### Abstract

- Top of page
- Abstract
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgments
- References
- Supporting Information

The pattern of genetic variances and covariances among characters, summarized in the additive genetic variance-covariance matrix, **G**, determines how a population will respond to linear natural selection. However, **G** itself also evolves in response to selection. In particular, we expect that, over time, **G** will evolve correspondence with the pattern of multivariate nonlinear natural selection. In this study, we substitute the phenotypic variance-covariance matrix (**P**) for **G** to determine if the pattern of multivariate nonlinear selection in a natural population of *Anolis cristatellus*, an arboreal lizard from Puerto Rico, has influenced the evolution of genetic variances and covariances in this species. Although results varied among our estimates of **P** and fitness, and among our analytic techniques, we find significant evidence for congruence between nonlinear selection and **P**, suggesting that natural selection may have influenced the evolution of genetic constraint in this species.

### Introduction

- Top of page
- Abstract
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgments
- References
- Supporting Information

The classic breeder’s equation, *R* = *h*^{2}*S*, predicts that the evolutionary response (*R*) to selection (*S*) will vary in direct proportion to the heritability of the trait (*h*^{2}; Falconer & MacKay, 1996; Roff, 1997). However, this picture is incomplete because selection usually acts simultaneously on multiple traits that are genetically correlated. Under these circumstances, it is the matrix of additive genetic variances and covariances (the **G** matrix) that determines the single generation response to linear selection (Lande, 1979; Lande & Arnold, 1983; Arnold *et al.*, 2001).

The single generation response to selection on a given trait is a function of the additive genetic variance for the trait, the linear selection gradient acting on it, **β**, the selection gradients on other traits, and the genetic covariance between them (Lande & Arnold, 1983). In fact, the predicted response to selection is simply the sum of these factors, i.e. for *n* traits, in which *V*_{A}(*i*) is the additive genetic variance for trait *i*, and *Cov*_{A}(*i*, *j*) is the additive genetic covariance between traits *i* and *j* (Lande & Arnold, 1983). **β**_{i}, the selection gradient, is just the partial regression coefficient of the *i*th phenotypic trait on fitness (Lande & Arnold, 1983). This equation can also expressed in matrix form as , in which **β** is a vector containing the linear selection gradients for all the traits in the analysis, and is a vector containing the changes in the mean phenotype for all traits (Lande & Arnold, 1983). When **β** does not align with an eigenvector of **G**, the effect of **G** is that it will tend to bias the course of evolution away from that which maximizes the increase in mean fitness in the population (Via & Lande, 1985; Schluter, 1996, 2000; Arnold *et al.*, 2001). By contrast, when the direction of selection, **β**, and a major eigenvector of **G** are aligned, evolution by natural selection can be greatly facilitated by **G** (Schluter, 1996).

What is obvious from this theory of quantitative trait evolution by natural selection is that **G** plays a central role, either in determining the amount of evolution in response to a particular level of selection (primarily via the additive genetic variances), or in determining the course of evolution in response to a particular direction of selection (via the additive genetic covariances between characters). Unfortunately and despite the importance of **G**, the factors that affect its evolutionary dynamics are poorly understood (Turelli, 1988; Jones *et al.*, 2003; Revell, 2007; Arnold *et al.*, 2008).

Theoretical and empirical studies both suggest that selection, particularly multivariate nonlinear selection, can play a very important role in the evolution of **G** (Lande, 1980; Cheverud, 1984; Turelli, 1988; Arnold *et al.*, 2001; Jones *et al.*, 2003). In particular, under natural selection, uncorrelated mutation, and genetic drift, **G** is expected to evolve in response to **γ**, the matrix of quadratic and correlational terms in the multivariate second order polynomial regression of phenotype on fitness (i.e. ). Here, *w* is individual fitness, *α* is the regression intercept, **z** is a vector of phenotypic trait values, and *ɛ* is random error. Specifically, we expect alignment between the eigenvectors of **G** and −**γ**^{−1} to evolve as the linear component selection flattens. At this point **G** should respond only to the pressure of mutational input, the loss of genetic variability due to drift, and the stabilizing or disruptive and correlational influence of quadratic selection (Cheverud, 1984).

