Transient kinetics of chemical reactions with bounded diffusion perpendicular to the reaction coordinate: Intramolecular processes with slow conformational changes

Intramolecular reactions inside macromolecules (e.g., binding of ligands to iron inside heme proteins) may often be coupled to slow random fluctuations in the reaction center geometry. This motion is "perpendicular" to the reaction coordinate. It can be described as bounded diffusion in the presence of a binding potential field and an intramolecular rate constant which depends on the perpendicular degree of freedom. The diffusion equation is solved under the appropriate reflective boundary conditions. The transient decay of the total population is multiexponential (power law) for small diffusivity, changing to monoexponential kinetics for large diffusivity. For large times or large diffusivity, direct integration is very tedious, but an eigenvalue expansion converges rapidly. It also allows the calculation of the "average survival time" (an extension of the "first passage time") a natural candidate for replacing the reciprocal rate constant in multiexponential kinetics. An example is given for electron transfer between two loosely bound sites in a macromolecule. The average survival time shows a non-Kramers dependence on diffusivity, of the type found in the binding kinetics in heme proteins.