Optimal Implementations for Reliable Circadian Clocks

Circadian rhythms are acquired through evolution to increase the chances for survival through synchronizing with the daylight cycle. Reliable synchronization is realized through two trade-off properties: regularity to keep time precisely, and entrainability to synchronize the internal time with daylight. We find by using a phase model with multiple inputs that achieving the maximal limit of regularity and entrainability entails many inherent features of the circadian mechanism. At the molecular level, we demonstrate the role sharing of two light inputs, phase advance and delay, as is well observed in mammals. At the behavioral level, the optimal phase-response curve inevitably contains a dead zone, a time during which light pulses neither advance nor delay the clock. We reproduce the results of phase-controlling experiments entrained by two types of periodic light pulses. Our results indicate that circadian clocks are designed optimally for reliable clockwork through evolution.

Figure 1

Examples of (a) one-light input pathway and (b) two-light input pathway cases. The insets describe the time course of molecular species that are affected by light stimuli. In (b) the light-sensitive molecular species correspond to $x_1$ and $x_2$, which exhibit peaks at different phase (this difference is given by $\nu$). (c) The phase difference $\nu$ is defined by the difference between the $k_1$th and $k_2$th molecular species.

Figure 2

(a) Normalized entrainability as a function of $\nu$ for $\alpha =0.5$ (solid line) and $\alpha =1$ (dashed line). The value is normalized by the entrainability at $\nu =0$ [i.e., $\mathcal{E}(\nu )/\mathcal{E}(\nu =0)$]. For $\alpha =1$, maximal is achieved at $\nu =0$ and 1.47. (b) Dead zone length $L$ as a function of the phase difference $\nu$ for $\alpha =0.5$ (solid line) and $\alpha =1$ (dashed line). (c) Distance between $Z_{j_1}(\varphi )$ and $Z_{j_2}(\varphi )$ as a function of $\nu$ for $\alpha =0.5$ (solid line) and $\alpha =1$ (dashed line).

Figure 3

(a)–(b) Optimal PRCs [$Z(\varphi )$, $Z_{j_1}(\varphi )$, and $Z_{j_2}(\varphi )$] for (a) $\alpha =1$ and $\nu =0$ and (b) $\alpha =1$ and $\nu =1.47$. In (a) and (b), $Z(\varphi )$, $Z_{j_1}(\varphi )$, and $Z_{j_2}(\varphi )$ obtained with the variational method, are plotted with thick solid lines, dashed lines, and dot-dash lines, respectively. $Z_{j_1}(\varphi )$ and $Z_{j_2}(\varphi )$ obtained with a numerical method (NM) are plotted with circles and triangles, respectively [19]. In (a), PRCs that are symmetric with respect to the horizontal axis or $\varphi =3\pi /2$ are also optimal. In (b), PRCs that are symmetric with respect to the horizontal axis are also optimal. (c)–(d) Time course of the molecular species concentration for (c) $\alpha =1$ and $\nu =0$ and (d) $\alpha =1$ and $\nu =1.47$. Two species $x_{\mathrm{LC},k_1}$ and $x_{\mathrm{LC},k_2}$ are plotted with dashed and dot-dash lines. In (a) and (c), dashed and dot-dashed lines are indistinguishable.

Figure 4

(a)–(c) Theoretical reproduction of a light entrainment experiment involving hamsters [15]. (a) Optimal pPRC $Z(\varphi )$ of the two-input case $M=2$ with $\alpha =1$ and $\nu =1.47$ [Fig. 3]. (b) The time course of the molecular species concentration of $x_{\mathrm{LC},k_1}$ (dashed lines) and $x_{\mathrm{LC},k_2}$ (dot-dash lines) for $\alpha =1$ and $\nu =1.47$. (c) Molecular implementation of the murine light entrainment mechanism. The model can be described by Eq. (4) with $x_{j_1}=[Per1]$, $x_{k_1}=[Bmal1]$, and $x_{j_2}=x_{k_2}=[Per2]$. (d) Observed (solid line) and inferred-intrinsic ($N_{\mu }=2$, $\chi =1$; dashed line) pPRCs of humans as a function of the onset of pulses. Observed pPRC is brought from Ref. [24], which measured pPRC with $\ell =6.7\mathrm{h}$ light pulses. The horizontal dashed line indicates anticipated phase delay [24].