A G matrix analogue to capture the cumulative effects of nongenetic inheritance

The genetic variance-covariance (G) matrix describes the variances and covariances of genetic traits under strict genetic inheritance. Genetically expressed traits often influence trait expression in another via nongenetic forms of transmission and inheritance, however. The importance of non-genetic influences on phenotypic evolution is increasingly clear, but how genetic and nongenetic inheritance interact to determine the response to selection is not well understood. Here, we use the 'reachability matrix' - a key analytical tool of geometric control theory - to integrate both forms of inheritance, capturing how the consequences of generation-lagged maternal effects accumulate. Building on the classic Lande and Kirkpatrick model that showed how nongenetic (maternal) inheritance fundamentally alters the expected path of phenotypic evolution, we make novel inferences through decomposition of the reachability matrix. In particular, we quantify how nongenetic inheritance affects the distribution (orientation and shape) of ellipses of phenotypic change and how these distributions influence subsequent evolution. This interweaving of phenotypic means and variances accumulates generation by generation and is described analytically by the reachability matrix, which acts as an analogue of G when genetic and nongenetic inheritance both act.