The resting membrane potential

Concept

The resting membrane potential

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Statement

The Nernst equation applied to ion gradients.

Why it matters

membrane-transport already established that ion pumps such as the Na⁺/K⁺-ATPase build and maintain steep concentration gradients of Na⁺ and K⁺ across the plasma membrane, at the direct cost of ATP. The resting membrane potential is the electrical consequence of those gradients: a stable, negative-inside voltage difference that exists across essentially every animal cell's membrane even with no external stimulus present. It is the electrical baseline against which every excitable event in the nervous system is measured, and the starting point that receptor-ligand-binding, signal-transduction and second-messengers all perturb, directly or indirectly, when a cell responds to a signal.

Quantifying this baseline requires the Nernst equation, which converts a known ion concentration gradient into the specific voltage at which that ion's net movement across the membrane would stop — the essential tool for predicting how any given change in ion permeability or concentration will shift a cell's membrane potential.

Hypotheses

At rest, the plasma membrane is far more permeable to K⁺ than to Na⁺ or other major ions, owing to open K⁺ leak channels.Without a dominant resting permeability to one particular ion, the membrane potential would not settle close to any single ion's equilibrium potential, and no stable, predictable resting value would exist. The Na⁺/K⁺-ATPase continuously pumps Na⁺ out and K⁺ in against their electrochemical gradients, using ATP, offsetting the slow leak of each ion down its gradient.The resting state is a steady state maintained by ongoing active transport, not a true thermodynamic equilibrium; without continued pumping, the gradients that the Nernst equation depends on would gradually dissipate and the resting potential would decay toward zero. The Nernst equation itself assumes the membrane is permeable to only the single ion being considered, at electrochemical equilibrium for that ion alone. Real membranes are simultaneously permeable to several ions at once, which is why the actual resting potential requires the multi-ion Goldman equation (Proof, Step 3) rather than a single ion's Nernst potential taken in isolation.

Proof

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

The Nernst equation gives the equilibrium potential for a single ion species: the membrane voltage at which the electrical force on that ion exactly balances its concentration-gradient (diffusive) force, so its net flux across the membrane is zero. \(R\) is the gas constant, \(T\) absolute temperature, \(z\) the ion's charge, and \(F\) the Faraday constant. B

\text{At body temperature, } \frac{RT}{F}\approx 26.7\,\text{mV (natural log form); equivalently } \frac{2.303RT}{F}\approx 61\,\text{mV per decade (log}_{10}\text{ form).}

These numerical prefactors let the Nernst equation be evaluated quickly for any physiological ion gradient once \(z\) is known; for a monovalent cation (\(z=+1\)), a tenfold concentration ratio across the membrane corresponds to roughly a 61 mV equilibrium potential at 37°C. A

V_m = \frac{RT}{F}\ln\frac{P_K[K^+]_{\text{out}} + P_{Na}[Na^+]_{\text{out}} + P_{Cl}[Cl^-]_{\text{in}}}{P_K[K^+]_{\text{in}} + P_{Na}[Na^+]_{\text{in}} + P_{Cl}[Cl^-]_{\text{out}}}

The Goldman (constant-field) equation extends the single-ion Nernst treatment to a membrane simultaneously permeable to several ions, weighting each ion's contribution by its relative membrane permeability \(P\); an ion with negligible permeability contributes negligibly to \(V_m\) regardless of how steep its own concentration gradient is. B

\text{Since } P_K \gg P_{Na} \text{ at rest (Hypotheses), } V_m \text{ lies close to, but not exactly at, } E_K.

Because K⁺ permeability dominates the Goldman sum, the resting potential is pulled strongly toward K⁺'s own equilibrium potential (typically around \(-90\,\text{mV}\)); the small but non-zero Na⁺ permeability pulls it slightly positive of \(E_K\), giving the characteristic measured resting value of roughly \(-70\,\text{mV}\) in a typical neuron. A

\text{The Na}^+/\text{K}^+\text{-ATPase (3 Na}^+\text{ out : 2 K}^+\text{ in per ATP hydrolysed) continuously restores the gradients dissipated by resting leak currents, maintaining a stable steady-state } V_m.

