The “independence principle” of biological membranes: Its misuse

J. Theoret. BioL (1969) 24, 159-170

The "Independence Principle" of Biological Membranes: Its Misuse JOHN R. SEGAL

Neurophysiology-Biophysics Research Unit, Veterans Administration Hospital, Boston, Massachusetts, U.S.A. (Received 19 November 1968, and in revised form 14 March 1969) An analysis is presented of the "independence principle"--Hodgkin & Huxley's (1952) theory of the relationship between total membrane electrical conductance and extramembrane ionic composition. By means of theoretical examples of the composition dependence of the conductance of ion-exchanger membranes it is demonstrated that use of the "independence principle" can lead to qualitatively and quantitatively false conclusions about the respective transport numbers of a membrane's current carriers. Thus, contrary to what has been thought, the "independence principle" is not applicable as a general principle of membrane behavior. In one particular, biologically analogous, example, the "independence principle" does make an accurate, correct identification of the current carriers of the ion-exchanger membrane. However, this cannot be used as evidence of the applicability of the underlying assumptions of the "independence principle" in the biologically analogous situation. To so argue is untenable in that it unjustifiably ignores the numerous experimental instances where the electrophysiological effect of an ionic species is completely inexplicable in terms of the "'independence principle". It is shown that previous attempts to subsume such phenomena within the framework of the "independence principle" negate the claimed significance of those effects which are consistent with it. 1. Introduction

A crucial element of the conceptual foundation of contemporary electrophysiology is the "independence principle" of membrane conductance (Hodgkin & Huxley, 1952). The "independence principle" is a theory, derived by Hodgkin and Huxley, o f the relationship between the transmembrane electrical current carried by an ionic species and its concentration on the two sides o f the membrane. They found the theory to be consistent with the effect of extracellular sodium concentration on the early transient of the voltage159

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clamp current of the squid giant axon. This consistency has been adduced as evidence that first, there is an early current transient due exclusively to the movement of sodium ions and that, second, this movement and a subsequent, later, component are independent of each other, in the sense that each can be altered without affecting the other. Application of the "independence principle" has been extended far beyond its original context: virtually every present interpretation of ionic substitution experiments performed on biological membranes makes implicit or explicit use of it (for example: Hutter & Padsha, 1959; Dodge & Frankenhaeuser, 1959; Carmeliet, 1961; Reuben, Werman & Grundfest, 1961; Frankenhaeuser, 1962; Chandler, Hodgkin & Meves, 1965). Most investigators hold this view: if the total membrane current or conductance is reduced by lowering the concentration of a particular species, this constitutes necessary and sufficient evidence that the species carries current; or, on the other hand, if the total current or conductance does not alter, the species is not a current carrier. This belief rests, as will be seen below, upon the implicit or explicit acceptance of the "independence principle". I shall demonstrate that the "independence principle" does not serve the scientific purpose for which it was formulated: it cannot be used to deduce the identity of the carriers of current through biological membranes. The analysis shows that the "independence principle" is not applicable as a general theory of membrane behavior and demonstrates how its use can lead to highly erroneous conclusions regarding the carriers of current through a membrane. It is not the purpose of this paper to discuss the important, separate question of the validity of the assumptions and derivation of the "independence principle". The sole concern here is the issue of the legitimacy of the inferences about electrical transport through membranes based upon use of the "independence principle". The following analysis consists of three major steps: (1) a brief description and definition of the "independence principle" (section 2); (2) the demonstration that it is not generally applicable via theoretical examples of the behavior of an ion-exchanger membrane (section 3); and (3) the demonstration that, even in those instances where the results of experiments performed on biological membranes are consistent with the "independence principle", the significance of this compatability is unknown (section 3).

