Some dynamical properties of delayed weakly reversible mass-action systems

This paper focuses on the dynamical properties of delayed complex balanced systems. We first study the relationship between the stoichiometric compatibility classes of delayed and non-delayed systems. Using this relation we give another way to derive the existence of positive equilibrium in each stoichiometric compatibility class for delayed complex balanced systems. And if time delays are constant, the result can be generalized to weakly reversible networks. Also, by utilizing the Lyapunov-Krasovskii functional, we can obtain a long-time dynamical property about $\omega$-limit set of the complex balanced system with constant time delays. An example is also provided to support our results

A global convergence result for processive multisite phosphorylation systems

Multisite phosphorylation plays an important role in intracellular signaling. There has been much recent work aimed at understanding the dynamics of such systems when the phosphorylation/dephosphorylation mechanism is distributive, that is, when the binding of a substrate and an enzyme molecule results in addition or removal of a single phosphate group and repeated binding therefore is required for multisite phosphorylation. In particular, such systems admit bistability. Here we analyze a different class of multisite systems, in which the binding of a substrate and an enzyme molecule results in addition or removal of phosphate groups at all phosphorylation sites. That is, we consider systems in which the mechanism is processive, rather than distributive. We show that in contrast with distributive systems, processive systems modeled with mass-action kinetics do not admit bistability and, moreover, exhibit rigid dynamics: each invariant set contains a unique equilibrium, which is a global attractor. Additionally, we obtain a monomial parametrization of the steady states. Our proofs rely on a technique of Johnston for using "translated" networks to study systems with "toric steady states", recently given sign conditions for injectivity of polynomial maps, and a result from monotone systems theory due to Angeli and Sontag.Comment: 23 pages; substantial revisio

A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics

Motivated by questions in mass-action kinetics, we introduce the notion of vertexical family of differential inclusions. Defined on open hypercubes, these families are characterized by particular good behavior under projection maps. The motivating examples are certain families of reaction networks -- including reversible, weakly reversible, endotactic, and strongly endotactic reaction networks -- that give rise to vertexical families of mass-action differential inclusions. We prove that vertexical families are amenable to structural induction. Consequently, a trajectory of a vertexical family approaches the boundary if and only if either the trajectory approaches a vertex of the hypercube, or a trajectory in a lower-dimensional member of the family approaches the boundary. With this technology, we make progress on the global attractor conjecture, a central open problem concerning mass-action kinetics systems. Additionally, we phrase mass-action kinetics as a functor on reaction networks with variable rates.Comment: v5: published version; v3 and v4: minor additional edits; v2: contains more general version of main theorem on vertexical families, including its accompanying corollaries -- some of them new; final section contains new results relating to prior and future research on persistence of mass-action systems; improved exposition throughou