Reaction networks
Reaction-network theory is a broadly applicable paradigm: it can represent population dynamics, epidemiological dynamics, chemical and biological reaction networks, and even social networks.
Deterministic models track continuous quantities such as concentrations and work well when the homogeneous-mixing assumption holds and species populations are large. They fit chemistry well, since the species involved are present in Avogadro numbers. SIR epidemiological models and Lotka-Volterra population models are both familiar examples. Their analysis includes stability of equilibria, persistence and permanence of trajectories, boundedness, deficiency theory, and absolute concentration robustness. Control and observation are interesting but relatively standard problems because the underlying objects are differential equations.
The stochastic regime
Stochastic models track actual species counts and the state takes integer values. They capture processes where populations of some species are small enough that randomness cannot be ignored, as in systems biology where reaction stochasticity is the source of intrinsic noise. Under standard assumptions, the process is a continuous-time Markov process whose distribution is governed by the forward Kolmogorov equation, the so-called Chemical Master Equation.
Solving the master equation directly is impossible in general because it consists of a countably infinite system of ODEs. When the network involves only monomolecular reactions, the moments of the probability distribution admit a closed, finite-dimensional system. Once bimolecular reactions or nonlinear propensities enter, the moments no longer close and the moment-closure problem appears.
Control of stochastic reaction networks
Cybergenetics asks how feedback control can be implemented inside or alongside biological systems described by stochastic reaction networks. Antithetic integral feedback, a chemical realization of integral control, can achieve robust perfect adaptation. Output controllability, ergodicity, and noise-suppression results form a mathematical foundation for designing controllers that act on cells. In silico moment control closes feedback loops experimentally by computing actions from measured cell populations.
This is the line of work covered in the Quanta Magazine feature on the mathematics of cellular feedback, the ETH News piece on integral feedback control for cells, and an Annual Review article on noise in biomolecular systems.
Related publications
For the full list, see the publications page (search for "antithetic", "stochastic", "reaction network", or "biology").