Sequence entropy of folding and the absolute rate of amino acid substitutions

  • Nature Ecology & Evolution 119231930 (2017)
  • doi:10.1038/s41559-017-0338-9
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Adequate representations of protein evolution should consider how the acceptance of mutations depends on the sequence context in which they arise. However, epistatic interactions among sites in a protein result in hererogeneities in the substitution rate, both temporal and spatial, that are beyond the capabilities of current models. Here we use parallels between amino acid substitutions and chemical reaction kinetics to develop an improved theory of protein evolution. We constructed a mechanistic framework for modelling amino acid substitution rates that uses the formalisms of statistical mechanics, with principles of population genetics underlying the analysis. Theoretical analyses and computer simulations of proteins under purifying selection for thermodynamic stability show that substitution rates and the stabilization of resident amino acids (the ‘evolutionary Stokes shift’) can be predicted from biophysics and the effect of sequence entropy alone. Furthermore, we demonstrate that substitutions predominantly occur when epistatic interactions result in near neutrality; substitution rates are determined by how often epistasis results in such nearly neutral conditions. This theory provides a general framework for modelling protein sequence change under purifying selection, potentially explains patterns of convergence and mutation rates in real proteins that are incompatible with previous models, and provides a better null model for the detection of adaptive changes.

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We thank B. Khatri for helpful discussions. We acknowledge the support of the Medical Research Council (UK) (MC_U117573805) and the Biotechnology and Biological Sciences Research Council (UK) (BB/P007562/1) to R.A.G. and the National Institutes of Health (NIH; GM083127 and GM097251) to D.D.P.

Author information


  1. Division of Infection and Immunity, University College London, London, WC1E 6BT, UK

    • Richard A. Goldstein
  2. Department of Biochemistry and Molecular Genetics, University of Colorado School of Medicine, Aurora, CO, 80045, USA

    • David D. Pollock


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R.A.G. and D.D.P. jointly designed the study, analysed the results and wrote the paper. R.A.G wrote the simulation software and performed all mathematical derivations.

Competing interests

The authors declare no competing financial interests.

Corresponding authors

Correspondence to Richard A. Goldstein or David D. Pollock.

Electronic supplementary material

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