References: |
[1] B. D. Aguda and B. L. Clarke, Bistability in chemical reaction networks: theory and appli- cation to the peroxidase-oxidase reaction, J. Chem. Phys., 87 (1987), 3461–3470. [2] B. D. Aguda and Y. Tang, The kinetic origins of the restriction point in the mammalian cell cycle, Cell Prolif., 32 (1999), 321–335. [3] D. Angeli and E. Sontag, Monotone control systems, IEEE Transactions on Automatic Control, 48 (2002), 185–202. [4] D. Angeli and E. Sontag, Multi–stability in monotone input/output systems, Systems & Control Letters, 51 (2004), 1684–1698. [5] D. Angeli, J. E. Ferrell Jr. and E. Sontag, Detection of multistability, bifurcations and hys- teresis in a large class of biological positive–feedback systems, PNAS, 101 (2004), 1822–1827. [6] J. E. Bailey, Complex biology with no parameters, Nature Biotech., 19 (2001), 503–504. [7] N. Barkai and S. Leibler, Robustness in simple biochemical networks, Nature, 387 (1997), 913–917. [8] M. Eiswirth, J. B¨urger, P. Strasser and G. Ertl, Oscillating Langmuir-Hinshelwood mecha- nisms, J. Phys. Chem., 100 (1996), 19118–19123. [9] E. Beretta, F. Ventrano, F. Solimano and C. Lazzari, Some results about nonlinear chemical systems represented by trees and cycles, Bull. Math. Biol., 41 (1979), 641–664. [10] B. L. Clarke, Theorems on chemical network stability, J. Chem. Phys., 62 (1975), 773–775. [11] B. L. Clarke, Stability of topologically similar chemical networks, J. Chem. Phys., 62 (1975), 3726–3738. [12] B. L. Clarke, Stability of complex reaction networks, in “Advan. Chem. Phys.” (eds. I. Prigogine and S. Rice), New York Wiley, 43 (1980), 1–216. [13] C. Conradi, J. Saez-Rodriguez, E.-D. Gilles and J. Raisch, Using chemical reaction network theory to discard a kinetic mechanism hypothesis, IEE Proc.-Syst. Biol., 152 (2005), 243–248. [14] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: I. The injectivity property, SIAM J. App. Math., 66 (2005), 1526–1546. [15] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: II. The species-reaction graph, SIAM J. App. Math., 65 (2006), 1321–1338. [16] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks: ex- tensions to entrapped species models, IEE Proc.-Syst. Biol., 153 (2006), 179–186. [17] G. Craciun,Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS, 30 (2006), 8697–8702. [18] P. Delattre, L’evolution des syst´emes mol´eculaires. Bases th´eoretiques, Applications `a la chimie et `a la biologie, Paris: Maloine, 1971. [19] M. Domijan and M. Kirkilionis, Bistability and oscillations in chemical reaction networks, Warwick Preprint 04/2007. Submitted to Journal of Mathematical Biology. [20] M. Domijan and M. Kirkilionis, Graph theoretical conditions for Turing bifurcation in chem- ical reaction networks, in preparation. [21] P. Ellison, M. Feinberg, M.-H. Yue and H. Saltsburg, How catalytic mechanisms reveal them- selves in multiple steady-state data: II. An ethylene hydrogenation example, J. Mol. Catal. A: Chem., 154 (2000), 164–184. [22] G. L. Ermakov, A theoretical graph method for search and analysis of critical phenomena in biochemical systems. I. Graphical rules for detecting oscillators, Biochemistry (Moscow), 68 (2003), 1109–1020. [23] G. L. Ermakov, A theoretical graph method for search and analysis of critical phenomena in biochemical systems. II. Kinetic models of biochemical oscillators including two and three substances, Biochemistry (Moscow), 68 (2003), 1121–1131. [24] G. L. Ermakov and B. N. Goldstein, Simplest kinetic schemes for biochemical oscillators, Biochemistry (Moscow), 67 (2002), 473–484. [25] M. Feinberg, Complex balancing in general kinetic systems, Arch. Rational Mech. Anal., 49 (1972), 187–194. [26] M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal., 132 (1995), 311–370. [27] M. Feinberg and P. Ellison, The chemical reaction network toolbox, http://www.chbmeng.ohio-state.edu/$\sim$feinberg/crnt [28] A. Fernandez, C. Gutierrez and R. Rabanal, On local diffeomorphisms of Rn that are injec- tive, Qual. Theory Dyn. Syst., 4 (2003), 255–262. [29] R. J. Field, E. K¨or¨os and R. M. Noyes, Oscillations in chemical systems. 