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.

Keywords:

• Network dynamics
• consensus dynamics
• chemical reaction networks

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Recently, also in Craciun and Feinberg (2010), the idea of adding a zero complex was followed up; however in the different context of preclusion of multi-stability.

2. Sometimes called a multigraph, since we allow for multiple edges between vertices.

3. In the literature sometimes also referred to as reaction graphs.

4. In van der Schaft et al. (2013a), Rao et al. (2013a), the matrix Z was called the ‘complex stoichiometric matrix’.

5. In Chapman and Mesbahi (2011), such a matrix L was called an out-degree Laplacian matrix.

6. In the special case $\mathrm{im}D\cap kerZ=\left\{0\right\}$ (deficiency zero in the terminology of Feinberg (1972)), complex-balancedness is equivalent to the existence of a positive equilibrium of (3(3) $$\dot{x}=ZDK\mathrm{Exp}\left(Z^T\mathrm{Ln}x\right).$$(3) ).

7. Strong connectedness of the connected components is in the CRN literature often referred to as weak reversibility (Horn, 1972)

8. It is generally believed that most reaction networks are persistent. However, up to now, this persistence conjecture has been only proved in special cases (cf. Anderson (2011), Angeli, De Leenheer, and Sontag (2011), Siegel and MacLean (2000), and the references quoted in there).

9. See e.g. Oster et al. (1973), van der Schaft et al. (2013a), van der Schaft et al. (2013b) for further information.

10. This theorem goes back to the classical work of Kirchhoff on resistive electrical circuits (Kirchhoff, 1847); see Bollobas (1998) for a succinct treatment. Nice accounts of the matrix tree theorem in the context of CRNs can be found in Mirzaev and Gunawardena (2013), Gunawardena (2014).

11. Thermodynamically the assumption of detailed-balancedness is well-justified; it corresponds to microscopic reversibility (Oster et al., 1973).

12. It can be shown (van der Schaft et al., 2013a) that the matrix ${\mathcal{K}}^r\left(x^{*}\right)$ is independent of the choice of the thermodynamic equilibrium x* up to multiplicative factor for every connected component of $\mathcal{H}$.

13. As shown in van der Schaft et al. (2015) the weakened Wegscheider conditions are also equivalent to the notion of formal balancing introduced in Dickenstein and Perez Millan (2011) as a formalization of the ‘circuit conditions’ of Feinberg (1989).