Multigraph conditions for multistability, oscillations and pattern formation in biochemical reaction networks

Proceedings of the IEEE

Volumn 96, Issue 8, 2008, Pages 1281-1291

  Mincheva, Maya ^a   Craciun, Gheorghe ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Biochemical reaction networks; Multigraph; Multistability; Oscillations; Pattern formation; Positive feedback

Indexed keywords

CELL DEATH; FEEDBACK;

BIOCHEMICAL REACTION NETWORK; MULTI STABILITIES; MULTIGRAPHS; OSCILLATIONS; PATTERN FORMATION;

DIRECTED GRAPHS;

EID: 47949108455     PISSN: 00189219     EISSN: None     Source Type: Journal    
DOI: 10.1109/JPROC.2008.925474     Document Type: Article

References (78)

1. H. Kitano, "Systems biology: A brief review," Science, vol. 295, pp. 1662-1664, 2002.
2. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon, "Network motifs: Simple building blocks of complex networks," Science, vol. 298, pp. 824-827, 2002.
3. B. Clarke, "Stability of complex reaction systems," Adv. Chem. Phys., vol. 43, pp. 1-213, 1980.
4. P. Smolen, D. A. Baxter, and J. H. Byrne, "Modeling transcriptional control in gene networks - Methods, recent results, and future directions," Bull. Math. Biol., vol. 62, pp. 247-292, 2000.
5. J. J. Tyson, K. C. Chen, and B. Novak, "Sniffers, buzzers, toggles and blinkers: Dynamics of regulatory and signaling pathways in the cell," Curr. Opin. Cell Biol., vol. 15, pp. 221-231, 2003.
6. J. Tyson, "Classification of instabilities in chemical reaction systems," J. Chem. Phys., vol. 62, pp. 1010-1015, 1975.
7. D. Angeli, P. De Leenheer, and E. D. Sontag, "On the structural monotonicity of chemical reaction networks," in 2006 45th IEEE Conference on Decision and Control, 2006, pp. 7-12.
8. D. Angeli, J. E. Ferrell, and E. D. Sontag, "Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems," Proc. Nat. Acad. Sci., vol. 101, pp. 1822-1827, 2004.
9. J. Tyson, "Modeling the cell division cycle: cdc2 and cyclin interactions, " Proc. Nat. Acad. Sci., vol. 88, pp. 7328-7332, 1991.
10. A. Goldbeter, "A model for circadian oscillations in the Drosophila period protein (PER)," Proc. R. Soc. Lord. B. Biol. Sci., vol. 261, pp. 319-324, 1995.
11. S. Schuster, M. Marhl, and T. Hofer, "Modelling of simple and complex calcium oscillations," Eur. J. Biochem., vol. 269, pp. 1333-1355, 2002.
12. E. Sel'kov, "Self-oscillations in glycolysis. 1, A simple model," Eur. J. Biochem., vol. 4, pp. 79-86, 1968.
13. C. P. Fall, E. S. Marland, J. M. Wagner, and J. J. Tyson, Eds., Computational Cell Biology, New York: Springer, 2002.
14. N. Markevich, J. Hoek, and B. Kholodenko, "Signaling switches and bistability arising from multiple phosphorylation in protein kinase cascades," J. Cell Biol., vol. 164, pp. 353-359, 2004.
15. E. Ozbudak, M. Thattai, H. Lim, B. Shraiman, and A. Van Oudenaarden, "Multistability in the lactose utilization network of Escherichia coli," Nature, vol. 427, pp. 737-740, 2004.
16. A. Becskei, B. Seraphin, and L. Serrano, "Positive feedback in eukaryotic gene networks: Cell differentiation by graded to binary response conversion," EMBO Journ., vol. 20, pp. 2528-2535, 2001.
17. C. Soule, "Mathematical approaches to differentiation and gene regulation," C.R. Biologies, vol. 329, pp. 13-20, 2005.
18. J. D. Murray, Mathematical Biology, 2nd ed. New York: Springer-Verlag, 1993.
19. G. Craciun, Y. Tang, and M. Feinberg, "Understanding bistability in complex enzyme-driven reaction networks," Proc. Natl. Acad. Sci., vol. 103, no. 23, pp. 8697-8702, 2006.
20. J. Pomerening, S. Kim, and J. Ferrell, "Systems-level dissection of the cell-cycle oscillator: Bypassing positive feedback produces damped oscillations," Cell, vol. 122, pp. 565-578, 2004.
21. R. Thomas, D. Thieffry, and M. Kaufman, "Dynamical behaviour of biological regulatory networks. I. Biological role of feedback loops and practical use of the concept of the loop-characteristic state," Bull. Math. Biol., vol. 57, pp. 247-276, 1995.
22. I. Epstein and J. Pojman, introduction to Nonlinear Chemical Dynamics. Oscillation, Waves, Patterns and Chaos. New York: Oxford University Press, 1998.
