The accuracy of telling time via oscillatory signals

Circadian clocks are the central timekeepers of life, allowing cells to anticipate changes between day and night. Experiments in recent years have revealed that circadian clocks can be highly stable, raising the question how reliably they can be read out. Here, we combine mathematical modeling with information theory to address the question how accurately a cell can infer the time from an ensemble of protein oscillations, which are driven by a circadian clock. We show that the precision increases with the number of oscillations and their amplitude relative to their noise. Our analysis also reveals that their exists an optimal phase relation that minimizes the error in the estimate of time, which depends on the relative noise levels of the protein oscillations. Lastly, our work shows that cross-correlations in the noise of the protein oscillations can enhance the mutual information, which suggests that cross-regulatory interactions between the proteins that read out the clock can be beneficial for temporal information transmission.