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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, United States |
2. | Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706 |
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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