# American Institute of Mathematical Sciences

February  2022, 27(2): 799-819. doi: 10.3934/dcdsb.2021065

## Single-target networks

 1 Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, 53706 2 Department of Mathematics, University of Wisconsin-Madison, 53706

* Corresponding author: Jiaxin Jin

Received  June 2020 Revised  December 2020 Published  February 2022 Early access  March 2021

Fund Project: The authors are supported by NSF grant DMS-1816238

Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. Single-target networks are reaction networks with an arbitrarily set of source vertices, but only one sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either (i) the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or (ii) the system has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, we show that global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

Citation: Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. Single-target networks. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 799-819. doi: 10.3934/dcdsb.2021065
##### References:
 [1] D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM Journal on Applied Mathematics, 71 (2011), 1487-1508.  doi: 10.1137/11082631X. [2] D. F. Anderson, J. D. Brunner, G. Craciun and M. D. Johnston, On classes of reaction networks and their associated polynomial dynamical systems, (2020). [3] D. Angeli, A tutorial on chemical reaction network dynamics, European Journal of Control, 15 (2009), 398-406.  doi: 10.3166/ejc.15.398-406. [4] M. W. Birch, Maximum likelihood in three-way contingency tables, Journal of the Royal Statistical Society. Series B (Methodological), 25 (1963), 220-233.  doi: 10.1111/j.2517-6161.1963.tb00504.x. [5] B. Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM Journal on Mathematical Analysis, 51 (2019), 435-449.  doi: 10.1137/17M115534X. [6] B. Boros and J. Hofbauer, Permanence of weakly reversible mass-action systems with a single linkage class, SIAM Journal on Applied Dynamical Systems, 19 (2020), 352-365.  doi: 10.1137/19M1248431. [7] M. L. Brustenga, G. Craciun and M-S Sorea, Disguised toric dynamical systems, (2020). [8] G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2015), arXiv: 1501.02860 [math.DS]. [9] G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 87-106.  doi: 10.1137/17M1129076. [10] G. Craciun, A. Dickenstein, B. Sturmfels and A. Shiu, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006. [11] G. Craciun, J. Jin and P. Y. Yu, An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems, SIAM Journal on Applied Mathematics, 80 (2020), 183-205.  doi: 10.1137/19M1244494. [12] G. Craciun, J. Jin and P. Y. Yu, Dynamical equivalence to complex balancing as an open condition in parameter space, in Preparation. [13] G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM Journal on Applied Mathematics, 73 (2013), 305-329.  doi: 10.1137/100812355. [14] G. Craciun and C. Pantea, Identifiability of chemical reaction networks, Journal of Mathematical Chemistry, 44 (2008), 244-259.  doi: 10.1007/s10910-007-9307-x. [15] A. Dickenstein and M. Pérez Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828.  doi: 10.1007/s11538-010-9611-7. [16] M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.  