23. G. Craciun and M. Feinberg, "Multiple equilibria in complex chemical reaction networks: I. The injectivity property," SIAM J. Appl. Math., vol. 65, pp. 1526-1546, 2005.
24. G. Craciun and M. Feinberg, "Multiple equilibria in complex chemical reaction networks: II. The species-reactions graph," SIAM J. Appl. Math., vol. 66, pp. 1321-1338, 2006.
25. 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 models," J. Math. Biol., vol. 55, pp. 61-86, 2007.
26. M. Feinberg, Lectures on Chemical Reaction Networks, Notes of lectures given at the Mathematics Research Center of the University of Wisconsin in 1979. [Online]. Available: http://www.che.eng.ohio-state.edu/~FEINBERG/ LecturesOnReactionNetworks
27. A. Volpert and A. Ivanova, "Mathematical models in chemical kinetics," (in Russian), in Mathematical Modeling. Moscow: Nauka, 1987, pp. 57-102.
28. A. Volpert and S. Hudyaev, Analyses in Classes of Discontinuous Functions and Equations of Mathematical Physics. Dordrecht: Martinus Nijhoff, 1985, ch. 12.
29. J. Keener and J. Sneyd, Mathematical Physiology. New York: Springer, 1998.
30. F. Harary, Graph Theory. Reading: Addison-Wesley, 1969.
31. D. Angeli, "Systems with counterclockwise input-output dynamics," IEEE Trans. Aut. Contr., vol. 51, pp. 1130-1143, 2006.
32. P. Maini, "How the mouse got its stripes?" Proc. Nat. Acad. Sci., vol. 100, pp. 9656-9657, 2003.
33. I. Prigogine and R. Lefever, "Symmetry breaking instabilities in dissipative systems. II," J. Chem. Phys., vol. 48, pp. 1695-1700, 1968.
34. B. P. Belousov, "A periodic reaction and its mechanism," in Collection of Short Papers on Radiation Medicine. Moscow: Medical Publishers, 1959, pp. 145-152.
35. A. M. Zhabotinsky, "Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics)," Biofizika, vol. 9, pp. 306-311, 1964.
36. M. Mincheva and M. R. Roussel, "A graph-theoretic method for detecting potential Turing bifurcations," J. Chem. Phys., vol. 125, p. 204 102, 2006.
37. B. Aguda and C. Algar, "A structural analysis of the qualitative networks regulating the cell cycle and apoptosis," Cell Cycle, vol. 2, pp. 538-544, 2003.
38. G. Craciun, B. Aguda, and A. Friedman, "Mathematical analysis of a modular network coordinating the cell cycle and apoptosis," Math. Biosc. Eng., vol. 2, pp. 473-485, 2005.
39. P. Lancaster and M. Tismenetsky, The Theory of Matrices. Orlando: Academic, 1985.
40. J. Guckenheimer, M. Myers, and B. Sturmfels, "Computing Hopf bifurcation I," SIAM J. Numer. Anal., vol. 34, pp. 1-21, 1997.
41. G. Engel and H. Schneider, "The Hadamard-Fisher inequality for a class of matrices defined by eigenvalue monotonicity," Lin. Multil. Alg., vol. 4, pp. 155-176, 1976.
42. B. Clarke and W. Jiang, "Method for deriving Hopf and saddle-node bifurcation hypersurfaces and application to a model of the Belousov-Zhabotinsky system," J. Chem. Phys., vol. 99, pp. 4464-4478, 1993.
43. M. Doob, "Applications of graph theory in linear algebra," Math. Mag., vol. 57, pp. 67-76, 1984.
44. S. Pinchuk, "A counterexample to the real Jacobian conjecture," Math. Z., vol. 217, pp. 1-4, 1994.
45. J. Maybee, D. Olesky, P. van den Driessche, and G. Wiener, "Matrices, digraphs and determinants," SIAM J. Matrix Anal. Appl., vol. 10, pp. 500-519, 1989.
46. J. Cherry and F. Adler, "How to make a biological switch?" J. Theor. Biol., vol. 203, pp. 117-133, 2000.
47. T. Gardner, C. Cantor, and J. Collins, "Construction of a genetic toggle switch in Escherichia coli," Nature, vol. 403, pp. 339-342, 2000.
48. A. Loinger, A. Lipshtat, N. Balaban, and O. Biham, "Stochastic simulations of genetic switch systems," Phys. Rev. E, vol. 75, p. 021904, 2007.
49. A. Lipshtat, A. Loinger, N. Balaban, and O. Biham, "Genetic Toggle switch without cooperative binding," Phys. Rev. Lett., vol. 96, p. 188 101, 2006.
50. P. Grindrod, Patterns and Waves. The Theory and Applications of Reaction-Diffusion Equations. Oxford: Clarendon Press, 1991.
51. B. Clarke, "Graph theoretic approach to the stability analysis of steady state chemical reaction network," J. Chem. Phys., vol. 60, pp. 1481-1492, 1974.
