Genetic regulatory network motifs constrain adaptation through curvature in the landscape of mutational (co)variance.

Systems biology is accumulating a wealth of understanding about the structure of genetic regulatory networks, leading to a more complete picture of the complex genotype-phenotype relationship. However, models of multivariate phenotypic evolution based on quantitative genetics have largely not incorporated a network-based view of genetic variation. Here we model a set of two-node, two-phenotype genetic network motifs, covering a full range of regulatory interactions. We find that network interactions result in different patterns of mutational (co)variance at the phenotypic level (the M-matrix), not only across network motifs but also across phenotypic space within single motifs. This effect is due almost entirely to mutational input of additive genetic (co)variance. Variation in M has the effect of stretching and bending phenotypic space with respect to evolvability, analogous to the curvature of space-time under general relativity, and similar mathematical tools may apply in each case. We explored the consequences of curvature in mutational variation by simulating adaptation under divergent selection with gene flow. Both standing genetic variation (the G-matrix) and rate of adaptation are constrained by M, so that G and adaptive trajectories are curved across phenotypic space. Under weak selection the phenotypic mean at migration-selection balance also depends on M.