward after-currents, indicated by the arrows on the right of the traces, recorded from a fiber at pH 7.3 (solution Be) following the stepping of the membrane potential back to the holding potential from a more negative potential (the numbers on the traces) at which current had reached a steady state. Resting and holding potential, -80 mV. Calibration: abscissa, 1 second; ordinate, 5 mV. 1 mV = 3.14 x 10 ^  Amp/cm . b) I n i t i a l after-currents plotted as a function of the membrane potential during the preceding step. Data from the same f i b e r as (a). Open c i r c l e s are from a run i n which the holding potential was -60 mV and f i l l e d c i r c l e s are from a run i n which i t was -80 mV (the same as the resting p o t e n t i a l ) . Notice that abscissa i s deviation from holding poten t i a l . Calibration f o r the ordinate: —6 2 1 mV = 3.14 x 10 Amp/cm . 44a 45 charge'• Such aftercurrents could be caused either by a depletion or enhancement of chloride i n a compartment i n series with the e l e c t r i c a l pathway (an analogy would be the potassium depletion i n the transverse tubular system with hyperpolarizing pulses (Adrian, Chandler and Hodgkin, 1970b; Aimers, 1972a,b; Barry and Adrian, 1973)), or could be due to a recovery of a membrane process that resulted i n the decay of current during the test pulse. The likel i h o o d of both explanations was investigated using fibers where the internal chloride concentration was altered and also using two pulse experiments. This w i l l be described l a t e r . I I . Experiments on permanently depolarized fibers  Experimental rationale In the preceding section, the pH dependence of chloride conductance was described and some evidence was presented to support the dependence of current transients on the size of the hyperpolarizing voltage step rather than the absolute membrane potential. The experiments described were conducted on fibers that had resting potentials between -70 and -90 m i l l i v o l t s . The chloride concentration r a t i o across the membrane i s CI given by Nernst's equation o/Cl^= exp(-FV/RT), and varies between 16.0 (-70 mV) and 35.3 (-90 mV) at 20°C. In t h i s section, experiments performed on fibers where the concentration r a t i o was lowered by permanently depolarizing the fibers w i l l be described. The inte r n a l chloride concentration was increased while the external chloride concentration was kept constant at the same 46 value as f i b e r s with high concentrat ion r a t i o s . These experiments were performed for severa l reasons: The dependence of conductance on concentrat ion provides us with information on the poss ib le mechanisms of c h l o r i d e permeation. For ins tance , Warner (1972) suggested that ch lor ide might permeate the mem-brane v i a mobile charged c a r r i e r s of the type s tudied by Sandblom, Eisenman and Walker (1967). A p r e d i c t i o n of t h i s model i s that the degree of current -vo l tage r e c t i f i c a t i o n depends on the asymmetrical d i s t r i b u t i o n of c h l o r i d e across the membrane. S i m i l a r l y , Hodgkin and Horowicz (1959) i n descr ib ing the dependence of c h l o r i d e currents on membrane p o t e n t i a l by the constant f i e l d equation, suggest that the r e c t i f i c a t i o n i s caused by the concentrat ion gradient across the membrane (Chapter IV). I f t h i s were the case i n n e u t r a l s o l u t i o n s , then i t i s important to determine whether there i s any c o n t r i b u t i o n of concentrat ion gradients to the current -vo l tage c h a r a c t e r i s t i c s i n a c i d and a l k a l i n e s o l u t i o n s . In order to understand the e f fec t of pH on conductance, the p o s s i b l e causes of r e c t i f i c a t i o n such as concentrat ion gradients ( i f there are any) and pH must be c l a r i f i e d . In f i b e r s with high r e s t i n g p o t e n t i a l s , because of contractures , only very small d e p o l a r i z i n g steps could be a p p l i e d . Hence there i s very l i t t l e information on the response of the c h l o r i d e permeation system to d e p o l a r i z i n g pu l ses . In permenently depolar ized f i b e r s , a much wider range of p o t e n t i a l s could be a p p l i e d because the contrac t ion mechanism i s i n a c t i v a t e d . Another reason to study permanently depo lar ized f i b e r s was to study 47 the nature of current t rans ients and t h e i r voltage dependence. Some evidence has been presented that k i n e t i c s depend on the h y p e r p o l a r i z i n g voltage step rather than on the absolute membrane p o t e n t i a l . F i n a l l y , i n connection with current t r a n s i e n t s , i t was found that for the long ' creep ing ' decays to the steady state i n n e u t r a l and a l k a l i n e s o l u t i o n s , the currents were independent of vol tage and pH. A p l a u s i b l e explanation for these currents i s that they could a r i s e as a r e s u l t of dep le t ion or enhancement of ch lor ide i n a compartment i n s e r i e s with the current pathway; i f t h i s were the case, then an a l t e r a t i o n of the c h l o r i d e concentrat ion would inf luence the slow current t r a n s i e n t s . The muscles were d i s sec ted as usual i n standard Ringer ' s so lu t ions (A) . The muscle was slowly depolar ized by gradual a d d i t i o n of de-p o l a r i z i n g s o l u t i o n ( s o l u t i o n D) . The d e p o l a r i z a t i o n was kept slow to prevent the f i b e r s from c o n t r a c t i o n . The muscle was f i n a l l y soaked for 4 hours i n s o l u t i o n D, then t r a n s f e r r e d to a s o l u t i o n in which the potassium i n the s o l u t i o n was replaced by rubidium ( so lu t ion E(a ,b ) ) and allowed to e q u i l i b r a t e for about 30 minutes before r e -cording began. Rest ing p o t e n t i a l s of depolar ized f i b e r s Hold ing p o t e n t i a l s for depolar ized f i b e r s , which were w i t h i n 5 m i l l i v o l t s of the r e s t i n g p o t e n t i a l , were genera l ly near -20 mV. However, i n some ins tances , e s p e c i a l l y for f i b e r s i n a l k a l i n e s o l u t i o n s , 48 r e s t i n g p o t e n t i a l s as high as -30 mV were recorded. In an i s o t o n i c so lu t ion conta in ing 117 mM K + , the i n t e r n a l potassium concentrat ion i s 195 mM (re fer to sec t ion on 'a comment on the s o l u t i o n s ' ) hence a membrane p o t e n t i a l of -13 mV i s a n t i c i p a t e d from the Nernst equat ion. Since s o l u t i o n D, which was used to depolar ize the f i b e r s , was not t r a d i t i o n a l l y used to depo lar ize muscle preparat ions ( th i s s o l u t i o n was used to maintain the ex terna l ch lor ide concentrat ion i n depolar ized f i b e r s the same as i n p o l a r i z e d f i b e r s ) membrane p o t e n t i a l s were measured i n a number of f i b e r s i n s o l u t i o n D and another s o l u t i o n D*, used by Hutter and Warner (1972). This acted as a check on the e f f ec t of s o l u t i o n D. The r e s u l t s , in pH 7.4, are summarized i n Table I I . 1.3. Table I I . 1 . 3 Rest ing p o t e n t i a l s of depolar ized f i b e r s s o l u t i o n r e s t i n g p o t e n t i a l s D -13.8 ± 1.1 (S .D.) mV; n=16 D* -17.0 ± 2.1 (S .D.) mV; n=21 The r e s u l t s of Table I I . 1 . 3 i n d i c a t e that the mean r e s t i n g p o t e n t i a l s 49 of fibers i n solutions D and D* d i f f e r by less than 5 mV and that the resting potentials are as predicted by the Nernst equation. In solution D, the inte r n a l concentrations of potassium and chloride (from the Nernst equation) are: K. = 205 mM , CI. = 70 mM I l and i n solution D*, K. = 193 mM , CI. = 112 mM i i The potentials shown i n Table II.1.3 for fibers i n solution D are s l i g h t l y less negative than some of those to be reported below, especially for experiments i n alkalin e solutions. When potassium chloride was replaced with 117 mM rubidium chloride s l i g h t l y higher resting potentials (around -18 mV) were obtained. When the depolarized fibers were placed i n normal Ringer's solution (solution A), repolarization of the membrane potential was very slow: even a f t e r 25 minutes of washing, fibers that had been depolarized ( i n 117 mM K +) did not repolarize to membrane potentials more negative than -25 mV. This observation has also been reported previously by Adrian (1964) who found that fibers loaded with chloride usually f a i l e d to repolarize or repolarized only very slowly when placed i n solutions of low K +. Value of the inte r n a l r e s i s t i v i t y (R^) i n depolarized fibers Calibration of membrane current density (measured as AV, as described i n Methods), requires knowledge of the longitudinal resistance 50 of the sarcoplasm (see equation 1 of Chapter I ) . As prev ious ly descr ibed , there i s ample evidence to suggest that a reasonable value for th i s quant i ty i n normally p o l a r i z e d f i b e r s ( re s t ing p o t e n t i a l , -80 mV) i s 170 ficm and we have obtained data from cable measurements i n d i c a t i n g that t h i s res i s tance i s independent of e x t r a c e l l u l a r pH. In depolar ized f i b e r s , however, s ince the i n t r a -c e l l u l a r concentrat ion of c h l o r i d e may be grea t ly increased , the sarcoplasmic res i s tance may be g r e a t l y a l t e r e d . A set of experiments was performed to determine the i n t e r n a l r e s i s t i v i t y (R^) i n depolar ized f i b e r s . Since i t was d i f f i c u l t to measure f i b e r radius a c c u r a t e l y , f i r s t an apparent radius was c a l c u l a t e d for normally p o l a r i z e d f i b e r s us ing a value of 170 Qcm. The muscle was A then depolar ized with e i t h e r s o l u t i o n D or so lu t ion D and a new set of cable data recorded (input re s i s tance and d . c . length constant) . Using the equivalent radius estimate from the f i r s t set of measurements, a value of R. i n the depolar ized cond i t ion was e s tab l i shed . It i s x * i n t e r e s t i n g to note (Table I I . 1 . 4 ) that i n so lu t ion D , in which i n t r a c e l l u l a r K + = CI i s 305 mM (see sec t ion on r e s t i n g p o t e n t i a l of depo lar ized f i b e r s ) , R^ i s s i g n i f i c a n t l y lower than i n s o l u t i o n D ( in which K + + CI i s 265 mM) . The value of 140 ficm has been r o u t i n e l y used to c a l i b r a t e membrane current in depolar ized f i b e r s i n so lut ions of high c h l o r i d e content. C o n t r i b u t i o n of c a t i o n currents The c o n t r i b u t i o n of the ca t ion currents to the t o t a l membrane current was again checked i n depolar ized f i b e r s to ensure that they 51 Table I I . 1.4 Cable constants in depolarized f i b e r s . Solution D 5 —6 —3 Muscle V (mV) R (xlO ohm) X(xlO m) a(xlO cm) R.(ohm.cm) m o l 7.6.78 -82.6±3.7 2.1±1.1 22001512 5 . 8 1 1.9 170 -14.2±2.8 0.43±0.12 8751380 (5.8) 126164 10.6.78 -85.614.8 1.57±0.18 27751380 6.910.9 (170) -14.9±2.2 0.41±0.13 1036+175 (6.9) 110169 12.6.78 -82.6±1.9 1.65±0.48 18801293 5.711 (170) -13.9+2.4 0.51±0.17 7061118 (5.7) 171145 Overall mean R. in depolarized c e l l s 140156 (n=21) l Solution D* 14.6.78 -80.614.1 1.6810.40 14671219 4.910.7 (170) -14.610.5 0.7510.18 838H45 (4.9) 86148 16.6.78 -84.613.4 1.3910.15 29961785 7.611.3 (170) -17.610.6 0.2510.05 10671166 (7.6) 88126 Overall me V = resting p o t e n t i a l , m a = f i b e r radius , n R. i n depolarized c e l l R = input resistance, X= o R.= internal r e s i s t i v i t y 87137 (n=27). d.c. space constant, 52 contributed no si g n i f i c a n t current. The muscles were bathed i n a solution containing potassium sulphate at different pH's (solutions D(a,d)). Results for 4 c e l l s from different pH's are shown i n Fig.2.1.9. The potassium currents were not influenced by pH, and showed t y p i c a l inward-going r e c t i f i c a t i o n (Katz, 1948; Adrian, 1969). When potassium was replaced by rubidium (solutions G(a,d)) currents obtained were very small but not time-independent. In current-clamp experiments the voltage record showed a large 'creep' and i n voltage-clamp experiments, although the currents were very small and i n i t i a l currents d i f f i c u l t to measure, they were of the order of 1.5 to 2.0 times the steady state current. Rubidium current did not exceed 10% of the chloride content. A comment on the solutions In i n i t i a l experiments on depolarized f i b e r s , due to the extremely low rubidium conductance, we considered i t unnecessary to add osmotic buffer to the solutions. The fi b e r s did not show any signs of swelling even after 8 to 12 hours i n the unbuffered solutions. The resting membrane potential was monitored continuously i n several experiments and i t was found that the resting membrane potential did not change for periods of up to 4 hours i n unbuffered solutions. However, the p o s s i b i l i t y exists that some of the data might be a r t i f a c t s of f i b e r swelling (Speralakis and Schneider, 1968), and consequently some experiments were repeated with the rubidium containing solutions osmotically buffered with sucrose (solutions E(a,b,c,d)). The current-53 Figure 2.1.9 Steady state current-voltage relations for fi b e r s bathed in 117 mM K 2 S 0 4 s o l u t l o n (solution D(a,d)). Very small currents were recorded when the pulses were positive-going from the holding p o t e n t i a l (+10 mV). The dashed l i n e indicates the envelope of records made in 117 mM Btt^SO^ (solutions F(a,d)). Calibration: 1 mV = 3.79 x 10" 6 Amp/cm2. 54 voltage r e l a t i o n s obtained were s i m i l a r to those obtained i n the unbuffered s o l u t i o n s . In the r e s u l t s to be presented, no d i s t i n c t i o n w i l l be placed on the f ibers that are e i t h e r i n osmot ica l ly buffered or unbuffered so lut ions unless otherwise d i scussed . The components of the d e p o l a r i z i n g so lu t ions were ca l cu la ted as fo l lows: At -90 mV the membrane p o t e n t i a l i s described by the Goldman-Hodgkin-Katz equation (Hodgkin and K a t z , 1949) us ing a sodium to potassium permeabi l i ty r a t i o of .01 (Hodgkin and Horowicz, 1959) the i n t e r n a l potassium concentrat ion i s : K. = exp(V F/RT)x(2 .5 + .01x120) l m = exp(.09x96500/8.3x295)x(2.5 + .01x120) =128 mM From the Nernst equation CI . = 3 mM . l I f an i n t r a c e l l u l a r sodium concentrat ion of 10 mM i s assumed, then the i n t r a c e l l u l a r f ixed negative charge concentrat ion i s 135 m.eq./j l . The osmola l i ty of normal Ringer ' s s o l u t i o n i s 227 m.osm/Z (measured by f reez ing point depress ion) , thus g i v i n g a mean osmotic c o e f f i c i e n t of 0.96. I f i n t r a c e l l u l a r inorganic ions have the same osmotic a c t i v i t y , then the osmotic a c t i v i t y of the i n t e r n a l f ixed charge would be 92 m.osm/A . In 117 KC1 ( so lu t ion D ) , from Donnan e q u i l i b r i u m , & K . = 195 mM and l C I . = 70 mM l and by assumption , Na. = 10 mM. 55 Using an osmotic coefficient of 0.96, this yields 265 m.osm/Jl and the internal fixed charge yields 92 m.osm/£ . Therefore the total osmolality of the intracellular compartment i s 357 m.osm/i . The ions in the external solution give an osmolality of 225 m.osm/Jl Therefore for isotonicity with Ringer's solution, 132 m.osm/it is required. The solutions D and E(a,b,c,d) were slightly hypertonic as 160 mM/ sucrose was added. In response to both hyperpolarizing and depolarizing voltage steps from the holding potential (usually -20 mV) the chloride :transient current waveforms in depolarized fibers in different solutions whose pH varied from 5.4 to 8.4 always decreased monotonically to the steady state. The current transients were found to be composed of two phases: an i n i t i a l decay with a time constant of 100 milliseconds and then a much slower 'creeping' f a l l (300-500 msec time constant) to the steady state. There were several differences between the instantaneous and steady state current-voltage relations and the current transients observed in depolarized fibers compared with those of polarized fibers. The steady state and instantaneous properties are described f i r s t , in this section. A. The steady state and instantaneous current-voltage relations in  fibers that have been permanently depolarized Fibers in acid solutions (pH 5.4, solution Ea) The average resting potential of the fibers studied in this pH was -14 m i l l i v o l t s (n=9). Both the instantaneous and steady state current-56 vol tage r e l a t i o n s were downwards concave for hyperpolar iz ing voltage pulses . In the depo lar i za t ion d i r e c t i o n , the behavior was of two types: i n some f i b e r s , mainly those e x h i b i t i n g very small current d e n s i t i e s , the instantaneous and steady s tate currents reached a l i m i t i n g value for the larges t d e p o l a r i z i n g voltages appl ied (+50 mV) whereas for f i b e r s that had higher current d e n s i t i e s , the instantaneous and steady s ta te currents continued to increase with depo lar i z ing vo l tage ( F i g . 2 . 1 . 1 0 ) . In sp i t e of these v a r i a t i o n s , i t was c l ear that i n a c i d s o l u t i o n s , the c h l o r i d e permeation system exhibi ted inward r e c t i f i c a t i o n — t h e conductance for inward current (ch lor ide e f f lux) was greater than for outward current ( ch lor ide i n f l u x ) . I t i s important to no t i ce the d i f f erence between the c u r r e n t -voltage r e l a t i o n s i n p o l a r i z e d and i n depolarized f i b e r s . In p o l a r -i zed f i b e r s , the instantaneous r e l a t i o n s were l i n e a r , and steady s tate r e c t i f i c a t i o n , although i n the same d i r e c t i o n as i n depolar ized f i b e r s , i s caused by the r i s i n g component of the current t rans ients (see F i g . 2 . 1 . 3 ) . In depolar ized f i b e r s , the downwards concavity of the steady s ta te r e l a t i o n s i s caused by the f a l l i n current . There was a very small r i s e i n current i n some f i b e r s for large hyperpo lar i za t ions (greater than -50 mV hyperpo lar i za t ion ) but the amplitude of the r i s e was l e ss than 5% of the t o t a l steady s tate c u r r e n t . In depolar ized f i b e r s , the magnitude of the current t rans ient s (the d i f f erence between instantaneous and steady s tate current) was much l a r g e r than i n p o l a r i z e d f i b e r s . F i b e r s i n so lu t ions of pH range (6 .7-7.3) ( s o l u t i o n E b , c ) In th i s pH range, the current vo l tage r e c t i f i c a t i o n i n permanently depolar ized f i b e r s changed from upwards concave ( ' a l k a l i n e ' behavior) to 57 Figure 2.1.10 Instantaneous (open c i r c l e s ) and steady state ( f i l l e d c i r c l e s ) current-voltage r e l a t i o n s i n a depolarized f i b e r i n acid solution (pH 5.4). Holding potential= -20 mV. In depolarized f i b e r s i n acid solutions the current always f e l l to the steady state. 9>% 09> O 1 O cm o + CM E — < C N C N 1 0> *6 • o • o o I O > o \ ° o • o 58 downwards concave ('acid' behavior). Fig.2.1.11 shows the current-voltage relations from two fibers with the same resting potential (-20 mv). One was bathed i n a solution of pH 6.7 (Fig.2.1.11a) and the other i n a solution of pH 7.3 (Fig.2.1.lib). The u n f i l l e d and f i l l e d symbols represent the instantaneous and steady state currents, respectively. I t i s of interest to note that as was the case for depolarized fibers i n acid solutions (see Fig.2.1.10) the instantaneous and steady state current-voltage curves r e c t i f y i n the same direction. Fibers i n solutions of pH 8.4-8.8 The current-voltage relations of depolarized fibers i n this pH range are d i f f i c u l t to interpret. The membrane current was continually reduced i n response to hyperpolarizing voltages from the holding p o t e n t i a l . As a r e s u l t , the current voltage curves exhibited several branches. This phenomenon, which we have called 'variable conducting states', was seen i n a l l c e l l s studied i n th i s pH range (average resting potential -19 mV, no. of c e l l s = 20). The steady state current voltage relations of the fibers had a region of negative slope which was much more pronounced than was ever seen i n polarized f i b e r s (Fig.2.1.Ic). Shown i n Fig.2.1.12 are the current-voltage relations from a fi b e r held at -30 mV. The points are numbered according to the order in which the pulses were applied. For large hyperpolarizations the membrane current was very small. In t h i s f i b e r , f i v e hyperpolarizing runs (not a l l shown) were applied. The slope conductance (both 59 Figure 2.1.11 Instantaneous (open symbols) and steady state ( f i l l e d symbols) current-voltage r e l a t i o n s f o r fi b e r s i n pH 6.7 (a, solution Eb) and pH 7.3 (b, solution Ec). The resting and holding potential of each f i b e r was -20 mV. —6 2 Calibration: ordinate, 1 mV = 3.79 x 10 Amp/cm . 59a 60 Figure 2.1.12 Instantaneous (open symbols) and steady state ( f i l l e d symbols) current-voltage r e l a t i o n s of a depolarized f i b e r i n a l k a l i n e solution (pH 8 . 4 ) . The numbers beside the points indicate the order in which the pulses were applied. Five hyperpolarizing runs were conducted on t h i s f i b e r , but not a l l of them are shown on t h i s graph. The dashed l i n e s are the instantaneous current-voltage r e l a t i o n s . 6 1 instantaneous and steady state) at the holding potential for the different states are tabulated i n Table II.1.5. For each state, the rat i o of the instantaneous and steady state conductance i s given. The constancy of th i s r a t i o i s suggestive that a variable fraction of the membrane's chloride channels may be operational at any time or that the conductance of each channel may vary. The experiments described i n Fig.2.1.12 were performed on osmotically unbuffered solutions. Because of the complex behavior, i t i s possible that some of these effects might be due to fi b e r swelling, as the fiber s were not i n osmotic equilibrium. Consequently, the experiments were repeated with fibers in isotonic solutions. Here, the instantaneous I-V relations appeared to be linea r and steady state relations saturated at large negative potentials and on repeated po l a r i z a t i o n , the membrane conductance was continually reduced. In these f i b e r s , the return to the holding potential after large hyperpolarizations re-activated the cont r a c t i l e system and l o c a l contractures sometimes occurred, indicated by sharp 'wobbles' on the current record. These contractions could not always be seen i n the stereomicroscope. To try to obviate any d i f f i c u l t y or error that might be introduced by contractions f i b e r s were placed i n a hypertonic solution (solution Ed*). In hypertonic solutions, the fibers were shrunken, e a s i l y damaged and impalement was d i f f i c u l t . The currents were too large for the preparations be adequately voltage clamped, so no results are available from these preparations. As we have repeatedly noted, the instantaneous current i n depolar-ized fibers, i n a l k a l i n e solutions (pH 8.4-8.8) i s a l i n e a r function of the voltage. This i s i n contrast to polarized f i b e r s (resting potential= 62 Table I I . 1 . 5 Instantaneous and steady state conductances at the holding potential for 5 conducting states i n a depolarized fi b e r at pH 8.4. The data i s given as mV: (current scale/mV (membrane potential)) -2 and the values i n brackets (mho.cm ). Run Instantaneous Steady state b conductance (a) conductance (b) ~a 1 0.6 (21. 7 X ic" 4) 0.5 (18.1 X 10 4 ) .83 2 0.425 (15. 4 X -4 10 ) 0.313 (11.3 X 10"*) .73 3 0.35 (10. 5 X IO-*) 0.30 (10.9 X 10"*) .86 4 0.40 (14. 5 X 10 4 ) 0.30 (10.9 X IO"*) .75 5 0.325 (11. 