Reciprocity Between Robustness of Period and Plasticity of Phase in Biological Clocks

Circadian clocks exhibit the robustness of period and plasticity of phase against environmental changes such as temperature and nutrient conditions. Thus far, however, it is unclear how both are simultaneously achieved. By investigating distinct models of circadian clocks, we demonstrate reciprocity between robustness and plasticity: higher robustness in the period implies higher plasticity in the phase, where changes in period and in phase follow a linear relationship with a negative coefficient. The robustness of period is achieved by the adaptation on the limit cycle via a concentration change of a buffer molecule, whose temporal change leads to a phase shift following a shift of the limit-cycle orbit in phase space. Generality of reciprocity in clocks with the adaptation mechanism is confirmed with theoretical analysis of simple models, while biological significance is discussed.

Figure 1

Reciprocity between the robustness of period and plasticity of phase in the PTO model. (a) Difference between periods at two temperatures (${\beta }_1=1.0$ and ${\beta }_2=1.5$) ($\mathrm{\Delta }T/T$, red circle) and the amplitude of the phase response curve against a transient jump of temperature from ${\beta }_1$ to ${\beta }_2$ ($\mathrm{\Delta }\varphi$, green square) plotted against different values of $E_{dp}$ while $E_p$ is fixed at 1.0. $\mathrm{\Delta }T$ is normalized by the period at ${\beta }_1$, and $\mathrm{\Delta }\varphi$ is normalized by the duration of stimulus and difference between ${\beta }_1$ and ${\beta }_2$. $\mathrm{\Delta }T/T$ and $\mathrm{\Delta }\varphi$ are negatively correlated across the entire range of $E_{dp}$. (b) Entrainability is plotted against various $E_{dp}$ with fixed $E_p=1.0$. (c) Phase response curve against transient increase in $\beta$. As a stimulus, the inverse temperature $\beta$ is increased from ${\beta }_1$ to ${\beta }_2$ for the duration of one unit of time. Lines of different colors represent the PRCs for different values of activation energy for the dephosphorylation reaction $E_{dp}$.

Figure 2

Reciprocity between the robustness of period and plasticity of phase in the TTO model. Difference between periods at two temperatures (${\beta }_1=0.0$ and ${\beta }_2=0.5$) ($\mathrm{\Delta }T/T$, red circle) and the amplitude of the phase response curve against a transient jump of temperature from ${\beta }_1$ to ${\beta }_2$ ($\mathrm{\Delta }\varphi$, green square) plotted against various $E_a$, while activation energies for other reactions are fixed at 1.0. $\mathrm{\Delta }T/T$ and $\mathrm{\Delta }\varphi$ are calculated similarly to how they are calculated in Fig. 1.

Figure 3

Schemes of the reciprocity between the robustness of period and plasticity of phase. (a) Schematic networks of a generic (bio)chemical oscillator exhibiting homeostasis of period. Pointed and flat arrowheads indicate positive and negative regulation, respectively. Correspondences with a simple feedforward adaptation motif are represented by green characters in parentheses. (b) Scheme of limit-cycle orbits with compensation of the period against environmental change. Blue dotted line is a stable limit-cycle orbit before environmental change. The green dashed line and magenta solid line are stable limit-cycle orbits after environmental change when the period is perfectly compensated and partially compensated, respectively.