To anticipate daily variations in the environment and coordinate biological activities into a daily cycle many organisms possess a circadian clock. In the absence of external time cues the circadian rhythm persists with a period of approximately 24 h. The clock phase can be shifted by single pulses of light, darkness, chemicals, or temperature and this allows entrainment of the clock to exactly 24 h by cycles of these zeitgebers. On the other hand, the period of the circadian rhythm is kept relatively constant within a physiological range of constant temperatures, which means that the oscillator is temperature compensated. The mechanisms behind temperature compensation and temperature entrainment are not fully understood, neither biochemically nor mathematically. Here, we theoretically investigate the interplay of temperature compensation and entrainment in general oscillatory systems. We first give an analytical treatment for small temperature shifts and derive that every temperature-compensated oscillator is entrainable to external small-amplitude temperature cycles. Temperature compensation ensures that this entrainment region is always centered at the endogenous period regardless of possible seasonal temperature differences. Moreover, for small temperature cycles the entrainment region of the oscillator is potentially larger for rectangular pulses. For large temperature shifts we numerically analyze different circadian clock models proposed in the literature with respect to these properties. We observe that for such large temperature shifts sinusoidal or gradual temperature cycles allow a larger entrainment region than rectangular cycles.
