Alpha helices are more evolutionarily robust to environmental perturbations than beta sheets: Bayesian learning and statistical mechanics for protein evolution

How typical elements that shape organisms, such as protein secondary structures, have evolved, or how evolutionarily susceptible/resistant they are to environmental changes, are significant issues in evolutionary biology, structural biology, and biophysics. According to Darwinian evolution, natural selection and genetic mutations are the primary drivers of biological evolution. However, the concept of ``robustness of the phenotype to environmental perturbations across successive generations," which seems crucial from the perspective of natural selection, has not been formalized or analyzed. In this study, through Bayesian learning and statistical mechanics we formalize the stability of the free energy in the space of amino acid sequences that can design particular protein structure against perturbations of the chemical potential of water surrounding a protein as such robustness. This evolutionary stability is defined as a decreasing function of a quantity analogous to the susceptibility in the statistical mechanics of magnetic bodies specific to the amino acid sequence of a protein. Consequently, in a two-dimensional square lattice protein model composed of 36 residues, we found that as we increase the stability of the free energy against perturbations in environmental conditions, the structural space shows a steep step-like reduction. Furthermore, lattice protein structures with higher stability against perturbations in environmental conditions tend to have a higher proportion of $α$-helices and a lower proportion of $β$-sheets. This result is qualitatively confirmed by comparing the histograms of the percentage of secondary structures of evolutionarily robust proteins and randomly selected proteins through an empirical validation using a protein database.