Entropy production constrains information throughput in gene regulation

Biological signal processing typically requires energy, leading us to hypothesize that a cell’s information processing capacity is constrained by its energy dissipation. Signals and their processing mechanisms are often modeled using Markovian chemical reaction networks (CRNs). To enable rigorous analysis, we review and reformulate stochastic thermodynamics for open CRNs, utilizing Kurtz’s process-based formulation. In particular, we revisit the identification of the energy dissipation rate with the entropy production rate (EPR) at the non-equilibrium steady state (NESS). We also highlight potential inconsistencies in traditional formulations for generic Markov processes when applied to open CRNs, which may lead to erroneous conclusions about equilibrium, reversibility, and the EPR. Additionally, we review the concepts of mutual information (MI) and directed information (DI) between continuous-time trajectories of CRNs, which capture the transmission of spatiotemporal patterns. We generalize existing expressions for the MI, originally accounting for transmission between two species, to now include transmission between arbitrary subnetworks. A rigorous derivation of the DI between subnetworks is presented. Based on channel coding theorems for continuous-time channels with feedback, we argue that directed information is the appropriate metric for quantifying information throughput in cellular signal processing. To support our initial hypothesis within the context of gene regulation, we present two case studies involving small promoter models: a two-state nonequilibrium promoter and a three-state promoter featuring two activation levels. We provide analytical expressions of the directed information rate (DIR) and maximize them subject to an upper bound on the EPR. The maximum is shown to increase with the EPR.