What Really Happens in Diffusion With Reversible Trapping, and What Are the Implications for Biological Diffusion Models?

A classical question in homogenization/coarse graining is, what are the effective (average) diffusion coefficient D_eff and rate coefficient k_eff characterizing the macroscopically observable transport and reaction of a solute molecules/particle that diffuse through an array (periodic or random) of perfectly or partially absorbing traps (reactive or sticky grains)?  Most analyses to date have addressed the irreversible case where—once trapped—the diffusing solute is effectively destroyed or bound forever (see representative references^1-12).  This case is directly relevant to such technological applications as aerosol deposition.  In biological systems, however, reversibility of binding/trapping is the rule rather than the exception.  Indeed, any irreversibly bound species would necessarily accumulate without bound, and spell disease or death in the absence of a mechanism for removal/clearance.  Interestingly, there is a dearth of homogenization theory for the reversible case.

This talk develops a comprehensive answer to the following question.  Suppose a solute diffuses through a medium with two-phase microstructure, so that its free concentration C(x,y,z,t) and bound concentration B(x,y,z,t) are governed by the equations

            ∂ C / ∂ t  =  D_a ∇²C – [(k_on)_a C – (k_off)_a B]

            ∂ B / ∂ t  =                   (k_on)_a C – (k_off)_a B                                                                        (1a)

in phase a, and

            ∂ C / ∂ t  =  D_b ∇²C – [(k_on)_b C – (k_off)_b B]

            ∂ B / ∂ t  =                   (k_on)_b C – (k_off)_b B                                                                        (1b)

in phase b at the microscopic scale.  What is the equation governing the macroscopically observable behavior?  The answer turns out not to be of the same form with homogenized coefficients, namely

            ∂ C / ∂ t  =  D_eff ∇²C – [(k_on)_eff C – (k_off)_eff B]

            ∂ B / ∂ t  =                    (k_on)_eff C – (k_off)_eff B,                                                                   (2)

as might be expected.  Rather, the correct macroscopic description generally involves a free concentration C(x,y,z,t) and two effective bound concentrations B^[1](x,y,z,t) and B^[2](x,y,z,t) that evolve collectively according to the more complex system of transport equations

            ∂ C    / ∂ t  =  D_eff ∇²C – [(k_on^[1])_eff C – (k_off^[1])_eff B^[1]] – [(k_on^[2])_eff C – (k_off^[2])_eff B^[2]]

            ∂ B^[1] / ∂ t  =                     (k_on^[1])_eff C – (k_off^[1])_eff B^[1]

            ∂ B^[2] / ∂ t  =                                                                    (k_on^[2])_eff C – (k_off^[2])_eff B^[2].       (3)

A general prescription is derived to calculate the effective coefficients (k_on^[1])_eff, (k_off^[1])_eff, (k_on^[2])_eff and (k_off^[2])_eff, involving an eigenvalue problem posed in a unit cell of the microstructure.  Asymptotic analysis yields an explicit formula that approximates the requisite eigenvalue very accurately, thereby allowing the effective binding rate constants to be determined without solving a diffusion problem.  Although the analysis is originally carried out for spatially periodic microstructures, the asymptotic reveals a broader applicability of the results, which are verified vis-à-vis a brute-force numerical calculation.  The analysis is generalized to an N-phase microstructure for any N ≥ 2.

The homogenization paradigm has important implications for diffusion models of biological tissues, for which: (ii) multiphase microstructure invariable produces two or more microscopic sets of on- and off binding rate constants  [(k_on)_a, (k_off)_a], [(k_on)_b, (k_off)_b], …; and (ii) Eq. (2) is invariably assumed to apply at the macroscopic scale.  Practical application of the analysis is demonstrated within the context of drug/chemical diffusion through the surface (barrier) layer of skin, the stratum corneum, in which reversible binding to keratin protein is known to occur.^13-16

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