A network dynamics approach to chemical reaction networks

Abstract

A treatment of a chemical reaction network theory is given from the perspective of nonlinear network dynamics, in particular of consensus dynamics. By starting from the complex-balanced assumption, the reaction dynamics governed by mass action kinetics can be rewritten into a form which allows for a very simple derivation of a number of key results in the chemical reaction network theory, and which directly relates to the thermodynamics and port-Hamiltonian formulation of the system. Central in this formulation is the definition of a balanced Laplacian matrix on the graph of chemical complexes together with a resulting fundamental inequality. This immediately leads to the characterisation of the set of equilibria and their stability. Furthermore, the assumption of complex balancedness is revisited from the point of view of Kirchhoff's matrix tree theorem. Both the form of the dynamics and the deduced behaviour are very similar to consensus dynamics, and provide additional perspectives to the latter. Finally, using the classical idea of extending the graph of chemical complexes by a 'zero' complex, a complete steady-state stability analysis of mass action kinetics reaction networks with constant inflows and mass action kinetics outflows is given, and a unified framework is provided for structure-preserving model reduction of this important class of open reaction networks

Coexistence in molecular communications

“…A related question is whether more complex production and absorption dynamics in the molecular communication link can be accounted for. Recent work in the control literature on advection dynamics [28] has provided results for constant inflows and mass action kinetics outflows, which provide an initial direction to address this question.…”

Bond Graph Representation of Chemical Reaction Networks

“…In parallel with the seminal work of Oster et al [7,8], the mathematical foundations of chemical reaction networks (CRN) were being laid by Feinberg [13], Horn and Jackson [14] and Feinberg and Horn [15]. This approach to chemical reaction network theory was further developed by Sontag [16], Angeli [17], and van der Schaft et al [18,19,20]. General results on stability of both closed and open systems of chemical reactions have been derived and applied to reveal dynamic features of complex (bio)chemical networks [21], dissipation in noisy chemical networks Polettini et al [22], metabolic networks [23] and multistability in interferon signalling Otero-Muras et al [24].…”

Hamiltonian Feedback Design for Mass Action Law Chemical Reaction Networks

“…Further work includes the generalization of the proposed method for the port-Hamiltonian description [24] where the more general complex balanced case could be covered.…”

“…For the special case of reversible chemical reaction networks it was shown [20] that they admit a generalized Hamiltonian description if the number of reversible reactions is less or equal than the number of species in the system. Very recently, a port-Hamiltonian description of close complex balanced chemical reaction networks have also been proposed [24].…”