A cell's resting potential: leaks, a pump, and what happens when the pump stops

A cell membrane leaks K⁺ out and Na⁺ in, and the Na⁺/K⁺ pump, spending ATP, puts them back. The membrane potential settles where the leaks' currents cancel the pump's, against the Mullins–Noda equation. Vary the K⁺ outside, how leaky the membrane is to Na⁺, and the free energy each ATP gives the pump; then stop the pump and watch the cell run down, over hours, to Donnan equilibrium. (More about the model below.)


  
About this simulation

  
A slice through the cell: 1 µm of unstirred fluid outside, the membrane, and 3.3 µm of cytoplasm, the
    volume per membrane area of a cell 20 µm across. The fluid outside is held at a fixed composition, like
    the blood; the cytoplasm is closed, so what's in it is set by what crosses the membrane. Both solutions
    are strictly neutral. Inside are 142 mM of anions that can't cross (proteins, phosphates, A⁻).

  
The membrane is a capacitor (1 µF/cm²) that K⁺, Na⁺ and Cl⁻ cross through channels, each by
    electrodiffusion in the field across the membrane, with a permeability P (Goldman–Hodgkin–Katz). The
    Na⁺/K⁺-ATPase moves 3 Na⁺ out and 2 K⁺ in per ATP, at a rate that saturates in each (half-saturated at
    10 mM Na⁺ inside and 1.5 mM K⁺ outside) and falls to zero as the work it does approaches the free energy
    of the ATP: its static head, 3Δμ̄Na − 2Δμ̄K = ΔGATP. In steady state no
    ion accumulates, so the Na⁺ and K⁺ leaks stand in the pump's 3 : 2 ratio, which gives the Mullins–Noda
    equation, Vm = VT ln((r PK[K]o + PNa[Na]o)/(r
    PK[K]i + PNa[Na]i)) with r = 3/2, whatever the pump's
    kinetics. Cl⁻ has no pump, so it sits at its Nernst potential.

  
The level diagram draws each ion's species voltage Vi = μ̄i/(ziF),
    against its own value outside, and φ the same way. Each ion's line is flat through each solution and
    steps at the membrane by Vm − Ei: the driving force physiologists use, which is its
    electrochemical potential difference per charge. An ion with no step (Cl⁻ here) is in equilibrium across
    the membrane, whatever φ does; the sign and size of a step say which way that ion leaks and how hard.

  
Stopping the pump, nothing holds the gradients: Na⁺ leaks in, K⁺ out, the cell depolarizes, and
    it slides, over hours (a cell's contents are large against what crosses), toward Donnan equilibrium,
    where every permeant ion's step is zero and the impermeant anions alone hold a potential, the dashed line.
    A real cell also takes in water as it goes (here the volume is fixed) and would swell; the pump is what
    keeps it from bursting.

  
Concentrations are typical of a mammalian neuron (Hille, Ion Channels of Excitable Membranes, 3rd
    ed.), PK = 10⁻⁸ m/s, and the pump's vmax is 5 × 10⁻⁷ mol/(m²·s), about 3000 pumps per
    µm² at 100 cycles a second.