Figure 3. Experiments and models of biological clocks show that external driving can be viewed as a switch between distinct day-time and night-time dynamics.

(a) Experiments on the Kai system at distinct ATP levels corresponding to day and night conditions reveal limit cycles shifted relative to each other in a phosphorylation space for Kai (reproduced from [Leypunskiy et al., 2017[). Similar behavior (Winfree, 2001) is seen in models of diverse biochemical oscillators studied in Figure 2. (b) We build a minimal model of such driven clocks as a limit cycle of radius $R$ whose center is shifted by a distance $L$ between day and night. In cycling conditions (see signal in (d)), an entrained clock’s state executes a trajectory that encompasses both limit cycles as shown (bottom). (c) For damped clocks (Saunders, 2002), phenomenology suggests that the day and night limit cycle dynamics are replaced by a point attractor whose position changes between day and night. The relaxation time between the day and night attractors is comparable to $\sim 12$ hours, giving rise to the trajectory shown in cycling conditions. (d) The plot shows cycling conditions of light intensities that couple to (b) and (c).