The prediction that the **G** matrix should evolve to align with the curvature of the individual selection surface, given by **γ**, arises as follows. Lande (1980) provided the following equation for change in the **G** matrix across a generation under linear and quadratic selection: Δ**G** = **G**(**γ **− **ββ**′)**G** + **U**. Here all terms have been defined except for **U** which contains the mutational inputs of genetic variation and covariation each generation. At equilibrium (when **G** is no longer changing, i.e. Δ**G** = **0**) we can assume that **β** = **0**, because the population will be at the optimum where linear selection is absent. Therefore, rearrangement of the previous expression gives: −**G**γ**G** = **U**. If we assume that mutational variance is equal and uncorrelated among traits (i.e. **U** = *k***I**, where **I** is the identity matrix), then we can solve for **G** at equilibrium as (Lande, 1980; Cheverud, 1984). In this study, we usually compare **G** to −**γ**^{−1} (as is common in similar studies, e.g. Blows *et al.*, 2004), rather than to , because if the latter can be evaluated it will have the identical eigenvectors in the same rank order as −*γ*^{−1} (which can nearly always be evaluated; note that−**γ**^{−1} also has the same eigenvectors as **γ**, although their order is rearranged). Alignment of the individual selection surface and **G** has been predicted in theory (e.g. Lande, 1980; Cheverud, 1984; Arnold *et al.*, 2001), found in simulation studies (e.g. Jones *et al.*, 2003; Revell, 2007), but tested for explicitly in few prior instance that we know of (Brodie, 1992, 1993; Conner & Via, 1993; Blows *et al.*, 2004; McGlothlin *et al.*, 2005; Hunt *et al.*, 2007).

The actual value of **G** found in a natural population will be a function not only of the strength and pattern of selection, but also of the mutation rates and mutational variances, particularly as these vary among traits, as well as of the mutational covariances and the effective population size (Lande, 1980; Arnold *et al.*, 2001; Jones *et al.*, 2003). These factors are generally quite difficult to measure empirically. The mutation rate has been calculated in some studies (e.g. Kimura, 1968), and the mutational variance-covariance matrix has been estimated in very few (e.g. Camara & Pigliucci, 1999; Camara *et al.*, 2000). Selection studies are more common (Endler, 1986; Kingsolver *et al.*, 2001), but few have looked for a correspondence between **G** and multivariate selection (e.g. Brodie, 1992; Blows *et al.*, 2004; McGlothlin *et al.*, 2005; Hunt *et al.*, 2007).

Iguanian lizards in the genus *Anolis*, commonly referred to as anoles, are one of the most diverse and interesting groups of taxa in the new world tropical region (Losos, 2009). With approximately 350 species in its ranks, *Anolis* is the second most species rich genus of vertebrates (behind the neotropical frog genus *Pristimantis*, formerly *Eleutherodactylus*; Hedges *et al.*, 2008), and the richest genus of amniotes. Even aside from its species richness, the genus *Anolis* has become a textbook example of ecological and morphological convergence in an adaptive radiation (Futuyma, 1998; Schluter, 2000; Losos, 2009). On the separate Greater Antillean islands of the Caribbean, anoles have independently diversified into similar ecological specialists with concomitant morphological adaptations (Williams, 1972; Losos, 1990a,b, 2009; Losos *et al.*, 1998). As a consequence, anoles have become among the best studied adaptive radiations of animals (Schluter, 2000). Nonetheless, little is known about the evolutionary quantitative genetics of anoles (but see Revell *et al.*, 2007).

The Puerto Rican crested anole, *Anolis cristatellus*, is among the most common and ubiquitous members of the vertebrate fauna of the Puerto Rican bank islands, which include the main island of Puerto Rico, and its close geographic neighbours. *Anolis cristatellus* is found at very high natural densities (e.g. Genet, 2002), particularly in the tropical lowland areas in which it is most common, and has been quite well studied (e.g. Huey & Webster, 1976; Fleishman *et al.*, 1993; Leal & Rodríguez-Robles, 1995; Perry *et al.*, 2003; Leal & Fleishman, 2004; Glor *et al.*, 2007; Revell *et al.*, 2007). Unfortunately, the natural habitat for *A. cristatellus*, lowland dry and mesic tropical forest, has been largely decimated by human activities on the main island which are concentrated in coastal lowland areas (Birdsey & Weaver, 1987; Thomlinson *et al.*, 1996).

