How do I compute the voltage accross a cell membrane given microstate of neuron (distribution of charged ions)?

Question

I am not a physicist, and I'm not sure what keywords to use to search for this.

I'm trying to understand how I could in theory quantitatively know, what the voltage across a cell membrane in a Neuron would be, given complete information about the microstate of the system at the atomic/molecular level (positions of all the ions and so on). I am not even sure which information one needs to determine the voltage. E.g. do we need to know the nr of charged ions on both sides of the membrane? Do we need to divide by the volume of the neuron? Or by the surface area, given that the ions will concentrate near the boundary (i.e. near the membrane)? Does the distribution of ions within the neuron matter? Do we have to make approximations to make a prediction, or is there a simple exact relation?

Ideally I'm looking for a simple quantitative formula, but I'm not sure what the name is.

Answer 1

The typical textbook start is the Nernst equation, which gives the reversal potential for a ion species. That is, is the electric potential that balances a given concentration gradient: $$E_{rev}=\frac{RT}{zF}\ln\left(\frac{[X]_{outside}}{[X]_{inside}}\right) $$ where $z$ is the ion charge, $R$ the gas constant, $T$ temperature, $F=96480$ Coulomb/mol, and $[X]$ is the ion concentration. For example, the reversal potential for potassium in the body is -80 mV.

The resting potential of the membrane is the sum of the individual potentials when they and their ion currents balance each other, producing a zero current across the membrane. In terms of conductances $g_i$ for each ion this can be expressed as $$E_{rest}=\frac{\sum_i g_i+E_i}{\sum_i g_i}$$

A more accurate equation than the above Nernst-based version (which assumes permeability is the same as conductance) is the Goldman-Hodgkin-Katz equation.

Note that the conductances in real cells can change due to voltage sensitive ion channels, which give rise to the actual dynamics. The Hodgkin-Huxley model models how the currents flow based on the voltage and capacitance, with terms modelling the potentially complex channel responses. A lot of fun stuff there.

Answer 2

As always in thermodynamics, the voltage (potential energy/charge) across a boundary between two differently charged fluids is the difference of the chemical potential per particle times the difference of particle number densities.

By the basics, the chemical potential is the energy to add one particle from a free ensemble of the same temperature.

Of course one may use other zeros of the chemical potential, taking the ion from a ground state of zero, heat it, infuse it into the fluid by a ion pump in the membrane; but only the difference counts over a membrane.