2. The "Independence Principle" Some confusion is possible between the "independence principle" and the assumption of independence of material flows (an example of the latter is

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the irreversible thermodynamic treatment of membrane phenomena when it is assumed that all phenomenological cross-coetfieients are zero). The two are not equivalent, although the derivation of the "independence principle" does assume independence of flows, among other assumptions. The difference is illustrated by the theory of fixed charge membranes, employed below, which also assumes that flows are independent, but it will be seen that the fixed charge theory can be completely inconsistent with the "independence principle". In order to preserve the important distinction between the assumption of independence of flows and the "independence prineiple", the latter is written with quotation marks. To be completely specific: "independence principle" refers only to the particular theory of membrane conductance postulated by Hodgkin and Huxley. The "independence principle" states that the electric current, lj, carried by the jth species, Cj, is given by =

k

zje(Ecj], e

EC ]o),

(1)

where Zj is the charge of the species, F is the faraday, [ ]~ denotes intracellular concentration, [ ]o denotes extracellular concentration, E is intracellular potential minus extracellular potential, R is the gas constant, T is absolute temperature, and k 1 depends on potential but is independent of the concentrations of Cj [equations (7), (8) and (11), Hodgkin & Huxley, 1952]. When the concentrations of thejth species are altered to [ ]'i and [ ]~, = (ECj], e

e

(2)

where lj is the current of the jth species in the new media. A derivation of equation (2) is given in Hodgkin & Huxley, 1952; a detailed discussion of the underlying assumptions can be found in Ussing (1949), Hoshiko & Lindley (1964) and Kedem & Essig (1965). Equations (1) and (2) alone are not useful because the individual lj's are inaccessible to direct measurement by existing techniques on a fast enough time scale. Thus, the individual 1Ss must be inferred from measurements of /, the total membrane current; I is, by definition, ~ Ij. In order to use the "independence principle" to deduce the individual 1j's from I and the effect upon 1 of alterations in [Cj], the important, fundamental assumption has been made (Hodgkin & Huxley, 1952; cf. references cited in section 1) that when [Ci] is altered the only component of I which changes is I i, and this alteration is given by equation (2). This is equivalent to assuming that k i [equation (1)] is independent of the concentrations of all species. (A relaxation of this assumption and the ensuing consequences are discussed in section 3.)

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Although the most detailed application o f the "independence principle" has been to experiments performed upon the squid giant axon (Hodgkin & Huxley, 1952),t the "principle" was presented as a general theory of the relationship between membrane conductance and the ionic composition of the surrounding solutions. There is nothing in its derivation which is known to restrict its application to either the squid giant axon or biological membranes, alone. Indeed, for the reason which follows, the "principle" must be generally applicable--to both biological and inanimate m e m b r a n e s - if it is to be applied to biological membranes. Present knowledge o f the properties o f biological membranes is only o f the total system: membrane-plus-surrounding solutions. The intrinsic, microscopic, physicochemical attributes of the membranes themselves are unknown. Until these are known, one cannot apply to studies o f macroscopic biological membrane phenomena a theory which makes, implicitly or explicitly, a particular, restricted set of assumptions about the intramembrane transport mechanism--it cannot be decided now if that mechanism is operative within the biological membrane. F o r the present, therefore, the only class of theory which may be properly applied to biological membranes is one which is applicable to all membranes as a general principle of membrane behavior (the irreversible thermodynamic treatment o f membrane phenomena, for example). If, in fact, the "independence principle" is not a generally applicable theory, then conclusions about biological membranes based upon it become open to question. To demonstrate that the "principle" is not generally applicable, it is sufficient to show that it is inapplicable to any one particular membrane. As shown in the following section, the ion-exchanger membrane suits this purpose. Consideration of it is germaine to the question of the general applicability of the "principle" to biological membranes unless, in the future, it can be shown that fixed charges are unrelated to the electrical properties of biological membranes [existing evidence suggests the opposite (see references cited in Segal, 1968)]. f Hodgkin and Huxley's application of the "independence principle" is in error. For a given voltage-clamp depolarization they compared peak values of I and I' rather than values of I and I" at the same time following initiation of the depolarization. Practically, for their situation the distinction is not too important, but it does seem that this is the cause of some of the discrepancies they noted. Chandler & Meves (1965) made the same error in interpreting the effect of [Na]~ on the early transient of voltage-clamp current. However, in their case the correct application of the "independence principle" will increase the already great discrepancies between it and the observed effect of reducing [Na]~ (e.g. they state that when [Na]~ is raised from zero to 150 mM the value of the early I' measured for a t 14 mv depolarization is only 0"47 of that predicted by the "principle").

