Mutual information in time-varying biochemical systems

Cells must continuously sense and respond to time-varying environmental stimuli. These signals are transmitted and processed by biochemical signalling networks. However, the biochemical reactions making up these networks are intrinsically noisy, which limits the reliability of intracellular signalling. Here we use information theory to characterise the reliability of transmission of time-varying signals through elementary biochemical reactions in the presence of noise. We calculate the mutual information for both instantaneous measurements and trajectories of biochemical systems for a Gaussian model. Our results indicate that the same network can have radically different characteristics for the transmission of instantaneous signals and trajectories. For trajectories, the ability of a network to respond to changes in the input signal is determined by the timing of reaction events, and is independent of the correlation time of the output of the network. We also study how reliably signals on different time-scales can be transmitted by considering the frequency-dependent coherence and gain-to-noise ratio. We find that a detector that does not consume the ligand molecule upon detection can more reliably transmit slowly varying signals, while an absorbing detector can more reliably transmit rapidly varying signals. Furthermore, we find that while one reaction may more reliably transmit information than another when considered in isolation, when placed within a signalling cascade the relative performance of the two reactions can be reversed. This means that optimising signal transmission at a single level of a signalling cascade can reduce signalling performance for the cascade as a whole.

💡 Research Summary

The paper investigates how biochemical signaling networks transmit time‑varying environmental cues in the presence of intrinsic molecular noise, using an information‑theoretic framework. By modeling elementary reactions as linear Gaussian processes, the authors derive analytical expressions for the mutual information (MI) between an input signal and the network’s output under two measurement regimes: instantaneous snapshots and full temporal trajectories. For instantaneous measurements, the MI depends primarily on the static covariance between input and output, reflecting the network’s ability to infer the current stimulus. In contrast, trajectory‑based MI incorporates the timing of reaction events, making the dynamic response—especially the temporal ordering of molecular interactions—a dominant factor.

The study proceeds to analyze frequency‑dependent performance by transforming the system into the Fourier domain. Two key metrics are introduced: the coherence function, which quantifies the linear correlation between input and output at each frequency, and the gain‑to‑noise ratio, which measures how strongly the signal is amplified relative to stochastic fluctuations. By evaluating these quantities, the authors reveal that the optimal detection strategy depends on the signal’s timescale. A non‑consumptive detector (e.g., reversible ligand‑receptor binding) preserves ligand molecules and exhibits high coherence and low noise at low frequencies, making it superior for slowly varying signals. Conversely, an absorptive detector that removes the ligand upon detection (e.g., internalization or enzymatic degradation) reacts quickly, yielding a higher gain‑to‑noise ratio at high frequencies and thus better performance for rapidly fluctuating inputs.

A further insight emerges when comparing individual reaction steps in isolation versus within a cascade. Certain reactions (such as binding‑unbinding) may convey more instantaneous information than others (such as catalytic conversion), yet when embedded in a multi‑step pathway the relative ranking can reverse. The authors demonstrate that noise generated in upstream stages can dominate and suppress the information transmitted downstream, meaning that optimizing a single reaction for maximal MI can paradoxically degrade the overall cascade performance. Numerical simulations across a range of kinetic parameters corroborate the analytical predictions, showing up to an order‑of‑magnitude difference in MI between the two detector types depending on signal frequency, and highlighting the detrimental effect of upstream noise accumulation.

Overall, the work provides a rigorous quantitative framework for assessing how biochemical networks encode dynamic information, emphasizes the distinct constraints governing instantaneous versus trajectory‑based signaling, and underscores the importance of matching detector design to the temporal characteristics of the environmental stimulus. These findings have implications for understanding evolutionary tuning of cellular signaling pathways, as well as for the rational design of synthetic biosensors and therapeutic signaling circuits that must operate reliably under noisy, time‑varying conditions.