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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 4: 1207-1226. doi: 10.3934/mbe.2013.10.1207
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Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks

  • 1.
    Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115
  • 2.
    Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706
  • Received: 01 July 2012 Accepted: 29 June 2018 Published: 01 June 2013

  • MSC : Primary: 92C15; Secondary: 80A30.

  • 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[J]. Mathematical Biosciences and Engineering, 2013, 10(4): 1207-1226. doi: 10.3934/mbe.2013.10.1207

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  • 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.


    [1] Comm. in Math. Sciences, 7 (2009), 867-900.
    [2] Adv. in Appl. Math., 44 (2010), 168-184.
    [3] Partial differential equations and dynamical systems, 85-133, Res. Notes in Math., 101, Pitman, Boston, MA, 1984.
    [4] Ph.D thesis, Ohio State University, 2002.
    [5] SIAM J. Appl. Math., 65 (2005), 1526-1546.
    [6] SIAM J. Appl. Math., 66 (2006), 1321-1338.
    [7] PNAS, 103 (2006), 8697-8702.
    [8] J. Math. Chem., 46 (2009), 322-339.
    [9] Arch. Rational Mech. Anal., 49 (1972) 187-194.
    [10] 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] Arch. Rational Mech. Anal., 132 (1995), 311-370.
    [12] Interscience, New York, 1960.
    [13] FEBS Lett., 217 (1987), 212-215.
    [14] FEBS Lett., 532 (2002), 295-299.
    [15] Arch. Rational Mech. Anal., 47 (1972), 81-116.
    [16] Academic Press, Orlando, 1985.
    [17] Proc. IEEE, 96 (2008), 1281-1291.
    [18] J. Math. Biol., 55 (2007), 61-86.
    [19] J. Chem. Phys., 125 (2006), 204102.
    [20] 2nd ed., Springer-Verlag, New York, 1993.
    [21] J. Math. Biol., 41 (2000), 493-512.
    [22] Lin. Alg. Appl., 398 (2005), 69-74.
    [23] Math. Biosci., 240 (2012), 92-113.
    [24] J. R. Soc. Interface, 2 (2005), 419-430.
    [25] Bull. Math. Biol., 57 (1995), 247-276.
    [26] Phil. Trans. R Soc. London B, 237 (1952), 37-72.
    [27] (Russian), Nauka, Moscow, 1987, 57-102.
    [28] J. Math. Anal. Appl., 254 (2001), 138-153.
    [29] arXiv:1202.3621, (2012).

  • This article has been cited by:

    1. Polly Y. Yu, Gheorghe Craciun, Mathematical Analysis of Chemical Reaction Systems, 2018, 58, 00212148, 733, 10.1002/ijch.201800003
    2. Marc R. Roussel, Talmon Soares, Graph-based, dynamics-preserving reduction of (bio)chemical systems, 2024, 89, 0303-6812, 10.1007/s00285-024-02138-0
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