Protein electron transfer reorganization energy spectrum from normal mode analysis. 1. Theory

We present an analytical model that describes the coupling of protein fluctuations to electron transfer, The model treats both the protein and the bulk solvent to couple to electron transfer, The protein is represented by a low-dielectric cavity containing explicit protein atoms, and the bulk solvent is represented by a high-dielectric continuum surrounding the cavity. Protein fluctuations are modeled by collective normal modes with solvation energies incorporated through explicit reaction field energies. The shifts of the equilibrium normal mode variables upon electron transfer, related to the mode-specific couplings and reorganization energies, are calculated assuming the difference of the potential energy surfaces before and after electron transfer by a hyper plane in the normal mode vector space. This linear coupling assumption allows only one set of normal mode vectors to span both the reactant and product equilibrium conformations. The model is equivalent to a reduced spin-boson formalism (protein only); however, unlike previous work within this formalism, the bath modes are not spatially anonymous in our treatment. They are associated with unambiguous frequency and spatial signatures allowing a spectral analysis of protein reorganization energy with one-to-one connection with actual protein fluctuation. This aspect of our model is very crucial since it allows, for the first time, to make a direct connection between actual protein motion and electron transfer, as demonstrated by a simulation presented in an accompanying paper (J. Phys. Chem. 1998, 102, XXX).