Allen Orr and the genetics of adaptation

Abstract

Over most of the 20th century, evolutionary biologists predominantly subscribed to a strong form of “micro-mutationism,” in which adaptive phenotypic divergence arises from allele frequency changes at many loci, each with a small effect on the phenotype. To be sure, there were well-known examples of large-effect alleles contributing to adaptation, yet such cases were generally regarded as atypical and unrepresentative of evolutionary change in general. In 1998, Allen Orr published a landmark theoretical paper in Evolution, which showed that both small- and large-effect mutations are likely to contribute to “adaptive walks” of a population to an optimum. Coupled with a growing set of empirical examples of large-effect alleles contributing to divergence (e.g., from QTL studies), Orr's paper provided a mathematical formalism that converted many evolutionary biologists from micro-mutationism to a more pluralistic perspective on the genetic basis of evolutionary change. We revisit the theoretical insights emerging from Orr's paper within the historical context leading up to 1998, and track the influence of this paper on the field of evolutionary biology through an examination of its citations over the last two decades and an analysis of the extensive body of theoretical and empirical research that Orr's pioneering paper inspired.

Adaptation, morphological evolution, mutations, pleiotropy, population genetics, selection – natural

When many of us were first introduced to evolutionary biology, we were confronted with three cautionary tales of faulty evolutionary thinking from the history of the field. We were told of Lamarck, whose evolutionary model of acquired characteristics and progressive drive toward perfection stands in stark contrast to the modern Darwinian theory of evolution as a blind interaction between random, heritable variation, and natural selection (Huxley 1942). We learned of Wynne-Edwards, the group-selection proponent, whose views of adaptation for the good of the species were loudly rebuked in favor of individual-level selection (Maynard Smith 1964; Williams 1966), as a far more powerful mechanism for generating adaptation. Finally, we were taught the folly of discredited macro-mutational theories of adaptation by early Mendelians, of Goldschmidt's “hopeful monsters,” and arguments by punctuated equilibrium enthusiasts that stasis and change in the fossil record are somehow incompatible with the neo-Darwinian view that evolution is gradual and attributable to allele frequency changes at many genes with small individual effects (Charlesworth et al. 1982).

The first two cautionary tales, with minor caveats, are justified given our current state of knowledge about how evolution works. Less obvious—at least at the close of the 20th century—was that the third tale rested on much weaker theoretical and empirical grounds than we were originally led to believe. Although extreme hypotheses of saltationism, in which complex adaptations emerge from single mutations, can justifiably be dismissed, there is surely a wide space of plausibility between the infinitesimal or “micro-mutationist” models of evolution, on the one hand, and hopeful monsters on the other. By the mid-20th century, ecological genetics had already yielded plenty of examples of simple Mendelian traits contributing to rapid adaptation in natural populations, although one could always claim that these specific cases were unrepresentative of adaptation in general, either because humans were agents of unusually strong selection (e.g., cases of industrial melanism) or because these cases involved “special” traits with unrepresentative genetic bases. Yet by the close of the century, data from (at the time) new QTL mapping studies showed that conspicuous patterns of divergence between populations or species could potentially be explained by a surprisingly small number of genes (e.g., Falconer and Mackay 1996, Ch. 21), once again raising the specter of a “weak version” of macro-mutationism (see Orr and Coyne 1992). Despite these new data, the field lacked a clear theoretical argument for when or why we might expect to observe alleles of large effect contributing to evolutionary change.

In 1998, Allen Orr published an extremely influential theoretical paper that satisfied this need for a new theory. The paper, motivated by the emerging QTL results, provided a compelling formal argument for why adaptation might often involve mutations with large phenotypic effects. But the influence of Orr's paper went much further than that. It helped convince a wide spectrum of evolutionary researchers that questions about the genetics of adaptation were broadly important yet surprisingly poorly resolved. Indeed, such questions are fundamental, as they cut to the very core of what evolution is about: the interaction between genetic variation that is random with respect to the needs of the organism, and the ability of selection to drive change that ultimately meets those needs. Orr's paper was remarkably accessible to both empirically and theoretically oriented researchers, inspiring a significant expansion of population genetic modeling of adaptation, and a slew of empirical studies that were directly motivated by Orr's predictions. In short, Orr's paper changed the way many in the field viewed genetic aspects of adaptation, to the extent that modern evolutionary biologists take a far more pluralistic view of the genetics of adaptation than they did in the 20th Century.

In honor of Orr's exceptional study and the 75th anniversary of the Society for the Study of Evolution (SSE), we explore the influence of Orr's (1998) paper on contemporary evolutionary biology research. We begin by revisiting the historical context of theoretical and empirical research in which Orr's paper was published. We then outline the mathematical framework and key results emerging from Orr's analysis. Finally, we analyze the influence of Orr's paper over the past two decades, as reflected in citation patterns up to the end of 2020, and the experimental and theoretical research that emerged in response to Orr's results.

Limitations of Classical Evolutionary Models of Adaptation

A sufficient theory for the genetics of adaptation must take into account the evolutionary dynamics of phenotypes expressed by members of a population as well as the frequencies and phenotypic effects of genotypes underlying the traits contributing to adaptation. Such theory should, ideally, render predictions that are empirically testable, including aspects of the genetic basis of evolutionary divergence between populations or species that are potentially measurable using experimental evolution or QTL mapping approaches. Nevertheless, classical evolutionary theory is composed of two dominant modeling traditions—an evolutionary quantitative genetics tradition focused almost exclusively on phenotypes, and a population genetics tradition in which phenotypes are largely ignored—that are poorly suited for addressing questions about the genetics of adaptation (Lewontin 1974; Hartl and Taubes 1998; Orr 1998, 2005b).

The quantitative genetics tradition focuses on changes in phenotypes and ignores their genetic basis beyond assuming that evolution involves allele frequency changes at very many small-effect loci throughout the genome. In the model's most idealized form, the heritable component of variation for each trait is “infinitesimal” (a limit in which the number of loci approaches infinity and each contributes infinitesimally to the trait variance; Bulmer 1980; Hill 2014). Genetic details pertinent to evolution are subsumed into a single parameter, the additive genetic variance for the trait, which to a first approximation remains constant over time, leading to elegantly simple predictions about the evolutionary dynamics of phenotypic adaptation (Gomulkiewicz and Holt 1995; Lande and Shannon 1996; Chevin 2013). The model cannot, however, tell us anything about the genetic basis of adaptation since that is assumed a priori.

