Linear conjugacy of chemical kinetic systems

Allen L. Nazareno; Raymond Paul L. Eclarin; Eduardo R. Mendoza; Angelyn R. Lao; Allen L. Nazareno; Raymond Paul L. Eclarin; Eduardo R. Mendoza; Angelyn R. Lao

doi:10.3934/mbe.2019421

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 8322-8355. doi: 10.3934/mbe.2019421

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Linear conjugacy of chemical kinetic systems

1.
Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, Laguna, 4031, Philippines
2.
Department of Mathematics, Mariano Marcos State University, Ilocos Norte, 2906 Philippines
3.
Mathematics and Statistics Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines
4.
Max Planck Institute of Biochemistry, 85152 Martinsried, Germany
5.
LMU Faculty of Physics, Geschwister -Scholl-Platz 1, 80539 Munich, Germany

Received: 19 April 2019 Accepted: 16 September 2019 Published: 19 September 2019

Two networks are said to be linearly conjugate if the solution of their dynamic equations can be transformed into each other by a positive linear transformation. The study on dynamical equivalence in chemical kinetic systems was initiated by Craciun and Pantea in 2008 and eventually led to the Johnston-Siegel Criterion for linear conjugacy (JSC). Several studies have applied Mixed Integer Linear Programming (MILP) approach to generate linear conjugates of MAK (mass action kinetic) systems, Bio-CRNs (which is a subset of Hill-type kinetic systems when the network is restricted to digraphs), and PL-RDK (complex factorizable power law kinetic) systems. In this study, we present a general computational solution to construct linear conjugates of any "rate constant-interaction function decomposable" (RID) chemical kinetic systems, wherein each of its rate function is the product of a rate constant and an interaction function. We generate an extension of the JSC to the complex factorizable (CF) subset of RID kinetic systems and show that any non-complex factorizable (NF) RID kinetic system can be dynamically equivalent to a CF system via transformation. We show that linear conjugacy can be generated for any RID kinetic systems by applying the JSC to any NF kinetic system that are transformed to CF kinetic system.
- linear conjugacy,
- chemical reaction network,
- chemical kinetic system,
- Johnston-Siegel Criterion,
- dynamical equivalence,
- rate constant-interaction function decomposable (RID)
Citation: Allen L. Nazareno, Raymond Paul L. Eclarin, Eduardo R. Mendoza, Angelyn R. Lao. Linear conjugacy of chemical kinetic systems[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 8322-8355. doi: 10.3934/mbe.2019421

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Abstract

Two networks are said to be linearly conjugate if the solution of their dynamic equations can be transformed into each other by a positive linear transformation. The study on dynamical equivalence in chemical kinetic systems was initiated by Craciun and Pantea in 2008 and eventually led to the Johnston-Siegel Criterion for linear conjugacy (JSC). Several studies have applied Mixed Integer Linear Programming (MILP) approach to generate linear conjugates of MAK (mass action kinetic) systems, Bio-CRNs (which is a subset of Hill-type kinetic systems when the network is restricted to digraphs), and PL-RDK (complex factorizable power law kinetic) systems. In this study, we present a general computational solution to construct linear conjugates of any "rate constant-interaction function decomposable" (RID) chemical kinetic systems, wherein each of its rate function is the product of a rate constant and an interaction function. We generate an extension of the JSC to the complex factorizable (CF) subset of RID kinetic systems and show that any non-complex factorizable (NF) RID kinetic system can be dynamically equivalent to a CF system via transformation. We show that linear conjugacy can be generated for any RID kinetic systems by applying the JSC to any NF kinetic system that are transformed to CF kinetic system.

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