Time-dependent diffusion coefficient as a probe of the permeability of the pore wall

The time dependence of the mean-square displacement (or equivalently of the diffusion coefficient) in the presence of a permeable barrier can be used as a probe of the surface-to-volume ratio and permeability of a membrane. An exact, universal, short-time asymptotics in a pack of cells, assuming that the surfaces are locally smooth, shows that the effects of nonzero permeability appear as a correction to the diffusion coefficient that is linear in time, whereas the surface-to-volume ratio enters as a square root in time. With κ as the permeability of the membrane, we find, for the particles released inside the cells, DR,eff(t)=DR[1−(SR/VR){4DRt/(9π)−κtDL(DL+DR)/(6DR)}]+⋯ . Here DR and DL are free (i.e., bulk) diffusion coefficients inside and outside of the cell, respectively, and SR/VR is the total internal surface divided by the total internal cell volume. The other terms linear in t that add to the right side of above equation are DR(SR/VR)[(1/6)ρt−(1/12)DRt〈(1/R1+1/R2)〉R], where ρ is a surface relaxation, which is generally negligible in biological samples, and 〈(1/R1+1/R2)〉R is the average of the principal radii of curvatures over the interior surface. An equivalent expression for the particles starting outside the cell is obtained by swapping L↔R. The NMR data on erthrocytes show that the effect of permeability can be significant within the time scales of measurement and hence κ is deducible from the data. The long-time behavior given previously [Proc. Natl. Acad. Sci. USA 92, 1229 (1994)] is augmented by giving a nonuniversal form that includes the rate of approach to this limit.