Identifying Hard Bounds on Molecular Fluctuation in Stochastic Reaction Systems

Abstract

Non-equilibrium stochastic reaction networks are common in biological systems, creating heterogeneity in genetically identical populations. Predicting the dynamics of such systems is challenging because small changes in the type of reaction rates or network topology can dramatically change the behavior, and because most sub-networks of interest are embedded in larger networks that affect their behavior. This dissertation is therefore dedicated to identifying broader principles for stochastic reaction networks that hold for large classes of systems where many properties can be left unspecified. That is possible by focusing on "impossibility proofs", i.e., not asking not what a particular system does, but what no member in a broader family of systems could ever do. By considering various classes of feedback control systems, we identified a new physical limit on fluctuations in connected components of reaction networks. Specifically, we found that it is impossible to simultaneously suppress fluctuations in all components of reaction networks, and that depending on the topology, a number of components must display large fluctuations in order for fluctuations in other components to be suppressed. In connected reaction networks it is thus impossible to reduce the statistical uncertainty in all components, regardless of the control mechanisms or energy dissipation. Suppressing fluctuations around a target value is important for some control systems, but for others the challenge is to achieve precise dynamics, such as regular oscillations. Many synthetic oscillators in biology were mathematically designed to maximize precision, but most still performed much worse than their natural counterparts, sometimes barely oscillating at all. This discrepancy may reflect the fact that most of the theory focused on deterministic conditions for oscillations, failing to account for intrinsically stochastic effects at low numbers of molecules. Here we propose a metric based on the integration of the autocovariance function to quantify the quality of oscillations, i.e., how noisy a stochastic oscillator is, and identify a fundamental limit on the quality of oscillation for a broad class of oscillator models.