Abstract
Siphons in a chemical reaction system are subsets of the species that have the potential of being absent in a steady state. We present a characterization of minimal siphons in terms of primary decomposition of binomial ideals, we explore the underlying geometry, and we demonstrate the effective computation of siphons using computer algebra software. This leads to a new method for determining whether given initial concentrations allow for various boundary steady states.
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Adleman, L., Gopalkrishnan, M., Huang, M.-D., Moisset, P., Reishus, D., 2008. On the mathematics of the law of mass action. arXiv:0810.1108.
Anderson, D., 2008. Global asymptotic stability for a class of nonlinear chemical equations. SIAM J. Appl. Math. 68, 1464–1476.
Anderson, D., Shiu, A., 2010. Persistence of deterministic population processes and the global attractor conjecture. SIAM J. Appl. Math. (to appear). arXiv:0903.0901.
Angeli, D., Sontag, E., 2006. Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles. Nonlinear Anal. Ser. B: Real World Appl. 9, 128–140.
Angeli, D., De Leenheer, P., Sontag, E., 2007. A Petri net approach to persistence analysis in chemical reaction networks. In: Queinnec, I., Tarbouriech, S., Garcia, G., Niculescu, S.-I. (Eds.), Biology and Control Theory: Current Challenges, Lecture Notes in Control and Information Sciences, vol. 357, pp. 181–216. Springer, Berlin.
Chavez, M., 2003. Observer design for a class of nonlinear systems, with applications to chemical and biological networks. Ph.D. Thesis, Rutgers University.
Cordone, R., Ferrarini, L., Piroddi, L., 2005. Enumeration algorithms for minimal siphons in Petri nets based on place constraints. IEEE Trans. Syst. Man Cybern. Part A 35(6), 844–854.
Cox, D., Little, J., O’Shea, D., 2007. Ideals, Varieties and Algorithms, 3rd edn., Undergraduate Texts in Mathematics. Springer, New York.
Craciun, G., Dickenstein, A., Shiu, A., Sturmfels, B., 2009. Toric dynamical systems. J. Symb. Comput. 44(11), 1551–1565.
Craciun, G., Pantea, C., Rempala, G., 2009. Algebraic methods for inferring biochemical networks: a maximum likelihood approach. Comput. Biol. Chem. 33(5), 361–367.
Diaconis, P., Eisenbud, D., Sturmfels, B., 1998. Lattice walks and primary decomposition. In: Sagan, B.E., Stanley, R.P. (Eds.), Mathematical Essays in Honor of Gian-Carlo Rota, pp. 173–193. Birkhäuser, Basel.
Dickenstein, A., Matusevich, L., Miller, E., 2010. Combinatorics of binomial primary decomposition. Math. Z. (to appear). arXiv:0803.3846.
Eisenbud, D., Sturmfels, B., 1996. Binomial ideals. Duke Math. J. 84, 1–45.
Gawrilow, E., Joswig, M., 2000. Polymake: a framework for analyzing convex polytopes. In: Polytopes—Combinatorics and Computation, Oberwolfach, 1997, DMV Sem., vol. 29, pp. 43–73. Birkhäuser, Basel.
Grayson, D., Stillman, M. Macaulay 2, a software system for research in algebraic geometry, www.math.uiuc.edu/Macaulay2/.
Kahle, T., 2009. Decompositions of binomial ideals. arXiv:0906.4873.
Manrai, A., Gunawardena, J., 2008. The geometry of multisite phosphorylation. Biophys. J. 95, 5533–5543.
Miller, E., Sturmfels, B., 2005. Combinatorial Commutative Algebra, Graduate Texts in Mathematics. Springer, New York.
Roune, B., 2009. The slice algorithm for irreducible decomposition of monomial ideals. J. Symb. Comput. 44(4), 358–381.
Siegel, D., MacLean, D., 2004. Global stability of complex balanced mechanisms. J. Math. Biol. 27, 89–110.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Shiu, A., Sturmfels, B. Siphons in Chemical Reaction Networks. Bull. Math. Biol. 72, 1448–1463 (2010). https://doi.org/10.1007/s11538-010-9502-y
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DOI: https://doi.org/10.1007/s11538-010-9502-y