Graphical requirements for multistationarity in reaction networks and their verification in BioModels

Baudier, Adrien; Fages, François; Soliman, Sylvain

doi:10.1016/j.jtbi.2018.09.024

Cited by 9 publications

(8 citation statements)

References 31 publications

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“…After choosing an order of the set of reactions, these vectors are gathered as columns of a matrix, called the stoichiometric matrix N ∈ R n×r . The stoichiometric matrix for network (1) is…”

Section: Reaction Networkmentioning

confidence: 99%

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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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Section: Reaction Networkmentioning

confidence: 99%

“…We illustrate this framework and computations with selected perturbations γ for our running example (1). First, note that due to the matrix of conservation laws in (7), the steady state equations for x 1 , x 2 and x 4 are redundant.…”

Section: Sign-sensitivitiesmentioning

confidence: 99%

Sign-sensitivities for reaction networks: an algebraic approach

Feliu

2019

Mathematical Biosciences and Engineering

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“…A CRN can be interpreted in a hierarchy of Boolean, discrete, stochastic and differential semantics [7,13] which is at the basis of a rich theory for the analysis of their dynamical properties [14,9,1], and more recently, of their computational power [7,6,11]. In particular, their interpretation by Ordinary Differential Equations (ODE) allows us to give a precise mathematical meaning to the notion of analog computation and high-level functions computed by cells [10,25,23], with the following definitions: Definition 1.…”

Section: Introductionmentioning

confidence: 99%

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

Hemery

Fages

Soliman

2021

Computational Methods in Systems Biology

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The Turing completeness result for continuous chemical reaction networks (CRN) shows that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equations (PODE), the dualrail encoding of real variables by the difference of concentration between two molecular species, and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants. In this paper, we present a polynomialization algorithm of quadratic time complexity to transform a system of elementary differential equations in PODE. This algorithm is used as a front-end transformation to compile any elementary mathematical function, either of time or of some input species, into a finite CRN. We illustrate the performance of our compiler on a benchmark of elementary functions relevant to CRN design problems in synthetic biology specified by mathematical functions. In particular, the abstract CRN obtained by compilation of the Hill function of order 5 is compared to the natural CRN structure of MAPK signalling networks.

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“…A CRN can be interpreted in a hierarchy of Boolean, discrete, stochastic and differential semantics [7,13] which is at the basis of a rich theory for the analysis of their dynamical properties [14,9,1], and more recently, of their computational power [7,6,11]. In particular, their interpretation by Ordinary Differential Equations (ODE) allows us to give a precise mathematical meaning to the notion of analog computation and high-level functions computed by cells [10,25,23], using the following definitions: Definition 1.…”

Section: Introductionmentioning

confidence: 99%

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

Hemery,

Fages,

Soliman

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The Turing completeness result for continuous chemical reaction networks (CRN) shows that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equations (PODE), the dualrail encoding of real variables by the difference of concentration between two molecular species, and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants. In this paper, we present a polynomialization algorithm of quadratic time complexity to transform a system of elementary differential equations to PODE. This algorithm is used as a front-end transformation to compile any elementary mathematical function, either of time or of some input species, into a finite CRN. We illustrate the performance of our compiler on a benchmark of elementary functions relevant to CRN design problems in synthetic biology specified by mathematical functions. In particular, the abstract CRN obtained by compilation of the Hill function of order 5 is compared to the natural CRN structure of MAPK signalling networks.

show abstract

Graphical requirements for multistationarity in reaction networks and their verification in BioModels

Cited by 9 publications

References 31 publications

Sign-sensitivities for reaction networks: an algebraic approach

Sign-sensitivities for reaction networks: an algebraic approach

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

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