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## Phylogenetic comparative analysis: A modeling approach for adaptive evolution. The American Naturalist 164:683–695 (2004)

Citations: | 77 - 1 self |

### BibTeX

@MISC{Butler04phylogeneticcomparative,

author = {Marguerite A Butler and Aaron A King},

title = {Phylogenetic comparative analysis: A modeling approach for adaptive evolution. The American Naturalist 164:683–695},

year = {2004}

}

### OpenURL

### Abstract

abstract: Biologists employ phylogenetic comparative methods to study adaptive evolution. However, none of the popular methods model selection directly. We explain and develop a method based on the Ornstein-Uhlenbeck (OU) process, first proposed by Hansen. Ornstein-Uhlenbeck models incorporate both selection and drift and are thus qualitatively different from, and more general than, pure drift models based on Brownian motion. Most importantly, OU models possess selective optima that formalize the notion of adaptive zone. In this article, we develop the method for one quantitative character, discuss interpretations of its parameters, and provide code implementing the method. Our approach allows us to translate hypotheses regarding adaptation in different selective regimes into explicit models, to test the models against data using maximum-likelihood-based model selection techniques, and to infer details of the evolutionary process. We illustrate the method using two worked examples. Relative to existing approaches, the direct modeling approach we demonstrate allows one to explore more detailed hypotheses and to utilize more of the information content of comparative data sets than existing methods. Moreover, the use of a model selection framework to simultaneously compare a variety of hypotheses advances our ability to assess alternative evolutionary explanations. Keywords: Ornstein-Uhlenbeck, Brownian motion, selective regime, adaptation, evolutionary model, Anolis lizards. We have stressed throughout the important role that models of evolutionary change play in our statistical methods. Brownian motion models have been put to use for characterizing * E-mail: mabutler@utk.edu. † E-mail: king@tiem.utk.edu. Am. Nat. 2004. Vol. 164, pp. 683-695. ᭧ 2004 by The University of Chicago. 0003-0147/2004/16406-40201$15.00. All rights reserved. change in continuously varying characters, as has a Markov model in the case of dichotomous characters. New models, based on undoubtedly wicked mathematics, will gradually emerge. In his seminal article introducing the method of independent contrasts, The use of BM is not limited to the method of independent contrasts. Although it is not always made explicit, BM is the underlying model of evolution in nearly all phylogenetic comparative methods for quantitative characters including phylogenetic autocorrelation Perhaps because the BM model does not adequately de-684 The American Naturalist scribe change in adaptive characters, several investigators have attempted to improve the fit of the BM model in comparative analyses. These modifications fall into two basic categories. The first weakens the strength of the BM to the point that the model becomes nonphylogenetic and selects the best-fitting model from among this class (e.g., Hansen (1997) made an innovative contribution to the modeling of adaptive hypotheses. Following the suggestion of The Ornstein-Uhlenbeck Model and Its Evolutionary Interpretation The OU model is the simplest mathematical expression for an evolutionary process with selection. It differs from BM in that it possesses a selective optimum. It is important to note, however, that it includes BM as a special case. As one varies its parameters, one obtains a variety of distributions that are collectively consistent with phenotypic evolution under both drift and selection. In order to better understand the OU process and the role it plays in our approach to comparative analysis, we will spend some time developing the model and discussing its evolutionary interpretation. Let us begin by considering the evolution of a quantitative character X along one branch of a phylogenetic tree. We can decompose the change in X into deterministic and stochastic parts. The former may be interpreted as the force of selection acting on the character, the latter as the effect of random drift and other, unmodeled, forces. Accordingly, the OU model has two terms: Equation The parameter j measures the intensity of the random fluctuations in the evolutionary process. Model-Based Comparative Analysis 685 This term is linear in X, so it is as simple as it might possibly be. It contains two additional parameters: a measures the strength of selection, and v gives the optimum trait value. The force of selection is proportional to the distance, , of the current trait value from the opv Ϫ X(t) timum. Thus, if the phenotype has drifted far from the optimum, the "pull" toward the optimum will be very strong, whereas if the phenotype is currently at the optimum, selection will have no effect until the stochasticity moves the phenotype away from the optimum again or there is a change in the optimum, v, itself. Because of its dependence on the distance from the optimum, the OU process can be used to model stabilizing selection. The effect of varying a can be seen in Because the OU model reduces to BM when , it a p 0 can be viewed as an elaboration of the BM model. As a statistical model, its primary justification is to be sought in the fact that it represents a step beyond BM in the direction of realism while yet remaining mathematically tractable. As a model of evolution, the OU process is consistent with a variety of evolutionary interpretations, two of which we mention here. Lande (1976) showed that under certain assumptions, evolution of the species' mean phenotype can take the form of an OU process. In Lande's formulation, both natural selection and random genetic drift are assumed to act on the phenotypic character; the OU process's optimum denotes the location of a local maximum in a fitness v landscape. Hansen (1997) raised questions concerning the timescale of the approach of a species' mean phenotype to its optimal value relative to that of macroevolution. Specifically, he