8 X -4 10 ) 0.288 (10.4 X -4 10 ) .87 63 -80 mV) where the instantaneous current-voltage relations were found to r e c t i f y i n the same direction as the steady state relations (Fig.1.1a). In a l k a l i n e solutions, the current-voltage relations appeared to i n f l e c t near the holding potential: i n general the outward-current conductance was less than the inward for small polarizations from the holding p o t e n t i a l . That i s , polarizations s u f f i c i e n t l y small that i n the hyperpolarizing d i r e c t i o n , the steady state current-voltage re l a t i o n was l i n e a r . Summary of the effect of pH on resting conductance of depolarized fibers Table II.1.6 summarizes the effect of pH on the resting conductance of depolarized f i b e r s . As with polarized f i b e r s , the magnitude of the resting conductance of depolarized fi b e r s exhibited large variations. However, i t can also be seen (Table II.1.6) that the resting conductance i s reduced by at least 50% when the pH i s reduced from 8.4 to pH 5.4 (as i n the case of polarized f i b e r s , Table II.1.1). An interesting observation i s that the resting conductance of fibers i n the neutral range (6.7-7.0) was similar to those i n alkaline solutions, suggesting that the pK of the pH t i t r a t a b l e s i t e might be lower i n depolarized fibers than i n polarized f i b e r s , and perhaps that there i s a dependence of pK on resting potential (Parker and Woodbury, 1976). This appears also to be true i n Rana temporaria (Hutter and Warner, 1972, Fig. 4 A,B). However the sample size of our experiment (in Table II.1.6) i s too small for this conclusion to be confirmed and in view of the large v a r i a b i l i t y between f i b e r s , experiments need to be 64 performed on the same fi b e r at different pH's. Comparison of the current-voltage relations i n polarized and depolarized  fibers We summarize the s i m i l a r i t i e s and differences i n the current-voltage relations between polarized and depolarized fibers i n solutions of different pH: acid solutions polarized depolarized neutral solutions polarized depolarized instantaneous li n e a r downwards concave instantaneous conductance limited conductance limited a l k a l i n e solutions instantaneous polarized saturating steady state downwards concave downwards concave steady state conductance lim i t e d conductance limited ( * ) steady state saturating; and i n some f i b e r s , negative slope conductance depolarized l i n e a r ; conduc- saturating; conductance tanee__edueed reduced with repeated po l a r i z a t i o n r e s u l t i n g negative slope conductance with repeated pol a r i z a t i o n (*) In depolarized f i b e r s i n neutral solutions, even though the instantaneous and steady state current-voltage relations were both conductance l i m i t e d as i n polarized f i b e r s i n the same pH, the relations i n the depolarized f i b e r s appeared to be l i n e a r for a greater hyperpolarizing voltage range. 65 Table I I . 1 . 6 E f f e c t of pH on r e s t i n g conductance i n depolar ized f i b e r s . pH r e s t i n g conductance from current -vo l tage r e l a t i o n s 5.4 3.0 x 10 mho cm -4 6.7-7.0 6.7 x 10 mho cm -4 8.4* 7.5 x 10 mho cm * Because f i b e r s i n a l k a l i n e s o l u t i o n s exh ib i t ed severa l conducting s ta te s , t h i s value i s the highest conducting s ta te . 66 B. Voltage Dependence of Current Transients i n Depolarized F ibers As we have described i n the preceding sec t ion , i n response to h y p e r p o l a r i z i n g vo l tage steps from the holding p o t e n t i a l , the current waveforms i n depolar ized f i b e r s i n so lut ions of d i f f e r e n t pH's ( so lut ions E) were q u a l i t a t i v e l y the same: current trans ients always f e l l to the steady s tate i n two phases—an i n i t i a l decay with a time constant of 100 msec and then a much slower ' creeping ' f a l l (300-500 msec) to the steady s ta te . The h y p e r p o l a r i z i n g current t rans ient s were resolved into two components us ing the same approximation procedure adopted for p o l a r i z e d f i b e r s . The slow component i n depolar ized f i b e r s , as we have found for p o l a r i z e d f i b e r s , was independent of vo l tage . Table I I . 1 . 7 summarizes the voltage dependence of the time constant of the fas t component. It can be seen that the vo l tage dependence of the time constants in so lu t ions of d i f f e r e n t pH i s very s i m i l a r . The current t rans i en t s i n depolar ized f i b e r s i n a l k a l i n e so lut ions were more complex as they exhib i ted a wide range of time constants for t h e i r r a p i d t r a n s i e n t s . These v a r i a t i o n s ex i s ted not only between f i b e r s , but between runs i n a s ing l e f i b e r when the f i b e r went from one conducting s tate to another. When a c e l l was i n a h i g h l y conducting s ta te , currents were large and the time course of the r a p i d t r a n s i e n t s was very b r i e f (as short as 10 msec for a 50 mV p u l s e ) . In lower conducting states the k i n e t i c s of the i n i t i a l phase of r e l a x a t i o n were slower. In F i g . 2 . 1 . 1 3 , the fas t time constants for two c e l l s ( c e l l 2.6-2 and c e l l 2.10.3) for which data for only one conducting s tate are 67 included, are plotted as a function of the amplitude of the voltage step. Despite the differences i n the values of the time constants, t h e i r voltage dependence remained constant. I t can be seen from Table II.1.7 that the mean of the voltage dependence of the fast decays of i n i t i a l current i n a l k a l i n e solutions i s similar to those recorded at other pH's. Depolarizing voltages For depolarizing pulses from the holding potential, current always f e l l to the steady l e v e l i n solutions of low to high pH's. A family of pulses i s shown inFig.2.1.14 from a f i b e r i n a solution of pH 8.4 (solution Ed). The time constants of current decay were 100 to 300 milliseconds. It was clear that the larger the depolarizing voltage, the more rapid the transient but the data obtained were too scattered for the voltage dependence of the time constant to be determined. Priming of the membrane conductance with conditioning pulses i n depolar- ized f i b e r s i n a l k a l i n e solutions Some depolarized f i b e r s i n a l k a l i n e solutions exhibited very complex time and voltage dependent current transients that could be induced by conditioning the membrane with very large hyperpolarizations. These conditioning pulses greatly influenced the membrane conductance. Fig.2.1.15 shows the results of one experiment i n whcih the membrane potential of a depolarized f i b e r was stepped from -30 mV to -118 mV. This large priming pulse (of duration 3 msec) was followed by a sequence of test pulses to -72 mV, applied 17 seconds apart. As the e f f e c t s of priming dissipated, the time course of the slow processes s e t t l e d back Summary of the voltage dependence of the fast time-constants of relaxation of chloride current i n depolarized f i b e r s . pH Voltage dependence (msec/mV) 5.4 ' -1.82 ± 0.70 * n=4 6.4 -1.67 ± 0.43 n=3 8.4 -1.52 ± 0.24 n=6 * Standard deviation. 69 Figure 2.1.13 Dependence of relaxation time constants on the size of the membrane potential step for two depolarized fi b e r s at pH 8.4 (solution Ed). Triangles are for c e l l 2.6.2, the open and closed symbols representing two different conducting states. A l l depolarized f i b e r s i n a l k a l i n e solution exhibited at least two conducting states, but data for only one are included here for c e l l 2.10.3 ( f i l l e d c i r c l e s ) . The slopes of the two regression l i n e s are not s i g n i f i c a n t l y d i f f e r e n t . 69a 70 Figure 2.1.14 Waveforms of outward currents i n response to depolarizing pulses i n a depolarized f i b e r at pH 8.4 (solution Ed). The numbers on the traces indicate the amplitudes of the voltage steps from the holding potential (-20 mV). Note the large inward aftercurrents following large pulses. Calibration: one second, 10 mV. 1 mV = 3.79 x 10 2 Amp/cm . 65 50 30 m 15 71 Figure 2.1.15 Currents recorded i n a priming sequence. The protocol i s described in the text. The upper panel shows the current recorded during the priming pulse. Note the large inward aftercurrent. Lower panel: currents recorded i n response to test steps applied 17 seconds apart i n the sequence 1 ,2,3,4. Note that even though the conductance i s s t i l l very high at the end of the f i r s t test pulse the aftercurrents are always outward. Cal i b r a t i o n : 500 msec, 5 mV. —6 2 1 mV = 3.79 x 10 Amp/cm . Holding po t e n t i a l , -30 mV. Solution Ed (pH 8.4). 71a 72 toward the i n i t i a l values. The time constants of both the i n i t i a l rapid and the slower, priming-sensitive phases are l i s t e d i n Table II.1.8 for each of the current records i n Fig.2.1.15. Trace 4 (the current recorded during the test step applied 68 seconds after the priming pulse) i s the same as was recorded before priming. Since the effects of i priming were so long-lasting and the resetting of i n i t i a l conditions involved very long waiting periods, we have not been able to establish the time and voltage dependence of the onset of priming nor the time and voltage dependence of i t s disappearance. Instantaneous after-currents i n depolarized fibers In depolarized f i b e r s , as i n polarized f i b e r s , the direction of the after-current after returning to the holding potential from a preceding hyperpolarization or depolarization was opposite to that during the preceding pulse. However, i t s magnitude was much larger than i n polarized f i b e r s . Regardless of the external pH, with depolarizing conditioning pulses, the instantaneous after-currents always reached a saturating value with large depolarizations (greater than 50 mV from the resting p o t e n t i a l ) . This observation i s especially of interest i n view of the different r e c t i f i c a t i o n behavior of the steady state current voltage relations of the preceding pulse, thus suggesting that the a f t e r -currents were independent of the preceding steady state current. With hyperpolarizing conditioning pulses, the after-currents i n fibers i n al k a l i n e solutions also showed a tendency to reach a maximum Table I I . 1.8 Time dependence of the recovery of time constants a f t e r pr iming . Time* Fast time constant 17 seconds 34 seconds 51 seconds 68 seconds 160 msec 160 msec 160 msec 160 msec Slow time constant 4.25 seconds 2.25 seconds 800 msec 550 msec * Time a f t e r the beginning of the (three second) c o n d i t i o n i n g pu l se . 74 value whereas In a c i d so lu t ions , the observations were v a r i a b l e : some f i b e r s showed sa tura t ion whereas others d i d not ( F i g . 2 . 1 . 1 6 ) . Discuss ion Steady s tate current -vo l tage r e l a t i o n s The steady s tate current -vo l tage r e l a t i o n for ch lor ide ions i n Xenopus muscle membrane i s s i m i l a r to that for Rana temporaria (Hutter and Warner, 1972): the steady state c h a r a c t e r i s t i c s depend on the e x t r a c e l l u l a r pH, with a t r a n s i t i o n near pH 7. The r e l a t i o n s show upward concavity i n the h y p e r p o l a r i z i n g segment for pH greater than 7 and upward convexity (inward-going r e c t i f i c a t i o n ) i n the same region i n a c i d s o l u t i o n s . This behavior has been seen to be independent of the r e s t i n g p o t e n t i a l s of the f i b e r s , except that when c h r o n i c a l l y depolar ized f i b e r s are inves t iga ted i n a l k a l i n e s o l u t i o n there i s a negative slope region for large h y p e r p o l a r i z a t i o n s that i s not as pronounced i n normally p o l a r i z e d c e l l s . Negative slope conductance has not been observed i n Rana temporaria (Warner, 1972; Hutter and Warner, 1972). The negative slope conductance observed i n our experiments i s not as pronounced as that reported by Palade and B a r c h i (1977) for c h l o r i d e conductance i n normally p o l a r i z e d f i b e r s from r a t diaphram. Replacement of c h l o r i d e by methylsulphate i n the mammalian muscle a l t e r s the degree of negat ive slope conductance: indeed f o r 75% replacement the current -vo l tage r e l a t i o n becomes l i n e a r (Palade and 75 B a r c h i , 1977). Our experiments were performed before the work of Palade and Barchi (1977) appeared in the l i t e r a t u r e and we have not attempted to confirm t h i s observat ion on our preparat ion: however we have observed that r e p l a c i n g most of the c h l o r i d e i n so lu t ion with sulphate i n depolar ized f i b e r s ne i ther s i g n i f i c a n t l y reduces steady s tate current nor abol i shes negative slope behavior . Instantaneous current -vo l tage r e l a t i o n s The instantaneous current -vo l tage r e l a t i o n s i n Xenopus l a e v i s d i f f e r from those reported in Rana temporaria (Warner, 1972) and for ra t diaphram (Palade and B a r c h i , 1977). In both the above works, the instantaneous current -vo l tage r e l a t i o n s were l i n e a r at a l l pH' s , whereas i n Xenopus, the instantaneous current -vo l tage r e l a t i o n s u s u a l l y r e c t i f y i n the same d i r e c t i o n as the steady state r e l a t i o n s i n p o l a r i z e d f i b e r s and become more l i n e a r i n depolar ized f i b e r s . It i s not c l e a r what the d i f f erence between Xenopus and Rana might be due to: whether i t i s due to an a c t u a l d i f f erence i n the membranes or i n the methods used to estimate the instantaneous c u r r e n t s . Re lat ions between the instantaneous and steady state proper t i e s i n  p o l a r i z e d and depolar ized f i b e r s The d i f f e r e n t r e c t i f i c a t i o n curves that are observed i n f i b e r s he ld at d i f f e r e n t r e s t i n g p o t e n t i a l s and at d i f f e r e n t pH's r a i s e some d i f f i c u l t i e s i n the i n t e r p r e t a t i o n of the e f f ec t s of v a r i a t i o n s i n the c h l o r i d e concentrat ion g r a d i e n t . In a c i d so lut ions the steady s ta te c u r r e n t - v o l t a g e r e l a t i o n s of 76 AV(mV) -40 Vm(mV) 40 — i 9«-Figure 2.1.16 Instantaneous aftercurrents recorded i n two depolarized f i b e r s i n acid solutions plotted as a function of the amplitude of the preceding voltage step. The f i b e r s were held at -20 mV. —6 2 Calibration: 1 mV = 3.79 x 10 Amp/cm . 77 both polarized and depolarized fibers r e c t i f y i n the same direction but the causes of the r e c t i f i c a t i o n appear to be different: i n polarized f i b e r s , i t i s caused by a r i s i n g current (Fig.2.1.3 of chapter II) whereas i n depolarized f i b e r s , the r e c t i f i c a t i o n i s caused by the i n i t i a l l y f a l l i n g transient current. In polarized fibers i n acid solutions, the current transients are composed of two components: an i n i t i a l f a l l and a subsequent r i s e , whereas i n depolarized f i b e r s , the r i s e i s negligible. Since the current transients of the f a l l have the same voltage dependence i n polarized as i n depolarized f i b e r s , i t i s reasonable to conclude that they represent the same component. The steady state current-voltage r e l a t i o n i n polarized fibers with the r i s i n g component removed i s linea r (Fig.2.1.3a), whereas i t i s downwards concave i n depolarized f i b e r s . The instantaneous current voltage relations are l i n e a r i n polarized fibers and downwards concave i n depolarized f i b e r s . In contrast i n alkaline solutions, the instantaneous and steady state relations r e c t i f y i n the same direction i n polarized f i b e r s , both reaching a saturating current and i n some instances negative slope conductance. In depolarized f i b e r s , the steady state current-voltage relations also exhibit saturating currents and in many f i b e r s , a pronounced negative slope conductance. In depolarized f i b e r s , the instantaneous current-voltage relations are li n e a r over a much larger voltage range than i n polarized f i b e r s . Dependence of chloride conductance on concentration In the experiment on Xenopus, the resting chloride conductance has 7 8 been found to be independent of the chloride concentration at a l l pH's. This i s a puzzling result as i t i s at variance with previous observations on the dependence of chloride currents on chloride concentration i n neutral solutions. Since the constant f i e l d theory has been found to approximate chloride currents i n neutral solutions (Hodgkin and Horowicz,1959a: Adrian, 1961; Adrian and Freygang, 1962: Harris, 1963; Hutter and Warner, 1972), the voltage dependence of the current-voltage r e c t i f i c a t i o n and the.magnitude of the resting conductance should increase s i g n i f i c a n t l y with a tenfold increase i n inte r n a l chloride concentration. This was not observed in Xenopus, and i t i s not clear what the causes of th i s difference might be. The only possible cause i s an inaccurate determination of the internal r e s i s t i v i t y (see equation 1, chapter I) and hence an incorrect c a l i b r a -tion of the current density. However, the data obtained on the inte r n a l r e s i s t i v i t y of depolarized fibers suggest that there i s a very small change i n the internal r e s i s t i v i t y with a ten-fold increase in i n t e r n a l chloride concentration. The pH s e n s i t i v i t y of fast transients The only aspect of chloride current transient behavior that i s pH-dependent i s that i n polarized fibers i n acid solutions there i s a r i s i n g component of the current i n response to a single hyperpolarizing voltage step. I t i s not seen elsewhere. The presence of t h i s component does not a l t e r the voltage dependence of the rate of any preceding f a l l . 79 Aftercurrents and slow transients Slow transients, or 'creep', in ionic currents under voltage clamp conditions and non-zero aftercurrents have been attributed to enhancement or depletion of ionic concentration in restricted or unstirred spaces in electrical continuity with the inside of the fiber and the bulk extracellular space. For example, potassium current creep in frog muscle fibers with hyperpolarization (Adrian, Chandler and Hodgkin, 1970b; Aimers, 1972a,b; Barry and Adrian, 1975) is thought to be due to a depletion of potassium in the lumen of the transverse tubular system. The available evidence in the literature suggests that there is no chloride conductance in the T-system (Hodgkin and Horowicz, 1959; Gage and Eisenberg, 1969a). Therefore i f such an explanation were to be invoked for the slow decay of chloride current and non-zero aftercurrents, the existence of another internal (external) space where local depletion (enhancement) of chloride concentration occurs, during transmembrane flux, would have to be demonstrated. Strong evidence against this comes from experiments in depolarized fibers in chloride-rich media. The changes in concentration gradients across membranes in these conditions would be minimal, yet the amplitudes of the i n i t i a l aftercurrents were at least as large in these as in normally polarized fibers. Additionally, aftercurrent amplitudes have seen to saturate under conditions where the preceding steady state currents are s t i l l increasing with membrane potential. 80 Experimental Section 2 . Two Pulse Voltage Clamp Experiments In t h i s section the nature of the f a l l in membrane conductance with hyperpolarizing potentials ('inactivation') i s investigated further by two pulse experiments. F i r s t , conditioning potentials (V^) of varying amplitude were applied and the conductance at the end of the conditioning pulse was examined by applying a fixed test potential (V,,). The instantaneous current at the beginning of the test potential, l 2(0) ;was used as an indication of the degree of 'inactivation' at the end of the conditioning pulse. I. Instantaneous current-voltage relations A. Experiments where the conditioning potential was varied The q u a l i t a t i v e behavior of a l l f i b e r s studied was independent of the external pH and the resting membrane potential. When V^ was more negative than V^, the test currents rose to the steady state values, whereas for V^ more positive than V^, the test currents f e l l ( F i g . 2 . 2 . l a ) . When V^ was large and negative (or p o s i t i v e ) , the i n i t i a l test current 1^(0) approached an asymptotic minimum (or maximum). For intermediate values of V^, instantaneous test current 12(0) was found to be sigmoidally related to the conditioning potential. This i s shown i n F i g . 2 . 2 . l b for a t y p i c a l c e l l i n a l k a l i n e solution (resting p o t e n t i a l , -80 mV, pH 8.4, solution Bd). The open c i r c l e s represent the steady state current during the test pulse and the f i l l e d c i r c l e s , 1 2 ( 0 ) . The abscissa i s the membrane potential during the conditioning step (V,). The observations 81 Figure 2.2.1 a) Traces from a two-pulse experiment in alkaline solution showing the experimental protocol and test current transients, I„. Holding potential = -70 mV. —6 Solution Bd. Current calibration: 1 mV = 3.14 x 10 Amp/cm^ . b) Instantaneous currents (I^CO)) ( f i l l e d circles) recorded at the onset of a step to -130 mV 0 ^ ) from various conditioning potentials (V^). The open circles indicate the steady state current at for each pulse. Resting potential - -80 mV. pH 8.4 (solution Ed). c) Normalized data from two cells plotted against the amplitude of the conditioning step. The f i l l e d circles are obtained from (b). c -140 V, (mV) -80 82 are similar to those made in alkaline solutions in Rana temporaria (Warner, 1972, Fig. 5A). Normalization procedures: comparison of data from different fibers Since c e l l s exhibited different maximum and minimum saturation currents, a normalization procedure was adopted to compare the effects of pH on different c e l l s . The rela t i o n (2.1) l 2 ( 0 ) n o r m a l i z e d = ( i 2 ( 0 ) - I 2 ( 0 ) m i n ) / ( I 2 ( 0 ) m a X - I 2 ( 0 ) m l n ) , -r /^ \E<ax . _ ,_.min , . . . . was used. I 2 (0) and I^{0) are the asymptotic maximum and minimum recorded values of I 2 ( 0 ) , respectively. The procedure i s equivalent to making the minimum asymptotic value zero and the maximum inward current, one. Fibers i n alk a l i n e solutions Normalized data for two polarized f i b e r s i n alk a l i n e solution i s plotted i n Fig.2.2;lc. The f i l l e d : c i r c l e s are derived from Fig.2.2.lb for which c e l l was -130 mV; the open c i r c l e s are for a fib e r i n which V 2 was -104 mV. In both cases, the normal holding potential was -80 mV. The measured currents dif f e r e d greatly i n amplitude, the steady state value of I 2 i n one (V2= -130 mV) being almost three times than i n the other (V2 = -104 mV). Similar differences were seen i n the measured value of I 2 ( 0 ) . The curves were f i t t e d to the data points i n Fig.2.2.1c using the d i s t r i b u t i o n function 8 3 ,~ „v _> .-.normalized , ., , . • , •„ (2.2) I (0) = 1 / ( 1 + exp(-k(V 1-V*))) where k=zF/RT (z=valence, F i s the Faraday constant, R i s the gas constant, and T i s the absolute temperature), k i s the shape parameter and V* i s the half-saturation voltage. A good f i t of the d i s t r i b u t i o n function to a l l data from a l l fibers, whether polarized or depolarized, i n a l k a l i n e solutions was obtained with k=.087 (z — 2 ) . Fibers i n acid solutions The only difference between the results i n alka l i n e and acid solutions was the value of the slope parameter k. Normalized data from three polarized f i b e r s i n acid solutions (pH 5.4) i s shown in Fig.2.2.2a. Here k=0.188 (z=£fe4) and this value gave good f i t to data from a l l c e l l s i n acid solutions that showed a f a l l i n g transient phase with hyperpolarization, whether polarized or depolarized . In one c e l l studied in which only r i s i n g transients were seen (with hyperpolarization, for instance, see Fig.2.0.Ic), the sign of k was found to be reversed although i t s magnitude was not changed ( i . e . the sigmoid curve was inverted and had the opposite slope). Further experiments i n these f i b e r s i s required for an understanding of the r i s e i n conductance with hyperpolarization. These observations suggest that at the onset of ^(O) i s pro-portional to the d i s t r i b u t i o n of a charged species, between two states that either enhance or impede the passage of ions through the chloride channel. The sign of t h i s e f f e c t i v e charge cannot be determined, as 84 formally i t i s a consequence of the way i n which the voltage term in the d i s t r i b u t i o n function i s written ((V-V ) or (V -V)). The assumption of positive valency i s consistent with the idea that i t could become protonated i n acid solution. The difference between acid and alkaline results i s c l e a r l y demonstrated i n Fig.2.2.2b i n which the derived relationships are superimposed. B. Experiments i n which the test potential was varied. We have observed that at the 'end' of a hyperpolarizing or depolarizing pulse, when the membrane potential was returned to the holding l e v e l , there was a t a i l of after-current, or equivalently, a s h i f t of zero current voltage had occurred. This phenomenon was investigated i n experiments i n which, a f t e r a period of conditioning at a constant voltage, the membrane potential was stepped to a variable test l e v e l . An estimate of the zero current voltage could then be made from the intersection of the instantaneous current 12(0) with the axis. Results from a depolarized f i b e r at pH 5.4 are depicted i n Fig.2.2.3a. The f i l l e d c i r c l e s represent steady state currents and the open c i r c l e s i n i t i a l currents recorded when the membrane potential was stepped from -80 mV to more positive voltages. The l i n e a r i t y of t h i s r e l a t i o n was t y p i c a l of a l l those for c e l l s bathed i n acid solutions and an example from an experiment on a polarized f i b e r i n a l k a l i n e solution (pH 8.4) i s shown i n Fig.2.3.3b. Non-linearity was always seen as upward concavity (current tending to saturate) as would be expected from the behavior 85 Figure 2.2.2 a) Normalized data from v a r i a b l e experiments for three p o l a r i z e d f i b e r s ( res t ing p o t e n t i a l = -70 mV) at pH 5.4 ( so lu t ion Ba) . b) Di f ference between the d i s t r i b u t i o n func t ion for f i b e r s i n a c i d so lu t ions (a) and i n a l k a l i n e so lu t ions (b) . 