Abstract A gene family is a set of evolutionarily related genes formed by duplication. Genes within a gene family can perform a range of different but possibly overlapping functions. The process of duplication produces a gene that has identical functions to the gene it was duplicated from with subsequent divergence over time. In this paper, we explore different models for the ongoing evolution of a gene family. First, we consider a detailed model with multi-dimensional state-space which consists of binary matrices where rows of a matrix correspond to genes, columns correspond to functions, and the ijth entries record whether or not gene i performs function j. The large state space of this model makes it unsuitable for numerical analysis, but by considering the behavior of this detailed model we can test the suitability of two alternative models with more tractable state-spaces. Next, we consider a quasi-birth-and-death process (QBD) with two-dimensional states (n,m). The state (n, m) records the number of genes in the family, and the number of so called redundant genes (which are permitted to be lost). We contrast this to a level-dependent QBD with three-dimensional states (n, m, k) that record additional information which affects the transition rates. We show that the model with two-dimensional states (n, m) is insufficient for meaningful analysis, while the model with the three-dimensional states (n, m, k) is able to capture the qualitative behavior of the detailed model. We illustrate the fit between the level-dependent QBD and the original, detailed model, with numerical examples.

中文翻译：

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用于基因重复家族进化的水平依赖 QBD 模型

摘要 基因家族是一组通过重复形成的进化相关基因。基因家族中的基因可以执行一系列不同但可能重叠的功能。复制过程产生的基因与复制它的基因具有相同的功能，随后随着时间的推移发生分歧。在本文中，我们探索了基因家族持续进化的不同模型。首先，我们考虑一个具有多维状态空间的详细模型，该模型由二进制矩阵组成，其中矩阵的行对应于基因，列对应于函数，第 ij 个条目记录基因 i 是否执行函数 j。该模型的大状态空间使其不适合数值分析，但是通过考虑这个详细模型的行为，我们可以测试两个具有更易处理状态空间的替代模型的适用性。接下来，我们考虑具有二维状态 (n,m) 的准生死过程 (QBD)。状态 (n, m) 记录了家族中基因的数量，以及所谓的冗余基因（允许丢失）的数量。我们将其与具有三维状态 (n, m, k) 的水平相关 QBD 进行对比，该 QBD 记录了影响跃迁率的附加信息。我们表明具有二维状态 (n, m) 的模型不足以进行有意义的分析，而具有三维状态 (n, m, k) 的模型能够捕获详细模型的定性行为。我们说明了水平相关的 QBD 与原始的详细模型之间的拟合，