Effect of Function Perturbation on the Steady-state Distribution of Genetic regulatory Networks

Abstract—The dynamics of a rule-based gene regulatory net-work are determined by the regulatory functions in conjunction with whatever probability distributions are involved in network transitions. In the case of Boolean networks (BNs) and, more gen-erally, probabilistic Boolean networks (PBNs), there has been a sig-nificant amount of investigation into the effect of perturbing gene states, in particular, the design of intervention strategies based on finite- or infinite-horizon control polices. This paper considers the less investigated issue of function perturbations. A single function perturbation affects network dynamics and alters the long-run dis-tribution, whereas any individual gene perturbation has only tran-sient effects and does not change the long-run distribution. We de-rive analytic results for changes in the steady-state distributions of PBNs resulting from modifications to the underlying regulatory rules and apply the derived results to find optimal structural in-terventions to avoid undesirable states. The results are applied to a WNT5A network and a mammalian cell cycle related network, re-spectively, to achieve more favorable steady-state distributions and reduce the risk of getting into aberrant phenotypes. Index Terms—Boolean networks (BNs), genetic regulatory net-works, Markov chains, metastasis, optimal structural intervention, probabilistic Boolean networks (PBNs), steady-state distribution. I