Decoding frequency-modulated signals increases information entropy in bacterial second messenger networks

Bacterial second messenger networks transmit environmental information through both amplitude and frequency modulation strategies. However, the mechanisms by which cells decode frequency-encoded signals remain poorly understood. By reconstructing the cyclic adenosine monophosphate second messenger system in Pseudomonas aeruginosa, we demonstrate that frequency-to-amplitude signal conversion emerges through three distinct filtering modules that decode frequency-encoded signals into gene expression patterns. Our mathematical framework predicts a range of frequency filtering regimes controlled by a dimensionless threshold parameter. We validated these using synthetic circuits and an automated experimental platform. Quantitative analysis reveals that under the given parameter conditions, frequency modulation expands the accessible state space more substantially than amplitude modulation alone. The total number of accessible states scales as the square of the number of regulated genes for frequency-enhanced control, compared with the power of 0.8 for amplitude-only control. This results in approximately two additional bits of information entropy in three-gene systems when using frequency-based control. Our findings establish the fundamental principles of frequency-based signal processing in bacterial second messenger networks, revealing how cells exploit temporal dynamics to regulate multiple genes and expand accessible state spaces. This provides insights into both cellular information physics and design principles for synthetic biology.