General method to find the attractors of discrete dynamic models of biological systems

Gan, Xiao; Albert, Réka

doi:10.1103/physreve.97.042308

Cited by 25 publications

(29 citation statements)

References 64 publications

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“…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 . When considered generally, the expanded network encodes causal links between regions of state-space.…”

Section: Discussionmentioning

confidence: 99%

“…Much of our analyses rely on the construction and properties of an expanded network 9,31,32,44,45 . The expanded network of a Boolean system encodes the causal relationships between node states reflected in the regulatory functions.…”

Section: ‫ݔ‬mentioning

confidence: 99%

“…The sequential lock-in of stable motifs restricts the system's dynamics until it reaches an attractor. A counterpart to stable motifs, oscillating motifs, identify node subsets that cannot achieve a stable state, and thus give rise to oscillating or complex attractors 31,32 .…”

Section: Introductionmentioning

confidence: 99%

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

Deritei

Rozum

Regan

et al. 2019

Preprint

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We perform logic-based network analysis on a model of the mammalian cell cycle. This model is composed of a Restriction Switch driving cell cycle commitment and a Phase Switch driving mitotic entry and exit. By generalizing the concept of stable motif, i.e., a self-sustaining positive feedback loop that maintains an associated state, we introduce the concept of conditionally stable motif, the stability of which is contingent on external conditions. We show that the stable motifs of the Phase Switch are contingent on the state of three nodes through which it receives input from the rest of the network. Biologically, these conditions correspond to cell cycle checkpoints. Holding these nodes locked (akin to a checkpoint-free cell) transforms the Phase Switch into an autonomous oscillator that robustly toggles through the cell cycle phases G1, G2 and mitosis. The conditionally stable motifs of the Phase Switch Oscillator are organized into an ordered sequence, such that they serially stabilize each other but also cause their own destabilization. Along the way they channel the dynamics of the module onto a narrow path in state space, lending robustness to the oscillation. Self-destabilizing conditionally stable motifs suggest a general negative feedback mechanism leading to sustained oscillations.

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Section: Discussionmentioning

confidence: 99%

Section: ‫ݔ‬mentioning

confidence: 99%

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

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Rozum

Regan

et al. 2019

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“…To the best of our knowledge, none of these methods have been implemented for general multistate systems. Interestingly, the concept of stable motifs has recently been generalized for multistate networks [13].…”

Section: Comparison To the Feedback Vertex Set (Fvs) Control Methodsmentioning

confidence: 99%

Control of Intracellular Molecular Networks Using Algebraic Methods

Vieira

Laubenbacher²,

Murrugarra

2019

Preprint

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Many problems in biology and medicine have a control component. Often, the goal might be to modify intracellular networks, such as gene regulatory networks or signaling networks, in order for cells to achieve a certain phenotype, such as happens in cancer. If the network is represented by a mathematical model for which mathematical control approaches are available, such as systems of ordinary differential equations, then this problem might be solved systematically. Such approaches are available for some other model types, such as Boolean networks, where structurebased approaches have been developed, as well as stable motif techniques.

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“…For example, the composite node x > 0 ∧ y > 0 is true 106 only when x and y are positive, and there are directed edges from x > 0 and y > 0 to 107 this composite node. In deterministic finite-level systems, it is possible to choose a finite 108 number of statements that fully characterize the state space [33], but in general, the 109 nodes of an expanded network embody partial information about the system. For a 110 given choice of virtual and composite nodes, the expanded network is unique, however, a 111 different choice of virtual nodes for the same system can lead to different expanded 112 networks.…”

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confidence: 99%

Identifying (un)controllable dynamical behavior in complex networks

Rozum

Albert

2017

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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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General method to find the attractors of discrete dynamic models of biological systems

Cited by 25 publications

References 64 publications

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

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

Control of Intracellular Molecular Networks Using Algebraic Methods

Identifying (un)controllable dynamical behavior in complex networks

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