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The Starzak membrane conductance model: A comment
J. theor. BioL (I974) 48, 477--479
The Starzak Membrane Conductance Model: A Comment We wish to point out that we have, quite independently of Starzak (1973a, b), but also employing constraints identical to those of Tille (1965), derived a model of potassium conductance based upon an interactive subunit membrane (Davis & Kerr, 1967; Kerr, 1971; Bretag, Davis & Kerr, 1974). From our consideration of these interactive models, the following facts emerge:
(1) The derivation by TiUe (1965) is not statistically rigorous, as probabilities regarding the state of sites are simply multiplied even though there is interaction between neighbouring sites. (2) Consequently, neither Starzak's nor our own model is statistically rigorous for similar reasons. However, Starzak makes no mention of this limitation in his derivation. (3) Nevertheless, these models are reasonable approximations, being of a kind used previously in the description of other interactive processes, e.g. Onsager & Dupuis (1962) for the migration of defects in ice. We are at present involved in making a more exact calculation, based on Tille's constraints, which must necessarily include correlations between different sites (e.g. Green & Hurst, 1964; Irvine, 1970). There are also certain specific criticisms which can be applied to Starzak's model: (1) Starzak invokes one trigger mechanism (nonspecific subunit array relaxation) to initiate both sodium and potassium conductance. He also invokes this as the mechanism for calcium ion- and hyperpolarization-induced parallel time shifts in both the sodium and potassium conductance kinetics. Yet it is well known that sodium kinetics have a time constant about an order of magnitude smaller than potassium kinetics. And it is well known that the potassium conductance delay due to conditioning hyperpolarization is large and obvious in squid axons (Cole & Moore, 1960), and in amphibian nodes of Ranvier (Hille, 1970; Bretag, unpublished observations), although delays in sodium conductance have previously gone unnoticed and, if they exist, must presumably be quite small. In fact, Starzak even states that in his model potassium conductance will be dominated by much slower processes subsequent to the initial triggering. This leaves him with a satisfactory mechanism for a possible sodium delay (although any such delay, at least following a 477
478
A . H . B R E T A G , C. A. H U R S T A N D D. I. B. K E R R
conditioning hyperpolarization, is not obvious) and no mechanism for the large and obvious potassium delay. Such an argument as this was important in the definition of our model, where we suggest that those subunits triggering potassium conductance increase are different from those triggering sodium conductance. Hence they can have different properties and especially different rate constants for their conformation changes in response to potential changes. A potassium triggering subunit may thus be rather like a ten times slower sodium triggering subunit, leaving delay systems intact for both. (2) The many papers describing optical changes in nerve membranes (see Cohen, 1973) all suggest that the major structural changes in axon membranes are voltage-linked and are not conductance-linked as Starzak's model would imply. There are recent indications (Singer & Nicholson, 1972) that only a fraction of the membrane surface is protein, the most likely candidate for conformational changes of the kind implied by the models of Starzak and of ourselves. Thus a diffusion over the entire membrane of a conductance-linked relaxation process involving interaction between subunits is unlikely. In contrast, we describe a model in which conductance-linked conformation changes need be associated only with discrete membrane patches occupying but a small percentage of the total membrane area. (3) Starzak bases his subunit size on an average distance between charged sites of 11 A, as calculated using Gouy Chapman theory by Gilbert & Ehrenstein (1969). There is, however, no evidence to indicate that a subunit may not have more than one charged site. Furthermore, although sodiumconducting channels may be, on average, some 6000 A apart in the squid axon, they are calculated to be ten times closer than this in the node of Ranvier (Hille, 1970), and, indeed, if there are 50 times as many potassium conducting channels (Hille, 1970), then these must be seven times closer still (assuming square arrays). This would place potassium pores less than 100 A apart in the node, which is not an unreasonable diameter for a single subunit, even assuming the entire area of the node to be covered by potassium conductance triggering subunits. With respect to such calculations of average pore separation, we would suggest that the conductive pores are really at least as closely arranged as calculated for the node and that they appear in condensations or "active patches" surrounded by inert membrane areas. The node is thus expected to be composed largely of active membrane, whereas the squid axon membrane could be at least 99 ~o inactive. Therefore, although there would be ditficulties associated with its small size and its wrapping of connective tissue, the node of Ranvier should be the obvious place to look for optical changes (and hence conformation changes) which are conductance linked.
