Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays

Abstract

Delay-differential equations are commonly used to model genetic regulatory systems with the delays representing transcription and translation times. Equations with delayed terms can also be used to represent other types of chemical processes. Here we analyze delayed mass-action systems, i.e. systems in which the rates of reaction are given by mass-action kinetics, but where the appearance of products may be delayed. Necessary conditions for delay-induced instability are presented in terms both of a directed graph (digraph) constructed from the Jacobian matrix of the corresponding ODE model and of a species-reaction bipartite graph which directly represents a chemical mechanism. Methods based on the bipartite graph are particularly convenient and powerful. The condition for a delay-induced instability in this case is the existence of a subgraph of the bipartite graph containing an odd number of cycles of which an odd number are negative.

Partial differential equations for self-organization in cellular and developmental biology

“…Qualitative techniques will be especially valuable for such studies. For example, graph theory can be applied to study the stability of large networks of ODEs [104,105]. Note that the applied study of the linear stability of PDEs typically involves the reduction to ODEs via an expansion in the eigenfunctions of the transport operator.…”

A graph-theoretic method for detecting potential Turing bifurcations

“…By the necessary condition given in the previous paragraph, if a r changes sign then a real root can change its sign as parameters vary, and if a k changes sign for some k r then the real part of a pair of complex roots can change its sign. 23 The connection of the bipartite graph G of the chemical mechanism ͑1͒ with instabilities in spatially homogeneous mass-action kinetics models has been studied elsewhere. 23,28 The bipartite graph can be constructed directly from the chemical mechanism ͑1͒.…”

“…23 The connection of the bipartite graph G of the chemical mechanism ͑1͒ with instabilities in spatially homogeneous mass-action kinetics models has been studied elsewhere. 23,28 The bipartite graph can be constructed directly from the chemical mechanism ͑1͒. It allows for multiple paths between the same substances ͑unlike, e.g., the digraph 23 ͒ which makes it suitable to represent more complex reactions.…”

“…This condition generalizes the concept of indirect catalysis. 29 The ordinary differential equation counterpart to this problem has been studied elsewhere 23,28 and many of the results therein are essential for understanding the theory in this paper. In Sec.…”

“…21 Biochemical networks are often represented by various types of graphs. 15,[22][23][24] Here we study mass-action kinetics reaction networks and will represent them as bipartite graphs. 25 The convenience of bipartite graph representation, especially for biochemical networks, has been emphasized elsewhere.…”