Hydrophobic forces and the length limit of foldable protein domains

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.