Department of Systematic and Evolutionary Botany (ISEB), University of Zurich, Zurich, Switzerland.
Department of Botany and Zoology, University of Stellenbosch, Stellenbosch, South Africa.
Syst Biol. 2018 Mar 1;67(2):304-319. doi: 10.1093/sysbio/syx075.
The evolution of quantitative characters over long timescales is often studied using stochastic diffusion models. The current toolbox available to students of macroevolution is however limited to two main models: Brownian motion and the Ornstein-Uhlenbeck process, plus some of their extensions. Here, we present a very general model for inferring the dynamics of quantitative characters evolving under both random diffusion and deterministic forces of any possible shape and strength, which can accommodate interesting evolutionary scenarios like directional trends, disruptive selection, or macroevolutionary landscapes with multiple peaks. This model is based on a general partial differential equation widely used in statistical mechanics: the Fokker-Planck equation, also known in population genetics as the Kolmogorov forward equation. We thus call the model FPK, for Fokker-Planck-Kolmogorov. We first explain how this model can be used to describe macroevolutionary landscapes over which quantitative traits evolve and, more importantly, we detail how it can be fitted to empirical data. Using simulations, we show that the model has good behavior both in terms of discrimination from alternative models and in terms of parameter inference. We provide R code to fit the model to empirical data using either maximum-likelihood or Bayesian estimation, and illustrate the use of this code with two empirical examples of body mass evolution in mammals. FPK should greatly expand the set of macroevolutionary scenarios that can be studied since it opens the way to estimating macroevolutionary landscapes of any conceivable shape. [Adaptation; bounds; diffusion; FPK model; macroevolution; maximum-likelihood estimation; MCMC methods; phylogenetic comparative data; selection.].
长期以来，定量性状的演化通常使用随机扩散模型进行研究。然而，宏观进化学生目前可用的工具仅限于两种主要模型：布朗运动和奥恩斯坦-乌伦贝克过程，以及它们的一些扩展。在这里，我们提出了一种非常通用的模型，可以推断在随机扩散和任何可能形状和强度的确定性力下演变的定量特征的动态，该模型可以适应有趣的进化情景，如定向趋势、破坏性选择或具有多个峰值的宏观进化景观。该模型基于在统计力学中广泛使用的一般偏微分方程：福克-普朗克方程，在群体遗传学中也称为柯尔莫哥洛夫前进方程。因此，我们将该模型称为 FPK，即福克-普朗克-柯尔莫哥洛夫。我们首先解释了如何使用该模型来描述定量性状进化的宏观进化景观，更重要的是，我们详细说明了如何将其拟合到经验数据。通过模拟，我们表明该模型在区分替代模型和参数推断方面都具有良好的性能。我们提供了 R 代码，可使用最大似然或贝叶斯估计来拟合模型到经验数据，并使用哺乳动物体质量演化的两个经验示例来说明该代码的使用。FPK 应该会极大地扩展可以研究的宏观进化情景的范围，因为它为估计任何可想象形状的宏观进化景观开辟了道路。[适应；边界；扩散；FPK 模型；宏观进化；最大似然估计；MCMC 方法；系统发育比较数据；选择。]
