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A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems

1 Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Dr, Madison, WI 53706, USA;
2 Faculty of Mathematics, University of Vienna, 1010 Vienna, Austria;
3 Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA;
4 Department of Biomolecular Chemistry, University of Wisconsin-Madison, 420 Henry Mall, WI 53706, USA

Special Issues: Mathematical analysis of reaction networks: theoretical advances and applications

Mass-action kinetics and its generalizations appear in mathematical models of (bio)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.
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Keywords reaction network; generalized Birch's theorem; generalized mass-action; vertex-balanced steady states

Citation: Gheorghe Craciun, Stefan Muller, Casian Pantea, Polly Y. Yu. A generalization of Birchs theorem and vertex-balanced steady states for generalized mass-action systems. Mathematical Biosciences and Engineering, 2019, 16(6): 8243-8267. doi: 10.3934/mbe.2019417

References

  • 1. M. Feinberg, Complex balancing in general kinetic systems, Arch. Ration. Mech. Anal., 49 (1972), 187-194.
  • 2. F. Horn and R. Jackson, General mass action kinetics, Arch. Ration. Mech. Anal., 47 (1972), 81-116.
  • 3. F. Horn, Necessary and Sufficient Conditions for Complex Balancing in Chemical-Kinetics, Arch. Ration. Mech. Anal., 49 (1972), 172-186.
  • 4. M. A. Savageau, Biochemical systems analysis. I. Some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol., 25 (1969), 365-369.
  • 5. E. O. Voit, Biochemical systems theory: a review, ISRN Biomath., 2013 (2013), Article ID 897658.
  • 6. S. Müller and G. Regensburger, Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents, (2014), Lecture notes in computer science, 8660 LNCS, Springer, 302-323.
  • 7. S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM J. Appl. Math., 72 (2012), 1-23.
  • 8. M. D. Johnston, Translated Chemical Reaction Networks, Bull. Math. Biol., 76 (2014), 1081-1116.
  • 9. L. Brenig, Complete factorization and analytic solutions of generalized Lotka-Volterra equations, Physics Letters, 133 (1989), 378-382.
  • 10. L. Brenig and A. Goriely, Universal canonical forms for time-continuous dynamical systems, Phys. Rev. A, 40 (1989), 4119-4122.
  • 11. M. Feinberg, Lectures on chemical reaction networks, 1979. Available from: https://crnt.osu.edu/LecturesOnReactionNetworks.
  • 12. J. Gunawardena, Chemical reaction network theory for in-silico biologists, 2003. Available from: http://vcp.med.harvard.edu/papers/crnt.pdf.
  • 13. G. Craciun, A. Dickenstein, A. Shiu, et al., Toric dynamical systems, J. Symbolic Comput., 44 (2009), 1551-1565.
  • 14. B. Boros, S. Müller and G. Regensburger, Complex-balanced equilibria of generalized mass-action systems: Necessary conditions for linear stability, ArXiv e-prints, arXiv:1906.12214 [math.DS].
  • 15. L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 8 (2005).
  • 16. G. Craciun, L. D. Gracía-Puente and F. Sottile, Some geometrical aspects of control points for toric patches, Mathematical methods for curves and surfaces, Springer, 111-135.
  • 17. J. D. Brunner and G. Craciun, Robust persistence and permanence of polynomial and power law dynamical systems, SIAM J. Appl. Math., 78 (2018), 801-825.
  • 18. G. Craciun, Polynomial dynamical systems as reaction networks and toric differential inclusions, SIAM J. Appl. Algebra Geom., 3 (2019), 87-106.
  • 19. P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Isr. J. Chem, 58 (2018), 733-741.
  • 20. M. D. Johnston, A computational approach to steady state correspondence of regular and generalized mass action systems, Bull. Math. Biol., 77 (2015), 1065-1100.
  • 21. C. Conradi, J. Saez-Rodriguez, E. D. Gilles, et al., Using chemical reaction network theory to discard a kinetic mechanism hypothesis, IEE Proc.-Syst. Biol., 152 (2005), 243-248.
  • 22. N. I. Markevich, J. B. Hoek and B. N. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell. Biol., 164 (2004), 353-359.
  • 23. D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508.
  • 24. M. Gopalkrshnan, E. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM J. Appl. Dyn. Syst., 13 (2013), 758-797.
  • 25. G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math., 73 (2013), 305-329.
  • 26. C. Pantea, On the persistence and global stability of mass-action systems, SIAM J. Math. Anal., 44 (2012), 1636-1673.
  • 27. G. Craciun, Toric Differential Inclusions and a Proof of the Global Attractor Conjecture, ArXiv e-prints, 1-41. arXiv:1501.02860. Available from: http://arxiv.org/abs/1501.02860.
  • 28. S. Müller, J. Hofbauer and G. Regensburger, On the bijectivity of families of exponential/generalized polynomial maps, SIAM J. Appl. Algebra Geom., 3 (2019), 412-438.
  • 29. V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974.
  • 30. A. R. Shastri, Elements of Differential Topology, CRC Press, 2011.
  • 31. B. Boros, J. Hofbauer and S. Müller, On global stability of the Lotka reactions with generalized mass-action kinetics, Acta Appl. Math., 151 (2017), 53-80.
  • 32. B. Boros, J. Hofbauer, S. Müller, et al., The center problem for the Lotka reactions with generalized mass-action kinetics, Qual. Theo. Dyn. Syst., 17 (2018), 403-410.
  • 33. B. Boros, J. Hofbauer, S. Müller, et al., Planar S-systems: Global stability and the center problem, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 707-727.    

 

This article has been cited by

  • 1. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu, An Efficient Characterization of Complex-Balanced, Detailed-Balanced, and Weakly Reversible Systems, SIAM Journal on Applied Mathematics, 2020, 80, 1, 183, 10.1137/19M1244494

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