Evolutionary-rate variation among sites within proteins depends on functional and biophysical properties that constrain protein evolution. It is generally accepted that proteins must be able to fold stably in order to function. However, the relationship between stability constraints and among-sites rate variation is not well understood. Here, we present a biophysical model that links the thermodynamic stability changes due to mutations at sites in proteins (ΔΔG) to the rate at which mutations accumulate at those sites over evolutionary time. We find that such a “stability model” generally performs well, displaying correlations between predicted and empirically observed rates of up to 0.75 for some proteins. We further find that our model has comparable predictive power as does an alternative, recently proposed “stress model” that explains evolutionary-rate variation among sites in terms of the excess energy needed for mutants to adopt the correct active structure (ΔΔG*). The two models make distinct predictions, though, and for some proteins the stability model outperforms the stress model and vice versa. We conclude that both stability and stress constrain site-specific sequence evolution in proteins.

Several recent works have shown that protein structure can predict site-specific evolutionary sequence variation. In particular, sites that are buried and/or have many contacts with other sites in a structure have been shown to evolve more slowly, on average, than surface sites with few contacts. Here, we present a comprehensive study of the extent to which numerous structural properties can predict sequence variation. The quantities we considered include buriedness (as measured by relative solvent accessibility), packing density (as measured by contact number), structural flexibility (as measured by B factors, root-mean-square fluctuations, and variation in dihedral angles), and variability in designed structures. We obtained structural flexibility measures both from molecular dynamics simulations performed on nine non-homologous viral protein structures and from variation in homologous variants of those proteins, where they were available. We obtained measures of variability in designed structures from flexible-backbone design in the Rosetta software. We found that most of the structural properties correlate with site variation in the majority of structures, though the correlations are generally weak (correlation coefficients of 0.1–0.4). Moreover, we found that buriedness and packing density were better predictors of evolutionary variation than structural flexibility. Finally, variability in designed structures was a weaker predictor of evolutionary variability than buriedness or packing density, but it was comparable in its predictive power to the best structural flexibility measures. We conclude that simple measures of buriedness and packing density are better predictors of evolutionary variation than the more complicated predictors obtained from dynamic simulations, ensembles of homologous structures, or computational protein design.

Several recent works have shown that protein structure can predict site-specific evolutionary sequence variation. In particular, sites that are buried and/or have many contacts with other sites in a structure have been shown to evolve more slowly, on average, than surface sites with few contacts. Here, we present a comprehensive study of the extent to which numerous structural properties can predict sequence variation. The quantities we considered include buriedness (as measured by relative solvent accessibility), packing density (as measured by contact number), structural flexibility (as measured by B factors, root-mean-square fluctuations, and variation in dihedral angles), and variability in designed structures. We obtained structural flexibility measures both from molecular dynamics simulations performed on nine non-homologous viral protein structures and from variation in homologous variants of those proteins, where they were available. We obtained measures of variability in designed structures from flexible-backbone design in the Rosetta software. We found that most of the structural properties correlate with site variation in the majority of structures, though the correlations are generally weak (correlation coefficients of 0.1-0.4). Moreover, we found that buriedness and packing density were better predictors of evolutionary variation than structural flexibility. Finally, variability in designed structures was a weaker predictor of evolutionary variability than buriedness or packing density, but it was comparable in its predictive power to the best structural flexibility measures. We conclude that simple measures of buriedness and packing density are better predictors of evolutionary variation than the more complicated predictors obtained from dynamic simulations, ensembles of homologous structures, or computational protein design.

Abstract. Evolutionary-rate variation among sites within proteins depends on functional and biophysical properties that constrain protein evolution. It is generally accepted that proteins must be able to fold stably in order to function. However, the relationship between stability constraints and among-sites rate variation is not well understood. Here, we present a biophysical model that links the thermodynamic stability changes due to mutations at sites in proteins (∆∆G) to the rate at which mutations accumulate at those sites over evolutionary time. We find that such a "stability model" generally performs well, displaying correlations between predicted and empirically observed rates of up to 0.75 for some proteins. We further find that our model has comparable predictive power as does an alternative, recently proposed "stress model" that explains evolutionary-rate variation among sites in terms of the excess energy needed for mutants to adopt the correct active structure (∆∆G * ). The two models make distinct predictions, though, and for some proteins the stability model outperforms the stress model and vice versa. We conclude that both stability and stress constrain site-specific sequence evolution in proteins.

Computational protein design attempts to create protein sequences that fold stably into pre-specified structures. Here we compare alignments of designed proteins to alignments of natural proteins and assess how closely designed sequences recapitulate patterns of sequence variation found in natural protein sequences. We design proteins using RosettaDesign, and we evaluate both fixed-backbone designs and variable-backbone designs with different amounts of backbone flexibility. We find that proteins designed with a fixed backbone tend to underestimate the amount of site variability observed in natural proteins while proteins designed with an intermediate amount of backbone flexibility result in more realistic site variability. Further, the correlation between solvent exposure and site variability in designed proteins is lower than that in natural proteins. This finding suggests that site variability is too uniform across different solvent exposure states (i.e., buried residues are too variable or exposed residues too conserved). When comparing the amino acid frequencies in the designed proteins with those in natural proteins we find that in the designed proteins hydrophobic residues are underrepresented in the core. From these results we conclude that intermediate backbone flexibility during design results in more accurate protein design and that either scoring functions or backbone sampling methods require further improvement to accurately replicate structural constraints on site variability.