Dynamics of diffusion with reversible binding in microscopically heterogeneous membranes: General theory and applications to dermal penetration

Essentially all biological membranes and tissues exhibit microscopic heterogeneity in the form of cellular, lamellar or other organization, and molecular diffusion in these materials is frequently slowed by binding to elements of the microstructure ("trapping"). This paper addresses situations where binding is describable as a linear reversible process at the microscale, with forward ("on") and reverse ("off") rate constants k_f(x) and k_r(x) that vary with position. Very commonly it is tacitly assumed that the macroscopically observable binding behavior should follow the same rate law with the substitution of appropriate effective (tissue-average) rate constants k-f and k-r. This assumption is probed theoretically for spatially periodic microstructures using a judicious application of numerical calculations and asymptotic analysis to prototypical one-dimensional transport problems. We find that smooth microscopic variations produce an anomalous macroscopic exchange between free and bound solute populations that is not well described by a single pair of forward and reverse rate constants, i.e., violates the usual paradigm. In contrast, discontinuous variations (as in two-phase composite media) are evidently well described by the usual paradigm. For the latter case we derive simple and general algebraic equations giving k-f and k-r, and generalize them to any three-dimensional unit cell representing the tissue microstructure. Validity of the formulas is demonstrated with reference to a concrete example describing molecular diffusion through the stratum corneum (barrier) layer of skin, comprising lipid (intercellular) and corneocyte (cellular) phases. Our analysis extends coarse-graining (homogenization, effective transport) theory for irreversible trapping systems to the reversible case.