2. Thorough analysis of temporal oscillation in bromate-cerium-malonic acid system, J. Am. Chem. Soc., 94 (1972), 8649–8664. [30] R. J. Field and R. M. Noyes, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877–1884. [31] D. Gale and H. Nikaidˆo, The Jacobian matrix and global univalence of mappings, Mathematische Annalen, 159 (1965), 81–93. [32] K. Gatermann and B. Huber, A family of sparse polynomial systems arising in chemical reaction systems, J. Symbolic Comput., 33 (2002), 275–395. [33] A. Goldbeter, “Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour,” Cambridge University Press, Cambridge, 1996. [34] B. N. Goldstein, Switching mechanism for branched biochemical fluxes: graph-theoretical analysis, Biophys Chem., 125 (2007), 314–319. [35] B. N. Goldstein, G. Ermakov , J. J. Centelles, H. V.Westerhoff and M. Cascante, What makes biochemical networks tick?, Eur J Biochem., 271 (2004), 3877–3887. [36] 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. [37] B. N. Goldstein and A. A. Maevsky, Critical switch of the metabolic fluxes by phosphofructo- 2-kinase:fructose-2,6-bisphosphatase. A kinetic model, FEBS Lett., 532 (2002), 295–299. [38] M. Golubitsky and I. Stewart, Patterns of oscillation in coupled cell systems, In “Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J.E. Marsden” (P. Holmes, P. Newton and A. Weinstein, eds.) Springer-Verlag, New York, (2002), 243–286. [39] J.-L. Gouz´e, Positive and negative circuits in dynamical systems, J. Biol. Syst., 6 (1998), 11–15. [40] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B ! 3B;B ! C, Chem. Eng. Sci., 39 (1994), 1087–1097. [41] J. Hadamard, Sur les transformations poncuelles, Bull. de la Soc. Math. de France, 34 (1906), 71–84. [42] F. J. M. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rational Mech. Anal., 49 (1972), 172–186. [43] F. J. M. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81–116. [44] K. L. C. Hunt, P. M. Hunt and J. Ross, Nonlinear dynamics and thermodynamics of chemical reactions far from equilibrium, Annu. Rev. Phys. Chem., 41 (1990), 409–439. [45] C. Hyver, Valuers propres des syst´emes de transformation r´epresentables par des graphes en arbres, J. Theor. Biol., 42 (1973), 397–409. [46] A. N. Ivanova, Conditions for uniqueness of stationary state of kinetic systems, related, to structural scheme of reactions, Kinet. Katal., 20(4) (1979), 1019–1023. [47] A. N. Ivanova and B. L. Tarnopolskii, One approcah of the determination of a number of qualitative features iin the behaviour of kinetic systems, and realization of this approach in a computer (critical conditions, autooscillations),Kinet. Katal., 20 (1979), 1541–1548. [48] H. Kitano, Systems biology: A brief overview, Science, 295 (2002), 1662–1664. [49] W. Klonowski, Simplifying principles for chemical and enzyme reaction kinetics, Biophys. Chem., 18 (1983), 73–87. [50] K. Krischer, M. Eiswirth, and G. Ertl, Oscillatory CO oxidation on Pt(110):Modeling of temporal self-organisation, J. Chem. Phys., 96 (1992), 9161–9172. [51] T. M. Leib, D. Rumschitzki and M. Feinberg, Multiple steady states in complex isothermal CFSTRS . I. General considerations, Chem. Engng. Sci., 43 (1988), 321–328. [52] E. R. Lewis, “Network Models in Population Biology,” Springer-Verlag, 1977. [53] N. Markevich, J. Hoek and B. