Generalized entropy in molecular biology and non-linear growth model of SARS-CoV-2 variant succession

Entropy measures are widely used in molecular biology to, among other, describe sequence variability, identify genetic polymorphisms and evolutionarily conserved sites, analyze and predict binding sites, distinguish coding from non-coding regions, and predict RNA and protein secondary structure. Although Shannon entropy is most frequently used, the generalized entropies, such for example, Tsallis entropy, offer alternative descriptions for systems with strong correlations or heavy tails. The analysis of time evolution of the cumulative tails derived from site-frequency spectra (SFS) computed from SARS-CoV-2 genomic data leads us to the discovery of a characteristic pattern, which includes a discontinuity, that can be traced down to a number of mutations, preserving identity over a time interval. In addition, the discontinuity moves, with time, towards higher frequencies, eventually dominating the whole population. We present a nonlinear two-phase growth model whose analytical solution takes the form of a q-exponential function—analogous to the functions that arise from Tsallis-entropy maximization—and show that it accurately reproduces the shape of the SARS-CoV-2 cumulative tails.