LETTERS TO THE E D I T O R
479
Furthermore, the importance of the "active patch" hypothesis should not be ignored in any consideration of the evolution of the node of Ranvier and saltatory conduction in vertebrates. For these reasons, we are inclined to discount entirely Starzak's interpretation of the initial triggering mechanism involving, as it does, unnecessary and unfounded conductance-linked conformational changes in the entire membrane, and, being as it is, incapable of explaining the Cole & Moore (1960) potassium conductance delays due to conditioning hyperpolarization. Tille's form of interaction and Starzak's involvement of calcium ions may, however, be valid for membrane subunits within discrete active patches.
School of Pharmacy, South Australian Institute of Technology Department of Mathematical Physics, University of Adelaide Department of Human Physiology and Pharmacology, University of Adelaide, Adelaide, South Australia
A. H. BRETAG
C. A. HURST D . I. B. KERR
(Received 29 January 1974) REFERENCES BRETAG,A. H., DAVIS,B. R. & KERR,D. I. B. (1974). J. membr. Biol. 16, 363. COHEN,L. B. (1973). Physiol. Rev. 53, 373. COLE, K. S. & MOORE,J. W. (1960). Biophys. 3". 1, 1. DAvis, B. R. & KERR,D. I. B. (1967). Seventh Annual Meeting of Physics in Medicine and Biology, Biophysics Group, Australian Institute of Physics. Abstract No. 1. GILBERT,D. L. & EHRENSTEIN,G. (1969). Biophys. J. 72, 685. GREEN, H. S. & Ht~ST, C. A. (1964). Order-Disorder Phenomena. London: Interscience. HILLE,B. (1970). Prog. Biophys. tnolec. Biol. 21, 1. IRVINE,R. D. (1970). dust. J. Phys. 23, 833. KERR, D. I. B. (1971). Biomathematics Symposium. 43rd Congress of the Australian and New Zealand Association for the Advancement of Science (ANZAAS), Brisbane. Abstract No. 1. OrSAGER,L. & DtrPUIS,M. (I 962). In Electrolytes--International Symposium (B. Pesce, ed.). New York: Pergamon. SINGER,S. J. & NICHOLSON,G. L. (1972). Science, IV. Y. 175, 720. STARZAK,M. E. (1973a). 3". theor. Biol. 39, 487. STARZAK,M. E. (1973b). J. theor. Biol. 39, 505. TmLE, J. (1965). Biophys. J. 5, 163.
The Starzak Membrane Conductance Model: A Comment We wish to point out that we have, quite independently of Starzak (1973a, b), but also employing constraints identical to those of Tille (1965), derived a model of potassium conductance based upon an interactive subunit membrane (Davis & Kerr, 1967; Kerr, 1971; Bretag, Davis & Kerr, 1974). From our consideration of these interactive models, the following facts emerge:
(1) The derivation by TiUe (1965) is not statistically rigorous, as probabilities regarding the state of sites are simply multiplied even though there is interaction between neighbouring sites. (2) Consequently, neither Starzak's nor our own model is statistically rigorous for similar reasons. However, Starzak makes no mention of this limitation in his derivation. (3) Nevertheless, these models are reasonable approximations, being of a kind used previously in the description of other interactive processes, e.g. Onsager & Dupuis (1962) for the migration of defects in ice. We are at present involved in making a more exact calculation, based on Tille's constraints, which must necessarily include correlations between different sites (e.g. Green & Hurst, 1964; Irvine, 1970). There are also certain specific criticisms which can be applied to Starzak's model: (1) Starzak invokes one trigger mechanism (nonspecific subunit array relaxation) to initiate both sodium and potassium conductance. He also invokes this as the mechanism for calcium ion- and hyperpolarization-induced parallel time shifts in both the sodium and potassium conductance kinetics. Yet it is well known that sodium kinetics have a time constant about an order of magnitude smaller than potassium kinetics. And it is well known that the potassium conductance delay due to conditioning hyperpolarization is large and obvious in squid axons (Cole & Moore, 1960), and in amphibian nodes of Ranvier (Hille, 1970; Bretag, unpublished observations), although delays in sodium conductance have previously gone unnoticed and, if they exist, must presumably be quite small. In fact, Starzak even states that in his model potassium conductance will be dominated by much slower processes subsequent to the initial triggering. This leaves him with a satisfactory mechanism for a possible sodium delay (although any such delay, at least following a 477
478
A . H . B R E T A G , C. A. H U R S T A N D D. I. B. K E R R
