The mutation matrix and the evolution of evolvability

Evolvability is a key characteristic of any evolving system, and the concept of evolvability serves as a unifying theme in a wide range of disciplines related to evolutionary theory. The field of quantitative genetics provides a framework for the exploration of evolvability with the promise to produce insights of global importance. With respect to the quantitative genetics of biological systems, the parameters most relevant to evolvability are the G-matrix, which describes the standing additive genetic variances and covariances for a suite of traits, and the M-matrix, which describes the effects of new mutations on genetic variances and covariances. A population's immediate response to selection is governed by the G-matrix. However, evolvability is also concerned with the ability of mutational processes to produce adaptive variants, and consequently the M-matrix is a crucial quantitative genetic parameter. Here, we explore the evolution of evolvability by using analytical theory and simulation-based models to examine the evolution of the mutational correlation, r(mu,), the key parameter determining the nature of genetic constraints imposed by M. The model uses a diploid, sexually reproducing population of finite size experiencing stabilizing selection on a two-trait phenotype. We assume that the mutational correlation is a third quantitative trait determined by multiple additive loci. An individual's value of the mutational correlation trait determines the correlation between pleiotropic effects of new alleles when they arise in that individual. Our results show that the mutational correlation, despite the fact that it is not involved directly in the specification of an individual's fitness, does evolve in response to selection on the bivariate phenotype. The mutational variance exhibits a weak tendency to evolve to produce alignment of the M-matrix with the adaptive landscape, but is prone to erratic fluctuations as a consequence of genetic drift. The interpretation of this result is that the evolvability of the population is capable of a response to selection, and whether this response results in an increase or decrease in evolvability depends on the way in which the bivariate phenotypic optimum is expected to move. Interestingly, both analytical and simulation results show that the mutational correlation experiences disruptive selection, with local fitness maxima at -1 and +1. Genetic drift counteracts the tendency for the mutational correlation to persist at these extreme values, however. Our results also show that an evolving M-matrix tends to increase stability of the G-matrix under most circumstances. Previous studies of G-matrix stability, which assume nonevolving M-matrices, consequently may overestimate the level of instability of G relative to what might be expected in natural systems. Overall, our results indicate that evolvability can evolve in natural systems in a way that tends to result in alignment of the G-matrix, the M-matrix, and the adaptive landscape, and that such evolution tends to stabilize the G-matrix over evolutionary time.