INVESTIGATION Fisher’s Geometrical Model Emerges as a Property of Complex Integrated Phenotypic Networks

Models relating phenotype space to fitness (phenotype–fitness landscapes) have seen important developments recently. They can roughly be divided into mechanistic models (e.g., metabolic networks) and more heuristic models like Fisher’s geometrical model. Each has its own drawbacks, but both yield testable predictions on how the context (genomic background or environment) affects the distribution of mutation effects on fitness and thus adaptation. Both have received some empirical validation. This article aims at bridging the gap between these approaches. A derivation of the Fisher model “from first principles” is proposed, where the basic assumptions emerge from a more general model, inspired by mechanistic networks. I start from a general phenotypic network relating unspecified phenotypic traits and fitness. A limited set of qualitative assumptions is then imposed, mostly corresponding to known features of phenotypic networks: a large set of traits is pleiotropically affected by mutations and determines a much smaller set of traits under optimizing selection. Otherwise, the model remains fairly general regarding the phenotypic processes involved or the distribution of mutation effects affecting the network. A statistical treatment and a local approximation close to a fitness optimum yield a landscape that is effectively the isotropic Fisher model or its extension with a single dominant phenotypic direction. The fit of the resulting alternative distributions is illustrated in an empirical data set. These results bear implications on the validity of Fisher’s model’s assumptions and on which features of mutation fitness effects may vary (or not) across genomic or environmental contexts. THE distribution of the fitness effects (DFE) of random mutations is a central determinant of the evolutionary fate of a population, together with the rate of mutation. Obviously, it determines the rate of adaptation by de novo mutations, by setting the mutational input of fitness variance. Furthermore, by setting the distribution of fitness at mutation–selection balance, the DFE also determines the amount of standing variance in populations at equilibrium and their potential for future adaptation. The DFE is therefore central to evolutionary theory, for both adapting and equilibrium populations. There is, however, no widely accepted model that predicts the distribution of fitness effects of randommutations and how it is affected by various environmental or genetic contexts. Yet, predicting what happens under changed conditions is a minimum requirement for many applications of evolutionary theory. “Phenotype–fitness landscapes” provide a general tool for such inference: by defining changed conditions (genetic background or environment) as explicit alternative “positions” in the landscape, their effects can be handled quantitatively. “Mechanistic” landscapes One such approach has seen considerable development in the past decade: models that explicitly describe the “direct” molecular effect of a mutation (on RNA secondary structure, on metabolic reactions, etc.) and integrate its effect on cellular yield or growth rate, through a network of phenotypic interaction. This approach, which can take various forms, is often dubbed “systems biology” (reviewed in Papp et al. 2011). It relies on a phenotype–fitness landscape that is parameterized from some empirical knowledge of the system, to describe part of the complex functional effect of given mutations. Probably the most popular and most advanced example of this approach is flux balance analysis (FBA). FBA has proved accurate in predicting, from first principles, the fitness effect of a wide variety of gene deletions (alone or in combination) in several model microbial species, mostly the bacterium Escherichia coli (Ibarra et al. 2002) and the yeast Saccharomyces Copyright © 2014 by the Genetics Society of America doi: 10.1534/genetics.113.160325 Manuscript received December 1, 2013; accepted for publication January 30, 2014; published Early Online February 28, 2014. Supporting information is available online at http://www.genetics.org/lookup/suppl/ doi:10.1534/genetics.113.160325/-/DC1. Address for correspondence: Institut des Sciences de l’Evolution–Montpellier, ISEM CNRS UMR 5554, Université Montpellier II, Pl. Eugène Bataillon, Bât. 22, 34090 Montpellier, France. E-mail: [email protected] Genetics, Vol. 197, 237–255 May 2014 237 cerevisiae (Papp et al. 2004; Segré et al. 2005). It relies on a description of the effect of the removal of a given gene on the full metabolic network of the cell and ultimately on cell yield and growth rate. These approaches are powerful in both predictivity and explanatory potential, as they provide hints on why a particular genetic change has a given fitness effect. Other landscape models focus on point mutations affecting particular metabolic pathways [e.g., the