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September  2012, 17(6): 2153-2170. doi: 10.3934/dcdsb.2012.17.2153

## Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks

 1 Department of Electrical and Electronic Engineering, Imperial College London, London, SW7 2AZ, United Kingdom 2 Automatic Control Laboratory, Department of Information Technology and Electrical Engineering, Swiss Federal Institute of Technology (ETH) Zürich, Zürich, ETL I26 8092, Switzerland 3 Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI 53706, United States

Received  June 2011 Revised  September 2011 Published  May 2012

Dynamical system models of complex biochemical reaction networks are high-dimensional, nonlinear, and contain many unknown parameters. The capacity for multiple equilibria in such systems plays a key role in important biochemical processes. Examples show that there is a very delicate relationship between the structure of a reaction network and its capacity to give rise to several positive equilibria. In this paper we focus on networks of reactions governed by mass-action kinetics. As is almost always the case in practice, we assume that no reaction involves the collision of three or more molecules at the same place and time, which implies that the associated mass-action differential equations contain only linear and quadratic terms. We describe a general injectivity criterion for quadratic functions of several variables, and relate this criterion to a network's capacity for multiple equilibria. In order to take advantage of this criterion we look for explicit general conditions that imply non-vanishing of polynomial functions on the positive orthant. In particular, we investigate in detail the case of polynomials with only one negative monomial, and we fully characterize the case of affinely independent exponents. We describe several examples, including an example that shows how these methods may be used for designing multistable chemical systems in synthetic biology.
Citation: Casian Pantea, Heinz Koeppl, Gheorghe Craciun. Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2153-2170. doi: 10.3934/dcdsb.2012.17.2153
##### References:
 [1] D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508. doi: 10.1137/11082631X.  Google Scholar [2] D. F. Anderson and A. Shiu, The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858. doi: 10.1137/090764098.  Google Scholar [3] M. Banaji, P. Donnell and S. Baigent, $P$ matrix properties, injectivity, and stability in chemical reaction systems, SIAM J. Appl. Math., 67 (2007), 1523-1547. doi: 10.1137/060673412.  Google Scholar [4] M. Banaji and G. Craciun, Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems, Adv. Appl. Math., 44 (2010), 168-184. doi: 10.1016/j.aam.2009.07.003.  Google Scholar [5] M. Banaji and G. Craciun, Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Comm. Math. Sci., 7 (2009), 867-900.  Google Scholar [6] G. Blekherman, Nonnegative polynomials and sums of squares,, preprint, ().   Google Scholar [7] S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.  Google Scholar [8] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546. doi: 10.1137/S0036139904440278.  Google Scholar [9] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. II. The species-reactions graph, SIAM J. Appl. Math., 66 (2006), 1321-1338. doi: 10.1137/050634177.  Google Scholar [10] G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS, 103 (2006), 8697-8702. doi: 10.1073/pnas.0602767103.  Google Scholar [11] G. Craciun, L. D. Garcia-Puente and F. Sottile, Some geometrical aspects of control points for toric patches, Mathematical Methods for Curves and Surfaces, Lecture Notes in Computer Science, 5862 (2010), 111-135. Google Scholar [12] G. Craciun, J. W. Helton and R. J. Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci., 216 (2008), 140-149. doi: 10.1016/j.mbs.2008.09.001.  