Multisite reversible geminate reaction.

We provide an analytic solution for diffusion-influenced geminate reaction with multiple (N) reversible binding sites (of which one may be irreversible). The solution obtained in the Laplace domain, for two different initial conditions, is valid for the case when the sites are overlapping spheres with no long-range interactions with the diffusing particle. The possibility to invert into the time domain is determined by a characteristic polynomial. When all its roots are distinct, it is possible to apply the Lagrange interpolation formula and obtain a partial-fraction expansion that can be termwise inverted. At long times the occupancy of all sites, and for all initial conditions, decays as t(-3/2). The behavior at short times depends on the initial condition: when starting from contact, the binding probability rises as t(1/2), but if the particle is initially bound to one of the sites, the occupancy of the others rises as t(3/2). In between these two power laws we observe an intermediate-time kinetics consisting of N decaying exponentials. Those which are slower than a characteristic diffusion time are in the reaction-control regime and fit a discrete-state kinetic approximation with no adjustable parameters, whereas the faster kinetic steps are diffusion controlled. The model solved herein may depict a wide range of physical situations, from multisite proton transfer kinetics to hydrogen-bond dynamics of liquid water.