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MISUSE

3. Analysis The inapplicability of the "independence principle" as a general theory is demonstrated by the following examples of the theoretical behavior of a fixed charge, ion-exchanger membrane. These examples illustrate (a) how use of the "principle" can provide false information about the current carriers, (b) how there can be effects of ionic substitution which are inexplicable by the "principle", although (c) in particular circumstances the "principle" can correctly identify the carriers. In addition, the analysis of the ion-exchanger examples in terms of the "principle" reveals the origin of the generally-held belief described in the second paragraph of section 1. Consider the case of a highly negatively charged membrane situated between two identical aqueous solutions of C molar NaCI plus C/IO molar KCI, where C is so small compared to the fixed charge concentration that there is complete co-ion (C1) exclusion from the membrane. Let all the NaCI of one solution phase (the "inside" phase) be removed. We will examine the effect of this removal on the limiting conductance obtained for large transmembrane potential differences when the "inside" phase is made positive with respect to that of the second, "outside", phase (the conductance is referred to hereafter as "limiting cathodal conductance", in conformity with electrophysiological convention). In the first of two cases to be explored (case I) the NaCI is replaced by a completely dissociated uni-univalent electrolyte, KaCI, where Ka is any cation other than Na or K, while in the second case (case II) the NaC1 is replaced by an osmotic equivalent of nonelectrolyte. CASE I

The limiting cathodal conductance of the membrane in the original situation, G, is G = A F 2 / d • (CNaUNa+ C'KUK) where the C"s are the intramembrane concentrations at the membrane"inside" solution interface, the u's are the intramembrane mobilities, subscripts denote the species to which the quantities refer, A is membrane area, and d is membrane thickness [equation (37), Teorell, 1953; Schlrgl, 1954]. Note that, of course, the relative portions of G due to the movements of Na or of K are determined by the relative values of the mobilityconcentration terms within the bracket. The boundary conditions are: C'Na+ C'K--N = 0 (local electroneutrality), where N is the fixed charge concentration, and ct~:~ = ~na Cr,/CK CNa, where the C's are concentrations in the bulk "inside" solution phases. The use of ~a, the separation factor of Na and K (p. 153, Helfferich, 1962), establishes a more general boundary relationT.B.

I1

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ship between intra- and extra-membrane concentrations than the familiar, simple Donnan distribution. Thus, the effect o f m e m b r a n e selectivity is included. (Note that ~ a = 1 for a D o n n a n distribution and that deviations from ~ a = 1 denote that the membrane prefers one species to the other. When N a is preferred to K, c~ ~ > 1, and when K is preferred to Na, c~ ~ < 1.) When the two boundary conditions are inserted into the expression for G, it follows that

G = NAF2/d'(uK + 10cq~a UN~)](I + 10ezra).

(3)

An analogous expression obtains for G', the limiting cathodal conductance when K a has completely replaced Na, and

G'/G = (1 + 10~'~)(uK + 1 0 ~ ~uK~)/(1 + 10~a)(u~ + 1 0 ~ ~urea).

(4)

There are three possible effects of substituting K a for N a : G'/G = 1, G'/G < 1, or G'/G > 1. Each of these will be interpreted now as they would be by the "independence principle" and the results contrasted with an actual physical situation as it might exist. In general, G can be written as G = gN~+gK+gc~ where the g's are the respective portions of the total limiting cathodal conductance due to the movements of Na, of K, and of C1 (gcJ is zero in the present ion-exchanger membrane example). According to the "independence principle", when K a is substituted for the "inside" Na, gN. vanishes, but neither gK nor gc, alters. Thus, the "principle" states that G' = gKa + gK+gct, where gKa is the portion of G' due to the movement of Ka. i f G ' / G = 1, then G ' - G = O , and according to the "independence principle" gN,,--gK~ = 0; therefore, TNa = rK., where the z's are the respective transport numbers of N a and of K a (no statement can be made about the value of zK+Zc~, the sum o f the transport numbers of K and CI). At best, this is not a useful result because the ~'s could be any one of all their possible values, and no information would be available as to the actual current carried by each species. At worst, the conclusion that zN~ = ~ , can be incorrect. F o r example, if ct~~ = 0 and c~ ~ = oo, then [equations (3) and (4)] ZtK t/Na, "CNa 1 and zK~ = 0.t Thus, the transport numbers of N a and K a =