Google Scholar [13] G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems,, preprint, ().   Google Scholar [14] G. Craciun, C. Pantea and E. D. Sontag, Graph-theoretic analysis of multistability and monotonicity for biochemical reaction networks, in "Design and Analysis of Biomolecular Circuits" (eds. H. Koeppl, G. Setti, M. di Bernardo and D. Densmore), Springer-Verlag, 2011. Google Scholar [15] G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551-1565. doi: 10.1016/j.jsc.2008.08.006.  Google Scholar [16] J. Davenport and J. Heintz, Real quantifier elimination is doubly exponential, J. Symb. Comput., 5 (1988), 29-35.  Google Scholar [17] P. De Leenheer and H. Smith, Feedback control for chemostat models, J. Math. Biol., 46 (2003), 48-70. doi: 10.1007/s00285-002-0170-x.  Google Scholar [18] R. J. Duffin and E. L. Peterson, Duality theory for geometric programming, SIAM J. Appl. Math., 14 (1966), 1307-1349. doi: 10.1137/0114105.  Google Scholar [19] R. J. Duffin, E. L. Peterson and C. Zener, "Geometric Programming: Theory and Application," John Wiley & Sons, Inc., New York-London-Sydney, 1967.  Google Scholar [20] J. G. Ecker, Geometric programming: Methods, computations and applications, SIAM Review, 22 (1980), 338-362. doi: 10.1137/1022058.  Google Scholar [21] M. Feinberg, Lectures on chemical reaction networks, written version of lectures given at the Mathematical Research Center, University of Wisconsin, Madison, WI, 1979. Available from: http://www.chbmeng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks. Google Scholar [22] M. Hafner, H. Koeppl, M. Hasler and A. Wagner, 'Glocal' robustness analysis and model discrimination for circadian oscillators, PLoS Computational Biology, 5 (2009), e1000534, 10 pp.  Google Scholar [23] J. W. Helton, V. Katsnelson and I. Klep, Sign patterns for chemical reaction networks, Journal of Mathematical Chemistry, 47 (2010), 403-429. doi: 10.1007/s10910-009-9579-4.  Google Scholar [24] J. W. Helton, I. Klep and R. Gomez, Determinant expansions of signed matrices and of certain jacobians, SIAM J. of Mat. Anal. Appl., 31 (2009), 732-754. doi: 10.1137/080718838.  Google Scholar [25] R. A. Horn and C. R. Johnson, "Topics in Matrix Analysis," Cambridge University Press, Cambridge, 1991.  Google Scholar [26] H. Koeppl, S. Andreozzi and R. Steuer, Guaranteed and randomized methods for stability analysis of uncertain metabolic networks, in "Advances in the Theory of Control, Signals and Systems with Physical Modeling," Lecture Notes in Control and Information Sciences, 407, Springer, Berlin, (2010), 297-307.  Google Scholar [27] R. McDaniel and R. Weiss, Advances in synthetic biology: On the path from prototypes to applications, Curr. Opin. in Biotech., 16 (2005), 476-483. doi: 10.1016/j.copbio.2005.07.002.  Google Scholar [28] C. Pantea, BioNetX, a software package for examining dynamical properties of biochemical reaction network models., Available from: \url{http://cap.ee.ic.ac.uk/~cpantea/}., ().   Google Scholar [29] C. Pantea, On the persistence and global stability of mass-action systems,, preprint, ().   Google Scholar [30] C. Pantea and G. Craciun, Computational methods for analyzing bistability in biochemical reaction networks, Proceedings of the IEEE International Symposium on Circuits and Systems, 2010. Google Scholar [31] S. Pinchuk, A counterexample to the strong real Jacobian conjecture, Math. Z., 217 (1994), 1-4. doi: 10.1007/BF02571929.  Google Scholar [32] G. Pòlya and G. Szegő, "Problems and Theorems in Analysis. I. Series, Integral Calculus, Theory of Functions," Corrected printing of the revised translation of the fourth German edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 193, Springer-Verlag, Berlin-New York, 1978.  