Generation of membrane potential beyond the conceptual range of Donnan theory and Goldman-Hodgkin-Katz equation

Abstract

Donnan theory and Goldman-Hodgkin-Katz equation (GHK eq.) state that the nonzero membrane potential is generated by the asymmetric ion distribution between two solutions separated by a semipermeable membrane and/or by the continuous ion transport across the semipermeable membrane. However, there have been a number of reports of the membrane potential generation behaviors in conflict with those theories. The authors of this paper performed the experimental and theoretical investigation of membrane potential and found that (1) Donnan theory is valid only when the macroscopic electroneutrality is sufficed and (2) Potential behavior across a certain type of membrane appears to be inexplicable on the concept of GHK eq. Consequently, the authors derived a conclusion that the existing theories have some limitations for predicting the membrane potential behavior and we need to find a theory to overcome those limitations. The authors suggest that the ion adsorption theory named Ling’s adsorption theory, which attributes the membrane potential generation to the mobile ion adsorption onto the adsorption sites, could overcome those problems.

Appendices

Appendix A

The left expression of (28) is derived by the equilibrium equation, COOK ⇌ COO⁻ + K⁺, where K _D is the dissociation constant. The left expression of (28) is transformed into the right expression of (28) using [Carboxyl] ≡ [COOK] + [COO⁻]. For the hydrogel synthesis, we used 1.44 g acrylic acid (M _w = 72.06) [19]. Assuming that the volume of hydrogel synthesized was the same as the volume of deionized water used for the gel synthesis (We used 0.05 L of deionized water.), and using the experimentally measured swelling ratio, 20.80 (see Table 2), [Carboxyl] in G-0.0001 is given by (1.44 ÷ 72.06) ÷ (0.05 ×20.80) = 0.019 M = 19 molm⁻³. The macroscopic electroneutrality represented by (29) establishes at $\mathrm {x} = +\infty$ under the condition [H⁺] and [OH⁻] are negligibly low.

$$ K_{D}=\frac{[COO^{-}][K^{+}]}{[COOK]} \rightarrow K_{D}=\frac{[COO^{-}][K^{+}]}{[Carboxyl]-[COO^{-}]} $$

(28)

$$\begin{array}{@{}rcl@{}} [K^{+}] - [COO^{-}] - [Cl^{-}] \sim 0 \end{array} $$

(29)

Employing Donnan theory and $\phi _{g}|_{x=+\infty }= -0.076$ V in Table 2, (30) is derived, where B ≡ e/kT. (30) gives $[\mathrm {K}^{+}(\mathrm {x}=+\infty )] = 2.02~\text {mol m}^{-3}$ and $[\text {Cl}^{-}(\mathrm {x}=+\infty )] = 0.005~\text {molm}^{-3}$. Using them and (29), $[\text {COO}^{-}]|_{x =+\infty } = 2.015$ molm⁻³ is obtained. Plugging [Carboxyl] = 19 molm⁻³, $[\mathrm {K}^{+}(\mathrm {x}=+\infty )] = 2.02~\text {mol m}^{-3}$ and $[\text {COO}^{-}]\mid _{x =+\infty } = 2.015$ molm⁻³ into the right expression of (28), K$_{D} = 0.24~\text {molm}^{-3}$ is obtained.

$$\begin{array}{@{}rcl@{}} [K^{+}] = 0.1N_{A}\exp(-B\phi_{g}|_{x=+\infty}) \ \ \ [Cl^{-}] = 0.1N_{A}\exp(+B\phi_{g}|_{x=+\infty}) \end{array} $$

(30)

Appendix B

K _A in (19) was estimated by the following experiment: A 2 mm-thick hydrogel containing -SO₃H groups was synthesized. The pregel solution consisted of 2-acrylamido-2-methylpropane sulfonic acid (20.7 g), N,N’-methylenebisacrylamide (0.077 g), N,N,N’,N’-tetra-methylethylenediamin (a few drops), ammonium persulfate (0.04g) and deionized water (50 g). The hydrogel synthesized was equilibrated in 0.01 M KCl solution. We assume that -SO₃H was converted into -SO₃K. We made measurement of potential, ψ, at the far inside of this hydrogel in reference to the bathing solution and observed ψ = 0.0165 V.

[K⁺] and [Cl⁻] at the far inside of the hydrogel is given by (31), where B ≡ e/kT. Because of the equilibrium equation SO$_{3}^{-}$ + K⁺ $\rightleftharpoons$ SO₃K, (32) is derived, where K _a is the association constant of -SO₃K of hydrogel.

$$\begin{array}{@{}rcl@{}} [K^{+}]=10N_{A}\exp(-B\psi) \ \ \ \ \ [Cl^{-}]=10N_{A}\exp(+B\psi) \end{array} $$

(31)

$$ K_{a}=\frac{[SO_{3}K]}{[SO_{3}^{-}][K^{+}]} $$

(32)

The total quantity of SO₃K + SO$_{3}^{-}$ is given by 20.7 ÷ 207 = 0.1 mol, where the molecular weight of 2-acrylamido-2-methylpropane sulfonic acid is 207. Assuming that the volume of gel synthesized was same as the volume of deionized water used for this synthesis, i.e. 0.05 L, and using the experimental swelling ratio of this hydrogel in 0.01 M KCl solution, 47, the total concentration of SO₃K + SO$_{3}^{-}$ in the hydrogel was given by 0.1 ÷ (0.05 ×47) = 0.043 M = 43 molm⁻³. Therefore, [Sulfonic] ≡ [SO$_{3}^{-}$] + [SO₃K] = 43 molm⁻³. (32) is rearranged into (33) using [Sulfonic] = [SO$_{3}^{-}$] + [SO₃K]. It is quite natural to believe that the electroneutrality is established at the far inside of hydrogel. Therefore, (34) establishes under the condition that [H⁺] and [OH⁻] are negligibly low. Using (31) and (34), [SO$_{3}^{-}$] at the far inside of hydrogel is given 13.7 molm⁻³. Therefore, using (31), (33), [Sulfonic] = 43 molm⁻³ and [SO$_{3}^{-}$] = 13.7 molm⁻³, K _a is given 0.11 mol⁻¹ m ³. Selemion CMV contains -SO₃H groups, and the hydrogel so far described contains -SO₃H groups, too. Therefore, we employed the K _a of hydrogel as K _A of the Selemion CMV.

$$ [K_{a}]=\frac{[Sulfonic]-[SO_{3}^{-}]}{[SO_{3}^{-}][K^{+}]} $$

(33)

$$ [K^{+}] - [SO_{3}^{-}] - [Cl^{-}] = 0 $$

(34)