=

"~ Note that upon substituting Ka for Na, zK increases from 0 to 1 ; since G'/G = 1, the substitution causes the actual current carried by K, at a given transmembrane potential difference, to increase. This example and those which follow are protinstances of interaction of flows, in the sense of irreversible thermodynamics, but are due to the alterations in the intramembrane concentrations of Na and K which occur at the "inside" membrane-solution interface when the "inside" NaCI is removed. The alterations are those necessitated by the separation factors and the maintenance of local electroneutrality within the membrane. The fixed charge theory employed here assumes that the Nernst-Planck equation describes the flux of each species and therefore assumes that flows are independent.

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are as different as is possible, but nevertheless the "principle" would state that they are equal.t The general implication of G'/G < 1 is, according to the "independence principle", that #Ka < gNa. F o r example, consider the concrete case o f G ' / G = 1/10. The "principle" states that #Na--gK~=9G/10; therefore, 9G/10 < #Na -< G, and 0 < #K, < G/IO. The respective transport numbers are: 9/10 < zNa < 1, 0 _< zK~ < 1, 0 < ZK+ZCl < 1/10 initially, and 0 < zK+Zci <_ 1 when K a has replaced Na. In contrast, if0t~~ = 0 and 0t~a = ~ , then [equations (3) and (4)] uK = 10UKa; hence, ZNa = 0, zKa = 1, %: = 1 initially and z~: = 0 in the second solution. Thus, the identification of the current carriers in the original NaC1-KC1 solution would be completely in e r r o r - - t h e actual values of the transport numbers o f N a and of K would be at the opposite extremes f r o m those deduced from the "principle". When G'/G > 1, gK, > gnu, according to the "independence principle". Consider the particular case o f G'/G = 10, for example. According to the "principle", gK,--gN~ = 9G and z~a = zNJlO+9/lO, i.e. "t'Ka>__"L'Na.However, let ~ = ~ and ct[" = 0, then [equations (3) and (4)] uN~ = uK/lO, zN~ = 1, and zKa = 0. T h a t is, zK, < zN~--diametrically opposite to that deduced f r o m the "principle". These three examples demonstrate that the "independence principle" is not a generally applicable theory of the relationship between total m e m b r a n e conductance and ionic concentrations. There can be changes in m e m b r a n e conductance which are superficially consistent with the "principle", but their analysis can yield, as shown, information about the m e m b r a n e ' s current carriers which is both quantitatively and qualitatively false. CASE 11

When all the "inside" NaCl is replaced by non-electrolyte, CK = N, and G " = uKN,4FE/d, where G" is the limiting cathodal conductance in the second solution. Thus,

G"/G = uK(1 + 10ct~)/(u~ + 10ct~a uN,).

(5)

As in case 1, there are three possible effects: G"/G = 1, G"/G > 1, or G"/G < I. According to the "independence principle" the sole relevant effect of removing the "inside" NaCI would be to eliminate gNa; gc~ and #K would not alter. Thus, G" would be given by G" = gK+#c~. t In a study of the hyperpolarizing resistance increase of lobster muscle fibers, Reuben, Werman & Grundfest (1961) replaced the NaCI of the bathing medium by a variety of salts; they found no alteration in the resistance. From this result they concluded that neither Na nor CI was a current carrier. No explicit theoretical justification for such an interpretation is given, but the entire context of the paper suggests that it is the "independence principle". However according to it, all that could be deduced from the experiment is that the electrical transport of all the species studied was identical.