52. B. Clarke, "Stability analysis of a model reaction network using graph theory," J. Chem. Phys., vol. 60, pp. 1493-1504, 1974.
53. B. Clarke, "Stoichiometric Network Analysis," Cell Biophys., vol. 12, pp. 237-253, 1988.
54. B. Clarke, "Stability of bromate-cerium-malonic acid network. 1. Theoretical formulation," J. Chem. Phys., vol. 64, pp. 4165-4178, 1976.
55. B. Clarke, "Stability of bromate-cerium-malonic acid network. 2. Steady-state formic acid case," J. Chem. Phys., vol. 64, pp. 4179-4191, 1976.
56. B. Aguda and B. Clarke, "Bistability in chemical reaction networks-theory and application to the peroxidase-oxidase reaction," J. Chem. Phys., vol. 87, pp. 3461-3470, 1987.
57. M. Eiswirth, A. Freund, and J. Ross, "Operational procedure towards the classification of chemical oscillators," J. Chem. Phys, vol. 95, pp. 1294-1299, 1991.
58. A. Ivanova, "Conditions for the uniqueness of the stationary states of kinetic systems, connected with the structures of their reaction mechanisms," Kinet. Katal., vol. 20, pp. 1019-1023, 1979.
59. A. Ivanova and B. Tarnopolskii, "One approach to the determination of a number of qualitative features in the behavior of kinetic systems, and realization of this approach in a computer (critical conditions, autooscillations)," Kinet. Ratal., vol. 20, pp. 1541-1548, 1979.
60. G. Ermakov and B. Goldstein, "Simplest kinetic schemes for biochemical oscillations," Biochemistry (Moscow), vol. 67, pp. 473-484, 2002.
61. B. Goldstein, "Switching mechanism for branched biochemical fluxes: Graph-theoretic analysis," Bioph. Chem., vol. 125, pp. 314-319, 2007.
62. J. Eisefeld and C. DeLisi, "On conditions for qualitative instability of regulatory circuits with applications to immunological control loops," in Mathematics and Computers in Biomedical Applications, Eisenfield and DeLisi, Eds. North Holland: Elsevier Publishers, 1985.
63. R. Thomas, "On the relation between the logical structure of the systems and their ability to generate multiple steady states or sustained oscillations," Springer Ser., Synergetics, vol. 9, pp. 180-193, 1981.
64. C. Soule, "Graphic requirements for multistationarity," Complexus, vol. 1, pp. 123-1333, 2003.
65. M. Kaufman, C. Soule, and R. Thomas, "A new necessary condition on interaction graphs for multistationarity," J. Theor. Biol., vol. 248, pp. 675-685, 2007.
66. T. Wilhelm, "Analysis of structures causing instabilities," Phys. Rev. E, vol. 76, p. 011911, 2007.
67. D. Angeli and E. Sontag, "Monotone control systems," IEEE Trans. Automat. Control, vol. 48, pp. 1684-1698, 2003.
68. U. Alon, M. Surette, N. Barkai, and S. Leibler, "Robustness in bacterial Chemotaxis," Nature, vol. 397, pp. 168-171, 1999.
69. J. Stelling, E. Gilles, and F. Doyle, "Robusteness properties of circadian clock architectures," PNAS, vol. 101, pp. 13 210-13 215, 2004.
70. 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., vol. 55, pp. 87-104, 2007.
71. R. Bliss, P. Painter, and A. Marr, "Role of feedback inhibition in stabilizing the classical operon," J. Theor. Biol., vol. 97, pp. 177-193, 1982.
72. F. Buchholtz and F. Schneider, "Computer simulation of T3/T7 phage infection using lag times," Biophys. Chem., vol. 26, pp. 171-179, 1987.
73. J. Mahaffy, "Cellular control models with linked positive and negative feedback and delays. I. The models," J. Theor. Biol., vol. 106, pp. 89-102, 1984.
74. N. Monk, "Oscillatory expression of Hesl, p53, and NF-κB driven by transcriptional time delays," Curr. Biol., vol. 13, pp. 1409-1413, 2003.
75. N. Rosenfeld, M. Elowitz, and U. Alon, "Negative autoregulation speeds the response times of transcription networks," J. Mol. Biol., vol. 323, pp. 785-793, 2002.
76. Y. Meda and M. Sano, "Regulatory dynamics of synthetic gene networks with positive feedback," J. Mol. Biol., vol. 359, pp. 1107-1124, 2006.
77. I. Zevedei-Oancea and S. Schuster, "A theoretical framework for detecting signal transfer routes in signaling networks," Computers & Chem. Eng., vol. 29, pp. 597-617, 2005.
78. M. Elowitz and S. Leibler, "A synthetic oscillatory network of transcriptional regulators," Nature, vol. 403, pp. 335-338, 2000.