Because the pump exports more positive charge than it imports (3 Na⁺ out for every 2 K⁺ in), it also makes a small direct (electrogenic) contribution to the negative resting potential, on top of its indirect role of maintaining the concentration gradients the Nernst/Goldman equations depend on. A

Result

V_m \approx \frac{RT}{F}\ln\frac{P_K[K^+]_{\text{out}}+P_{Na}[Na^+]_{\text{out}}}{P_K[K^+]_{\text{in}}+P_{Na}[Na^+]_{\text{in}}} \approx -70\,\text{mV}

Reading. The resting membrane potential is a permeability-weighted average of the individual ions' Nernst equilibrium potentials, dominated by K⁺ because resting K⁺ permeability greatly exceeds that of Na⁺, and actively maintained against dissipation by the Na⁺/K⁺-ATPase.

Scope. Applies to any cell at electrochemical steady state with a dominant resting permeability to one ion; the same Nernst/Goldman framework, with different dominant permeabilities and gradients, underlies the depolarisation phase of the action potential and the resting potentials of non-neuronal excitable and non-excitable cells alike.

Corollaries & converses

membrane-transport's account of active ion pumping is the mechanism that this result depends on directly (Hypotheses); the resting potential is, in effect, the measurable electrical signature of that transport activity.
signal-transduction and receptor-ligand-binding events that open or close specific ion channels act by locally changing the relevant permeability terms \(P_K\), \(P_{Na}\), etc. in the Goldman equation, shifting \(V_m\) away from its resting value — the resting potential computed here is the baseline every such signalling event displaces the cell from.
Converse: from a measured resting potential and known ion concentrations, the relative permeabilities \(P_K:P_{Na}\) can be estimated by fitting the Goldman equation, a standard technique for characterising a cell's channel population electrophysiologically.

Fails without

Drop K⁺ permeability dominance (Hypotheses): if resting Na⁺ and K⁺ permeabilities were comparable, the Goldman equation's weighted sum would place \(V_m\) close to \(0\,\text{mV}\), between \(E_K\) (negative) and \(E_{Na}\) (positive), eliminating the large, stable negative resting potential that excitable cells require as a baseline from which to depolarise.
Drop continuous Na⁺/K⁺-ATPase activity: without ongoing active transport to counter resting leak currents, the concentration gradients underlying \(E_K\) and \(E_{Na}\) would run down over time; \(V_m\) would drift away from its steady-state value and eventually toward \(0\,\text{mV}\) as all ions approached uniform concentration across the membrane.

Common errors

Assuming the resting membrane potential equals \(E_K\) exactly, rather than a permeability-weighted value that lies close to, but slightly positive of, \(E_K\) because \(P_{Na}\) is small but not zero (Step 4).
Getting the sign or ratio inverted in the Nernst equation — the equation uses outside-over-inside concentration in the logarithm, and swapping the ratio flips the sign of the computed potential.
Forgetting to use the ion's actual charge \(z\) (e.g. \(z=-1\) for Cl⁻), which flips the sign of that ion's equilibrium potential relative to a cation with the same concentration ratio.
Treating the resting state as a true equilibrium (zero net ion flux for every ion, no energy expenditure required) rather than a steady state actively maintained against continuous leak by ATP-consuming pumps (Hypotheses).

Discussion

The Nernst equation is named for Walther Nernst, whose late-19th-century work on electrochemistry established the general relationship between concentration gradients and electrode/membrane potentials well before its application to living cells. Its extension to a membrane permeable to multiple ions simultaneously, the constant-field (Goldman) equation, was derived by David Goldman in 1943 and subsequently applied directly to nerve membranes by Alan Hodgkin and Bernard Katz, whose broader work on the ionic basis of the action potential (with Andrew Huxley) is one of the foundational results of modern neuroscience.

The specific stoichiometry of the Na⁺/K⁺-ATPase (3 Na⁺ exported for every 2 K⁺ imported, per ATP hydrolysed) makes the pump directly electrogenic: it moves net positive charge out of the cell with every cycle, contributing a small additional hyperpolarising component to \(V_m\) beyond its indirect role of sustaining the concentration gradients that the Nernst and Goldman equations depend on.

Common misconception: that the resting potential is a fixed, universal number ("−70 mV") true of all cells. It is specific to each cell type's particular ion concentrations and relative channel permeabilities; skeletal muscle, cardiac, and different classes of neuron all have somewhat different resting potentials, all computable from the same underlying Goldman-equation framework with cell-type-specific parameters.

