arXiv subjects

Matthew D. Johnston

Publications and source records attributed to Matthew D. Johnston.

25 records · Page 2Linked to original sources

arXiv2013

Translated Chemical Reaction Networks

Many biochemical and industrial applications involve complicated networks of simultaneously occurring chemical reactions. Under the assumption of mass action kinetics, the dynamics of these chemical reaction networks are governed by systems of polynomial ordinary differential equations. The steady states of these mass action systems have been analysed via a variety of techniques, including elementary flux mode analysis, algebraic techniques (e.g. Groebner bases), and deficiency theory. In this paper, we present a novel method for characterizing the steady states of mass action systems. Our method explicitly links a network's capacity to permit a particular class of steady states, called toric steady states, to topological properties of a related network called a translated chemical reaction network. These networks share their reaction stoichiometries with their source network but are permitted to have different complex stoichiometries and different network topologies. We apply the results to examples drawn from the biochemical literature.

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arXiv2012

Computing Weakly Reversible Linearly Conjugate Chemical Reaction Networks with Minimal Deficiency

Mass-action kinetics is frequently used in systems biology to model the behaviour of interacting chemical species. Many important dynamical properties are known to hold for such systems if they are weakly reversible and have a low deficiency. In particular, the Deficiency Zero and Deficiency One Theorems guarantee strong regularity with regards to the number and stability of positive equilibrium states. It is also known that chemical reaction networks with disparate reaction structure can exhibit the same qualitative dynamics. The theory of linear conjugacy encapsulates the cases where this relationship is captured by a linear transformation. In this paper, we propose a mixed-integer linear programming algorithm capable of determining weakly reversible reaction networks with a minimal deficiency which are linearly conjugate to a given reaction network.

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arXiv2011

Dynamical Equivalence and Linear Conjugacy of Chemical Reaction Networks: New Results and Methods

In the first part of this paper, we propose new optimization-based methods for the computation of preferred (dense, sparse, reversible, detailed and complex balanced) linearly conjugate reaction network structures with mass action dynamics. The developed methods are extensions of previously published results on dynamically equivalent reaction networks and are based on mixed-integer linear programming. As related theoretical contributions we show that (i) dense linearly conjugate networks define a unique super-structure for any positive diagonal state transformation if the set of chemical complexes is given, and (ii) the existence of linearly conjugate detailed balanced and complex balanced networks do not depend on the selection of equilibrium points. In the second part of the paper it is shown that determining dynamically equivalent realizations to a network that is structurally fixed but parametrically not can also be written and solved as a mixed-integer linear programming problem. Several examples illustrate the presented computation methods.

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arXiv2011

A Linear Programming Approach to Weak Reversibility and Linear Conjugacy of Chemical Reaction Networks

A numerically effective procedure for determining weakly reversible chemical reaction networks that are linearly conjugate to a known reaction network is proposed in this paper. The method is based on translating the structural and algebraic characteristics of weak reversibility to logical statements and solving the obtained set of linear (in)equalities in the framework of mixed integer linear programming. The unknowns in the problem are the reaction rate coefficients and the parameters of the linear conjugacy transformation. The efficacy of the approach is shown through numerical examples.

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arXiv2011

Weak Dynamic Non-Emptiability and Persistence of Chemical Kinetics Systems

A frequently desirable characteristic of chemical kinetics systems is that of persistence, the property that if all the species are initially present then none of them may tend toward extinction. It is known that solutions of deterministically modelled mass-action systems may only approach portions of the boundary of the positive orthant which correspond to semi-locking sets (alternatively called siphons). Consequently, most recent work on persistence of these systems has been focused on these sets. In this paper, we focus on a result which states that, for a conservative mass-action system, persistence holds if every critical semi-locking set is dynamically non-emptiable and the system contains no nested locking sets. We will generalize this result by introducing the notion of a weakly dynamically non-emptiable semi-locking set and making novel use of the well-known Farkas' Lemma. We will also connect this result to known results regarding complex balanced systems and systems with facets.

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arXiv2011

Linear Conjugacy of Chemical Reaction Networks

Under suitable assumptions, the dynamic behaviour of a chemical reaction network is governed by an autonomous set of polynomial ordinary differential equations over continuous variables representing the concentrations of the reactant species. It is known that two networks may possess the same governing mass-action dynamics despite disparate network structure. To date, however, there has only been limited work exploiting this phenomenon even for the cases where one network possesses known dynamics while the other does not. In this paper, we bring these known results into a broader unified theory which we call conjugate chemical reaction network theory. We present a theorem which gives conditions under which two networks with different governing mass-action dynamics may exhibit the same qualitative dynamics and use it to extend the scope of the well-known theory of weakly reversible systems.

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arXiv2010

A Stratum Approach to Global Stability of Complex Balanced Systems

It has long been known that complex balanced mass-action systems exhibit a restrictive form of behaviour known as locally stable dynamics. This means that within each compatibility class $\mathcal{C}_{\mathbf{x}_0}$---the forward invariant space where solutions lies---there is exactly one equilibrium concentration and that this concentration is locally asymptotically stable. It has also been conjectured that this stability extends globally to $\mathcal{C}_{\mathbf{x}_0}$. That is to say, all solutions originating in $\mathcal{C}_{\mathbf{x}_0}$ approach the unique positive equilibrium concentration rather than $\partial \mathcal{C}_{\mathbf{x}_0}$ or $\infty$. To date, however, no general proof of this conjecture has been found. In this paper, we approach the problem of global stability for complex balanced systems through the methodology of dividing the positive orthant into regions called strata. This methodology has been previously applied to detailed balanced systems---a proper subset of complex balanced systems---to show that, within a stratum, trajectories are repelled from any face of $\mathbb{R}_{\geq 0}^m$ adjacent to the stratum. Several known global stability results for detailed balanced systems are generalized to complex balanced systems.

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