Population genetics models generally focus on alleles and genotypes within a population, yet are curiously disconnected from the phenotypes targeted by natural selection or the phenotypic states that we might recognize as adaptive (Hartl and Taubes 1998; Orr 2005b). In most population genetic models, the only phenotype specified is fitness. And although the evolutionary dynamical properties of allele frequencies and quantitative traits do not fundamentally differ (e.g., within-generation changes of quantitative traits and allele frequencies are each a special case of the Robertson-Price identity; Walsh and Lynch 2018, ch. 6), there is obviously something missing when the predictions of population genetic models are solely framed in terms of fitness or fitness variance. Questions about the genetics of adaptation clearly require some amalgamation of the genotype- and trait-centric modeling traditions. Various hybrid models that straddle population and quantitative genetics have popped up over the last century (e.g., Wright 1935; Robertson 1960; Hill 1982; Turelli and Barton 2004), yet these are exceptions to the vast majority of theoretical work that falls entirely within one conceptual silo or the other.

Fisher's Geometric Model and the Genetics of Adaptive Variation

The limited capacity of classical evolutionary theory to address questions about the genetic basis of adaptive divergence is not a big problem if the essence of the infinitesimal perspective on evolutionary change is largely correct, at least as an approximation of reality. That is, if we can safely assume that mutations of small effect are all that matters in adaptation, and that most evolutionary change is highly polygenic (caused by cumulatively many genes with individually small effects), then there is no pressing need to go much further.

The infinitesimal perspective is essentially a reworking of Darwin's theory of gradualism (Darwin 1859; Huxley 1942) and the origin of the idea is, therefore, pre-Mendelian (Provine 1971; Orr and Coyne 1992; Orr 2005a). Mathematical justification for this perspective came much later, and is widely credited to R. A. Fisher, whose 1918 paper provided the theoretical foundation for the infinitesimal model: a description of trait variation and covariances between relatives that became the cornerstone of modern quantitative genetics and animal breeding (Fisher 1918; Bulmer 1980; Hill and Kirkpatrick 2010; Visscher and Goddard 2018). Fisher later offered a formal justification for why the infinitesimal view should apply to the genetic basis of adaptive divergence. In a short passage from The Genetical Theory of Natural Selection (Fisher 1930), Fisher introduced what is now known as “Fisher's geometric model” (Fig. 1). He imagined that adaptation required the evolutionary adjustment of many different features of an organism toward an optimal trait combination. Effects of mutations on a set of traits could be represented by a vector within multidimensional phenotypic space, with beneficial mutations altering multivariate trait expression to be closer to the optimal state. Fisher “showed” (i.e., presented a correct result with no derivation; Leigh 1987; Orr 2005a) that a random mutation had an approximately 50% chance of being favorable in the infinitesimal limit, whereas large-effect mutations were almost certain to be harmful because their tendency to carry some traits closer to their optima would typically be offset by negative pleiotropic effects on other traits (the concept of universal pleiotropy is implicit in Fisher's model). Fisher reasoned that because large-effect mutations are likely to be harmful, they must therefore be unimportant for adaptation. Although the belief that small-effect mutations are what matters in evolution was already widely held before 1930 (including by Fisher; Fisher 1922; Turner 1985), the geometric model justified the belief, which proceeded to dominate the thinking of 20th Century evolutionary biologists (although some remained circumspect; e.g., Maynard Smith 1983; Turner 1985).

Figure 1

Fisher's geometric model. The left-hand panel shows a two-dimensional example (n = 2 traits), where a population at position A in phenotypic space is displaced from the optimum at position O. The distance between A and O is d, and r is the magnitude of the effect of a random mutation that alters the phenotype to the point at the tip of the arrow. Mutations are randomly oriented in phenotypic space; for example, with two dimensions, mutations of size r will be uniformly distributed on the surface of the grey circle. The top right-hand panel shows Fisher's high-dimensional approximation (Fisher 1930, pp. 38–41; valid for n > 10 traits), in which small-effect mutations (i.e., for the infinitesimal limit: r → 0) have a nearly 50% chance of being beneficial, whereas large-effect mutations are predominantly deleterious. The bottom right-hand panel shows Kimura's (1983) prediction for the probability distribution of fixed mutations (the dotted line denotes the mean, following eq. (8) of Orr 1998), which assumes that new mutations follow a uniform size distribution.

An important strength of Fisher's geometric model is that it explicitly maps the relation between genotype, phenotype, and fitness—a big improvement over traditional quantitative genetic and population genetic models, which limit their focus to genotypes or phenotypes rather than both. At first glance, it seems odd that Fisher's geometric model, which dates to around the same period of time that these classical modeling traditions were developed, failed to inspire further theoretical work for over 50 years, prior to its resurrection by Kimura in his 1983 treatise on the Neutral Theory (Kimura 1983, pp. 135–137, 154–156). Population and quantitative genetics theory flourished during this time span, yet it was only by the end of the 1990s that theory using Fisher's geometric model became common (early studies include Leigh 1987; Rice 1990; Hartl and Taubes 1996, 1998; Peck et al. 1997; Orr 1998).

Why did it take so long for evolutionary biologists to realize and take advantage of the utility of Fisher's model? One reason for the gap is that the limitations of classical evolutionary theory were not particularly obvious to many researchers who might have been inclined to extend Fisher's original model to study adaptation. It is only in retrospect that great ideas and insights seem obvious. A second possible reason for delay pertains to the complexity of Fisher's model. A genetically explicit theoretical study of partly stochastic evolutionary change involving multiple genetic loci and high-dimensional phenotypes is a daunting proposition, particularly in an era where computer simulations were far less prevalent than they are today.

As it turned out, the onset of the molecular evolution era, from the 1960s to the 1980s, spurred the development of new population genetics theory that would ultimately prove to be extremely useful for later analyses of adaptation in the geometric model framework. New “origin-fixation” models of molecular evolution focused on substitution rates of amino acid or DNA sequences, which were modeled as the product of the number of new mutations entering a population at each generation (e.g., 2Nu for a diploid gene in a population of size N, with u representing the gene's mutation rate) and the fixation probability of new mutations entering the population (see McCandlish and Stoltzfus 2014 for a superb overview of the history, assumptions, and applications of origin-fixation models). An important feature of this new theory is that evolution is mutation-limited, with individual substitutions (adaptive or otherwise) spaced out over time rather than temporally overlapping. Sequences of substitutions could then be modeled as a series of non-overlapping evolutionary steps, with each step modeled using the origin-fixation mathematical machinery. This simplifying assumption greatly facilitates mathematical analyses of long-term evolutionary dynamics.

Coincidentally or not, Fisher's model resurfaced shortly after the development and popularization of origin-fixation models of evolution (McCandlish and Stoltzfus 2014). By nesting Fisher's model of random mutation and selection within the origin-fixation framework that he and others developed, Kimura (1983, pp. 154–156) showed that mutations of intermediate size were the most likely to contribute to adaptation (Fig. 1). Kimura noted that, while small mutations were the most likely to be beneficial, such mutations were also the most likely to be lost by drift. Mutations of intermediate size balance the probability of being beneficial, which declines with mutation size, against fixation probabilities, which for beneficial mutations increase with mutation size.