86 Figure 2 .2 .3 Steady s tate and instantaneous current -vo l tage r e l a t i o n s in two-pulse experiments. The f i l l e d c i r c l e s represent steady state c u r r e n t -vo l tage curves and the open c i r c l e s , the instantaneous currents recorded when the membrane p o t e n t i a l was suddenly stepped to a v a r i a b l e tes t vo l tage from a prev ious ly f ixed cond i t ion ing p o t e n t i a l (V^). a) Depolarized f i b e r (res t ing p o t e n t i a l , -20 mV) at pH 5.A ( so lu t ion E a ) . Vj= -80 mV. b) P o l a r i z e d f i b e r ( res t ing p o t e n t i a l , -80 mV) at pH 8.4 ( so lu t ion E d ) . V = -118 mV. —6 2 Current c a l i b r a t i o n : a) 1 mV = 3.79 x 10 Amp/cm . b) 1 mV = 3.14 x 10~ 6 Amp/cm 2 . 87 of currents at the onset of test steps from the holding potential. A s h i f t of zero current potential i n a negative-going direction from the resting potential after hyperpolarizing conditioning has been noted by Warner (1972). In our experiments, t h i s s h i f t ranged from 0 to 25 m i l l i v o l t s . The dependence of the magnitude of the voltage s h i f t on the conditioning voltage was investigated i n depolarized fibers in alkaline solutions. Results are displayed i n Fig.2.2.4 for an experiment i n which three different values of were used over a range of from -58 to +7 mV i n a fiber i n osmotically buffered solution. The variable relations are plotted i n Fig.2.2.4a (open c i r c l e s , V2=-38 mV; f i l l e d c i r c l e s , V^=-^7 mV; f i l l e d squares, V^=-57 mV) i n which the l i n e s are drawn according to equation 2.2 scaled between the minimum and maximum currents for each V T h e value of k that was used was the same as for a l l other c e l l s i n a l k a l i n e solution (.087). Notice that the separation of the relations i s less at the l e f t hand end (V^ most negative) than at the right-hand end (V^ most p o s i t i v e ) , that the separation of the upper-most l i n e (least negative value of V^) from the middle r e l a t i o n i s almost the same as that of the lowest r e l a t i o n (largest negative value of V^). This implies that for equal steps of V^, the relations are equally spaced on the current axis: there i s no l a t e r a l s h i f t i n g of these r e l a t i o n s , as i n each case V* i s 9 mV more negative than the holding potential ( i . e . -27 mV). In Fig. 2.2.4b the points for V^-58 mV (smallest currents), V^-40 mV, V^=-33 mV, Vj=-13 mV and V^=-3 mV (largest currents) are replotted on the V 0 versus current (I o ( 0 ) ) plane. The l i n e s through the points are drawn 88 Figure 2.2.4 a) Family of 12(0) versus r e l a t i o n s obtained from a depolar ized f i b e r ( res t ing p o t e n t i a l , -18 mV) in osmot ica l ly buffered a l k a l i n e s o l u t i o n ( so lut ion E d ) . The numbers on the l i n e s i n d i c a t e the absolute membrane p o t e n t i a l at V^. The l i n e s were f i t t e d according to equation 2.2, scaled to the maximum and minimum values of I 2 ( 0 ) . b) Family of 12(0) versus r e l a t i o n s r e p l o t t e d from (a) . The numbers adjacent to the l i n e s i n d i c a t e the absolute membrane p o t e n t i a l at V ^ . The l i n e s were f i t t e d by eye. 89 by eye and appear to meet i n a point. The importance of th i s observation w i l l become apparent i n Chapter I I I . The rationale for drawing straight l i n e s was that i n an e a r l i e r run on the same c e l l , for a of -55 mV and a range of from -68 mV to +12 mV the relations was seen to be linea r . I t i s not shown here because the currents recorded were s l i g h t l y less than those for the runs shown and the r e l a t i o n f e l l outside the envelope of those included i n Fig.2.2.4b. I I . The voltage dependence of chloride current transients i n two pulse  experiments. The voltage dependence of current transients i n response to hyper-polarizing voltage pulses from the holding potential has been found to depend only on the size of the voltage step. In this section, the voltage dependence of current transients i n two pulse experiments i s described. I t w i l l be seen that the voltage dependence of r i s i n g currents (when i s more negative than V^) and the f a l l i n g currents (when i s less negative than Vy) are both dependent on the size of the voltage step. A. Dependence of time constants on the voltage step lV2~^_l Depolarized fibers When was less negative than V^, the decay of the test current 12(0)was resolvable into either one or two time constants for f i b e r s i n both a l k a l i n e (pH 8.4) and acid (pH 5.4) solutions. Fig.2.2.5 shows the time constants of the i n i t i a l transients from a f i b e r with a two time constant decay. The open symbols represent the fast time constants for the i n i t i a l f a l l of current i n experiments i n which the conditioning pulse (V^) was more positive than V 2 - There was a slow time constant (not shown i n the Figure) whose magnitude was 100 msec and was voltage independent. In other f i b e r s , t h i s slow time constant has been found to vary between 100 and 250 msec. The f i l l e d symbols represent the converse situation (V^ more negative than V^) i n which r i s i n g currents were observed. The regression l i n e r e l a t i n g the time constant to the absolute voltage difference has a slope of -1.46 msec/mV. This i s similar to the voltage dependence i n one pulse experiments (Table II.1.7). For the fibers that exhibited only one time constant with current decay, the voltage dependence was greater. This i s shown i n Fig. 2.2.6 for a f i b e r at pH 5.4 and held at -20 mV. When was more positive than V^, the f a l l i n g transient I ^ was f i t t e d by a single exponential with a time constant of 250-350 msec and a voltage dependence of -2.5 msec/mv. When the test currents rose (V^ more negative than V 2) the regression l i n e r e l a t i n g the time constants to the absolute voltage difference l V 2 ~ V ] J ^ a c i a s^°P e °f -1-78 msec/mV for l V 2 _ V i l 8 r e a t e r than 15 mV (otherwise the transients were too small for the time constants to be accurately resolved). Polarized f i b e r s In acid solutions, the current transients were too small for accurate analysis. Otherwise the conclusion about the dependence of current trans-ients on the size of the voltage step was also found to hold i n neutral and alkalin e solutions. 91 Figure 2.2.5 Time constants of relaxation of the test currents (I^) in a two-pulse experiment, plotted as a function of the absolute value of the difference between the voltage steps. The data was obtained from two depolarized f i b e r s (resting and holding p o t e n t i a l , -30 mV) in alka l i n e solution (solution Ed). F i l l e d triangles: fast time constant for r i s i n g transients (V more negative than V^). Open symbols: time constants of f a l l i n g currents (V^ less negative than V^)• The triangles (open and f i l l e d ) are from the same f i b e r . 150 T msec A O O 4 > 0 0 v 2 - V l mV 92 Figure 2.2.6 Top panel: data derived from a depolarized fi b e r (resting p o t e n t i a l , -20 mV) at pH 5.4 (solution Ea) in an experiment similar to that of Figure 2.2.5. F i l l e d symbols are again the time constants of r i s i n g transients and open c i r c l e s are for f a l l i n g transients. Bottom panel: f a l l i n g transients from an experimental sequence (from the same f i b e r as the top panel) i n which the relaxation was resolved into only one time constant. Summary of the voltage dependence of current transients 93 One and two pulse experiments on polarized and depolarized fibers i n solutions of different pH's (except i n acid solutions when the cur-rent transients exhibit only a ris e ) have revealed that the voltage dependence of current transients may be summarized as follows: When the conditioning and the test potentials are both more negative than the resting potential the fast time constant depends on the size of the voltage step, whether the currents r i s e (V^ i s more negative than V^) or f a l l (V^ more positive than V ). For potentials that are more positive than the resting potential, the current transients are two to three times longer than their corresponding hyperpolarizations but the a n a l y t i c a l expression for their voltage dependence has not been obtained. Appearance of a delayed (non-exponential) f a l l i n g current When the test current i n i t i a l l y rose (V^ more negative than V^) a delayed f a l l to the steady state was often seen i n depolarized fibers at a l l pH's and was p a r t i c u l a r l y prominent i n those fibers that exhibited priming i n alkalin e solutions. The time-course of the delayed f a l l was non-exponential and i t exhibited complex voltage and time dependence behavior. As was made increasingly negative and held constant, the i n i t i a l rate of r i s e of current became more rapid and the delay before the f a l l to the steady state was reduced (Fig.2.2.7). The i n i t i a l r i s e i n current was apparently due to chloride because the time constant (of the r i s e i n i n i t i a l current) was found to have the same voltage dependence on |V7~V |. The magnitude of the 94 Figure 2.2.7 Changes i n the waveform of I i n an experiment i n which was held constant at -35 mV. The amplitude of V^, i n m i l l i v o l t s , i s indicated on the l e f t of the traces. The horizontal l i n e above each trace indicates the holding current. Calibration: abscissa; 1 second; ordinate; 5 mV (for the current —6 2 traces). 1 mV = 3.79 x 10 Amp/cm . 82 ' non-exponential component was found to increase with increasing (when the conditioning potential was held constant). I t i s possible that the delayed non-exponential f a l l might be due to another ionic component. This w i l l be discussed i n experimental section 3. Discussion The results presented i n this section show that instantaneous currents i n Xenopus muscle membrane depend on the membrane potential at which a steady state has been reached just prior to the application of a voltage step as well as the magnitude of the step. Observations that the minimal instantaneous test currents (1^(0)) are not zero at a l l pH and for very large negative conditioning poten-t i a l s indicate that inactivation i s not complete and that the probabi-l i t y of a chloride channel opening at any transmembrane potential at any time does not reach negligibly small values. S h i f t i n g of the S-curves A curious observation with the sigmoid curves i n polarized and depolarized solutions was that the mean of the d i s t r i b u t i o n was always between -9 and -30 mV with respect to the holding p o t e n t i a l . This suggests that some cont r o l l i n g moiety within the channel finds a minimum free energy state under conditions of electro-chemical equilibrium of the ions (chloride) normally conducted by the channel. Relation between instantaneous and steady current values C l a s s i c a l descriptions of ion transport across c e l l membranes consider the driving force on an ion the deviation of the transmem-brane potential from i t s equilibrium or Nernst potential. I f this view were adopted for chloride i n Xenopus, then the current would be expressed as: I p , = g.(V -V . ) , here g i s the chord conductance with L* X HI v>X respect to the equilibrium, ( V ^ ) , or holding potential (when the two are the same). We have noted that the steady state current-voltage relations r e c t i f y as a function of external pH. I t i s interesting to note that the sigmoid curves (1^(0) versus relations) were observed regardless of the previous steady state conductance. A quantitative comparison of the differences can be made by defining the conditioning steady state conductance g(V^ ,°°) by: g C v ^ o o ) = K v 1 , » ) / ( v 1 - v c l ) and the instantaneous test conductance g(V"2,0) g ( v 2 , o ) = i 2 ( v 2,0 ) / ( v 2 - v c l ) where I(V^ ,°°) = steady state conditioning current density I(V 2,0) = instantaneous test current density For f i b e r s bathed i n acid solutions, g(V^ ,°°) increased with increasingly negative conditioning pulses whereas i t decreased i n alka-l i n e solutions. Since the instantaneous I 2(0) (as a function of V^) relations were always sigmoid regardless of the slope of the steady state current-voltage relations for currents that i n i t i a l l y decrease, i t i s q u a l i t a t i v e l y clear that steady state conductances are not equal to the subsequent instantaneous conductances. Two quantitative com-parisons of g(v^ ,°°) and g(V2»0) can be made. F i r s t , the instantaneous test current I(V 1,V o,0) i s predicted from the conditioning steady state current I(V^ ,°o) using the re l a t i o n i (v l t v 2 ,o ) - (v 2-v c l ) i (v l f»)/(v 1-v c l ) and i s compared with the experimentally measured instantaneous I 2 from variable experiments. Conversely, the steady state (V^) current-voltage r e l a t i o n i s predicted from the measured instantaneous 1^ v i a i ( v l f « ) - (v 1-v c l)i(v 1,v 2,o)/(v 2-v c l) i . The f i l l e d c i r c l e s of Fig.2.2.8 are the measured currents and the open c i r c l e s are the corresponding values predicted using the appropriate I(V 1, 0 0) or I 2 ( V 2 , 0 ) . Several conclusions can be drawn from these quantitative com-parisons : For both polarized and depolarized fibers i n acid solutions, the measured instantaneous I 2 i s always less than predicted from the previous steady state conditions (Fig.2.2.8b,d). In contrast, measured instantaneous I 2 i s always greater than predicted for polarized fibers i n alkaline solutions (Fig.2.2.8f). Conversely, measured I(V2»0) predict increasing steady state currents with increasingly negative rather than the l i m i t i n g currents that were observed (Fig.2.2.8e). For depolarized fi b e r s i n alkalin e solutions, measured instanta-neous I 2 i s always less than predicted from the previous steady conditions (Fig.2.2.8h). F i n a l l y , i t i s of interest to note that since the measured minimal instantaneous I 0 i s near zero (holding current) for depolarized 98 f i b e r s in both a c i d and a l k a l i pH's , the predic ted steady state (V-^ ) current -vo l tage r e l a t i o n s always show a negative slope conductance ( F i g . 2 . 2 . 8 c , g ) , whereas i n po lar i zed f i b e r s , the measured minimal I(V2>0) i s much larger and under these c a l c u l a t i o n s , increas ing currents are always predic ted with increas ing V^'s ( F i g . 2 . 2 . 8 a , e ) . The voltage dependence of the k i n e t i c s of current t rans i en t s The c h l o r i d e permeation system responds asymmetrical ly to p o s i t i v e -and negative -going vol tage steps from the r e s t i n g p o t e n t i a l . In the p o s i t i v e or d e p o l a r i z i n g d i r e c t i o n , the current t rans i en t s are approximately three times slower than t h e i r corresponding hyperpo lar i z ing vol tage pulses . Th i s asymmetric response i s not predic ted from the movement of a charged p a r t i c l e wi th in the membrane phase. Relaxat ion rates have been shown to have the same dependence on the vo l tage d i f f erence l ^ - V j | at a l l pH's and i n p o l a r i z e d and depolar ized f i b e r s , at l eas t when and are more negative than the r e s t i n g p o t e n t i a l . Th i s i s so i n sp i t e of the steady s tate c u r r e n t -vol tage r e l a t i o n s being very d i f f e r e n t under the d i f f e r e n t condi t ions imposed. The valence of a charged group assoc iated with the c o n t r o l of the instantaneous currents in the c h l o r i d e channel was der ived from the slope of the sigmoid r e l a t i o n s (12(0) versus V ^ ) . I f t h i s group were to re lax from one s tate to another when the membrane p o t e n t i a l i s changed, one would expect the rate of r e l a x a t i o n (and therefore the speed of the current t rans ient s ) to change by a fac tor of two in going from a l k a l i n e to a c i d s o l u t i o n s . Th i s was not observed i n experiments (despite the large v a r i a b i l i t y of the r e l a x a t i o n r a t e s ) . 99 We do not know whether t h i s i s due to the 'groups' involved in the control of i n i t i a l currents and current transients being different or to the method of time constant extrapolation or approxi-mation being inappropriate for th i s kind of analysis (for example, perhaps we should have been measuring an i n i t i a l extremely fast transient that we could not resolved). F i n a l l y , i f current transients r e f l e c t a change of state of some controlling group or s i t e , within the chloride channel, the present results indicate the s i t e has no preferred, or minimum potential energy, state, and the rate of tr a n s i t i o n from one state to another depends l i n e a r l y on the i n i t i a l force applied (voltage step), over a wide range. 100 Figure 2.2.8 In this figure the f i l l e d c i r c l e s represent data obtained experimentally and open c i r c l e s represent points computed as described i n the text. In (a) the f i l l e d c i r c l e s represent the steady state current-voltage r e l a t i o n for a polarized c e l l i n acid solution (holding p o t e n t i a l , -70 mV; solution Ba). Data was used to compute the open c i r c l e s i n (b). Conversely, the f i l l e d c i r c l e s i n (b) represent experimentally derived values of I (0) (AV) as a function of V,(V ) 2 1 m and these data were used to obtain the open c i r c l e s i n (a). Data i n (c) and (d) are from a depolarized f i b e r at pH 5.4 (holding p o t e n t i a l , -20 mV; solution Ea) and those i n (e), (f) and (g), (h) are from polarized (holding p o t e n t i a l , -80 mV; solution Bd) and depolarized fib e r s (holding potential,-30 mV; solution Ed) respectively at pH 8.4. 1 0 0 a -120 V m ( "V) _Q -120 Vm (™V) 8 0 0 ,8 ' * 8 « AV i mV o _ r S T 8 8, 4V e -140 oo8» f -140 o o •• 8 ° . [mV h H . t I I mV 8 » 7 7 101 Experimental Section 3: The Inactivation and Recovery of I n i t i a l  Currents; Dependence of Kinetics on I n i t i a l Conditions I. Recovery of Conductance When a hyperpolarizing step was applied to a muscle fib e r held at the resting potential the current decreased to the steady state i n a l k a l i n e solutions and increased to the steady state i n acid solutions. Upon return of the hyperpolarizing voltage to the resting potential transient aftercurrents were noted; these aftercurrents, opposite i n direction to the hyperpolarizing currents, r e f l e c t the recovery of the membrane conductance to i t s o r i g i n a l resting condition. This recovery was studied by applying a second hyperpolarizing constant (amplitude) test pulse at a variable period jt after the return of the f i r s t pulse to the holding poten t i a l . Fig.2.3.1 shows the experimental protocol. The f i r s t pulse set the membrane to a certain 'conditioned' state. When the voltage was returned to the holding p o t e n t i a l , the mechanisms responsible for the observed changes i n conductance began to be restored to th e i r i n i t i a l , or 'unconditioned' state. The current at the beginning of the second pulse 1 2 ( 0 ) , and the timecourse of relaxation of t h i s current (I^) were used as indicators of the degree to which i n i t i a l conditions had been restored. From observations that have described, some of the results may be anticipated: For no time separation between the conditioning and test pulses (conditioning step more negative than test step) the test current ^ rose to a steady state; i f the two steps were equal then the test current was constant (I _ (0)=I 9 (°°)) . When the 102 Figure 2'.3". 1 Protocol of recovery experiments. Two voltage pulses about three seconds long were a p p l i e d , the f i r s t , V ^ , set to a more negative p o t e n t i a l than the second,V^. The two pulses are separated by a b r i e f and v a r i a b l e re turn to the ho ld ing p o t e n t i a l ( t ) . 103 test potential was separated from the conditioning potential after a long period of recovery at the holding potential, then the test currents f e l l to the steady value regardless of the size of the con-di t i o n i n g potential. Thus for t^O a r i s e i n test current i s found and for long _t a f a l l i s observed (with different time constants for the two cases). Therefore as the time period of recovery was varied, the i n i t i a l currents and the time constants of their relaxations to the steady value would be expected to depend on the duration of the recovery at the holding'potential. A. Recovery experiments i n polarized fibers The results from fibers i n neutral and alkaline solutions were si m i l a r ; the dependence of i n i t i a l currents and dependence of time constants on duration of recovery are i l l u s t r a t e d i n Fig.2.3.2. Open c i r c l e s are from a f i b e r i n pH 7.3 and f i l l e d c i r c l e s i n pH 8.4. The dependence of recovery of the i n i t i a l currents on _t appeared to be sigmoid (S-shape) (Fig.2.3.2b). The rates of recovery of both time constants (Fig.2.3.2a) and i n i t i a l currents (Fig.2.3.2b) for fibers i n neutral and al k a l i n e solutions were s i m i l a r . These results w i l l be described i n greater d e t a i l for depolarized fibers (as they are similar to polarized f i b e r s ) . In acid solutions (pH 5.4-6.7) for the fibers that exhibited a biphasic response with hyperpolarizations (Fig.2.Ob) the transients associated with the test pulses were very small and i t was not possible to analyze for time constants. Fig.2.3.3 shows the waveforms from a f i b e r i n acid solution (pH 5.4) whose current rose with hyperpolarization. With increasing 1 0 4 durations of recovery at the holding potential, the magnitude of the current transient increased (the amplitude of 1 ^ ( 0 ) decreased) and the rate of r i s e to the steady state increased. Cl e a r l y , i n contrast to situations where a f a l l i n g current transient accompanied hyperpolarization from the holding potential, the monotonic r i s e i n current during conditioning caused an enhancement of ^ ( O ) . One experiment from these fibers showed that when was fixed and was variable, i n i t i a l depended sigmoidally on V^, but the sigmoid curves were inverted when compared with the result when conditioning by negative-going voltages caused a diminution of 1 2 ( 0 ) . The occurrence of sigmoid relations i n both cases suggests that they are under the control of sim i l a r mechanisms. This i s reinforced by the present observations that recovery of i n i t i a l current and ki n e t i c s both depend on i n i t i a l conditions for f i b e r s exhibiting normal or inverted sigmoid ( 1 2 ( 0 ) ) dependence on conditioning potential V^. However, a difference does exist between them as the r i s i n g currents have a different voltage dependence from the f a l l . No attempt was made to investigate these d e t a i l s any further. B. Recovery experiments i n depolarized f i b e r s The recovery of conductance from a hyperpolarizing prepulse was studied i n more d e t a i l i n depolarized than i n polarized f i b e r s because the difference between the i n i t i a l .test current 1 2 ( 0 ) and steady state test currents ^ ( " O was much larger; consequently the kin e t i c s could be more accurately resolved. In the experiments the 105 test potential was always chosen more positive than the conditioning potential. Recovery of i n i t i a l current was then characterized by the duration (t^) of the period at the holding potential at which the i n i t i a l current had recovered to a value equal to the steady state current (l^C00))* although i n i t i a l and steady states were sometimes separated by non-steady currents. For t^  less than t^, 12(0) was less than I^ O") and for _t greater than t ^ , 12(0) was greater than ^("O (Fig.2.3.A). Both the dependence of i n i t i a l current 12(0) on _t and the dependence of current transients of on _t w i l l be described i n the following. Fibers i n acid solution (pH 5.4) The recovery of i n i t i a l current showed both exponential dependence on t^  (time constant varying from 15 to 25 msec) and sigmoid dependence (Fig.2.3.5). Because of the sigmoid and exponential dependence, the parameter _t^ was found to be an adequate indicator of the rate of recovery i n different f i b e r s . This rate of recovery appeared to depend on the amplitude of : the larger the conditioning pulse the smaller the t ^ and when recovery time-course followed an exponential, the smaller the time-constant. These observations can only be considered as preliminary as an i n s u f f i c i e n t number of experiments was performed on the same f i b e r with different conditioning potentials for the nature of t h i s dependence to be resolved. The dependence of the k i n e t i c s of transients on jt can be q u a l i t a t i v e l y observed i n Fig.2.3.4. The f i b e r was i n a solution 106 Figure 2.3.2 a) Recovery of relaxation time constants i n two polarized f i b e r s . Open c i r c l e s are from a fib e r studied at pH 7.3 (resting potential, -80 mV, solution Be). F i l l e d c i r c l e s are from a f i b e r i n a solution of pH 8.4 (resting potential, -80 mV, solution Bd). In t h i s experiment, both pulses (V^ and V^) were of the same amplitude and hence only f a l l i n g current transients were observed. b) 1 2 ( 0 ) plotted as a function of _t (period of recovery at the holding potential (-80 mV) ) in the same experiment. The right hand ordinate i s for data from the fib e r studied at pH 7.3 and l e f t hand ordinate i s for the fib e r studied at pH 8.4. The abscissa i s equivalent to the steady state current, 9.25 mV and 6.25 mV on the AV scale for the f i b e r at pH 7.3 and that at pH 8.4 respectively. 