# A Computational Approach to Steady State Correspondence of Regular and Generalized Mass Action Systems

@article{Johnston2015ACA, title={A Computational Approach to Steady State Correspondence of Regular and Generalized Mass Action Systems}, author={Matthew D. Johnston}, journal={Bulletin of Mathematical Biology}, year={2015}, volume={77}, pages={1065-1100} }

It has been recently observed that the dynamical properties of mass action systems arising from many models of biochemical reaction networks can be characterized by considering the corresponding properties of a related generalized mass action system. The correspondence process known as network translation in particular has been shown to be useful in characterizing a system’s steady states. In this paper, we further develop the theory of network translation with particular focus on a subclass of…

## 12 Citations

### Network Translation and Steady-State Properties of Chemical Reaction Systems

- Computer ScienceBulletin of mathematical biology
- 2018

It is shown that network translation can often provide a method for deriving the steady-state value of the robust species and is presented with a MILP algorithm for the identification of translated chemical reaction networks that improves on previous approaches, allowing for easier application of the theory.

### Analysis of mass-action systems by split network translation

- Computer ScienceJournal of Mathematical Chemistry
- 2021

We introduce the notion of corresponding a chemical reaction network to a split network translation , and use this novel process to extend the scope of existing network-based theory for…

### A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems.

- MathematicsMathematical biosciences and engineering : MBE
- 2019

A generalization of Birch's theorem is presented, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states of generalized mass-action kinetics.

### A linear programming approach to dynamical equivalence, linear conjugacy, and the Deficiency One Theorem

- MathematicsJournal of Mathematical Chemistry
- 2016

A mixed-integer linear programming framework capable of determining whether a given mass action system has a dynamically equivalent or linearly conjugate representation which has an underlying network satisfying the Deficiency One Theorem is presented.

### A Deficiency-Based Approach to Parametrizing Positive Equilibria of Biochemical Reaction Systems

- EconomicsBulletin of mathematical biology
- 2019

Conditions which guarantee a parametrization of the set of positive equilibria of a generalized mass-action system are presented and an easy check for the occurrence of absolute concentration robustness is demonstrated for the EnvZ–OmpR pathway.

### Computing Weakly Reversible Deficiency Zero Network Translations Using Elementary Flux Modes

- Computer ScienceBulletin of mathematical biology
- 2019

A computational method for performing structural translation is presented, which has been studied recently in the context of analyzing the steady states and dynamical behavior of mass-action systems derived from biochemical reaction networks, and how it can be incorporated into recently proposed algorithms for establishing mono- and multistationarity in biochemical reaction systems.

### A note on "MAPK networks and their capacity for multistationarity due to toric steady states"

- Computer Science
- 2014

It is shown that the capacity for toric steady states in the three networks analyzed in that paper can be derived using the process of network translation, which corresponds the original mass action system to a generalized massaction system with the same steady states.

### Linear conjugacy of chemical kinetic systems.

- Mathematics, Computer ScienceMathematical biosciences and engineering : MBE
- 2019

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.

### Complex Balanced Equilibria of Weakly Reversible Poly-Pl Systems: Existence, Stability, and Robustness

- MathematicsMATCH - Communications in Mathematical and in Computer Chemistry
- 2022

Poly-PL kinetic systems (PYK) are kinetic systems consisting of nonnegative linear combinations of power law functions. In this contribution, we analyze these kinetic systems using two main…

### A computational approach to linear conjugacy in a class of power law kinetic systems

- ChemistryJournal of Mathematical Chemistry
- 2017

This paper studies linear conjugacy of PL-RDK systems, which are kinetic systems with power law rate functions whose kinetic orders are identical for branching reactions, i.e. reactions with the same…

## References

SHOWING 1-10 OF 30 REFERENCES

### Translated Chemical Reaction Networks

- Computer ScienceBulletin of mathematical biology
- 2014

This paper presents a novel method for characterizing the steady states of mass action systems that explicitly links a network’s capacity to permit a particular class of steady states to topological properties of a generalized network called a translated chemical reaction network.

### Linear conjugacy of chemical reaction networks

- Mathematics
- 2011

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…

### Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiency.

- MathematicsMathematical biosciences
- 2013

### Generalized Mass Action Systems: Complex Balancing Equilibria and Sign Vectors of the Stoichiometric and Kinetic-Order Subspaces

- MathematicsSIAM J. Appl. Math.
- 2012

A notion of generalized mass action systems that admits arbitrary power-law rate functions and serves as a more realistic model for reaction networks in intracellular environments is suggested.

### Chemical Reaction Systems with Toric Steady States

- MathematicsBulletin of mathematical biology
- 2012

The main result gives sufficient conditions for a chemical reaction system to have toric steady states, and the capacity of such a system to exhibit positive steady states and multistationarity is analyzed.

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

- Mathematics
- 2011

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…

### The existence and uniqueness of steady states for a class of chemical reaction networks

- Chemistry
- 1995

My purpose here is to draw some general relationships between the structure of a chemical reaction network and the nature of the set of equilibrium states for the corresponding system of nonlinear…

### Necessary and sufficient conditions for complex balancing in chemical kinetics

- Mathematics
- 1972

SummaryIn a recent publication (Horn & Jackson [1]) it was shown that complex balancing together with mass action type rate laws ensures certain stability properties of a kinetic system, thereby…

### Complex-linear invariants of biochemical networks.

- Computer ScienceJournal of theoretical biology
- 2012