lactose utilization pathway (Perfeito et al. 2011)]. These studies test whether given mechanistic models can be accurately fitted to observations. FBA, on the contrary, seeks to predict, from first principles and independent calibration data, the effect of a set of deletions. Finally, a mechanistic approach has recently been proposed at the scale of multicellular organisms, with a developmental model (based on tooth morphology) predicting how mutations affect morphology and subsequently fitness (Salazar-Ciudad and Jernvall 2010; Salazar-Ciudad and MarinRiera 2013). However, this model is intended as illustrative rather than quantitatively predictive, and it has not been empirically tested. All these mechanistic approaches come at a cost, almost by definition: they require more or less extensive empirical descriptions of the genotype–phenotype–fitness relationship, and they are bound to describe only the particular mechanism considered. Therefore, they are mostly applied in species/ strains where this relationship has been characterized empirically or can be “guessed” (a minimum requirement for FBA is a full genome sequence plus good knowledge of the growth medium). Mechanistic models are designed to describe a given aspect of a mutation’s effect (e.g., metabolic effect, secondary structure and RNA stability, etc.), typically at a cellular level. It is challenging to extend these predictions to mutations of unknown type (indels, gene duplications, transposon inserts, or single-nucleotide substitutions) that affect various functions and that modify an unknown aspect of the organism’s fitness (expression levels, behavior, etc.). The scale of the prediction also typically limits applications to multicellular organisms (where the model must be integrated over many differentiated cells) or viruses (where it is the host phenotype that must be modeled). Overall, the unprecedented refinement of these mechanistic models has clearly provided key information, some of which is used here. However, their very precision limits their ability to predict the effect of random mutations, in less wellcharacterized species and environments, and hence their potential application in medicine, agronomy, or ecology. “Heuristic” landscapes A different approach has also been used for decades to predict the DFE: more heuristic landscapes like Fisher’s (1930) geometrical model (FGM) (reviewed in Orr 2005). In this model, which may take various forms according to the starting assumptions, adaptation is characterized by stabilizing selection (quadratic or Gaussian), on a set of unspecified traits. Pleiotropic mutations jointly modify these traits, forming smooth (typically normal) distributions. The most predictive version is the isotropic FGM, where all traits are equivalent with respect to selection or mutation. In principle, the FGM can be used to predict how the DFE is affected by any environmental or genotypic context (epistasis), with any type of nonsilent mutation. Empirical support of the model’s predictions has recently accumulated (Martin and Lenormand 2006a,b; Martin et al. 2007; MacLean et al. 2010; Sousa et al. 2011; Weinreich and Knies 2013), although it was sometimes relatively indirect. The most quantitative tests (Martin et al. 2007; MacLean et al. 2010; Sousa et al. 2011), and hence those with most statistical power, used the model to predict how the DFE is affected by epistasis. More generally, the FGM indeed predicts both the pervasiveness and the diminishing-return form of epistasis, as documented repeatedly in experimental evolution (e.g., MacLean et al. 2010; Chou et al. 2011; Khan et al. 2011; Sousa et al. 2011). A model of pleiotropic mutations affecting the distance to an optimum is also qualitatively consistent with the prevalence of antagonistic pleiotropic effects affecting unused functions during long-term adaptation (as observed in Cooper and Lenski 2000). Finally, note that the model has been applied to various types of mutations (random point mutations, transposon inserts, and antibiotic resistance mutations) and in several species, although most were model microbial species, for logistic reasons. In spite of this potential, Fisher’s model is typically considered merely heuristic and too simplified to quantitatively capture the complex processes relating mutations to fitness components (growth rate, viability, fertility, etc.). Indeed, to date, the model has proved to be predictive only in a small number of tests. Compared to mechanistic models, it also does not predict the effect of particular mutations or their functional underpinnings, but only distributions among sets of mutants (but see Weinreich and Knies 2013). In any case, it remains unclear why such a simple model should capture features of highly complex processes. Therefore, even if further tests confirmed its quantitative predictivity, we would still be unable to tell under what conditions it should break down, which would limit its usefulness in forecasting.