1 mV corresponds to a current —6 2 density of 3.79 x 10 Amp/cm . 200 T msec o o 50 L i _ 0 600 t (msec) b 3r § 1 o o • l2(0) mV 0 600 t (msec) o 107 Figure 2.3.3 Current traces recorded during a recovery experiment on a polarized fiber (resting potential, -70 mV) at pH 5.4 (solution Ba). Note that as t_ increased, the i n i t i a l current 1 2 ( 0 ) decreased. Both voltage steps were of the same amplitude, -70 mV (absolute membrane potential was -140 mV). The current traces were inverted by the d i g i t a l averager. The currents are inward. Calibration: 5 mV. 1 mV = 3.14 x 10 ^  Amp/cm . 107a 1 0 8 r i ibo Figure 2.3.4 Waveforms seen as t_ i s v a r i e d : as _t i s made longer the i n i t i a l current at the onset of the second pulse ( I 2 (0 ) ) increases . When _t = t^ , I 2 (0) = I 2 ( « ) . The numbers on the traces i n d i c a t e t_ i n m i l l i s e c o n d s . For the f i b e r for which r e s u l t s are depicted here ( res t ing p o t e n t i a l , -20 mV, pH 5.4, s o l u t i o n Ea) t^ was a b o u t . 14 m i l l i s e c o n d s (middle t r a c e ) . —6 2 C a l i b r a t i o n : 5 mV. 1 mV = 3.79 x 10 Amp/cm . 109 Figure 2.3 .5 1 2 ( 0 ) as a funct ion of _t for two depolar ized f i b e r s at pH 5.4. Resting p o t e n t i a l s of the f i b e r s were -20 mV. So lu t ion Ea . In one f i b e r ( f i l l e d c i r c l e s ) recovery of i n i t i a l current appeared to fo l low a sigmoid timecourse, whereas i n the other ( f i l l e d squares) i t appeared to r i s e according to the funct ion X 2 ( t ) - I 2 ( 0 ) m a x ( 1 " e x p ( - t / 8 » . The s o l i d l i n e i s a p lo t of the l a t t e r funct ion with s= 15 m i l l i s e c o n d s . The ordinate scale i s normal ized. 109a 1 1 0 Figure 2.3.6 The dependence of the instantaneous test current 1^(0), and the time constant of the relaxation of the test current to i t s steady state on the period of recovery, t_ , at the holding potential. Open c i r c l e s indicate the time constants of current during the test pulse (right hand ordinate). When _t was less than 14 milliseconds, current i n i t i a l l y rose (1^(0) <I^(^)) and the time constant decreased with increasing durations of recovery. The f i l l e d c i r c l e s indicate the i n i t i a l currents ( 1 2 ( 0 ) ) . pH 5.4; holding potential, -20 mV; solution Ea. The conditioning potential, , was -60 mV (absolute membrane potential) and the test potential, \^ , was -40 mV (absolute membrane p o t e n t i a l ) . —6 2 Current c a l i b r a t i o n : 1 mV = 3.79 x 10 Amp/cm . 110a I l l of pH 5.4 and had a resting potential of -20 mV. As the duration of recovery _t approaches t ^ ( t ^ = 14 msec i n this instance), the rate of r i s e of the i n i t i a l test current decreases. Increasing the duration of recovery beyond i n turn increases the rate of relaxation (now an exponential decay) as shown i n Fig.2.3.4. The dependence of the time constants of the test current 1^ o n the duration of recovery at the holding potential (-20) from the fi b e r of Fig.2.3.4 i s shown i n Fig.2.3.6. The open c i r c l e s are the time constants and also shown are the instantaneous or i n i t i a l test currents (^(O) ( f i l l e d c i r c l e s ) ) . The time constants for _t greater than 14 msec (t^) a r e for f a l l i n g currents. Fibers i n alkalin e solution (pH 8.4) The recovery of i n i t i a l currents for fibe r s i n alkalin e solutions was sim i l a r to those i n acid solutions: f i b e r s exhibited both exponential and sigmoid dependence on t_ with approximately the same rates of recovery. However, i n a l k a l i n e solutions, the rate of recovery i n a group of fibe r s was found to be unusually rapid. An example i s shown i n Fig.2.3.7 for current waveforms corresponding to steps from the holding potential (-30mV) to -126 mV and then to -74 mV and f i n a l l y back to the holding p o t e n t i a l . The steps to -126 and to -74 mV were separated by variable duration steps (t) back to the holding potential. When t was less than 5 msec the i n i t i a l current (12(0)) was very small and rose to a maximum after which current f e l l non-exponentially. When t_ exceeded 6 msec the amplitude of I 9(0) 112 increased dramatically (relative to the amplitude at 5 msec) and decayed rapidly to a value about 2.5 times the steady l e v e l ; there-after the decay was slow and non-exponential. This waveform was similar to that described previously for following a conditioning pulse that 'primed' the membrane conductance. The sigmoid dependence of the i n i t i a l currents on _t for the f i b e r of Fig.2.3.7 i s shown i n Fig.2.3.8. In f i b e r s bathed at pH 8.4 that showed transitions from near-minimal 1 2 ( 0 ) to near maximal i n i t i a l currents over a range of _t of a few milliseconds, the r i s i n g transients ( t < t ^ could often be "resolved into two time constants: an i n i t i a l time constant of about 10 msec that did not vary with t^  and a slower time constant which became longer with J t , up to _t = 5 msec (Fig.2.3.9). Despite the rapid t r a n s i t i o n s , the dependence of time constants on _t was similar to that for fibers i n which the i n i t i a l current t r a n s i t i o n was very much slower (cf. Fig.2.3.5). When t exceeded t (7 msec i n Fig.2.3.8) — —o transient 1^ was a non-exponential f a l l . I I . The Rate of Inactivation of Chloride Currents: Experiments i n which the Duration of the Conditioning Step was varied. Upon application of a hyperpolarizing pulse from the holding po t e n t i a l , the membrane conductance inactivates or f a l l s . The objec-tiv e of the experiments described i n t h i s section was to study the rate of th i s i n a c t i v a t i o n by applying a second test pulse before steady state for the conditioning pulse i s reached. The i n i t i a l current during the test pulse can be used as an ind i c a t i o n of the state 113 Figure 2.3.7 Current waveforms recorded in a recovery experiment on a depolarized f i b e r at pH 8.4 (resting potential, -30 mV, solution Ed). As shown on the Figure, t_ was 0, 5, 6 and 7 milliseconds i n the selected traces. was 96 mV (to an absolute membrane potential of -126 mV) and was 44 mV (to an absolute membrane potential of -74 mV). Calibration: current = 5 mV. —6 2 1 mV = 3.79 x 10 Amp/cm . abscissa = one second. 1 1 3 a -114 20 r "2(o) mV 0 t ' . 0 t ( m s e c ) 10 Figure 2.3.8 Dependence of 12(0) on jt for the f i b e r on which the waveform of Figure 2.3.7 were measured. Note that at very short times the i n i t i a l current was outward; the t r a n s i t i o n to maximal inward current was extremely rapid. —6 2 Current c a l i b r a t i o n : 1 mV = 3.79 x 10 Amp/cm . 115 180r T m s e c 100L 2 0 r ° 0 t (msec) Figure 2.3.9 Time constants for r i s i n g currents i n a f i b e r that showed a rapid t r a n s i t i o n trom r i s i n g to f a l l i n g currents as a function of t_ i n a recovery experiment at pH 8.4. There were two time constants, one of about 10 msec that was independent of t and a longer one that was time dependent. 1 1 6 Figure 2.3.10 a) Current traces from a fi b e r held at -20 mV at pH 8.4 (solution Ed). In this experiment the dura-tion of was altered: the duration of ( m i l l i -seconds) i s shown on each trace. The amplitudes of and V 2 were 52 and 26 mV, respectively. When was maintained for longer than 125 msec I 2 ( 0 ) was less thant I^(a>). Notice the large af ter-currents. —6 2 Calibration: 5 mV 1 mV=3.79 x 10~ Amp/cm . b) The open c i r c l e s show I 2(0) as a function of the duration of V^. The horizontal l i n e running across the plot indicates I 2 ( e o ) and 0 indicates the holding current. The two uppermost plotted points actually indicate outward current. The f i l l e d c i r c l e s indicate the recovery of I 2(0) as a function of _t, when i s held for a s u f f i c i e n t time for 1^ to have reached a steady state. At short times both the diminution and the recovery processes have timecourses that closely approxi-mate exponentials with time constants of 100 milliseconds (plotted l i n e s ) but slow considerably at l a t e r times. 0 100 500 1000 time (msec) 117 Figure 2.3.11 Effects of duration of the conditioning potential (Vj) on the ki n e t i c time constant of the test current (I^) i n a fiber at pH 8.4 (the same fiber as Fig.2.3.10: holding potential, -20 mV, solution Ed). Abscissa i s the duration of V^ and ordinate i s the time constant of the test current. Open c i r c l e s denote f a l l i n g currents and f i l l e d c i r c l e s , r i s i n g currents. When the duration of V^ was very b r i e f , the test current (I^) f e l l to the steady stat with an i n i t i a l l y fast time constant (open c i r c l e s ) and then a slower one (open squares). The fast time constant was dependent on the duration of V^ whereas the slower one was independent. As the duration of V^ increased, the test current became biphasic: f i r s t r i s i n g , with a time constant indicated by the f i l l e d c i r c l e s , and then f a l l i n g toward the steady state (the uppermost c i r c l e s ) . When the durations of V^ was lyin g between 200 and 400 milliseconds, the biphasic responses were too small for accurate resolution of the time constants. 117a 500 r T m s e c 100 0 J L J 200 400 1000 d u r a t i o n o f V , ( m s e c ) 1 1 8 of the conductance prior to the application of the test potential. The experiments were conducted on depolarized fibers i n alkaline solutions (solution Ed, pH 8.4). Fig.2.3.10a shows the current traces from such an experiment on a fi b e r held at -20 mV. and were hyperpolarizing pulses of 52 and 26 mV, respectively (absolute membrane potentials were -72 and -40 mV). 1^ f e l l from the i n i t i a l to the steady l e v e l when was less than 125 msec i n duration, whereas 12(0) was less than l^C00) for longer than t h i s duration. The i n i t i a l current became outwards i n this f i b e r for exceeding 260 msec. The open c i r c l e s of Fig.2.3.10b show the dependence of ^ (O) on the duration of the conditioning pulse. The dependence of the time constants of the test current 1^ i s shown i n Fig.2.3.11. When the duration of was increased from 10 msec to 125 msec, the rate of f a l l of 1^ decreased. When the step to exceeded 125 msec the rate of r i s e of increased with increasing conditioning time. The current relaxations were resolved for durations less than 30 msec into an i n i t i a l f a l l whose time constant depended sigmoidally on the duration of (open c i r c l e s ) and delayed f a l l that was independent of the conditioning period (open squares). As the conditioning period approached 1 second, the i n i t i a l f a l l reached minimum rate. Comparison of the rates of ina c t i v a t i o n and recovery. To further probe the underlying processes involved i n current 119 relaxation both the time dependence of the inactivation and recovery of i n i t i a l test currents, and the time dependence of the inactiva-tion and recovery of the test current time constants were compared. The dependence of i n i t i a l test current 1^(0) on _t (from the same fi b e r as Fig.2.3.10a) i s plotted as s o l i d c i r c l e s on Fig.2.3. 10b. The dashed and s o l i d lines are exponential f i t s to the i n i t i a l currents from inactivation and recovery experiments. The lines have the same time constants (100 msec). I t thus appears that the i n i t i a l phase of the recovery and inactivation of i n i t i a l currents follow the same time course. Waveforms of the test current as t. was increased are shown i n Fig.2.3.12a. When _t was within 60 msec, the recovery of the i n i t i a l time constant (Fig.2.3.12b) appeared sigmoid, although the maximum asymptote was not always seen because the r i s i n g transient became too small for time constants to be resolved at longer times (Fig.2.3.12a). For periods between 60 and 200 msec, the waveform was biphasic, current r i s i n g to a maximu, then f a l l i n g toward the steady state. The amplitudes of the transients were too small for accurate analy-s i s . For longer times the waveform had only a decaying phase that was seen to be a single exponential (open c i r c l e s , Fig.2.3.12b). Qu a l i t a t i v e l y , the recovery of the time constants i n Fig.2.3.12b and the i n i t i a l segment ( f i r s t 60 msec) of Fig.2.3.11 are s i m i l a r . This observation, together with the s i m i l a r i t y of the rates of i n a c t i -vation and recovery of instantaneous currents, suggests a symmetry 120 Figure 2.3.12 a) Current transients i n a recovery experiment performed on the same f i b e r as Fig. 2.3.10. The duration of recovery _t , i n milliseconds, i s indicated on top of :each sweep. The straight l i n e s on the l e f t indicate the r e l a t i v e position of the holding currents for each of the successive sweeps. In the waveforms corresponding to t = 200 and 100 m i l l i -seconds, the current was outwards when the membrane potential was stepped back to the holding potential. b) This figure shows the dependence of the relaxation time constants of the test current on _t. The s o l i d c i r c l e s are time constants for r i s i n g currents and the open c i r c l e s are for f a l l i n g currents. 120a O O O 2_> (A J J o o E o CO 121 between the i n a c t i v a t i o n and recovery processes. Discussion Dependence of k i n e t i c s on i n i t i a l conditions One of the main conclusions i n t h i s section i s that the rate of diminution of i n i t i a l current during hyperpolarizing conditioning i s approximately equal to the rate of i t s recovery at the resting poten t i a l . Moreover, the dependence of instantaneous current on i n i t i a l conditions i s paralleled by a similar dependence of the relaxation rate constants on i n i t i a l conditions. A dependence of conductance k i n e t i c s on previous history has also been reported i n the potassium channel i n Xenopus myelinated nerve fibers (Frankenhaeuser, 1963) and more recently i n the potassium channel i n frog node ( P a l t i , Ganot and Stampfli, 1976). The mechanism responsible for t h i s i s obscure as i t i s not predicted by f i r s t order (cf. Hodgkin and Huxley, 1952) k i n e t i c s . We have shown, i n the preceding experimental section, that k i n e t i c s of current transients depend on the voltage step between conditioning and test voltages l V 2 _ ^ i l ' This dependence on ! V2 -V]J (the amount of energy input into the transport system) i s the most puzzling and d i f f i c u l t aspect of the chloride conductance to under-stand. I t w i l l be shown i n the following chapter that t h i s i s so even i n non-steady conditions. That i s , even during non-stationary conditions, i t i s possible to define an equivalent *V ' upon which 122 k i n e t i c s depend. Slow currents under 'primed' conditions Throughout the three experimental sections, references have been made to a current that arose as the result of 'priming' i n chronically depolarized f i b e r s . The necessary condition for i t s appearance was a step from a large negative potential to potentials more positive than about -50 mV; i t was p a r t i c u l a r l y noticeable i n fibers i n a l k a l i n e solutions. The observation that t h i s component i s absent i n negative-going steps from the holding potential suggests that the channel mechanism con t r o l l i n g t h i s current exists i n three states: (1) an inactivated state i n which i t cannot conduct. At potentials near -20 mV (the resting potential for depolarized fibers i n alkaline solutions) most channels are inactivated. (2) an activated, but non-conducting state. The t r a n s i t i o n from inactivated to activated state requires less than 50 msec for potentials more negative than -100 mV. At -50 mV a s i g n i f i c a n t f r a c t i o n of the channels i s reactivated, since steps from the steady state at th i s voltage to the holding potential e l i c i t e d inward aftercurrents. (3) A conducting state. Activated channels are made to conduct by applying a positive-going step from (depolarization). A simple k i n e t i c scheme r e l a t i n g these states i s depicted i n Fig.2.3.13. At large negative potentials inactivated channels are rapidly activated and conducting channels made non-conducting, but 1 2 3 activated (dashed arrows). Depolarization (solid arrows) would cause rapid opening of activated channels (conducting state) as well as very slow in a c t i v a t i o n that would presumably proceed with and without a transient conducting state ( t o t a l inactivation has been seen to require more than one minute). No attemptwas made to test whether these currents were due to chloride. However, the experimental evidence does not support the other candidates, rubidium and calcium. A contribution by rubidium ions seems u n l i k e l y since there would be no net driving force on ions that permeate K +-channels, at the resting potential. The evidence supporting calcium i s c o n f l i c t i n g : Beaty and Stefani (1976) have presented evidence for inward calcium current triggered by depolarization, that inactivates slowly. Such a current would normally be short-c i r c u i t e d by a much larger chloride shunt (Beaty and Stefani, 1976). However, i n our experiments the 'primed' currents were as large as those recorded i n the absence of priming. 124 Figure 2.3.13 Schematic r e l a t i n g the postulated states of the conductance mechanism of the 'primed' current to membrane potenti a l . Solid arrows indicate the dir e c t i o n of changes of the state of the mechanism i n response to depolarization; dashed arrows show the d i r e c t i o n of changes when the membrane potential i s made more negative. For more explanation, see text. inactivated activated (non-conducting) A conducting 1 2 5 Chapter I I I The I n i t i a l Current-Voltage Manifold 126 In the preceding chapter, data from experiments on fibers at different pH's and under different resting potentials were presented. Among various aspects of chloride conductance that were described and summarized i n the appropriate sections were: the steady state conductance as a function of voltage, pH, and resting pote n t i a l ; the instantaneous conductance as a function of pH, the voltage of the conditioning (V ) and test potentials (V^), and concentration; the dependence of current transients on pH and voltage; and the time course of inactivation and recovery of conductance as well as the dependence of ki n e t i c s on i n i t i a l conditions. An essential observation was that the instantaneous chloride currents and relaxation ki n e t i c s depend on the magnitude and 'state' of the prepulse (condition-ing p o t e n t i a l ) . In this chapter, we w i l l show that an overall perspective of much of the experimental data can be obtained. The idea i s to represent the dependence of instantaneous test currents 1^(0) as a function of the conditioning potential (V ) and test potential (V^) i n a three dimensional rectangular coordinate system where the independent variables are and and the dependent variable i s i n i t i a l test current 12(0). The result of the representation, called a manifold (two dimen-si o n a l ) , provides us with an o v e r a l l perspective of the experimental results and enables us to compare and relate diverse experimental protocols such as the i n a c t i v a t i o n and recovery of conductance and the dependence of current transients on a voltage step and on i n i t i a l conditions. 1 2 7 The representation requires that values of 1 2 ( 0 ) D e specified on families of mutually perpendicular planes—the I (0)xV (when V2 is constant) and the IjWxV^ (when is constant) planes. The forms of these relations are as described in the results (for example, Fig. 2.2.4a and Fig. 2.2.4b). In the event that experimental data are available, these should be obtained from a single c e l l . In other words, detailed analytical expressions for the dependence of 12(0) on and are required. We were unable to obtain such exten-sive data from a single fiber to construct an experimental manifold but we did obtain sufficient experimental evidence to provide a strong framework for a theoretical synthesis. Since 1 2 ( 0 ) depends on both V\j and V 2 , information from variable V (amplitude of held fixed) experiments provides information on variable (amplitude of held fixed) experiments. More precisely, any specified current, I^iO) CV^Y^) is on the intersection of the planes ^(COxV^ (at constant V°) and I 2(0)xV 2 (at constant V°). Assumptions The approach used in developing the manifold relies heavily on the sigmoid dependence of 1 2 ( 0 ) on (at a constant v 2 ) . We w i l l focus our construction on fibers in alkaline solutions (using a shape parameter of .087, z=2, Fig. 2.2.18c). Linear ^ ( 0 ) versus V2 (for constant V^) relations w i l l be used throughout. These relations are highly non-linear (even saturating, see Fig. 2.2.3b) in polarized fibers in alkaline solutions (pH 8.4) but in depolarized fibers 128 at the same pH, were much more l i n e a r . However, s ince a n a l y t i c a l expressions descr ib ing them over a wide voltage range were not obtained, the assumption of l i n e a r i t y was adopted. F i n a l l y an a d d i t i o n a l r e l a t i o n r e q u i r i n g the dependence of the instantaneous tes t currents 12(0) on for d i f f e r e n t condi t ion ing voltages i s r e q u i r e d . The assumption that we used was that when these r e l a t i o n s (the dependence of 12(0) on V^) are p l o t t e d i n the same plane , regard-less of (as i n F i g . 