Goldman-Hodgkin-Katz

In the presence of several different ions, the equilibrium of the cell depends on the relative permeability of the ions. For this, we use the Goldman-Hodgkin-Katz equation:  

$$V_{rest} = \frac{RT}{F} \ln \frac{P_K [K^+]_{out} +P_{Na} [N... ...{in}}{P_K [K^+]_{in} +P_{Na} [Na^+]_{in} + P_{Cl} [Cl^-]_{out}}$$ (2)

Permeability of an ion is dependent on a number of factors such as the size of the ion, its mobility, etc. During rest in the squid giant axon, the permeabilities have the ratio P_K:P_Na:P_Cl = 1:0.03:0.1 so that

$$V_{rest} = 58 \log \frac{1(10)+0.03(460)+0.1(40)}{1(400)+0.03(50)+0.1(540)} = -70 mV$$

Since P_K dominates, this is close to E_K. During an action potential the ratio is P_K:P_Na:P_Cl =1:15:.1 so that

$$V_{m} = 58 \log \frac{1(10)+15(460)+0.1(40)}{1(400)+15(50)+0.1(540)} = +44 mV$$

Later on we will approximate the GHK equations by a linearized version:

$$V_{eq} = \frac{g_{Na} E_{Na} + g_{K} E_K + g_{Cl} E_{Cl}}{g_{Na}+g_{K}+g_{Cl}}$$

where the conductances g are proportional to the permeabilities.

HOMEWORK

1.

Suppose the external potassium in a mammalian cell is increased by a factor of 10. What is the new value of E_K?

2.

At $10^\circ C$ a cell contains 80 mM sodium inside and has only 100 mM sodium outside. What is the equilibrium potential for sodium?

3.

Using the same permeabilities for the mammalian cell as were used for the squid axon, compute V_rest,V_m using the table below.