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2013, 10(4): 1207-1226. doi: 10.3934/mbe.2013.10.1207

## Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks

 1 Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115, United States 2 Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706

Received  July 2012 Revised  March 2013 Published  June 2013

We describe a necessary condition for zero-eigenvalue Turing instability, i.e., Turing instability arising from a real eigenvalue changing sign from negative to positive, for general chemical reaction networks modeled with mass-action kinetics. The reaction mechanisms are represented by the species-reaction graph (SR graph), which is a bipartite graph with different nodes representing species and reactions. If the SR graph satisfies certain conditions, similar to the conditions for ruling out multiple equilibria in spatially homogeneous differential equations systems, then the corresponding mass-action reaction-diffusion system cannot exhibit zero-eigenvalue Turing instability for any parameter values. On the other hand, if the graph-theoretic condition for ruling out zero-eigenvalue Turing instability is not satisfied, then the corresponding model may display zero-eigenvalue Turing instability for some parameter values. The technique is illustrated with a model of a bifunctional enzyme.
Citation: Maya Mincheva, Gheorghe Craciun. Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1207-1226. doi: 10.3934/mbe.2013.10.1207
##### References:
 [1] M. Banaji and G. Craciun, Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Comm. in Math. Sciences, 7 (2009), 867-900. [2] M. Banaji and G. Craciun, Graph theoretic approaches to injectivity in general chemical reaction systems, Adv. in Appl. Math., 44 (2010), 168-184. doi: 10.1016/j.aam.2009.07.003. [3] E. D. Conway, Diffusion and predator-prey interaction: Pattern in closed systems, Partial differential equations and dynamical systems, 85-133, Res. Notes in Math., 101, Pitman, Boston, MA, 1984. [4] G. Craciun, "Systems of Nonlinear Equations Deriving from Complex Chemical Reaction Networks," Ph.D thesis, Ohio State University, 2002. [5] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: I. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546. doi: 10.1137/S0036139904440278. [6] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: II. The Species-Reaction graph, SIAM J. Appl. Math., 66 (2006), 1321-1338. doi: 10.1137/050634177. [7] G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS, 103 (2006), 8697-8702. doi: 10.1073/pnas.0602767103. [8] P. Donnell, M. Banaji and S. Baigent, Stability in generic mitochondrial models, J. Math. Chem., 46 (2009), 322-339. doi: 10.1007/s10910-008-9464-6. [9] M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal., 49 (1972) 187-194. [10] M. Feinberg, "Lectures on Chemical Reaction Networks," Written Version of Lectures Given at the Mathematical Research Center, University of Wisconsin, Madison, WI, 1979. Available at http://www.chbmeng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks. [11] M. Feinberg, Existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal., 132 (1995), 311-370. doi: 10.1007/BF00375614. [12] F. R. Gantmakher, "Applications of the Theory of Matrices," Interscience, New York, 1960. doi: 10.1063/1.3062774. [13] B. N. Goldstein and A. N. Ivanova, Hormonal regulation of 6-phosphofructo-2-kinase fructose-2.6-bisphosphatase: Kinetic models, FEBS Lett., 217 (1987), 212-215. doi: 10.1016/0014-5793(87)80665-8. [14] B. N. Goldstein and A. A. Maevsky, Critical switch of the metabolic fluxes by phosphofructo-2-kinase: Fructose-2, 6-bisphosphatase, FEBS Lett., 532 (2002), 295-299. doi: 10.1016/S0014-5793(02)03639-6. [15] F. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81-116. [16] P. Lancaster and M. Tismenetsky, "The Theory of Matrices," Academic Press, Orlando, 1985. doi: 10.5802/aif.1029. [17] M. Mincheva and G. Craciun, Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, Proc. IEEE, 96 (2008), 1281-1291. doi: 10.1109/JPROC.2008.925474. [18] M. Mincheva and M. R. Roussel, Graph-theoretic methods for the analysis of chemical and chemical networks I. Multistability and oscillations in mass-action kinetics models, J. Math. Biol., 55 (2007), 61-86. doi: 10.1007/s00285-007-0099-1. [19] M. Mincheva and M. R. Roussel, A graph-theoretic approach for detecting Turing bifurcations, J. Chem. Phys., 125 (2006), 204102. [20] J. D. Murray, "Mathematical Biology," 2nd ed., Springer-Verlag, New York, 1993. doi: 10.1007/b98869. [21] R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512. doi: 10.1007/s002850000056. [22] R. A. Satnoianu and P. van den Driessche, Some remarks on matrix stability with application to Turing instability, Lin. Alg. Appl., 398 (2005), 69-74. doi: 10.1016/j.laa.2004.04.003. [23] G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92-113. doi: 10.1016/j.mbs.2012.05.004. [24] E. de Silva and M. P. H. Stumpf, Complex networks and simple models in biology, J. R. Soc. Interface, 2 (2005), 419-430. [25] R. Thomas, D. Thieffry and M. Kaufman, Dynamical behaviour of biological regulatory networks, Bull. Math. Biol., 57 (1995), 247-276. [26] A. Turing, The chemical basis of morphogenesis, Phil. Trans. R Soc. London B, 237 (1952), 37-72. [27] A. Volpert and A. Ivanova, "Mathematical Modeling," (Russian), Nauka, Moscow, 1987, 57-102. [28] L. Wang and M. Y. Li, Diffusion-driven instability in reaction-diffusion systems, J. Math. Anal. Appl., 254 (2001), 138-153. doi: 10.1006/jmaa.2000.7220. [29] C. Wiuf and E. Feliu, A unified framework for preclusion of multiple steady states in networks of interacting species, arXiv:1202.3621, (2012).