Orr's Model for the Genetics of Adaptation

Orr's (1998) landmark paper is notable for two reasons. First, it presents an exceptionally lucid description of an important, unresolved problem in evolutionary theory, and one that was not well appreciated at the time. The issue was whether it was reasonable to assume that the genetic basis of adaptation was infinitesimal, and if not, could we make any predictions at all about what the genetic basis of adaptation is likely to be? Although Orr and Coyne (1992) raised this question several years earlier, Orr (1998) clarified the issue by presenting a technical yet accessible analysis of the theoretical basis of the prevailing argument that adaptation should have an infinitesimal genetic basis. As pointed out by Kimura (1983), Fisher's original analysis made predictions about beneficial mutations, yet it neglected evolution. Kimura addressed this oversight by modeling the first step of adaptation to an optimum (i.e., the first adaptive substitution to fix during an “adaptive walk”). But because Kimura did not consider substitutions beyond the first, questions about the dynamics and genetic basis of adaptive walks to an optimum were left unanswered.

Orr's paper is also notable because it pushed the analysis of Fisher's model much further than previous studies. It illustrated, by argument and example, the utility of Fisher's model for addressing questions about the genetics of adaptation, and generated several completely new and empirically relevant predictions about the genetics of adaptation. Orr's analysis focused on two related questions about the genetics of adaptation. The first involves the dynamics of a population's approach to its optimum during an adaptive walk, that is, the effect of a series of adaptive substitutions on the distance (or lag) between the population and its optimum. Orr's second and more influential prediction concerns the phenotypic effect sizes of mutations that contribute to adaptation. We elaborate below on both of these results after first outlining the most important assumptions of Orr's model.

ASSUMPTIONS OF ORR'S MODEL

Orr's model relies on a number of simplifying assumptions (as all useful models do) that are important to keep in mind. Adaptation involves n independent traits united through universal pleiotropy (i.e., each mutation affects the expression of all n traits). There is a single multivariate trait optimum (a point in the n-dimensional phenotypic space, denoted O in Fig. 1). The population is displaced from the optimum following an abrupt change in the environment, and the optimum subsequently remains constant as the population adapts toward it. Selection and mutation are “isotropic” (each trait is subject to equally strong stabilizing selection and there is no correlational selection between pairs of traits; each trait has the same marginal distribution of mutational effects and there are no mutational correlations between traits). The population is haploid, which ensures that each mutation will experience either positive or purifying selection and balancing selection is ruled out. The population is mutation-limited, which follows the “origin-fixation” framework that Kimura (1983) applied in his analysis of the first substitution to fix in Fisher's model. As a result, adaptation uses new mutations rather than standing genetic variation, and selective sweeps are staggered over time rather than temporally overlapping. Finally, only adaptive mutations are permitted to become fixed in the population (mildly deleterious substitutions are ignored). Each of these assumptions has since been relaxed by more recent theoretical models—a point that we return to further below.

EVOLUTIONARY APPROACH TO THE OPTIMUM

During adaptation toward a fixed optimum, quantitative genetics model with infinitesimal genetic architecture predict that the evolutionary lag between the population and its optimum will decline geometrically over time (Gomulkiewicz and Holt 1995; Chevin 2013). The temporal dynamics of adaptive walks in Orr's model differ in two notable ways from those of quantitative genetic models. First, the timescale of evolutionary change is in units of substitutions rather than generations, which reflects the assumption that evolution is mutation-limited (Orr 1998; McCandlish and Stoltzfus 2014). Second, whereas the evolutionary approach to the optimum is both predictable and smooth in quantitative genetic models (drift can, of course, be included in such models, but the theoretical machinery does not require it; Lande 1976), adaptation in Fisher's geometric model includes two sources of stochasticity that lead to an uneven evolutionary approach of an adapting population to its optimum. Unpredictability in the dynamics of adaptation is caused by randomness in the mutations that happen to arise within the population during the adaptive walk, as well as random chance in whether a given beneficial mutation is eventually fixed or lost via drift.

Despite uncertainty for individual adaptive walks, Orr was able to show that the average distance of a population to its optimum follows a geometric rule reminiscent of classical quantitative genetics theory (e.g., eq. (5) of Gomulkiewicz and Holt 1995). Imagine a large (effectively infinite) number of identical populations, each evolving in the same environment, and each having an initial displacement of $d_0$ from the optimum. While stochasticity causes the dynamics of adaptation to vary among populations, the average distance to the optimum after k substitutions will be:

$$\overline{d_k}=d_0{(1-M/n)}^k,$$

1

where n is the number of traits, and M is a positive number that depends on aspects of the phenotypic effect size distribution of new mutations entering the population (Fig. 2). While Orr presented an approximation for M under the assumption that new mutation had a uniform distribution of sizes, he later showed that it will overestimate the rate of approach to the optimum when a substantial fraction of mutations have large phenotypic effects (Orr 2000; we find that $\overline{d_k}=d_0{(1-3/2n)}^k$ closely approximates the approach to the optimum when mutation sizes are uniformly distributed and dimensionality is reasonably high, i.e., n > 10).

Figure 2

Orr's model for the genetics of adaptation. The left-hand panel shows the distance to the optimum during 100 adaptive walks with n = 25 traits (each adaptive walk is plotted in grey), as well as the average distance to the optimum following each adaptive substitution (red curve: average among simulation runs; black curve: mean distance predicted by Orr 1998, with an improved approximation based on later results by Orr 2000). The right-hand panels compare the average size of the first adaptive substitution (based on Orr's eq. (9)) and the largest adaptive substitution fixed during an adaptive walk (filled circles are averages of 1000 simulated adaptive walks; open circles are based on Orr's eq. (17)). The top-right panel presents results for the absolute mutation size relative to the initial distance from the optimum (i.e., r/d₀); the bottom-right panel shows mutation size in Fisher's scale (i.e., $x=r\sqrt{n}/2d_0$⁠). Results assume an initial displacement from the optimum of d₀ = 1, with new mutation sizes following a uniform distribution.

Orr's analysis also implies that there is a “cost of complexity” in which greater degrees of pleiotropy (higher n) decrease the magnitude of approach to the optimum following each adaptive substitution. Because each step during an adaptive walk decreases the lag to the optimum by a factor of (1 – M/n), pleiotropy can be seen to limit the amount of evolutionary movement towards the optimum, per adaptive substitution (the approach is proportional to 1/n). This constraint is further compounded by decreasing rates of adaptive substitution with increasing pleiotropy (Orr 2000), although later work has shown that the severity and scope of such costs of complexity are highly sensitive to assumptions about the relation between mutation size and n (see below; Wang et al. 2010).