suggested that the macroevolutionary OU process he proposed could only operate on far too slow a timescale to be identical with the Landean OU process (cf. Lande 1980). He proposed a different interpretation based on the supposition that at any point in its history, a given phenotypic character is subject to a large number of conflicting selective demands (genetic and environmental) so that its present value is the outcome of a compromise among 686 The American Naturalist them. Under this interpretation, evolutionary changes on the macroevolutionary timescale occur as the balance among these selective forces shifts as the individual selective forces themselves undergo small, independent (or nearly independent) random changes. In other words, Hansen interprets the OU process as a qualitative model of the dynamics of peaks in an adaptive landscape. Although questions of its interpretation remain open, it is clear that the OU process can be used to describe the evolution of a single lineage. One can blend in phylogenetic information by assuming that each lineage in the tree evolves according to its own OU process, that is, that there is one optimum per branch of the phylogeny. Complex evolutionary scenarios can be modeled by allowing different branches of the phylogeny to have different optima ( Nuts and Bolts In this section, we demonstrate how the OU process can be integrated with phylogenetic information and biological hypotheses to give specific predictions on the distribution of trait values. Three components are needed: (1) a set of data on the distribution of a quantitative character across species, (2) a phylogeny with branch lengths showing the evolutionary history of the species in question, and (3) one or more hypotheses regarding the selective regimes operative on each of the branches in evolutionary time. In this framework, and in keeping with other comparative methods, components We comment here on two issues involving the phylogeny: on polytomies and on the units in which branch lengths are reported. First, for the application of Hansen's model, it is not necessary that the phylogeny be fully resolved: polytomies pose no difficulties in the computations. We stress, however, that in this approach, phylogenies are assumed to accurately reflect the evolutionary history of the system in question. Thus, polytomies are assumed to reflect true radiation and not phylogenetic uncertainty. The effect of phylogenetic uncertainty on comparative hypotheses is an important topic; in this article, we give only a brief indication of how phylogenetic uncertainty can be incorporated into the model-selection procedure. Second, the Hansen model requires phylogenetic branch lengths to be on a common timescale. Because the units of a and j are directly related to time, inter- species evolve according to a Brownian motion process with a speciation event at time s, after which they evolve independently. B, Two species evolving under an Ornstein-Uhlenbeck process. After the speciation event at time s, species 2 entered a new adaptive regime specified by (white v 2 bar), whereas the lineage leading to species 1 has been evolving under the same adaptive regime with optima (black bar) for its entire history. Model-Based Comparative Analysis 687 v 1 pretation of these model parameters will be difficult if the phylogeny is not clocklike. v 0 We can write the quantitative character in vector format, with separate entries for each lineage, [ ] Under the BM model, the two lineages are supposed to have evolved together according to a purely random drift from to (so that for ). Thereafter, they continued to drift independently. The distribution of under the BM model is bivariate normal X(T) with expectation and variance-covariance matrix [ ] s T Because the mean and variance-covariance matrix completely determine the distribution of , one can readily X(T) apply ML methods for the estimation of the parameters and j. v 0 Similarly, the Hansen model gives a multivariate normal distribution for . For illustrative purposes, let us make X(T) the hypothesis that species 2, after its divergence from species 1 at , evolved under a new selective regime, t p s characterized by the optimum trait value X(T) The expected mean trait values at the end of each evolutionary lineage can be computed from equation (A2) in the appendix in the online edition of the American Naturalist: It is important to note that both the expectation and the variance-covariance matrix tend to equation (4), as a r . Hence, the BM model is nested within the class of 0 Hansen models. As before, because is distributed X(T) multivariate normally, it is easy to apply ML methods to estimate the parameters, which include a, , and in v v The only new wrinkle that arises with the Hansen model is due to the fact that a enters into equation (6) in a nonlinear fashion, and hence nonlinear optimization is needed to estimate this parameter. Full mathematical and computational details are given in the appendix; computer code is available at the authors' Web site (http:// www.tiem.utk.edu/∼king). Examples In this section, we provide a guide to the implementation of the Hansen model by means of examples; a technical description of the method can be found in the appendix. For each evolutionary hypothesis, we will obtain a model by assigning an optimum to each branch of the phylogeny. Therefore, we examine our hypotheses and determine how many optima each requires. One optimum will be required for each hypothesized selective regime. Next, we make our hypotheses phylogenetically explicit by "painting" the optima on the appropriate branches of the phylogeny. This association of hypothesized optima to branches is translated into a mathematical model in which, as we have seen above, the expected value of a species' trait is a weighted average where the weights depend on how long each lineage has evolved under each regime. Likelihood maximization then fits the parameter values for each model to the data. Finally, we compare the fit of the alternative models using standard model selection methods including the likelihood ratio test and information criteria (Akaike Information Criterion [AIC] and Schwarz Information