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It has been widely acknowledged that the Goldman-Hodgkin-Katz (GHK) equation fully explains membrane potential behavior. The fundamental facet of the GHK equation lies in its consideration of permeability of membrane to ions, when the membrane serves as a separator for separating two electrolytic solutions. The GHK equation describes that: variation of membrane permeability to ion in accordance with ion species results in the variation of the membrane potential. However, nonzero potential was observed even across the impermeable membrane (or separator) separating two electrolytic solutions. It gave rise to a question concerning the validity of the GHK equation for explaining the membrane potential generation. In this work, an alternative theory was proposed. It is the adsorption theory. The adsorption theory attributes the membrane potential generation to the ion adsorption onto the membrane (or separator) surface not to the ion passage through the membrane (or separator). The computationally obtained potential behavior based on the adsorption theory was in good agreement with the experimentally observed potential whether the membrane (or separator) was permeable to ions or not. It was strongly speculated that the membrane potential origin could lie primarily in the ion adsorption on the membrane (or separator) rather than the membrane permeability to ions. It might be necessary to reconsider the origin of membrane potential which has been so far believed explicable by the GHK equation.

Potential generation between two electrolytic solutions separated with an ion-permeable membrane is one of fundamental phenomena in biology field known as membrane potential. Membrane potential has been for decades studied by a number of researchers [

The leading scientist at present who challenges the current membrane theory is a physiologist Dr. Gilbert Ling. He has advocated his own theory for explaining the origin of the membrane potential. Ling insists that the membrane permeability to the ions has nothing to do with the potential generation and the ions adsorption on the membrane surface generates the membrane potential. His theory is in harmony with all the past reports about the membrane potential generation, and even the reports which have been explicable by the GHK equation are within the range of his theory [

An electrochemist, Dr. Cheng, decades ago proposed a similar theory to the Ling’s for explaining the potential generation of the glass electrode [

Our study of membrane potential that will be described in this paper also brought us a conclusion that the membrane potential might have almost nothing to do with the membrane permeability to the ions, and the ion adsorption on the membrane surface appears to have dominant influence on the potential generation. Based on our experimental observations, we derived an equation for explaining the potential we experimentally observed. The equation never takes into consideration the passage of ions through the membrane but needs to take into consideration the ion adsorption on the membrane surface. The computational potential behavior obtained by employing the equation was in good agreement with the experimental results.

Potential generated between two KCl aqueous solutions was measured using an electrometer of HE-104A (HOKUTO DENKO CO., Tokyo, Japan) and Ag/AgCl electrodes. All the KCl solution used in this research were prepared by dissolving KCl with highly deionized water. These two KCl solutions were in contact with each other through the intermediary of a sheet of Selemion AMV (Asahi Glass Co., Ltd., Tokyo, Japan) which is an anion exchange membrane. The anion exchange membrane is quite permeable to anions and less permeable to cations, since it contains fixed cations and mobile anions in the hydrated state. Whole experimental system is illustrated in _{L} while that of right solution is represented by _{R}.

Experimental setup for measuring the potential between two KCl aqueous solutions separated through the intermediary of (_{L} and _{R}, respectively, for all the setups.

Potential generated between two separately placed KCl aqueous solutions was measured using a setup illustrated in

Photo of an Ag wire and an Ag wire coated with AgCl.

Potential generated between two KCl aqueous solutions separated with a surface-modified glassy carbon plate was measured as illustrated in

The surface-modified glassy carbon plate we use had –COOH groups on its surface. Its fabrication procedure is explained here. A glassy carbon plate was placed in mixed acid consisting of HNO_{3} and H_{2}SO_{4} weight ratio 1:3 [

GC1: A glassy carbon plate was placed in a mixed acid for 8 h at 302 K. After this treatment, the glassy carbon plate was washed with deionized water and dried in atmosphere.

GC2: A glassy carbon plate was placed in a mixed acid for 36 h at 302 K. After this treatment, the glassy carbon plate was washed with deionized water and dried in atmosphere.

GC3: A glassy carbon plate was placed in a mixed acid for 36 h at 318 K in hot water. After this treatment, the glassy carbon plate was washed with deionized water and dried in atmosphere.

From the view point of chemistry, the concentration of –COOH on the surface of the glassy carbon plates increases in the order of GC1–GC2–GC3.

