Protein Folding, the Levinthal Paradox and Rapidly Mixing Markov Chains

In [20, 21], A. Šali, E. Shakhnovich and M. Karplus modeled protein folding using a 27-bead heteropolymer on a cubic lattice with normally distributed contact energies. Using a Monte-Carlo folding algorithm with a local move set between conformations, Šali et al. attempted to answer the Levinthal paradox [13] of how a protein can fold rapidly, i.e. within milliseconds to seconds, despite the magnitude of the conformation space (e.g. approximately 526 ≈ 1018 for the 27mer). Letting t0(P ) denote the folding time (i.e. first passage time) and ∆(P ) denote the energy gap between the lowest energy Ei0(native state) and second lowest energy Ei1 of protein P with normally distributed contact energy, Šali, Shakhnovich and Karplus observed that ∆(P ) is large exactly when t0(P ) is small. Using Sinclair’s notion of rapid mixing [17] and his modification of the Diaconis-Stroock [6] bound on relative pointwise distance in terms of the subdominant eigenvalue, we provide the first theoretical basis for the principal observation of Šali, Shakhnovich and Karplus. Specifically, we show that the mean first passage time is bounded above by c1πi0πi1 + c2, where πi0 [resp. πi1 ] is the Boltzmann probability of the system being in the native minimum energy state [resp. second minimum]. It follows that this upper bound decreases iff the energy gap Ei1 −Ei0 increases. Our result is actually proved for pivot moves (rotations) with multiple occupancy, rather than local moves, but it seems clear that our technique can be extended to cover a variant of the model of [20, 21].