PHENOTYPIC EFFECTS OF MUTATIONS CONTRIBUTING TO ADAPTATION

Fisher argued that adaptation is a multivariate problem, in which a large number of traits are selected to conform to a set of trait optima determined by many environmental variables. A notable feature of multivariate models of evolution is that the evolutionary response of a population to selection can easily become misaligned with the direction of selection (Schluter 1996; Hansen and Houle 2008). In evolutionary quantitative genetics theory, one can distinguish between the overall multivariate response of a set of traits (“respondability”) and the projection of this response on the direction of selection (“evolvability”; see Hansen and Houle 2008). These two measures of evolutionary change diverge, with respondability exceeding evolvability, when the multivariate orientation of directional selection differs from the phenotypic axis with the highest genetic variability.

In Fisher's geometric model, each mutation causes a multivariate phenotypic deviation (in each of the n trait axes) from the prevailing “wild-type” genotype. The absolute magnitude of a mutation (denoted r; Fig. 1) is equivalent to the Euclidean norm of the vector of changes to each of the n traits. Fisher (1930) emphasized that a mutation's scaled size, $x=r\sqrt{n}/2d$⁠, where d represents the current distance to the optimum, determined its probability of being beneficial (Fig. 1). Random mutations in Fisher's geometric model are hardly ever expected to align perfectly with the direction of selection (e.g., Fig. 1, where the mutation depicted by the red arrow is angled away from the orientation of selection from A and O). Indeed, Hartl and Taubes (1996) showed that the angles between vectors of selection and random mutations converge (with high n) to a normal distribution with a mean angle of 90° (π/2), implying that mutations are, on average, orthogonal to the direction of selection.

Orr (1998) illustrated an important consequence of this general misalignment between mutation and selection. The phenotypic effect sizes of mutations that contribute to adaptation are often large relative to the degree to which individual substitutions move the population toward the optimum, similar to the evolvability/respondability distinction described above. Orr specifically showed that the distribution of phenotypic effect sizes of adaptive substitutions is roughly exponential, a result that is robust to details about: (1) the relation between fitness and the distance to the optimum, (2) the distribution of new mutation sizes, and (3) whether one focuses on the total (n-dimensional) effects of substitutions or their effects on individual traits (see also Orr 1999).

For the case of uniformly distributed mutation sizes, Orr demonstrated that the size distribution of substitutions in Fisher's scale will be exponential with parameter $\lambda \approx 2.9$⁠. Moreover, the average size of the first substitution during an adaptive walk will be $\mathrm{E}(x_1)\approx 1.06$⁠; the average size of the largest substitution during a walk becomes:

$$\mathrm{E}\left(x_{\max }\right)\approx \frac{1}{\lambda }\left[\ln \left(\frac{\lambda n}{2}\right)+\gamma \right]$$

2

where $\gamma$ ≈ 0.5772 is Euler's constant (see Orr's eq. (17)). These predictions correspond to the following absolute measures of magnitude: $\mathrm{E}(r_1)\approx 2.12d_0\mathrm{E}(x_1)/\sqrt{n}$ and $\mathrm{E}(r_{\max })\approx 2d_0\mathrm{E}(x_{\max })/\sqrt{n}$⁠. Orr's analysis demonstrated that, in contrast to the infinitesimal view of adaptation, we should expect the largest mutations fixed during adaptation to be substantial enough to be detectable in plausible genetic mapping experiments. Indeed, the average size of the largest fixed factor is, on average, nearly as large as the initial displacement of the population from its optimum (Fig. 2), particularly in scenarios involving modest pleiotropy (e.g., $\mathrm{E}(r_{\max })$ approaches 80% of the initial displacement, d₀, when n = 10; see the upper-right panel of Fig. 2).

The Response

Orr's paper, along with the many cases of large-effect loci that have been shown to contribute to evolutionary divergence, has led to a shift in opinion among evolutionary biologists since the 1990s. Mutations of large effect are now widely acknowledged as being important in adaptation, and Orr's influence in this respect is apparent from the space devoted to his model in major textbooks of evolutionary genetics (Charlesworth and Charlesworth 2010; Walsh and Lynch 2018), and from reactions against what some regard to be an over-enthusiasm for large-effect alleles in genetic studies of population and species divergence. For example, where Orr and Coyne (1992) once pushed back against the prevailing “micro-mutationist” views that predominated among 20th Century evolutionary biologists, Rockman (2012) suggests that the pendulum has swung too far in the other direction, with too many researchers expecting to see large-effect substitutions wherever they look.

We were interested in more formally evaluating the influence that Orr's paper had on both theoretical and empirical research within the field. The influence of theoretical models on empirical research is often weak, or indirect (see Provine 1977; Fitzpatrick et al. 2018; Rosenberg 2020). Theory papers on sexual selection and speciation, for example, are more often cited by other theory papers (∼56-59% of the time) than by empirical papers (∼19-23% of the time, with the remainder coming from reviews); these patterns do not align with the proportions of theory versus empirical papers in these disciplines (empirical papers represent the vast majority; see Fitzpatrick et al. 2018). In some cases, mathematical models may ultimately prove to be highly influential, but only after their major findings are translated and popularized by influential reviews and by researchers who excel at communicating quantitative insights. For example, Provine (1977) argued that classical evolutionary theory by Wright, Fisher, and Haldane, was profoundly influential in the evolutionary synthesis period of the mid 20th Century, though transmission of this work owed much to widely read popularizers, such as Dobzhansky (i.e., through Dobzhansky's 1937 book). Rosenberg (2020) provides a more recent example of the popularization of an important model by Chesson (1994) by a highly cited review (Chesson 2000). Fitzpatrick et al. (2018) come to a similar conclusion regarding the influence of reviews on the transmission of theoretical results.

To evaluate the breadth, consistency, and nature of Orr's (1998) influence on evolutionary research during the past two decades, we downloaded references and abstracts from the Web of Science database, including all research papers citing Orr (1998) through the end of 2020. We classified these papers into three research categories: primarily empirical, primarily theoretical, and reviews (including perspectives, commentaries, etc.). Most papers fell cleanly into one of the three categories (a small number of “hybrid” theoretical and empirical papers was excluded from subsequent analyses).

Citation rates of Orr (1998) were high almost immediately following its publication (there is no notable lag), and have remained consistent up to the present. This pattern also holds true when citing papers are divided into categories of citation (although pre-2000 citations are enriched for reviews, which might have spread word about the paper). Interestingly, most citing papers are empirical, which makes Orr (1998) an exception to what seems to be a more general rule for evolutionary theory (i.e., theory papers are more often cited by other theory papers; Fitzpatrick et al. 2018). Of course, citations can be a relatively rough proxy for influence (see Servedio 2020), but this does not appear to be the case for Orr (1998). A closer look at citing theory and empirical papers confirms that Orr (1998) had a profound effect on both research categories, with many references genuinely engaging with Orr's (1998) predictions. We discuss the influence of the paper on empirical and theoretical work, below.