To our knowledge, this habitat destruction has thus far produced no serious impact on population densities of *A. cristatellus*, as there is no evidence of population declines in this species in anthropogenically modified and disturbed environments. However, habitat destruction and alteration has probably substantially altered the selective regime for populations of this species. Consequently, in our efforts to measure linear and nonlinear selection potentially reflective of that experienced during the majority of the evolutionary history of this species, we sought out an environment relatively free from recent human modification. This led us to the off-shore island of Vieques. Vieques is a moderate sized Caribbean island – approximately 32 km long at its longest, and approximately 7 km wide at its widest. Although much of the island was once cleared for sugar cane farming, the 62 year U.S. Navy presence on the island (finally ending in 2003) prevented extensive anthropogenic development and thus allowed natural vegetation to re-grow in many areas. Whether or not this secondary forest well approximates the historic conditions on the isle is unknown. However, we feel that it probably represents a selective milieu more consistent with historical conditions than that which is often found in the highly developed lowland areas of the main island. We chose the intermittently insular/peninsular cay, Cayo de Tierra, off the southern coast of Vieques as our study area for several reasons including its complete lack of human habitation, its isolation from adjacent populations of *Anolis cristatellus*, and its relatively manageable dimensions. Figure 1 shows aerial and terrestrial views of the cay.

In this study, we detail efforts to (i) elucidate multivariate linear and nonlinear natural selection in a population of *A. cristatellus* and (ii) compare the structure of nonlinear selection to that of genetic variances and covariances. In so doing, we will examine the hypothesis that multivariate nonlinear selection has influenced the evolution of quantitative genetic parameters in this species.

### Supporting Information

- Top of page
- Abstract
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgments
- References
- Supporting Information

**Appendix S1** Methods and results for canonical analyses of multivariate selection.

**Table S1.1 **Canonical analysis of **γ** from the quadratic regression selection analysis on external data is the *i*th column of **M**, the matrix of eigenvectors in the canonical analysis of the curvature of the selection surface, **γ** (Table S2.3).

**Table S1.2 **Canonical analysis of **γ** from the quadratic regression selection analysis on internal (x-ray) data.

**Table S1.3 **Canonical analysis of **γ** from the selection differential analysis on external data.

**Table S1.4 **Canonical analysis of **γ** from the selection differential analysis on internal data.

**Table S2.1 **Eigenvectors (**P**_{i}) and eigenvalues (λ_{i}) of **P**_{ex}, the phenotypic variance covariance matrix calculated from external measurements.

**Table S2.2 **Eigenvectors (**P**_{i}) and eigenvalues (λ_{i}) of **P**_{in}, the phenotypic variance covariance matrix calculated from internal x-ray measurements.

**Table S2.3 **Quadratic regression selection analysis; external data.

**Table S2.4 **Eigenvectors and eigenvalues of −**γ**^{−} and **γ**. Left-right rank order of the eigenvectors is based on −**γ**^{−}.

**Table S2.5 **Eigenvectors and eigenvalues of −**γ**^{−} and **γ**. Rank order of the eigenvectors is based on −**γ**^{−}. Internal data; least-squares regression selection analysis.

**Table S2.6 **Eigenvectors and eigenvalues of −**γ**^{−} and **γ**. Rank order of the eigenvectors is based on −**γ**^{−}. External data; selection differential analysis.

**Table S2.7 **Eigenvectors and eigenvalues of −**γ**^{−} and **γ**. Rank order of the eigenvectors is based on −**γ**^{−}. Internal data; selection differential analysis.

**Table S2.8 **Analysis using the method of Hunt *et al.* (2007); external data.

**Table S2.9 **Analysis using the method of Hunt *et al.* (2007); internal (x-ray) data.

**Table S2.10 **Variability of **P** evaluated over the eigenvectors of **γ**; external data.

**Table S2.11 **Variability of **P** evaluated over the eigenvectors of **γ**; internal (x-ray) data.

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