The relationship between dN/dS and scaled selection coefficients

Numerous computational methods exist to assess the mode and strength of natural selection in protein-coding sequences, yet how distinct methods relate to one another remains largely unknown. Here, we elucidate the relationship between two widely-used phylogenetic modeling frameworks: dN/dS models and mutation-selection (MutSel) models. We derive a mathematical relationship between dN/dS and scaled selection coefficients, the focal parameters of MutSel models, and use this relationship to gain deeper insight into the behaviors, limitations, and applicabilities of these two modeling frameworks. We prove that, if all synonymous changes are neutral, standard MutSel models correspond to dN/dS < 1. However, if synonymous codons differ in fitness, dN/dS can take on arbitrarily high values even if all selection is purifying. Thus, the MutSel modeling framework cannot necessarily accommodate positive, diversifying selection, while dN/dS cannot distinguish between purifying selection on synony...