The Influence of Orr (1998) on Empirical Evolutionary Research

Although originally inspired by QTL analysis of adapting traits, Orr's paper motivated much greater empirical research into adaptation genetics. In the decades that followed, advances in sequencing technologies facilitated a large influx of studies attempting to identify the genetic basis of adaptation in both natural and experimental settings. Orr's paper was ideally timed, placing itself at the forefront of a wave of empirical investigations into the genetics of adaptation. The expectation that at least some large effect loci will contribute to adaptation provided an important theoretical justification for this empirical work. Importantly, the paper also developed specific theoretical predictions that could be tested experimentally leading to a number of studies that sought to quantify the effect sizes of mutations contributing to adaptation in real organisms.

EXPERIMENTAL EVOLUTION STUDIES

Experimental evolution studies offer a tractable approach for addressing predictions about the genetics of adaptation (Long et al. 2015). They typically take isogenic microorganisms (but not exclusively), expose them to a novel environment and then attempt to measure the fitness effects of mutations contributing to adaptation to the new environment (e.g., by carrying out competitive growth assays of evolved lines relative to ancestral genotypes in the novel environment). It should be noted that microbial experimental evolution studies essentially measure fitness rather than the distribution of phenotypic effects, as modeled by Orr (1998). Microbial systems have many advantages, however, including short generation times and the ability to easily sequence the evolved genomes to identify adaptive substitutions. In one of the first experimental evolution studies to test Orr's predictions, Holder and Bull (2001) challenged two strains of bacteriophage with a novel temperature environment, and found in both instances that the first two substitutions accounted for over half of the fitness gains during experimental evolution. This pattern agrees with Orr's (1998) qualitative prediction that adaptation should follow a pattern of diminishing returns whereby larger effect mutations are more commonly substituted early (and smaller ones later) during adaptive walks to an optimum. Of the many similar experimental evolution studies subsequently carried out, the predominant finding is that large-effect mutations fix early during adaptive walks, followed by a series of smaller-effect substitutions (e.g., Rozen et al. 2002; Betancourt 2009; Collins and de Meaux 2009; MacLean et al. 2010; Estes et al. 2011).

Perhaps not unexpectedly, the dynamics of adaptation in real systems are often more complex than those described under the simplifying assumptions of Orr's model, even in microbial populations evolving in carefully controlled laboratory settings. Microbial populations are large and typically not mutation limited. Strongly beneficial mutations are more likely to escape loss through clonal interference (Levy et al. 2015; Gerrish and Lenski 1998; Orr 2010), which should inflate their contributions to divergence, while small effect loci are less likely to be observed experimentally (Barrett et al. 2006; Levy et al. 2015). Indeed, Orr pointed out that many of his assumptions (see above) could be violated in adapting populations (Orr 1998). Experimental evolution studies have specifically sought to identify how changes to some of these assumptions may influence effect size distributions (e.g., Collins and de Meaux 2009; Burke et al. 2014; Morley and Turner 2017). In a study using Chlamydomonas, populations adapting to a sudden environmental change showed evidence of substitutions of large effect contributing early on to adaptation, while populations responding to slow rates of change showed patterns consistent with adaptation via substitutions of small effect (Collins and de Meaux 2009). Overall these experimental evolution studies have provided rich insight into the genetics of adaptation, and many of them were directly or indirectly inspired by Orr's 1998 model.

QTL STUDIES IN NATURAL POPULATIONS

In natural populations, “top-down” approaches, where associations between genetic variants and phenotypes are identified, have been used to illuminate the genetic basis of adaptation in many species. The empirical data that inspired Orr's (1998) paper were largely derived from linkage mapping populations using parental lines that had diverged in key traits likely influencing fitness (Shrimpton and Robertson 1988; Doebley and Stec 1991; Dorweiler et al. 1993; Tanksley 1993; Bradshaw et al. 1995, 1998). In opposition to the gradualist views of many evolutionary biologists at the time, several of these early studies identified large effect QTL. The success of these early QTL mapping studies alongside Orr's paradigm shifting theory spurred a proliferation of research examining the genetics of adaptation, made easier by the increasing availability of inexpensive sequencing. Although empirical studies are inarguably biased towards finding large-effect loci, a major conclusion of decades of this research is that large-effect loci are certainly not uncommon (Schluter et al. 2021).

The best empirical systems to test Orr's model are those whose conditions share many of Orr's assumptions, such as adaptation to an abrupt environmental change via de novo mutations (i.e., successive selective sweeps). Domesticated species potentially satisfy many of these conditions, since domestication traits are unlikely to be favored in the wild and were subject to recent and strong artificial selection (see Lande 1983). For instance, maize is highly differentiated from its wild progenitor (teosinte) for a number of traits including plant architecture, ear, kernel, and nutritional characteristics. Large effect QTL, alongside smaller effect loci, contribute to many of these phenotypic differences (Doebley and Stec 1991; Dorweiler et al. 1993; Doebley 2004; Lemmon and Doebley 2014; Fang et al. 2020) and several of these regions show the footprint of recent positive selection (Wright et al. 2005). Resistance to pesticides and herbicides is another class of traits having been strongly selected. Large effect resistance loci have been frequently identified, typically in genes that code for proteins that are the target of the insecticide or herbicide (Ffrench-Constant 2013; Baucom 2019).

Examples of large-effect loci contributing to adaptation are not limited to species experiencing abrupt anthropogenic changes to their environment. There is an ever-growing list of studies that have identified large effect loci contributing to adaptive divergence in natural populations, some of which have become textbook examples (e.g., Colosimo et al. 2005; Bradshaw et al. 1998; Steiner et al. 2007; Reed et al 2011). In a number of cases, specific functional variants have been identified. For instance, in beach mice (Peromyscus polionotus) a single polymorphism in the coding region of the Mc1r gene is involved in adaptive coat color shifts (Hoekstra et al. 2006). Some of these genes contributing to large effect changes have even been involved repeatedly in adaptation, both within and between lineages, which implies a limited set of possible genetic pathways to adaptation (e.g., the number of types of beneficial mutations is small; Orr 2005c; Yeaman et al. 2018). The Mc1r gene has been implicated in adaptive color change in a diverse array of species, from mice to lizards to birds (Rosenblum et al. 2004; Hoekstra et al. 2006; Hubbard et al. 2010; Uy et al. 2016).

