Phylogenetics uses alignments of molecular sequence data to learn about evolutionary trees relating species. Along branches, sequence evolution is modelled using a continuous-time Markov process characterized by an instantaneous rate matrix. Early models assumed the same rate matrix governed substitutions at all sites of the alignment, ignoring variation in evolutionary pressures. Substantial improvements in phylogenetic inference and model fit were achieved by augmenting these models with multiplicative random effects that describe the result of variation in selective constraints and allow sites to evolve at different rates which linearly scale a baseline rate matrix. Motivated by this pioneering work, we consider an extension using a quadratic, rather than linear, transformation. The resulting models allow for variation in the selective coefficients of different types of point mutation at a site in addition to variation in selective constraints. We derive properties of the extended models. For certain non-stationary processes, the extension gives a model that allows variation in sequence composition, both across sites and taxa. We adopt a Bayesian approach, describe an MCMC algorithm for posterior inference and provide software. Our quadratic models are applied to alignments spanning the tree of life and compared with site-homogeneous and linear models.

中文翻译：

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在统计系统发育学中概括跨位点的速率异质性

系统发生学使用分子序列数据的比对来了解与物种相关的进化树。沿着分支，使用以瞬时速率矩阵为特征的连续时间马尔可夫过程对序列演化进行建模。早期模型假设相同的速率矩阵控制对齐的所有位点的替换，忽略进化压力的变化。系统发育推断和模型拟合的实质性改进是通过用乘法随机效应增强这些模型来实现的，这些模型描述了选择性约束变化的结果，并允许位点以不同的速率进化，线性缩放基线速率矩阵。受这项开创性工作的启发，我们考虑使用二次而非线性变换进行扩展。所得模型除了选择性约束的变化外，还允许一个位点上不同类型点突变的选择性系数发生变化。我们推导出扩展模型的属性。对于某些非平稳过程，扩展提供了一个模型，该模型允许跨站点和分类群的序列组成发生变化。我们采用贝叶斯方法，描述用于后验推理的 MCMC 算法并提供软件。我们的二次模型应用于跨越生命树的比对，并与站点同质和线性模型进行比较。跨站点和分类群。我们采用贝叶斯方法，描述用于后验推理的 MCMC 算法并提供软件。我们的二次模型应用于跨越生命树的比对，并与站点同质和线性模型进行比较。跨站点和分类群。我们采用贝叶斯方法，描述用于后验推理的 MCMC 算法并提供软件。我们的二次模型应用于跨越生命树的比对，并与站点同质和线性模型进行比较。