Identifying (un)controllable dynamical behavior in complex networks

Abstract

Detailed features of complex systems (e.g., precise variable values) are sometimes of secondary importance when considering system interventions. This is especially true in systems that are subject to a large degree of uncertainty. Instead, qualitative features (e.g., whether a set of variables is "high" or "low") often determine realworld control strategies, particularly in biological systems. This has inspired a great deal of work in qualitative modeling. Nevertheless, quantitative dynamical models can be required to accurately predict qualitative behavior. In these cases, a method for determining which qualitative features are robust to external controls and internal perturbations is desired. We present a new method to meet this need that generalizes a concept known as stable motifs from the analysis of Boolean networks to a large class of dynamical systems. This new method uses an auxiliary expanded network to help identify self-sustaining behavior and inform system control strategies. We demonstrate how to implement this approach on an example and illustrate its utility for two published biological network models: the segment polarity gene network of Drosophila melanogaster, and the T-cell signaling network. Expanded NetworkA key goal in the study of complex dynamical systems is to extract important qualitative information from models of varying specificity (e.g., (1, 2)). This has been approached via the construction of qualitative models (e.g., discrete models (3-6)) and also by analytic techniques (e.g., enumerating qualitatively distinct regions of parameter space for generalized mass-action systems (7)). In this work, we describe a generalization of a technique for qualitative models that is applicable in any dynamical framework.Many of the qualitative features one might wish to identify in the repertoire of a dynamical system arise from feedback loops in the associated regulatory network. Indeed, recent research has highlighted the importance of feedback loops in the control of dynamical systems (8-12). For example, control of any feedback vertex set (i.e., a set of variables whose removal eliminates all feedback) and of any source variables is sufficient to drive an ODE-described system into any of its natural attractors, provided these ODEs meet some relatively permissive criteria relating to continuity and boundedness (8,10,13).Positive feedback loops in particular are associated with important qualitative information about a system, such as the presence of multistability (14-17). Multistability has been of particular interest in biomolecular systems because it is necessary for cell-fate branching and decision making (4,(18)(19)(20)(21). Two approaches to identifying the effects of positive feedback loops are especially relevant here. The first of these is the methods put forth by Angeli and Sontag for studying monotone input-output systems (MIOS) (22). Their approach identifies steady states in sign-consistent monotone systems (those lacking negative cycles or incoherent feed-forward loops) and de...