# Dynamical properties of Discrete Reaction Networks

- 336 Downloads
- 9 Citations

## Abstract

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of *Discrete Reaction Networks* (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a proof of non-reachability in a given DRN has immediate implications for any concrete stochastic model based on that DRN, independent of the choice of kinetic laws and constants. Moreover, if we assume that stochastic kinetic rates are given by the mass-action law (or any other kinetic law that gives non-vanishing probability to each reaction if the required number of interacting substrates is present), then reachability properties are equivalent in the two settings. The analysis of two types of global dynamical properties of DRNs is addressed: *irreducibility*, i.e., the ability to reach any discrete state from any other state; and *recurrence*, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The necessary and sufficient conditions obtained involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.

## Mathematics Subject Classification

37N25 92C42 92D25 92C45 80A30## Notes

### Acknowledgments

We thank Andrei Caldararu for helpful comments and suggestions. The work of LP was supported by the Swiss SystemsX.ch project. The work of GC was supported by NIH grant R01GM086881. The work of HK was supported by the Swiss National Science Foundation, grant no. PP00P2 128503.

## References

- Anderson D, Craciun G, Kurtz T (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bull Math Biol 72:1947–1970CrossRefMATHMathSciNetGoogle Scholar
- Angeli D, Leenheer PD, Sontag ED (2007) A petri net approach to the study of persistence in chemical reaction networks. Math Biosci 210(2):598–618CrossRefMATHMathSciNetGoogle Scholar
- Cohen H (1993) A course in computational algebraic number theory. Springer, BerlinGoogle Scholar
- Craciun G, Nazarov F, Pantea C (2013) Persistence and permanence of mass-action and power-law dynamical systems. SIAM J Appl Math 73(1):305–329Google Scholar
- Craciun G, Tang Y, Feinberg M (2006) Understanding bistability in complex enzyme-driven reaction networks. Proc Natl Acad Sci 103(23):8697–8702CrossRefMATHGoogle Scholar
- Fages F, Soliman S (2008) Abstract interpretation and types for systems biology. Theor Comput Sci 403(1):52–70CrossRefMATHMathSciNetGoogle Scholar
- Feinberg M (1979) Lectures on chemical reaction networks. In: Notes of lectures given at the Mathematics Research Center of the University of Wisconsin. http://www.chbmeng.ohio-state.edu/feinberg/LecturesOnReactionNetworks/
- Feinberg M (1987) Chemical reaction network structure and the stability of complex isothermal reactors-I. The deficiency zero and deficiency one theorems. Chem Eng Sci 42(10):2229–2268CrossRefGoogle Scholar
- Johnston M, Siegel D, Szederkényi G (2012) A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks. J Math Chem 50:274–288CrossRefMATHMathSciNetGoogle Scholar
- Lawler GF (2006) Introduction to stochastic processes, 2nd edn. Chapman & Hall/CRC, LondonGoogle Scholar
- Le Novère N, Bornstein B, Broicher A, Courtot M, Donizelli M, Dharuri H, Li L, Sauro H, Schilstra M, Shapiro B, Snoep JL, Hucka M (2006) BioModels database: a free, centralized database of curated, published, quantitative kinetic models of biochemical and cellular systems. Nucl Acids Res 34(Database issue):D689–D691Google Scholar
- Leloup JC, Goldbeter A (1999) Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in
*Drosophila*. J Theor Biol 198(3):445–459CrossRefGoogle Scholar - Murata T (1989) Petri nets: properties, analysis and applications. Proc IEEE 77(4):541–580CrossRefGoogle Scholar
- Petri CA (1962) Kommunikation mit Automaten. PhD thesis, University of BonnGoogle Scholar
- Shinar G, Feinberg M (1987) Concordant chemical reaction networks. Math Biosci 240(2):92–113CrossRefMathSciNetGoogle Scholar
- Shiu A, Sturmfels B (2010) Siphons in chemical reaction networks. Bull Math Biol 72(6):1448–1463CrossRefMATHMathSciNetGoogle Scholar
- Wilkinson DJ (2006) Stochastic modelling for systems biology. Chapman and Hall/CRC, LondonGoogle Scholar