2.2 .4 ) , they i n t e r s e c t i n the same point (see step 4 below). I t i s i n t e r e s t i n g at t h i s moment to note that despite the approximate nature of the l a s t two assumptions, the q u a l i t a t i v e p r e d i c t i o n s of the representat ion w i l l be seen to account for much of our experimental data . The synthesis i s perfomed as fo l lows: 1. A reference sigmoid r e l a t i o n (I^ (0) (V ,V°) ) i s placed i n the plane (open c i r c l e s , F i g . 3 .1a) . The choice of V° and the p o s i t i o n of the r e l a t i o n i n the plane are a r b i t r a r y . The value of V°= (Vji~70)mV was se lec ted for convenience, s ince i t i s the l arges t value used i n our experiments. (When V - L = V 2 ' t * i e instantaneous tes t current 12(0) i s equal to the steady state c o n d i t i o n i n g current and the point at A represents the steady s tate current corresponding to a h y p e r p o l a r i z a t i o n voltage step of 70 mV). There would have been no loss of g e n e r a l i t y had any other been chosen. The slope of the r e l a t i o n s i s appropriate to currents i n a l k a l i n e so lut ions ( k = . 0 8 7 , F i g . 2 . 2 . l b , c ) . The q u a l i t a t i v e 129 proper t i e s of the construct would not be a l t e r e d i f the steeper slope of the a c i d - d e r i v e d data had been used. 2. The points (open c i r c l e s of F i g . 3 . l a ) are r e p l o t t e d on the l2(0)xV2 plane (open c i r c l e s , F i g . 3 . l b and c ) . 3. A family of ^WxV^ r e l a t i o n s ( f ixed V ) ( F i g . 3.1c) i s generated by f i r s t p r o j e c t i n g a l i n e from the smallest current value p l o t t e d (point A, F i g . 3 . l a and c) through the ho ld ing -current axis at a vol tage negative to the ho ld ing p o t e n t i a l . A value of -20 mV was used i n F i g . 3 . 1 c as t h i s value was approximately the l arges t value obtained for the s h i f t i n e q u i l i b r i u m p o t e n t i a l with h y p e r p o l a r i z a t i o n . Secondly, a l i n e i s then drawn through the reference point (V, ,1, ) and open c i r c l e B ,a t which V =V . This l i n e i s n n 1 h the instantaneous current -vo l tage r e l a t i o n for steps from the ho ld ing p o t e n t i a l . 4. The family of I (0)xV2 r e l a t i o n s i s completed by l o c a t i n g a l l ( F i g . 3 . l b ) i n the p lane . Lines are then drawn through AP and BQ to i n t e r s e c t i n 0 (the point of i n t e r s e c t i o n of a l l the rays i n F i g . 3 . 1 c , t h i s point i s not shown i n F i g . 3 . 1 c ) . OAB and OPQ are s i m i l a r t r i a n g l e s and PP^/AA^ = PQ/AB. PP. i s the only unknown. Once a l l P. are l o c a t e d , the l i n e s through A_.P^ complete the family ( F i g . 3 . 1 c ) . 5. A family of ^ ( O ^ V r e l a t i o n s i s now obtained by repeatedly revers ing step 2: for each of a number of chosen values of V„ data points are r e p l o t t e d from the I (0)xV plane onto 130 the I„(0)xV1 plane. The f i l l e d c i r c l e s (P ) are given as 2 1 3 an example. It i s seen by construction that the I 2(0)xV^ curve passing through the holding potential at the holding current (V, ,1, ) gives outward h h currents for more negative than the holding potential,(when i s equal to the holding potential),the recurring observation i n a l l our experiments. The manifold i s drawn i n Fig.3.2. The left-hand abscissa i s V^, the right-hand abscissa V^, and the v e r t i c a l axis, current. When v i = V 2 > instantaneous and steady state currents are equal, indicated by f i l l e d c i r c l e s i n Fig.3.2. They w i l l be referred to as stationary points. Contrary to the sense of our other figures, current plotted i n the upward direction here i s inward. The plane containing the abscissae represents holding current: when the manifold ri s e s above i t , currents are inward r e l a t i v e to the holding current, and when the manifold f a l l s beneath the plane, currents are outwards. Applications of the Manifold Prediction of the instantaneous currents The surface enables us to predict the instantaneous current that w i l l be recorded when the voltage i s changed from one value, at which a steady state has been attained, to another. An example i s given i n Fig.3.3. In a_ three of the S-relations of Fig.3.2 have been replotted. The l i n e s with arrows indicate the dir e c t i o n of a Figure 3.1 Steps i n the construction of the current-voltage manifold (see t e x t ) . 132 Figure 3.2 The current-voltage manifold. For description, see text. 133 Figure 3.3 Use of the manifold to predict the responses to a sequence of voltage steps. For d e t a i l s , see text. 134 Figure 3.4 (a) and (b) demonstrate the use of the manifold i n predicting currents when voltage steps are made after steady state conditions have been reached. (c) and (d) simulate a case i n which the voltage step from to ' i s made before the current at has decayed to the steady state (2'). From data obtained on the voltage dependence of time constants, i t i s possible to predict the time constants of the t r a n s i t i o n a l currents. 1 3 4 a 135 cycle of voltage changes and i n lj the currents observed at the onset of the steps of voltage are plotted, measured d i r e c t l y from a, and joined by 'transient currents' with similar timecourses to those observed i n alkal i n e solution. When the membrane potential i s suddenly stepped to and held at a new value Ov^) the current changes with time u n t i l i t s e t t l e s at the stationary point of the S-curve corresponding to that V2-It can be seen from Fig.3.1c that as (the holding potential) i s made more negative the slope of the instantaneous (V^) current-voltage r e l a t i o n of negative-going steps decreases. This i s consistent with the observed reduction i n a one pulse experiment when the holding potential i s made more negative. I t i s also clear that as i s made more negative the s h i f t of the I 2 ( 0 ) x V 2 intercept on the voltage axis approaches a l i m i t i n g value. The dependence of instantaneous currents on i n i t i a l conditions in non-steady states can be explained q u a l i t a t i v e l y using the instantane-ous current manifold. We s h a l l f i r s t describe the results of an experiment i n which the membrane potential i s stepped from the holding potential (V, ) to a h more negative voltage 0^)» then to an intermediate voltage (V^). Two conditions are compared: one i n which the i n i t i a l pulse i s s u f f i c i e n t l y long for a steady state to have been reached before the step i s made, the other i n which the step to the intermediate voltage i s made before the steady state i s reached (an experimental example i s given i n Fig.3.4). In both cases, the second pulse i s s u f f i c i e n t l y 136 long that steady conditions are attained before the step back to the holding poten t i a l . These transitions are shown i n Fig.3.4a. The same manifold segment i s used as was employed i n Fig.3.3a. For the f i r s t case, the membrane po t e n t i a l , M^ , i s kept constant and the current declines u n t i l the stationary point i s reached, at 2. The step to i s associated with an i n i t i a l current, 3; current r i s e s to 4 (stationary point) and the return to e l i c i t s the outward current at 5, and so on. The i n i t i a l currents are joined by a r b i t r a r i l y drawn transients in Fig.3.4b. For the second sequence the i n i t i a l current at 1 i s the same, but at 2' i t has not declined to the stationary value. At th i s point, a step from V 2 to gives the current at 3' that decays to the steady state at 4, and so on as shown i n Fig.3.4d. The results of recovery experiments (eg. Fig.2.3.12) can also be accounted for using the manifold as seen i n Fig.3.5. The same i n i t i a l step from to V 2 i s used as i n Fig.3.4. After the current has reached steady state at 2 the membrane potential i s returned to the holding potential for a period _t before being stepped to V^. The i n i t i a l current at the holding potential (3) i s followed by a decay to 4. The membrane voltage i s now stepped to and the current at 5, which i s a stationary point as the manifold i s con-structed, i s recorded. The i n i t i a l and steady state currents are i d e n t i c a l ; the potential i s then switched back to V, (6). The h currents are sketched i n Fig.3.5b. I f the duration of the transient 137 step to the holding potential were less than t ^ there would be a r i s i n g transient during the next step, a f a l l i n g transient i f i t were longer than _t^. Dependence of ki n e t i c s on i n i t i a l conditions We have seen experimentally that the rate of relaxation from i n i t i a l to steady state currents i s dependent on the size of the voltage step. That i s , i n Fig.3.6, the rate of relaxation from 1 to 2 depends on the distance between t h e i r projections on the V^V^ plane: precisely | V v i | • A simple extension of th i s p r i n c i p l e i s to predict the time constants of relaxations when voltage steps are made from non-steady states, using the manifold and the measured dependence of time constants on J^11 * ^ e ^° n o t t a ^ e i n t o account the contributions of voltage independent transients. Linear regression analysis of the dependence of relaxation time constants on | V 2 _ V i | ^ v) showed that time constants (x) can be closely approximated by (1) 1 = -1.5 AV + 125 (msec) A 60 mV step from V^ to i n Fig.3.6 gives an instantaneous current at 1 that f a l l s toward the steady condition at 2 with a time constant of 35 msec (from equation 1). After one time constant current has decayed to 3 and the membrane potential i s suddenly stepped to 30 mV more positive p o t e n t i a l , V^. Current, i n i t i a l l y at 4, ri s e s to the steady state at 5. The distance between the pro-138 jections of 4 and 5 i n the V^, V plane i s approximately 9 mV. Alte r n a t i v e l y , point 5 could be reached from the stationary point on a trajectory intermediate between and V , 9 mV more negative than . This potential 0^-9) mV could be considered as the effective conditioning voltage i n an experiment i n which the conditioning pulse i s s u f f i c i e n t l y long for steady state current to have been reached. Thus (V^-9) mV may be considered an 'equivalent V j ' for purposes of interpretation of the manifold. Current w i l l r i s e to the steady state with a time constant of -1.5 x 9 + 125 msec = 111 msec Had the step to V' been made d i r e c t l y from V, (a 30 mV step) the / h f a l l i n g transients would have had a time constant of 80 msec. These results are i n quali t a t i v e agreement with experimental observations. Discussion Relationships between steady state and instantaneous current-voltage  relations In the discussion to section 2 of Chapter 2, (Fig.2.2.8), the discrepancy between the steady state current-voltage relations and the family of sigmoid (^(0) versus V^) relations was quantitatively analyzed using the standard formulation: ( 1 ) hi = 8-(Vn-Vcl) I t was observed that t h i s description of chloride conductance was inadequate to relate the steady state conductance to the subsequent 139 conductance (or v i c e versa) i n two pulse voltage clamp experiments. The reason for t h i s i n a b i l i t y to r e l a t e the two conductances was that i t d id not take in to account the s h i f t of the zero current p o t e n t i a l V with c o n d i t i o n i n g voltage V^. In t h i s d i s c u s s i o n , i t w i l l be shown that the steady state r e l a t i o n s and the family of instantaneous current -vo l tage r e l a t i o n s are r e l a t e d , but the r e l a t i o n s h i p i s much more complex than the simple expression (1) . Moreover, i t w i l l be shown that one cannot understand the steady s tate ch lor ide c u r r e n t -voltage c h a r a c t e r i s t i c s without an o v e r a l l comprehensive understanding of a l l the c h a r a c t e r i s t i c s (both instantaneous and steady s t a t e ) . F i g . 3 . 7 shows the s ta t ionary points on a manifold generated by a family of c h a r a c t e r i s t i c s with the slope parameter k=.087, the common value obtained i n a l k a l i n e s o l u t i o n s . The dashed l i n e connecting the s ta t ionary points i s the steady state current -vo l tage r e l a t i o n . I t can be seen that t h i s steady state current -vo l tage r e l a t i o n i s governed by the s h i f t i n g or spacing of the S-curves i n the plane (current vol tage p l a n e ) . When the d i spers ion (or separat ion between the sigmoid curves) increases with i n c r e a s i n g l y negative V2» the steady s tate current -vo l tage r e l a t i o n shows downward concavi ty , as in a c i d s o l u t i o n s . As the d i spers ion decreases with i n c r e a s i n g l y negative V^, sa turat ion or a region of negative slope may be observed i n the current -vo l tage r e l a t i o n s , as i n a l k a l i n e s o l u t i o n s . 140 Figure 3.5 Representation of a recovery experiment on the current-voltage manifold. Since the amplitude of the step from the holding potential to 1 S t n e same as that from V 2 back to the holding p o t e n t i a l , the time constants of relaxation from 1 to 2 and from 3 to 4 should be the same. For d e t a i l s , see text. 141 Figure 3.6 Prediction of instantaneous currents and relaxation time constants from the current-voltage manifold when steps are made during non-steady conditions. An i n i t i a l step from V, to (V, -60 mV) i s held for one time constant h h (35 msec) before the membrane potential i s stepped back to (V -30 mV). The time constant of the relaxation from h 4 to 5 (a stationary point) i s dependent on the distance between their projections on the voltage scale. For d e t a i l s , see text. 1 4 2 Figure 3.7 Relationship between the steady state current-voltage r e l a t i o n and the dispersion of the S-curves. The points on the S-curves are the stationary points and the broken l i n e s joining them are the steady state r e l a t i o n s . When the dispersion increases with increasingly negative V^, as i n the top panel, the steady state current-voltage r e l a t i o n shows downward concavity, as i n acid solutions. With uniform dispersion (center panel) the r e l a t i o n tends toward saturation and as dispersion decreases with increasingly negative (bottom panel) there i s a rapid approach to saturation or possibly a region of negative slope. Note the effect of dispersion on the resting conductance. 1 4 2 a 1 4 3 Chapter IV Theoretical Considerations: Models of Membrane Permeation Process 144 Introduction The purpose of t h i s chapter i s to present a number of simple models of ion movement through membranes to determine i f they share any of the properties that have been observed for chloride permeation. P a r t i c u l a r attention i s given to relations predicted between steady-state currents and voltage and the dependence of conductance changes on time and voltage. As mentioned in Chapter 1 , current-voltage relations that are non-linear and conductance changes depending on v o l -tage and time are the rule rather than the exception i n studies from a wide variety of b i o l o g i c a l membranes. Many different kinds of mechanisms have been proposed to account for steady state r e c t i f i c a t i o n i n membranes. Two of these mechanisms we have mentioned i n Chapter 1 involve changes in the number of conducting channels or i n the conductance of a single channel. On the other hand, satisfactory theories for the voltage dependence of conductance changes for ionic channels in c e l l membranes (for instance, the sodium and potassium channels i n nerve) are lacking, and . t h i s remains one of the least well understood aspects of ion transport across membranes. Of the models that have been proposed for chloride transport i n amphibian fib e r s (Hodgkin and Horowicz, 1959; Spurway, 1970; Venosa, Ruarte, and Horowicz, 1972; Warner, 1972; Woodbury and Miles, 1973) only two types have been considered: c l a s s i c a l e l e c t r o d i f f u s i o n models as exemplified by constant f i e l d theory (Hodgkin and Horowicz, 1959), and channel models (Spurway, 1970; Woodbury and Miles, 1973). Electro-d i f f u s i o n systems have been much studied (Cole, 1968) and their steady-145 state conductance properties are well known (Adrian, 1969; Hope, 1971). Moreover, the pH-dependence of the steady state chloride current-voltage relations cannot be accounted for by electrodiffusion theory i n the manner of i t s formulation i n the past (Hutter and Warner, 1972). But in our view, i t i s of interest to review the behavior of the electro-d i f f u s i o n model with regards to r e c t i f i c a t i o n r a t i o or the dependence of conductance on concentration; to determine whether the chloride permeation system i s mimicked by electrodiffusion i n any aspect other than the steady state I-V r e l a t i o n near n e u t r a l i t y (Hodgkin and Horowicz, 1959; Adrian, 1961; Hutter and Warner, 1972). The chloride system has been seen to exhibit voltage dependent transients and time and voltage depen-dent recovery of conductance after experiencing changes of membrane potential (conditioning steps); the transient properties of constant f i e l d theory under voltage clamp during stationary and non-stationary conditions w i l l be investigated for comparison. The evidence for chloride translocation v i a ionic channels or pores has been reviewed in Chapter 1. Coir aim i n the present chapter i s not to perform an exhaustive analysis of the properties of ionic channels but only to point out some of the more salient features of channel transport and discuss these features i n view of the data on chloride transport. Most of the material i n the chapter i s well known, as our present under-standing of transport processes across membranes involves very simple concepts. I t w i l l be seen that some of the models considered are too idealized and are c l e a r l y inadequate to account for the observed behavior of chloride conductance, but only through the understanding 146 gained from a succession of simple models can we eventually arrive at a model which could perhaps account for our observations. Throughout the discussion, ionic a c t i v i t i e s w i l l be assumed to be equal to concentrations. Because physiological concentrations of 2 chloride are low ( from 3-10 mM/£) and no estimate of chloride a c t i v i t y c o e f f i c i e n t s inside c e l l s i s available, there i s no loss of generality i n setting the chloride a c t i v i t y coefficients at one. Models of membrane permeation process A. Electrodiffusion models Definitions and symbols; t y p i c a l dimensions d = membrane thickness, t y p i c a l dimension 50-80 A°. x = distance coordinate within the membrane, x l i e s between 0 and d. z = valence of the permeating ion. c(x) = concentration of the permeating ion within the membrane phase. Cj . , c^j. are the concentration of ions i n the e x t r a c e l l u l a r and i n t r a c e l l u l a r compartments respectively. Physiological concentrations are c^= 120 mM/& ; = 4 mM/1 u(x) = electrochemical potential of the permeating ion within the membrane phase, u = electrophoretic mobility of the permeating ion within the membrane phase, u has the dimensions cm sec *volt u w i l l be assumed to be related to d i f f u s i o n c o e f f i c i e n t vi a Einstein's r e l a t i o n D = U R T / | Z | F 147 e = d i e l e c t r i c constant of the l i p i d phase of the membrane, approximately (4-5). 3 = oil/water p a r t i t i o n c o e f f i c i e n t . V(x) = potential p r o f i l e within the membrane, i n Volts. V = (V ~Vj) i s the difference in potential between i n t r a c e l l u l a r and extracellular compartments. In experimental situations V can vary between +100 and -200 mV without d i e l e c t r i c breakdown of the membrane. V = zero current potential, o zF/RT : F i s the Faraday; R i s the gas constant and T i s absolute temperature. For z=l, at room temperature (293°K), zF/RT = 39.6 V o l t s " 1 . _2 I ( t ) = transmembrane current density (Amp.cm ) measured i n voltage clamp experiments, t i s usually referenced from the onset of a voltage clamp step. P ^ = chloride permeability c o e f f i c i e n t , defined by p c i - P u c i R T / d l z c i l F » z c i = - 1 • Consider a membrane of thickness d separating two w e l l - s t i r r e d solutions, designated as compartment I and I I . The membrane w i l l be assumed to be the rate l i m i t i n g step for the flow of ions. This flow i s proportional to the gradient of the electro-chemical potential u(x) of the ions. *. The electrochemical potential i s given by the sum of £he e l e c t r i c a l potential zFV(x) and a concentration potential RT*log(c(x)). 148 The transport equations relating the concentration of permeant ions i n the membrane and the potential are obtained using the conser-vation of mass: (1) 3c = 3_• , uRT 8c + 3V . 3t 3x zF 3x 3x and the e l e c t r o s t a t i c (Poisson's) r e l a t i o n : (2) 32V _ 4TT 0 (zFc) 3 x At the boundaries between the aqueous and membrane phases, the concentrations of ions w i l l be assumed, for s i m p l i c i t y , to obey the p a r t i t i o n r e l a t i o n (Hodgkin and Katz,1948): (3) C ( 0 ) = 3 C I C ( d ) = e c l ; [ g i s a p a r t i t i o n c o e f f i c i e n t independent of voltage, c^ and c^j are the concentrations in the ex t r a c e l l u l a r and i n t r a c e l l u l a r compartments. Equations (1-3) are the c l a s s i c a l e l e c t r o - d i f f u s i o n equations for a single permeant ion. I t i s e x p l i c i t that the time scale of the equilibrium at the boundaries i s much shorter than that for d i f f u s i o n within the membrane. In order to integrate equations (1-3), the approximate assumption that has frequently been made i s that the potential gradient throughout the membrane i s approximately constant (Goldman, 1943; Hodgkin and Katz, 1949). The r e l a t i v e l y low d i e l e c t r i c constant of c e l l membranes suggests that the intramembrane 149 concentrat ion of mobile ions would have to be small for the assumption of constant f i e l d to be a good approximation (Adrian, 1969), or e q u i -v a l e n t l y , space charge e f fec t s are n e g l i g i b l e . The transmembrane current that i s measured i n voltage clamp experiments i s the average current : d (4) I ( t ) = ^ l f ( ^ T | c u c | V d x d / zF 8x 3x 0 It can be expressed as a sum of currents due to d i f f u s i o n and migrat ion: d c(x)dx (5) I ( t ) = 3uRT( ° I I C I ) + zFu -d d Z J 0 V i s the transmembrane vo l tage . Steady state proper t i e s The steady state proper t i e s of constant f i e l d theory have been w e l l reviewed (Adrian, 1969; Hope, 1971). At a given t r a n s -membrane p o t e n t i a l V, there i s a steady s tate concentrat ion p r o f i l e (maintained by the energy stored i n the e l e c t r i c f i e l d ) given by FV/RT_ (6) c(x) = r ° l e ° I I , r C I I " ° I v FVx/RTd , P 1 FV/RT . V FV/RT / 1 e - 1 e - 1 This i s i l l u s t r a t e d in F i g . 4 . l a , b for various values of V . The membrane conductance according to equation 5, i s the average charge of ions wi th in the membrane phase. The steady s tate c u r r e n t - v o l t a g e r e l a t i o n 150 Figure 4.1 a) and (b) are concentration p r o f i l e s of permeant anion within the membrane for various values of the membrane potential, shown on the relations. In (a) the resting (zero current) potential i s -90 mV and i n (b) i t i s -30 mV. The scales are normalized so that at the extracellular boundary the concentration i s 1. c) shows current-voltage relations for different zero current potentials (shown on the r e l a t i o n s ) . The dotted l i n e s represent the independent inward (below the zero current axis) and outward (above the axis) currents. The s o l i d l i n e s are t o t a l currents obtained by the addition of the inward and outward currents. d) The r e c t i f i c a t i o n r a t i o , defined by equation 10, in constant f i e l d e l e c t r o d i f f u s i o n theory . The numbers on the l i n e s indicate the zero current potentials. e) , (f) and (g) are current-voltage relations of modified constant f i e l d theory incorporating surface charge (equation 11). In (e) the resting potential i s -90 mV, in (f) i t i s -30 mV and i n (g) i t i s 0 mV. The (outside) surface charge, i n mV, i s indicated on each r e l a t i o n . 