Counting Subnetworks Under Gene Duplication in Genetic Regulatory Networks

Gene duplication is a fundamental evolutionary mechanism that contributes to biological complexity and diversity (Fortna et al. 2004). Traditionally, research has focused on the duplication of gene sequences (Zhang 1914). However, evidence suggests that the duplication of regulatory elements may also play a significant role in the evolution of genomic functions (Teichmann and Babu 2004; Hallin and Landry 2019). In this work the evolution of regulatory relationships belonging to gene-specific-substructures in a GRN are modeled. In the model, a network grows from an initial configuration by repeatedly choosing a random gene to duplicate. The likelihood that the regulatory relationships associated with the selected gene are retained through duplication is determined by a vector of probabilities. That is to say that each gene family has its own probability of retaining regulatory relationships. Occurrences of gene-family-specific substructures are counted under the gene duplication model. In this work gene-family-specific substructures are referred to as subnetwork motifs. These subnetwork motifs are motivated by network motifs which are patterns of interconnections that recur more often in a specialized network than in a random network (Milo et al. 2002). Subnetwork motifs differ from network motifs in the way that subnetwork motifs are instances of gene-family-specific substructures while network motifs are isomorphic substructures. These subnetwork motifs are counted under Full and Partial Duplication, which differ in the way in which regulation relationships are inherited. Full duplication occurs when all regulatory links are inherited at each duplication step, and Partial Duplication occurs when regulation inheritance varies at each duplication step. Note that Full Duplication is just a special case of Partial Duplication. Moments for the number of occurrences of subnetwork motifs are determined in each model. In the end, the results presented offer a method for discovering gene-family-specific substructures that are significant in a GRN under gene duplication.