Theory of measurement for site-specific evolutionary rates in amino-acid sequences

In the field of molecular evolution, we commonly calculate site-specific evolutionary rates from alignments of amino-acid sequences. For example, catalytic residues in enzymes and interface regions in protein complexes can be inferred from observed relative rates. While numerous approaches exist to calculate amino-acid rates, it is not entirely clear what physical quantities the inferred rates represent and how these rates relate to the underlying fitness landscape of the evolving proteins. Further, amino-acid rates can be calculated in the context of different amino-acid exchangeability matrices, such as JTT, LG, or WAG, and again it is not well understood how the choice of the matrix influences the physical inter-pretation of the inferred rates. Here, we develop a theory of measurement for site-specific evolutionary rates, by analytically solving the maximum-likelihood equations for rate inference performed on sequences evolved under a mutation-selection model. We demonstrate that for realistic analysis settings the measurement process will recover the true expected rates of the mutation-selection model if rates are measured relative to a naive exchangeability matrix, in which all exchangeabilities are equal to 1/19. We also show that rate measurements using other matrices are quantitatively close but in general not mathematically equivalent. Our results demonstrate that insights obtained from phylogenetic-tree inference do not necessarily apply to rate inference, and best practices for the former may be deleterious for the latter.\n\nSignificance StatementMaximum likelihood inference is widely used to infer model parameters from sequence data in an evolutionary context. One major challenge in such inference procedures is the problem of having to identify the appropriate model used for inference. Model parameters usually are meaningful only to the extent that the model is appropriately specified and matches the process that generated the data. However, in practice, we dont know what process generated the data, and most models in actual use are misspecified. To circumvent this problem, we show here that we can employ maximum likelihood inference to make defined and meaningful measurements on arbitrary processes. Our approach uses misspecification as a deliberate strategy, and this strategy results in robust and meaningful parameter inference.