Convergence Analysis and Network Properties of Wagner's Artificial Gene Regulatory Network Model

The gene regulatory network (GRN) is essential to life, as it governs all levels of gene products that enable cell survival and numerous cellular functions. Wagner's GRN model, which has mathematical roots origin from the Ising model and neural networks, is a powerful computation tool that helps integrate network thinking into biology, and motivated a new research theme focusing on the evolution of genetic networks. However, except the formal mathematical
foundation described in [1], few papers have focused on providing further mathematical analysis of the model. Moreover, network characteristics of Wagner's GRN model when varying key parameters are unclear. Therefore, in this paper, I present a convergence analysis of Wagner's GRN model by using the Markov chain theory. I show mathematically that if we consider the evolution process as an optimisation process, then the probability of nding the optimal conguration (a certain target phenotype) converges to probability one. In addition, I investigate network characteristics such as stability, robustness and path length in initial populations. I find that generally small networks with a sparse connectivity have a higher initial stability.The robustness is also observed to be higher in initial stable networks with a low network connectivity. These results are partly explained by the pattern, as shown in this paper, that small networks with a sparse connectivity generally have a shorter path length and, therefore, they are not only able to quickly reach equilibrium phenotypic states but also more likely to resist genetic perturbations.