Anomalous properties of a simple small-volume membrane transport system at steady state

We investigate the transport of a single ligand L, between two macroscopic baths with different concentrations of L, by a simple membrane system. The center of this system is a very small volume V that might contain N = 0, 1, 2,... molecules of L. Access to V from either bath is by means of a single binding site for L. The steady-state properties of this simple system are studied exactly by a discrete-N master-equation approach. It is found that the mean concentration of L in V, c = N/V, is not a significant quantity either kinetically or thermodynamically. Further, the chemical potential of L in V is not defined and hence the overall thermodynamic force on L between the two baths cannot be subdivided at V. In fact, because V is small, the transport system must be treated as an indivisible system with interdependent parts. In the limit when V and N are very large, V becomes in effect a third macroscopic bath and two well-defined transport subsystems emerge (between V and either bath).