Global dynamics for steep nonlinearities in two dimensions

Gedeon, Tomáš; Harker, Shaun; Kokubu, Hiroshi; Mischaikow, Konstantin; Oka, Hiroe

doi:10.1016/j.physd.2016.08.006

Cited by 29 publications

(43 citation statements)

References 19 publications

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“…Nevertheless, the main contribution tof (x) comes from the values of f s whose argument s is close to x. Due to the contributions of neighbouring terms, any sharp features of f s are "mollified" in the functionf (x); for example, a step function (Gedeon et al, 2017;Crawford-Kahrl et al, 2019)…”

Section: Qss Approximationmentioning

confidence: 99%

Mixture distributions in a stochastic gene expression model with delayed feedback

Bokes

Borri

Palumbo

et al. 2019

Preprint

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Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a quasi-steady-state (QSS) approximation, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable steady state of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable steady states; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

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Section: Qss Approximationmentioning

confidence: 99%

Mixture distributions in a stochastic gene expression model with delayed feedback

Bokes

Borri

Palumbo

et al. 2019

Preprint

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“…There has been a lot of discussion about the technical challenges switching systems present and whether and how these models represent the dynamics of nearby perturbed continuous models. Systems of ODEs similar to the class of L-systems have been studied as continuous perturbations of the class of S-systems [28,22]. In this interpretation, the interval [\vargammj,i , \vargamm + j,i ] in the definition of \sigma L j,i contains the threshold \theta j,i of \sigma S j,i , and has length \epsilon , a small number.…”

Section: Examplesmentioning

confidence: 99%

“…The central question that we address in this paper is the comparison of the global dynamics of S-and L-systems of ODEs for a given regulatory network, as captured by the state transition graphs associated with the maps \scrF S and \scrF L . This investigation is a global version of a question that has been explored before where a piecewise constant nonlinearity is perturbed around a threshold to form a continuous function [17,28,22]. In this paper we address this question by comparing the Morse graphs associated with \scrF L and \scrF S .…”

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confidence: 99%

Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models

Crawford-Kahrl

Cummins

Gedeon

2019

SIAM J. Appl. Dyn. Syst.

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Modeling the dynamics of biological networks introduces many challenges, among them the lack of first principle models, the size of the networks, and difficulties with parameterization. Discrete time Boolean networks and related continuous time switching systems provide a computationally accessible way to translate the structure of the network to predictions about the dynamics. Recent work has shown that the parameterized dynamics of switching systems can be captured by a combinatorial object, called a Dynamic Signatures Generated by Regulatory Networks (DSGRN) database, that consists of a parameter graph characterizing a finite parameter space decomposition, whose nodes are assigned a Morse graph that captures global dynamics for all corresponding parameters. We show that for a given network there is a way to associate the same type of object by considering a continuous time ODE system with a continuous right-hand side, which we call an L-system. The main goal of this paper is to compare the two DSGRN databases for the same network. Since the L-systems can be thought of as perturbations (not necessarily small) of the switching systems, our results address the correspondence between global parameterized dynamics of switching systems and their perturbations. We show that, at corresponding parameters, there is an order preserving map from the Morse graph of the switching system to that of the L-system that is surjective on the set of attractors and bijective on the set of fixed-point attractors. We provide important examples showing why this correspondence cannot be strengthened.context of cell biology, the problems of nonlinear dynamics are compounded by the lack of first principles that would determine appropriate nonlinearities, the difficulty in obtaining precise experimental data needed to determine parameters, and the need to analyze the dynamics of 5--10 dimensional systems over 30--50 parameters.Recently, we introduced in [16, 15] a new approach to this problem that assigns two finite objects to any network with positive and negative edges. First is a parameter graph whose nodes are in 1-1 correspondence with regions in the parameter space, where these regions form a decomposition of the parameter space. To each region of the parameter space, i.e., a node of the parameter graph, there is an associated state transition graph that characterizes allowable transitions between well-defined states that partition the phase space. Since state transition graphs can be large, a useful description of the recurrent trajectories is a Morse graph, which is the graph of strongly connected path components of the state transition graph. This information is compiled into a database of Dynamic Signatures Generated by Regulatory Networks (DSGRN database), where to each node of the parameter graph there is an associated Morse graph that captures the recurrent dynamics valid for all parameters in the corresponding parameter region.The advantages of such a description of global dynamics is its finiteness and the resulting computa...

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“…We hasten to add (this is made clear in the sections that follow) that we do not view (1) as a mathematical model for the biological process; rather (1) is only used to motivate the combinatorial computations that are the focus of this paper. To emphasize this point we recall that the main result of [22] is that applying the methods described in this paper to 2-dimensional switching systems results in a representation of the global dynamics that is valid for a much wider family of nonlinearities, i.e. for a system of the form (2)…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Representation of Parameter Space for Switching Networks

Cummins¹,

Gedeon²,

Harker³

et al. 2016

SIAM J. Appl. Dyn. Syst.

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158

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We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network ( ) database. The motivation stems from urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and decades of parameters. The input to the database computations is a regulatory network, i.e. a directed graph with edges indicating up or down regulation. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate), and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piece-wise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamical signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type of information stored in the database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5 nodes and a 29-dimensional parameter space.

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Global dynamics for steep nonlinearities in two dimensions

Cited by 29 publications

References 19 publications

Mixture distributions in a stochastic gene expression model with delayed feedback

Mixture distributions in a stochastic gene expression model with delayed feedback

Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models

Combinatorial Representation of Parameter Space for Switching Networks

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