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Constant-Field Equation” in Membrane Ion Transport

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Constant-Field Equation” in Membrane Ion Transport Milestone in Physiology JGP 100th Anniversary The enduring legacy of the “constant-field equation” in membrane ion transport Osvaldo Alvarez1 and Ramon Latorre2 1Departamento de Biología, Facultad de Ciencias, Santiago, Chile 2Centro Interdisciplinario de Neurociencia de Valparaíso, Facultad de Ciencias, Universidad de Valparaíso, Valparaíso, Chile In 1943, David Goldman published a seminal paper in The Journal of General Physiology that reported a concise expression for the membrane current as a function of ion concentrations and voltage. This body of work was, and still is, the theoretical pillar used to interpret the relationship between a cell’s membrane potential and its exter- nal and/or internal ionic composition. Here, we describe from an historical perspective the theory underlying the constant-field equation and its application to membrane ion transport. 2 2 z i FV /RT ______P i ​​ ​z i ​ ​F V _____________C i ,in​​ ​e ​ − C i ,out Introduction I i = ​ ​​ , (1) RT ( e z i FV /RT​ − 1 ) The Goldman (1943) paper is, perhaps, one of the most well-known articles published in The Journal of General where Ii is the current density carried by ion i, Pi is the Physiology. Because of the great impact that this article permeability coefficient, zi the valence of ion I, V is the still has on the study of transport of ions across mem- membrane voltage, Ci,in and Ci,out are the ion concentra- branes, the JGP editorial board thought that it deserved tions in the solutions bathing either side of the mem- some discussion and comments. brane, and R, T, and F have their usual meaning. In Goldman was interested in explaining the phenom- this equation, a positive current carries charge from the enon of electrical rectification found at the time in inside to the outside and the membrane voltage is the several different biological preparations such as Valo- internal minus the external electric potential. Goldman nia, a species of algae found in the oceans, the squid concluded that the theory could account reasonably giant axon, and the frog muscle (Blinks, 1930a,b; well for the variation of the resting potential of the giant Cole and Curtis, 1941; Katz, 1942). This phenome- axon of the squid with external K+ concentration as de- non, in which the current is large when flowing in termined by Curtis and Cole (1942). one direction and small in the reverse direction, The Second World War interrupted the work of most was hard to understand. Thermodynamic equations scientists, and it was not until 1949 that Goldman’s are not applicable, because ion transport in cells is analysis of the movement of ions across membranes out of equilibrium. Goldman’s experimental work reappeared in the literature. This time, it was invoked was an attempt to reproduce some of the electrical to show that Bernstein’s membrane hypothesis (Bern- properties of biological membranes in artificial sys- stein, 1902), although correct in the identification of The Journal General of Physiology tems. However, his work is mostly remembered (and the origin of the resting potential, was mistaken regard- cited) for the section in which he considered that ing the ionic basis of the action potential. Bernstein’s ions move according to their concentration (diffu- theory proposed that membrane excitation consisted of sion) and voltage gradients and that the electric field a large increase in membrane permeability to all ions, through the membrane is constant, hence the name in which case the membrane potential would fall nearly “constant-field equation.” Under these assumptions, to zero. Although in the late 1930s there were already and considering that the ion concentrations at the squid axon experiments indicating that there was a internal and external membrane boundaries are di- transient reversal of the membrane potential during rectly proportional to those in the aqueous solutions, nerve activity (Hodgkin and Huxley, 1939), there was he was able to arrive at an explicit solution of the no clear theoretical basis to explain this result. Hod- Nernst–Planck (NP) equation (Nernst, 1888; Planck, gkin and Katz (1949) took advantage of the Goldman 1890). The NP equation describes the contribution current equation and obtained an expression for the of concentration and voltage gradients to the current membrane potential, Vr, in terms of the internal and density carried by single ionic species. Today, we know external ion composition and the ion permeabilities: it as the constant-field current equation (Eq. 1): © 2017 Alvarez and Latorre This article is distributed under the terms of an Attribution– Noncommercial–Share Alike–No Mirror Sites license for the first six months after the publication date (see http ://www .rupress .org /terms /). After six months it is available under Correspondence to Ramon Latorre: [email protected] a Creative Commons License (Attribution–Noncommercial–Share Alike 4.0 International Abbreviations used: GHK, Goldman–Hodgkin–Katz; NP, Nernst–Planck. license, as described at https ://creativecommons .org /licenses /by -nc -sa /4 .0 /). The Rockefeller University Press J. Gen. Physiol. 