Biochemical communication with noisy feedback under energy constraints

Abstract

Biochemical systems process signals through stochastic reaction dynamics that are inherently continuous in time and often exhibit memory, feedback, and nonequilibrium driving. At the same time, they are frequently modeled by effective reactions, e.g., multi-step processes such as transcription are treated as single events, while energetic bookkeeping is commonly omitted. Moreover, mesoscopic dissipation estimates are highly sensitive to whether coarse-graining and reservoir coupling are performed in a thermodynamically consistent way. Together, these features complicate the direct application of classical Shannon information theory and stochastic thermodynamics "as is" to biochemical reaction networks. This paper provides a self-contained route from first principles to a practically usable framework for studying information transmission through chemical reaction networks (CRNs) under energetic constraints. In particular, we discuss and extend the notions of classical information theory, methodically progressing to a level of generality that is necessary for the theme of causal communication through general CRNs. We then derive expressions for mutual information and directed information between bipartite CRN trajectories of disjoint sets of molecular species and show that the MI diverges without bipartiteness. These expressions account for cases in which different reactions are indistinguishable after projection to the respective subnetworks or where multiple driving mechanisms produce the same observable effect. We finally introduce a rigorous, operational Shannon-style continuous-time chemical communication model: messages are encoded by time-dependent chemostat protocols for a set of signaling molecules, the causal channel law is an immutable property of the reaction dynamics, and channel capacity is posed as an optimization over causal chemical encoders subject to thermodynamic costs of encoding and transmission. Trajectory information measures and the operational channel capacity are related by a Fano-type converse theorem. Complementary, we formulate the dual perspective of minimum-energy-per-bit necessary for reliable communication. A tractable promoter-switching example illustrates the practical application. Our work provides a formal and general framework to obtain universal energetic bounds for reliable communication in biochemical systems.