Figure 1.

Fisher’s geometric model with and without O×E and G×E effects: an example in two dimensions. The left panel shows the classic model, which excludes O×E and G×E effects. The wild-type genotype expresses phenotype at position A; a mutant homozygote expresses at position m; the optimal phenotype is at position o. Gray vectors (arrows, of length z and r) describe the displacements of the optimum (z) and the mutant homozygote (r) away from the wild-type phenotype. Mutant displacements that land within the gray circle (as shown in the example mutation) improve fitness; those landing outside of the circle reduce fitness. O×E and G×E effects (shown in the right panel) randomly shift the phenotypic values of the trait optimum and the mutant phenotype. Example O×E and G×E shifts are depicted by the dashed vectors, with respective magnitudes of z_E and r_E. The environment-specific locations of the optimum and mutant genotype (i.e., for the ith environment, as shown) are represented by points O_i and M_i. In the ith environment, the total displacements away from the wild-type phenotype are now represented by the bold, black vector that points toward O_i (with magnitude z_i). All extant genotypes will experience this same shift in the optimum. A new mutation would have had phenotype m, but now in the new environment its displacement from the initial wild-type phenotype is M_i (with magnitude r_i). Mutant displacements that land within the black circle are beneficial; those that land outside (as shown) are deleterious. Additional discussion is provided in the main text.