Effective Potential Energy Expression for Membrane Transport

All living cells transport molecules and ions across membranes, often against concentration gradients. This active transport requires continual energy expenditure and is clearly a nonequilibrium process for which standard equilibrium thermodynamics is not rigorously applicable. Here we derive a nonequilibrium effective potential that evaluates the per particle transport energy invested by the membrane. A novel method is used whereby a Hamiltonian function is constructed using particle concentrations as generalized coordinates. The associated generalized momenta are simply related to the individual particle energy from which we identify the effective potential. Examples are given and the formalism is compared with the equilibrium Gibb's free energy.