Membrane Potential

Most cells have a negative voltage inside (about -70 mV in neurons at rest). This membrane potential powers signaling in neurons and muscle, drives secondary active transport, and keeps membranes ready to respond to stimuli.

Why the Inside Is Negative

Two main reasons:

1. The Na+/K+ ATPase pumps 3 positive charges out per 2 positive charges in, directly making the inside more negative (electrogenic).
2. K+ has high cytoplasmic concentration and is the most permeable ion at rest (because K+ leak channels are open). K+ tends to flow down its gradient out of the cell, leaving behind a net negative charge inside. This sets the resting potential close to the K+ equilibrium potential.

The Nernst Equation

For a single ion at equilibrium, the voltage difference across the membrane that would stop net ion flow is the equilibrium potential (Eion). The Nernst equation calculates it:

$E_{\text{ion}} = \frac{RT}{zF} \ln\frac{[\text{ion}]_{\text{out}}}{[\text{ion}]_{\text{in}}}$

• R = gas constant, T = temperature in K, z = ion charge, F = Faraday constant.
• At 37°C, this simplifies to E (mV) = (61.5 / z) × log([out]/[in]) for a single ion.

At typical cellular concentrations:

• $E_{K}$ ≈ -90 mV
• $E_{\text{Na}}$ ≈ +60 mV
• $E_{\text{Cl}}$ ≈ -65 mV
• $E_{\text{Ca}}$ ≈ +120 mV

The Goldman Equation

The Nernst equation handles one ion at a time. The actual resting potential depends on all permeable ions. The Goldman-Hodgkin-Katz equation weighs each ion by its permeability:

$V_m = \frac{RT}{F} \ln\frac{P_K[\text{K}]_o + P_{\text{Na}}[\text{Na}]_o + P_{\text{Cl}}[\text{Cl}]_i}{P_K[\text{K}]_i + P_{\text{Na}}[\text{Na}]_i + P_{\text{Cl}}[\text{Cl}]_o}$

At rest, K+ has the highest permeability, so Vm is close to $E_{K}$ but not exactly equal because some Na+ and Cl- leak through too. When Na+ channels open (action potential), permeability shifts toward Na+, and Vm swings toward $E_{\text{Na}}$.