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If G"/G = I, then the "independence principle" states that the transport number of Na, %a, must be zero. However, if e~ a = 1, then [equations (3) and (5)] u~ = uNa and ZN, = 10/11. Thus, the "principle" would fail to reveal the fact that, when present, Na was virtually the exclusive carrier of membrane current--further evidence of the inapplicability of the "principle" as a general theory. The effect G"/G > I is inexplicable in terms of the "independence principle". According to it, there is no mechanism by which replacing an ionic species by a non-electrolyte can lead to a conductance increase. The manner in which such phenomena have been explained by users of the "principle" is discussed below. For the present purpose, note only that G"/G> i is perfectly consistent with the behavior of an ion-exchanger membrane. It would be obtained if, for example, aNa = 1 and u~: > uNa. Finally, consider G"/G< 1. According to both the "independence principle" and equations (3) and (5) %a > 0, if G"/G < 1. In the particular case of G"/G = 1/i0, for example, the "principle" states that gsa = 9G/10 and ZN, = 9/10; and equations (3) and (5) state that zN, > 9/10 for all values of ~N~, USa and uj¢ commensurate with G"/G = 1/10. Thus, there is at least one instance in which the "independence principle" can be used to make an accurate identification of the current carriers of an ion-exchanger membrane. This raises the possibility that certain biological instances of consistency between the "principle" and the effect of removing an ionic species occur because the current carried by the species is indeed that which has been inferred from such analyses (e.g. Adelman & Taylor, 1964). It might be argued that such consistency is evidence that those conditions exist which are required for the "principle" to be applicable. However, to reason so is to drop, without justification, the context of the numerous experimental situations where the electrophysiological effect of removing an ionic species or substituting another for it is inexplicable in terms of the "independence principle". Consider the following four examples of the behavior of the squid axon, each of which is inconsistent with the "principle". First, Chandler, Hodgkin & Meves (1965) reduced the [K]~ of a squid axon by replacing most of the KCI of an internal perfusate by sucrose; they found a reduction of total steady-state, outward, voltage-clamp current which follows the "independence principle" if potassium is the exclusive carrier of steady-state, outward current.t However, when one-half of the original intraceUular potassium was replaced not by sucrose but by rubidium, the t This use of the "independence principle" overlooks the experimental evidence that potassium influx and e~ux are not independent (Hodgkin & Keynes, 1955) and that therefore equation (1) does not apply to potassium.

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steady-state current decreased by 71 ~o (Chandler & Meves, 1965). This is in contrast to the maximum decrease of 50 ~o which could occur according to the "principle". The maximum decrease would be obtained if potassium were the exclusive carrier of steady-state current and rubidium carried none. The discrepancy becomes even more severe when the rubidium is replaced by cesium: the steady-state, outward, voltage-clamp current becomes completely negligible. Second, the observed effect of [K]o on the steady-state current-voltage relationship cannot be accounted for by the "independence principle". When [K]o is increased by substituting potassium for sodium (or for choline, when choline has replaced all the sodium), thereby maintaining a fixed concentration of extracellular anions, the limiting cathodal conductance decreases (Segal, 1959; Ehrenstein & Gilbert, 1966) [a 10-fold linear increase upon raising [K]o from 10"6 to 200 mM (Segal, 1959; unpublished observations)]. Contrariwise, the "principle" states that there should be no alteration--no matter what the respective transport number of all the species present. The limiting cathodal conductance should depend upon [K]~ but should be independent of [K]o.t Third, when the extracellular sodium chloride of a squid axon is replaced by choline chloride (Hodgkin & Huxley, 1952) or by sucrose (Adelman & Taylor, 1964), the ensuing alterations in voltage-clamp current are consistent with the "independence principle" if the electrical transport o f choline is negligible and the early current transient is due to sodium. However, following intracellular injection of tetraethylammonium chloride the "principle" is no longer applicable to the effect of [Na]0: reduction o f [Na]o leads to an increase in the conductance due to sodium, calculated according to the "principle"---contradictory to the "principle" which demands a decrease (Tasald & Hagiwara, 1957). The fourth and final example is the most familiar one: the effect of divalent ions. The electrical transport due to the movement of calcium through the I" Ehrenstein & Gilbert (1966) have proposed that the decreased cathodal conductanc~ in 440 m i [K]o is a consequence of the prolonged depolarization of the membrane by the elevated [K]o. They showed that following a hyperpolarizing voltage-clamp pulse in elevated [K]o, the cathodal conductance temporarily increases to its value in normal [K]o. However, the converse effect has not been demonstrated: there is no evidence that with normal [K]o the limiting cathodal conductance of squid axons depolarized for extended periods by (outward) current is the same as when elevated [Klo is the depolarizing agent. Ehrenstein and Gilbert appear to have failed to obtain this result when they performed the experiment [a positive related result was obtained for the node of Ranvier (Frankenhaeuser & Waltman, 1959)]. Thus, the failure of the "independence principle" to account for the effect of [K]o cannot be attributed to the depolarizing action of an elevated [K]o. It might be argued that the deviation is related to the fact that equation (1) does not apply to potassium (see footnote on p. 166). But this is to offer further, independent, evidence of the inapplicability of the "principle".