1 5 0 a 1 5 1 -VF/RT P F 2 V 1 - ( C I I / C I ) 6 ( 7 ) I(V) = - — cz -^j^ obeys the independence p r i n c i p l e : Ionic current inwards (anion e f f lux) I . (V) = + P F 2 V e - ™ ' * 1 m v ™ — c RT II , -VF/RT 1 - e and outwards (anion in f lux ) I fc (V) = - P F 2 V out — — c RT I , -VF/RT 1 - e are independent. A s y m p t o t i c a l l y , for large transmembrane p o t e n t i a l s , the current i s p r o p o r t i o n a l to vol tage: T ~ T = p p 2 l n RT C I I V V F / R T < < 0 (8) I = T = - P F 2 out — c V VF/RT » 0 This property of l i m i t i n g conductances being p r o p o r t i o n a l to the i n t e r n a l and externa l concentrat ions of permeating ions w i l l be r e f e r r e d to as ' r e c t i f i c a t i o n of the e l e c t r o d i f f u s i o n t y p e ' . The terminology w i l l a lso be a p p l i e d to channels . An experimental s i t u a t i o n of s p e c i a l i n t e r e s t to us i s when the ex terna l c h l o r i d e concentrat ion i s maintained while i t s i n t e r n a l concentrat ion i s v a r i e d . By the independence p r i n c i p l e , only the outward movement of c h l o r i d e i s a f f e c t e d . The r e s u l t a n t c u r r e n t -152 voltage relations are shown i n Fig.4.1c Linearization of the current-voltage relations with decreasing concentration gradients i s a characteristic of ' r e c t i f i c a t i o n of the electrodiffusion type'. An alternate quantitative method of studying the nature of r e c t i f i c a t i o n i s to investigate the r e c t i f i c a t i o n r a t i o (Lauger, 1973) defined as the r a t i o of the slope conductance at a given membrane potential V to the zero current potential V Q : — ( V ) (9) X(V) = % — ( V ) dV^ oJ Experimentally, t h i s i s a useful device in comparing experiments from different c e l l s that exhibit large variations in current density. Moreover, i f there are uncertainties i n the c a l i b r a t i o n of membrane current densities because of uncertainties i n cable parameters such as the internal r e s i s t i v i t y (see, for instance, equation 1, chapter i ) , then they are removed as the r e c t i f i c a t i o n r a t i o i s independent of cable parameters. In the chloride system in Xenopus, we find experimen-t a l l y that the r e c t i f i c a t i o n r a t i o i s independent of the resting potential as X(V) for both polarized and depolarized f i b e r s are sim i l a r . However, i t i s highly dependent on external pH: X(V) i s greater than 1 in acid solutions (downwards concave) in the hyperpolarizing range, and i s less than 1 i n alkalin e solutions (upwards concave). The expression for X(V) from constant f i e l d e l e c t r o d i f f u s i o n theory 153 (10) X(V) = (-!)(! - e - F V o / R V { ( x _ ^ - V ) ^ _ (FV / R T ) e - F V o / R T ( l - e " F V / R T ) o (FV/RT) e " F V / R T ( 1 - e F V o / R T ) , ( 1 - e " ^ 1 ) It can be seen from Fig.4.Id that A(V) i s always less than one when the internal chloride concentration i s less than the external. Hence i t c l e a r l y f a i l s to account for experimental observations. Electrodiffusion theory may be modified to change the r e c t i f i c a t i o n r a t i o at a given resting potential. An e x p l i c i t assumption of the model i s that the rates for entry into the membrane are much :greatertthan the membrane limited d i f f u s i o n . Hutter and Warner (1972) have attempted to adapt Frankenhaeuser 1s (1960) modification of the constant f i e l d equation incorporating surface charges to account for constant f i e l d r e c t i f i c a t i o n at different pH's. Here, the steady state current voltage r e l a t i o n i s modified by a surface voltage term V : s (11) I(V)=- PF (V-V )c { — } RT S . - ( V - V )F/RT I — e s The direc t i o n of the current voltage r e c t i f i c a t i o n depends on the difference between potential V G and the zero current potential V q= (RT/F) l o g g ( c I I / c I ) (Frankenhaeuser , 1960; Adrian, 1969; Hutter and Warner, 1972). Asymptotically, 154 (12) I(V) - -||i(V-V s) FV/RT « 0 RT x 1 I(V) = -PF2(V-V ) c T e F V s / R T FV/RT » 0 — s l RT If (V -V ) <0, r e c t i f i c a t i o n i s i n the direction that the conductance o s for outward current i s greater than for inward current. If ( v 0~V s)= 0, there i s no r e c t i f i c a t i o n (the current-voltage relations are l i n e a r ) . I f (V -V ) >0, r e c t i f i c a t i o n i s i n the direction that the conductance o s for inward current i s greater than the conductance for outward current. I t i s seen (Fig.4.l e,f,and g) that the direction of the r e c t i f i c a t i o n changes depending on the surface potential. However, the current-voltage characteristics do not cross or intersect each other i f the conductances are appropriately scaled so that the effects of pH on the resting conductance are taken into account (Hutter and Warner, 1972) . It i s also noteworthy that the current-voltage relations are always conductance limited (equation 12), and therefore cannot account for observations i n alkalin e solutions, where saturation and negative slope conductance i s sometimes observed. An extension of ele c t r o d i f f u s i o n model An alternate formulation which we have studied i s the generalization of the Nernst-Planck equations (equations 1-3) by the introduction of an invariant barrier function w(x) into the expression for the 155 e lec trochemica l p o t e n t i a l u(x) ( H a l l , Mead and Szabo, 1973; Neumcke and Lauger, 1969): w(x) represents the energetic cost of moving an ion from place to place ins ide the membrane. The form of such a b a r r i e r funct ion can e i ther be c a l c u l a t e d from the energy required to move an ion from a region of high d i e l e c t r i c constant to one of low d i e l e c t r i c constant (Neumcke and Lauger, 1969) or i t can be e m p i r i c a l l y f i t t e d from a study of current -vo l tage c h a r a c t e r i s t i c s ( H a l l , Mead and Szabo, The steady state current -vo l tage r e l a t i o n i s obtained by i n t e -grat ing the current r e l a t i o n I(x) = -zFuc(x)vy(x) The subscr ip t s I and II denote the values of the parameters c, V, and w in the compartments I and II r e s p e c t i v e l y . The I -V r e l a t i o n (equation 14) was studied assuming d i f f e r e n t p o t e n t i a l energy p r o f i l e s w(x) ( t r i a n g u l a r , t r a p e z o i d a l , parabo l i c ) and var ious asymmetric p r o f i l e s that may represent membrane s i t e s which c h l o r i d e might i n t e r a c t with as i t crosses the membrane. Our conclusion.(assuming constant e l e c t r i c f i e l d wi th in the membrane) i s that the current -vo l tage r e l a t i o n s are always conductance l i m i t e d i n an asymptot ical fash ion , regardless of the p o t e n t i a l energy p r o f i l e v ( x ) . The transmembrane e l e c t r i c f i e l d (13) y ( x ) = RT log(c (x) ) + zFV(x) + w(x) 1973). (14) I(V) = - | z | F u 0 156 w i l l ultimately be the dominant force and the l i m i t i n g conductances are proportional to the internal and external concentrations of the permeating ions. The current-voltage relations are highly dependent on the ionic concentration gradients across the membrane and the I-V r e l a t i o n i s the sum of two effects: one due to concentration gradients across the membrane ('electrodiffusion' effect) and the other to the asymmetric barrier function. Transient properties When a test voltage V^ i s applied across a membrane previously i n a steady state under conditioning voltage V^, the concentration of permeant ions within the membrane w i l l redistribute to a new e q u i l i -brium p r o f i l e . The time course of this r e d i s t r i b u t i o n i s obtained from the solution of the i n i t i a l value problem: 3c ,uRT.,32c , FV 3c, ^ _ 3t" = ( ^ F ~ ) { ~ 2 + ZRT 3x"} 0 < X < d OA (15) c(x,0) = c o(x) c(0,t) = Bc T , c(d,t) = Bc i ; t c (x) i s the i n i t i a l concentration p r o f i l e p rior to the onset o of the voltage at t=0, and i s given by equation (6) with V replaced by V^ i f the membrane i s i n i t i a l l y i n a steady state. Equation (15) describes d i f f u s i o n i n a constant force f i e l d or d i f f u s i o n with a constant d r i f t v e l o c i t y . The formal solution can be obtained by many standard methods. The relaxation spectrum i s most e a s i l y obtained from the eigenvalue problem associated with equation (15) by separat ion of v a r i a b l e s . The so lu t ion , wr i t ten as a sum of a steady s tate c e o ( x ) and a t rans ient term c t r ( x , t ) i s ; (16) c ( x , t ) = C o o (x) + c t r ( x , t ) F V / R T c ( x ) - 3 " C H + 3 C H " C I FV x/RTd F V 2 / R T _ l e F V 2 / R T _ x (17) c t r ( x , t ) = y A n s i n n*(x/d) e~ n C A ' = (1/2) I e z F V 2 x / R T d (c (x) - c (x)) s i n mr( x /d )dx n / 0 0 o 0 (18) X 2 = ( n i r ) 2 + < z F V R T ) 2 D , „ > i n ^ : —2 a It i s we l l known that the p r i n c i p a l c h a r a c t e r i s t i c time constant (n=l, equation 18) of d i f f u s i o n with no d r i f t i n a f i n i t e domain i s 2 2 4d / IT D . Under the inf luence of d r i f t (the imposed transmembrane p o t e n t i a l V^), the c h a r a c t e r i s t i c time constant i s modif ied to 2 2 2 4d / ( TT + (FV 2 /RT) )D . The macroscopic current measured i n vo l tage clamp experiments, defined by equation (5), i s then expressed as: (19) I ( t ) = 1 + 1 ( t ) , where oo t r d d - I I -I> * " — 1 n"RT (20) I - B r ^ T T " CT> + z F u - ~ I c e o (x)dx d 2 I (21) I (t) = z F u - § d i n=l 00 f V _ x2 A e n s i n nir(x/d)dx n 158 The conclusion that can be drawn from equation (21) i s that the ki n e t i c s of current relaxations depend only on the absolute membrane potential, and are independent of the i n i t i a l state (that i s , the conditioning potential v ^ ) -The p r i n c i p a l time constant for chloride may be estimated from P values obtained from steady state current-voltage r e l a t i o n (7). p n = ~f^T = 4 - D • Therefore CI d zF d T = Ad 2 B _ _ TT2 + (FV 2/RT) 2 Pd If d = 60 A° P c l = 8.0 X 10~ 6 cm sec' 1 (Table I I . 1.1) V 2 = -90 mV B = 10" 5 then x = 1.3 X 10~ 7 sec The above estimate of the time constant (T) may not be v a l i d since i t depends on a p r i o r i knowledge of the p a r t i t i o n c o e f f i c i e n t 3 , a value d i f f i c u l t to obtain for c e l l membranes. However, an alternate estimate , of the r a t i o of the p r i n c i p l e time constants at V=0 (T q) and V= -90 mV ( t _ 9 0 ) , i s independent of B : . T 4d 2/(7r 2 + (3.8) 2) = . 64 T ( ) 4 d 2 / ( T T 2 ) This implies that i f chloride transport were to be described be el e c t r o d i f f u s i o n , as has been shown i n neutral solutions by many authors (Hodgkin and Horowicz, 1959; Adrian, 1961; Adrian and 159 Freygang, 1962a.; H a r r i s , 1963, 1965; Hutter and Warner, 1972), then the time constants for current decay for small hyperpo lar i z ing pulses from the holding p o t e n t i a l in p o l a r i z e d and depolar ized f i b e r s would d i f f e r by a fac tor of 2. This i s not observed in experiments in Xenopus l a e v i s . Instantaneous currents and a f t e r c u r r e n t s The expression for the instantaneous current I ( V 2 > 0 ) corresponding to a test vo l tage appl i ed to a membrane which had been i n a steady state corresponding to a cond i t ion ing pulse Vj i s : (22) I ( V 2 , 0 ) = guRT ( cIT - C j . ) + zFuV 2 j c ( x , V 1 , « > ) dx d d 2 C ( x , V 1 , c o ) Jo This instantaneous current i s an a f f i n e funct ion of the tes t vol tage <v 2). On re turn to the zero current p o t e n t i a l V q a f t er an exploratory test step V j , there i s an a f t e r - c u r r e n t due to the r e s t o r a t i o n of the concentrat ion p r o f i l e to the e q u i l i b r i u m p o s i t i o n . Th i s t rans ient a f t e r c u r r e n t , given by < 2 3 ) X t r ( t ) = TVolZ n=l - J r 2 (c- . (V. ,x) - c (V ,x)) e ^ n ^ o ^ s in nir(x/d)dx U I oo o 0 i s the mirror image of the decay of the current s t a r t i n g from the zero current p o t e n t i a l except that the d r i v i n g force i s V ( 4- 0) . The magnitude of the s h i f t in e q u i l i b r i u m p o t e n t i a l V ^ ^ j . can be 160 obtained by setting the instantaneous current to be zero: j d (0) = 0 = BuRT d (c II 2 s h i f t c(x,V ,«?) dx 0 to obtain (24) V s h i f t RT zF )dx The s h i f t of zero current p o t e n t i a l , V , i s proportional to the concentration differences across the membrane. Hence i n the case of decreasing concentration gradients across the membrane (as in depolarized f i b e r s ) , the magnitudes of the s h i f t in zero-current Some aspects of electrodiffusion and experimental results can be related while others cannot. When a voltage i s applied to a membrane in a steady state, the instantaneous current i s proportional to the applied voltage. Current transients then relax to the steady state with i n f i n i t e l y many time constants depending on the absolute membrane potential. The steady state current-voltage r e l a t i o n always exhibits a l i m i t i n g conductance that depends on the in t e r n a l and external ionic concentrations. On return to the zero current potential after a test pulse, there i s an aftercurrent opposite i n di r e c t i o n to the test current. The magnitude of t h i s aftercurrent depends l i n e a r l y on the difference between the ionic concentration between the two sides of potential should disappear. Experimentally, this was not observed. Summary 1 6 1 the membrane. Kinetics of current relaxations depend absolutely on membrane potential even when the i n i t i a l conditions are not steady. B. Channel or pore models Apart from a geometric or structural description, a channel may be characterized by a series of potential energy barriers. When an ion enters a channel, i t undergoes a series of random walks or discrete jumps over successive potential barriers. The forward and backward jump rates over any barrier depend on the e l e c t r i c f i e l d within the membrane. When the transmembrane voltage i s changed, there are current relaxations that depend on the relaxation of the potential p r o f i l e within a single channel to a new equilibrium. The time scales for these processes are very fas t . If the jump rates are also time dependent, since they are probably due to macromolecular movements within the membrane, they are expected to be slower. The i n i t i a l current-voltage manifold may be used to i l l u s t r a t e a possible relationship between the above and experimental results as follows: each point (current density) i n the manifold i s described by the number of open channels and the conductance of a single channel (for s i m p l i c i t y , channels w i l l be assumed either 'open' or 'closed'). Coordinates with the same conditioning potential have the same number of open channels and those with the same have the same conductance for each channel. Starting from a stationary state, when a voltage across the membrane i s altered to a new value, the i n i t i a l current depends on the activating 162 barrier p r o f i l e at the previous steady state. Since the relaxation of the conductance of a single channel i s very fast (faster than the resolution of our experiments), the instantaneous current measured i s a product of the number of open channels during steady and the single channel conductance at V 2. In time, the current undergoes a relaxation due to changes in the number of open channels. Alternatively, the current transients may be due to a time dependent molecular relaxation within the channel and consequently the conductance of a single channel i s changed. The purpose of t h i s section i s to explore some qualitative predictions of the ideas expressed above. The f i r s t objective of the present section i s to describe the general formulation for a pore model. General formulation for a pore model 2 I t i s assumed that the membrane contains N i d e n t i c a l pores/cm . The pores are independent (do not interact with each other), and only ions of the same sign (anions) are able to enter the pore. A potential energy diagram for the sit u a t i o n of an ion d i f f u s i o n across a pore i s th shown i n Fig.A.2. The rate constants for the jump from the n minimum to the right and to the l e f t i s denoted by and k . According to the Theory of Absolute Rates (Glasstone, Laidler and Eyring, 1941) the following expressions for the rate constants are obtained: (25) k n - <n(kT/h) e x p ( - ( f — - f — ) ) n m 0 t l t 2,...n. k_n= l n ( k T / h ) . e x p ( - ( f ^ X - f f n ) ) 1 6 3 Figure 4.2 A potential energy diagram for d i f f u s i o n within an ionic channel. Energy i s plotted v e r t i c a l l y and distance horizontally. Symbols are as defined i n the text. 164 min max f , and f n are the non-dimensionalized free energies (free energy th divided by RT) i n the n minimum and at the top of the barrier to the right of the n1"*1 minimum, k, T , R, and h are the Boltzmann constant, absolute temperature, the gas constant and Planck's constant respectively. The term (kT/h), which i s an attempt rate (Stephens, 1978) i s 6.2 X 12 X o 10 sec (298 K) and has the dimensions of a frequency, the trans-l a t i o n a l v i b r a t i o n a l frequency of the activated complex (Glasstone, Laidler and Eyring, 1941). The transmission coefficient K and K n -n w i l l be set equal to one. The * j ' s (Eig-4.2) may be thought of as the mean jump distances for the ion. I f c. i s the concentration of ions per cubic centimeter 3 at the j minimum, then N., the number of molecules which are candi-3 dates to jump across the barrier i n a square centimeter of area normal to the direc t i o n of d i f f u s i o n , i s N.= X.c. . 3 3 3 In the derivation of the equations, the pore w i l l be assumed to contain no more than one ion. However, even though the model i s based on single occupancy, i t i s not a r e s t r i c t i v e assumption as the an a l y t i c a l expressions derived reduce to the si t u a t i o n when more than one ion i s allowed inside the pore. Moreover by studying the model under what appears i n i t i a l l y as unnecessarily r e s t r i c t i v e assumptions, the required conditions for possible saturation and blocking effects can be derived. The pore I s assumed to be accessible to two ionic species A,and B. Let A. and B. represent states of the channel with ionic species A and B at the j minimum (j = l , 2,.. n). The state 0 represents the channel when 165 A B A B i t i s free of an occupied ion. k , k. , k . and k . are the forward n J -J -3 and backward rate constants of A and B respectively at the minimum. A B Let Pj and p^ . be the pr o b a b i l i t i e s that a single channel i s occupied by A or B at the j 1 " * 1 minimum ( 1 ^n) . p^ i s the probability that the channel i s unoccupied, and i s equal to (26a) p Q = ( 1 " X ^ P j + P j } } N Q , the number of unoccupied channels per square cm i s (26b) N Q = Np Q The outer or inner minimum (at positions 0 and n+1, Fig.4.2) w i l l not be considered as pore positions, and we assume that the concentrations N^ . and N i n these positions to be proportional to the bulk concentrations c^ . and c : N I = V C I , N I I = V C I I The constant of proportionality V i s equal to the volume from which an ion may enter the pore i n a single jump (Lauger, 1973). I t can be expressed as the product (v=Xa) of the distance X between the f i r s t energy b a r r i e r i n the outer (or inner) mouth of the pore and the entrance bar r i e r to the membrane, and a , the ef f e c t i v e cross sectional area of the pore (Lauger, 1973). A l l the transitions and states of the channel are connected by the diagram shown i n Fig.4.3 where the rate constants have equivalent interpretations as conditional p r o b a b i l i t i e s . 1 6 6 For the present discussion, the rate constants k . are assumed to be independent of time but depend on the e l e c t r i c f i e l d within A B the membrane. N and N^. , the number of channels per square centimeter with ions A or B i n the j 1 " * 1 minimum.are = Np^ and = Np^ and 3 J 3 3 also depend on the membrane potential. When the transmembrane potential i s changed, as i n voltage clamp experiments, the occupational densities A B ,A B p_. and p_. and consequently N? and N_. relax to new stationary values depending on the membrane potential. The time evolution of t h i s relaxation i s described by the matrix equation: (27) dt — — PQ A p l A P l A P„ n B P l B P2 ' B P„ n ,. A,. A . B . B ( k 0 + k - ( n + l ) + k0 + k-(n+l) A -(n+1) B 0 B C-(n+l) -1 -(k^+kj) k^2 , A A —3 3 * * • The rows are not independent since . . . A, . . . A , (28) > Q(t) + p*(t) + ... + p n ( t ) + P l ( t ) + . + P°(t) - 1 1 6 7 The matrix equation (27) i s the Chapman-Kolmogorov equation for a temporally homogeneous and s p a t i a l l y inhomogeneous b i r t h and death process. The equations for the fluxes J\ of A across the potential energy ..max ,„ . , maximum f (0$ n) are: A , A , A A J o = A o k o p o - * i k - i p i A , A A . A A J ! = M k l P l " X2 k-2 P2 (29) J n = An kn Pn " * (n+l) k-(n+l) P0 There are similar equations for B. Steady state properties The steady state current-voltage relations for a single ionic species can be obtained either by solving the flux equations (29) by successive elimination (Lauger, 1973) or by graph theoretic means (King and Altman, 1956; H i l l , 1966; Heckmann et a l , 1972; Macey and Oliver, 1967). For our purposes, the r e l a t i o n as f i r s t expressed by Lauger (1973) i s adequate: ( C - K C ) I = zFNvk, 0 n n + 1 ~ , V C I Q I + V C H K Q I I k . -3 i + 7 s . + J- k J £ = 1 K £ 1 6 8 Figure 4.3 A schematic diagram i l l u s t r a t i n g a l l the states of a channel which permits the entrance of one of two ionic species A or B into the channel but only one ion i s allowed to occupy a channel at one time. 169 R. -tr k,i-:< n j QX - 2 sjII R u j= l u=l n n j - 1 J = l 3=2 u=l The I-V r e l a t i o n (30) was der ived on the r e s t r i c t i v e assumption of s i n g l e occupancy. However, i n the event that most of the pores are empty of ions , that i s , n (32) v (c Q + c Q K ) « (1 + £ s ) j = l the r e l a t i o n reduces to C I " K C I I (33) I = NzFvk Q n -Equation (33) was f i r s t derived by Zwol insk i , Eyr ing and Reese (1949) and was f i r s t extended to equation (30) by Lauger (1973). Spec ia l cases of the equation for s p e c i f i c p r o f i l e s of the rate constants K^'s have been extens ive ly analyzed (Johnson , Eyr ing and P o l i s s a r , 1954; P a r l i n and E y r i n g , 1954; Woodbury, 1971). A l t e r n a t i v e l y , current -vo l tage r e l a t i o n s may be obtained i n the case of i n t e r a c t i o n s of ions wi th in channels (Hladky, 1965; Hladky and H a r r i s , 1967) or s ing le f i l e d i f f u s i o n of ions wi th in channels (Sandblom et a l , 1977; Eisenman et a l , 1976, 1978). Recent ly , evidence (French and Adelman, J r . , 1976) has been gathered for m u l t i p l e occupancy and in ter ference of ions i n the 170 sodium ( H i l l e et a l , 1973; Woodhull, 1973; H i l l e , 1975b; Cahalan and Begenisich,1976)and potassium channels i n nerve and muscle (Cahalan and Armstrong, 1972;Gay and Stanfield, 1977; H i l l e and Schwarz, 1978) and in compounds that form channels (Gramicidin A) i n l i p i d bilayers (Bamberg and Lauger, 1977; Sandblom et a l , 1977; Eisenman et a l , 1978; Schragina et a l , 1978). At the present , there i s some evidence supporting the p o s s i b i l i t y of single occupancy of chloride channels, as w i l l be discussed in the following section. However, to consider possible multiple occupancy or interactions within the chloride channel would be premature. Specifications for a possible chloride channel In the preceding section, the equations for a channel model were derived on very general assumptions. Single occupancy of ionic channels was considered because of the possible relevance to the chloride channel. I t i s plausible that the interference of chloride movement by Br , I , NO^  , and SCN may be due to the occupancy of a s i t e within the channel excluding the entrance or passage of chloride. In addition the independence of resting conductance on inte r n a l chloride concentration could also be due to a saturation of the channels by chloride even at very low concentrations. This would be the case i f the chloride channel densities within the membrane are low (equation 32). A f i r s t approximation f o r a simple model of a chloride channel i s a symmetric pore with i d e n t i c a l i n t e r n a l barriers but diff e r e n t surface barriers. There are several reasons why these approximations 171 might be pertinent: Woodbury and Miles (1972) have suggested that there i s only one rate l i m i t i n g step at the outer surface (Parker and Woodbury, 1976) of the membrane i n the chloride channel. The rate l i m i t i n g step at the surface w i l l be represented by a surface barrier on the external side of the channel and the i n t e r i o r step in jumping from the cytoplasm into the membrane i s also represented by a surface barrier. Since the available experimental evidence on the internal pH has shown that i t i s f a i r l y constant ( H i l l , 1955; Caldwell, 1956, 1958; Kostyuk and Sorokina, 1960; Waddel and Bates, 1969) while the external pH and transmembrane potential can have large variations, one of the effects of changing the external pH could be a change of the external barrier potential to anion permeation. The near 'constant f i e l d l i k e ' current-voltage relations at neutral solutions suggests that perhaps the channel might exhibit some form of regularity with respect to the in t e r n a l potential energy p r o f i l e s . That i s , under some conditions, the model exhibits constant f i e l d behavior. Hence the model envisages n potential energy minima equally spaced within the membrane. The distance between successive maximum and minimum i s denoted by l/2(n+l). Let v = FV/RT be the normalized transmembrane potential and v = v/(n+l) • The voltage dependence of the rate constants k^ and k n i s a function of the e l e c t r i c f i e l d within the membrane. The simplifying assumption of a constant e l e c t r i c f i e l d w i l l be used. Then the voltage dependence of the rate constants are expressed i n the form: 172 k = k ° exp(v/2(n+l)) = k ° exp(v /2 ) (34) n 3 J k_ n = k ° exp(-v/2(n+l)) = k ° exp( -v /2 ) Notice that with h y p e r p o l a r i z a t i o n , the backward rate constants (rate constants for c h l o r i d e e f f lux) increase . The voltage independent terms k ° = (kT/h) e x p ( - ( f m a x - f m i n ) ) = (kT/h) e x p ( - ( f m a X - f m ± n ) ) (35) 3 2 m 1<J*P k°_. = (kT/h) e x p ( - ( f m a x - f m i n ) ) = (kT/h) e x p ( - ( f m a X - f m i n ) ) At the inner and outer mouths of the channel: (36) k Q = k ° , e x p ( v / 2 ) k - ( n + l ) = k - ( n + l ) e x p ( - ^ 2 ) The non-voltage dependent terms k ° and k ° ( n + j j are expressed as: k ° = (kT/h) exp(—f ) (There i s no l o s s of g e n e r a l i t y (37) in assuming that the free energies i n th zero) When the channels are mostly unoccupied, the current -vo l tage r e l a t i o n (33) becomes: = ( k T / h ) e x p ( - f 1 1 )  e compartments I and II to be (38) I - M * ° c I ( 1 - l . ^ ) . * " ) e V / 2 , .