Mean residence time of molecules diffusing in a cell bounded by a semi-permeable membrane: Monte Carlo simulations and an expression relating membrane transition probability to permeability.

The rapid exchange of water across erythrocyte membranes is readily measured using an NMR method that entails doping a suspension of cells with a moderately high concentration of Mn(2-) and measuring the rate of transverse relaxation of the nuclear magnetisation. Analysis of the data yields an estimate of the rate constant for membrane transport, from which the membrane permeability can be determined. It is assumed in the analysis that the efflux rate of the water is solely a function of the rate of membrane permeation and that the time it takes for intracellular water molecules to diffuse to the membrane is relatively insignificant. The limits of this assumption were explored by using random-walk simulations of diffusion in cells modelled as parallel planes, spheres, and biconcave discs. The rate of membrane transport was specified in terms of a transition probability but it was not initially clear what the relationship should be between this parameter and the diffusional membrane permeability P(d). This relationship was derived and used to show that the mean residence time for a water molecule is determined by P(d) when the diffusion coefficient is above a certain threshold value; it is determined by the distance to the membrane below that value.