Modelling reaction norms with few assumptions

Reaction norms describe the phenotypic expression of a genotype over a range of environments. In evolutionary theory, these are typically modelled using very specific shapes, such as straight lines or logistic (S-shaped) curves. This may lead to wrong conclusions when the assumptions being made are too restrictive. Here we present a novel method to model smooth non-linear reaction norms with very few a priori assumptions about shape, using restricted cubic splines – a type of function defined piecewise by polynomials. We present an example of an optimal non-linear reaction norm obtained analytically, where a hypothetical single celled organism is able to vary its metabolic investment into processing different nutrients which themselves vary in digestibility. To simulate this example reaction norm, we use a restricted cubic spline based on k independent gene values. Each simulation consisted of a starting population of 1,000 individuals with a horizontal reaction norm defined by 5, 10 or 20genes, which were then allowed to evolve freely for 300,000 generations. All populations achieved a shape very similar to the optimal reaction norm, independently of the number of genes used.Moreover, a lower number of genes does not lead to faster adaptation. Here we show that modelling smooth non-linear reaction norms is possible without constricting their shape.