Graphtheoretic conditions for zeroeigenvalue Turing instability in general chemical reaction networks
Pages: 1207  1226,
Issue 4,
August
2013
doi:10.3934/mbe.2013.10.1207 Abstract
References
Full text (429.5K)
Related Articles
Maya Mincheva  Department of Mathematical Sciences, Northern Illinois University, Dekalb, IL 60115, United States (email)
Gheorghe Craciun  Department of Mathematics and Department of Biomolecular Chemistry, University of WisconsinMadison, Madison, WI 53706, United States (email)
1 
M. Banaji and G. Craciun, Graphtheoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Comm. in Math. Sciences, 7 (2009), 867900. 

2 
M. Banaji and G. Craciun, Graph theoretic approaches to injectivity in general chemical reaction systems, Adv. in Appl. Math., 44 (2010), 168184. 

3 
E. D. Conway, Diffusion and predatorprey interaction: Pattern in closed systems, Partial differential equations and dynamical systems, 85133, 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), 15261546. 

6 
G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: II. The SpeciesReaction graph, SIAM J. Appl. Math., 66 (2006), 13211338. 

7 
G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzymedriven reaction networks, PNAS, 103 (2006), 86978702. 

8 
P. Donnell, M. Banaji and S. Baigent, Stability in generic mitochondrial models, J. Math. Chem., 46 (2009), 322339. 

9 
M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal., 49 (1972) 187194. 

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.ohiostate.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), 311370. 

12 
F. R. Gantmakher, "Applications of the Theory of Matrices," Interscience, New York, 1960. 

13 
B. N. Goldstein and A. N. Ivanova, Hormonal regulation of 6phosphofructo2kinase fructose2.6bisphosphatase: Kinetic models, FEBS Lett., 217 (1987), 212215. 

14 
B. N. Goldstein and A. A. Maevsky, Critical switch of the metabolic fluxes by phosphofructo2kinase: Fructose2, 6bisphosphatase, FEBS Lett., 532 (2002), 295299. 

15 
F. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81116. 

16 
P. Lancaster and M. Tismenetsky, "The Theory of Matrices," Academic Press, Orlando, 1985. 

17 
M. Mincheva and G. Craciun, Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks, Proc. IEEE, 96 (2008), 12811291. 

18 
M. Mincheva and M. R. Roussel, Graphtheoretic methods for the analysis of chemical and chemical networks I. Multistability and oscillations in massaction kinetics models, J. Math. Biol., 55 (2007), 6186. 

19 
M. Mincheva and M. R. Roussel, A graphtheoretic approach for detecting Turing bifurcations, J. Chem. Phys., 125 (2006), 204102. 

20 
J. D. Murray, "Mathematical Biology," 2nd ed., SpringerVerlag, New York, 1993. 

21 
R. A. Satnoianu, M. Menzinger and P. K. Maini, Turing instabilities in general systems, J. Math. Biol., 41 (2000), 493512. 

22 
R. A. Satnoianu and P. van den Driessche, Some remarks on matrix stability with application to Turing instability, Lin. Alg. Appl., 398 (2005), 6974. 

23 
G. Shinar and M. Feinberg, Concordant chemical reaction networks, Math. Biosci., 240 (2012), 92113. 

24 
E. de Silva and M. P. H. Stumpf, Complex networks and simple models in biology, J. R. Soc. Interface, 2 (2005), 419430. 

25 
R. Thomas, D. Thieffry and M. Kaufman, Dynamical behaviour of biological regulatory networks, Bull. Math. Biol., 57 (1995), 247276. 

26 
A. Turing, The chemical basis of morphogenesis, Phil. Trans. R Soc. London B, 237 (1952), 3772. 

27 
A. Volpert and A. Ivanova, "Mathematical Modeling," (Russian), Nauka, Moscow, 1987, 57102. 

28 
L. Wang and M. Y. Li, Diffusiondriven instability in reactiondiffusion systems, J. Math. Anal. Appl., 254 (2001), 138153. 

29 
C. Wiuf and E. Feliu, A unified framework for preclusion of multiple steady states in networks of interacting species, arXiv:1202.3621, (2012). 

Go to top
