Diffusion with binding in tissues

Ligand molecules must diffuse to the cell surface before they can participate in the reactions shown in Equation 4-52. In the previous chapter, the concentration gradient in the vicinity of a spherical cell was determined for the situation in which the solute concentration disappears rapidly at the surface (Equation 3-46) rapid disappearance provided a boundary condition (ca = 0 at the cell surface). In the present situation, the reaction at the surface is not instantaneous therefore, the rate of diffusion at the cell surface must be balanced by the rate of ligand disappearance due to the binding reaction  

The solution to Equation 4-54 subject to the two boundary conditions (Equations 4-53 and 4-55) is  

The overall rate of binding at the cell surface—which accounts for a balance between diffusion of ligand to the surface and a finite rate of association with receptors at the surface—can be calculated from the flux at the cell surface  

In Chapter 3, the rate of diffusion of solute to a cell surface was determined for the special case of very rapid reaction at the surface. In that special case, the rate of diffusion determines the rate of reaction at the surface (recall Equation 3-48) the rate constant for the diffusion-limited reaction is  

The overall forward rate constant can, therefore, be written in terms of the rate constant for the diffusion-hmited reaction (/c, ) and the rate constant for the intrinsic binding reaction (cr q )  

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