show all references

##### References:
 [1] M. Banaji and G. Craciun, Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Comm. in Math. Sciences, 7 (2009), 867-900. [2] M. Banaji and G. Craciun, Graph theoretic approaches to injectivity in general chemical reaction systems, Adv. in Appl. Math., 44 (2010), 168-184. doi: 10.1016/j.aam.2009.07.003. [3] E. D. Conway, Diffusion and predator-prey interaction: Pattern in closed systems, Partial differential equations and dynamical systems, 85-133, Res. Notes in Math., 101, Pitman, Boston, MA, 1984. [4] G. Craciun, "Systems of Nonlinear Equations Deriving from Complex Chemical Reaction Networks," Ph.D thesis, Ohio State University, 2002. [5] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: I. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546. doi: 10.1137/S0036139904440278. [6] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: II. The Species-Reaction graph, SIAM J. Appl. Math., 66 (2006), 1321-1338. doi: 10.1137/050634177. [7] G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS, 103 (2006), 8697-8702. doi: 10.1073/pnas.0602767103. [8] P. Donnell, M. Banaji and S. Baigent, Stability in generic mitochondrial models, J. Math. Chem., 46 (2009), 322-339. doi: 10.1007/s10910-008-9464-6. [9] M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal., 49 (1972) 187-194. [10] M. Feinberg, "Lectures on Chemical Reaction Networks," Written Version of Lectures Given at the Mathematical Research Center, University of Wisconsin, Madison, WI, 1979. Available at http://www.chbmeng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks. [11] M. Feinberg, Existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal., 132 (1995), 311-370. doi: 10.1007/BF00375614. [12] F. R. Gantmakher, "Applications of the Theory of Matrices," Interscience, New York, 1960. doi: 10.1063/1.3062774. [13] B. N. Goldstein and A. N. Ivanova, Hormonal regulation of 6-phosphofructo-2-kinase fructose-2.6-bisphosphatase: Kinetic models, FEBS Lett., 217 (1987), 212-215. doi: 10.1016/0014-5793(87)80665-8. [14] B. N. Goldstein and A. A. Maevsky, Critical switch of the metabolic fluxes by phosphofructo-2-kinase: Fructose-2, 6-bisphosphatase, FEBS Lett., 532 (2002), 295-299. doi: 10.1016/S0014-5793(02)03639-6. [15] F. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81-116. [16] P. Lancaster and M. Tismenetsky, "The Theory of Matrices," Academic Press, Orlando, 1985. doi: 10.5802/aif.1029. [17] M. Mincheva and G. Craciun, Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, Proc. IEEE, 96 (2008), 1281-1291. doi: 10.1109/JPROC.2008.925474. [18] M. Mincheva and M. R. Roussel, Graph-theoretic methods for the analysis of chemical and chemical networks I. Multistability and oscillations in mass-action kinetics models, J. Math. Biol., 55 (2007), 61-86. doi: 10.1007/s00285-007-0099-1. [19] M. Mincheva and M. R. Roussel, A graph-theoretic approach for detecting Turing bifurcations, J. Chem. Phys., 125 (2006), 204102. [20] J. D. Murray, "Mathematical Biology," 2nd ed., Springer-Verlag, New York, 1993. doi: 10.1007/b98869. [21] R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493-512. doi: 10.1007/s002850000056. [22] R. A. Satnoianu and P. van den Driessche, Some remarks on matrix stability with application to Turing instability, Lin. Alg. Appl., 398 (2005), 69-74. doi: 10.1016/j.laa.2004.04.003. [23] G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92-113. doi: 10.1016/j.mbs.2012.05.004. [24] E. de Silva and M. P. H. Stumpf, Complex networks and simple models in biology, J. R. Soc. Interface, 2 (2005), 419-430. [25] R. Thomas, D. Thieffry and M. Kaufman, Dynamical behaviour of biological regulatory networks, Bull. Math. Biol., 57 (1995), 247-276. [26] A. Turing, The chemical basis of morphogenesis, Phil. Trans. R Soc. London B, 237 (1952), 37-72. [27] A. Volpert and A. Ivanova, "Mathematical Modeling," (Russian), Nauka, Moscow, 1987, 57-102. [28] L. Wang and M. Y. Li, Diffusion-driven instability in reaction-diffusion systems, J. Math. Anal. Appl., 254 (2001), 138-153. doi: 10.1006/jmaa.2000.7220. [29] C. Wiuf and E. Feliu, A unified framework for preclusion of multiple steady states in networks of interacting species, arXiv:1202.3621, (2012).
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