# Links between topology of the transition graph and limit cycles in a two-dimensional piecewise affine biological model

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## Abstract

A class of piecewise affine differential (PWA) models, initially proposed by Glass and Kauffman (in J Theor Biol 39:103–129, 1973), has been widely used for the modelling and the analysis of biological switch-like systems, such as genetic or neural networks. Its mathematical tractability facilitates the qualitative analysis of dynamical behaviors, in particular periodic phenomena which are of prime importance in biology. Notably, a discrete qualitative description of the dynamics, called the transition graph, can be directly associated to this class of PWA systems. Here we present a study of periodic behaviours (i.e. limit cycles) in a class of two-dimensional piecewise affine biological models. Using concavity and continuity properties of Poincaré maps, we derive structural principles linking the topology of the transition graph to the existence, number and stability of limit cycles. These results notably extend previous works on the investigation of structural principles to the case of unequal and regulated decay rates for the 2-dimensional case. Some numerical examples corresponding to minimal models of biological oscillators are treated to illustrate the use of these structural principles.

## Keywords

Biological regulatory networks Piecewise affine differential models Limit cycles Structural principles Transition graph## Mathematics Subject Classification (2000)

34K13 92B05## Notes

### Acknowledgments

We would like to thank Denis Thieffry for critical reading of the manuscript, and also on of the reviewers for his careful reading and many useful suggestions. W. Abou-Jaoudé was supported in part by the LabEx MemoLife (http://www.memolife.biologie.ens.fr). M. Chaves and J. L. Gouzé were supported in part by the projects GeMCo (ANR 2010 BLANC020101), ColAge (Inria-Inserm Large Scale Initiative Action), RESET (Investissements dAvenir, Bioinformatique), and also by the LABEX SIGNALIFE (ANR-11-LABX-0028-01).

