Folding kinetics of proteins and copolymers

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Abstract
We model the kinetic processes by which globular proteins and other heteropolymers fold to compact states. We perform Monte Carlo dynamics simulations on short self-avoiding copolymer chains on two-dimensional square lattices. The driving force for collapse is the aversion of nonpolar monomers for water. The chain monomers are of two types, H and P; favorable interactions occur among HH contacts. One respect in which this study differs from previous Monte Carlo folding studies is that the chains are sufficiently short that: (i) we can know unequivocally which conformations are at global minima (the ‘‘native’’ states) and which are at local minima of free energy, (ii) we can explore ‘‘pathway space’’ densely to determine the relative probabilities of all the possible pathways, and thus we establish that the model is ergodic and gives the equilibrium distribution in the long-time limit. We find that any individual molecule passes through a wide range of conformational states, often many times. Nevertheless, this meandering is not inconsistent with the observation that proteins fold through specific pathways involving particular sequences of events. Some pathways are strongly favored; i.e., folding is ‘‘cooperative’’ in that a ‘‘nucleating’’ HH contact acts as a constraint that restricts local conformational freedom and speeds the ‘‘zipping up’’ of other contacts nearby. Which pathways are favored depends on the sequence and the solvent. How does a chain fold to its native state so quickly? For these short chains of fixed length, we observe three regimes of folding kinetics. In the ‘‘Levinthal limit,’’ as the HH attraction approaches zero, folding is slow because the molecule searches randomly through the large ensemble of open conformations. In the ‘‘multiple minima’’ limit, when the HH attraction is very strong, folding is also slow because the chain becomes stuck in local minima of wrong compact conformations. However, we find that folding is relatively fast for intermediate HH attraction because the driving force ‘‘directs’’ the molecule toward a small ensemble of compact states, and yet incorrect potential wells are not too deep to slow this process.