It may be unsurprising that discrete traits, such as color polymorphisms, have a relatively simple genetic basis, with a large effect QTL contributing to adaptive variation (Rockman 2012). However, even classic quantitative traits adapting in natural populations can have contributions of large-effect loci. In Darwin's finches, large effect loci underlie beak shape (Alx1) and beak size (Hmga2), two traits that facilitate adaptation to different food resources and were likely instrumental in the adaptive radiation of the group (Lamichhaney et al. 2015, 2016; Han et al. 2017). Another well-known example is that of threespine stickleback, where the evolution of freshwater forms (from a marine ancestor) has occurred repeatedly following post-glaciation invasion of lakes. Adaptation to freshwater involves a reduction in body armor which is advantageous in high-predation marine environments. Fixation of an ancient large-effect, low-armor Ectodysplasin (Eda) allele in freshwater populations of stickleback has been observed repeatedly (Colosimo et al. 2005; Jones et al. 2012).

Both Alx1 and Hmga2 in Darwin's finches and Eda in stickleback highlight another complexity that was not addressed by Orr (1998): preexisting genetic variation. Orr's model assumes de novo beneficial variants arising and fixing one at a time in an adaptive walk towards a fixed optimum. However, evidence across an increasing number of studies points to hybridization, or ancient standing variation, as the origin of many large-effect ecologically important alleles in an increasing number of species (Nelson and Cresko 2018; Marques et al. 2019; Todesco et al. 2020; Lamichhaney et al. 2015, 2016; Han et al. 2017; Meier et al. 2017). This is not, of course, the only source of large-effect loci involved in the adaptation of natural populations. For instance, in stickleback, the repeated evolution of pelvic reduction in freshwater is caused by independent large-effect deletions in an enhancer of Pitx1 (Chan et al. 2010; Xie et al. 2019). However, large-effect loci that have been previously tested by selection in other lineages, perhaps even evolving multiple beneficial substitutions in a single region or reduced deleterious effects on other traits, appear to be an important source of adaptive variation.

QTL from linkage mapping studies tend to be large and can harbor many genes potentially containing multiple functional substitutions. This may lead to an overestimation of the contribution of individual large-effect mutations to adaptation (Remington 2015). The problem of multiple mutations combining to form a single large-effect QTL is lessened in GWAS studies due to the more rapid reduction in linkage disequilibrium, but not completely eliminated, especially in regions that may have experienced a recent sweep or reduced recombination (e.g., an inversion). Further, the net effect of a genomic region on a phenotype can be missed in an SNP by SNP GWAS analysis, which raises additional analytical challenges (Remington 2015). Although it can be difficult to distinguish the individual effects of mutations within such complex alleles from one another, transgenic approaches are increasingly enabling this (Evans et al. 2021). There are several explanations for why such complex alleles might evolve, including mutational processes that generate linked beneficial mutations (e.g., tandem gene duplications; inversions) as well as theoretical reasons why such mutations could be favored in some circumstances, such as divergent selection with gene flow (Yeaman and Whitlock 2011; Yeaman 2013).

Testing the exponential prediction using QTL identified in natural populations is challenging for a number of reasons, some of which are outlined above. In addition, linkage mapping populations are typically derived from a small number of parental lines grown in controlled conditions, and may not be representative of allele frequencies in natural populations or their trait effects in the local environment. Therefore the magnitude of the QTL effect may be dependent on the conditions or the individuals used in the experiment (Slate 2005). Genome-wide association studies mitigate the bias introduced by a small number of parental lines, but population structure can present a challenge when comparing divergent ecotypes or nascent species. QTLs tend to be estimated for traits with fitness consequences that are not fully understood, where the shape of the fitness landscape is uncertain. Similarly, the high trait dimensionality of mutations is an important assumption in Orr's model, yet the level of pleiotropy of QTL is usually not well understood. For example, low armor Eda alleles in stickleback have frequently been selected for in freshwater lakes (Colosimo et al. 2005; Jones et al. 2012; Schluter et al. 2021). The production of body armor requires resources and comes with costs associated with body size and fecundity, contributing to its advantage in low predation freshwater lakes. However, even when controlling for body size, a major QTL for fitness in experimental freshwater ponds has been identified near Eda suggesting that the beneficial fitness effects of this QTL were also caused by selection on other traits (Schluter et al. 2021). Eda has known effects on traits besides armor and body size, such as schooling behavior (Greenwood et al. 2016), but the genotype-trait-fitness map remains a mystery even in this well-studied case.

Another complication of the top-down approach is that small effect loci will be difficult to identify and their effects may be overestimated when detected and sample sizes are small. Unbiased estimates of small effect QTL require large sample sizes (Beavis 1998), and most evolutionary studies are too underpowered to estimate a significant proportion of the effect size distribution. Nevertheless, several studies have attempted to test aspects of Orr's model by examining QTL effect size distributions. Because of issues surrounding the detectability of small effect loci, truncated effect size distributions are expected (Otto and Jones 2000). Consequently, the general prediction is that these distributions should be right skewed (i.e., a small number of loci of large effect, but most loci should be the intermediate or small effect). In plants, multiple meta-analyses of QTL studies have revealed right-skewed PVE (phenotypic variation explained) distributions for many ecologically important traits, from disease resistance to growth to phenology (Louthan and Kay 2011; Hall et al. 2016). Many of these traits are thought to be under spatially varying selection. In stickleback, the distribution of effect sizes for 1034 QTL from 28 published studies examining ecologically important traits (e.g., feeding, defense, body shape) similarly matched the general expectation of a right-skewed PVE distribution (Peichel and Marques 2017). QTL distributions have also been used to test a second prediction stemming from Orr (1998), which is that increased distance from the optimum leads to the fixation of larger effect mutations. In stickleback, freshwater lakes that were more divergent from the ancestral environment (i.e., the lakes lacked predators while marine environments have predators) had populations with larger effect QTL compared to those in lakes that were more similar to the ancestral environment (i.e., lakes with predators), consistent with this prediction (Rogers et al. 2012).

The examples above suggest that the contribution of large-effect loci to adapting traits is the norm, but our understanding of the genetic architecture of adaptation is incomplete owing to the various biases we have discussed. Particularly in humans, large GWAS studies have identified a highly complex polygenic basis to many traits (Zeng et al. 2018; Sella and Barton 2019). Human height is perhaps the most well studied polygenic trait with a large number of small effect loci contributing to trait variation (McEvoy and Visscher 2009; Yang et al. 2010). Debate exists regarding the signature of polygenic local adaptation of height in Europe, with some studies identifying an excess of tall alleles in northern populations (Berg and Coop 2014; Robinson et al. 2015). More recently this finding has been called into question and may reflect population structure (Berg et al. 2019; Sohail et al. 2019), and so the genetic variation observed may largely reflect the confines of stabilizing instead of divergent selection across Europe. The inverse relationship between effect size and allele frequency is consistent with stabilizing selection (Marouli et al. 2017; Zeng et al. 2018,). However, even for this highly polygenic trait, in certain African populations, some GWAS candidates for height show sweep signatures indicative of recent positive selection for short stature (Perry et al. 2014).