\ £maXv r i i v j . 2v , 3v (n-1 v. -(f -f ), [ 1 + (e + e + e + . . + e ) e -t-.SL I I . -max nv - ( f - f ) f e e e ] 173 The dependence of the current -vo l tage r e l a t i o n (38) on the number of i n t e r n a l b a r r i e r s , the concentrat ion of the permeant anion, and on the d i f f erences between f , f and f i s shown i n F i g . 4 .4. F i g . 4 . 4 a shows the l i n e a r i z a t i o n e f fect of increas ing the number of i n t e r n a l b a r r i e r s : the current -vo l tage r e l a t i o n s are more l i n e a r for a greater p o l a r i z i n g range (Woodbury, 1971; Lauger, 1973). The current -vo l tage r e l a t i o n s r e c t i f y e i t h e r inwards or outwards depending on the d i f f erence between the a c t i v a t i o n energies of the surface step and the i n t e r n a l p o t e n t i a l energy maxima. T h i s i s i l l u s t r a t e d i n F i g . 4 . 4 b for a channel with symmetrical i n t e r n a l (5 i n t e r n a l minima) and surface b a r r i e r s (inner and outer a c t i v a t i o n energies are equal ) . The degree of r e c t i f i c a t i o n i s dependent on a s ing le parameter, which i s expressed as a func t ion of the d i f f erence between the surface a c t i v a t i n g energy and the a c t i v a t i n g emergy of any one of the i n t e r n a l maxima ^max (39) q = e x p ( - ( f T - f m a X ) ) I ID. 3.X When q >1 (f < f ) , i e . i n t e r n a l l y ra te l i m i t e d , the current -vo l tage r e l a t i o n bends downwards for hyperpo lar i za t ions and upwards for I nicLx d e p o l a r i z a t i o n s . Conversely when q <1 (f > f , the rate l i m i t i n g Step i s at the s u r f a c e ) , the current -vo l tage curves are concave upwards for hyperpo lar i za t ions and bend towards the vol tage ax i s with d e p o l a r i z a t i o n . In the extrme case when q = 0, the surface rate l i m i t i n g step becomes completely dominant. The r e c t i f i c a t i o n reaches a l i m i t i n g degree as the r a t i o q approaches zero. In F i g . 4 . 4 b these r e c t i f i c a t i o n s are not complicated by a concentrat ion component because the r e s t i n g 174 or zero current p o t e n t i a l i s zero . The dependence of the current -vo l tage r e c t i f i c a t i o n on concen-t r a t i o n gradients across the membrane i s shown in F i g . 4 .4c . The l i n e a r current -vo l tage r e l a t i o n of F i g . 4 . 4 b becomes a constant f i e l d r e c t i f i c a t i o n . Depending on whether the surface or i n t e r i o r of the membrane i s the rate l i m i t i n g s tep, the e f fec t of concentrat ion v a r i e s ( F i g . 4 . 4 d and 4.4e) . When the zero current p o t e n t i a l (or concentrat ion gradient across the membrane) i s decreased i n the s i t u a t i o n where the i n t e r i o r i s the rate l i m i t i n g step, the degree of r e c t i f i c a t i o n increases . In contras t , when the surface i s the rate l i m i t i n g step, decreasing the concentrat ion gradient of the permeant ion g r e a t l y l i n e a r i z e s the I-V r e l a t i o n s . In F i g . 4 . 4 d and 4.4e the current -vo l tage c h a r a c t e r i s t i c s are nor -malized so that the chord conductances at the r e s t i n g p o t e n t i a l are equal . In F i g . 4 . 4 f the e f fect of changing the surface p o t e n t i a l f ^ on the ins ide mouth of the channel i s shown. Here, the energy of the f i r s t maximum i s decreased. It can be seen that the hyperpo lar i z ing current i s grea t ly increased . Instead of the l i n e a r current -vo l tage r e l a t i o n when q = 1.0, i t becomes downwards concave whereas i n the d e p o l a r i z a t i o n d i r e c t i o n , there i s no change in the c h a r a c t e r i s t i c s . V a r i a t i o n s in the number of conducting channels Steady s tate current -vo l tage r e l a t i o n s measured from a c o l l e c t i o n of channels could be complicated by vol tage dependence of the number of conducting channels . 175 In t h i s s e c t i o n , the current-vo l tage r e l a t i o n s of some cases w i l l be examined. The current-vo l tage r e l a t i o n i s i n the form: (AO) I ( V , » ) = n ( V , « . ) I (V,oo) where n(V,oo) i s the dependence of the number of open channels on vol tage ( in the steady state) and I ^ ^ ' 0 0 ) i s the steady state current -vo l tage r e l a t i o n of a s ing l e channel . The a n a l y s i s of the steady s tate r e l a t i o n s a lso requires a hypothesis on the dependence of the channel parameters on pH to account for the pH dependence of the steady s tate current -vo l tage r e c t i f i c a t i o n . Moreover, i t i s c l e a r that reduct ion of s ing le channel conductances by a constant amount that i s pH dependent or reduct ion of the number of channels by a constant amount with pH i s inadequate to account f o r the degree of r e c t i f i c a t i o n s ince they would on ly y i e l d f a m i l i e s of p a r a l l e l c h a r a c t e r i s t i c s . Parker and Woodbury (1976) have reported a s h i f t of the S-curves of r e s t i n g conductance with pH as a funct ion of r e s t i n g p o t e n t i a l , and have suggested that a t i t r a t a b l e group located about one-quarter of the distance from the outer mouth of the channel c o n t r o l s r e s t i n g conductance. The equation: 1 (Ala) g(V) = -(pH-pK) 1 + 10 exp(zFV/A.RT) s a t i s f i e s t h i s c o n d i t i o n ( F i g . A . 5 a , b ) . An i n t e r p r e t a t i o n i s that such a group d i s t r i b u t e s two s tates to c o n t r o l the open and c losed 176 states of the channel . At a f ixed membrane p o t e n t i a l , changing the pH s h i f t s the h a l f - p o t e n t i a l of the d i s t r i b u t i o n . In order to obta in the required current-vo l tage r e l a t i o n s , a knowledge of the d r i v i n g force i s necessary. In the absence of such knowledge, the s implest assumption i s that current i s obtained from conductance by the m u l t i p l i c a t i o n of a l i n e a r f a c t o r i z ^ ^ - V ^ ) . Then (41b) I = g_ n o ( V - V c l ) g ( V ) where g(V - V _ . ) i s the conductance of a s i n g l e channel and n i s the t o t a l °- m CI o number of channels . V ^ i s zero current p o t e n t i a l . Current -vo l tage r e l a t i o n s are shown i n F i g . 4 . 5 c for pH 5 and 8. These r e l a t i o n s are always upwards concave at the zero current p o t e n t i a l v c r I f the exponent ia l term i s made to increase more r a p i d l y with vo l tage (eg. exp(zFV/2RT rather than exp(zfV/4RT)) , a sa turat ing current -vo l tage r e l a t i o n , or one that e x h i b i t s negative s lope, i s generated ( F i g . 4 . 5 d ) . The model us ing the e m p i r i c a l expression (41a) c l e a r l y cannot account for c h l o r i d e currents because the I-V r e l a t i o n s pred ic ted are not downwards concave (inward r e c t i f i c a t i o n ) at any pH. The r e l a t i o n s of F i g . 4 . 5 c and 4.5d were obtained assuming that a s i n g l e channel conductance i s p r o p o r t i o n a l to (V -V •.) (g_ i s the constant of p r o p o r t i o n a l i t y ) . in L » I We have studied the model us ing the more complex expressions f o r s ing le channel conductance def ined by equation 38 and i l l u s t r a t e d in F i g . 4 . 4 . Since the asymptotic behavior of the I -V r e l a t i o n s i s governed by the exponent ia l express ion i n g(V) (equation 41a), i n order for 177 inward r e c t i f i c a t i o n to occur (as i n ac id so lut ions) the growth rate of the s ing l e channel conductance (^^(V)) with voltage needs to be greater than t h i s exponent ia l . Th i s i s implausible^based on the model of channel behavior expressed by equation (38) and i l l u s t r a t e d in F i g . 4 . A . An a l t e r n a t e hypothesis , suggested by the curves of F i g . A . 5 d i s that as the pH of the external s o l u t i o n i s lowered, the groups do not sense as much of the e l e c t r i c f i e l d as they do i n more a l k a l i n e so lu t ions . I t i s even conceivable that the group changes valence and sign as a funct ion of the l o c a l pH. Such an empir ica l r e l a t i o n s h i p between current and vol tage i s : 1 (42a) I = g • (Vm - y c l ) 1 + exp(pK-pH)FV/RT g i s a constant of p r o p o r t i o n a l i t y . A s p e c i a l i n t e r p r e t a t i o n may be given to equation (A2a). The f r a c t i o n 1/(1+exp(pK-pH)FV/RT)) may be in terpre ted as the f r a c t i o n of channels 'b locked' by a p a r t i c l e w i th in i t . In a l k a l i n e s o l u t i o n s , the channel i s blocked during h y p e r p o l a r i z a t i o n s and i s 'unblocked' in a c i d s o l u t i o n s . P l o t s of I -V r e l a t i o n s (from equation A2a) are shown i n F i g . A . 6 a and b for zero current p o t e n t i a l s of -90 mV and -30 mV r e s p e c t i v e l y . The curves are normalized at the zero current p o t e n t i a l so that the r e s t i n g conductance at pH 8.0 i s three times the r e s t i n g conductance at pH 6.0 ( F i g . 2 . 1 . A ) . In agreement with experimental observat ions: the curvature 178 of the I -V r e l a t i o n s depends continuously on pH, from upwards concave and negative slope conductance in a l k a l i n e so lut ions to downwards con-c a v i t y in a c i d so lu t ions ; the r e l a t i o n s do not depend on concentration gradients ( re s t ing or zero current p o t e n t i a l ) to any appreciable extent . The expression (42a) may be modified by. the more complex expression: 1 (42b) I = 1 + exp(pK-pH)FV/RT where I (V) i s the s ing le channel current -vo l tage r e l a t i o n of equation 38. However, the q u a l i t a t i v e behavior of equations 42a and 42b remain unchanged. Trans ient proper t i e s In t h i s s ec t ion , the voltage and time dependence of channel transport i s cons idered. When the transmembrane p o t e n t i a l i s suddenly perturbed, the evo lut ion of the occupation d e n s i t i e s N^, N 2 , . . . , N n to a s ta t ionary d i s t r i b u t i o n with a s ing l e channel i s described by the matrix equation: (43) d _ _ dt N l - ( k j + k ^ ) k_ 2 . . . k o N o N 2 = kj (k 2 +k_ 2 ) k _ 3 . • N 2 + 0 N n N n k N -(n+1) n+1 179 F igure 4.4 The q u a l i t a t i v e behavior of the current-vol tage equation (equation 38). a) I f the number (n) of i n t e r n a l b a r r i e r s i s increased, the current -vo l tage r e l a t i o n i s l i n e a r i z e d . A l l the other parameters remain f i x e d . The numbers ind icate the number of b a r r i e r s . b) Thi s graph i l l u s t r a t e s the ef fect of d i f f e r e n t ra te l i m i t i n g steps i n the case of a 'homogeneous' channel with symmetrical surface b a r r i e r s at the inner and outer mouths of the channel. The numbers beside each curve ind icate the value of q (equation 39). c) Th i s graph shows the inf luence of concentrat ion gradients on current -vo l tage r e c t i f i c a t i o n . Note that the d i r e c t i o n of the (I-V) r e c t i f i c a t i o n remains un-changed, but the magnitude of the normalized current densi ty for hyperpo lar i za t ion (as compared to (b)) i s reduced. d) The purpose of these two graphs i s to i l l u s t r a t e the e f fec t of concentrat ion gradients on (I-V) r e c t i f i c a t i o n when the r a t e - l i m i t i n g step i s e i ther at the i n t e r i o r (d) or surface (e) of the membrane. Both graphs are normalized so that a l l curves have equal chord conductances at the zero current p o t e n t i a l . In both graphs, the i n t e r i o r surface step i s assumed to be equal to the ex terna l step. The r e s u l t s with asymmetrical b a r r i e r s at the i n t e r f a c e s are s i m i l a r except that the r e c t i f i c a t i o n i n the inward d i r e c t i o n (with hyperpo lar i za t ion ) i s much l a r g e r with a smaller i n t e r n a l b a r r i e r . In (d) i t i s seen that with decreasing zero -current p o t e n t i a l , the degree of r e c t i f i c a t i o n increases . In (e) , as the zero current i s decreased, the degree of r e c t i f i c a t i o n decreases. The numbers on the l i n e s are the z e r o -current p o t e n t i a l s . In (d) the curves have a l l been s h i f t e d to superimpose the -50 mV r e l a t i o n s h i p at 180 zero current and i n (e) they have a l l been sh i f ted to superimpose the 0 mV r e l a t i o n s h i p . f) T h i s f igure i l l u s t r a t e s the e f fect of asymmetrical p o t e n t i a l b a r r i e r s at the inner and outer mouths of H13.X the channel (with the i n t e r n a l b a r r i e r heights f kept at the same value i n a l l the curves ) . The small f i gure inset beside each curve i l l u s t r a t e s the r e l a t i v e heights of the a c t i v a t i n g energy b a r r i e r s . Curve 0 (q=0). The value of e = .4 (see text , equations 37, 38). That i s , the b a r r i e r at the external surface i s l arger than at the i n t e r n a l surface , i s a l so greater than f i n t h i s graph (hence the downwards concavi ty of the curve ) . f ^ f 1 1 Curve 1 (e = 1 . 0 ) . The i n t e r n a l b a r r i e r height f 1 i s the same as the external b a r r i e r height f** and ^11 _ ^max f l - f " Curve 50 ( e = 50.0) . The i n t e r n a l b a r r i e r hexght i s very small (f > f > f ) . 181 with i n i t i a l d i s t r i b u t i o n N (0) n Set the matrix A to be - ( k l + k _ l ) •2) (44) A = - (k 3 +k_ 3 ) k_ 4 In the fo l lowing d i s c u s s i o n , most of the channels w i l l be assumed to be unoccupied so that s ing le occupancy e f fec t s are not present . The b a r r i e r s w i th in the channel w i l l a l so be assumed to be symmetric and r e g u l a r . The c h a r a c t e r i s t i c times for the r e l a x a t i o n (of occupation dens i t i e s ) i n such a channel i s obtained by so lv ing the c h a r a c t e r i s t i c polynomial det(A - XI) = 0. Seshadri (1971), using the method of generat ing funct ions and Frehland&Lauger (1974) , us ing orthogonal polynomials , have solved the problem when there i s no vol tage gradient across the membrane. We have extended the s o l u t i o n of Frehland and Lauger (1973) to include non-zero p o t e n t i a l gradients across the membrane. However, the conclus ions that can drawn i s exact ly analogous to the extension of d i f f u s i o n to incorporate d r i f t . T h i s i s not s u r p r i s i n g as the channel model with regular i n t e r n a l b a r r i e r s i s a d i s c r e t i z a t i o n of continuous e l e c t r o - d i f f u s i o n (Woodbury, 1971; 1 8 2 Figure 4.5 a) Fami l i e s of S-shaped curves sh i f ted by vol tage (equation 41a). b) P l o t s of the conductance from (a) when the pH i s remained constant and the membrane p o t e n t i a l i s v a r i e d . The numbers beside each curve i n d i c a t e the pH. c) Current -vo l tage r e l a t i o n s using equation 41a and a l i n e a r channel conductance vol tage (equation 41b)• The currents at pH 5.0 have been m u l t i p l i e d by 100 to make them v i s i b l e against those at pH 8.0 d) Current -vo l tage r e l a t i o n s with the 'group' at var ious dis tances w i t h i n the membrane. The number beside each curve i n d i c a t e s the d is tance wi th in the membrane. 182a 183 Frehland and Lauger, 1973). Hence the d e t a i l s w i l l not be presented and we w i l l consider only the q u a l i t a t i v e behavior: When the vo l tage across a membrane (with n i n t e r n a l p o t e n t i a l energy minima) i s changed, the current relaxes with a spectrum of n c h a r a c t e r i s t i c time constants depending on the absolute membrane p o t e n t i a l . The magnitudes of these r e l a x a t i o n time constants depend on whether the surface or the i n t e r i o r of the membrane i s the ra te l i m i t i n g step. This i s summarized i n Table IV.1 for the largest and smallest time constants . Our d i s cuss ion has been based on the assumption of no s ing le occupancy e f f e c t s , p r i m a r i l y for s i m p l i c i t y . I f s ing le occupancy i s incorporated in to t h i s model with regu lar i n t e r n a l b a r r i e r s , i t s e f f ec t i s to introduce a delay in to the r e l a x a t i o n t rans ient s of the occupation d e n s i t i e s and hence membrane currents (Macey and O l i v e r , 1967). This may be q u a l i t a t i v e l y observed as fo l lows: a channel with s ing l e occupancy may be viewed as a chain of react ions N * ^ N„ ==^N„ ==* — 5=iN 5 = = ^ 1SL , where N denote the s tate of 1 I i n 0 j th the channel with an ion i n the j minimum, and i s the channel free of ions . Hence a delay would be expected between the entrance of an ion in to a channel and i t s eventual e x i t . However, i n sp i t e of t h i s de lay , s ince the r e l a x a t i o n s are i o n i c i n nature , observed k i n e t i c s are expected to be much f a s t e r (microseconds) then experimental ly observed r e l a x a t i o n s ( m i l l i s e c o n d s ) . When the ra te constants k . 's are time dependent as a r e s u l t of r e l a x a t i o n of membrane bound molecules w i th in a channel , the matrix 184 equation (43) becomes non-autonomous (equation of a temporally inhomogenous b i r t h and death process ) . Without fur ther experimental evidence on the l o c a l f i e l d ins ide the membrane and the nature of these movements, i t w i l l not be f r u i t f u l to pursue these matters f u r t h e r . But i t i s of i n t e r e s t to note that i f the matrix A ( t ) (equation 44) i s decomposed i n t o : (45) A(t ) = A(0) + A i ( t ) A ( t ) = A ( 0 ) + ( A ( t ) " A ( 0 ) ) 3 = 1 2 where A.(t) i s the matrix whose en tr i e s a 3 are 3 i k r when k 4 j - ( k ( t ) -k (0)) , i = j - 1 , k = j - ( ( k ( t ) -k (0)) + (k ( t ) -k (0)) ) , i= j , k=j (k ( t ) -k (0)) , 3=3+1 , k=j then the s o l u t i o n of (43) i s (46) N(t) =N(0) e A ( 0 ) t ^ " E J A . ( t ) d t 3 = 1 The expression (46) i n d i c a t e s that formal ly , current re laxat ions wi th in a s ing l e channel may be equ iva len t ly viewed as changes i n the number of conducting channels . Here the product *j-i e A j ( t ^ d t i s equivalent to changes in the number of channels . Hence subsequent d i scuss ions w i l l be r e s t r i c t e d to changes i n the number of channels . 185 Table IV.1 Dominant fas t and slow time constants for channel model with n i d e n t i c a l i n t e r n a l b a r r i e r s (from Frehland and Lauger, 1974). Rate l i m i t i n g step Smallest time constant Largest time constant i n t e r i o r ra tes are the same as the s u r -face rates <"°-x!r&i> 1/k (l+cosir/(n+l)) l / 2 k (1+cos nir/(n+1)) i n t e r i o r ra te l i m i t e d ( k ° « k _ 1 ; k ° « k n + 1 ) l / 2 k (1+cos 27r/n) l / 2 k (1+cos(n-l)ir/n) surface rate l i m i t e d ( k ° » k ° 1 ; k ° » k ' + 1 ) l / 2 k (1+cosir/n) For d e f i n i t i o n of the symbols, r e f e r to F i g . 4 . 2 and equation 35 ( k ° = k ° for l £ i <n). k ° , k ° and k ° . are vo l tage independent. 186 The dependence of the number n ( V , t ) of conducting channels on vol tage and time has been found to be very v a r i e d . In the sodium and potassium channels of nerve and muscle (Hodgkin and Huxley, 1952; A d r i a n , chandler and Hodgkin, 1970a) i t i s described by f i r s t order k i n e t i c s but s a t i s f a c t o r y theories accounting for the vol tage dependence of conductance changes i n both of these channels are l a c k i n g . Understanding of the t rans ient propert ie s of the c h l o r i d e permeation poses severa l problems. The most outstanding problem i s the dependence of k i n e t i c s on d i f ferences between condi t ion ing and test p o t e n t i a l s (or d i f f erences between i n i t i a l and the newly imposed v o l t a g e ) . In C h a p t e r l l l when the manifold was constructed, t h i s a lso was extended to non-s tat ionary i n i t i a l condi t ions as i t was shown that when vol tage steps are made during recovery or i n a c t i v a t i o n of conductance, there i s an equivalent voltage from which the current t rans i en t s may be considered as s t a r t i n g . A second problem i s the independence of current t rans ient s on external pH in sp i t e of the wide v a r i a t i o n s in steady state r e c t i f i c a t i o n . Since the e q u i l i b r i a of chemical reac t ions depend on the rate constants i t i s d i f f i c u l t to r e c o n c i l e the steady state and t rans i en t proper t i e s of c h l o r i d e conductance. In view of these d i f f i c u l t i e s , only a b r i e f d i scuss ion i s cons idered. I t w i l l be c l e a r that t r a d i t i o n a l models that have been proposed for other i o n i c channels (such as sodium and potassium channels i n nerve and muscle) are inadequate to account for the observat ions for c h l o r i d e current t r a n s i e n t s . 187 The simplest model i s that of a channel d i s t r i b u t i n g between two s ta tes , open and c losed , depending on the transmembrane vol tage . When the vo l tage i s changed, a r e d i s t r i b u t i o n in to a new equi l ibr ium occurs . I f the r e d i s t r i b u t i o n i s a f i r s t order (unimolecular) process: k l Open »- Closed then the r e l a x a t i o n k i n e t i c s fo l low an exponential time course c ( t ) = (c(0) - c ( ~ ) ) e " t / T + c(~) (c(t) i s the concentrat ion of e i ther open or c losed channels) . The ra te constant = 1/(k^(V) + k_^(V)) depends on voltage but i s independent of the i n i t i a l concentrat ion c ( 0 ) . The independence of k i n e t i c s of i n i t i a l condi t ions a l so app l i e s to l i n e a r chains of f i r s t order reac t ions : Open I , I - — - - — I —•» Closed — 1-— 2 -=— •<— n ^— K i n e t i c s depend as w e l l on the absolute membrane p o t e n t i a l s ince the eigenvalues of the t r a n s i t i o n matrix for such a chain depend only on the rate constants k (1 < j < n ) . For instance , the sodium and potassium channel k i n e t i c s of nerve have been modeled p r i m a r i l y by a sequence of f i r s t order steps (Hoyt, 1963; Goldman, 1964,1965; Armstrong, 1969; J a i n et a l , 1970; Fishman et a l , 1971; Moore and Jakobsson, 1971; Moore and Cox, 1976). 