Understanding the Robustness of Gene Regulation via Derrida Values

It has been discovered that gene regulation is a strongly stochastic, yet still robust process. One well-known indicator for the robustness of gene regulatory networks is the so-called Derrida plot. For a fixed number of perturbations, the Derrida value is the expected Hamming distance after one time step. This value is generally small for networks that exhibit stable behavior and large for networks with more chaotic behavior. Thus far, attainment of these values has depended on time-consuming Monte Carlo simulations.

However, for networks based on multi-state nested canalizing functions – a class of functions found to be prevalent in gene regulation – explicit formulas for the Derrida values can be found; both for general multi-state nested canalizing functions and for Boolean nested canalizing functions of a particular Hamming weight.

Having actual formulas for Derrida values of networks governed by any nested canalizing function precludes simulation, which simplifies the use of Derrida plots for robustness investigations of complex networks. Thus, this research contributes to a better understanding of the robustness of gene regulation.