Fig 1.
Modeling light responses of the circadian clocks.
(A), (B) Schematic representation of the degradation and induction responses of the circadian clock. (A) Degradation response in Drosophila and (B) induction response in mammals. In (B), light signals increase Ca2+ levels. Subsequently, the elevation of Ca2+ increases the levels of phosphorylated CREB. See also the main text. (C) Negative feedback loop in the three-variable Goodwin model. x, y and z are the levels of repressor mRNA, cytoplasmic protein and nuclear protein, respectively. Nuclear protein represses mRNA transcription as indicated by a line with a perpendicular bar. (D) Change in a biochemical reaction by a light signal γl(t) described by Eq (4). (E) Quantification of phase shift Δϕ. The green solid and black broken lines indicate time series of nuclear protein z with or without light perturbation, respectively. Time in the horizontal axis is normalized by the period of oscillation Tp. We measure the difference in peak times after transient behaviors.
Fig 2.
Dead zone generated by the saturation of repressor mRNA transcription in the degradation response.
(A) Time series of the levels of mRNA x, cytoplasmic protein y and nuclear protein z in Eqs (1)–(3) without light signals. (B) Normalized phase response curves (PRC) to light signals. PRCs for different values of the rate of light-induced degradation εl are shown. (C) Time series of the transcription rate 1/(1+(z/K1)n) in Eq (1). The PRC for εl = 0.3 is also plotted (right y-axis) as a reference. (D) Phase sensitivity −Zz (blue solid line). In (B) and (D), the time series of z (green broken line) is also plotted (right y-axis). In all panels, time is normalized by the period of oscillation Tp = 24. Values of reaction parameters in Eqs (1)–(3) are listed in S1 Table. In (B), Td = 0.5Tp/24 = 0.5.
Fig 3.
Dependence of the dead zone length on the Michaelis constant for nuclear protein degradation Km.
(A) Phase sensitivity –Zz and (B) time series of the levels of nuclear protein z for different values of Km. (C) Dead zone length Ld (top) and the maximum and minimum values of the phase sensitivity –Zz(max) and –Zz(min) (bottom) as a function of Km. In (A) and (C), the phase sensitivity is normalized by the period of oscillation Tp. (D) Time series of transcription rate 1/(1+(z/K1)n) in Eq (1) for different values of Km. Values of reaction parameters are listed in S1 Table.
Fig 4.
Phase response curves with no extended dead zone in the negative feedback loop model Eqs (8)–(10) with the induction response.
(A) Time series of the levels of mRNA x, cytoplasmic protein y and nuclear protein z in Eqs (8)–(10) in the absence of a light signal. (B) Phase shift Δϕ as a function of the onset of light signals tl. Results for different values of εl are shown. (C) Phase sensitivity Zx. In (B) and (C), the time series of x (red broken line) is plotted (right y-axis) as a reference. Time is normalized by the period of oscillation Tp. Values of reaction parameters are listed in S1 Table. Tp = 12.67. In (B), Td = 0.5Tp/24 = 0.3.
Fig 5.
Dead zone generated by the saturation of repressor translation in the induction response.
(A) Time series of the levels of mRNA x, cytoplasmic protein y and nuclear protein z in Eqs (8), (10) and (11) in the absence of a light signal. (B) Phase shift Δϕ as a function of the onset of light signals tl. Results for different values of εl are shown. (C) Time series of the saturation index sx = x/(Kt+x). The PRC with εl = 0.3 is also plotted (right y-axis). (D) Phase sensitivity Zx. In (B) and (D), the time series of x (red broken line) is plotted (right y-axis) as a reference. Values of reaction parameters are listed in S1 Table. Tp = 24. In (B), Td = 0.5Tp/24 = 0.5.
Fig 6.
Dependence of dead zone length on the Michaelis constant for translation Kt.
(A) Phase sensitivity Zx for different values of Kt. (B) Time series of the saturation index sx = x/(Kt+x) for different values of Kt. (C) Dead zone length Ld as a function of Kt. (D) Dependence of the maximum and minimum values of the phase sensitivity Zx(max) (red circles) and Zx(min) (blue squares), respectively, on Kt. Tp is the period of oscillation (Tp = 24). Values of reaction parameters are listed in S1 Table.
Fig 7.
Dead zone generated by the Hill function for translation in the induction response.
(A) Phase shift Δϕ as a function of the onset of light signals tl for different values of the Hill coefficient h in Eq (12). (B) Dependence of the dead zone length Ld on h. (C) Time series of mRNA x for different values of the Hill coefficient h. (D) Dependence of the maximum and minimum values of phase sensitivity Zxmax and Zxmin on h. See S1 Table for values of reaction parameters. We set Tp = 24 by tuning τ for each value of h. In (A) Td = 0.5Tp/24 = 0.5 and εl = 0.1.