Steady-State Description of Cell Membrane Potential and IV Curves

Driving force on ions across membranes- generate simple IV curves

I've already mentioned IV curves in a previous post, but here's a couple more random tidbits about them:

• The x-intercept of the curve is where the current = 0 (i.e. no flow of that ion into or out of the cell). If the concentrations of the two ions are equal to each other, the x-intercept will occur when V_m = 0. Otherwise, the x-intercept will occur somewhere else.
• The x-intercept is also called the reversal potential (I think). This is because this is the point where the current reverses direction (if you go slightly past the reversal potential in one direction, current is going out of the cell, and if you go in the other direction, current is going into the cell).

Describe the steady state situation for multiple ion permeability

In my first post for PHYL3001, I talked about electrochemical equilibrium. Unfortunately, membrane potentials are actually more complicated than that, as there are multiple ions moving back and forth across the membrane. This results in a "steady state," rather than true equilibrium.

Aside from K⁺, another important ion involved in the "steady state" potential is Na⁺. At first glance, Na⁺ has a lot going for it: it has a concentration gradient going into the cell (there is more Na⁺ outside than inside the cell) and an electrical gradient going in the same direction (it is more negative inside than outside of the cell, and the negative charge attracts Na⁺. However, permeability to Na⁺ is much less than permeability to K⁺. Hence, the flow of Na⁺ into the cell more or less balances out the flow of K⁺ out of the cell, resulting in a "steady state."

An important caveat to note is that, if this was allowed to proceed without any intervention, eventually the concentrations of these ions would be out of whack. That's why the Na⁺/K⁺ pump, which you've probably heard of a lot, exists: it helps to maintain ion concentrations and thus also helps maintain steady state.

Explain principles of derivation of the steady state (Goldman) equation for membrane potential

Nernst-Planck Electrodiffusion Theory

The first thing I'm going to cover here is the Nernst-Planck Electrodiffusion Theory, which, to my understanding, suggests the parameters that govern the rate of ion movement through the membrane. Essentially, the main factors that control the rate of ion movement through the membrane are concentration gradient, V_m and P_x, which is the permeability coefficient for ion x. P_x can be calculated with the following equation:

P_x = D_xβ/a where D_x is the diffusion constant, β is the partition coefficient (the ability of the ion to dissolve in the membrane) and a is the membrane thickness.

This model relies on a few assumptions: that the membrane is homogenous throughout, the transmembrane voltage changes linearly across the membrane, ions move independently of each other and the permeability coefficient is constant.

Goldman-Hodgkin-Katz (GHK) Equation

Anyway now that you know what a permeability coefficient is, we're going to move on to the Goldman-Hodgkin-Katz current equation, which I'm not going to write here because a) we don't need to memorise it (thank goodness) and b) it's kinda complicated and would probably be more of a distraction than a help. Essentially, it defines the current of an ion in terms of valency, voltage, permeability coefficient, temperature and so on. This creates a curved line, unlike the linear IV curves that I've been talking about up until now.

One cool thing about this curve is that the outward current and inward current appear to be affected by the concentrations in different areas. Generally, outward currents of the ion tend to be more reflective of intracellular concentration of that ion, and vice versa.

Goldman Equation

Now we're finally on to talking about the Goldman equation! I've talked for a bit about current and steady states and so forth, and now it finally comes together with this equation. In the steady state, the net membrane current is zero, but there is still movement of some ions because the membrane isn't at equilibrium. However, these currents cancel each other out, so the sum of all of the ionic currents equals zero.

A little bit of equation rearranging later, and you end up with the Goldman equation:

V_m = -60log₁₀[(P_KK_in + P_NaNa_in + P_ClCl_out)/(P_KK_out + P_NaNa_out + P_ClCl_in)]

You might notice this as an extended version of the Nernst equation in which multiple ions, as well as the permeability for each ion, are included.

Illustrate use of the Goldman equation

There is also a variant of this equation in which the relative membrane permeability between two ions, rather than the absolute permeability, is considered:

V_m = -60log₁₀[(K_in + αNa_in)/(K_out + αNa_out)]

where α = relative membrane permeability of Na to K (i.e. an α value of 0.1 means that the membrane is 10x more permeable to K than to Na).

You can fiddle around with different values of α to try and find a value that fits your data points the best, and I think there are also computer programs that can do that for you too.