Hydrophobic forces and the length limit of foldable protein domains

Abstract

To find the native conformation (fold), proteins sample a subspace that is typically hundreds of orders of magnitude smaller than their full conformational space. Whether such fast folding is intrinsic or the result of natural selection, and what is the longest foldable protein, are open questions. Here, we derive the average conformational degeneracy of a lattice polypeptide chain in water and quantitatively show that the constraints associated with hydrophobic forces are themselves sufficient to reduce the effective conformational space to a size compatible with the folding of proteins up to approximately 200 amino acids long within a biologically reasonable amount of time. This size range is in general agreement with the experimental protein domain length distribution obtained from approximately 1,200 proteins. Molecular dynamics simulations of the Trp-cage protein confirm this picture on the free energy landscape. Our analytical and computational results are consistent with a model in which the length and time scales of protein folding, as well as the modular nature of large proteins, are dictated primarily by inherent physical forces, whereas natural selection determines the native state.levinthal paradox | lattice model | kinetics | folding funnel E ver since the discovery that proteins can spontaneously selfassemble into unique three-dimensional shapes (folds) (1), the mechanism of this folding process has been a focus of biology. It has been shown that random sequences can fold into a unique ground state which is separated from other folds by an energy gap (2). However, assuming the existence of a unique native fold, there is no assurance that the protein can efficiently parse the fold space to find it. In particular, how nature is able to search the exponentially increasing number of folds accessible to proteins of nontrivial length has not been explicitly elucidated; Levinthal famously estimated that for a protein consisting of 150 amino acids, in which each degree of freedom is discretized into only ten possible values, it would take much longer than the age of the universe to sample all the folds even at the limit of molecular motion (3).The qualitative resolution of the Levinthal paradox has been the concept of the folding funnel, whereby a global bias in the multidimensional energy landscape channels the protein toward the subspace containing the native fold (4). For example, by introducing an artificial search bias in favor of native contacts (5) or designing sequences favoring specific secondary structures (6), fast folding was computationally observed. This suggests that the funnel arises from evolutionary tuning of the intramolecular interactions via sequence mutation and that feasible protein folding times are the result of natural selection.In contrast to this picture, we show below that a general effect, namely the hydrophobic force, is sufficient to account for the kinetics of fast folding without sequence evolution. This force, which is the global tendency for the chain to collapse to a co...

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