Robustness leads close to the edge of chaos in coupled map networks: toward the understanding of biological networks

Abstract

Dynamics in biological networks are, in general, robust against several perturbations. We investigate a coupled map network as a model motivated by gene regulatory networks and design systems that are robust against phenotypic perturbations (perturbations in dynamics), as well as systems that are robust against mutation (perturbations in network structure). To achieve such a design, we apply a multicanonical Monte Carlo method. Analysis based on the maximum Lyapunov exponent and parameter sensitivity shows that systems with marginal stability, which are regarded as systems at the edge of chaos, emerge when robustness against network perturbations is required. This emergence of the edge of chaos is a self-organization phenomenon and does not need a fine tuning of parameters.

Emergence of cooperative bistability and robustness of gene regulatory networks

“…There has been some numerical evidence of a correlation between mutational robustness and noise robustness [8,19,36]. However, in the present study, while the robustness against noise was a direct consequence of bistability, the relationship between bistability and mutational robustness was not clear.…”

“…The importance of noise, which originated from the finiteness of the number of molecules such as transcription factors, in the gene regulation systems has been emphasized by previous studies [8,11,36,61,63,64,80]. We found that, although robustness against noise was not taken into account in the definition of fitness, robustness against both the input noise and the internal noise was acquired automatically as a byproduct of bistability.…”

“…By regarding quantities other than the physical energy as energy, this method can be applied to a variety of problems. It is particularly suitable for counting rare states or for estimating the appearance probability of rare events [33][34][35][36][37][38]. Ref.…”

“…Ref. [36] used this method to investigate the mutational robustness of biologically-inspired networks.…”

Asymptotic transversal stability for synchronized attractors in a metapopulation model

“…Indeed, each patch corresponds to a vertex of a graph; the local reproduction function f operates at each vertex, and there is an interaction along the edges of the graph due to the dispersal mechanisms. Although we are restricted to a metapopulation model, it is worth noticing that CMNs appear in several areas as electronics, neurology, chemistry, cryptography and others [36][37][38][39][40]. In particular, CMNs where the network dynamic and the coupling are time varying have an intrinsic interest [41].…”