The Goldman-Hodgkin-Katz Equation Including Calcium

In Section 9.6 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I derive the Goldman-Hodgkin-Katz equation. It accounts for both diffusion and electrical forces acting on ions in the membrane (presumably passing through ion channels spanning the lipid bilayer). If only one ion were present, its concentration on each side of the membrane would determine the equilibrium, or reversal, potential. For instance more potassium is inside a cell than outside, so diffusion pushes the positively charged potassium ions out. As the outside becomes positive, the resulting electric field in the membrane pushes potassium back in. The reversal potential, v_rev, is the potential across the membrane when diffusion and electrical forces balance.

Mathematically, we can derive the reversal potential for any ion C by starting with an expression for its current density, J_C

where z is the valence, e is the elementary charge, v is the potential, ω_C is the permeability, N_A is Avogadro's number, k_B is the Boltzmann constant, T is the absolute temperature, and [C₁] and [C₂] are the concentrations outside and inside the membrane. (See IPMB for a derivation of this complicated equation.) To find the reversal potential, we set J_C to zero and solve for v.

When more than one ion can cross the membrane, the situation is more complicated. Russ and I examined a membrane that can pass three ions: sodium, potassium, and chloride. The resulting equation for the reversal potential—also known as the Goldman-Hodgkin-Katz equation—is

We then write

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.

The citation is to

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.

Below is a new homework problem, based on Appendix A of Lewis’s paper, analyzing a more complicated GHK equation that includes calcium along with sodium and potassium.

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.

What’s the lesson to be learned from this homework problem? First, the GHK expression including calcium has the potential on the left side of the equation, but also on the right side, inside a logarithm. No simple way exists to calculate v_rev. My first thought is to use an iterative method, but I haven’t looked into this in detail. Second, notice how a small modification to the problem—changing chloride to calcium—made a major change in how difficult the problem is to solve. Adding the negative chloride ion to positive sodium and potassium resulted in a trivial change to the GHK equation (the inside chloride concentration appears in the numerator rather than the outside concentration). However, adding the divalent cation calcium totally messes up the equation, making it difficult to solve except with numerical methods.

I advocate for simple models. They provide tremendous insight. However, the moral of this story is if you push a toy model too hard, it can become complicated; it’s no longer a toy.