doi: 10.1007/BF00255665. [17] M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. The Deficiency Zero and the Deficiency One Theorems, Chemical Engineering Science, 42 (1987), 2229-2268. [18] M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827. [19] M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, Springer International Publishing, 2019. [20] M. Gopalkrishnan, E. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM Journal on Applied Dynamical Systems, 13 (2014), 758-797.  doi: 10.1137/130928170. [21] J. Gunawardena, Chemical reaction network theory for in-silico biologists, (2003), http://vcp.med.harvard.edu/papers/crnt.pdf, [22] F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.  doi: 10.1007/BF00255664. [23] F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225. [24] M. D. Johnston, Translated chemical reaction networks, Bulletin of Mathematical Biology, 76 (2014), 1081-1116.  doi: 10.1007/s11538-014-9947-5. [25] M. D. Johnston and E. Burton, Computing weakly reversible deficiency zero network translations using elementary flux modes, Bulletin of Mathematical Biology, 81 (2019), 1613-1644.  doi: 10.1007/s11538-019-00579-z. [26] M. D. Johnston, D. Siegel and G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiencys, Mathematical Biosciences, 241 (2013), 88-98.  doi: 10.1016/j.mbs.2012.09.008. [27] G. Lipták, G. Szederkényi and K. M. Hangos, Computing zero deficiency realizations of kinetic systems, Systems & Control Letters, 81 (2015), 24-30.  doi: 10.1016/j.sysconle.2015.05.001. [28] S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM Journal on Applied Mathematics, 72 (2012), 1926-1947.  doi: 10.1137/110847056. [29] L. Onsager, Reciprocal relations in irreversible processes I., Physical Review, 37 (1931), 405-426. [30] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511610684.007. [31] C. Pantea, On the persistence and global stability of mass-action systems, SIAM Journal on Mathematical Analysis, 44 (2012), 1636-1673.  doi: 10.1137/110840509. [32] J. Rudan, G. Szederkényi, K. M. Hangos and T. Péni, Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks, Journal of Mathematical Chemistry, 52 (2014), 1386-1404.  doi: 10.1007/s10910-014-0318-0. [33] S. Schuster and R. Schuster, A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation, Journal of Mathematical Chemistry, 3 (1989), 25-42.  doi: 10.1007/BF01171883. [34] G. Szederkényi, Comment on "Identifiability of chemical reaction networks" by G. Craciun and C. Pantea, Journal of Mathematical Chemistry, 45 (2009), 1172-1174.  doi: 10.1007/s10910-008-9499-8. [35] G. Szederkényi, J. R. Banga and A. A. Alonso, CRNreals: A toolbox for distinguishability and identifiability analysis of biochemical reaction networks, Bioinformatics, 28 (2012), 1549-1550. [36] G. Szederkényi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks, Journal of Mathematical Chemistry, 49 (2011), 1163-1179.  doi: 10.1007/s10910-011-9804-9. [37] A. I. Vol'pert, Differential equations on graphs, Math. USSR-Sb, 88 (1972), 578-588. [38] R. Wegscheider, Über simultane gleichgewichte und die beziehungen zwischen thermodynamik und reactionskinetik homogener systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 22 (1901), 849-906. [39] P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.

