Fisher's geometrical model of evolutionary adaptation—Beyond spherical geometry

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Abstract

Fisher's geometrical model of evolutionary adaptation has recently been used in a variety of contexts of interest to evolutionary biologists. The renewed interest in this model strongly motivates generalizations that make it a more realistic description of evolutionary adaptation. Previously, the distribution of mutant effects has, for analytical tractability, rather than biological realism, been taken as spherically symmetric. Here we substantially extend Fisher's model, by allowing a wider class of mutational distributions that incorporate mutational bias and more general deviations from spherical symmetry such as correlations between mutant effects. We also incorporate work on generalized fitness landscapes, thereby reducing the number of artificial assumptions underlying the model. The generalized model exhibits a substantially increased flexibility and a far richer underlying geometry. We find that the distribution characterizing selection coefficients of new mutations is expressed in terms of a number of geometrical invariants associated with mutation, selection and the parental phenotype.

Introduction

In his famous book The Genetical Theory of Natural Selection, Fisher outlined a view of evolutionary adaptation in terms of intuitive, geometrical considerations (Fisher, 1930). An organism was described as having n quantitative traits (i.e. n characters with effectively continuous variation). Examples of nine such characters that have been investigated in Drosophila melanogaster are viability, fecundity, hatchability, development time, longevity, mating speed, phototaxis, body length and abdominal bristle number (Keightley and Ohnishi, 1998).

Fisher viewed the quantitative characters of an organism as the Cartesian coordinates in an n-dimensional “space of characters,” and a particular organism, with its particular set of n characters, was then geometrically represented as a point in this space. In the original formulation of Fisher, the level of adaptation of an organism was determined from its distance from a fixed point in the n-dimensional character space: the closer an organism is to this fixed point, the higher is its fitness. This fixed point was thus implicitly taken as a fitness optimum and since only the distance from this point is of significance, surfaces of constant fitness are hyperspheres surrounding the optimum, i.e. circles, if there are only n=2 characters (see Fig. 1), spheres when there are n=3 characters, . The intention of Fisher was not obviously to provide a realistic model of adaptation, but rather to illustrate how adaptation is determined by a number of different features of an organism acting in concert.

In Fisher's geometric description, the change in characters associated with a mutation corresponds to a mutant offspring lying at a different position (in the character space), compared with that of its parent (we are assuming an asexual population). Such a change is beneficial—or adaptive—if it results in an increase in some measure of the organism's viability/reproductive success, i.e. its fitness. A mutation is adaptive if an individual carrying a newly arisen mutation is closer to the location of the fitness optimum than that of its parent—see Fig. 1. The mutational changes considered by Fisher were taken to have the simplest distribution, namely that of being equally likely to occur in all directions in the character space (spherically symmetric).

Fisher's considerations amount to an explicit model of evolutionary adaptation, with analytical or quantitative results derivable for results such as the proportion of beneficial mutations.

Quite recently, there has been renewed interest in this model because, despite being highly simplistic, there is the implicit belief that certain features it exhibits may be robust to modifications of the underlying assumptions and hence allow its conclusions to have wider applicability. The recent work, which uses Fisher's model in its original form, includes investigation of the size of mutations contributing to adaptation (Orr, 1998, Orr, 1999, Hartl and Taubes, 1998, Burch and Chao, 1999), topics such as drift load (Hartl and Taubes, 1996; Peck et al., 1997; Poon and Otto, 2000), hybridization (Barton, 2001) and evolutionary rates (Orr, 2000; Welch and Waxman, 2003). Generalizations of Fisher's model have also been considered (Rice, 1990, Whitlock et al., 2003, Waxman and Welch, 2005).

The renewed interest in this model strongly motivates generalizations that make it a more realistic description of evolutionary adaptation. Here we make some progress in this direction, by not only incorporating recent work on generalized fitness functions of a stabilizing form (Waxman and Welch, 2005) but, more importantly, by incorporating a wider class of distributions of mutational effects, beyond the spherically symmetric ones that have been considered to date. Thus, with the ultimate aim of setting out a somewhat more general framework for Fisher's geometrical model, we consider distributions of mutant effect that incorporate mutational bias and allow correlations between the mutational changes on different traits. In the framework presented, the distribution of mutant effects has surfaces of constant probability density that are ellipses or their higher dimensional analogues (ellipsoids) and the distribution has a functional form that includes a normal distribution as a special case. The present work therefore reduces some of the artificial assumptions about mutation that have been present in Fisher's geometrical model to date, and provides a useful tool for subsequent work employing the model.

The generalized model, outlined above, exhibits a substantially increased flexibility and a far richer underlying geometry. The present work concentrates on a fundamental quantity; a distribution characterizing new mutations and exposes the way the richer geometry manifests itself in quantities of interest associated with such mutations.

Section snippets

Model

Consider a population of asexual organisms that are subject to selection and mutation on the values of n quantitative characters, z1,z2,,zn, which make up the relevant phenotype of an individual. Each of the different characters continuously ranges from - to and we neglect any environmental component of the characters. It is convenient to collect all n characters into the column vector z=(z1,z2,,zn)T where the superscript T denotes the transpose of a matrix.

Results/methods

In the present work we determine a distribution characterizing the selection coefficients of new mutations. This distribution is derived for mutations characterized by Eq. (1). Any notions about the size of mutations that contribute to the distribution characterizing selection coefficients (however size is defined) need not be addressed since all mutations can make a contribution, irrespective of any of their attributes.

To proceed, we note that a random mutational change r of parental phenotype

Discussion

In this work we have presented further theoretical developments of Fisher's geometrical model of evolutionary adaptation. In particular, we have extended Fisher's model to incorporate a distribution of mutant effects that includes (i) correlations between mutational effects on different traits, (ii) mutational biases on different traits and (iii) a class of distributions of mutant effects that have the property that surfaces of constant probability density are ellipsoids. This includes a

Acknowledgements

It is a pleasure to thank John Welch for stimulating discussions and an anonymous referee for helpful suggestions that have improved this paper. I thank Guillaume Martin and Thomas Lenormand for a preprint of their recent paper. This work was supported by the Leverhulme Trust.

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