Analysis of gene regulatory network models with graded and binary transcriptional responses

Veflingstad, Siren R.; Plahte, Erik

doi:10.1016/j.biosystems.2006.09.036

Cited by 26 publications

(35 citation statements)

References 45 publications

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“…The aim of the present paper is to continue and generalize analysis initiated in [8] and [15]. Exactly as in these papers, our starting point is the system (1) with Boolean response functions, which is regarded as an established model of gene regulatory networks.…”

Section: Introductionmentioning

confidence: 99%

“…[3], [6]), while the other utilizes singular perturbation analysis (see e.g. [8], [13], [15]), where smooth response functions of sigmoid-type replace the step functions.…”

Section: Introductionmentioning

confidence: 99%

“…The main novelty of our paper compared to the works [8], [15] consists in replacing multilinear regulatory functions with general nonlinear functions. A mathematical motivation of our choice is closely related to specific properties of piecewise-linear systems.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Tafintseva¹,

Machina²,

Ponosov³

2013

Nonlinear Analysis: Real World Applications

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We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.

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Section: Introductionmentioning

confidence: 99%

“…[3], [6]), while the other utilizes singular perturbation analysis (see e.g. [8], [13], [15]), where smooth response functions of sigmoid-type replace the step functions.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Tafintseva¹,

Machina²,

Ponosov³

2013

Nonlinear Analysis: Real World Applications

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“…(1) could be applied to GRNs of any size and complexity, and the generic analysis developed in [7,14] is fully applicable to networks of any size. However, as the network becomes large, there is a combinatorial explosion of phase-space domains and parameter combinations that need to be investigated, and a computerized, algorithmic approach becomes a necessity.…”

Section: A Modeling Framework For the Study Of Grn Dynamicsmentioning

confidence: 99%

“…(5) is confined to the so-called Z-cube Z(D S ) = [0, 1] σ associated with D S . In the limit q → 0, the interior of the Z-cube describes the motion of the singular Z-variables, while the faces describe the switching variables in the adjacent switching domains, and the vertices represent the adjacent boxes [14]. Each Z-cube has a set of entrance and exit points where trajectories can enter, respectively exit from, the interior of the Z-cube.…”

Section: A Modeling Framework For the Study Of Grn Dynamicsmentioning

confidence: 99%

An Algorithm for Qualitative Simulation of Gene Regulatory Networks with Steep Sigmoidal Response Functions

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Abstract. A specific class of ODEs has been shown to be adequate to describe the essential features of the complex dynamics of Gene-Regulatory Networks (GRN). But, the effective exploitation of such models to predict the dynamics of specific GRNs by classical numerical schemes is greatly hampered by the current lack of precise and quantitative information on regulation mechanisms and kinetic parameters. Due to the size and complexity of large GRNs, classical qualitative analysis could be very hard, or even impracticable, to be carried out by hand, and conventional qualitative simulation approaches rapidly lead to an exponential growth of the generated behavior tree that, besides all possible sound behaviors, may also contain spurious ones. This paper discusses the work-in-progress of a research effort aiming at the design and implementation of a computational framework for qualitative simulation of the dynamics of a class of ODE models of GRNs. The algorithm we propose results from a set of symbolic computation algorithms that carry out the integration of qualitative reasoning techniques with singular perturbation analysis methods. The former techniques allow us to cope with uncertain and incomplete knowledge whereas the latter ones lay the mathematical groundwork for a sound and complete algorithm capable to deal with regulation processes that occur at different time scales.

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Assigning probabilities to qualitative dynamics of gene regulatory networks

Ironi

Lanzarone

2014

J. Math. Biol.

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Mathematical and computational modeling frameworks play the leading role in the analysis and prediction of the dynamics of gene regulatory networks. The literature abounds in various approaches, all of which characterized by strengths and weaknesses. Among the others, Ordinary Differential Equations (ODE) models give a more general and detailed description of the network structure. But, analytical computations might be prohibitive as soon as the network dimension increases, and numerical simulation could be nontrivial, time-consuming and very often impracticable as precise and quantitative information on model parameters are frequently unknown and hard to estimate from experimental data. Last but not least, they do not account for the intrinsic stochasticity of regulation. In the present paper we consider a class of ODE models with stochastic parameters. The proposed method separates the deterministic aspects from the stochastic ones. Under specific assumptions and conditions, all possible trajectories of an ODE model, where incomplete knowledge of parameter values is symbolically and qualitatively expressed by initial inequalities only, are simulated in a single run from an initial state. Then, the occurrence probability of each trajectory, characterized by a sequence of parameter inequalities, is computed by associating probability density functions with network parameters. As demonstrated by its application to the gene repressilator system, the method seems particularly promising in the design of synthetic networks.

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Analysis of gene regulatory network models with graded and binary transcriptional responses

Cited by 26 publications

References 45 publications

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

An Algorithm for Qualitative Simulation of Gene Regulatory Networks with Steep Sigmoidal Response Functions

Assigning probabilities to qualitative dynamics of gene regulatory networks

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