Answer to Question #210411 in Electricity and Magnetism for Jay

Question

Assume that frog muscle cell membrane is permeable to Na and K only. The membrane permeabilities

for Na and K are 𝑃𝐾 and 𝑃𝑁𝑎, respectively. At 180C the resting membrane potential is Δ𝑉 = -85 mV. What

is ratio of permeability 𝑃 / 𝑃 ? Suppose that 𝐶𝐾 = 124 mmole/liter, 𝐶𝐾 = 2.2 mmole/liter and 𝐾𝑁𝑎 𝑖𝑛 𝑜𝑢𝑡

𝐶𝑁𝑎 = 4 mmole/liter, 𝐶𝑁𝑎 = 109 mmole/liter.

Answer 1

We will assume we're under the conditions that also satisfy the Goldman-Hodgkin-Katz constant field equation: this expression gives, at equilibrium state, the transmembrane potential in terms of the specific membrane permeabilities for each ion and their intra- and extracellular concentrations:

$V_m=\frac{RT}{F}\ln(\large{\frac{\sum P_iC_{out}^{M+}+P_iC_{in}^{N-}}{\sum P_iC_{in}^{M+}+P_iC_{out}^{N-}}})$

Then we consider that frog muscle cell membrane is permeable to Na and K only and this makes the equation become:

$V_m=\frac{RT}{F}\ln(\large{\frac{P_{K^+}[K^+]_{out}+P_{{Na}^+}[{Na}^+]_{out}}{P_{K^+}[K^+]_{in}+P_{{Na}^+}[{Na}^+]_{in}}})$

Then, we can work on the equation to obtain $P_{{Na}^+}/P_{K^+}$ as:

$e^{\frac{V_mF}{RT}}={\frac{[K^+]_{out}+ \large{\frac{P_{{Na}^+}}{P_{K^+}} } [{Na}^+]_{out}}{[K^+]_{in}+\large{\frac{P_{{Na}^+}}{P_{K^+}}}[{Na}^+]_{in}}}$

$\implies \frac{P_{{Na}^+}}{P_{K^+}}= \large{ \frac{e^{\frac{V_mF}{RT}}[K^+]_{in}-[K^+]_{out} } { [{Na}^+]_{out} - e^{\frac{V_mF}{RT}}[{Na}^+]_{in}} }$

Then we substitute all the values (T=18 °C; F = 96485 C/mol; V_m=-0.085 V; [K⁺]_in=124 mmol/L; [K⁺]_out=2.2 mmol/L; [Na⁺]_in=4 mmol/L; [Na⁺]_out=109 mmol/L) and find:

$\frac{P_{{Na}^+}}{P_{K^+}}= \frac{e^{\frac{(-0.085\,V)(96485\frac{C}{mol})}{(8.314\frac{J}{molK})(291.15\,K)}}[124\frac{mmol}{L}]-[2.2\frac{mmol}{L}] } { [109\frac{mmol}{L}] - e^{\frac{(-0.085\,V)(96485\frac{C}{mol})}{(8.314\frac{J}{molK})(291.15\,K)}}[4\frac{mmol}{L}]}$

$\implies \frac{P_{{Na}^+}}{P_{K^+}}= 0.01826$

In conclusion, the rate between the permeabilities $P_{{Na}^+}/P_{K^+}$ is equal to 0.01826 under these conditions at the frog muscle cell membrane.

Reference:

• Bergethon, P. R., & Simons, E. R. (2012). Biophysical chemistry: molecules to membranes. Springer Science & Business Media.