We made measurement of potential,

Potential across the anion exchange membrane separating two KCl solutions was measured at the environmental temperature _{L} was increased from 10^{−5} M to 3.4 M, while that of _{R} was maintained constant at 0.1 M. According to the current membrane theory, the membrane potential is given by Equation (1), which is based on the GHK Equation.
_{K} and _{Cl} represent the permeability of membrane to K^{+} and Cl^{−}, respectively; [K^{+}]_{L} and [K^{+}]_{R} represent the K^{+} concentration in the left and right compartments, respectively; and [Cl^{−}]_{L} and [Cl^{−}]_{R} represent the Cl^{−} concentration in the left and right compartments, respectively;

Assuming the anion exchange membrane is virtually impermeable to the cation, K^{+}, Equation (1) is reduced into Equation (2).
^{−}]_{R}/[Cl^{−}]_{L}. Since [Cl^{−}]_{R} = _{R} and [Cl^{−}]_{L} = _{L}, [Cl^{−}]_{R}/[Cl^{−}]_{L} = _{R}/_{L}.

_{10}([Cl^{−}]_{R}/[Cl^{−}]_{L}). The computational potential data was obtained by considering ion activity of KCl [

Potential generated between two KCl solutions ^{−}]_{R}/[Cl^{−}]_{L}). ○: an anion exchange membrane separator; ×: Ag wire separator; •: potential computed by employing the Goldman-Hodgkin-Katz (GHK) equation.

Experimentally observed potential was obtained, once the potential reached the stable potential, and usually such a stable potential was achieved tens seconds after supplying two KCl solutions in both left and right compartments of setup illustrated in _{L}. However, the deviation of each experimental potential data from the average potential in _{L} was as low as 10^{−5} M. Another potential, when _{L} was 3.4 M, was 72 ± 1 mV.

Although the discrepancy between the experimental and computational potential becomes larger at lower −log_{10}([Cl^{−}]_{R}/[Cl^{−}]_{L}), they are still in good agreement with each other. As long as the concept of the GHK equation is valid, the discrepancy could be amended by considering the concentration of ions not taken into account for the potential computation of _{L}. Hereafter, the experimentally observed potential across the anion exchange membrane shown in _{aem}, and the computational data set shown in _{com}.

To sum up, these experimental results are in line with the prediction of GHK equation, and nothing unusual was observed. So, the experimental result in

Potential across the Ag wire generated by two KCl solutions was measured using the setup _{Ag}. _{Ag} behavior is quite similar to _{aem} and even in better agreement with _{com}. Similarly to the procedure of potential measurements across the anion exchange membrane described in the previous section, once the potential reached the stable potential, experimental potential data was obtained. Compared with the potential measurement described in the previous section, the potential was quite stable and such stable potential was achieved much shorter period after supplying two KCl solutions in the left and right compartments of setup illustrated in _{Ag} in _{L}, which is similar trend to the trend described in the previous section. However, the deviation of each experimental potential data from the average potential becomes less compared with those of the potential data _{aem}, and the deviation of most of the _{Ag} from the average potential shown in _{L} was as low as 10^{−5} M. Another potential, when _{L} was 3.4 M was 91 ± 1 mV.

Surface of AgCl coated wire looks quite inhomogeneous as seen in _{Ag} shown in

Ag wire is undoubtedly impermeable to ions. However, the potential across the Ag wire was almost same as the potential across the anion exchange membrane which is permeable to ions. Is membrane permeability a primary factor for determining how high membrane potential is generated?

Ling and Cheng independently advocate their own but mutually similar theories for explaining the membrane potential generation mechanism [_{Ag} considering ion adsorption on the surface of AgCl coat on the wire.

Cheng reported that Cl^{−} adsorbs onto AgCl. Hence, we assume that AgCl at the ends of Ag wire illustrated in ^{−}. Under this assumption, we derived an Equation explaining the potential behavior of _{Ag}. ^{−}. Hence, Equations (3)–(5) are derived using the concept of Langmuir isotherm, where K is an association constant.

The interface between AgCl coat at the Ag wire end and KCl solution.

