1.5 Membrane Potential

Resting Potential, Nernst Equation, and Goldman-Hodgkin-Katz Theory

Learning Objectives

  • •Explain the ionic basis of the resting membrane potential
  • •Apply the Nernst equation to calculate equilibrium potentials for individual ions
  • •Use the Goldman-Hodgkin-Katz equation to predict membrane potential with multiple ions
  • •Describe how Na⁺/K⁺-ATPase maintains the resting potential

The Resting Membrane Potential

All living cells maintain a voltage difference across their plasma membrane, with the inside typically negative relative to the outside. This resting membrane potential (Vm) is essential for electrical signaling in excitable cells and drives many transport processes.

Typical Resting Potentials

−70 mV

Neurons

−90 mV

Skeletal muscle

−85 mV

Cardiac muscle

−40 mV

Smooth muscle

Ion Distribution Across the Membrane

The membrane potential arises from unequal distribution of ions across the plasma membrane, maintained by the Na⁺/K⁺-ATPase and the selective permeability of ion channels.

IonIntracellular (mM)Extracellular (mM)Ratio (out/in)Eion (mV)
K⁺14040.03−94
Na⁺1214512+67
Cl⁻411629−89
Ca²⁺0.0001220,000+129

Key Points

  • • K⁺: High inside, low outside → diffuses outward
  • • Na⁺: High outside, low inside → diffuses inward
  • • Large anions (proteins, ATP, organic phosphates): Trapped inside, cannot cross membrane
  • • At rest, membrane is ~100× more permeable to K⁺ than Na⁺

The Nernst Equation

The Nernst equation calculates the equilibrium potential (Eion) for a single ion—the voltage at which the electrical driving force exactly balances the concentration gradient, so there is no net flux of that ion.

Nernst Equation

Eion = (RT/zF) ln([ion]out/[ion]in)

At 37°C, simplified to:

Eion = (61.5 mV / z) log₁₀([ion]out/[ion]in)

R = 8.314 J/mol·K
T = Temperature (K)
z = Ion valence
F = 96,485 C/mol

Example: Potassium Equilibrium Potential

Given: [K⁺]out = 4 mM, [K⁺]in = 140 mM, z = +1

EK = (61.5 mV / 1) × log₁₀(4/140)

EK = 61.5 × log₁₀(0.0286)

EK = 61.5 × (−1.54)

EK = −94.7 mV

Interpretation: If the membrane were permeable only to K⁺, the membrane potential would be −94.7 mV.

Goldman-Hodgkin-Katz (GHK) Equation

Real membranes are permeable to multiple ions. The GHK equation extends Nernst to account for the relative permeabilities of different ions.

GHK Voltage Equation

Vm = (RT/F) ln (PK[K⁺]out + PNa[Na⁺]out + PCl[Cl⁻]in) / (PK[K⁺]in + PNa[Na⁺]in + PCl[Cl⁻]out)

Note: Cl⁻ concentrations are inverted because it's an anion (negative charge).

At Rest (typical neuron)

  • PK : PNa : PCl = 1 : 0.04 : 0.45
  • K⁺ dominates → Vm ≈ −70 mV
  • Close to EK but not quite
  • Small Na⁺ permeability pulls it positive

During Action Potential Peak

  • PK : PNa : PCl = 1 : 20 : 0.45
  • Na⁺ dominates → Vm ≈ +40 mV
  • Approaches ENa (+67 mV)
  • Voltage-gated Na⁺ channels open

Role of the Na⁺/K⁺-ATPase

The resting potential is not at equilibrium—it's a steady state maintained by continuous pump activity.

Pump Function

  • • 3 Na⁺ out, 2 K⁺ in per ATP
  • • Net outward current (electrogenic)
  • • Contributes ~−5 to −10 mV directly
  • • Main role: maintain gradients
  • • Uses 30-40% of cell's ATP

What Happens if Pump Stops?

  • • Gradients run down over minutes-hours
  • • Vm slowly depolarizes
  • • Cells swell (osmotic imbalance)
  • • Excitable cells become unresponsive
  • • Ouabain/digitalis block the pump

Electrochemical Driving Force

The driving force on an ion depends on how far the membrane potential is from that ion's equilibrium potential.

Driving Force Equation

DF = Vm − Eion

DF < 0: Cations driven inward, anions outward

DF > 0: Cations driven outward, anions inward

Example: Ion Flow at Rest (Vm = −70 mV)

K⁺DF = −70 − (−94) = +24 mV
→ Small outward leak
Na⁺DF = −70 − (+67) = −137 mV
→ Strong inward drive
Cl⁻DF = −70 − (−89) = +19 mV
→ Small inward (anion!)

Gibbs-Donnan Equilibrium

The presence of impermeant intracellular anions (proteins, nucleic acids, phosphates) creates an asymmetric distribution of permeable ions and a small negative potential—even without active transport.

Donnan Effect Consequences

  • • [K⁺]in > [K⁺]out (to balance trapped anions)
  • • [Cl⁻]in < [Cl⁻]out
  • • Creates osmotic gradient favoring water entry
  • • Na⁺/K⁺-ATPase prevents cell swelling by pumping Na⁺ out
  • • The pump is an "osmotic safety valve"

Clinical Relevance

Hyperkalemia

  • • Elevated [K⁺]out → less negative EK
  • • Membrane depolarizes (Vm → 0)
  • • Cardiac effects: peaked T waves → VF
  • • Treatment: calcium gluconate, insulin+glucose, dialysis
  • • Life-threatening above 6.5 mM

Hypokalemia

  • • Low [K⁺]out → more negative EK
  • • Membrane hyperpolarizes
  • • Muscle weakness, cramps
  • • Cardiac: U waves, arrhythmias
  • • Common with diuretics, vomiting

Local Anesthetics

Lidocaine, bupivacaine, and related drugs block voltage-gated Na⁺ channels, preventing action potential generation. They preferentially block channels in the inactivated state (use-dependent block), making rapidly firing neurons most susceptible.

Mechanism: Drug enters channel from cytoplasmic side, binding to residues in S6 helix.

Key Equations Summary

Nernst Equation (37°C):

Eion = (61.5 mV / z) × log₁₀([ion]out/[ion]in)

GHK Equation:

Vm = 61.5 × log₁₀[(PK[K]o + PNa[Na]o + PCl[Cl]i) / (PK[K]i + PNa[Na]i + PCl[Cl]o)]

Driving Force:

DF = Vm − Eion

Ohm's Law for Ion Current:

Iion = gion × (Vm − Eion)

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