conditioning hyperpolarization, is not obvious) and no mechanism for the large and obvious potassium delay. Such an argument as this was important in the definition of our model, where we suggest that those subunits triggering potassium conductance increase are different from those triggering sodium conductance. Hence they can have different properties and especially different rate constants for their conformation changes in response to potential changes. A potassium triggering subunit may thus be rather like a ten times slower sodium triggering subunit, leaving delay systems intact for both. (2) The many papers describing optical changes in nerve membranes (see Cohen, 1973) all suggest that the major structural changes in axon membranes are voltage-linked and are not conductance-linked as Starzak's model would imply. There are recent indications (Singer & Nicholson, 1972) that only a fraction of the membrane surface is protein, the most likely candidate for conformational changes of the kind implied by the models of Starzak and of ourselves. Thus a diffusion over the entire membrane of a conductance-linked relaxation process involving interaction between subunits is unlikely. In contrast, we describe a model in which conductance-linked conformation changes need be associated only with discrete membrane patches occupying but a small percentage of the total membrane area. (3) Starzak bases his subunit size on an average distance between charged sites of 11 A, as calculated using Gouy Chapman theory by Gilbert & Ehrenstein (1969). There is, however, no evidence to indicate that a subunit may not have more than one charged site. Furthermore, although sodiumconducting channels may be, on average, some 6000 A apart in the squid axon, they are calculated to be ten times closer than this in the node of Ranvier (Hille, 1970), and, indeed, if there are 50 times as many potassium conducting channels (Hille, 1970), then these must be seven times closer still (assuming square arrays). This would place potassium pores less than 100 A apart in the node, which is not an unreasonable diameter for a single subunit, even assuming the entire area of the node to be covered by potassium conductance triggering subunits. With respect to such calculations of average pore separation, we would suggest that the conductive pores are really at least as closely arranged as calculated for the node and that they appear in condensations or "active patches" surrounded by inert membrane areas. The node is thus expected to be composed largely of active membrane, whereas the squid axon membrane could be at least 99 ~o inactive. Therefore, although there would be ditficulties associated with its small size and its wrapping of connective tissue, the node of Ranvier should be the obvious place to look for optical changes (and hence conformation changes) which are conductance linked.
LETTERS TO THE E D I T O R
479
Furthermore, the importance of the "active patch" hypothesis should not be ignored in any consideration of the evolution of the node of Ranvier and saltatory conduction in vertebrates. For these reasons, we are inclined to discount entirely Starzak's interpretation of the initial triggering mechanism involving, as it does, unnecessary and unfounded conductance-linked conformational changes in the entire membrane, and, being as it is, incapable of explaining the Cole & Moore (1960) potassium conductance delays due to conditioning hyperpolarization. Tille's form of interaction and Starzak's involvement of calcium ions may, however, be valid for membrane subunits within discrete active patches.
School of Pharmacy, South Australian Institute of Technology Department of Mathematical Physics, University of Adelaide Department of Human Physiology and Pharmacology, University of Adelaide, Adelaide, South Australia
A. H. BRETAG
C. A. HURST D . I. B. KERR
(Received 29 January 1974) REFERENCES BRETAG,A. H., DAVIS,B. R. & KERR,D. I. B. (1974). J. membr. Biol. 16, 363. COHEN,L. B. (1973). Physiol. Rev. 53, 373. COLE, K. S. & MOORE,J. W. (1960). Biophys. 3". 1, 1. DAvis, B. R. & KERR,D. I. B. (1967). Seventh Annual Meeting of Physics in Medicine and Biology, Biophysics Group, Australian Institute of Physics. Abstract No. 1. GILBERT,D. L. & EHRENSTEIN,G. (1969). Biophys. J. 72, 685. GREEN, H. S. & Ht~ST, C. A. (1964). Order-Disorder Phenomena. London: Interscience. HILLE,B. (1970). Prog. Biophys. tnolec. Biol. 21, 1. IRVINE,R. D. (1970). dust. J. Phys. 23, 833. KERR, D. I. B. (1971). Biomathematics Symposium. 43rd Congress of the Australian and New Zealand Association for the Advancement of Science (ANZAAS), Brisbane. Abstract No. 1. OrSAGER,L. & DtrPUIS,M. (I 962). In Electrolytes--International Symposium (B. Pesce, ed.). New York: Pergamon. SINGER,S. J. & NICHOLSON,G. L. (1972). Science, IV. Y. 175, 720. STARZAK,M. E. (1973a). 3". theor. Biol. 39, 487. STARZAK,M. E. (1973b). J. theor. Biol. 39, 505. TmLE, J. (1965). Biophys. J. 5, 163.