Toric dynamical systems

Journal of Symbolic Computation

Volumn 44, Issue 11, 2009, Pages 1551-1565

  Craciun, Gheorghe ^a   Dickenstein, Alicia ^b   Shiu, Anne ^c   Sturmfels, Bernd ^c  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b UNIVERSIDAD DE BUENOS AIRES   (Argentina)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Birch's Theorem; Chemical reaction network; Complex balancing; Deficiency zero; Detailed balancing; Matrix tree theorem; Moduli space; Polyhedron; Toric ideal; Trajectory

Indexed keywords

EID: 68549135061     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jsc.2008.08.006     Document Type: Article

References (29)

1. Akin E. The Geometry of Population Genetics. Lecture Notes in Biomathematics vol. 31 (1979), Springer, New York
2. Akin E. Cycling in simple genetic systems. J. Math. Biol. 13 (1982) 305-324
3. Anderson D. Global asymptotic stability for a class of nonlinear chemical equations. SIAM J. Appl. Math. 68 5 (2008) 1464-1476
4. Beerenwinkel N., Pachter L., and Sturmfels B. Epistasis and the shape of fitness landscapes. Statist. Sinica 17 (2007) 1317-1342
5. Cox D., Little J., and O'Shea D. Ideals, Varieties and Algorithms. third edition. Undergraduate Texts in Mathematics (2007), Springer Verlag
6. Craciun G., Tang Y., and Feinberg M. Understanding bistability in complex enzyme-driven reaction networks. Proc. Natl. Acad. Sci. 103 23 (2006) 8697-8702
7. De Leenheer P., Angeli D., and Sontag E. Monotone chemical reaction networks. J. Math. Chem. 41 (2007) 295-314
8. Feinberg M. Complex balancing in general kinetic systems. Arch. Ration. Mech. Anal. 49 3 (1972) 187-194
9. Feinberg, M., 1979. Lectures on chemical reaction networks. Notes of Lectures Given at the Mathematics Research Center of the University of Wisconsin. http://www.che.eng.ohio-state.edu/~FEINBERG/LecturesOnReactionNetworks
10. Feinberg M. Chemical reaction network structure and the stability of complex isothermal reactors I. The deficiency zero and deficiency one theorems. Chem. Eng. Sci. 42 10 (1987) 2229-2268
11. Feinberg M. Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity. Chem. Eng. Sci. 44 9 (1989) 1819-1827
12. Feinberg M. Existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Ration. Mech. Anal. 132 (1995) 311-370
13. Gatermann K. Counting stable solutions of sparse polynomial systems in chemistry. In: Green E., et al. (Ed). Contemporary Mathematics. Symbolic Computation: Solving Equations in Algebra, Geometry and Engineering vol. 286 (2001) 53-69
14. Gatermann K., and Huber B. A family of sparse polynomial systems arising in chemical reaction systems. J. Symbolic Comput. 33 3 (2002) 275-305
15. Gatermann K., and Wolfrum M. Bernstein's second theorem and Viro's method for sparse polynomial systems in chemistry. Adv. Appl. Math. 34 (2005) 252-294
16. Gnacadja G., Shoshitaishvili A., Gresser M., Varnum B., Balaban D., Durst M., Vezina C., and Li Y. Monotonicity of interleukin-1 receptor-ligand binding with respect to antagonist in the presence of decoy receptor. J. Theoret. Biol. 244 (2007) 478-488
17. Gunawardena, J., 2003. Chemical reaction network theory for in-silico biologists. Technical Report. http://vcp.med.harvard.edu/papers/crnt.pdf
18. Horn F., and Jackson R. General mass action kinetics. Arch. Ration. Mech. Anal. 47 2 (1972) 81-116
19. Horn F. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Ration. Mech. Anal. 49 3 (1972) 172-186
20. Horn F. Stability and complex balancing in mass-action systems with three complexes. Proc. Roy. Soc. A 334 (1973) 331-342
21. Horn F. The dynamics of open reaction systems. Mathematical aspects of chemical and biochemical problems and quantum chemistry. SIAM-AMS Proceedings vol. VIII (1974) 125-137
22. Huber B., Rambau J., and Santos F. The Cayley trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings. J. Eur. Math. Soc. 2 (2000) 179-198
23. Kuepfer L., Sauer U., and Parrilo P. Efficient classification of complete parameter regions based on semidefinite programming. BMC Bioinform. 8 12 (2007)
24. Pachter L., and Sturmfels B. Algebraic Statistics for Computational Biology (2005), Cambridge University Press, Cambridge
25. Rabinovich, Y., Sinclair, A., Wigderson, A., 1992. Quadratic dynamical systems. In: Proc. 33rd Annual Symposium on Foundations of Computer Science, FOCS, pp. 304-313
26. Siegel D., and Chen S.F. Global stability of deficiency zero chemical networks. Canad. Appl. Math. Quart. 2 (1994) 413-434
27. Sontag E. Structure and stability of certain chemical networks and applications to the kinetic proofreading model of T-cell receptor signal transduction. IEEE Trans. Automat. Control 46 (2001) 1028-1047
28. Stanley R. Enumerative Combinatorics, vol. 2 (1999), Cambridge University Press
29. Sturmfels B. Gröbner Bases and Convex Polytopes. University Lectures Series vol. 8 (1996), American Mathematical Society, Providence, Rhode Island