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


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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Fig. 1: Relative stabilities of amino acid pairs.
Fig. 2: Comparison of predicted and observed substitution rates.
Fig. 3: Example of a trajectory before and after a substitution from glutamic acid to lysine.
Fig. 4: Accuracy of site-specific stability and evolutionary Stokes shift predictions.


  1. 1.

    Breen, M. S., Kemena, C., Vlasov, P. K., Notredame, C. & Kondrashov, F. A. Epistasis as the primary factor in molecular evolution. Nature 490, 535–538 (2012).

  2. 2.

    Usmanova, D. R., Ferretti, L., Povolotskaya, I. S., Vlasov, P. K. & Kondrashov, F. A. A model of substitution trajectories in sequence space and long-term protein evolution. Mol. Biol. Evol. 32, 542–554 (2015).

  3. 3.

    Sarkisyan, K. S. et al. Local fitness landscape of the green fluorescent protein. Nature 533, 397–401 (2016).

  4. 4.

    Ashenberg, O., Gong, L. I. & Bloom, J. D. Mutational effects on stability are largely conserved during protein evolution. Proc. Natl Acad. Sci. USA 110, 21071–21076 (2013).

  5. 5.

    Gong, L. I. & Bloom, J. D. Epistatically interacting substitutions are enriched during adaptive protein evolution. PLoS Genet. 10, e1004328 (2014).

  6. 6.

    Pollock, D. D., Thiltgen, G. & Goldstein, R. A. Amino acid coevolution induces an evolutionary Stokes shift. Proc. Natl Acad. Sci. USA 109, E1352–E1359 (2012).

  7. 7.

    Pollock, D. D. & Goldstein, R. A. Strong evidence for protein epistasis, weak evidence against it. Proc. Natl Acad. Sci. USA 111, E1450 (2014).

  8. 8.

    Shah, P., McCandlish, D. M. & Plotkin, J. B. Contingency and entrenchment in protein evolution under purifying selection. Proc. Natl Acad. Sci. USA 112, E3226–E3235 (2015).

  9. 9.

    Pollock, D. D., Taylor, W. R. & Goldman, N. Coevolving protein residues: maximum likelihood identification and relationship to structure. J. Mol. Biol. 287, 187–198 (1999).

  10. 10.

    Muse, S. V. & Gaut, B. S. A likelihood approach for comparing synonymous and nonsynonymous nucleotide substitution rates, with application to the chloroplast genome. Mol. Biol. Evol. 11, 715–724 (1994).

  11. 11.

    Nielsen, R. & Yang, Z. Likelihood models for detecting positively selected amino acid sites and applications to the HIV-1 envelope gene. Genetics 148, 929–936 (1998).

  12. 12.

    Tamuri, A. U., dos Reis, M., Hay, A. J. & Goldstein, R. A. Identifying changes in selective constraints: host shifts in influenza. PLoS Comput. Biol. 5, e1000564 (2009).

  13. 13.

    Castoe, T. A. et al. Evidence for an ancient adaptive episode of convergent molecular evolution. Proc. Natl Acad. Sci. USA 106, 8986–8991 (2009).

  14. 14.

    Goldstein, R. A., Pollard, S. T., Shah, S. D. & Pollock, D. D. Nonadaptive amino acid convergence rates decrease over time. Mol. Biol. Evol. 32, 1373–1381 (2015).

  15. 15.

    Kondrashov, A. S., Sunyaev, S. & Kondrashov, F. A. Dobzhansky–Muller incompatibilities in protein evolution. Proc. Natl Acad. Sci. USA 99, 14878–14883 (2002).

  16. 16.

    Halpern, A. L. & Bruno, W. J. Evolutionary distances for protein-coding sequences: modeling site-specific residue frequencies. Mol. Biol. Evol. 15, 910–917 (1998).

  17. 17.

    Lartillot, N. & Philippe, H. A Bayesian mixture model for across-site heterogeneities in the amino-acid replacement process. Mol. Biol. Evol. 21, 1095–1109 (2004).

  18. 18.

    Tamuri, A. U., dos Reis, M. & Goldstein, R. A. Estimating the distribution of selection coefficients from phylogenetic data using sitewise mutation-selection models. Genetics 190, 1101–1115 (2012).

  19. 19.

    Tamuri, A. U., Goldman, N. & dos Reis, M. A penalized-likelihood method to estimate the distribution of selection coefficients from phylogenetic data. Genetics 197, 257–271 (2014).

  20. 20.

    Rodrigue, N. On the statistical interpretation of site-specific variables in phylogeny-based substitution models. Genetics 193, 557–564 (2013).

  21. 21.

    Spielman, S. J. & Wilke, C. O. Extensively parameterized mutation-selection models reliably capture site-specific selective constraint. Mol. Biol. Evol. 33, 2990–3002 (2016).

  22. 22.

    Goldstein, R. A. & Pollock, D. D. The tangled bank of amino acids. Protein. Sci. 25, 1354–1362 (2016).

