Multistationarity in Structured Reaction Networks

Dickenstein, Alicia; Millán, Mercedes Pérez; Shiu, Anne; Tang, Xiaoxian

doi:10.1007/s11538-019-00572-6

Cited by 53 publications

(67 citation statements)

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“…Such parametrizations have been shown in recent years to be indispensable for analyzing multistationarity (multiple steady states, which are necessary for bistability) and oscillations [22,31,45]. Indeed, here we build on results in [10,12,17]. Specifically, following [12], we investigate oscillations by employing a steady-state parametrization together with a criterion of Yang [47] that characterizes Hopf bifurcations in terms of determinants of Hurwitz matrices.…”

Section: Erkmentioning

confidence: 99%

“…in which the (k, l)-th entry is b 2k−l as long as n ≥ 2k − l ≥ 0, and 0 otherwise. 2 As noted earlier, here we consider parametrizations of the form φ(â; x), while [17] allowed those of the form φ(â;x). Also, "conservative" in Proposition 2.4 can be generalized to "dissipative" [17].…”

Section: Hopf Bifurcationsmentioning

confidence: 99%

“…A steady-state parametrization for the full ERK network was given in [17, Examples 3.1 and 3.7]. That parametrization, however, can not specialize to accommodate irreversible versions of the network (in the effective parameters given in [17], two of the denominators are k on and on , so we can not set those rate constants to 0). So, in the next subsection, we give an alternate steady-state parametrization that, although quite similar to the one in [17], specializes when considering irreversible versions of the network (see Proposition 3.1).…”

Section: The (Full) Erk Networkmentioning

confidence: 99%

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Oscillations and bistability in a model of ERK regulation

et al. 2019

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This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust -this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network -namely, the fully processive dual-site network -which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.

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Section: Erkmentioning

confidence: 99%

Section: Hopf Bifurcationsmentioning

confidence: 99%

Section: The (Full) Erk Networkmentioning

confidence: 99%

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Oscillations and bistability in a model of ERK regulation

et al. 2019

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“…Toric varieties have aided in characterizing the central 459 CRNT concept of complex-balancing [25]. They have also enabled systematic 460 determination of kinetic parameters that give rise to multistability for large classes of 461 networks [46], including biologically relevant networks such as the MAPK pathway [47]. 462 In the context of sc-data, we leverage toric geometry to study the reactions underlying 463 cellular phenotypes without having to perform simulations, which can be difficult with 464 sparse and complex data.…”

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confidence: 99%

Inferring reaction network structure from single-cell, multiplex data, using toric systems theory

Wang

Lin

Sontag

et al. 2019

Preprint

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The goal of many single-cell studies on eukaryotic cells is to gain insight into the biochemical reactions that control cell fate and state. In this paper we introduce the concept of effective stoichiometric space (ESS) to guide the reconstruction of biochemical networks from multiplexed, fixed time-point, single-cell data. In contrast to methods based solely on statistical models of data, the ESS method leverages the power of the geometric theory of toric varieties to begin unraveling the structure of chemical reaction networks (CRN). This application of toric theory enables a data-driven mapping of covariance relationships in single cell measurements into stoichiometric information, one in which each cell subpopulation has its associated ESS interpreted in terms of CRN theory. In the development of ESS we reframe certain aspects of the theory of CRN to better match data analysis. As an application of our approach we process cytomery-and image-based single-cell datasets and identify differences in cells treated with kinase inhibitors. Our approach is directly applicable to data acquired using readily accessible experimental methods such as Fluorescence Activated Cell Sorting (FACS) and multiplex immunofluorescence. Author summaryWe introduce a new notion, which we call the effective stoichiometric space (ESS), that 1 elucidates network structure from the covariances of single-cell multiplexed data. The 2 ESS approach differs from methods that are based on purely statistical models of data: 3 it allows a completely new and data-driven translation of the theory of toric varieties in 4 geometry and specifically their role in chemical reaction networks (CRN). In the 5 process, we reframe certain aspects of the theory of CRN. As illustrations of our 6 approach, we find stoichiometry in different single-cell datasets, and pinpoint 7 dose-dependence of network perturbations in drug-treated cells. 8 Introduction 9Single-cell, multiplexed datasets have become prevalent [1,2], and include data on 10 transcript levels measured by sc-RNAseq [3], protein levels measured by flow 11 cytometry [4], or cell morphology and protein localization measured by multiplex 12 imaging [5-8]. An obvious advantage of such data is that it makes possible the detection 13 August 6, 2019 1/23 and quantification of differences among cells in a population, including those arising 14 from cyclic processes such as cell division and differentiation programs that are 15 asynchronous between cells [9, 10]. A more subtle advantage of single-cell data is that 16 they report on relationships among measured features, phosphorylation state of 17 receptors and nuclear localization of transcription factors for example. Because such 18 features are subject to natural stochastic fluctuation across a population of cells [11], 19 measuring the extent of correlation between otherwise independently fluctuating 20 features makes it possible to infer the topologies of biological networks [12, 13]. 21 A wide variety of tools have been developed for visualization of s...

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“…As it has been used in several works, e.g. [3,6], the set of all positive steady states is studied by means of a parametrization ϕ: U → R n >0 , such that the image of ϕ is the set of positive steady states (see [3] for strategies to find parametrizations).…”

Section: Sign-sensitivitiesmentioning

confidence: 99%

Sign-sensitivities for reaction networks: an algebraic approach

Feliu

2019

Mathematical Biosciences and Engineering

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This paper presents an algebraic framework to study sign-sensitivities for reaction networks modeled by means of systems of ordinary differential equations. Specifically, we study the sign of the derivative of the concentrations of the species in the network at steady state with respect to a small perturbation on the parameter vector. We provide a closed formula for the derivatives that accommodates common perturbations, and illustrate its form with numerous examples. We argue that, mathematically, the study of the response to the system with respect to changes in total amounts is not well posed, and that one should rather consider perturbations with respect to the initial conditions. We find a sign-based criterion to determine, without computing the sensitivities, whether the sign depends on the steady state and parameters of the system. This is based on earlier results of so-called injective networks. Finally, we address systems with multiple steady states and the restriction to stable steady states.

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Multistationarity in Structured Reaction Networks

Cited by 53 publications

References 50 publications

Oscillations and bistability in a model of ERK regulation

Oscillations and bistability in a model of ERK regulation

Inferring reaction network structure from single-cell, multiplex data, using toric systems theory

Sign-sensitivities for reaction networks: an algebraic approach

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