Phase Diagram of Random Heteropolymers: Replica Approach and Application of a New Monte Carlo Algorithm

In this contribution we study the phase diagram of the Random Bond Heteropolymer model, introduced ten years ago as a simplified model for Protein Folding. In its lattice version, only non-bonded monomers on neighboring sites interact. The interactions at each pair of neighbors are independent Gaussian variables. Two phase transitions are expected: the usual collapse transition and a freezing transition after which only a finite number of different contact maps are relevant. The two transition lines cross at a triple point. For larger disorder, the collapse happens inside the frozen phase. In the frozen phase the entropy of the set of contact maps is non-extensive, and the specific entropy of the system is dominated by the configurational entropy at the contact maps kept fixed. At low density this is finite. At very high density contact maps determine their configuration, and the entropy is zero. Thus we conjecture the existence of a new phase transition where the latter entropy vanishes. In the phase with zero entropy the freezing is abrupt: the ground state structures are the relevant ones at freezing and remain stable as temperature is lowered. In the other phase the freezing is gradual and the ground state structures are stable only at zero temperature. But we cannot exclude that abrupt freezing takes place only for maximally compact structures, thus only in the limit in which all interactions are attractive and T ! 0. Simulations confirm fully some features of this picture (for instance, the collapse transition before the freezing seems to be predicted exactly by the annealed approximation), while not much can be said about the existence of the conjectured phase transition. The Monte Carlo method that we used, the Pruned Enriched Rosenbluth Method (PERM), has proved to be very efficient. Its principles and its implementation are described in an Appendix.