Robustness of Clocks to Input Noise

To estimate the time, many organisms, ranging from cyanobacteria to animals, employ a circadian clock which is based on a limit-cycle oscillator that can tick autonomously with a nearly 24 h period. Yet, a limit-cycle oscillator is not essential for knowing the time, as exemplified by bacteria that possess an “hourglass”: a system that when forced by an oscillatory light input exhibits robust oscillations from which the organism can infer the time, but that in the absence of driving relaxes to a stable fixed point. Here, using models of the Kai system of cyanobacteria, we compare a limit-cycle oscillator with two hourglass models, one that without driving relaxes exponentially and one that does so in an oscillatory fashion. In the limit of low input noise, all three systems are equally informative on time, yet in the regime of high input-noise the limit-cycle oscillator is far superior. The same behavior is found in the Stuart-Landau model, indicating that our result is universal.

Figure 1

Overview of different timekeeping systems. (a) A push-pull network (PPN). Each protein can switch between a phosphorylated and an unphosphorylated state, and the input signal enhances the phosphorylation rate. In the absence of driving, the PPN relaxes exponentially to a steady state (middle panel). Yet, in the presence of an oscillatory input, e.g., sunlight, the system exhibits oscillations from which the time can be inferred (lower panel). (b) The uncoupled-hexamer model (UHM), inspired by the Kai system of P. marinus. It consists of KaiC hexamers which can switch between an active state in which the phosphorylation level tends to rise and an inactive one in which it tends to fall. The phosphorylation rate is, via changes in the ATP/ADP ratio, enhanced by the light input [11, 12]. The system is akin to a harmonic oscillator, with an intrinsic frequency ${\omega }_0$, resulting from the hexamer phosphorylation cycle. However, the hexamers are not coupled via KaiA as in the CHM shown in panel (c), so it cannot sustain autonomous oscillations; in the absence of driving, it relaxes in an oscillatory fashion to a stable fixed point (middle panel). (c) The coupled-hexamer model (CHM), inspired by the Kai system of S. elongatus. Like the UHM, it consists of KaiC hexamers, which tend to be phosphorylated cyclically. However, in contrast to the UHM, the hexamers are synchronized via KaiA, such that the system can exhibit limit-cycle oscillations in the absence of driving (middle panel). In all models, time is estimated from the protein phosphorylation fraction $p(t)$.

Figure 2

The mutual information $I(p;t)$ as a function of the input-noise strength ${\sigma }_s^2$, for the push-pull network, the uncoupled-hexamer model, and the coupled-hexamer model, see Fig. 1. In the limit of low input noise, all systems are equally informative on time, but in the high-noise regime the CHM is most accurate. The parameters have been optimized to maximize $I(p;t)$; since these are (nearly) independent of ${\sigma }_s^2$ (Figs. S1–S3 [14]), they are fixed (Table S1 [14]).

Figure 3

The mutual information $I(p;t)$ as a function of $\alpha$ of the Stuart-Landau model [Eq. (2)], for different strengths of the input noise ${\sigma }_s^2$. Clearly, $I(p;t)$ rises as the system is changed from a damped oscillator like the UHM ($\alpha <0$) to a limit-cycle oscillator like the CHM ($\alpha >0$). Moreover, the increase is most pronounced when ${\sigma }_s^2$ is large, as also observed for the UHM and CHM, see Fig. 2. Parameters: $\nu =0$; $\beta =\omega$; $\epsilon =0.5\omega$; ${\sigma }_s^2$ in units of ${\omega }^{-1}$.