Fisher’s Microscope and Haldane’s Ellipse

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Abstract
Fisher's geometrical model was introduced to study the phenotypic size of mutations contributing to adaptation. However, as pointed out by Haldane, the model involves a simplified picture of the action of natural selection, and this calls into question its generality. In particular, Fisher's model assumes that each trait contributes independently to fitness. Here, we show that Haldane's concerns may be incorporated into Fisher's model solely by allowing the intensity of selection to vary between traits. We further show that this generalization may be achieved by introducing a single, intuitively defined quantity that describes the phenotype prior to adaptation. Comparing the process of adaptation under the original and generalized models, we show that the generalization may bias results toward either larger or smaller mutations. The applicability of Fisher's model is then discussed.