show all references

##### References:
 [1] D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM Journal on Applied Mathematics, 71 (2011), 1487-1508.  doi: 10.1137/11082631X. [2] D. F. Anderson, J. D. Brunner, G. Craciun and M. D. Johnston, On classes of reaction networks and their associated polynomial dynamical systems, (2020). [3] D. Angeli, A tutorial on chemical reaction network dynamics, European Journal of Control, 15 (2009), 398-406.  doi: 10.3166/ejc.15.398-406. [4] M. W. Birch, Maximum likelihood in three-way contingency tables, Journal of the Royal Statistical Society. Series B (Methodological), 25 (1963), 220-233.  doi: 10.1111/j.2517-6161.1963.tb00504.x. [5] B. Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM Journal on Mathematical Analysis, 51 (2019), 435-449.  doi: 10.1137/17M115534X. [6] B. Boros and J. Hofbauer, Permanence of weakly reversible mass-action systems with a single linkage class, SIAM Journal on Applied Dynamical Systems, 19 (2020), 352-365.  doi: 10.1137/19M1248431. [7] M. L. Brustenga, G. Craciun and M-S Sorea, Disguised toric dynamical systems, (2020). [8] G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2015), arXiv: 1501.02860 [math.DS]. [9] G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 87-106.  doi: 10.1137/17M1129076. [10] G. Craciun, A. Dickenstein, B. Sturmfels and A. Shiu, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006. [11] G. Craciun, J. Jin and P. Y. Yu, An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems, SIAM Journal on Applied Mathematics, 80 (2020), 183-205.  doi: 10.1137/19M1244494. [12] G. Craciun, J. Jin and P. Y. Yu, Dynamical equivalence to complex balancing as an open condition in parameter space, in Preparation. [13] G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM Journal on Applied Mathematics, 73 (2013), 305-329.  doi: 10.1137/100812355. [14] G. Craciun and C. Pantea, Identifiability of chemical reaction networks, Journal of Mathematical Chemistry, 44 (2008), 244-259.  doi: 10.1007/s10910-007-9307-x. [15] A. Dickenstein and M. Pérez Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828.  doi: 10.1007/s11538-010-9611-7. [16] M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.  doi: 10.1007/BF00255665. [17] M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. The Deficiency Zero and the Deficiency One Theorems, Chemical Engineering Science, 42 (1987), 2229-2268. [18] M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827. [19] M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, Springer International Publishing, 2019. [20] M. Gopalkrishnan, E. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM Journal on Applied Dynamical Systems, 13 (2014), 758-797.  doi: 10.1137/130928170. [21] J. Gunawardena, Chemical reaction network theory for in-silico biologists, (2003), http://vcp.med.harvard.edu/papers/crnt.pdf, [22] F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.  doi: 10.1007/BF00255664. [23] F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225. [24] M. D. Johnston, Translated chemical reaction networks, Bulletin of Mathematical Biology, 76 (2014), 1081-1116.  doi: 10.1007/s11538-014-9947-5. [25] M. D. Johnston and E. Burton, Computing weakly reversible deficiency zero network translations using elementary flux modes, Bulletin of Mathematical Biology, 81 (2019), 1613-1644.  doi: 10.1007/s11538-019-00579-z. [26] M. D. Johnston, D. Siegel and G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiencys, Mathematical Biosciences, 241 (2013), 88-98.  doi: 10.1016/j.mbs.2012.09.008. [27] G. Lipták, G. Szederkényi and K. M. Hangos, Computing zero deficiency realizations of kinetic systems, Systems & Control Letters, 81 (2015), 24-30.  doi: 10.1016/j.sysconle.2015.05.001. [28] S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM Journal on Applied Mathematics, 72 (2012), 1926-1947.  doi: 10.1137/110847056. [29] L. Onsager, Reciprocal relations in irreversible processes I., Physical Review, 37 (1931), 405-426. [30] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511610684.007. [31] C. Pantea, On the persistence and global stability of mass-action systems, SIAM Journal on Mathematical Analysis, 44 (2012), 1636-1673.  doi: 10.1137/110840509. [32] J. Rudan, G. Szederkényi, K. M. Hangos and T. Péni, Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks, Journal of Mathematical Chemistry, 52 (2014), 1386-1404.  doi: 10.1007/s10910-014-0318-0. [33] S. Schuster and R. Schuster, A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation, Journal of Mathematical Chemistry, 3 (1989), 25-42.  doi: 10.1007/BF01171883. [34] G. Szederkényi, Comment on "Identifiability of chemical reaction networks" by G. Craciun and C. Pantea, Journal of Mathematical Chemistry, 45 (2009), 1172-1174.  doi: 10.1007/s10910-008-9499-8. [35] G. Szederkényi, J. R. Banga and A. A. Alonso, CRNreals: A toolbox for distinguishability and identifiability analysis of biochemical reaction networks, Bioinformatics, 28 (2012), 1549-1550. [36] G. Szederkényi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks, Journal of Mathematical Chemistry, 49 (2011), 1163-1179.  doi: 10.1007/s10910-011-9804-9. [37] A. I. Vol'pert, Differential equations on graphs, Math. USSR-Sb, 88 (1972), 578-588. [38] R. Wegscheider, Über simultane gleichgewichte und die beziehungen zwischen thermodynamik und reactionskinetik homogener systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 22 (1901), 849-906. [39] P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.