## References

- Abou-Jaoudé W, Ouattara D, Kaufman M (2009) From structure to dynamics: frequency tuning in the p53-mdm2 network i. Logical approach. J Theor Biol 258:561–577CrossRefGoogle Scholar
- Abou-Jaoudé W, Chaves M, Gouzé JL (2011) A theoretical exploration of birhythmicity in the p53-mdm2 network. PLoS One 6:e17,075Google Scholar
- Bar-Or R, Maya R, Segel L, Alon U, Levine A et al (2000) Generation of oscillations by the p53-mdm2 feedback loop: a theoretical and experimental study. Proc Natl Acad Sci USA 97:11,250–11,255CrossRefGoogle Scholar
- Casey R, de Jong H, Gouzé JL (2005) Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J Math Biol 52:27–56CrossRefGoogle Scholar
- Dayarian A, Chaves M, Sontag E, Sengupta A (2009) Shape, size, and robustness: feasible regions in the parameter space of biochemical networks. PLoS Comput Biol 5:e1000, 256Google Scholar
- Decroly O, Goldbeter A (1982) Birhythmicity, chaos, and other patterns of temporal self-organization in a multiply regulated biochemical system. Proc Natl Acad Sci USA 79:6917–6921CrossRefzbMATHMathSciNetGoogle Scholar
- de Jong H, Gouzé JL, Hernandez C, Page M, Sari T et al (2004) Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bull Math Biol 66:301–340CrossRefMathSciNetGoogle Scholar
- Edwards R (2000) Analysis of continuous-time switching networks. Physica D 146:165–199CrossRefzbMATHMathSciNetGoogle Scholar
- Edwards R, Kim S, van den Driessche P (2011) Control design for sustained oscillation in a two-gene regulatory network. J Math Biol 62:453–478CrossRefzbMATHMathSciNetGoogle Scholar
- Farcot E (2006) Geometric properties of a class of piecewise affine biological network models. J Math Biol 52:373–418CrossRefzbMATHMathSciNetGoogle Scholar
- Farcot E, Gouzé JL (2009) Periodic solutions of piecewise affine gene network models with non uniform decay rates: the case of a negative feedback loop. Acta Biotheo 57:429–455CrossRefGoogle Scholar
- Farcot E, Gouzé JL (2010) Limit cycles in piecewise-affine gene network models with multiple interaction loops. Int J Syst Sci 41:118–130CrossRefGoogle Scholar
- Ferrell J (1996) Tripping the switch fantastic: how a protein kinase cascade can convert graded inputs into switch-like outputs. Trends Biochem Sci 21:460–466CrossRefGoogle Scholar
- Filippov A (1988) Differential equations with discontinunos righthand sides. Kluwer Academic Publishers, DordrechtGoogle Scholar
- Gedeon T (2000) Global dynamics of neural nets with infinite gain. Physica D 146:200–212CrossRefzbMATHMathSciNetGoogle Scholar
- Glass L (1975a) Classification of biological networks by their qualitative dynamics. J Theor Biol 54:85–107CrossRefGoogle Scholar
- Glass L (1975b) Combinatorial and topological methods in nonlinear chemical kinetics. J Chem Phys 63:1325–1335CrossRefGoogle Scholar
- Glass L (1977a) Statistical mechanics and statistical methods in theory and application. In: Landman U (ed) Combinatorial aspects of dynamics in biological systems. Springer, USAGoogle Scholar
- Glass L (1977b) Global analysis of nonlinear chemical kinetics. In: Berne B (ed) Statistical mechanics, part B: time-dependent processes. Plenum Press, New York, pp 311–349CrossRefGoogle Scholar
- Glass L, Kauffman S (1973) The logical analysis of continuous, non-linear biochemical control networks. J Theor Biol 39:103–129CrossRefGoogle Scholar
- Glass L, Pasternack J (1978a) Prediction of limit cycles in mathematical models of biological oscillations. Bull Math Biol 40:27–44CrossRefMathSciNetGoogle Scholar
- Glass L, Pasternack J (1978b) Stable oscillations in mathematical models of biological control systems. J Math Biol 6:207–223CrossRefzbMATHMathSciNetGoogle Scholar
- Goldbeter A (1996) Biochemical oscillations and cellular rythms. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Goldbeter A (2002) Computational approaches to cellular rhythms. Nature 420:238–245CrossRefGoogle Scholar
- Gouzé JL, Sari T (2002) A class of piecewise linear differential equations arising in biological models. Dyn Syst 17:299–316CrossRefzbMATHMathSciNetGoogle Scholar
- Keizer J, Li Y, Stojilkovic S, Rinzel J (1995) Insp3-induced \(\text{ Ca }^{2+}\) excitability of the endoplasmic reticulum. Mol Biol Cell 6:945951CrossRefGoogle Scholar
- Kim D, Kwon Y, Cho K (2007) Coupled positive and negative feedback circuits form an essential building block of cellular signaling pathways. Bioessays 29:85–90CrossRefGoogle Scholar
- Lewis J, Glass L (1992) Nonlinear and symbolic dynamics of neural networks. Neural Comput 4:621–642CrossRefGoogle Scholar
- Lu L, Edwards R (2010) Structural principles for periodic orbits in glass networks. J Math Biol 60:513–41CrossRefMathSciNetGoogle Scholar
- Lu L, Edwards R (2011) Structural principles for complex dynamics in glass networks. Int J Bifurcat Chaos 21:237–254CrossRefzbMATHMathSciNetGoogle Scholar
- McCulloch W, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biophys 5:115–133CrossRefzbMATHMathSciNetGoogle Scholar
- Mestl T, Plahte E, Omholt S (1995) Periodic solutions in systems of piecewise-linear differential equations. Dyn Stab Syst 10:179–193CrossRefzbMATHMathSciNetGoogle Scholar
- Omholt S, Kefang X, Andersen O, Plahte E (1998) Description and analysis of switchlike regulatory networks exemplified by a model of cellular iron homeostasis. J Theor Biol 195:339–350CrossRefGoogle Scholar
- Perko L (1991) Differential equations and dynamical systems. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Plahte E, Mestl T, Omholt S (1995) Stationary states in food web models with threshold relationships. J Biol Syst 3:569–577CrossRefGoogle Scholar
- Ptashne M (1992) A genetic switch: phage lambda and higher organisms. Cell Press and Blackwell Science, CambridgeGoogle Scholar
- Ropers D, de Jong H, Page M, Schneider D, Geiselmann J (2006) Qualitative simulation of the carbon starvation response in
*Escherichia coli*. Biosystems 84:124–152CrossRefGoogle Scholar - Smith H (1986) Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal 10:1037–52CrossRefzbMATHMathSciNetGoogle Scholar
- Snoussi E (1989) Qualitative dynamics of piecewise-linear differential equations: a discrete mapping approach. Dyn Stab Syst 4:189–207CrossRefzbMATHMathSciNetGoogle Scholar
- Strogatz S (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, BoulderGoogle Scholar
- Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42:563–585CrossRefGoogle Scholar
- Thomas R, d’Ari R (1990) Biological feedback. CRC Press, FloridazbMATHGoogle Scholar
- Tsai T, Choi Y, Ma W, Pomerening J, Tang C, Ferrell J (2008) Robust, tunable biological oscillations from interlinked positive and negative feedback loops. Science 321:126–129CrossRefGoogle Scholar