Abstract
It is well known that oscillations in models of biochemical reaction networks can arise as a result of a single negative cycle. On the other hand, methods for finding general network conditions for potential oscillations in large biochemical reaction networks containing many cycles are not well developed. A biochemical reaction network with any number of species is represented by a simple digraph and is modeled by an ordinary differential equation (ODE) system with non-mass action kinetics. The obtained graph-theoretic condition generalizes the negative cycle condition for oscillations in ODE models to the existence of a pair of subnetworks, where each subnetwork contains an even number of positive cycles. The technique is illustrated with a model of genetic regulation.
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References
Clarke B.L.: Stability of complex reaction networks. Adv. Chem. Phys. 43, 1–215 (1980)
Clarke B.L.: Stoichiometric network analysis. Cell Biophys. 12, 237–253 (1988)
Craciun G., Feinberg M.: Multiple equilibria in complex chemical reaction networks: I. The injectivity property. SIAM J. Appl. Math. 65, 1526–1546 (2005)
Craciun G., Feinberg M.: Multiple equilibria in complex chemical reaction networks: II. The species-reactions graph. SIAM J. Appl. Math. 66, 1321–1338 (2006)
de Silva E., Stumpf M.P.H.: Complex networks and simple models in biology. J. R. Soc. Interface 2, 419–430 (2005)
Fallat S.: Bidiagonal factorizations of totally nonnegative matrices. Am. Math. Mon. 108, 697–712 (2001)
Feinberg M.: Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49, 187–194 (1972)
Forger D., Peskin C.: A detailed predictive model of the mammalian circadian clock. Proc. Natl. Acad. Sci. 100, 14806–14811 (2003)
Gantmakher F.R.: Applications of the Theory of Matrices. Interscience, New York (1959)
Gatermann K.: Counting stable solutions of sparse polynomial systems in chemistry. Contemp. Math. 286, 53–69 (2001)
Goldbeter A.: A model for circadian oscillations in the Drosophila period protein (PER). Proc. R. Soc. Lond. B Biol. Sci. 261, 319–324 (1995)
Goldbeter A.: Biochemical Oscillations and Cellular Rhythms: The Molecular Basis of Periodic and Chaotic Behaviour. Cambridge University Press, Cambridge (1996)
Goodman B.: Oscillatory behavior in enzymatic control processes. Adv. Enzym. Regul. 3, 425–439 (1965)
Griffith J.: Mathematics of cellular control processes. I. Negative feedback to gene. J. Theor. Biol. 20, 202–208 (1968)
Harary F.: Graph Theory. Addison-Wesley, Reading (1968)
Hartwel L., Hopfield J., Leibler S., Murray A.: From molecular to modular cell biology. Nature 402, C47–C52 (1999)
Jacob F., Monod J.: Genetic regulatory mechanisms in the synthesis of proteins. J. Mol. Biol. 3, 318–356 (1961)
Keener J., Sneyd J.: Mathematical Physiology. Springer, New York (1998)
Kitano H.: Systems biology: a brief overview. Science 295, 1662–1664 (2002)
Kuznetsov Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (1998)
Lancaster P., Tismenetsky M.: The Theory of Matrices. Academic Press, Orlando (1985)
Liu W.M.: Criterion of Hopf bifurcations without using eigenvalues. J. Math. Anal. Appl. 182, 250–256 (1994)
Mahaffy J.: Genetic control models with diffusion and delays. Math. Biosci. 90, 519–533 (1988)
Mahaffy J., Pao C.V.: Models of genetic control with time delays and spatial effects. J. Math. Biol. 20, 39–57 (1984)
Maybee J., Olesky D., van den Driessche P., Wiener G.: Matrices, digraphs and determinants. SIAM J. Matrix Anal. Appl. 10, 500–519 (1989)
Milo R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U.: Network motifs: simple building blocks of complex networks. Science 298, 824–827 (2002)
Mincheva M.: Graph-theoretic condition for oscillations arising from pairs of subnetworks. Bull. Math. Biol 73, 2277–2304 (2011)
Mincheva M., Craciun G.: Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks. Proc. IEEE 96, 1281–1291 (2008)
Mincheva M., Roussel M.R.: Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation models. J. Math. Biol. 55, 61–86 (2007)
Mincheva M., Roussel M.R.: A graph-theoretic method for detecting potential Turing bifurcations. J. Chem. Phys. 125, 204102 (2006)
Murray J.D.: Mathematical Biology. 2nd edn. Springer, New York (1993)
Prigogine I., Lefever R.: Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48, 1695–1703 (1968)
Saithong T., Painter K., Millar A.: The contributions of interlocking loops and extensive nonlinearity to the properties of the circadian clocks models. PLOS One 5, e13867 (2010)
Smolen P., Baxter D.A., Byrne J.H.: Modeling transcriptional control in gene networks—methods, recent results, and future directions. Bull. Math. Biol. 62, 247–292 (2000)
Strumfels B., Myers M., Guckenheimer J.: Computing Hopf bifurcations. SIAM J Numer. Analy. 34, 1–21 (1997)
Thomas R., Thieffry D., Kaufman M.: Dynamical behaviour of biological regulatory networks. Bull. Math. Biol. 57, 247–276 (1995)
Tyson J.J.: Classification of instabilities in chemical reaction systems. J. Chem. Phys. 62, 1010–1015 (1975)
Tyson J.J.: Modeling the cell division cycle: cdc2 and cyclin interactions. Proc. Natl. Acad. Sci. 88, 7328–7332 (1991)
Tyson J.J., Chen K.C., Novak B.: Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol. 15, 221–231 (2003)
A. Volpert, A. Ivanova, in Mathematical Modeling (Russian), (Nauka, Moscow, 1987), pp. 57–102
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Mincheva, M. Oscillations in non-mass action kinetics models of biochemical reaction networks arising from pairs of subnetworks. J Math Chem 50, 1111–1125 (2012). https://doi.org/10.1007/s10910-011-9955-8
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DOI: https://doi.org/10.1007/s10910-011-9955-8