Mutation load and mutation-selection-balance in quantitative genetic traits

Haldane (1937) showed that the reduction of equilibrium mean fitness in an infinite population due to recurrent deleterious mutations depends only on the mutation rate but not on the harmfulness of mutants. His analysis, as well as more recent ones (cf. Crow 1970), ignored back mutation. The purpose of the present paper is to extend these results to arbitrary mutation patterns among alleles and to quantitative genetic traits. We derive first-order approximations for the equilibrium mean fitness (and the mutation load) and determine the order of the error term. For a metric trait under mutation-stabilizing-selection balance our result differs qualitatively from that of Crow and Kimura (1964), whose analysis is based on a Gaussian assumption. Our general approach also yields a mathematical proof that the variance under the usual mutation-stabilizing-selection model is, to first order, micro/s (the house-of cards approximation) as micro/s tends to zero. This holds for arbitrary mutant distributions and does not require that the population mean coincide with the optimum. We show how the mutant distribution determines the order of the error term, and thus the accuracy of the house-of-cards approximation. Upper and lower bounds to the equilibrium variance are derived that deviate only to second order as micro/s tends to zero. The multilocus case is treated under the assumption of global linkage equilibrium.