The Goldman-Hodgkin-Katz Equation Including Calcium

When ions have different valences, the GHK equation becomes more complicated. Lewis (1979) has derived an analogous equation for transport of sodium, potassium, and calcium.

Lewis CA (1979) “Ion-concentration dependence of the reversal potential and the single channel conductance of ion channels at the frog neuromuscular junction.” Journal of Physiology, Volume 286, Pages 417–445.

Section 9.6

Problem 20 ½. Derive an expression for the Goldman-Hodgkin-Katz equation when you have three ions that can pass through the membrane: sodium, potassium, and calcium.

(a) Write down an expression like Eq. 9.53 for the current density for each ion:J_Na, J_K, and J_Ca. Hint: be careful to include the valence z properly.

(b) Assume the amount of charge in the cell does not change with time, so J_Na + J_K + J_Ca = 0. Try to solve the resulting equation for the reversal potential, v_rev. You should find it difficult, because the expression for J_Ca has a different denominator than do J_Na and J_K.

(c) Define a new permeability for calcium,

Now derive an expression for v_rev. Your result should look similar to Eq. 9.55, except for some factors of four, and in the numerator the new calcium permeability will be multiplied by a voltage-dependent factor.

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