Solvent-quality-dependent contact formation dynamics in proteins

The mean time of contact formation between two ends of a protein chain shows power law dependence with respect to the number of residues, τCF ∼ Nα. Fluorescence quenching measurements based on triplet-triplet energy transfer show variation in the value of scaling exponent α for different protein-solvent systems. This points to the relevance of the protein-solvent interactions (solvent quality) and hydrodynamic interactions in determining the time scale of contact formation. Here, starting from a non-Markovian diffusion equation supplemented with an exponential sink term that accounts for the energy transfer reaction between the donor and acceptor groups, we calculate the mean time of contact formation using the Wilemski-Fixman closure approximation. The non-Markovian diffusion-reaction equation includes the effects of solvent quality and hydrodynamic interaction in a mean-field fashion. It shows that the contact formation dynamics is mainly governed by two time scales, the reciprocal of the intrinsic rate of quenching (k0ET)-1, and the relaxation time τ0 of the coarse-grained residue of size b. A comparison of these two time scales yields reaction-controlled and diffusion-controlled kinetic regimes with different values of the scaling exponents. In between these regimes the increase in the number of residues switches the kinetics from RC at low N to DC at large N. These general results suggest that experimental estimates of the scaling exponents reflect solvent-quality dependence of the RC kinetics.