Modeling Stabilizing Selection: Expanding the Ornstein-Uhlenbeck Model of Adaptive Evolution

Abstract

Comparative methods used to study patterns of evolutionary change in a continuous trait on a phylogeny range from Brownian motion processes to models where the trait is assumed to evolve according to an Ornstein-Uhlenbeck (OU) process. Although these models have proved useful in a variety of contexts, they still do not cover all the scenarios biologists want to examine. For models based on the OU process, model complexity is restricted in current implementations by assuming that the rate of stochastic motion and the strength of selection do not vary among selective regimes. Here, we expand the OU model of adaptive evolution to include models that variously relax the assumption of a constant rate and strength of selection. In its most general form, the methods described here can assign each selective regime a separate trait optimum, a rate of stochastic motion parameter, and a parameter for the strength of selection. We use simulations to show that our models can detect meaningful differences in the evolutionary process, especially with larger sample sizes. We also illustrate our method using an empirical example of genome size evolution within a large flowering plant clade. K E Y W O R D S : Brownian motion, comparative method, continuous characters Hansen model, Ornstein-Uhlenbeck.Single-rate Brownian motion works reasonably well as a model for evolution of traits. It models drift, drift-mutation balance, and even stabilizing selection toward a moving optimum (Hansen and Martins 1996). However, a single parameter model can certainly not explain the evolution of traits across all life. There have been extensions to the model, such as a single Ornstein-Uhlenbeck (OU) process that has a constant pull toward an optimum value, a multiple mean OU process with different possible means for different groups (Hansen 1997;Butler and King 2004), and multiple rate Brownian motion processes allowing different rates of evolution on different branches (O'Meara et al. 2006;Thomas et al. 2006). These models, while useful, still do not cover all the scenarios biologists want to examine. For example, existing models with a value toward which species are being pulled all have a fixed strength of pull over the entire history of the group. It is possible to allow the rate of stochastic motion to vary, or the value of the attractor to vary, but not for both to vary. Such restrictions on model complexity may make sense when phylogenies are limited to a few dozen taxa. However, in an era where phylogenies can have over 55,000 taxa (Smith et al. 2011), we may be so bold as to attempt to fit models that vary both rates and means of the evolutionary process. This article develops and implements such models. Hansen (1997) described a model where quantitative characters are assumed to evolve according to an OU process. The Hansen model, as it has become known, expresses the amount of change in a quantitative trait along each branch in a phylogeny and is given by the stochastic differential equation: Generalizing the Hansen Model(1) Equation ...

How predation shaped fish: the impact of fin spines on body form evolution across teleosts

“…As our preliminary analyses indicated that the estimates of u for spiny and spineless lineages were stabilized by assuming u at the root was distributed according to the stationary distribution of the OU process, we set root.station ¼ TRUE for all analyses. We checked the results of the OUWIE analyses to ensure that the eigenvalues of the Hessian were positive, as this is an indicator that the parameters were reliably estimated [58]. When a negative eigenvalue was found, we removed the results for that model and tree combination from the final dataset.…”

“…We test our predictions using generalized Ornstein -Uhlenbeck (OU) models [57][58][59], which allow the presence and absence of spines to influence the primary optima of body depth or width (u), the stochastically constant rate of depth or width evolution (s 2 ), and/or the strength of pull towards the primary optima (a). u is the trait value reached by an 'infinite number of populations identical to the common ancestor evolving independently' [57, p. 1342] with spines in a fixed state, and thus quite different from the realized trait optimum in real populations.…”

A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes

“…OU processes have been widely studied and used in diverse areas of applied mathematics like biological modeling [23], [24], mathematical finance [25], [26] and statistical physics [27]. Using known results for the statistical distribution of the first time at which an OU process is above certain threshold [28], [27], [29], we propose a closed-form expression to estimate the error probability of the SC-LDPC code as a function of M , L and the gap to the ensemble's BP threshold.…”

Comparative tests of the role of dewlap size in Anolis lizard speciation

“…We fitted the new multi-c model in which s 2 t is constant but c varies between states. Finally, we fitted a multiple-optimum OU model (multi-OU) implemented in OUwie [42] that allows the optimum dewlap area to vary between geographical states, whereas s 2 and the strength of attraction to the optimum are constant. All six models account for intraspecific variability using species-specific standard errors.…”