Total adsorption site concentration [s]_{T} is given by Equation (4).
_{T} = [s] + [sCl^{−}]

Concentration of sCl^{−} is given by Equation (5) using Equations (3) and (4).
^{−}] =

Charge density in KCl solution, ρ, is given by Equation (6) because of the Boltzmann distribution of ions.
_{o}: concentration of K^{+} and Cl^{−} at

Owing to the charge neutrality, Equation (7) establishes.
^{−}] +

Employing Poisson-Boltzmann equation and Equation (6), Equation (8) is derived.
_{o}: vacuum permittivity, ɛ: relative permittivity of water.

Equation (9) is derived by the use of Equations (5), (7) and (8), where the boundary condition d

Equation (10) is derived by the use of Equations (6) and (8) under the boundary conditions of

Solving Equation (10) with respect to

The association constant ^{−}] one another. Hence, [Cl^{−}] in Equation (11) can be given as _{o}. Therefore, Equation (11) is written as Equation (12).

Electric field within the range 0 < _{sur}. Hence, Equation (13) establishes.
_{sur}
_{sur} is given by Equation (14) owing to the Gauss’s law, and ɛ_{low} in the Equation (14) represents the permittivity of water within the range 0 < _{low} is expected to be much lower than ɛ due to the dielectric saturation of water [_{sur} =

Equation (15) is derived by the use of Equations (9), (12)–(14).

It is speculated that there exists nonzero potential in the bulk phase of KCl solution, when the potential at _{L}(_{R}(_{L}(_{R}(_{L}(_{R}(

Graphical correspondence between experimental setup and adsorption model.

Substituting the Equations (16) and (17) into ∆ϕ = ϕ_{L}(_{R}(

As described in the _{Ag} was measured by increasing _{L} from 10^{−5} M to 3.4 M, while maintaining _{R} at 0.1 M. By the use of same condition, theoretically expected potential was obtained by computing ∆ϕ. For the computation, four factors should have been determined experimentally but were unable to be determined. They were [s]_{T}, _{low}. Hence, we assumed quite plausible quantities for them as follows: [s]_{T} was assumed to be 1.56 × 10^{18} m^{−2} by considering molecular dimension. It was guessed that K is quite low, since the experimentally obtained potential did not saturate by the increase of C_{L} even up to the highest concentration as seen in ^{−25} m^{−3}. d was assumed to be 5 Å by considering interfacial structure illustrated in _{low}. However, some literature shows that hydrated ions permittivity is only a bit lower than the relative permittivity of water [_{low}. The results are shown in _{10}_{L}. Computational potential represented by □ in _{10}_{L} and the computational potential was obtained by considering ion activity of KCl [

Although a good agreement between the experimental and computational potentials is seen in

Potential across the surface-modified glassy carbon plate separating two KCl solutions was measured using the setup _{10}[K^{+}]_{L} (=log_{10}_{L}) as shown in _{GC}. We take log_{10}[K^{+}]_{L} as

×: potential generated between two KCl solutions using a separator of Ag wire _{10}_{L} (=log_{10}[Cl^{−}]_{L}); □: computational potential ∆ϕ based on the adsorption theory _{10}_{L} (=log_{10}[Cl^{−}]_{L}); Data represented by • in this diagram is created by rearranging the data in

Potential generated between two KCl solutions using a separator of GC1 (GC2, GC3) _{10}_{L} (=log_{10}[K^{+}]_{L}), where offset potentials are added to some data for decongesting the diagram. Open and closed marks represent the experimental and computational potentials, respectively. Circle marks: in case a GC1 used Triangle marks: in case a GC2 separator used (+25 mV offset potential is added to the actual data) Square marks: in case a GC3 separator used (+50 mV offset potential is added to the actual data).

^{+} and H^{+} are in the adsorbed state to –COO^{−}. Equations (18) and (19) are derived for the dissociation of –COOH and that of –COOK, where _{h} and _{k} are dissociation constants for them, respectively.

Interface between the surface-modified glassy carbon plate and KCl solution. Some potassium ions are in the state of adsorption to –COO^{−}. Coordinate system is set in the same way as

_{h}=

_{k}=

[–COO_{T} represents the total concentration of –COO^{−}, –COOK and –COOH, and it is given by Equation (20).
_{T} = [–COO^{−}

Equation (21) is derived by the use of Equations (18)–(20).
^{−}

In the same manner as described in the ^{−}] +

The same Poisson-Boltzmann equation expression as Equation (8) establishes for the system in question. By the use of the Poisson-Boltzmann equation, Equations (21) and (23), Equation (24) is derived as Equation (9) was derived, where the boundary condition d

By the same procedure for deriving Equations (11), Equation (25) is derived.

