What makes biochemical networks tick?

Abstract

In view of the increasing number of reported concentration oscillations in living cells, methods are needed that can identify the causes of these oscillations. These causes always derive from the influences that concentrations have on reaction rates. The influences reach over many molecular reaction steps and are defined by the detailed molecular topology of the network. So-called Ôautoinfluence pathsÕ, which quantify the influence of one molecular species upon itself through a particular path through the network, can have positive or negative values. The former bring a tendency towards instability. In this molecular context a new graphical approach is presented that enables the classification of network topologies into oscillophoretic and nonoscillophoretic, i.e. into ones that can and ones that cannot induce concentration oscillations. The network topologies are formulated in terms of a set of uni-molecular and bi-molecular reactions, organized into branched cycles of directed reactions, and presented as graphs. Subgraphs of the network topologies are then classified as negative ones (which can) and positive ones (which cannot) give rise to oscillations. A subgraph is oscillophoretic (negative) when it contains more positive than negative autoinfluence paths. Whether the former generates oscillations depends on the values of the other subgraphs, which again depend on the kinetic parameters. An example shows how this can be established. By following the rules of our new approach, various oscillatory kinetic models can be constructed and analyzed, starting from the classified simplest topologies and then working towards desirable complications. Realistic biochemical examples are analyzed with the new method, illustrating two new main classes of oscillophore topologies.

Chemical interpretation of oscillatory modes at a Hopf point

“…[2][3][4] A general approach to the identification of oscillatory mechanisms has previously been discussed by Eiswirth and coworkers 18 (see also ref. 19). Their approach does not depend on the presence of a Hopf bifurcation, but instead on knowledge of the full reaction mechanism, which is often not available.…”

Graph theory and qualitative analysis of reaction networks

“…The directed bipartite graph (DB graph) was developed by Ivanova [46,47,79], and more rigorously analysed by Mincheva and Roussel in [55,56]. It has been applied to some real mechanisms by Ermakov, Goldstein et al [23,34,36,37], and analysed for small networks with up to four species [22,24,35]. The directed bipartite graph G bip has the same vertex and edge set as constructed for the SR graph, except that outflow and inflow reactions are included in the reaction vertex set (V 2 ).…”

Graph-theoretic analysis of a model for the coupling between photosynthesis and photorespiration

“…[18][19][20][21][22] We can sometimes relate some particular behavior (e.g., multistability or oscillations) to properties of a suitable graphical representation of the network. 5,10,16,[23][24][25][26][27][28][29][30][31][32][33][34][35][36] In particular, we recently developed a graphical analysis that provides a necessary condition for the generation of delay-induced instabilities in delayed massaction systems. 33 Delayed terms have a number of uses in the modeling of biochemical systems.…”