Level-dependent QBD models for the evolution of a family of gene duplicates

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 n=1,2,… of genes in the family, and the number m=0,1,…,n 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 k=1,…,K 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.