Influence of diffusion on the kinetics of multisite phosphorylation

When an enzyme modifies multiple sites on a substrate, the influence of the relative diffusive motion of the reactants cannot be described by simply altering the rate constants in the rate equations of chemical kinetics. We have recently shown that, even as a first approximation, new transitions between the appropriate species must also be introduced. The physical reason for this is that a kinase, after phosphorylating one site, can rebind and modify another site instead of diffusing away. The corresponding new rate constants depend on the capture or rebinding probabilities that an enzyme-substrate pair, which is formed after dissociation from one site, reacts at the other site rather than diffusing apart. Here we generalize our previous work to describe both random and sequential phosphorylation by considering inequivalent modification sites. In addition, anisotropic reactive sites (instead of uniformly reactive spheres) are explicitly treated by using localized sink and source terms in the reaction-diffusion equations for the enzyme-substrate pair distribution function. Finally, we show that our results can be rederived using a phenomenological approach based on introducing transient encounter complexes into the standard kinetic scheme and then eliminating them using the steady-state approximation.