Gradualism, natural selection, and the randomness of mutation-fisher, Kimura, and Orr, connecting the dots

Evolutionary gradualism, the randomness of mutations, and the hypothesis that natural selection exerts a pervasive and substantial influence on evolutionary outcomes are pair-wise logically independent. Can the claims about selection and mutation be used to formulate an argument for gradualism? In his Genetical Theory of Natural Selection, R.A. Fisher made an important start at this project in his famous "geometric argument" by showing that a random mutation that has a smaller effect on two or more phenotypes will have a higher average fitness than a random mutation that has a larger phenotypic effect. Motoo Kimura's demonstration that a gene's probability of fixation depends on both the selection coefficient and the effective population size shows that Fisher's argument for gradualism was mistaken. Here we analyze Fisher's argument and explain how Kimura's theory leads to a conclusion that Fisher did not anticipate. We identify a fallacy that reasoning about fitness differences and their evolutionary consequences should avoid. We then distinguish forward-directed from backward-directed versions of gradualism. The backward-directed thesis about a single mutation may be correct, but the forward-directed thesis is not. After that we consider a sequence of random mutations that all affect the same cluster of phenotypes and Allen Orr's idea that there is an optimal sequence of mutation sizes, moving from larger to smaller as the population approaches a fixed optimum. This provides a likelihood justification for a forward-directed version of gradualism when there is a fixed optimum, but it also provides an argument against forward-directed gradualism if the optimum shifts substantially as the population evolves. Finally, we consider whether genome-wide association studies (GWASs) can furnish empirical evidence that bears on the truth of gradualism. The version of gradualism that GWASs directly bear on concerns the phenotypic effect size of a gene after it arises by mutation and has reached an appreciable frequency, not the phenotypic size of the mutation event itself.