Stochastic models of Chemical Reaction Networks: multiscale approximations and convergence

Chemical reaction networks are mathematical models widely used to describe the dynamical behaviour of systems in biology, epidemiology, chemistry. In such models, individual units (e.g. molecules), belonging to different groups (e.g. chemical species), interact with each other according to specific laws, named reactions, which may be represented in a graph. Reactions are usually modelled as stochastic counting processes, so that the number of molecules in the system for each species is described by a continuous time Markov chain. When the number of molecules is very high, species dynamics may be suitably described in terms of chemical concentrations, and the stochastic model is well approximated by a deterministic, continuous dynamical system. However, in some specific cases, such continuous limit proves to be unsatisfactory, and the intrinsic discreteness of the model cannot be disregarded. For instance, in signal transduction processes, a single molecule may trigger a biochemical cascade which causes a transition in the cell state. An example of a chemical reaction system displaying such peculiar behaviour was suggested by Togashi and Kaneko: in this model, biochemical cascades are driven by fast autocatalytic reactions, while inflows and outflows of single species happen at much slower rates. For this class of systems, the classical theory of convergence for chemical reaction networks cannot always be applied, since many of the underlying assumptions fail. In this thesis, classical theory is first analysed in detail, stating the main results concerning convergence on different time-scales. A simplified version of the Togashi-Kaneko example is studied in such framework and convergence to non-degenerate limit models is verified, whenever possible. The main criticalities arise when the time-scale is accelerated so that trajectories of the stochastic process describing species concentrations display sharp peaks or rapid switches to different stable states, both induced by the fast autocatalytic reactions and corresponding to failed or completed transitions, respectively. The second part of the thesis is devoted to the theory of weak convergence in metric and non-metric spaces, with focus on the Skorohod space of cadlag functions. It is shown that the simplified version of the Togashi-Kaneko model considered above does not converge to a naturally arising limiting model in the Skorohod topology. In particular, the sequence of probability measures on the Skorohod space is not relatively compact. The last part of the work presents an alternative sequential topology on the Skorohod space, proposed by Jakubowski, which is weaker than the Skorohod topology and not metrizable. In this topology, sharp peaks corresponding to failed transitions coalesce and cancel out, and therefore it seems reasonable for the simplified Togashi-Kaneko model to converge in the Jakubowski topology to the above-mentioned limit. In view of proving such convergence, it is shown that, under suitable conditions, autocatalytic cascades may be separated from inflow and outflow reactions, in the sense that no inflows and outflows happen during an autocatalytic cascade. Moreover, uniform tightness of the sequence of probability measures is verified in this topology.