Google Scholar [33] P. E. M. Purnick and R. Weiss, The second wave of synthetic biology: From modules to systems, Nature Reviews Molecular Cell Biology, 10 (2009), 410-422. doi: 10.1038/nrm2698.  Google Scholar [34] J. B. Rawlings and J. G. Ekerdt, "Chemical Reactor Analysis and Design Fundamentals," Nob Hill Publishing, Madison, 2004. Google Scholar [35] R. Swan, Tarski's Principle and the elimination of quantifiers., Available from: \url{http://www.math.uchicago.edu/~swan/expo/Tarski.pdf}., ().   Google Scholar [36] A. Tarski, "A Decision Method for Elementary Algebra and Geometry," RAND Corporation, Santa Monica, 1948.  Google Scholar [37] V. Weispfenning, The complexity of linear problems in fields, J. Symb. Comput., 5 (1988), 3-27.  Google Scholar [38] C. Zener, A mathematical aid in optimizing engineering designs, PNAS USA, 47 (1961), 537-539. doi: 10.1073/pnas.47.4.537.  Google Scholar

show all references

##### References:
 [1] D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71 (2011), 1487-1508. doi: 10.1137/11082631X.  Google Scholar [2] D. F. Anderson and A. Shiu, The dynamics of weakly reversible population processes near facets, SIAM J. Appl. Math., 70 (2010), 1840-1858. doi: 10.1137/090764098.  Google Scholar [3] M. Banaji, P. Donnell and S. Baigent, $P$ matrix properties, injectivity, and stability in chemical reaction systems, SIAM J. Appl. Math., 67 (2007), 1523-1547. doi: 10.1137/060673412.  Google Scholar [4] M. Banaji and G. Craciun, Graph-theoretic criteria for injectivity and unique equilibria in general chemical reaction systems, Adv. Appl. Math., 44 (2010), 168-184. doi: 10.1016/j.aam.2009.07.003.  Google Scholar [5] M. Banaji and G. Craciun, Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements, Comm. Math. Sci., 7 (2009), 867-900.  Google Scholar [6] G. Blekherman, Nonnegative polynomials and sums of squares,, preprint, ().   Google Scholar [7] S. Boyd and L. Vandenberghe, "Convex Optimization," Cambridge University Press, Cambridge, 2004.  Google Scholar [8] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. I. The injectivity property, SIAM J. Appl. Math., 65 (2005), 1526-1546. doi: 10.1137/S0036139904440278.  Google Scholar [9] G. Craciun and M. Feinberg, Multiple equilibria in complex chemical reaction networks. II. The species-reactions graph, SIAM J. Appl. Math., 66 (2006), 1321-1338. doi: 10.1137/050634177.  Google Scholar [10] G. Craciun, Y. Tang and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS, 103 (2006), 8697-8702. doi: 10.1073/pnas.0602767103.  Google Scholar [11] G. Craciun, L. D. Garcia-Puente and F. Sottile, Some geometrical aspects of control points for toric patches, Mathematical Methods for Curves and Surfaces, Lecture Notes in Computer Science, 5862 (2010), 111-135. Google Scholar [12] G. Craciun, J. W. Helton and R. J. Williams, Homotopy methods for counting reaction network equilibria, Math. Biosci., 216 (2008), 140-149. doi: 10.1016/j.mbs.2008.09.001.  Google Scholar [13] G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems,, preprint, ().   Google Scholar [14] G. Craciun, C. Pantea and E. D. Sontag, Graph-theoretic analysis of multistability and monotonicity for biochemical reaction networks, in "Design and Analysis of Biomolecular Circuits" (eds. H. Koeppl, G. Setti, M. di Bernardo and D. Densmore), Springer-Verlag, 2011. Google Scholar [15] G. Craciun, A. Dickenstein, A. Shiu and B. Sturmfels, Toric dynamical systems, J. Symb. Comput., 44 (2009), 1551-1565. doi: 10.1016/j.jsc.2008.08.006.  Google Scholar [16] J. Davenport and J. Heintz, Real quantifier elimination is doubly exponential, J. Symb. Comput., 5 (1988), 29-35.  Google Scholar [17] P. De Leenheer and H. Smith, Feedback control for chemostat models, J. Math. Biol., 46 (2003), 48-70. doi: 10.1007/s00285-002-0170-x.  