  23. 23.

    Kimura, M. The role of compensatory neutral mutations in molecular evolution. J. Genet. 64, 7 (1985).

  24. 24.

    Goldstein, R. A. The evolution and evolutionary consequences of marginal thermostability in proteins. Proteins 79, 1396–1407 (2011).

  25. 25.

    Williams, P. D., Pollock, D. D., Blackburne, B. P. & Goldstein, R. A. Assessing the accuracy of ancestral protein reconstruction methods. PLoS Comput. Biol. 2, e69 (2006).

  26. 26.

    Privalov, P. L. Stability of proteins: small globular proteins. Adv. Protein. Chem. 33, 167–241 (1979).

  27. 27.

    Privalov, P. L. & Gill, S. J. Stability of protein-structure and hydrophoboc interaction. Adv. Protein. Chem. 39, 191–234 (1988).

  28. 28.

    Taverna, D. M. & Goldstein, R. A. Why are proteins marginally stable? Proteins 46, 105–109 (2002).

  29. 29.

    Zeldovich, K. B. & Shakhnovich, E. I. Understanding protein evolution: from protein physics to Darwinian selection. Annu. Rev. Phys. Chem. 59, 105–127 (2008).

  30. 30.

    Iwasa, Y. Free fitness that always increases in evolution. J. Theor. Biol. 135, 265–281 (1988).

  31. 31.

    Sella, G. & Hirsh, A. E. The application of statistical physics to evolutionary biology. Proc. Natl Acad. Sci. USA 102, 9541–9546 (2005).

  32. 32.

    Shenkin, P. S., Erman, B. & Mastrandrea, L. D. Information-theoretical entropy as a measure of sequence variability. Proteins 11, 297–313 (1991).

  33. 33.

    Crow, J. F. & Kimura, M. An Introduction to Population Genetics Theory (Harper & Row, New York, 1970).

  34. 34.

    Kimura, M. Some problems of stochastic processes in genetics. Ann. Math. Stat 28, 882–901 (1957).

  35. 35.

    Kimura, M. On the probability of fixation of mutant genes in a population. Genetics 47, 713–719 (1962).

  36. 36.

    Goldstein, R. A. Population size dependence of fitness effect distribution and substitution rate probed by biophysical model of protein thermostability. Genome Biol. Evol. 5, 1584–1593 (2013).

  37. 37.

    Cherry, J. L. Should we expect substitution rate to depend on population size? Genetics 150, 911–919 (1998).

  38. 38.

    Eyring, H. The activated complex in chemical reactions. J. Chem. Phys. 3, 107–115 (1935).

  39. 39.

    Fisher, R. The Genetic Theory of Natural Selection (Oxford Univ. Press, Oxford, 1930).

  40. 40.

    Wylie, C. S. & Shakhnovich, E. I. A biophysical protein folding model accounts for most mutational fitness effects in viruses. Proc. Natl Acad. Sci. USA 108, 9916–9921 (2011).

  41. 41.

    Izaguirre, J. A. et al CompuCell, a multi-model framework for simulation of morphogenesis. Bioinformatics 20, 1129–1137 (2004).

  42. 42.

    Miyazawa, S. & Jernigan, R. L. Estimation of effective interresidue contact energies from protein crystal structures: quasi-chemical approximation. Macromolecules 18, 534–552 (1985).

  43. 43.

    Lindqvist, Y., Johansson, E., Kaija, H., Vihko, P. & Schneider, G. Three-dimensional structure of a mammalian purple acid phosphatase at 2.2 Å resolution with a mu-(hydr)oxo bridged di-iron center. J. Mol. Biol. 291, 135–147 (1999).

  44. 44.

    Gillespie, D. T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977).

  45. 45.

    Kimura, M. A simple method for estimating evolutionary rate of base substitution through comparative studies of nucleotide sequences. J. Mol. Evol. 16, 111–120 (1980).

  46. 46.

    Forgy, E. Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometrics 21, 768–780 (1965).

  47. 47.

    Lloyd, S. Least squares quantization in PCM. IEEE Trans. Inf. Theory 28, 129–137 (1982).

  48. 48.

    Khatri, B. S. & Goldstein, R. A. A coarse-grained biophysical model of sequence evolution and the population size dependence of the speciation rate. J. Theor. Biol. 378, 56–64 (2015).

  49. 49.

    Khatri, B. S., McLeish, T. C. & Sear, R. P. Statistical mechanics of convergent evolution in spatial patterning. Proc. Natl Acad. Sci. USA 106, 9564–9569 (2009).

Download references


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




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.

Corresponding author

Correspondence to David D. Pollock.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Supplementary Information

Supplementary figures

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Goldstein, R.A., Pollock, D.D. Sequence entropy of folding and the absolute rate of amino acid substitutions. Nat Ecol Evol 1, 1923–1930 (2017).

Download citation

Further reading