(a) A single-target network that is globally stable under mass-action kinetics. (b)–(c) Single-target networks with no positive steady states. (d) Not a single-target network
Consider subnetworks of (a) under mass-action kinetics, whose associated dynamics is given by (9). If the coefficient of ${\boldsymbol{x}}^{ {\boldsymbol{y}}_1}$ in $\dot{x}$ is positive and the coefficient of ${\boldsymbol{x}}^{ {\boldsymbol{y}}_3}$ in $\dot{x}$ is negative, then the system (9) can be realized by a single-target network, determined by the sign of ${\boldsymbol{x}}^{ {\boldsymbol{y}}_2}$ in $\dot{x}$. If the net directions are as shown in (b), then (9) can be realized by the single-target network in (c). Similarly, if the net directions appear as in (d), then (9) can be realized by the network in (e)
Reversible systems in (a) Example 3.12 and (b) Example 3.13 that are dynamically equivalent to detailed-balanced systems. Each undirected edge represents a pair of reversible edges
Geometric argument for dynamically equivalence to single-target network in Example 3.12. Shown are the edges with $\mathrm{{X}}_1 + \mathrm{{X}}_2$ as their source. The centre $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right)^\top$ of the tetrahedron is marked in blue. With rate constants given in the example, the weighted sum of reaction vectors points from the source to the centre
(a) The mass-action system with two target vertices from Example 4.2, which is dynamically equivalent to a complex-balanced system using a subnetwork of (b) if and only if $\frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25$
The system in Figure 5(a) is dynamically equivalent to a complex-balanced system if and only if $\frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25$. The system is equivalent to (a) when $\kappa_2 \kappa_4 = 25 \kappa_1 \kappa_3$ and (b) when $25 \kappa_2 \kappa_4 = \kappa_1 \kappa_3$. For $\frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25$, the dynamically equivalent system is an appropriate convex combination of (a) and (b)
 [1] Eugene Kashdan, Dominique Duncan, Andrew Parnell, Heinz Schättler. Mathematical methods in systems biology. Mathematical Biosciences & Engineering, 2016, 13 (6) : i-ii. doi: 10.3934/mbe.201606i [2] Monique Chyba, Benedetto Piccoli. Special issue on mathematical methods in systems biology. Networks and Heterogeneous Media, 2019, 14 (1) : i-ii. doi: 10.3934/nhm.20191i [3] Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159 [4] N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59 [5] Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895 [6] Avner Friedman. PDE problems arising in mathematical biology. Networks and Heterogeneous Media, 2012, 7 (4) : 691-703. doi: 10.3934/nhm.2012.7.691 [7] Avner Friedman. Free boundary problems arising in biology. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013 [8] Pei Zhang, Siyan Liu, Dan Lu, Ramanan Sankaran, Guannan Zhang. An out-of-distribution-aware autoencoder model for reduced chemical kinetics. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 913-930. doi: 10.3934/dcdss.2021138 [9] Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091 [10] Vaughn Climenhaga. Multifractal formalism derived from thermodynamics for general dynamical systems. Electronic Research Announcements, 2010, 17: 1-11. doi: 10.3934/era.2010.17.1 [11] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [12] F. R. Guarguaglini, R. Natalini. Global existence and uniqueness of solutions for multidimensional weakly parabolic systems arising in chemistry and biology. Communications on Pure and Applied Analysis, 2007, 6 (1) : 287-309. doi: 10.3934/cpaa.2007.6.287 [13] Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255 [14] Charles Wiseman, M.D.. Questions from the fourth son: A clinician reflects on immunomonitoring, surrogate markers and systems biology. Mathematical Biosciences & Engineering, 2011, 8 (2) : 279-287. doi: 10.3934/mbe.2011.8.279 [15] Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081 [16] Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic and Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042 [17] Eduard Feireisl. Relative entropies in thermodynamics of complete fluid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3059-3080. doi: 10.3934/dcds.2012.32.3059 [18] Elena Shchepakina, Olga Korotkova. Canard explosion in chemical and optical systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 495-512. doi: 10.3934/dcdsb.2013.18.495 [19] Jacky Cresson, Bénédicte Puig, Stefanie Sonner. Stochastic models in biology and the invariance problem. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2145-2168. doi: 10.3934/dcdsb.2016041 [20] Jean-Pierre Françoise, Hongjun Ji. The stability analysis of brain lactate kinetics. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2135-2143. doi: 10.3934/dcdss.2020182

2020 Impact Factor: 1.327