The Kinetic Space of Multistationarity in Dual Phosphorylation
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The Kinetic Space of Multistationarity in Dual Phosphorylation Elisenda Feliu1
· Nidhi Kaihnsa2 · Timo de Wolff3 · Oguzhan ˘ Yürük3
Received: 14 May 2020 / Revised: 13 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Multistationarity in molecular systems underlies switch-like responses in cellular decision making. Determining whether and when a system displays multistationarity is in general a difficult problem. In this work we completely determine the set of kinetic parameters that enable multistationarity in a ubiquitous motif involved in cell signaling, namely a dual phosphorylation cycle. In addition we show that the regions of multistationarity and monostationarity are both path connected. We model the dynamics of the concentrations of the proteins over time by means of a parametrized polynomial ordinary differential equation (ODE) system arising from the mass-action assumption. Since this system has three linear first integrals defined by the total amounts of the substrate and the two enzymes, we study for what parameter values the ODE system has at least two positive steady states after suitably choosing the total amounts. We employ a suite of techniques from (real) algebraic geometry, which in particular concern the study of the signs of a multivariate polynomial over the positive orthant and sums of nonnegative circuit polynomials. Keywords Two-site phosphorylation · Multistationarity · Chemical reaction networks · Real algebraic geometry · Cylindrical algebraic decomposition · Circuit polynomials Mathematics Subject Classification 92Bxx · 14Pxx · 37N25 · 52B20 · 90C26
1 Introduction Multistationarity, that is the existence of multiple steady states in a system, has been linked to cellular decision making and switch-like responses to graded input [27,30,42]. In the context of chemical reaction networks, there exist numerous methods to decide whether multistationarity arises for some choice of parameter values [7,11,12,15–17,33,41]. How-
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10884-02009889-6) contains supplementary material, which is available to authorized users.
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Elisenda Feliu [email protected]
Extended author information available on the last page of the article
123
Journal of Dynamics and Differential Equations
ever, determining for which parameter values this is the case, is a very difficult problem with complicated answers. Some recent progress in understanding the parameter region of multistationarity has eased the problem by focusing on subsets of parameters, and providing regions that guarantee or exclude that the other parameters can be chosen in such a way that multistationarity arises [1,5]. Here, we completely characterize the region of multistationarity in terms of kinetic parameters for a simple model of phosphorylation and dephosphorylation, which is a building block of the MAPK cascade involved ubiquitously in cell signaling [23,24,34]. Phosphorylation processes are central in the
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