Consequences of stabilising selection for polygenic variation

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Abstract
A number of earlier authors have investigated the consequences for fitness of gene differences affecting the expression of a primary character where stabilising selection is acting in favour of an optimal expression. Assuming that the mean expression must be at or close to the optimum, deviations from the mean have been taken as measures of the deviation from the optimum, and the conclusion reached that, random drift apart, stabilising selection must ultimately lead to fixation of the commoner allele. It is now shown that this approach is incorrect: in an illustrative example the mean cannot be taken as synonymous with the optimum except in the trivial case where they have precisely the same value. Where the mean departs from the optimum, even by sampling variation only, continuing selection for it is effectively self-propagating and directional away from the optimum. Deviation from the optimum itself must be used in investigating the consequences of stabilising selection. The model used here is based on the biometrical description of the effects of a gene difference on a quantitative character as used by Mather and Jinks (1982). This includes a parameter, h, which allows dominance of any magnitude in either direction to be taken into account, and is adapted for the present purpose by introducing an additional parameter, m, measuring the departure from the optimum of the mid-point, or mid-parent, which is used biometrically as the origin for measurement of the gene effects. Assuming random mating and the absence of epistasis in the effects on the primary character, it is shown that stabilising selection acting on a pair of alleles (A and a) can have any one of three possible outcomes depending on the relative values of the m and h which characterise the effects of the gene difference on the primary character, viz: (i) a stable equilibrium in the population where in respect of the primary character Aa is nearer to the optimum than both homozygotes; (ii) fixation of the fitter allele where Aa is intermediate between the homozygotes in its departure from the optimum; (iii) a theoretical unstable equilibrium leading to fixation of the commoner allele, where Aa departs further from the optimum than both homozygotes. Only outcome (i) can lead to the conservation of variation in the population. Preliminary consideration is also given to the interaction in effect on fitness of two gene pairs affecting the same character and segregating simultaneously in the population. Published data from two experiments on the properties of the conserved variation for two chaeta characters in the Texas population of Drosophila melanogaster, as revealed by half-diallel sets of crosses among 11 and 16 inbred lines respectively, are shown to agree with theoretical expectations derived from the present analysis. Some consequences of differences of overall magnitude in the effects of the mutations by which the variation originated are discussed.