2017 Vol. 149 No. 10 911–920 https://doi.org/10.1085/jgp.201711839 911 the Goldman–Hodgkin–Katz (GHK) voltage equation proteins. Having only Na+ outside and K+ inside of the (Eq. 2, which is formally equivalent to Eq. 18 in Gold- axon reduced the GHK to a simpler expression known man’s 1943 paper). as the bi-ionic potential, which give us a direct measure- ment of the PNa+/PK+ permeability ratio when the con- ___RT ______________________P K ​​ ​​[K ] out + ​PNa ​​ ​​[Na ] out + ​PCl ​​ ​​[Cl ] in V r = ​ ​ ln ​ ​ . (2) F P K ​​ ​​[K ] in + ​PNa ​​ ​​[Na ] in + ​PCl ​​ ​​[Cl ] out centrations on both sides of the membrane are equal. In fact, when reduced to the bi-ionic case, the GHK The use of this equation allowed Hodgkin and Katz to equation is the most convenient way of determining the show that under resting conditions, the K+ permeability ratio between the permeabilities of the permeant ions. is 20 times larger than Na+ permeability. During nerve We should mention here that in the case of a mem- activity, the situation is reversed, and Na+ permeability brane that is selective to either positive or negative ions, becomes 20 times greater than the K+ permeability at the constant-field assumption is not required and the the peak of the action potential (Hodgkin and Katz, voltage equation for the bi-ionic case can be obtained 1949). Notably, the experimental strategy used by Hod- through integration, using the NP equation and the ap- gkin and Katz (1949) to determine the changes in active propriate boundary conditions (see Appendix B; e.g., membrane potential with external Na+ is essentially the Sten-Knudsen, 1978). same as the one used by Hille almost 20 yr later to de- The group of Hille (Hille, 1971, 1972, 1973; Dwyer termine the ion selectivity of different ion channels (see et al., 1980; Wollmuth and Hille, 1992) determined the Eq. 2 in Hodgkin and Katz, 1949). ion selectivity of several ion channels, including Na+, K+, The sodium hypothesis could not explain, however, HCN (Ih), and ACh receptor channels. Using a series the fact that crustacean muscle fibers elicit action po- of inorganic and organic ions of different size and by tentials after replacement of the external sodium by determining the permeability ratios relative to the ref- tetraethylammonium (Fatt and Katz, 1953). Some years erence ions (Na+, K+) by applying the GHK equation, later, Fatt and Ginsborg (1958) found that an isotonic they were able to obtain the approximate dimensions of solution of Sr2+ or Ba2+ could sustain large crustacean the selectivity filter of these channels. Because the inter- muscle action potentials. To interpret the data obtained nal concentration of the reference ions was unknown in crustacean muscle fibers, Fatt and Ginsborg (1958) in the different preparations, the experimental strategy expanded the constant-field theory so as to include di- used was to determine first the reversal potential in the valent cations, arriving at the conclusion that divalent presence of the reference ion and then in the presence cations determined the peak of the action potential in of the test cation. They obtained the permeability ratio these muscle fibers. The GHK voltage equation that between the test ion and the reference ion from the dif- incorporates a mixture of monovalent and divalent ference of these potentials because the GHK equation cations predicted that in this type of preparation the contains only the permeability coefficients and concen- permeability of Ba2+ at the peak of the action potential trations of the test and reference ions. was 50 times larger than that of K+. This was the first The GHK voltage equation can also be used to calcu- indication of the existence of Ca2+ action potentials and late the membrane voltage when there is a contribution hence voltage-dependent Ca2+ channels. to the total current from electrogenic pumps, as is the Hodgkin and Huxley’s dissection of the ionic currents case, for example, of the Na+/K+ pump or of electro- that underlie the action potential (Hodgkin and Hux- genic exchangers such as the Na+/Ca2+ exchanger (e.g., ley, 1952) identified voltage-dependent Na+ and K+ cur- Sjodin, 1980; Mullins, 1981; Armstrong, 2003). rents in the giant axon of the squid. On the other hand, A quick review of the JGP archives revealed that the in the early 1960s, Baker et al. (1962) were able to inter- GHK equation has also been used to determine the rel- nally perfuse giant squid axons. This made it possible to ative ion permeability in mechanotransducer channels learn about the selectivity of the Na+ conductance as the (Beurg et al., 2015), in Ca2+-activated Cl− channels (Jeng ionic composition of the internal medium could now et al., 2016), and in T-type Ca2+ channels (Smith et al., be controlled.
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