Increasingly, a complex, but perhaps more biologically realistic, polygenic view of trait adaptation is being adopted both in theoretical and in empirical studies, aided by individual based simulations and more powerful genomic datasets (e.g., Yeaman and Whitlock 2011; Yeaman et al. 2016; Stetter et al. 2018). The level of redundancy between the genotype and phenotype and between the phenotype and fitness can have profound effects on the genetic architecture of adaptation (Yeaman et al. 2018; Barghi et al. 2020). Low levels of redundancy can lead to a simple genetic basis involving a small number of genes contributing to adaptation. High levels of redundancy, where there are many interchangeable loci of small effect that contribute to trait variation, can lead to a situation more reminiscent of the infinitesimal model, where covariance among loci (shifts) and not large changes in individual allele frequencies (sweeps), leads to adaptive phenotypic divergence among populations. Under this scenario, loci do not sweep to fixation sequentially. Orr's 1998 model was transformational for the study of adaptation, where the extremes of infinitesimal and macro-mutationist models were superseded by a theory that proposed that both large and small effect mutations could be important for adaptation. This polygenic view follows from Orr's insight that neither model (sequential sweeps versus shifts) is sufficient to explain the full diversity of genetic architectures of adaptation that have been uncovered (Yeaman et al. 2018; Barghi et al. 2020). A range of outcomes are apparent and explaining the major biological causes of effect size distribution variation for adapting traits remains an important challenge (Dittmar et al. 2016).

The Influence of Orr (1998) on Evolutionary Theory

Orr (1998) inspired the development of two branches of evolutionary theory that emerged in earnest during the last two decades. First, we now know much more about the theoretical population genetics of phenotypic adaptation owing to those who have taken up the challenge of extending Orr's model beyond the set of conditions that he originally considered. Second, Fisher's geometric model, and modifications of it, has been widely used to address evolutionary questions that go well beyond the domain of Orr's (1998) model. These studies nevertheless exhibit the clear imprint of Orr's (1998) compelling example of the utility of Fisher's model in evolutionary biology. We discuss both legacies below.

THEORETICAL STUDIES OF THE GENETIC BASIS OF ADAPTIVE PHENOTYPIC DIVERGENCE

Orr has consistently emphasized (see Orr 1998, 2005b) that his model applies to a highly idealized scenario of adaptation. In his model, each population first experiences a single, abrupt environmental change, leading to displacement from its optimum. The population then proceeds through an adaptive walk to the new optimum by a series of discrete (temporally non-overlapping) selective sweeps, where substitutions are drawn from new, beneficial mutations rather than standing genetic variation. The population is assumed to be stable in size, well-mixed (i.e., with no population subdivision), and haploid. More recent models of the genetic basis of adaptation have since relaxed most of these assumptions.

A series of papers has asked how the genetics of adaptation are likely to change when the environment (and optimum) changes gradually over time rather than abruptly (Collins et al. 2007; Kopp and Hermisson 2007, 2009a, b; Matuszewski et al. 2014, 2015). The phenotypic effect size distribution of substitutions now depends on the magnitude of the rate of environmental change, which causes a lag between the population and the moving optimum, relative to the population's evolutionary potential (the latter depends on the product of the mutation rate, the variance of phenotypic effects of new mutations, and strength of selection to the optimum; see Kopp 2009b; Matuszewski et al. 2014). When the rate of environmental change is small relative to the population's evolutionary potential (i.e., evolution in an “environmentally limited” regime), adaptation proceeds by fixation of mutations with relatively small phenotypic effects, and the distribution of their sizes reflects the shape of the fitness surface. More rapid environmental change can shift adaptation to a “genetically limited” regime (though extinction is also possible), in which large-effect mutations can contribute depending on how often they arise by mutation. In both cases, the sizes of substitutions are no longer exponentially distributed (as in Orr 1998), and instead, have distributions with an intermediate mode.

For reasons of mathematical tractability, most models of the genetics of adaptation assume that mutation rates are sufficiently weak that adaptation largely uses new mutations rather than standing genetic variation. While it is clear that both classes of genetic variation contribute to adaptation, the relative contribution of new mutations versus standing genetic variation is currently unknown (for discussion, see: Messer and Petrov 2013; Jensen 2014; Hermisson and Penning 2017). Population genetic models of adaptation from standing genetic variation only became prevalent during the early 2000s (Orr and Betancourt 2001; Hermisson and Pennings 2005; Barrett and Schluter 2008; Messer and Petrov 2013). As such, only recently have such scenarios been integrated into theoretical models for the genetic basis of adaptive phenotypic divergence. In particular, Matuszewski et al. (2015) considered adaptation using new mutations and standing genetic variation in a model involving a single trait adapting to a continuously moving optimum in one direction. They showed that adaptation from standing genetic variation led to the fixation of mutations with smaller effects, compared to adaptation using new mutations (the distributions for both scenarios have intermediate modes). It remains unclear whether these results can be extrapolated to scenarios involving abrupt shifts of an optimum (as in Orr 1998), although future modeling can resolve this question (see p. 268 of Matuszewski et al. 2015).

While scenarios of directional change in the optimum and adaptation using standing genetic variation tend to decrease the phenotypic effect sizes of adaptive substitutions (particularly when the rate of change in the optimum is slow), two other factors have been shown to increase the effect sizes of alleles contributing to adaptation. First, scenarios of local adaptation under sustained gene flow limit the scope for divergence at loci with small-effect alleles, many of which may not overcome effects of swamping, and leading to few loci with large phenotypic effects contributing to a genetically stable basis of population differentiation (Griswold 2006; Yeaman and Whitlock 2011; for exceptions, see Yeaman 2015; Hodgins and Yeaman 2019). Second, under scenarios of adaptation during the evolutionary rescue, environmental stress initially causes the population to decline, which limits the potential for small-effect mutations to spread (Otto and Whitlock 1997), leading to enrichment of large-effect substitutions among populations that are able to adapt and persist in the stressful environment (Osmond et al. 2020; also see Anciaux et al. 2018).

THEORETICAL RESEARCH APPLYING FISHER'S GEOMETRIC MODEL TO OTHER QUESTIONS

A large proportion of theoretical studies citing Orr (1998) makes use of Fisher's geometric model to address evolutionary genetic questions unrelated to the genetic basis of adaptive phenotypic divergence. By providing a compelling case study, Orr (1998) appears to have popularized Fisher's geometric model, though uptake of Fisher's model was certainly reinforced by contemporary theory by Hartl and Taubes (1996, 1998) and subsequent reviews that further highlighted the model's utility (e.g., Orr 2005a, 2005b; Tenaillon 2014). Indeed, whereas only a handful of studies made use of Fisher's model prior to 1998 (e.g., Fisher 1930, pp 38–41; Haldane 1932, pp 174–176; Kimura 1983 pp. 135–137, 154–156; Leigh 1987; Rice 1990; Hartl and Taubes 1996; Peck et al. 1997), it has since become a widely used tool in models of evolution. Below, we outline the most prominent strands of theoretical work making use of Fisher's geometric model.

