The Influence of Network Topology on the Dynamics of Gene Regulatory Networks

Abstract

Over the last decades, studies have shown that the gene regulation of a wide range of organisms can be described with networks [Jeong H., Tombor B., Albert R., Ottval Z.N., Barab´asi A.L., 2000] [Jeong H., Mason S.P., Barab´asi A.L., Oltvai Z.N., 2001]. The interactions between the mRNA strands and proteins form links in the network, while these molecules form the nodes of the network. Numerical models for the dynamics of such a network, through solving a minimally nonlinear stochastic differential equation show nontrivial dynamics. This behavior is caused by non linearity introduced by the positivity condition, this is due to the fact that molecular concentrations cannot cannot be negative. Whether these dynamics are stable, oscillatory or chaotic seems to depend on the average connectivity of the network. There appears to be a region of networks that can transition from chaotic behavior to stable behavior. For a simple differential equation this would not be a region, but just one value. This region, referred to as the “Edge-of-Chaos” region, is therefore rather interesting. In this report we analyse the Lyapunov spectrum in order to quantify these dynamics. This is a way to measure the stability of the solution of out differential equation. We look at how different parameters in the minimally nonlinear differential equation and the generation of the network affect these dynamics. The investigated parameters include different edge weights, different noise levels, different equilibrium values and different network types. After studying the dynamics of a single network, we study the dynamics of populations of networks using an evolutionary algorithm and a co-evolutionary algorithm. We found that an edge of chaos region exists universally, that is for all types of networks we looked at, for all feasible noise levels, for all equilibrium values and for all network sizes. Whether or not a network lies in this region is determined by how much self interaction its nodes have compared to the strength of the interactions between the nodes. Additionally, the size of the network determines how well this “Edge-of-Chaos” plateau forms. Larger networks in this region behave more stable than smaller ones. We also came to the conclusion that both the evolutionary and co-evolutionary algorithm, using the Kullback-Leibler divergence in accordance with a probability distribution defined by a Hamiltonian to compare the fitness of individuals, do not yield the expected results. That is, they do not show any correlation between the connectivity of a network and the development of the population over time.