Worked examples

\text{Typical mammalian neuron: } [K^+]_{\text{out}}=5\,\text{mM},\ [K^+]_{\text{in}}=140\,\text{mM}.\quad E_K = 26.7\,\text{mV}\times\ln\!\left(\frac{5}{140}\right)

\(\ln(5/140)=\ln(0.0357)\approx-3.33\); \(E_K\approx26.7\times(-3.33)\approx-89\,\text{mV}\), matching the standard textbook range quoted for K⁺'s equilibrium potential in neurons. A

[Na^+]_{\text{out}}=145\,\text{mM},\ [Na^+]_{\text{in}}=15\,\text{mM}.\quad E_{Na}=26.7\,\text{mV}\times\ln\!\left(\frac{145}{15}\right)

\(\ln(145/15)=\ln(9.67)\approx2.27\); \(E_{Na}\approx26.7\times2.27\approx+61\,\text{mV}\). The measured resting potential (roughly \(-70\,\text{mV}\)) lies between these two individual equilibrium potentials, much closer to \(E_K\) than to \(E_{Na}\), exactly as expected from K⁺ permeability dominating the Goldman sum (Step 4). A

E_K\approx-89\,\text{mV},\quad E_{Na}\approx+61\,\text{mV},\quad V_m\approx-70\,\text{mV}\ (\text{close to }E_K\text{, not }E_{Na})

Reading. The two individually computed Nernst potentials bracket the actual measured resting potential, with the Goldman-weighted result falling much nearer the ion (K⁺) to which the resting membrane is more permeable.

Scope. The same two-line calculation, repeated for any ion given its inside/outside concentrations, predicts that ion's individual equilibrium potential; only the full permeability-weighted Goldman sum predicts the actual resting \(V_m\).

Problems

Chloride is distributed at \([Cl^-]_{\text{out}}=110\,\text{mM}\), \([Cl^-]_{\text{in}}=10\,\text{mM}\) in a typical neuron. Compute \(E_{Cl}\), remembering \(z=-1\) for chloride.
Solution
\(E_{Cl}=\frac{RT}{zF}\ln([Cl^-]_{\text{out}}/[Cl^-]_{\text{in}}) = \frac{26.7\,\text{mV}}{-1}\times\ln(110/10) = -26.7\times\ln(11) = -26.7\times2.40\approx-64\,\text{mV}\). Note the negative \(z\) flips the sign relative to what the same concentration ratio would give a cation; this value happens to sit close to the typical resting potential, consistent with chloride often being close to passive equilibrium at rest in many neurons.
A drug blocks the Na⁺/K⁺-ATPase. Predict, qualitatively, what happens to the resting membrane potential over the following minutes, and explain using the Goldman equation and Hypotheses.
Solution
Without pump activity, resting leak currents (K⁺ out, Na⁺ in, each flowing down its own electrochemical gradient) are no longer offset, so the K⁺ and Na⁺ concentration gradients gradually run down: \([K^+]_{\text{in}}\) falls, \([Na^+]_{\text{in}}\) rises. By the Goldman equation, both \(E_K\) and \(E_{Na}\) move toward each other (and toward \(0\,\text{mV}\)) as the concentration ratios flatten, so \(V_m\) depolarises (becomes less negative) over time, eventually approaching \(0\,\text{mV}\) as all gradients dissipate — this is exactly the "Fails without" scenario for continuous pump activity.
Explain why raising extracellular K⁺ concentration (hyperkalaemia) depolarises resting neurons, using the Nernst equation directly.
Solution
Raising \([K^+]_{\text{out}}\) reduces the outside/inside concentration ratio's magnitude (the ratio moves closer to \(1\)), so \(\ln([K^+]_{\text{out}}/[K^+]_{\text{in}})\) becomes less negative, making \(E_K\) itself less negative (closer to \(0\,\text{mV}\)). Since \(V_m\) sits close to \(E_K\) at rest (Step 4, K⁺ permeability dominance), a less negative \(E_K\) pulls the whole resting potential less negative as well — a depolarisation, exactly as observed clinically in hyperkalaemia.