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squid axon membrane is extremely small, compared to that due to sodium and potassium, for example (Hodgkin & Keynes, 1957). But nevertheless, alterations in [Ca]o have profound electrophysiological manifestations: both the kinetics and magnitudes of the conductance changes measured under voltage-clamp are affected by [Ca]0. For most of the phenomena observed, the effect of [Ca]o can be summarized by the statement, " . . . a fivefold increase in [extracellular] calcium is equivalent to a hyperpolarization of l0 to 15 my" (Frankenhaeuser & Hodgkin, 1957). This effect is quite significant as membrane conductance can increase 50-fold for a 10 mv change in membrane potential (Fig. 3, Frankenhaeuser & Hodgkin, 1957). The effect of calcium is inconsistent with the "independence principle": according to it, the current carried by calcium is too slight for an alteration in [Ca]o to have so great an effect on total membrane conductance. What, then, is the significance of those instances where the experimental data is consistent with the "independence principle"? The answer to this question requires a complete explanation for a//the experimental findings. Only when that has been found will it be known whether (a) it is simply coincidental that certain phenomena are in accord with the "'principle", or (b) the phenomena arise because the physical mechanism(s) of electrical transport underlying the theory is operative within the membrane in those particular instances. Only then will it be known whether phenomena inconsistent with the "principle" arise because (a) the transport mechanism is transformed from one subsumable by the "principle" to one which is not, or (b) the transport mechanism is invariant but not subsumed by the "principle". Until such questions have been resolved there is no justification for ignoring the evidence which is incompatible with the "principle" and focusing on only that which is consistent with it. An attempt has been made to subsume apparently incompatible data, as that above, within the formulation of the "independence principle". The logical consequence of this procedure is to completely undercut the claimed significance of those instances of consistency with the "principle". Effects such as those of the four examples above have been described as the "pharmacological" (Grundfest, 1961) (my quotation marks) or "inhibitory" (Chandler & Meves, 1965) (my quotation marks) actions of the ionic species under study. This terminology is offered as an explanation for the failure of the "independence principle" to account for the effect of altering the concentration of an ionic species. The only evidence for the separate class of actions referred to as "pharmacological" or "inhibitory" is the failure of the "principle". That is, according to the "'independence principle" the observed effect of altering the concentration of an ionic species cannot be due solely to an alteration in a current carried by that species.

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Within the context of the "principle" the "pharmacological" or "inhibitory" species would therefore be viewed--necessarily--as acting on the current carried by another species. But this contradicts the "principle", which assumes that alterations in the concentration of a species affect only the current carried by that species. It might be thought that this contradiction is resolved satisfactorily by the following modification of the "independence principle". If the kj's in equation (1) are no longer assumed to be independent of the concentration of a// species, then it can be said that the "pharmacological" and "inhibitory" species act by modifying the kj of one or more of the other species. In this manner "pharmacological" and "inhibitory" effects appear to be subsumable by the "principle". Note, however, that the physicochemical origin of the k / s has not been specified (Hodgkin & Huxley, 1952); thus it is not possible to evaluate the physical plausibility of such an explanation within the framework of the "'principle" alone--another theory is required. Without such a theory, no matter what the effect of the concentration of a species it can always be described, ad hoc, as a suitably tailored "pharmacological" or "inhibitory" action; the "explanation" can be neither denied nor confirmed within the context of the "principle". In addition to this problem of validation, the hypothesis of "pharmacological" and "inhibitory" phenomena, per se, as defined within the framework of the "independence principle", has a crucial bearing on the epistemological status of the "principle". Once it is stated that ions can have "pharmacological" or "inhibitory" actions then the meaning of those instances where the membrane's behavior is consistent with the "principle" becomes unknown within the context of the "principle" alone. How cart one distinguish between the "pharmacological" or "inhibitory" action of altering an ion's concentration and the effect being due to the fact that it is a current carrier ? How does one know that adherence of an ionic species to the "'independence principle" is not merely its particular form of "'pharmacological" or "'inhibitory" action on the movement of another species ? Thus, if it is claimed that ions can have "pharmacological" or "inhibitory" actions, as defined by the "principle", then adherence to the "independence principle" (in the original sense of Hodgkin & Huxley, 1952) cannot be claimed as evidence that a species is a current carrier. 4. Conclusion