Employing Equation (24) and the same derivation procedure for Equations (16) and (17), Equations (26) and (27) are derived.

Again employing the same procedure described in the _{L}(_{R}(_{GC}. For the computation, [H^{+}] was assumed to be 2.41 × 10^{21} m^{−3} (=4 × 10^{−6} M), since this experiment was carries out in the atmosphere. _{h} was assumed to be 2.41 × 10^{22} m^{−3} (=4 × 10^{−5} M) based on the work described in the reference [_{k}, _{low} and [–COO_{T} should have been determined experimentally but we could not determine them. Hence, we assumed quite plausible quantities for the first three as follows: Since –COOK is a salt, it is expected the dissociation constant of _{k} is by far higher than _{h}. _{k} was assumed to be 10-fold of _{h}, _{k} = 2.41 × 10^{22} m^{−3}. _{low}. However, the relative permittivity of water around proteins is often assumed to be quite low for computer chemistry field or so [_{low}. As to [–COO_{T}, the following quantities were assumed: [–COO_{T} = 1.38 × 10^{16} (≡[–COO_{1} for GC1), 4.00 × 10^{16} (≡[–COO_{2} for GC2), 25.0 × 10^{16} (≡[–COO_{3} for GC3) m^{−2}. All the computationally obtained potentials relatively well reproduce the experimental results. According to the fabrication process of the surface-modified glassy carbon plates, it is expected that the density [–COO_{1} < [–COO_{2} < [–COO_{3} as expected. The computational potential in

Experimental results _{GC} exhibit unexpected behavior in low K^{+} concentration regime, that is, all the experimental diagrams have the highest peak of potential around at log_{10}_{L} = −4 or the highest plateau at and below log_{10}_{L} = −4. Even such unexpected potential behavior is reproducible theoretically. Hence, the adsorption theory as a generation mechanism for the membrane potential is strongly validated, although some physical quantities needed for the potential computation based on the adsorption theory were given by assumption. However, still one might emphasize that the potential across an impermeable membrane is generated by ion adsorption onto the membrane but the potential across a permeable membrane is generated by ion transport through the membrane. In order to defy this emphasis, we carried out another experiment shown in the next section.

We made measurement of the potential across the impermeable ion exchange membrane. The procedure is described as below.

The impermeable ion exchange membrane was fabricated by gluing two sheets of Selemion AMV. As described in the

Structure of SES membrane and the experimental setup for measuring the potential across the SES membrane.

Potential generated between two KCl solutions ^{−}]_{R}/[Cl^{−}]_{L}). ○: Potential across the permeable membrane of Selemion AMV (same data as that shown in

It was observed that the potential generated by two KCl solutions separated even by impermeable membrane (or separator) was almost the same as the membrane potential observed using an ion exchange membrane permeable to ions. This observation gave rise to a doubt as to the current concept of the membrane theory. Our experimental and theoretical analysis revealed that membrane potential behavior is explicable by adsorption theory instead of the GHK equation. The adsorption theory was capable of explaining the potential generation across the separators to the ions irrespective of their permeability to ions.

Based on the experimental and theoretical results, the authors believe that the origin of membrane potential so far believed explicable by GHK equation should be reconsidered and the GHK equation might be replaced with the adsorption theory.

We would like to express our gratitude to Tsutomu Mori (Tokyo Institute of Technology) for teaching us physical chemistry. We would like to extend our gratitude to Lion Corp. (Tokyo) for providing us with Ketjen black free sample. This research was conducted under the financial support of the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Challenging Exploratory Research, 26650032, 2014.

Hirohisa Tamagawa was a supervisor of Sachi Morita who was a senior student in Gifu university. Experimental part of this work was conducted largely by Morita under the guidance of Tamagawa. Theoretical part of this work was performed by Tamagawa.

The authors declare no conflict of interest.

^{+}and Na

^{+}in pH glass electrode