title: Chemical reaction networks in random environments
Creators: name: Sinzger-D'Angelo, Mark
Publisher: name: UNSPECIFIED
Year: 2024

description: Cellular processes operate in context, rather than in isolation. To account for this heterogeneity, we consider the model class of chemical reaction networks (CRNs) with extrinsic noise (the environment), manifesting as additional stochasticity in the reaction rate constants. A joint description of subnetwork and environment is computationally heavy. Here, we attempt to marginally describe the subnetwork as if it was still embedded. This comes with the merit of attributing stochastic properties of the subnetwork to features of the environment. Additionally, the marginal description enables the estimation of information measures. To puzzle out how the cell makes reliable decisions in the presence of the environmental noise, we computationally target the path mutual information between environment and subnetwork. Namely, we consider the minimal Poisson channel, motivated by the reaction counters that sufficiently describe the CRN. For closed-form expressions and computational purposes, we require simplifying assumptions. Hence, a particular focus of this thesis lies on linear subnetworks in discrete state Markov environments and on general subnetworks in linear environments.

We contribute at different levels of the stochastic description: (i) At the level of moments, we generalize results from queuing theory about the exact stationary mean evaluation of linear CRNs in a discrete state Markov environments. Our analytic expression depends on the reaction rate constants of the linear subsystem, as well as the generator and stationary distribution of the Markov environment. We extend spectral decomposition results on intrinsic and extrinsic noise to the case of correlated environment components. (ii) We present Liouville master equations with boundary conditions for the probability evolution of a marginal CRN description via auxiliary statistics. Our method of the backward recurrence time parametrization (BReT-P) for piecewise-deterministic Markov processes (PDMP) introduces a standard form for the marginal description of CRNs via approximate filters. We derive generalized master equations for examples with a low number of species.
(iii) At the process level, we formalize a subclass of PDMPs having Dirac measures at jump times, which we call Dirac-PDMPs. We offer an approximate marginal simulation algorithm based on optimal linear filtering.

For CRNs in a linear stationary environment, i.e., with exponential auto/cross-covariance function, we provide an approximate filter that is based on Snyder's optimal linear filtering for counting processes. By regarding the chemical reactions as events, we establish a link between CRNs in a linear environment and Hawkes processes, a class of self-exciting counting processes widely used in event analysis. We show that the Hawkes approximation is equivalently obtained via moment closure scheme or as the optimal linear approximation under the mean-square error. Furthermore, we use martingale techniques to provide results on the agreement of the Hawkes process and the exact marginal process in their second order statistics, i.e., covariance, auto/cross-correlation. Taking the Hawkes model as a reference, we attribute stochastic properties of the subnetwork to the linear or non-linear dynamics of the environment, respectively. We introduce an approximate marginal simulation algorithm and illustrate it in case studies.

The empirical results focus on structure switching of macromolecules and excursions. Macromolecules that are abundant in different conformations have been studied for gene expression models using thermodynamic ensembles. However, we include switching dynamics as the environmental component. With our model reduction approach, this inclusion does not increase the number of species. Thereby, we bridge a gap between structure kinetics and gene expression models, which can further improve our understanding of gene regulatory networks and facilitate genetic circuit design.
For the modulation of translation by mRNA, we provide a method to quantify how mRNA structure dynamics contributes to translational heterogeneity. In a further set of applications we focus on the phenomenon of excursions, near-to-linear accumulation of a species in periods of a vanishing decay rate. We examine how this increases the stationary mean, attributing the increase to the zero state. We discuss a controller that mitigates the effect and evaluate the ability of the Hawkes model to capture excursions.

For estimating information-measures of the Poisson channel, we present an analytic approach that circumvents Monte Carlo sampling. This simulation-free estimation method is enabled by establishing the conditional intensity of counting processes and its asymptotic distribution as central quantitites. For the Poisson channel with binary Markov input, we express the mutual information as a Riemann integral. While in the classical result the capacity-achieving input favors the Off state, we in contrast report On-favoring regimes for lower-bounded average soujourn time in the On and Off states. Additionally, we provide evidence that among the binary inputs the exponential sojourn times are not optimal. Relaxing the exponential constraint of the binary input by allowing for multiple Off states, we discuss the information-theoretic advantage of cycling through several Off states.

Throughout, stochastic conditioning is used as the tool for model reduction. We take the two routes of conditioning on the environment and on the subnetwork. We present the equivalence of both routes in a unifying approach that uses the tower property on the Kolmogorov backward equation. Comparing different approximate filters for the structure switching environment, we shed light on their orthogonal strengths. We present doubts as to whether the presented model reduction strategies can resolve the curse of dimensionality of CRNs compared to the Doob-Gillespie algorithm. Overall, we contribute to the model reduction, marginal simulation, computation of information-measures and attribution theory for CRNs in random environments.