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell. Biol., 164 (2004), 353–359. [54] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii and U. Alon, Network motifs: Simple building blocks of complex networks, Science, 298 (2002), 824–827 . [55] M. Mincheva and M. R. Roussel, Graph-theoretic methods for the analysis of chemical and biochemical networks. I. Multistability and oscillations in ordinary differential equation mod- els, J. Math. Biol., 55 (2007), 61–68. [56] M. Mincheva and M. R. Roussel, A graph-theoretic method for detecting potential Turing bifurcations, J. Chem. Phys., 125 (2006). [57] M. Mincheva and M. R. Roussel, Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays, J. Math Biol., 55 (2007), 87–104. [58] M. Orb`an, P. de Kepper and I. R. Epstein, An iodine-free chlorite-based oscillator. The chlorite-thiosulfate reaction in a continuous flow stirred tank reactor, J. Phys. Chem., 86 (1982), 431–433. [59] M. Orb`an and I. R. Epstein, A new halogen-free chemical oscillator. Te reaction between sulfide ion and hydrogen peroxide in CSTR, J. Am. Chem. Soc., 107 (1985), 2302–2304. [60] A. S. Perelson and D. Wallwork, The arbitrary dynamic behavior of open chemical reaction systems. J. Chem. Phys., 66 (1977), 4390–4394. [61] S. Pinchuk, A counterexample to the real Jacobian conjecture, Math Z., 217 (1994), 1–4. [62] E. Plahte, T. Mestl and W. S. Omholt, Feedback circuits, stability and multistationarity in dynamical systems, J. Biol. Syst., 3 (1995), 409–413. [63] D. Rumschitzki and M. Feinberg, Multiple steady stattes in complex isothermal CFSTRs. II. Homogeneous reactors, Chem. Eng. Sci., 43 (1988), 329–337. [64] L. Sbano and M. Kirkilionis, Multiscale analysis of reaction networks, submitted to: Networks and Heterogeneous Media. [65] P. Schlosser and M. Feinberg, A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions, Chem. Eng. Sci., 49 (1994), 1749–1767. [66] P. Schuster, Landscape and molecular evolution, Phys. D, 107 (1997), 351–365. [67] E. E. Selkov, Self-oscillations in glycolysis. 1. A simple kinetic model, Eur. J. Biochem., 4 (1968), 79–86. [68] B. M Slepchenko and M. Terasaki, Cyclin aggregation and robustness of bio-switching, Mol. Biol. Cell., 14 (2003), 4695–4706. [69] E. H. Snoussi, Necessary conditions for multistationarity and stable periodicity, J. Biol. Syst., 6 (1998), 3–9. [70] E. Sontag, Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction, IEEE TRans. Automa. Control, 46 (2001), 1028–1047. [71] C. Soul´e, Graphic requirements for multistationarity, ComplexUs, 1 (2003), 123–133. [72] H. L. Smith, “Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,” American Mathematical Society, Providence, Rhode Island, 1995. [73] B. Smyth and F. Xavier, Injectivity of local diffeomorphisms from nearly spectral conditions, J Diff. Eq., 130 (1996), 406–414. [74] R. Thomas, On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations, Springer Ser. Synergetics, 9 (1981), 180–193. [75] R. Thomas, Deterministic chaos seen in terms of feedback circuits: analysis, synthesis, “labyrinth chaos”, Int. J. Bifurcations Chaos, 9 (1999), 1889–1905. [76] M. Kaufman, C. Soul´e and R. Thomas, A new necessary condition on interaction graphs for multistationarity, J. Theor. Biol., 248 (2007), 675–685. [77] R. Thomas and M. Kaufman, Multistationarity, the basis of cell differentiation and memory. I. Structural conditions of multistationarity and other nontrivial behaviour, Chaos, 11 (2001), 170–179. [78] B. Toni, D. Thieffry and D. Bulajich, Feedback loops analysis for chaotic dynamics with an application to Lorenz system, Fields Inst. Commun., 21 (1999), 473–483. [79] A. Volpert and A. N. Ivanova, Mathematical models in chemical kinetics, in “Mathematical modeling” (Russian), Nauka, Moscow, (1987), 57–102. [80] A. Volpert and S. Hudyaev, “Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics,” (Chapter 12), Martinus Nijhoff, Dordrecht, 1985. [81] W. Walter, “Ordinary Differential Equations,” Springer-Verlag. New York, 1998. |