Impact
The objectives of the project will provide new light on pivotal questions in systems biology, with radically different interpretations depending on the specific application domain:
- Biochemistry. Interpreting particles in the system as molecules in a cell provides one of the most fundamental motivations of this project. In this context, the results of the project would ensure sufficient stability of the Stochastic Reaction Network for the existence and the characterization of the long-time behavior of (a subset of) the reactions in a cell, independently of the fine-scale biological parameters (often impossible to infer with the desired precision). The impact of this theoretical result is multifold. First, it gives simple structural conditions ensuring the possibility of studying the long-time behavior of large cell populations. Second, it allows the application of averaging techniques for the study of slow-fast systems, ubiquitous in systems biology, to investigate the dynamics at slower timescales by integrating against the averaged behavior of the fast components, represented by its invariant measure. Finally, this result allows the extension of Wentzell-Freidlin estimates on spontaneous transitions between attractors of the limiting, deterministic dynamics to the more realistic, non-compact state space setting. On the other hand, the results obtained through addition of a spatial component to the analysis would allow for the study of the formation of spatial patterns (Turing patterns) and traveling waves within the cell or over a population of cells. This phenomenon is of extreme interest in biology as it is believed to be the mechanism underlying morphogenesis, a process that is still poorly understood.
- Epidemiology. In the context of epidemiology, the various particles represent different individuals, and the type of particles represent their infectious state. This application domain has sadly been of critical importance over recent years: an understanding of the infectious disease dynamics could have allowed us to predict and limit the spread of COVID-19 on our planet. In particular, the study of the spatial propagation of infectious diseases is an active domain of research, and the development of realistic graphical models of community contact would have tremendous implications in our ability to correctly model and therefore control the propagation of such infectious diseases. In this setting, the various spatial structures considered in the project reflect different confinement levels (i.e. the number of direct contacts of an individual with other members of the population) and the individuals’ freedom to move spatially.
- Synthetic biology. The quantitative estimates on the long-time behavior of large networks of chemical reactions would further significantly boost our ability to control such systems. This aspect plays a critical role in the context of systems biology, where researchers attempt to modify the behavior of complex biochemical systems such as the cell by minimally changing its structure via DNA editing. The implications of a more precise understanding of these complex systems would have far-reaching consequences in this context, shifting the experimental methodology from trial-and-error-based to a more principled one.
- Mathematics. Importantly, the impact of this project is not limited to its main application domains. Indeed, the methods developed to solve the problems this project aims to address are expected to have an equally important impact within the mathematical community. Several proof techniques utilized will be either completely new or applied in different contexts than the classical ones.