It has been amply demonstrated that the "independence principle" cannot be used to identify the carriers of current through biological membranes. The fundamental basis for this rejection is that the "principle" is not a generally applicable theory. In the case of biological membranes it is unknown

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if the particular conditions required for the applicability of the "principle" exist.Without such knowledge, use of the "principle" to interpret data which are apparently consistent with it can lead to a completely erroneous description of the intramcmbrane transport numbers. The "independence principle" is inadequate for the general purpose which underlies its use: to infer microscopic intramembrane processes from measurements of macroscopic electricalbehavior. This problem remains one of the central tasks of electrophysiology--its correct solution requires a membrane theory which is far more explicit and encompassing than the "independence principle". REFERENCES ADELMAN,W. J., JR. • TAYLOR,R. E. (1964). Biophys, J. 4, 451. CARMELmT,E. E. (1961). J. Physiol., Lond. 156, 375. C'r~.~,rDLER,W. K., HODGr,.n%A. L. & MEVES, H. (1965). J. Physiol., Lond. 180, 821. C-'HAN'DLER,W. K. & MEw.S, H. (1965). J. Physiol., Lond. 180, 788. DODGE, F. A. & FRANKENHAEUSER,B. (1959). J. Physiol., Lond. 148, 188. ErnteNs~n~, G. & GILBERT,D. L. (1966). Biophys. J. 6, 553. FRANKENHA~SER,B, (1962). 3". PhysioL, Lond. 160, 40. FRANKEN~ArUSER,B. & HODGKIN,A. L. (1957). J. PhysioL, Lond. 137, 218. FRANKENHAEUSER,B. & WAL'rMAN,B. (1959). J. PhysioL, Lond. 148, 677. GRUI,CDrrST, H. (1961). Ann. N. Y. Acad. Sci. 94, 405. HELrrE~CH, F. (1962). "Ion Exchange." New York: McGraw-Hill Book Co., Inc. HODGION, A. L. & HUXLEY, A. F. (1952). J. PhysioL, Lond. 116, 449. HODGKn% A. L. & K~Yt,r~s, R. D. (1955). d'. PhysioL, Lond. 128, 61. HODGKrN, A. L. & I~Yl,ms, R. D. (1957). J. PhysioL, Lond. 138, 253, Hosmxo, T. & Ln,n)t~Y, B. D. (1964). Biochim. biophys. Aeta 79, 301. HUTrER, O. F. & PADSrY.A,S. M. (1959). J. PhysioL, Lond. 146, 117. Y,~EM, O. & ESSIG,E. (1965). J. gen. PhysioL 48, 1047. R~UBEN, J. P., WERMAN,R. & GRUNDr~Sr, H. (1961). J. gen. PhysioL 45, 243. SCrn.6OL, R. (1954). Z. physik. Chem., Frankf. Ausg. 1, 305. SEOAL, L R. (1959). Doctoral Dissertation, Massachusetts Institute of Technology. SEGAL,J. R. (1968). Biophys. 3'. 8, 470. TASAKI,I. & HAOIWARA,S. (1957). jr. gen. PhysioL 40, 859. TEORELL,T. (1953). In "Progress in Biophysics and Biophysical Chemistry", vol. 3, p. 305. (J. A. V. Butler & J. T. Randell, eds.). New York: Academic Press, Inc. USS~G, H. H. (1949). Acta physioL scand. 19, 43.