×

zbMATH — the first resource for mathematics

Global stability of a class of futile cycles. (English) Zbl 1361.92027
Summary: In this paper, we prove the global asymptotic stability of a class of mass action futile cycle networks which includes a model of processive multisite phosphorylation networks. The proof consists of two parts. In the first part, we prove that there is a unique equilibrium in every positive compatibility class. In the second part, we make use of a piecewise linear in rates Lyapunov function in order to prove the global asymptotic stability of the unique equilibrium corresponding to a given initial concentration vector. The main novelty of the paper is the use of a simple algebraic approach based on the intermediate value property of continuous functions in order to prove the uniqueness of equilibrium in every positive compatibility class.

MSC:
92C40 Biochemistry, molecular biology
34D23 Global stability of solutions to ordinary differential equations
92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Al-Radhawi, MA; Angeli, D, New approach to the stability of chemical reaction networks: piecewise linear in rates Lyapunov functions, IEEE Trans. Autom. Control, 61, 76-89, (2016) · Zbl 1359.93440
[2] Anderson, DF, A proof of the global attractor conjecture in the single linkage class case, SIAM J. Appl. Math., 71, 1487-1508, (2011) · Zbl 1227.92013
[3] Angeli, D; Sontag, ED, Translation-invariant monotone systems and a global convergence result for enzymatic futile cycles, Nonlinear Anal. Real World Appl., 9, 128-140, (2008) · Zbl 1401.92086
[4] Blanchini, F; Giordano, G, Piecewise-linear Lyapunov functions for structural stability of biochemical networks, Automatica, 50, 2482-2493, (2014) · Zbl 1301.93125
[5] Conradi, C; Shiu, A, A global convergence result for processive multisite phosphorylation systems, Bull. Math. Biol., 77, 126-155, (2015) · Zbl 1334.92169
[6] Feinberg, M, The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Ration. Mech. Anal., 132, 311-370, (1995) · Zbl 0853.92024
[7] Horn, F; Jackson, R, General mass action kinetics, Arch. Ration. Mech. Anal., 47, 81-116, (1972)
[8] Khalil HK (2014) Nonlinear Systems, 3rd edn. Pearson Education Limited, Essex
[9] LaSalle, JP, Some extensions of liapunov’s second method, IRE Trans. Circuit Theory, CT-7, 520-527, (1960)
[10] Maeda, H; Kodama, S; Ohta, Y, Asymptotic behavior of nonlinear compartmental systems: nonoscillation and stability, IEEE Trans. Circuits Syst., CAS-25, 372-378, (1978) · Zbl 0382.93041
[11] Maeda, H; Kodama, S, Some results on nonlinear compartmental systems, IEEE Trans. Circuits Syst., CAS-26, 203-204, (1979) · Zbl 0395.93020
[12] Murray RM, Li Z, Sastry SS (1994) A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton · Zbl 0858.70001
[13] Salazar, C; Høfer, T, Multisite protein phosphorylation—from molecular mechanisms to kinetic models, FEBS J., 276, 3177-3198, (2009)
[14] Siegel, D; MacLean, D, Global stability of complex balanced mechanisms, J. Math. Chem., 27, 89-110, (2000) · Zbl 1012.92046
[15] Vol’pert AI, Hudjaev SI (1985) Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Martinus Nijhoff Publishers, Dordrecht
[16] Wang, L; Sontag, ED, On the number of steady states in a multiple futile cycle, J. Math. Biol., 57, 29-52, (2008) · Zbl 1141.92022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.