Configurational Diffusion on a Locally Connected Correlated Energy Landscape; Application to Finite, Random Heteropolymers

We study the time scale for diffusion on a correlated energy landscape using models based on the generalized random energy model (GREM) studied earlier in the context of spin glasses (Derrida B. and Gardner E., J. Phys. C 19 (1986) 2253) with kinetically local connections. The escape barrier and mean escape time are significantly reduced from the uncorrelated landscape (REM) values. Results for the mean escape time from a kinetic trap are obtained for two models approximating random heteropolymers in different regimes, with linear and bi-linear approximations to the configurational entropy versus similarity $q$ with a given state. In both cases, a correlated landscape results in a shorter escape time from a meta-stable state than in the uncorrelated model (Bryngelson J.D. and Wolynes P.G., J. Phys. Chem 93 (1989) 6902). Results are compared to simulations of the diffusion constant for 27-mers. In general there is a second transition temperature above the thermodynamic glass temperature, at and above which kinetics becomes non-activated. In the special case of an entropy linear in $q$, there is no escape barrier for a model preserving ultrametricity. However, in real heteropolymers a barrier can result from the breaking of ultrametricity, as seen in our non-ultrametric model. The distribution of escape times for a model preserving microscopic ultrametricity is also obtained, and found to reduce to the uncorrelated landscape in well-defined limits