Variable Elimination in Chemical Reaction Networks with Mass-Action Kinetics

AbstractBackgroundTheoretical analysis of signaling pathways can provide a substantial amount of insight into their function. One particular area of research considers signaling pathways capable of assuming two or more stable states given the same amount of signaling ligand. This phenomenon of bistability can give rise to switch-like behavior, a mechanism that governs cellular decision making. Investigation of whether or not a signaling pathway can confer bistability and switch-like behavior, without knowledge of specific kinetic rate constant values, is a mathematically challenging problem. Recently a technique based on optimization has been introduced, which is capable of finding example parameter values that confer switch-like behavior for a given pathway. Although this approach has made it possible to analyze moderately sized pathways, it is limited to reaction networks that presume a uniterminal structure. It is this limited structure we address by developing a general technique that applies to any mass action reaction network with conservation laws.ResultsIn this paper we developed a generalized method for detecting switch-like bistable behavior in any mass action reaction network with conservation laws. The method involves 1) construction of a constrained optimization problem using the determinant of the Jacobian of the underlying rate equations, 2) minimization of the objective function to search for conditions resulting in a zero eigenvalue 3) computation of a confidence level that describes if the global minimum has been found and 4) evaluation of optimization values, using either numerical continuation or directly simulating the ODE system, to verify that a bistability region exists. The generalized method has been tested on three motifs known to be capable of bistability.ConclusionsWe have developed a variation of an optimization-based method for discovery of bistability, which is not limited to the structure of the chemical reaction network. Successful completion of the method provides an S-shaped bifurcation diagram, which indicates that the network acts as a bistable switch for the given optimization parameters.

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On the geometric formulation of non-isothermal mass action chemical reaction networks

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2019.07.001 ◽

2019 ◽

Vol 52 (7) ◽

pp. 1-6

Author(s):

Arjan van der Schaft ◽

Bernhard Maschke

Keyword(s):

Chemical Reaction ◽

Chemical Reaction Networks ◽

Mass Action ◽

Reaction Networks ◽

Geometric Formulation ◽

Isothermal Mass

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Hamiltonian Feedback Design for Mass Action Law Chemical Reaction Networks

IFAC-PapersOnLine ◽

10.1016/j.ifacol.2015.10.232 ◽

2015 ◽

Vol 48 (13) ◽

pp. 158-163 ◽

Cited By ~ 1

Author(s):

György Lipták ◽

Gábor Szederkényi ◽

Katalin M. Hangos

Keyword(s):

Chemical Reaction ◽

Chemical Reaction Networks ◽

Mass Action ◽

Reaction Networks ◽

Feedback Design ◽

Mass Action Law

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A numerical approach for detecting switch-like bistability in mass action chemical reaction networks with conservation laws

BMC Bioinformatics ◽

10.1186/s12859-021-04477-x ◽

2022 ◽

Vol 23 (1) ◽

Author(s):

Brandon C Reyes ◽

Irene Otero-Muras ◽

Vladislav A Petyuk

Keyword(s):

Conservation Laws ◽

Signaling Pathways ◽

Chemical Reaction ◽

Reaction Network ◽

Chemical Reaction Networks ◽

Mass Action ◽

Numerical Continuation ◽

Rate Equations ◽

Reaction Networks ◽

Bistable Switch

Abstract Background Theoretical analysis of signaling pathways can provide a substantial amount of insight into their function. One particular area of research considers signaling pathways capable of assuming two or more stable states given the same amount of signaling ligand. This phenomenon of bistability can give rise to switch-like behavior, a mechanism that governs cellular decision making. Investigation of whether or not a signaling pathway can confer bistability and switch-like behavior, without knowledge of specific kinetic rate constant values, is a mathematically challenging problem. Recently a technique based on optimization has been introduced, which is capable of finding example parameter values that confer switch-like behavior for a given pathway. Although this approach has made it possible to analyze moderately sized pathways, it is limited to reaction networks that presume a uniterminal structure. It is this limited structure we address by developing a general technique that applies to any mass action reaction network with conservation laws. Results In this paper we developed a generalized method for detecting switch-like bistable behavior in any mass action reaction network with conservation laws. The method involves (1) construction of a constrained optimization problem using the determinant of the Jacobian of the underlying rate equations, (2) minimization of the objective function to search for conditions resulting in a zero eigenvalue, (3) computation of a confidence level that describes if the global minimum has been found and (4) evaluation of optimization values, using either numerical continuation or directly simulating the ODE system, to verify that a bistability region exists. The generalized method has been tested on three motifs known to be capable of bistability. Conclusions We have developed a variation of an optimization-based method for the discovery of bistability, which is not limited to uniterminal chemical reaction networks. Successful completion of the method provides an S-shaped bifurcation diagram, which indicates that the network acts as a bistable switch for the given optimization parameters.

Download Full-text