Translated Chemical Reaction Networks

Johnston, Matthew D.

doi:10.1007/s11538-014-9947-5

Cited by 48 publications

(182 citation statements)

References 35 publications

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“…x 10 x 8 , a 5 = x 1 x 9 x 2 , a 6 = x 3 x 10 x 4 . Substituting (25) into det Jac(h c,a ) -that is, considering the parametrization x → (φ a (x), x) where φ a (x) is given by (25) -yields the following critical function C(x), as in (20):…”

Section: Procedures 47 (Witness To Multistationarity For Linearly Binmentioning

confidence: 99%

Multistationarity in Structured Reaction Networks

et al. 2019

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Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems. First, we build on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain "critical function" changes sign. Here, we allow for more general parametrizations, which make it much easier to determine the existence of a sign change. This is particularly simple when the steady-state equations are linearly equivalent to binomials; we give necessary conditions for this to happen, which hold for many networks studied in the literature. We also give a sufficient condition for multistationarity of networks whose steady-state equations can be replaced by equivalent triangular-form equations. Finally, we present methods for finding witnesses to multistationarity, which we show work well for certain structured reaction networks, including those common to biological signaling pathways. Our work relies on results from degree theory, on the existence of explicit rational parametrizations of the steady states, and on the specialization of Gröbner bases.

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Section: Procedures 47 (Witness To Multistationarity For Linearly Binmentioning

confidence: 99%

Multistationarity in Structured Reaction Networks

et al. 2019

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“…Our work joins a growing number of works that harness steady-state parametrizations. Such results include criteria for when such parametrizations exist [26,40] and methods for using them to determine whether a network is multistationary [25,29,32,34]. Going further, steady-state parametrizations can also be used to find a witness to multistationarity or even the precise parameter regions that yield multistationarity [4,5].…”

Section: Connection To Related Workmentioning

confidence: 99%

Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System

2019

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This work investigates the emergence of oscillations in one of the simplest cellular signaling networks exhibiting oscillations, namely, the dual-site phosphorylation and dephosphorylation network (futile cycle), in which the mechanism for phosphorylation is processive while the one for dephosphorylation is distributive (or vice-versa). The fact that this network yields oscillations was shown recently by Suwanmajo and Krishnan. Our results, which significantly extend their analyses, are as follows. First, in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface defined by the vanishing of a single Hurwitz determinant. Second, this surface consists generically of simple Hopf bifurcations. Next, simulations suggest that when the steady state is unstable, oscillations are the norm. Finally, the emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system generically admits a Hopf bifurcation. Our proofs are enabled by the Routh-Hurwitz criterion, a Hopf-bifurcation criterion due to Yang, and a monomial parametrization of steady states.

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“…The deficiency of a CRN is a nonnegative parameter defined by δ = dim(ker(Y )∩im(I a )). Alternatively, the deficiency can be computed by the formula δ = n − − dim(S) (see [21]). The deficiency was first introduced in [10,18] and has been used extensively since in the context of steady states of mass-action systems [19,11,13,14,15,21,31].…”

Section: Chemical Reaction Networkmentioning

confidence: 99%

“…We introduce the following structural notion of network translation, which is weaker than those presented in [21,22,37,23].…”

Section: Structural Translationmentioning

confidence: 99%

Computing Weakly Reversible Deficiency Zero Network Translations Using Elementary Flux Modes

Johnston

Burton

2019

Bull Math Biol

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We present a computational method for performing structural translation, which has been studied recently in the context of analyzing the steady states and dynamical behavior of massaction systems derived from biochemical reaction networks. Our procedure involves solving a binary linear programming problem where the decision variables correspond to interactions between the reactions of the original network. We call the resulting network a reaction-toreaction graph and formalize how such a construction relates to the original reaction network and the structural translation. We demonstrate the efficacy and efficiency of the algorithm by running it on 508 networks from the European Bioinformatics Institutes' BioModels database. We also summarize how this work can be incorporated into recently proposed algorithms for establishing mono and multistationarity in biochemical reaction systems. Figure 1: A chemical reaction network (left) corresponding to a histidine kinase network where X and Y are two signaling proteins and p is a phosphate group [4]. This CRN has elementary flux modes {r 1 , r 2 , r 4 } and {r 2 , r 3 } which correspond to the directed cycles in the reaction-toreaction graph (center). The structural translation (right) has the same elementary flux modes and stoichiometric vectors as the CRN but the elementary flux modes correspond to cycles.Further connections between the deficiency and the steady states of mass-action systems have been established [11,12,13,14,15,7,6].The study of the deficiency was recently initiated in generalized chemical reaction networks (GCRNs) [31,32]. In a GCRN, each vertex in the reaction graph is associated with two potentially distinct complexes, one for the stoichiometry and one for the kinetic rate of the reaction. Surprisingly, for weakly reversible generalized mass-action systems which have a stoichiometric and kinetic-order deficiency of zero, we still obtain a simple monomial parametrization of the steady state set. A process for relating CRNs and GCRNs, called network translation, was furthermore established in [21]. Network translation consists of restructuring a given CRN in such a way that the resulting network (a GCRN) can be used to guarantee dynamical and steady state properties of the original CRN. The process has been utilized to establish connections between chemical reaction network theory [9], the algebraic study of toric varieties [6,29,8], and biochemical reaction modeling [22,37,5]. Recent work has also established a deficiency-based method for constructing rational parametrizations of steady state sets for a broad class of mass-action systems [23].In this paper, we focus on computational methods for performing the structural component of network translation, which we call structural translation. In general, given a biochemical reaction network of realistic scale, it is challenging to determine a suitable (e.g. weakly reversible, deficiency zero) structural translation. We extend the recent computational work of [22,37] by introducing an elementary flux m...

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Translated Chemical Reaction Networks

Cited by 48 publications

References 35 publications

Multistationarity in Structured Reaction Networks

Multistationarity in Structured Reaction Networks

Emergence of Oscillations in a Mixed-Mechanism Phosphorylation System

Computing Weakly Reversible Deficiency Zero Network Translations Using Elementary Flux Modes

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