Embedding a native state into a random heteropolymer model: the dynamic approach

We study a random heteropolymer model with Langevin dynamics, in the supersymmetric formulation. Employing a procedure similar to one that has been used in static calculations, we construct an ensemble in which the affinity of the system for a native state is controlled by a "selection temperature" T0. In the limit of high T0, the model reduces to a random heteropolymer, while for T0-->0 the system is forced into the native state. Within the Gaussian variational approach that we employed previously for the random heteropolymer, we explore the phases of the system for high and low T0. For high T0, the system exhibits a (dynamical) spin-glass phase, like that found for the random heteropolymer, below a temperature T(g). For low T0, we find an ordered phase, characterized by a nonzero overlap with the native state, below a temperature T(n) proportional to 1/T(0)>T(g). However, the random-globule phase remains locally stable below T(n), down to the dynamical glass transition at T(g). Thus, in this model, folding is rapid for temperatures between T(g) and T(n), but below T(g) the system can get trapped in conformations uncorrelated with the native state. At a lower temperature, the ordered phase can also undergo a dynamical glass transition, splitting into substates separated by large barriers.