Robustness of biomolecular networks suggests functional modules far from the edge of chaos

A common feature of complex systems is their ability to balance the flexibility needed to adapt to their environment with the rigidity required for robust function. It has been conjectured that living systems accomplish this by existing at the "edge of chaos", i.e., the critical boundary between ordered and disordered dynamics. Simple toy models of gene regulatory networks lend support to this idea, and mathematical tools developed for these toy models yield similar results when applied to experimentally-supported models of specific cellular regulatory mechanisms (functional modules). Here, however, we demonstrate that a deeper inspection of 72 experimentally-supported discrete dynamical models of functional modules reveals previously unobserved order in these systems on long time scales, suggesting greater rigidity in these systems than was previously conjectured. Our analysis relies on new measures that quantify the tendency of perturbations to spread through a discrete dynamical system. A benefit of our new approach is that it accounts for how system trajectories are mapped to phenotypes in practice. Because these measures are computationally expensive to estimate, existing tools were insufficient for the ensemble of models considered here. To simulate the tens of millions of trajectories required for convergence, we developed a multipurpose CUDA-based simulation tool, which we have made available as the open-source Python library cubewalkers. We find that in experimentally-supported models of biomolecular functional modules, perturbation propagation is more transitory than previously thought, and that even in cases where large perturbation cascades persist, their phenotypic effects are often minimal. Moreover, by examining the impact of update scheme on experimentally-supported models, we find evidence that stochasticity and desynchronization can lead to increased recovery from regulatory perturbation cascades in functional modules and uncover previously unreported population-level robustness to even timing perturbations in these systems. We identify specific biological mechanisms underlying these dynamical behaviors and highlight them in experimentally-supported regulatory networks from the systems biology literature. Based on novel measures and simulations, our results suggest that-contrary to current theory-functional modules of biological systems are ordered and far from the edge of chaos.