The combined flux technique for diffusion—reaction problems in partial equilibrium: Application to the facilitated transport of carbon dioxide in aqueous bicarbonate solutions

Abstract

A systematic method is derived for the simplification of a system of diffusion—reaction equations in which the kinetics fall into two distinct classes, E fast reactions and R — E slow reactions. The method has its origins in the “combined flux” technique introduced for the modelling of laminar flames by Dixon-Lewis et al. (1975, Proc. R. Soc. A 346, 261–278). In using this method the E fast reactions are assumed to be in local equilibrium throughout some region of space, and this allows a reduction by E in the number of differential equations which must be simultaneously solved to compute multicomponent concentration profiles and fluxes. By eliminating the fast reactions this approach alleviates the stiffness problem that typically accompanies a reaction network in which the characteristic timescales vary by several orders of magnitude and provides what is essentially the lowest order term of the asymptotic expansion in the outer region,. Once derived, this technique is applied to the problem of the facilitated transport of carbon dioxide in aqueous bicarbonate solution to determine, in particular, the effect of induced electric fields on membrane permeability. The theoretical results obtained show that diffusion potentials in the uncatalyzed membrane in partial reaction equilibrium, i.e. R ⪢ E, are several orders of magnitude lower than those predicted for the carbonic anhydrase catalyzed membrane which attains full reaction equilibrium, i.e. R = E. Furthermore, in the example studied the induced electric field is shown to have a negligibly small influence on the net flux of CO 2 regardless of the assumed kinetic rates.