Goldman-Hodgkin-Katz Equation and Driving Forces

The Goldman-Hodgkin-Katz (GHK) equation predicts the steady-state membrane voltage when more than one ion is permeant, weighting each ion's contribution by its permeability. Together with the idea of electrochemical driving force, it explains where the resting potential settles and how strongly each ion tends to move at any given voltage.

Definition

The GHK voltage equation expresses the steady-state membrane potential as a function of the permeabilities and inside/outside concentrations of the permeant ions; an ion's electrochemical driving force is the difference between the actual membrane voltage and that ion's equilibrium potential, which sets the direction and magnitude of its net flux.

Scope

This topic covers the constant-field voltage equation that combines the contributions of potassium, sodium, and chloride into a single predicted membrane potential, and the related concept of driving force, the difference between the membrane voltage and an ion's equilibrium potential. It builds on the single-ion equilibrium potentials treated in the permeability topic and explains the multi-ion steady state.

Core questions

• How is the resting potential determined when several ions are permeant at once?
• What assumption (the constant field) underlies the GHK equation?
• What is an ion's driving force, and how does it relate to its equilibrium potential?

Key concepts

• Constant-field assumption
• Permeability-weighted membrane potential
• Relative permeabilities of K+, Na+, Cl-
• Electrochemical driving force
• Reversal potential
• Steady-state versus equilibrium voltage

Key theories

Constant-field (GHK) theory of membrane potential

Treating the electric field across the membrane as constant, Goldman derived an expression for the steady-state voltage as a permeability-weighted balance of the permeant ions; Hodgkin and Katz applied it to nerve, explaining the resting potential and its shift when relative permeabilities change.

Mechanisms

When several ions can cross the membrane, no single equilibrium potential is reached; instead the membrane settles at a steady-state voltage where the inward and outward charge flows balance. The GHK equation, derived by Goldman (1943) under the assumption of a constant electric field within the membrane, gives this voltage as a logarithmic function of each ion's permeability multiplied by its concentration on each side. Because resting potassium permeability greatly exceeds sodium permeability, the predicted potential lies near the potassium equilibrium potential, shifting toward sodium when relative sodium permeability rises. The driving force on any ion is the gap between the present membrane voltage and that ion's equilibrium potential: the larger the gap, the stronger the net push on the ion, and net flux reverses direction when the voltage crosses the equilibrium potential. Hodgkin and Katz (1949) confirmed the framework by showing the membrane voltage tracked predicted values as external sodium was varied.

Clinical relevance

The GHK framework explains why altering relative ion permeabilities or extracellular concentrations changes the resting potential and excitability, which is the conceptual basis for understanding how electrolyte disturbances and channel-modifying agents affect excitable tissue. The entry is mechanistic reference material and gives no treatment guidance.

Evidence & guidelines

The equation is a theoretical result validated by direct membrane-potential measurements and is standard content in physiology and biophysics texts; it is reference material rather than guideline content.

History

David Goldman published the constant-field derivation in 1943 while working on the biophysics of membranes. Hodgkin and Katz adopted and applied it to the squid axon in 1949, and the combined result became known as the Goldman-Hodgkin-Katz equation, a cornerstone of membrane physiology.

Key figures

• David E. Goldman
• Alan Hodgkin
• Bernard Katz

Seminal works

• goldman-1943
• hodgkin-katz-1949

Frequently asked questions

How does the GHK equation differ from the Nernst equation?

The Nernst equation gives the equilibrium potential of a single ion, while the GHK equation gives the steady-state membrane voltage when several ions are permeant, weighting each by its permeability.

What is the driving force on an ion?

It is the difference between the membrane voltage and the ion's equilibrium potential; this gap determines how strongly and in which direction the ion tends to move, and net flux reverses when the voltage equals the equilibrium potential.