Canalization and Control in Automata Networks: Body Segmentation in Drosophila melanogaster

Abstract

We present schema redescription as a methodology to characterize canalization in automata networks used to model biochemical regulation and signalling. In our formulation, canalization becomes synonymous with redundancy present in the logic of automata. This results in straightforward measures to quantify canalization in an automaton (micro-level), which is in turn integrated into a highly scalable framework to characterize the collective dynamics of large-scale automata networks (macro-level). This way, our approach provides a method to link micro- to macro-level dynamics – a crux of complexity. Several new results ensue from this methodology: uncovering of dynamical modularity (modules in the dynamics rather than in the structure of networks), identification of minimal conditions and critical nodes to control the convergence to attractors, simulation of dynamical behaviour from incomplete information about initial conditions, and measures of macro-level canalization and robustness to perturbations. We exemplify our methodology with a well-known model of the intra- and inter cellular genetic regulation of body segmentation in Drosophila melanogaster. We use this model to show that our analysis does not contradict any previous findings. But we also obtain new knowledge about its behaviour: a better understanding of the size of its wild-type attractor basin (larger than previously thought), the identification of novel minimal conditions and critical nodes that control wild-type behaviour, and the resilience of these to stochastic interventions. Our methodology is applicable to any complex network that can be modelled using automata, but we focus on biochemical regulation and signalling, towards a better understanding of the (decentralized) control that orchestrates cellular activity – with the ultimate goal of explaining how do cells and tissues ‘compute’.

The Influence of Canalization on the Robustness of Boolean Networks

“…As in physics, mathematical modeling and analysis is an enabling technology for the drive to connect structural properties of dynamic biological networks to their dynamics. The study of robustness, as instantiated in the concept of canalization, is no exception, and many such studies have been published, see, e.g., [10,17,31,29].…”

Robustness and Evolvability of the Human Signaling Network

“…One might consider that the concept of ‘robust neighbor link’ is similar to that of ‘redundant link’ in the context of canalizing function, which is a function of multiple input variables with the property that one of its inputs can solely determine the output value regardless of other inputs [20], [42], [43] (see Fig. S12A).…”

A feedback loop of conditionally stable circuits drives the cell cycle from checkpoint to checkpoint

“…The expanded network framework is part of a broader effort to characterize and represent as a network the causal relationships between variables of a dynamical system. Related concepts include signed interaction hypergraphs 21 and dynamics canalization maps 48 . The logic domain of influence of a node state is conceptually similar to the threevalued (0, 1, unknown) logical steady state that results from fixing a node state 21 and to the dynamical modules of dynamics canalization maps, which represent the states inexorably stabilized by an input configuration 48 .…”

“…Related concepts include signed interaction hypergraphs 21 and dynamics canalization maps 48 . The logic domain of influence of a node state is conceptually similar to the threevalued (0, 1, unknown) logical steady state that results from fixing a node state 21 and to the dynamical modules of dynamics canalization maps, which represent the states inexorably stabilized by an input configuration 48 . The concepts of expanded network and stable motif have been generalized and implemented in multi-level discrete systems and continuous-variable systems described by ordinary differential equations 32,44 .…”

“…The average in-degree of the expanded network is less than two, markedly smaller than the average in-degree of the original Phase Switch Oscillator network, which is 3.5. This illustrates that regulators need to cooperate to induce state changes in target nodes 48,49 . The whole expanded network is an oscillating motif: it is strongly connected, it is composite-closed, it contains the complementary of each virtual node, and it does not have any stable motifs.…”