Minimum energy compact structures of random sequences of heteropolymers

We enumerate the number of minimum energy compact structures (MECS) for a two dimensional heteropolymer model consisting of hydrophobic (H) and hydrophilic beads averaged over all possible sequences as a function of the total number of beads N and the number of H beads NH. The analysis of the average number of compact structures, which grows exponentially with N, gives no indication of logarithmic corrections to the free energy. For intermediate values of the ratio NH/N, we estimate that the average number of MECS---which has a minimum at NHᵐⁱⁿ/N=0.6±{}0.1---does not grow with N. Our results suggest that the late stages of protein folding should be restricted to the sampling of MECS only.