Concentration equations for the diffusion of a reversibly reacting substance in gel-like media

Abstract

The concentration of a diffusible substanceA(x, t) in a semi-infinite geometry is studied for the set of reversible reactionsA+B _i ⇆C _i ;i=1...n, whereB _i andC _i are assumed to be associated with non-diffusible biological structures. Assuming chemical equilibrium prevails throughout for each reaction, it is shown that a single uncoupled partial differential equation is sufficient to specifyA(x, t) and indirectlyB _i (x, t) andC _i (x, t) as well:

$$\left[ {1 + \sum\limits_i {\frac{{K_i \beta _i }}{{\left( {1 + K_i A} \right)^2 }}} } \right]\frac{{\partial A}}{{\partial t}} = D_A \frac{{\partial ^2 A}}{{\partial x^2 }}$$

whereK _i is the chemical equilibrium constant of theith reaction, β₁ is concentration of binding sites of theith species (i.e.B _i+C _i) andD _A is the usual diffusion constant forA. Numerical solutions for boundary conditions amenable to the Boltzman transformation are presented and the range of parameters established over which the uniqueness and convergence of the solutions can be proven.