Notice Board

Chemical kinetics is governed by the law of mass-action, which generates rate equations of a very particular form with polynomial right-hand sides. Despite the relatively simple form of these equations, chemical systems are capable of arbitrarily complex behavior. Not every model displays interesting behaviors (oscillations, bistability, chaos, etc.), and even for those that do, these behaviors are typically found in only part of the parameter space. Finding the regions of interesting behavior generally involves a study of bifurcations, which are points where the behavior changes. For instance, in an Andronov-Hopf bifurcation, a steady state becomes unstable as an oscillatory solution emerges. Many elementary bifurcations can be detected by studying the Jacobian matrix of the differential equations, which tells us how fast the rates change with respect to the concentrations. For mass-action systems, this Jacobian has a very particular form. Nevertheless, for larger chemical systems, the analysis required to find bifurcations is all but impossible. It can be done numerically, but this raises the question of where to start looking in the parameter space, which can also be large. And of course, a priori, we may not know if a given model even has a sought-for behavior. It turns out that a class of bipartite graphs (graphs with two types of vertices) has a direct correspondence to the Jacobian matrix of the mass-action differential equations. The analysis of the Jacobian is then replaced by a search for subsets of the bipartite graph with certain properties, which turns out to be an easier problem. These subsets, known as fragments, are associated with a necessary but, unfortunately, not sufficient condition for specific bifurcations. This is still useful for two reasons: First, if the condition is met, then we know there is at least a chance of finding a given behavior among the model's solutions; otherwise, there is no point looking. Second, the graph-theoretic analysis suggests where in the parameter space one might start looking for the corresponding bifurcations.