Effective connectivity determines the critical dynamics of biochemical networks

Abstract

Living systems comprise interacting biochemical components in very large networks. Given their high connectivity, biochemical dynamics are surprisingly not chaotic but quite robust to perturbations—a feature C.H. Waddington named canalization. Because organisms are also flexible enough to evolve, they arguably operate in a critical dynamical regime between order and chaos. The established theory of criticality is based on networks of interacting automata where Boolean truth values model presence/absence of biochemical molecules. The dynamical regime is predicted using network connectivity and node bias (to be on/off) as tuning parameters. Revising this to account for canalization leads to a significant improvement in dynamical regime prediction. The revision is based on effective connectivity , a measure of dynamical redundancy that buffers automata response to some inputs. In both random and experimentally validated systems biology networks, reducing effective connectivity makes living systems operate in stable or critical regimes even though the structure of their biochemical interaction networks predicts them to be chaotic. This suggests that dynamical redundancy may be naturally selected to maintain living systems near critical dynamics, providing both robustness and evolvability. By identifying how dynamics propagates preferably via effective pathways, our approach helps to identify precise ways to design and control network models of biochemical regulation and signalling.

The nonlinearity of regulation in biological networks

“…This paper introduces the concept of regulatory nonlinearity as a new measure of characterization for Boolean networks. There are several other related characterizations of Boolean networks such as canalization [27], effective connectivity [10], symmetry [28] and controllability [29]. It has been previously reported that the levels of canalization (a measure of the extent to which fewer inputs influence the outputs of a Boolean function) and the mean effective connectivity (a measure of collective canalization) are high in biological networks [2, 10].…”

The Nonlinearity of Regulation in Biological Networks

“…This paper introduces the concept of regulatory nonlinearity as a new measure of characterization for Boolean networks. There are several other related characterizations of Boolean networks such as canalization [27], effective connectivity [10], symmetry [28] and controllability [29]. It has been previously reported that the levels of canalization (a measure of the extent to which fewer inputs influence the outputs of a Boolean function) and the mean effective connectivity (a measure of collective canalization) are high in biological networks [2,10].…”

“…There are several other related characterizations of Boolean networks such as canalization [27], effective connectivity [10], symmetry [28] and controllability [29]. It has been previously reported that the levels of canalization (a measure of the extent to which fewer inputs influence the outputs of a Boolean function) and the mean effective connectivity (a measure of collective canalization) are high in biological networks [2,10]. It has also been found that biological networks need few inputs to reprogram [30] and are relatively easier to control [3].…”

“…Looking at such questions from an even broader perspective it becomes evident that they are only instances of the ultimate puzzle of complex systems, namely what connects the structure and the function of a system. Even though recent work has attempted to answer this question from the perspective of the rules or dynamical laws that govern the system [39,10,38], more tools are needed [40]. In that regard, our framework of regulatory nonlinearity could be a novel addition to this burgeoning toolkit in that it could also be applied to continuous models of biological networks such as those based on differential equations.…”

“…Previous studies have found that certain characteristic features of the biological Boolean models, such as the mean in-degree, output bias, sensitivity and canalization, tend to assume an optimal range of values that support optimal function [9,10]. Here we study a new but generic feature of complex systems, namely, nonlinearity.…”