  The circadian clocks keeping time of day in many living organisms rely on
self-sustained biochemical oscillations which can be entrained by external
cues, such as light, to the 24-hour cycle induced by Earth rotation. However,
environmental cues are unreliable due to the variability of habitats, weather
conditions or cue-sensing mechanisms among individuals. A tempting hypothesis
is that circadian clocks have evolved so as to be robust to fluctuations in
daylight or other cues when entrained by the day/night cycle. To test this
hypothesis, we analyze the synchronization behavior of weakly and periodically
forced oscillators in terms of their phase response curve (PRC), which measures
phase changes induced by a perturbation applied at different phases. We
establish a general relationship between, on the one side, the robustness of
key entrainment properties such as stability and phase shift and, on the other
side, the shape of the PRC as characterized by a specific curvature or the
existence of a dead zone. This result can be applied to computational models of
circadian clocks where it accounts for the disparate robustness properties of
various forcing schemes. Finally, the analysis of PRCs measured experimentally
in several organisms strongly suggests a case of convergent evolution toward an
optimal strategy for maintaining a clock that is accurate and robust to
environmental fluctuations.
