Viscosity Dependence of the Folding Rates of Proteins

Abstract

The viscosity $(\ensuremath{\eta})$ dependence of the folding rates for four sequences (the native state of three sequences is a $\ensuremath{\beta}$ sheet, while the fourth forms an $\ensuremath{\alpha}$ helix) is calculated for off-lattice models of proteins. Assuming that the dynamics is given by the Langevin equation, we show that the folding rates increase linearly at low viscosities $\ensuremath{\eta}$, decrease as $1/\ensuremath{\eta}$ at large $\ensuremath{\eta}$, and have a maximum at intermediate values. The Kramers' theory of barrier crossing provides a quantitative fit of the numerical results. By mapping the simulation results to real proteins we estimate that for optimized sequences the time scale for forming a four turn $\ensuremath{\alpha}$-helix topology is about 500 ns, whereas for $\ensuremath{\beta}$ sheet it is about $10\ensuremath{\mu}\mathrm{s}$.