One strand of theory is concerned with predicting the distribution of fitness effects of new mutations (hereafter the “DFE”), from which the rate of evolution and the fitness effects of substitutions can be derived (Tenaillon 2014). Fisher (1930) used the original version of his model to calculate the probability that a mutation of a given size was beneficial (Fig. 1). Predictions for the full DFE among random mutations emerge naturally from Fisher's model and its later extensions (e.g., those relaxing isotropy assumptions: Waxman and Welch 2005; Martin and Lenormand 2006), once the distribution of mutant phenotypic effects and the relation between fitness and distance to the optimum are specified. With many independent trait dimensions (large n), the DFE typically converges, approximately, to a normal distribution, with mutations decreasing fitness, on average, and beneficial mutations occupying the upper tail of the fitness distribution (Waxman and Welch 2005; Martin and Lenormand 2006; Orr 2006; Tenaillon 2014). Most of the initial models for the DFE considered haploid populations with a single optimum, whereas more recent versions have considered the effects of diploidy (Sellis et al. 2011; Manna et al. 2011; Connallon and Clark 2014b) and multiple phenotypic optima on the DFE (Moorad and Promislow 2008; Moorad and Hall 2009; Martin and Lenormand 2015; Connallon and Clark 2014a, 2014b, 2015).

The second strand of theory is centered around the rate of adaptation and the “cost of complexity.” Building on results from his 1998 paper, Orr (2000) showed that pleiotropy (i.e., n) constrains adaptation in two ways: the distance the population moves toward its optimum following each adaptive substitution declines with n (as implied by eq. (1), above), and the average time between adaptive substitutions increases with n. Both factors ensure that the rate of adaptation declines with n, which Orr (2000) referred to as the “cost of complexity.” Although intuition suggests that n should often be high in complex organisms such as humans and many other eukaryotes, several subsequent studies have shown that costs of complexity might be more modest than implied by Orr's model. First, relaxing the assumptions of universal pleiotropy and isotropy substantially reduces the “effective dimensionality” of an organism, and thereby lessens evolutionary constraints imposed by pleiotropy (Welch and Waxman 2003; Waxman and Welch 2005; Martin and Lenormand 2006; Chevin et al. 2010; Lourenço et al. 2011; see Orr and Coyne 1992 for discussion anticipating some of these later theoretical results). Second, assumptions about the relation between mutation size (r) and pleiotropy (n) can lead to different expectations for when costs of complexity should arise. Wang et al. (2010) showed that, if mutation sizes increase in proportion to n^b, where b is a non-negative constant, then costs of complexity are guaranteed when b < 1/2, whereas pleiotropy can increase the rate of adaptation when b > 1/2.

A third set of theoretical models is concerned with patterns of molecular evolution, including rates of adaptive, deleterious, and compensatory substitutions (Hartl and Taubes 1996; Razeto-Barry et al. 2011, 2012; Gordo and Campos 2013; Lourenço et al. 2013; McCandlish et al. 2014), and the evolutionary limits of adaptation imposed by genetic drift. Several studies have used Fisher's geometric model to calculate the “drift load”—the steady-state mean reduction of population fitness due to fixation of deleterious alleles (Hartl and Taubes 1998; Poon and Otto 2000)—which increases with organismal complexity and decreases with effective population size (e.g., E[L] ∼ n/(n + 2N_e) in the haploid, isotropic version of Fisher's model, where N_e is the population's effective size; see Poon and Otto 2000; Tenaillon et al. 2007; Sella 2009).

Finally, many studies have addressed macroevolutionary questions using Fisher's geometric model. These include models of speciation and hybrid fitness (Barton 2001; Chevin et al. 2014; Fraïsse et al. 2016; Simon et al. 2018; Thompson et al 2019; Yamaguchi and Otto 2020; Schneemann et al. 2020), evolutionary models of extinction (e.g., by mutational meltdown: Poon and Otto 2000; through displacements of a population from a moving optimum: Kopp et al. 2018), and models of evolutionary rescue (Anciaux et al. 2018; Osmond et al 2020).

Communication of Theory in Evolutionary Biology

Orr's 1998 paper had a profound influence on evolutionary research as we have discussed above. The study is noteworthy because it is an example of a contribution that has bridged the gap between the theoretical and empirical silos (Fig. 3). Another metric highlights how exceptional his theory paper was. Papers with a high density of equations are typically associated with low citation numbers (Fawcett and Higginson 2012). Orr (1998) is an outlier in this respect, having a high density of equations yet also being highly cited (Kollmer et al. 2015) (Fig. 4). Why might this be? There are likely many reasons, including, as we argued above, that his key biological insight came at the right moment in time, when technological innovation made these questions empirically accessible.

Figure 3

Citation data for Orr (1998). ISI Web of Science data were originally downloaded on Dec 10, 2020 and later updated on May 11, 2021. Citing papers between 1998 and 2020 include 243 empirical papers, 189 reviews, perspectives, or commentaries, and 134 theoretical papers. Hybrid theoretical/empirical papers were not included in the analysis.

Figure 4

Relation between equation density (the number of equations per page of text) and cumulative citations for articles published in the 1998 issues of Evolution, The American Naturalist, and Proceedings of the Royal Society of London Series B. Data were compiled and published in Fawcett and Higginson (2012). Two outlier data points—Orr's (1998)  Evolution paper and an American Naturalist paper by Clark (1998)—are highlighted, each having a high citation rate given its equation density.

Another important reason for the success of the paper is the way in which Orr communicated his ideas. He embedded his question in a rich history of theory. He provided one of the most perspicuous explanations of Fisher's geometric model to date, while making a powerful case that the evolutionary question was important yet hitherto overlooked. In the development of his model, Orr provided enough mathematical detail to establish a fruitful foundation for later theory. However, he also laid out explicit verbal conclusions (there are 7 in total), with each punchline progressively building on previous ones, making the ideas accessible to those who did not wish to delve into the equations. Finally, he uncovered a compelling and perhaps counterintuitive result: that large-effect alleles can be important for the adaptation of continuous traits, not just discrete polymorphisms, motivating decades of research into the topic.

ACKNOWLEDGMENTS

The authors thank Allen Orr for discussing his paper with us early on in the process, and Akane Uesugi and two anonymous reviewers for helpful comments on the manuscript. K.H. and T.C. are supported by funds from the Australian Research Council and Monash University's School of Biological Sciences.

AUTHOR CONTRIBUTIONS

The authors contributed jointly to all stages of manuscript development.

Associate Editor: T. Chapman

Handling Editor: T. Chapman