# Injectivity, multiple zeros and multistationarity in reaction networks

@article{Feliu2015InjectivityMZ, title={Injectivity, multiple zeros and multistationarity in reaction networks}, author={Elisenda Feliu}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, year={2015}, volume={471} }

Polynomial dynamical systems are widely used to model and study real phenomena. In biochemistry, they are the preferred choice for modelling the concentration of chemical species in reaction networks with mass-action kinetics. These systems are typically parametrized by many (unknown) parameters. A goal is to understand how properties of the dynamical systems depend on the parameters. Qualitative properties relating to the behaviour of a dynamical system are locally inferred from the system at…

## 17 Citations

Multistationarity in Structured Reaction Networks

- MathematicsBulletin of mathematical biology
- 2019

This work builds on work of Conradi, Feliu, Mincheva, and Wiuf, who showed that for certain reaction networks whose steady states admit a positive parametrization, multistationarity is characterized by whether a certain “critical function” changes sign, and allows for more general parametic changes, which make it much easier to determine the existence of a sign change.

Analyzing multistationarity in chemical reaction networks using the determinant optimization method

- MathematicsAppl. Math. Comput.
- 2016

Some Results on Injectivity and Multistationarity in Chemical Reaction Networks

- MathematicsSIAM J. Appl. Dyn. Syst.
- 2016

The goal of this paper is to gather and develop some necessary and sufficient criteria for injectivity and multistationarity in vector fields associated with a chemical reaction network under a…

A survey of methods for deciding whether a reaction network is multistationary

- Computer Science
- 2014

This work surveys developments in embedding-minimal multistationary networks, with an emphasis on recent results that connect the capacity forMultistationarity of one network to that of another.

Influence of Space in the Steady States of Chemical Reaction Networks

- Biology
- 2019

The results prove that space can influence the steady states of chemical reaction networks, and suggest that when adding compartments, some networks gain the capacity for multistationarity, while others lose it.

Parameter region for multistationarity in $n-$site phosphorylation networks

- Mathematics
- 2022

This work addresses the problem of understanding the parameter region where a family of networks relevant in cell signalling admits at least two steady states, a property termed multistationarity.…

Transport in Unsteady Flows: from Deterministic Structures to Stochastic Models and Back Again (17w5048)

- Mathematics
- 2017

Infinite-Dimensional Systems Infinite-dimensional system theory is an area where the goal is to axiomatize certain properties in control theory within the context of infinite-dimensional theory with…

Bistability and Bifurcation in Minimal Self-Replication and Nonenzymatic Catalytic Networks.

- BiologyChemphyschem : a European journal of chemical physics and physical chemistry
- 2017

It is demonstrated that the onset of bistability requires at least second-order catalysis and results from a mismatch between the various forward and reverse processes.

Robustness and parameter geography in post-translational modification systems

- MathematicsbioRxiv
- 2019

A mathematical model is used to study how the molecular mechanism of protein modification exhibits the property of multiple internal states, which has been suggested to underlie memory and decision making, and finds that bistability is compromised by product rebinding and that unrealistic assumptions on enzyme mechanisms have obscured its parametric rarity.

Bistability of sequestration networks

- MathematicsDiscrete & Continuous Dynamical Systems - B
- 2021

We solve a conjecture on multiple nondegenerate steady states, and prove bistability for sequestration networks. More specifically, we prove that for any odd number of species, and for any production…

## References

SHOWING 1-10 OF 53 REFERENCES

Simplifying the Jacobian Criterion for Precluding Multistationarity in Chemical Reaction Networks

- MathematicsSIAM J. Appl. Math.
- 2012

A procedure is presented which simplifies the application of the Jacobian criterion, and as a result, a new class of networks for which multiple steady states is precluded: those in which all chemical species have total moleculari...

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.

Preclusion of switch behavior in reaction networks with mass-action kinetics

- Mathematics
- 2011

It is shown that a network is injective if and only if the determinant of the Jacobian of a certain function does not vanish, and relates injectivity of a chemical reaction network to that of the chemical Reaction network obtained by adding outflow, or degradation, reactions for all species.

Atoms of multistationarity in chemical reaction networks

- Computer ScienceJournal of Mathematical Chemistry
- 2012

This work enumerate and classify all 386 bimolecular and reversible two-reaction networks and proves that if a subnetwork admits multiple steady states, then these steady states can be lifted to a larger network, provided that the two networks share the same stoichiometric subspace.

Preclusion of switch behavior in networks with mass-action kinetics

- MathematicsAppl. Math. Comput.
- 2012

Power-Law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species

- Computer ScienceSIAM J. Appl. Dyn. Syst.
- 2013

The approach embraces and extends previous work on multistationarity, such as work in relation to chemical reaction networks with dynamics defined by mass-action or noncatalytic kinetics, and also work based on graphical analysis of the interaction graph associated with the system.

Multiple Equilibria in Complex Chemical Reaction Networks: Semiopen Mass Action Systems

- BiologySIAM J. Appl. Math.
- 2010

Results of the earlier papers to semiopen reactors are extended and show that very similar results obtain, provided that the network of chemical reactions satisfies certain weak structural conditions; weak reversibility is sufficient but not necessary.

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.