Corentin Briat

Reaction Networks

Reactions network theory is a very powerful and broadly applicable paradigm that can be used to represent population dynamics, epidemiological dynamics, chemical and biological reaction networks, social networks, etc.

Deterministic models consider the evolution of continuous quantities such as concentrations. These models are well-suited when the homogeneous mixing property is strongly satisfied and the species populations are large (so that the definition of concentration is well-defined and makes perfect sense). These assumptions are generally satisfied in some fields. For instance, this is the case in chemistry since species involved there are in Avogadro numbers. Deterministic models are very common in epidemiology, the so-called SIR-models, or even population models described, for instance, by Lotka-Volterra equations. Analysis of these models include stability of equilibrium points, persistence and permanence of trajectories, boundedness of trajectories, deficiency theory, absoute concentration robustness analysis, etc. Control and observation of these processes are also interesting and important problems but are quite standard as those proceses are represented by differential equations, possibly including delays and/or diffusion terms.

Stochastic models, on the other hand, keep track of the actual count of individuals in the reaction network and the state takes integer values. Such models are better able to characterize processes where population of certain species can be small and for which stochastic effects cannot be neglected. This is, for instance, the case in systems biology where randomness in the reactions is the source of intrinsic noise. Under some assumptions, the process is a contnuous-time Markov process that can be exactly described by the forward Kolmogorov-equation (also called the Chemical Master Equation). Solving the master equation is, in general, not possible since it consists of a countable infinite set of ordinary differential equations. When the network only involves monomolecular (or monoindividual) reactions, the dynamics of the moments of the probability distribution admit an explicit and closed-form that is finite-dimensional. When some reactions are however bimolecular or the propensity function is nonlinear, the moments do not admit any closed-form anymore and the moment closure problem arises. The analysis of stochastic reaction networks mostly involves open problems, that are the stochastic analogs of the ones in the deterministic setting. Furthermore, not only the actual trajectories of the stochastic process are important, the moments are also critical from a practical viewpoint.

Analysis of Stochastic Reaction Networks

It has been shown that ergodicity is the natural stability concept to consider to characterize the long term behavior of those networks. The reaction network is ergodic if the Chemical Master Equation has a globally attracting fixed point. Several conditions have been obtained in order to establish several properties of stochstic reaction networks such as the ergodicity of the network, the light-tailedness of the stationary distribition, or the boundedness of the stationary moments. Many of those conditions take the form of linear programs that scale linearly with the number of species involves in the network. Those results have been extended to address the case of uncertain networks and networks with delays with a very limited increase in the computational complexity through the exploitation of the structure of the problem.

In-silico control of the moment equation

The main difficulty when controlling the moment equation is the potential presence of unknown higher-order moments perurbing the lower-order moments dynamics. Note that this only arises when the controlled reaction network is bimolecular and mass-action. When the network is unimolecular, the moment equation is closed and when the propensity functions are non-mass-action, there is no solution available so far. It was shown that we can both control the mean and the variance of protein species in a gene expression network using simple PI control laws and that a similar controller structure could be used to control a bimolecular newtwork involving a dimerization reaction.

In-vivo control of single cells and cell-populations

Unlike in-silico control, the problem of in-vivo control consists of implementing controllers inside cells using synthetic biology and a DSA implementation of controllers. There are strong constraints that limit what can be done: the current technology does not allow to implement arbitrarily complex synthetic networks and we need to focus on simple and implementable solutions, the controller needs to have a structure that is amenable to chemical reactions, the controller needs to be energy-aware in the sense that it should not deplete the cell energy, and it needs to be robust since biological systems are poorly characterized in general. A satisfying solution to the problem or constant setpoint tracking and disturbance rejection was proposed through the concept of Antithetic Integral Controller that has been shown to work both in the deterministic and the stocahstic cases. Interestingly, the controller is shown to exploit noise to achieve its function in the stochastic case through the noise-induced property called innocuousness that stipulates that of the network is ergodic and output controllable, then the controlled network will be necessarily ergodic. This has to be contrasted with the deterministic case where an integrator can lead to oscillations and instabilities if not properly tuned.