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Evolutionary interplay between structure, energy and epistasis in the coat protein of the $ϕχ174$ phage family

Viral capsids are structurally constrained by interactions amongst the amino acids of the constituting proteins. Therefore, epistasis is expected to evolve amongst sites engaged in physical interactions, and to influence their substitution rates. In order to study the distribution of structural epistasis, we modeled \emph{in silico} the capsid of 18 species of the \phix \, family, including the wild type. \phix \, is amongst the simplest organisms, making it suitable for experimental evolution and \emph{in silico} modeling. We found nearly 40 variable amino acid sites in the main capsid protein across the 18 species. To study how epistasis evolved in this group, we reconstructed the ancestral sequences using a Bayesian phylogenetic framework. The ancestral states include 8 variable amino acids, for a total of 256 possible haplotypes. The $dN/dS$ ratio is low, suggesting strong purifying selection, consistent with the idea that the structure is constrained by some form of stabilizing selection. For each haplotype in the ancestral node and for the extant species we estimated \emph{in silico} the distribution of free energies and of epistasis. We found that free energy has not significantly increased but epistasis has. We decomposed epistasis up to fifth order and found that high-order epistasis can sometimes compensate pairwise interactions, often making the free energy seem additive. By synthesizing some of the ancestral haplotypes of the capsid gene, we measured their fitness experimentally, and found that the predicted deviations in the coat protein free energy do not significantly affect fitness, which is consistent with the stabilizing selection hypothesis.