Mutation and selection induce correlations between selection coefficients and mutation rates

The joint distribution of selection coefficients and mutation rates is a key determinant of the genetic architecture of molecular adaptation. Three different distributions are of immediate interest: (1) the nominal distribution of possible changes, prior to mutation or selection, (2) the de novo distribution of realized mutations, and (3) the fixed distribution of selectively established mutations. Here, we formally characterize the relationships between these joint distributions under the strong selection, weak mutation (SSWM) regime. The de novo distribution is enriched relative to the nominal distribution for the highest rate mutations, and the fixed distribution is further enriched for the most highly beneficial mutations. Whereas mutation rates and selection coefficients are often assumed to be uncorrelated, we show that even with no correlation in the nominal distribution, the resulting de novo and fixed distributions can have correlations with any combination of signs. Nonetheless, we suggest that natural systems with a finite number of beneficial mutations will frequently have the kind of nominal distribution that induces negative correlations in the fixed distribution. We apply our mathematical framework, along with population simulations, to explore joint distributions of selection coefficients and mutation rates from deep mutational scanning and cancer informatics. Finally, we consider the evolutionary implications of these joint distributions together with two additional joint distributions relevant to parallelism and the rate of adaptation. ### Competing Interest Statement The authors have declared no competing interest.