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Recent analyses of amino acid mutations in proteins reveal that mutations at many pairs of sites are epistatic-i.e., their effects on fitness are non-additive-the combined effect of two mutations being significantly larger or smaller than the sum of their effects considered independently. Interestingly, epistatic sites are not necessarily near each other in the folded structure of a protein, and may even be located on opposite sides of a molecule. However, the mechanistic reasons for long-range epistasis remain obscure. Here, we study long-range epistasis in proteins using a previously developed model in which off-lattice polymers are evolved under ligand binding constraints. Epistatic effects in the model are qualitatively similar to those recently reported for small proteins, and many are long-range. We find that a major reason for long-range epistasis is conformational change-a recurrent theme in both positive and negative epistasis being the transfer, or exchange of material between the ordered nucleus, which supports the binding site, and the liquid-like surface of a folded molecule. These local transitions in phase and folded structure are largely responsible for long-range epistasis in our model., Competing Interests: The authors have declared that no competing interests exist.

We investigate protein evolution using an off-lattice polymer model evolved to imitate the behavior of small enzymes. Model proteins evolve through mutations to nucleotide sequences (including insertions and deletions) and are selected to fold and maintain a specific binding site compatible with a model ligand. We show that this requirement is, in itself, sufficient to maintain an ordered folding domain, and we compare it to the requirement of folding an ordered (but otherwise unrestricted) domain. We measure rates of amino acid change as a function of local environment properties such as solvent exposure, packing density, and distance from the active site, as well as overall rates of sequence and structure change, both along and among model lineages in star phylogenies. The model recapitulates essentially all of the behavior found in protein phylogenetic analyses, and predicts that amino acid substitution rates vary linearly with distance from the binding site.

Protein evolution is frequently explored using minimalist polymer models, however, little attention has been given to the problem of structural drift, or diffusion. Here, we study neutral evolution of small protein motifs using an off-lattice heteropolymer model in which individual monomers interact as low-resolution amino acids. In contrast to most earlier models, both the length and folded structure of the polymers are permitted to change. To describe structural change, we compute the mean-square distance (MSD) between monomers in homologous folds separated by n neutral mutations. We find that structural change is episodic, and, averaged over lineages (for example, those extending from a single sequence), exhibits a power-law dependence on n. We show that this exponent depends on the alignment method used, and we analyze the distribution of waiting times between neutral mutations. The latter are more disperse than for models required to maintain a specific fold, but exhibit a similar power-law tail.

The biological function of a protein often depends on the formation of an ordered structure in order to support a smaller, chemically active configuration of amino acids against thermal fluctuations. Here we explore the development of proteins evolving to satisfy this requirement using an off-lattice polymer model in which monomers interact as low resolution amino acids. To evolve the model, we construct a Markov process in which sequences are subjected to random replacements, insertions, and deletions and are selected to recover a predefined minimum number of solid-ordered monomers using the Lindemann melting criterion. We show that polymers generated by this process consistently fold into soluble, ordered globules of similar length and complexity to small protein motifs. To compare the evolution of the globules with proteins, we analyze the statistics of amino acid replacements, the dependence of site mutation rates on solvent exposure, and the dependence of structural distance on sequence distance for homologous alignments. Despite the simplicity of the model, the results display a surprisingly close correspondence with protein data.

We consider a nonstatistical, computationally fast experiment to identify important topological constraints in folding small globular proteins of about 100-200 amino acids. In this experiment, proteins are expanded mechanically along a path of steepest increase in the free space around residues. The pathways are often consistent with folding scenarios reported in kinetics experiments and most accurately describe obligatory or mechanic folding proteins. The results suggest that certain topological "defects" in proteins lead to preferred, entropically favorable channels down their free energy landscapes.

It is shown that the sequence-ordering tendencies induced by design into different fast-folding, thermally stable native structures interfere. This interference results in a type of quasiorthogonality between optimal native structures, which divides sequence space into fast-folding, thermally stable families surrounded by slow-folding, low stability shells. A concrete example of this effect is provided by using a simple alpha carbon type model in which a complete correspondence is established between sequence and structure. It is speculated that gaps can occur in the space of protein-like sequences separating the sequence families and resulting in a mechanism for stability and diversity of protein sequence information.