Correspondence of Trap Spaces in Different Models of Bioregulatory Networks

2018

PLoS Comput Biol

We present a technique applicable in any dynamical framework to identify control-robust subsets of an interacting system. These robust subsystems, which we call stable modules, are characterized by constraints on the variables that make up the subsystem. They are robust in the sense that if the defining constraints are satisfied at a given time, they remain satisfied for all later times, regardless of what happens in the rest of the system, and can only be broken if the constrained variables are externally manipulated. We identify stable modules as graph structures in an expanded network, which represents causal links between variable constraints. A stable module represents a system “decision point”, or trap subspace. Using the expanded network, small stable modules can be composed sequentially to form larger stable modules that describe dynamics on the system level. Collections of large, mutually exclusive stable modules describe the system’s repertoire of long-term behaviors. We implement this technique in a broad class of dynamical systems and illustrate its practical utility via examples and algorithmic analysis of two published biological network models. In the segment polarity gene network of Drosophila melanogaster, we obtain a state-space visualization that reproduces by novel means the four possible cell fates and predicts the outcome of cell transplant experiments. In the T-cell signaling network, we identify six signaling elements that determine the high-signal response and show that control of an element connected to them cannot disrupt this response.

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Section: Resultssupporting

confidence: 89%

“…An analogous concept in ODE models is that of positive invariant sets (also called “trap spaces”) [16, 17]. These are regions of state space that system trajectories may enter but not exit.…”

Section: Introductionmentioning

confidence: 99%

Identifying (un)controllable dynamical behavior in complex networks

2018

PLoS Comput Biol

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“…418 In this example, the stable motif we have identified coincides with a global steady 419 state of the system. This observation is in agreement with [17], in which this system is 420 analyzed by application of theorems regarding the conservation of certain positive 421 invariant sets when a system is described by both a Boolean and an ODE model with 422…”

supporting

confidence: 77%

“…Several methods 26 for identifying the causal structure of state-space in discrete models have been proposed, 27 including hierarchical transition graphs [14] and prime implicant graphs [15]. 28An analogous concept in ODE models is that of positive invariant sets (also called 29 "trap spaces") [16,17]. These are regions of state space that system trajectories may 30 enter but not exit.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Identifying (un)controllable dynamical behavior in complex networks

2017

Preprint

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...

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Self-sustaining positive feedback loops in discrete and continuous systems

Journal of Theoretical Biology

2018