A note on Monotone Systems with Positive Translation Invariance

Abstract

We show that strongly monotone systems of ordinary differential equations which have a certain translation-invariance property are so that all solutions converge to a unique equilibrium. The result may be seen as a dual of a well-known theorem of Mierczynski for systems that satisfy a conservation law. An application to a reaction of interest in biochemistry is provided as an illustration.

Chemical networks with inflows and outflows: A positive linear differential inclusions approach

“…We will illustrate this by means of a case study of an important example arising in systems biology, known as the 2-step futile cycle. [7][8][9][10][11] In fact, the study of this particular example triggered the development of our main theoretical results. The 2-step futile cycle is one of the basic building blocks of various biochemical networks, for instance as a 2-step phosphorylation-dephosphorylation reaction.…”

“…The 2-step futile cycle is one of the basic building blocks of various biochemical networks, for instance as a 2-step phosphorylation-dephosphorylation reaction. We have previously studied the futile cycle in isolation, [7][8][9] and despite the fact that it is governed by a relatively large system of nonlinear differential equations for which traditional techniques such as the quasi-steady state approximation only yield partial results (as they are based on simpler approximations), its global dynamical behavior is now fairly well understood, mainly because of monotonicity properties, see Refs. 7 and 8.…”

Translation-invariant monotone systems II: Almost periodic/automorphic case

“…Recently Angeli and Sontag [4,5] and Angeli, Leenheer and Sontag [6] have contributed a new type of global convergence condition, named positive translation invariance, which is motivated by a chemical reaction network. A standard form for representing (well-mixed and isothermal) chemical reactions by ordinary differential equations is n is a function which provides the vector of reaction rates for any given vector of concentrations.…”

“…A standard form for representing (well-mixed and isothermal) chemical reactions by ordinary differential equations is n is a function which provides the vector of reaction rates for any given vector of concentrations. Choosing σ ∈ R m + and using the reaction coordinates x, S = σ + Γx, instead of the traditional species coordinates S, Angeli and Sontag [4,5] and Angeli, Leenheer and Sontag [6] have investigated the monotonicity and global behavior of systems in the reaction coordinates, (1.2)ẋ = f σ (x) = R(σ + Γx), evolving on the state space X σ = {x ∈ R n : σ + Γx ≥ 0}.…”

“…Motivated by the study of Angeli and Sontag [4,5] and Angeli, Leenheer and Sontag [6] and our previous work [31], we shall investigate the nonautonomous chemical reaction network. Suppose that the reaction rates depend on time almost periodically/automorphically:…”