A Criterion That Determines Fast Folding of Proteins: A Model Study

We consider the statistical mechanics of a full set of two-dimensional protein-like heteropolymers, whose thermodynamics is characterized by the coil-to-globular ($T_{\theta }$) and the folding ($T_f$) transition temperatures. For our model, the typical time scale for reaching the unique native conformation is shown to scale as ${\tau }_f\sim F(M)\exp (\sigma \mathrm{/}{\sigma }_0)$, where $\sigma =1-T_f\mathrm{/}{T}_{\theta }$, $M$ is the number of residues, and $F(M)$ scales algebraically with $M$. We argue that $T_f$ scales linearly with the inverse of entropy of low energy non-native states, whereas $T_{\theta }$ is almost independent of it. As $\sigma \to 0$, non-productive intermediates decrease, and the initial rapid collapse of the protein leads to structures resembling the native state. Based solely on {\it accessible} information, $\sigma$ can be used to predict sequences that fold rapidly.