Google Scholar [18] R. J. Duffin and E. L. Peterson, Duality theory for geometric programming, SIAM J. Appl. Math., 14 (1966), 1307-1349. doi: 10.1137/0114105.  Google Scholar [19] R. J. Duffin, E. L. Peterson and C. Zener, "Geometric Programming: Theory and Application," John Wiley & Sons, Inc., New York-London-Sydney, 1967.  Google Scholar [20] J. G. Ecker, Geometric programming: Methods, computations and applications, SIAM Review, 22 (1980), 338-362. doi: 10.1137/1022058.  Google Scholar [21] M. Feinberg, Lectures on chemical reaction networks, written version of lectures given at the Mathematical Research Center, University of Wisconsin, Madison, WI, 1979. Available from: http://www.chbmeng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks. Google Scholar [22] M. Hafner, H. Koeppl, M. Hasler and A. Wagner, 'Glocal' robustness analysis and model discrimination for circadian oscillators, PLoS Computational Biology, 5 (2009), e1000534, 10 pp.  Google Scholar [23] J. W. Helton, V. Katsnelson and I. Klep, Sign patterns for chemical reaction networks, Journal of Mathematical Chemistry, 47 (2010), 403-429. doi: 10.1007/s10910-009-9579-4.  Google Scholar [24] J. W. Helton, I. Klep and R. Gomez, Determinant expansions of signed matrices and of certain jacobians, SIAM J. of Mat. Anal. Appl., 31 (2009), 732-754. doi: 10.1137/080718838.  Google Scholar [25] R. A. Horn and C. R. Johnson, "Topics in Matrix Analysis," Cambridge University Press, Cambridge, 1991.  Google Scholar [26] H. Koeppl, S. Andreozzi and R. Steuer, Guaranteed and randomized methods for stability analysis of uncertain metabolic networks, in "Advances in the Theory of Control, Signals and Systems with Physical Modeling," Lecture Notes in Control and Information Sciences, 407, Springer, Berlin, (2010), 297-307.  Google Scholar [27] R. McDaniel and R. Weiss, Advances in synthetic biology: On the path from prototypes to applications, Curr. Opin. in Biotech., 16 (2005), 476-483. doi: 10.1016/j.copbio.2005.07.002.  Google Scholar [28] C. Pantea, BioNetX, a software package for examining dynamical properties of biochemical reaction network models., Available from: \url{http://cap.ee.ic.ac.uk/~cpantea/}., ().   Google Scholar [29] C. Pantea, On the persistence and global stability of mass-action systems,, preprint, ().   Google Scholar [30] C. Pantea and G. Craciun, Computational methods for analyzing bistability in biochemical reaction networks, Proceedings of the IEEE International Symposium on Circuits and Systems, 2010. Google Scholar [31] S. Pinchuk, A counterexample to the strong real Jacobian conjecture, Math. Z., 217 (1994), 1-4. doi: 10.1007/BF02571929.  Google Scholar [32] G. Pòlya and G. Szegő, "Problems and Theorems in Analysis. I. Series, Integral Calculus, Theory of Functions," Corrected printing of the revised translation of the fourth German edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 193, Springer-Verlag, Berlin-New York, 1978.  Google Scholar [33] P. E. M. Purnick and R. Weiss, The second wave of synthetic biology: From modules to systems, Nature Reviews Molecular Cell Biology, 10 (2009), 410-422. doi: 10.1038/nrm2698.  Google Scholar [34] J. B. Rawlings and J. G. Ekerdt, "Chemical Reactor Analysis and Design Fundamentals," Nob Hill Publishing, Madison, 2004. Google Scholar [35] R. Swan, Tarski's Principle and the elimination of quantifiers., Available from: \url{http://www.math.uchicago.edu/~swan/expo/Tarski.pdf}., ().   Google Scholar [36] A. Tarski, "A Decision Method for Elementary Algebra and Geometry," RAND Corporation, Santa Monica, 1948.  Google Scholar [37] V. Weispfenning, The complexity of linear problems in fields, J. Symb. Comput., 5 (1988), 3-27.  Google Scholar [38] C. Zener, A mathematical aid in optimizing engineering designs, PNAS USA, 47 (1961), 537-539. doi: 10.1073/pnas.47.4.537.  Google Scholar
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