Continuous attractor-based clocks are unreliable phase estimators

Statistical estimation theory determines the optimal way of estimating parameters of a fluctuating noisy signal. However, if the estimation is performed on unreliable hardware, a sub-optimal estimation procedure can outperform the previously optimal procedure. Here, we compare classes of circadian clocks by viewing them as phase estimators for the periodic day-night light signal. We find that continuous attractor-based free running clocks, such as those found in the cyanobacterium Synechococcus elongatus and humans, are nearly optimal phase estimators since their flat attractor directions efficiently project out light intensity fluctuations due to weather patterns (`external noise'). However, such flat directions also make these continuous limit cycle attractors highly vulnerable to diffusive 'internal noise'. Given such unreliable biochemical hardware, we find that point attractor-based damped clocks, such as those found in a smaller cyanobacterium with low protein copy number, Prochlorococcus marinus, outperform continuous attractor-based clocks. By interpolating between the two types of clocks found in these organisms, we demonstrate a family of biochemical phase estimation strategies that are best suited to different relative strengths of external and internal noise.