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 kappa as the permeability of the membrane, we find, for the particles released inside the cells, D(R,eff)(t)=D(R)[1-(S(R)/V(R)){4rootD(R)t/(9rootpi) - kappatrootD(L) (rootD(L) + rootD(R))/(6D(R))}]+.... Here D(R) and D(L) are free (i.e., bulk) diffusion coefficients inside and outside of the cell, respectively, and S(R)/V(R) 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 D(R)(S(R)/V(R))[(1/6)rhot - (1/12)D(R)t<(1/R(1)+1/R(2))>(R)], where rho is a surface relaxation, which is generally negligible in biological samples, and <(1/R(1)+1/R(2))>(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 kappa 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. (C) 2003 American Institute of Physics.