  Circadian clocks must be able to entrain to time-varying signals to keep
their oscillations in phase with the day-night rhythm. On the other hand, they
must also exhibit input compensation: their period must remain about one day in
different constant environments. The post-translational oscillator of the Kai
system can be entrained by transient or oscillatory changes in the ATP
fraction, yet is insensitive to constant changes in this fraction. We study in
three different models of this system how these two seemingly conflicting
criteria are met: the Van Zon model (Van Zon et al., PNAS, 2007), the Rust
model (Phong et al., PNAS, 2013), and our new model (Paijmans et al.,
arXiv:1612.02715). We find that the new model exhibits the best trade-off
between input compensation and entrainability: on the footing of equal
phase-response curves, it exhibits the strongest input compensation. Performing
stochastic simulations at the level of individual hexamers allows us to
identify a new mechanism, which is employed by the new model to achieve input
compensation: At lower ATP fraction, the individual hexamers make a shorter
cycle in the phosphorylation state space, which compensates for the slower pace
at which they traverse the cycle.
