Analysis and control of stochastic reaction networks - Applications to biology

Reaction networks are systems in which the populations of a finite number of species evolve according to predefined interactions. Such networks are found as modeling tools in many disciplines (spanning biochemistry, epidemiology, pharmacology, ecology and social networks). Traditionally, reaction networks are mathematically analyzed by expressing the dynamics as a set of ordinary differential equations. Such a deterministic model is reasonably accurate when the number of network participants is large. However, when this is not the case, the discrete nature of the interactions becomes important and the dynamics is inherently noisy. This random component of the dynamics cannot be ignored as it can have a significant impact on the macroscopic properties of the system. This is the reason why stochastic models for reaction networks are necessary for representing certain reaction networks. The tools for analyzing them, however, still lag far behind their deterministic counterparts.

In this talk, a short introduction to biological mechanisms and biological problems are first presented via simple examples in order to set up the ideas. Different modeling techniques will be briefly discussed together with their applicability domain, their benefits and their drawbacks. A constructive framework for examining the long-term behavior and stability properties of the reaction dynamics in a stochastic setting is then proposed. In particular, we will address the problem of determining ergodicity of the reaction dynamics, which is analogous to having a globally attracting fixed point for deterministic dynamics.  We will then demonstrate that stability properties of a wide class of networks can be assessed from theoretical results that can be recast as efficient and scalable linear programs, well-known for their tractability. It is notably shown that the computational complexity is often linear in the number of species, and worst-case quadratic. We illustrate the applicability of the results on several reaction networks arising in fields such as biochemistry, epidemiology and ecology. Finally, two control problems will be discussed. The first one is in-silico population control where the aim is to control the average number of proteins present in a cell-population. Variance control can also be performed this way. The second one is concerned with in-vivo single-cell and population control meaning that the goal is to control the average number of a protein in a single-cell (and/or over the cell population). In this latter framework, the controller is implemented in terms of reactions. Several properties of the closed-loop system (robustness, non-fragility, tracking, disturbance rejection, etc) will be emphasized and illustrated through several examples.