Counting Subnetworks Under Gene Duplication in Genetic Regulatory Networks

by Ashley Scruse, Jonathan Arnold, and Robert Robinson

Gene families help us understand how gene regulation evolves. While sequence evolution is well-studied, regulatory evolution is not. Members of gene families form genetic regulatory networks (GRNs) where gene A regulates gene B. We develop a model counting regulatory subnetwork motifs within gene families as they evolve through duplication. The result is a count |M(n)| of how these motifs evolve to a stage with n total genes. The key parameter is the probability that a regulatory relation is preserved when a gene is duplicated. For Full Duplication, the mean and variance of motif counts can be computed exactly, enabling significance tests to identify targets for directed evolution. Partial and Mixed Duplication models are also presented.

This graphical abstract illustrates the mathematical framework for counting subnetwork motifs in genetic regulatory networks (GRNs) under gene duplication. The left panel depicts the gene duplication process: before duplication, gene A regulates gene B; after duplication of B, the duplicate gene B' may inherit the regulatory relationship from A with probability π. The right panel presents the key analytical results for the Full Duplication model (π = 1), including closed-form expressions for the expected motif count E(|M|) and the second moment E(|M|²). The bottom section shows the statistical framework for identifying significant motifs, with formulas for variance and Z-score calculation, along with a workflow for applying these methods to identify targets for directed evolution. The paper also derives analogous results for Partial Duplication (0 ≤ π ≤ 1), where the inheritance probability vector π = (π₁, ..., πₖ) allows each gene family to have its own probability of inheriting regulatory relationships, and discusses the Mixed Duplication model, where with probability 1−q all regulatory links are retained and with probability q each link is retained independently with probability p. See the full article for these additional duplication models.