Critical dynamics in biological Boolean networks follows from symmetric response to input genes

Abstract

A recent observation on an extensive collection of biological gene regulatory networks suggests that the regulatory dynamics is tuned to remain close to the order-chaos boundary in the Lyapunov sense [1]. We here investigate, from a mathematical perspective, the structural/functional constraints which give rise to such accumulation around criticality in these systems. While the role of canalizing functions in this respect is well established, we find that critical sensitivity to small input variations also follows from an over-abundance of symmetrical inputs, i.e. regulatory genes invoking identical or complementary responses on their common target. A random network ensemble constructed to have the same distribution of symmetric inputs as in the above collection of biological networks captures the dependence of the sensitivity on mean activity bias, a nontrivial characteristic which the canalizing ensemble fails to fully reproduce.