Robustness in Genetic Threshold Boolean Random Regulatory Networks: A Mathematical Approach

On one hand, different indices of complexity have been introduced in biological networks: the number of their components, their connectance (or connectivity) or the number of the strong connected components of their interaction graph. On the other hand, indices of robustness for a biological network characterize its ability to recover from dynamical or parametric disturbance and are linked to the spectrum of the operator defining its dynamics. Here complexity will be quantified by the evolutionary entropy, which describes the way the asymptotic presence distribution of the corresponding dynamical system is spread over the state space and robustness will be characterized by the stability rate at which the system returns to this equilibrium distribution after a perturbation. The paper shows mathematical relationships between entropy and stability rate, in the general framework of Markov chains and in the specific case of Markov chains related to the genetic threshold Boolean random regulatory networks (getBren).