The Utility of Fisher's Geometric Model in Evolutionary Genetics

Abstract

The accumulation of data on the genomic bases of adaptation has triggered renewed interest in theoretical models of adaptation. Among these models, Fisher Geometric Model (FGM) has received a lot of attention over the last two decades. FGM is based on a continuous multidimensional phenotypic landscape, but it is for the emerging properties of individual mutation effects that it is mostly used. Despite an apparent simplicity and a limited number of parameters, FGM integrates a full model of mutation and epistatic interactions that allows the study of both beneficial and deleterious mutations, and subsequently the fate of evolving populations. In this review, I present the different properties of FGM and the qualitative and quantitative support they have received from experimental evolution data. I later discuss how to estimate the different parameters of the model and outline some future directions to connect FGM and the molecular determinants of adaptation.

Keywords: Fisher’s Geometric model; adaptive landscape; distribution of fitness effects; drift load; epistasis; phenotypic complexity; pleiotropy; robustness.

Figure 1

A) FGM fitness landscape, with fitness on the vertical axes and below the projection on a 2D plot. Fitness decays as the phenotype moves away from the optimal phenotype. Mutation of phenotypic size r moves the phenotype on a hyper-sphere. The same mutation (arrow) may have different effects depending on the initial phenotype. B) Fraction of beneficial mutations as a function of the scaled effect of mutation $x=\frac{r\sqrt{n}}{2d}$. The points represent the situations illustrated in A, assuming a 100 dimensions. A large r (blue) is less likely to be beneficial than a small r from the same phenotype (purple) or a mutation of similar r affecting a less optimal phenotype (green). C) The distribution of mutation effects on fitness depends on the initial position of the phenotype. D) The distribution is presented for the points presented in C assuming n = 20, $\stackrel{‒}{s}=-0.01$ and initial log-fitness of −0.01 (purple), −0.1 (blue) and −0.5 (green). E) Ten adaptive walks moving log-fitness from −0.1 to −0.01 are presented. F) Different distribution of mutation effects produced from a phenotype of log-fitness −0.1 are presented (other parameters as in D): total mutations (Gray), beneficial mutations (Blue), beneficial mutations surviving drift (Purple), mutations fixed during adaptive walks starting from that phenotype (Black).

Figure 1

A) FGM fitness landscape, with fitness on the vertical axes and below the projection on a 2D plot. Fitness decays as the phenotype moves away from the optimal phenotype. Mutation of phenotypic size r moves the phenotype on a hyper-sphere. The same mutation (arrow) may have different effects depending on the initial phenotype. B) Fraction of beneficial mutations as a function of the scaled effect of mutation $x=\frac{r\sqrt{n}}{2d}$. The points represent the situations illustrated in A, assuming a 100 dimensions. A large r (blue) is less likely to be beneficial than a small r from the same phenotype (purple) or a mutation of similar r affecting a less optimal phenotype (green). C) The distribution of mutation effects on fitness depends on the initial position of the phenotype. D) The distribution is presented for the points presented in C assuming n = 20, $\stackrel{‒}{s}=-0.01$ and initial log-fitness of −0.01 (purple), −0.1 (blue) and −0.5 (green). E) Ten adaptive walks moving log-fitness from −0.1 to −0.01 are presented. F) Different distribution of mutation effects produced from a phenotype of log-fitness −0.1 are presented (other parameters as in D): total mutations (Gray), beneficial mutations (Blue), beneficial mutations surviving drift (Purple), mutations fixed during adaptive walks starting from that phenotype (Black).

Figure 1

A) FGM fitness landscape, with fitness on the vertical axes and below the projection on a 2D plot. Fitness decays as the phenotype moves away from the optimal phenotype. Mutation of phenotypic size r moves the phenotype on a hyper-sphere. The same mutation (arrow) may have different effects depending on the initial phenotype. B) Fraction of beneficial mutations as a function of the scaled effect of mutation $x=\frac{r\sqrt{n}}{2d}$. The points represent the situations illustrated in A, assuming a 100 dimensions. A large r (blue) is less likely to be beneficial than a small r from the same phenotype (purple) or a mutation of similar r affecting a less optimal phenotype (green). C) The distribution of mutation effects on fitness depends on the initial position of the phenotype. D) The distribution is presented for the points presented in C assuming n = 20, $\stackrel{‒}{s}=-0.01$ and initial log-fitness of −0.01 (purple), −0.1 (blue) and −0.5 (green). E) Ten adaptive walks moving log-fitness from −0.1 to −0.01 are presented. F) Different distribution of mutation effects produced from a phenotype of log-fitness −0.1 are presented (other parameters as in D): total mutations (Gray), beneficial mutations (Blue), beneficial mutations surviving drift (Purple), mutations fixed during adaptive walks starting from that phenotype (Black).

Figure 2

Populations evolve toward some mutation-selection-drift-equilibrium depending on their population size. Here the adaptive walks are presented starting from a non-optimal phenotypes, one with a population size of 5 (Blue) and another one of 100 (Red).