188 The dependence of k i n e t i c s on i n i t i a l condit ions cannot be explained by f i r s t order r e a c t i o n s . Chemical reac t ions whose rates show a dependence on i n i t i a l concentrat ions u s u a l l y involve higher order (mult i -molecular) react ions or c a t a l y t i c reac t ions (Benson, 1960). But the k i n e t i c s of higher order reac t ions i n i n general non-exponential . For instance , the simple combination of two Gramic idin A molecules G + G 2G to form a conducting channel has a sigmoid dependence on time (Bamberg and Lauger, 1974) and the rate constants of current re laxat ions i n response to a step change i n transmembrane voltage depend on the d i f f e r e n c e between i n i t i a l and f i n a l current s . In the c h l o r i d e channel , the dependence of k i n e t i c s on a step change i n vo l tage i s more d i f f i c u l t to understand. The best approximate model that we have been able to f i n d and accounting q u a l i t a t i v e l y for some of the experimental observations i s that conversion from open to c losed s tates of a channel(or from a higher to a lower conducting state in a s i n g l e channel) involves the p a r t i c i p a t i o n of a molecule A which acts i n a c a t a l y t i c r o l e c o n t r o l l i n g the ra te of the convers ion. Such a r o l e has been postu lated for calcium i n the sodium channel in nerve (Moore and Cox, 1976). The a v a i l a b i l i t y or concentrat ion of A depends on membrane p o t e n t i a l . For ins tance , i n the k i n e t i c schemes (1) and (2): k . A (1) Open v Closed k (2) A c t i v e * Inac t ive k - 2 189 Figure 4.6 The current -vo l tage r e l a t i o n s according to equation (42a). The curves are normalized so that the r e s t i n g conductance at pH 8.0 i s 3 times greater than at pH 6.0 . The numbers beside each curve ind ica te the pH. The zero current p o t e n t i a l s for (a) and (b) are -90 mV and -30 mV r e s p e c t i v e l y . H 8 9 a 190 The concentrat ion C Q ( t ) of open channels a f ter a change i n vol tage from a steady s tate at V (condit ioning voltage) to a new vol tage V 2 at t = 0 i s - k (A (V )-A (V ) )e" ( t / T + k l A - ( V 2 ) t ) c (t) = c (0)e W V A « , ^ v 2 ; ; e o o A'-.(V,) and A (V„) are the i n i t i a l and steady state concentrat ions 0 1 0 0 2 of the a c t i v e form of the molecule A, and T = l / ( ( k 2 ( V 2 ) + k _ 2 ( V 2 ) ) i s the rate constant for r e a c t i o n (2). and shows a dependence on the i n i t i a l and f i n a l concentrat ions of A. The k i n e t i c s do not depend simply on the voltage step l ^ 2 _ V l ^ We have inves t iga ted more complex schemes than (1) and (2) , but they have a l l f a i l e d to account for the simple dependence of k i n e t i c s on the vo l tage step. Discuss ion In t h i s chapter , we have reviewed some of the q u a l i t a t i v e behavior of e l e c t r o d i f f u s i o n and channel transport with the hope of f i n d i n g some proper t i e s of the c h l o r i d e permeation system that might be descr ibed by standard models for ion t r a n s p o r t . The h e u r i s t i c view of the c h l o r i d e permeation system that we have assumed i s that i n s t a n -taneous conductances r e f l e c t s ing l e channel conductance whereas steady state conductances are the r e s u l t of changes e i t h e r i n the number of of conducting channels or i n the conductance of a s ing l e channel or p o s s i b l y both s i t u a t i o n s . In t h i s d i s c u s s i o n we w i l l consider in more 191 d e t a i l the a p p l i c a b i l i t y of t h i s hypothesis . Instantaneous current -vo l tage r e l a t i o n s The instantaneous conductance i s postulated to r e f l e c t the conductance of a s i n g l e channel . In p o l a r i z e d f i b e r s , the instantaneous c u r r e n t -vol tage r e l a t i o n s are l i n e a r in a c i d so lu t ions , conductance l i m i t i n g i n n e u t r a l s o l u t i o n s , and exh ib i t sa turat ion and negative slope conductance i n a l k a l i n e s o l u t i o n s . When the i n t e r n a l ch lor ide concentrat ion i s increased, and the c h l o r i d e concentrat ion gradient i s decreased ( in depolar ized f i b e r s ) , the instantaneous current -vo l tage r e l a t i o n s become downwards concave i n a c i d s o l u t i o n s , and l i n e a r for a greater vo l tage range in n e u t r a l and a l k a l i n e s o l u t i o n s . T h i s l i n e a r i z a t i o n behavior ind ica te s the conductance of s ing le channels may be descr ibable by the ' e l e c t r o d i f f u s i o n ' type of channel models (For instance , F i g . 4 . 4 of t h i s chapter ) . However, the observat ion that r e s t i n g conductances of both p o l a r i z e d and depolar ized f i b e r s are of the same magnitude i s not compatible with these views s ince s ing l e channel conductances are h i g h l y concentrat ion dependent i n ' e l e c t r o d i f f u s i o n ' models. Perhaps an explanat ion for the independence of concentrat ion would be that channels are saturated with permeating anions i n p o l a r i z e d f i b e r s , and any fur ther increase of the i n t e r n a l concentrat ion would have no s i g n i f i c a n t e f fec t on the c u r r e n t s . I f t h i s were the case, then the negative slope conductance observed i n some p o l a r i z e d f i b e r s in a l k a l i n e so lu t ions would be a t t r i b u t e d to mutual in t er f erence of i o n s . I t i s 192 s u r p r i s i n g then, that with a t en - fo ld increase i n concentrat ion i n depolar ized f i b e r s , there i s no observation of negative slope conductance. Ef f ec t of pH on current-rvoltage r e l a t i o n s In order to understand the ac t ion of external pH, a s a t i s f a c t o r y explanat ion would have to be given to the r e c t i f i c a t i o n of the steady state current -vo l tage c h a r a c t e r i s t i c s . Amongst a l l the simple models that we have s tudied , some of which are reviewed in t h i s chapter, the one that q u a l i t a t i v e l y y i e l d s the current -vo l tage r e l a t i o n s most s i m i l a r to those observed i n experiments i s the model of F i g . A . 6 where changing pH changes the e l e c t r i c a l s ign and charge of a group that contro l s ch lor ide permeation. However, the appearance of a more pronounced negative slope conductance i n depolar ized f i b e r s than i n p o l a r i z e d f i b e r s ( in the steady state current -vo l tage r e l a t i o n s i n a l k a l i n e s o l u t i o n s ) , i f i t i s not an a r t e f a c t , i s not p r e d i c t e d . In i o n i c channels , s e l e c t i v i t i e s depend on dynamic parameters such as the ra te constants for surmounting the a c t i v a t i n g energy b a r r i e r s ( B e z a n i l l a and Armstrong, 1972; Lauger, 1973; Armstrong, 1975a; H i l l e , 1975a). I f there i s a change i n the e f f e c t i v e charge of the group c o n t r o l l i n g c h l o r i d e permeation, then a change in the s e l e c t i v i t y pat tern of the anion channel would be a n t i c i p a t e d with changes i n external pH and membrane p o t e n t i a l . S h i f t s i n zero-current p o t e n t i a l S h i f t s of zero-current p o t e n t i a l are pred ic ted (see equation 2A) 193 in e l e c t r o d i f f u s i o n and ' e l e c t r o d i f f u s i o n ' channel models. But the magnitude of these s h i f t s depend on the steady state channel conductance of the preceding pulse . This i s contrary to experimental observat ions . The source of these s h i f t s remains unresolved. Dependence of current t rans ients on voltage and on i n i t i a l condi t ions The most p l a u s i b l e explanation for the dependence of k i n e t i c s on i n i t i a l condi t ions i s that current t rans ient s involve the mediation of a moiety whose presence determines the rate of a r e a c t i o n , i n t h i s ins tance , the t r a n s l o c a t i o n of c h l o r i d e across the membrane. I f t h i s moiety d i s t r i b u t e s a l so i n two states such that only one state can mediate the channel i n a c t i v a t i o n process , then the rate constants w i l l depend on the d i f f erence between the i n i t i a l and f i n a l concentrat ions of t h i s moiety. I f t h i s dependence i s on the membrane e l e c t r i c f i e l d , then the time constants w i l l depend on the d i f f erence between i n i t i a l and f i n a l vo l tage , but t h i s dependence i s more complex than the dependence of J V 9 ~ V 1 | that we have described from experiments. F i n a l Discuss ion 195 F i n a l Discuss ion Channel versus c a r r i e r transport One of the objec t ives of these experiments was to inves t iga te the c a r r i e r and channel hypotheses for c h l o r i d e transport in Xenopus l a e v i s muscle membrane. The r e s u l t s obtained i n t h i s study, l i k e those that we have reviewed i n Chapter I , support a channel hypothesis more favourably than a c a r r i e r for c h l o r i d e t ransport . Although there i s no compell ing s ing le piece of evidence that completely excludes a c a r r i e r , var ious observations tend to make the adoption of t h i s hypothesis implaus ib le : The c a r r i e r hypothesis as suggested by Warner (1972) based on the model studied by Sandblom, Eisenman and Walker (1967) cannot account for the dependence of r e c t i f i c a t i o n on concentrat ion grad ient s , nor the negative slope conductances observed in a l k a l i n e s o l u t i o n s . Perhaps an a l t e r n a t e formulat ion of a c a r r i e r model based e i t h e r on p o s i t i o n dependent m o b i l i t i e s of the c a r r i e r wi th in the membrane (Agin, 1972) or on more ad hoc assumptions such as the c a r r i e r concentrat ion on one s ide of the membrane being constant (Adrian, 1969) w i l l account for the negative slope conductance, but i n t r o d u c t i o n of these assumptions at the present time would be l a r g e l y e m p i r i c a l . The nature of the c h l o r i d e channel Diverse experimental evidence i n d i c a t e s that the i n t e r n a l s t ruc ture of the c h l o r i d e channel may be very complex: as we have mentioned, the b locking a c t i o n of Br , NO^ , I and SCN and the non- inter ference 196 of F and the fami ly of 'benzoate- l ike ' anions (Woodbury and M i l e s , 1973) suggests that the rate l i m i t i n g steps for d i f f e r e n t anions may d i f f e r . A d d i t i o n a l evidence on the poss ib le multi-component nature of the c h l o r i d e channel i s obtained from studies on the reduct ion of c h l o r i d e and permeabi l i ty by the d i su lphonic s t i lbene d e r i v a t i v e , SITS (4-a c e t a m i d o - 4 ' - i s o t h i o c y a n o - 2 , 2 ' - s t i l b e n e d i su lphonic a c i d ) , which has been found to i n h i b i t c h l o r i d e f l u x i n mammalian red blood c e l l s (Knauf and Roths te in , 1971) and barnacle muscle f i b e r s (Russel and Bro'dwick, 1976). At concentrat ions of .2 tnM and 1 mM, SITS has been found to reduce r e s t i n g c h l o r i d e conductance by 50% and 75% r e s p e c t i v e l y regardless of the ex terna l pH (Vaughan and Fong, 1978). Unl ike the i r r e v e r s i b l e ac t i on of SITS on red blood c e l l s and barnacle f i b e r s , i n Xenopus the reduct ion of c h l o r i d e conductance by SITS i s r e v e r s i b l e . The pH independent ac t ions of SITS suggest that the membrane s i t e a f fec ted by SITS i s d i f f e r e n t from the pH dependent s i t e . Parker and Woodbury (1976) have found two temperature dependent but c o r r e l a t e d components of c h l o r i d e conductance and have suggested that the pH dependent s i t e and the conductance a c t i v a t i n g s i t e are c lose together. A study of the temperature dependence of the a c t i o n of SITS w i l l a i d us in determining whether the SITS b inding s i t e i s the conductance a c t i v a t i n g s i t e . The reduct ion of c h l o r i d e conductance wi th repeated p o l a r i z a t i o n i n permanently depo lar ized f i b e r s i n a l k a l i n e s o l u t i o n s a l so suggests a multi-component channel for c h l o r i d e permeation. But the observat io pose two unsolved problems. The f i r s t i s whether t h i s i s a vol tage 197 or current e f f e c t . That i s , i s the conductance reduced by current passing through the channel or do channels d i s in tegra te or become less s table by repeated changes i n transmembrane po tent ia l? The second problem i s whether a v a r i a b l e number of channels i s i n a c t i v a t e d or the conductance of a s ing le channel i s reduced. Some unsolved problems Although we have c l a r i f i e d the phenomenon of instantaneous a f t e r c u r r e n t s , t h e i r source and mechanism remain to be reso lved . Another i n t e r e s t i n g and important aspect of c h l o r i d e conductance i s i t s independence of absolute membrane p o t e n t i a l . In our experiments, f a m i l i e s of S-shaped curves (the manifold) are t rans la ted i n the 1^(0) versus plane withchanges in r e s t i n g p o t e n t i a l . The k i n e t i c s of current t rans i en t s (as descr ibed i n Chapter III ) remain i n v a r i a n t r e l a t i v e to t h i s family of S-curves as i t i s sh i f ted along the plane. An e l u c i d a t i o n of t h i s t r a n s l a t i o n would g r e a t l y a i d us in understanding c h l o r i d e permeation. An understanding of the current t rans i en t s was one of the main objec t ives of our experiments. Although the dependence of k i n e t i c s on vol tage and pH i s c l a r i f i e d , the molecular mechanisms of the current t rans i en t s remain unresolved. The cons truc t ion of the i n i t i a l current -vo l tage manifo ld and i t s a b i l i t y to account for d iverse experimental protoco l s (such as instantaneous and steady s tate current -vo l tage r e c t i f i c a t i o n , the dependence of current t rans ient s on the d i f f e r e n c e between c o n d i t i o n i n g and t e s t vo l tages , and the dependence of k i n e t i c s on i n i t i a l cond i t ions ) suggests that there i s a fundamental uni ty i n the molecular mechanism governing c h l o r i d e transport at d i f f e r e n t external pH and d i f f erent ch lor ide concentrat ion gradients across the membrane. 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G . and Woodbury, J .W. (1976). T i t r a t a b l e s i t e s of C l ~ channels of frog muscle are near the outs ide of the membrane. Biophys. J . 16, 157a. P a r l i n , B. and E y r i n g , H . (1954). Membrane permeab i l i ty and e l e c t r i c a l p o t e n t i a l In: Ion Transport Across Membranes (Clarke , H . T . , E d . ) , pp.103-118, Academic Pres s , New York. 209 Planck, M. (1890). Uber die Erregung von E l e k t r i c i t a t und Warme i n E l e k t r o l y t e n . Ann. Phys ik . Chem. 39: 161-186. Russe l , J . M . and Brodwick, M.S. (1976). Chlor ide f luxes i n the d ia lyzed barnacle muscle f i b e r and the e f fec t of SITS. Biophys. J . 16, 156a. Sandblom, J . , Eisenman, G. and Neher, E . (1977). Ionic s e l e c t i v i t y , s a t u r a t i o n and block i n gramic id in A channels. I . Theory for the e l e c t r i c a l propert ie s of ion s e l e c t i v e channels having two p a i r s of b inding s i t e s and m u l t i p l e conductance s ta tes . J . 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Transport processes and e l e c t r i c a l phenomena i n i o n i c membranes. Progress i n Biophys. Molec. B i o l . 3,351-375. V a l d i o s e r a , R . , Clausen, C. and Eisenberg, R . S . (1974a). Measurement of the impedance of frog s k e l e t a l muscle f i b e r s . Biophys. J . 14, 295-315. V a l d i o s e r a , R . , Clausen, C . and Eisenberg , R . S . (1974b). C i r c u i t models of the passive e l e c t r i c a l propert ie s of frog s k e l e t a l muscle f i b e r s . J . Gen. P h y s i o l . 63, 432-459. V a l d i o s e r a , R . , Clausen, C. and Eisenberg , R . S . (1974c). Impedance of frog s k e l e t a l muscle f i b e r s i n var ious s o l u t i o n s . Gen. P h y s i o l . 63, 460-491. Vaughan, P . C . (1975). Muscle membrane. Prog. Neurobio l . 3, 217-250. Vaughan, P . C . and Fong, C . N . (1978). E f f e c t of SITS on c h l o r i d e permeation i n Xenopus l a e v i s . Can. J . P h y s i o l . Pharm.( in p r e s s ) . Venosa, R . A . , Ruarte , A . C . and Horowicz, P . (1972). Ch lor ide and potassium movements from f r o g ' s s a r t o r i u s muscle i n the presence of aromatic anions. J . Mem. B i o l . 9, 37-56. Waddel, W . J . and Bates, R . G . (1969). I n t r a c e l l u l a r pH. P h y s i o l . Rev. 49, 285-329. Warner, Anne E . (1972). K i n e t i c proper t i e s of the c h l o r i d e conductance of frog muscle. J . P h y s i o l . 227, 291-312. Woodbury, J .W. (1971). E y r i n g ra te theory model of the c u r r e n t -vo l tage r e l a t i o n s h i p s of ion channels i n e x c i t a b l e membranes. In: Chemical Dynamics: Papers i n Honor of Henry E y r i n g . H i r s c h f e l d e r , J . O . and Henderson,D. ed. Advances i n Chemical Phys ic s . V o l 21, 601-617. Woodbury, J .W. and M i l e s , P . R . (1973). Anion conductance of frog muscle membranes: one channel , two kinds of pH dependence. J . of Gen. P h y s i o l . 62, 324-353. Woodhull , Anne M. (1973). Ionic blockage of sodium channels i n nerve. J . Gen. P h y s i o l . 61, 687-708. Wright , E . M . and Diamond, J . M . (1977). Anion s e l e c t i v i t y i n b i o l o g i c a l systems. P h y s i o l . Rev. 57, 109-156. 211 Z w o l i n s k i , B . J . , E y r i n g , H . and Reese, C . E . (1949). D i f f u s i o n and membrane permeabi l i ty . J . Phys. Chem. 53, 1426-1453. 212 Appendix 213 The objec t ive of t h i s appendix i s to describe i n more d e t a i l , the i n t e r p r e t a t i o n of membrane current measurements using the three-microe lectrode vol tage clamp technique. E l e c t r i c a l l y , the equivalent c i r c u i t of the muscle membrane i s a complex branching d i s t r i b u t i v e network (Falk and F a t t , 1964; F a t t , 1964; Schneider, 1970; Va ld iosera et a l , 1974a, b , c ; Mathias et a l , 1977). The complexity a r i s e s from the geometry of the transverse tubular system, which have has success ive ly been modelled by a lumped RC ( p a r a l l e l res i s tance and capacitance) network (Fatt and Katz , 1951; F a l k and F a t t , 1964); d i s t r i b u t i v e d i sk (Falk and F a t t , 1964; F a t t , 1964; A d r i a n , Chandler and Hodgkin, 1969; Schneider, 1970; V a l d i o s e r a et a l , 1974a,b,c) and mesh model (Mathias et a l , 1977). In t h i s appendix, we w i l l use the d i sc model for i l l u s t r a t i o n i n the i n t e r p r e t a t i o n of the membrane current from 3 micro-e lec trode vol tage clamp. The e l e c t r i c a l parameters of the muscle membrane, viewed as a 3 dimensional c y l i n d r i c a l cable ( F i g . A l ) are: 2 ^ R = res i s tance i n cm of surface membrane m -1 -2 = G , G i s the conductance i n mho.cm m m -2 C = capaci tance i n Fcm of surface membrane, m R_j = sarcoplasmic r e s i s t i v i t y i n °.cm. -2 G = conductance of the wa l l s of the tubular membrane ( in mho.cm ) . w _2 C = capac i ty of the wa l l s of the tubular membrane (F.cm ) . w GT = lumina l r e s i s t i v i t y of the T-system (ft.cm). J L i _2 I = membrane current dens i ty (Amp.cm ) . m 214 Let a be the f i b e r radius and b the radius of the lumen of the T-system. The three dimensional parameters are reduced to one dimensional parameters v i a the r e l a t i o n s : r = R ( 2 T r a ) _ 1 m m = r e s i s t i v i t y of the membrane (ficm) = g * (g 1 S the membrane conductance(mho.cm m m c = ( 2 i r a)C m m = membrane capacitance (Fcm ) r = R i ( 2 T r a ) " 1 sarcoplasmic r e s i s i t i v i t y (ftcm )^ g = G (2irb) = conductance of the wal ls of the T-system (mho.cm 6 w w c = C (2irb) = tubular membrane capacitance (Fcm ) . W W g = G (2irb) 1 = luminal r e s i s t i v i t y of the T system i n ficm i = 1 (2-rra) = membrane current densi ty (Amp.cm m m 2 Let Y(s) be the in s ide -out s ide admittance per cm of membrane. Y ( s ) i s the p a r a l l e l sum of the surface admittance ^ m ( s ) a u ( ^ t n e tubular admittance Y , r ( s ) , Y ( s)=Y (s) + Y_ ( s ) . 1 m l 2 Y (s) = surface admittance per cm of membrane. It i s modeled by a m p a r a l l e l RC network, Y (s)= g + c s . m m m 2 Y^,(s) = c o n t r i b u t i o n to the membrane admittance per cm from transverse tubular system. T h i s admittance w i l l be taken to be of the form (Schneider, 1970; A d r i a n , Chandler and Hodgkin, 1969; V a l d i o s e r a et a l , 1974a, b ,c ) 215 G L I 1 ( a / X T ( s ) ) Y T ( s ) = A T ( s ) I 0 ( a / X T ( s ) ) where a / X T ( s ) = a ( ( g w + s c w ) / g L ) -1/2 1^ and 1^ are the modified Bessel functions of the f i r s t k i n d . The r e s i s t i v i t y of the e x t r a c e l l u l a r f l u i d i s usua l ly assumed to be n e g l i g i b l e . In the domain of the Laplace transformed v a r i a b l e s, the cable equation i s : and the transformed boundary condi t ion (no a x i a l current at the end of the f i b e r , x=0) The other boundary cond i t ion i s obtained by assuming that at the voltage (clamp) c o n t r o l p o s i t i o n (see F i g . 1 . 1 ) . V(Jt,t) = V U ) H ( t ) H i s the Heavis ide function? V ( £ ) i s the command or membrane p o t e n t i a l - r ± Y ( s ) V ( x , s ) = 0 dV(x ,s ) = 0 dx x=0 at x= £ . The s o l u t i o n for the vo l tage p r o f i l e V ( x , t ) i s : - (1) 216 -I i s the inverse Laplace transform operator. Hence the d i f f erence i n vol tage between the e lectrodes at A and B (F ig .1 .1 ) i s (2) V<t) - V ( 2 1 , t ) - v a , t ) . / W i i ^ ' r f . i M 1 s cosh t ( r Y ( s ) ) A / Steady state measurements In the steady s ta te , the d i f ference between the e lectrodes at A and B ( F i g . 1 . 1 ) i s : ( 3 ) A V ( . j = V U ) f c o s ^ r . Y j O ) ) ^ ! | cosh A(r Y ( 0 ) ) ' Under condi t ions of experimental i n t e r e s t , the tubular membrane conductance g i s the sum of the tubular w a l l i o n i c conductances for w sodium g^ a , potassium g™ and c h l o r i d e g ^ : gw= g ^ + g™ + g ^ . g^ a = 0 because Tetrodotoxin was present in the experimental w s o l u t i o n s , and g was n e g l i g i b l e because rubidium replaced potassium and rubidium conductance i s low. Experimental evidence (Hodgkin and Horowicz, 1959, 1960a; Eisenberg and Gage, 1969a,b) have ind ica ted that there i s no c h l o r i d e conductance wi th in the transverse tubular w system g c ^ = 0* Therefore , i n the steady s ta te , there i s no c o n t r i b u t i o n to the t o t a l membrane admittance from the transverse tubular system, and we have: (4) A V ( - ) = v q ) J C ° S h 2 * ( r i g m ) 1 / 2 - 1  cosh Jl (r .g ) l m 217 At the command p o t e n t i a l p o s i t i o n x= £ ( a t A, F i g . 1 . 1 ) the membrane % conductance i 0 0 = g V(A) and since r . g =1/X , we have from equation m m l m ( A ) : • / N » I T / N 1 J c o s h O i / X )  l (oo) = A V(«°) ^ <( m r i X I c o s h ( 2 £ / x ) " c o s h ( £ / x ) = ( 2 / 3 r . £ ) A V ( l ) ( 1 + Q ( * / X ) 2 } I f the (one dimensional) sarcoplasmic r e s i s t i v i t y r_^  i s converted 2 to R . , v i a R.= Tr(d/2) r . , where d i s the f i b e r diameter, then i i i (5) i m U ) = — V ( £ ) ( 1 + 0 ( £ / X ) ) m 6R . r and s ince I = i /ird , we have equation (1) of Chapter I . m m V "6RTf A V ( o 0 ) ( 1 +0(AA>A) Trans ient proper t i e s A n a l y t i c a l expressions for the t rans ient problem (equation 1) have not been obtained. A d r i a n , Chandler and Hodgkin, 1970a and Chandler and Schneider, 1976 have obtained a n a l y t i c a l s o l u t i o n s by d i s c r e t i z i n g the tubular model. A rough estimate of the time required to charge the membrane capacitance network may be obtained by assuming a lumped parameter p a r a l l e l RC model with Cm=(2-3) F . c m - 2 (Vaughan, 1975) and R = 3 2 = 10 flcm then the time constant i s (2-3) m i l l i s e c o n d s . -1 F i g u r e A l D i a g r a m m a t i c r e p r e s e n t a t i o n o f m u s c l e f i b and t h e e l e c t r i c a l p a r a m e t e r s d e f i n e d i n t h e A p p e n d i x . s u r f a c e p a r a m e t e r s t u b u l a r s y s t e m p a r a m e t e r s U ±c- '"bJ' t T — h -

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