Difficulty with Understanding a Transformation using the Goldman Equation (Electrochemistry)

[Paul Atreides]

Hello everyone. I have been steadily reading through a book on human physiology and I have come across a problem related to the Goldman Equation that I am trying to understand, which is the following:

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If the membrane potential is -90 mV at rest and +30 mV at the peak of the action potential what changes in the relative permeability of Na⁺ and K⁺ take place in a chloride-free medium?

Because of the fact that we are working with a chloride-free medium, chloride ions are ignored from the Goldman Equation which gives the following:

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E = RT/F * ln((P_Na[Na⁺]_o + P_K[K⁺]_o)/(P_Na[Na⁺]i + P_K[K⁺]_i))

Where E is the membrane potential, RT/F is a constant that is equal to 26.7 at 37 degrees Celsius, P with respective subscripts represent coefficients of different ions, and the ions in square brackets are the concentrations in mmol/L. Subscripts "o" and "i" represent extracellular and intracellular concentrations respectively.

Intracellular Sodium Ion Concentration = 20 mmol/L

Extracellular Sodium Ion Concentration = 145 mmol/L

Intracellular Potassium Ion Concentration = 150 mmol/L

Extracellular Potassium Ion Concentration = 4 mmol/L

I understand that in order to answer this question I need to show that the membrane potential for the sodium-potassium ion coefficient (Na⁺¹/K⁺¹) is higher at +30 mV than to -90 mV to illustrate that the membrane will be more permeable to sodium during the action potential since +30 mV is closer to the equilibrium potential of sodium at +53 mV and vice versa for potassium. But the solution given is the following:

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-90 = 26.7 * ln((a[145] + [4])/(a[20] + [150]) where "a" equals the sodium-ion coefficient (Na⁺¹/K⁺¹)

I understand most of the substitutions that take place, except I do not understand how the Goldman Equation is transformed into the solution equation for one membrane potential. Therefore, I would be much obliged if someone could indicate to me how it was accomplished.

Edited December 16, 2017 by Paul Atreides
typo, "in" should be "if"

[Paul Atreides]

Dear all, I think I have found the solution to my problem. Firstly, the sodium-potassium coefficient is not Na⁺¹/K⁺¹ but rather P_Na/P_K. From then on, it is a simple case of multiplying the numerator and the denominator of the original equation by 1/P_K. Thus, the following steps are obtained where P_Na/P_K can be represented by the arbitrary variable "a" to simplify when solving:

E = RT/F * ln((P_Na[Na⁺]_o + P_K[K⁺]_o)/(P_Na[Na⁺]i + P_K[K⁺]_i))

E = RT/F * ln((P_Na[Na⁺]_o * 1/P_K + P_K[K⁺]_o * 1/P_K)/(P_Na[Na⁺]i  * 1/P_K + P_K[K⁺]_i * 1/P_K))

E = RT/F * ln((P_Na/P_K[Na⁺]_o + [K⁺]_o)/(P_Na/P_K[Na⁺]i + [K⁺]_i))

E = RT/F * ln((a[Na⁺]_o + [K⁺]_o)/(a[Na⁺]i + [K⁺]_i))