Fig 1.
The Molecular network of circadian rhythms with feedback loops.
(A) The genes and RNA's are shown in italics, while the proteins and protein complexes are shown in capitals. The dynamical variables that are used in the model are given in the brackets. Light is the external zeitgeber. The protein BMAL1 positively regulates per1, per2, and Rev-erbα. per1 and per2 produce the corresponding proteins PER1 and PER2 and these proteins interact separately with BMAL1 to form a complex PER1-BMAL1 and PER2-BMAL1 to complete the negative feedback loop. Rev-erbα negatively regulates Bmal1 transcription to complete the second negative feedback loop. PER2 coregulates Bmal1 positively [32] and closes the only indirectpositive feedback loop. (B) Summary of the complete network shown in (A) that captures three negative and one positive feedback loops.
Fig 2.
Free-running circadian oscillations under constant darkness and light pulse at certain circadian time.
Blue curves are from simulation, and red circles are the experimental data points. Simulation results were obtained by integrating the model equations (Eqs 1–13) with the estimated parameters, which are given in the S2 Table (parameters used for DD, LD). For comparison, the individual time series were normalized to maximum 1 and the minimum 0. Experimental data points of per1, per2, and Bmal1 mRNA were extracted from [7], Rev-erbα from [35] and PER1, PER2 protein from [34]. (G) per expression in response light. A 30 min light pulse with amplitude of 0.2 is applied at CT14 (early night), It shows a phase delay (green broken line). The same light pulse applied at CT 22 (late night), induce phase advance (green solid line), and agrees with the experimental results [36]. Inset shows perturbed time series came back into the original limit cycle with a delay ('D') or with an advance ('A') with respect to unperturbed time series.
Table 1.
Comparison of different peaking time of different clock variables in CT.
Fig 3.
Bifurcation analysis with light L as the parameter.
(A) Oscillation amplitude is shown in green with L = 0 represents the DD conditions for the variable Mp1 (per1 mRNA). Black broken lines are unstable steady states. As light intensity increases, sustained oscillation disappeared via supercritical Hopf bifurcation (HB) and the system enters the stable steady state (red lines). (B) Period variation as a function of light intensity. Period increases very modestly with the increase in light intensity. We used Xppaut [39] for simulating bifurcation diagrams. Simulations are obtained by integrating the model equations (Eqs 1–13) with the estimated parameters, which are given in the S2 Table (parameters used for DD, LD simulation, WT).
Fig 4.
Bifurcation analysis of the effect of Bmal1- PER2 positive feedback loop.
(A) Hopf bifurcation obtained for the range of vs4 that modulates the positive feedback loop. (B) The period of oscillation is shown to increase with the increase in vs4. per2 mutant mice showed period decrement in LL condition [38], here we observed that the positive regulation of Bmal1 mRNA by PER2 protein enhance the period of oscillation of the system under LL condition. Simulations are obtained by integrating the model equations (Eqs 1–13) with the estimated parameters, which are given in the S2 Table (parameters used for DD, LD simulation, WT).
Fig 5.
Simulation of the rhythmic behavior of per1 and per2 mutant mice.
(A-C) are the simulations of per1 mutant. Sustained oscillation of per2 mRNA, PER2 protein, and Bmal1 mRNA are shown in blue lines. The simulated results are in good agreement with the experimental data (red circle) that are observed for per1ldc mutant mice [34]. (D-F) are the simulation shown for per2 mutant. Similar to per1 mutant, sustained oscillations of per1 mRNA, PER1 protein, and Bmal1 mRNA are observed in per2 mutant and are in good agreement with the experimental results that observed in per2ldc mutant mice [34]. For comparison, each time series were normalized between 0 and 1. For per1 mutant, parameter values adjusted are vs1 = 0 nMh-1, v4 = 0.43 nMh-1, and kp4 = 0.19 nMh-1. For per2 mutant, parameter values varied are vs1 = 0.7 nMh-1, vs2 = 0 nMh-1, vs3 = 4.5 nMh-1, vs5 = 0.5 nMh-1,v1 = 0.44 nMh-1, v2 = 1.38 nMh-1, v3 = 1.67 nMh-1,k1 = 1.44 h-1, kd3 = 0.08 h-1,kp1 = 0.11 nM-1h-1, and kp2 = 0.18 nM-1h-1. Remaining parameters are same as that of in Fig 2. per2 mRNA that peaks at CT8 is taken as the reference point in the case of per1 mutant, and PER1 protein that peaks at CT12 is taken as the reference point for per2 mutant.
Table 2.
Period of molecular phenotype of mutants in mammalian SCN.
Table 3.
Peaking time of different oscillator components in mutants given in CT.
Fig 6.
Gating variables, phase response curves of wild type, per1, and per2 mutants with and without a gating variable.
In the left of the figure is shown the gating variable changes the light intensity at different phases. (A) Constant and varying gating variables are used for per1 (blue) and per2 (red) respectively. Gating variable for per1 is constant throughout the whole circadian period, whereas gating variable for per2 is changed with circadian time, and reaches the maximum value between CT15 and CT20. It is at this maximum value of gating variable, a maximum delay in PRC is observed experimentally [41]. (B) An example of single light pulse with unit amplitude applied at CT14 for 30 min duration is shown. (C) The light pulse for per1 and per2 after multiplying with the gating variable. Light pulse for per1 is scale down to half and the light pulse for per2 have different values at CT14. Simulated phase response curves without the gating variable for (E) wild- type (F) per1 mutant and (G) for per2 mutant. Experimental data points were extracted from [41] and they are shown in red circles and a continuous line was drawn for readability. The blue lines are simulated PRC curves. To simulate PRC, light pulse L in the model was applied for duration of 30 mins with an amplitude value 0.35, and phase difference is measured after 10 cycles. For WT,per1 mRNA (Mp1) peaking at CT 6 is taken as the reference point. per2 mRNA peaks at CT 8 is taken as the reference point when simulating per1 mutant, and PER1 protein (P1c) that peaks at CT 12 is taken as the reference point when simulating per2 mutant. For clarity, two horizontal lines are drawn; one to show the zero phases (in black) and the other magenta line in the bottom indicates the maximum phase delay of wild type. In all the experimental PRC, CT15 is the phase at which maximum delay occurs. (F) per1 mutant PRC for which the maximum phase shift occurs at CT 15 and it is much higher than the wild type. (G) per2 mutant PRC for which, the maximum phase shift occurs also at CT 15, but it is lower than the wild type. Simulated PRC with the gating variable for (H) wild- type (I) per1 mutant and (J) for per2 mutant. Compared to the PRC without a gating variable, per2 mutant shows suppressed phase delay. Parameters for WT are as in (Fig 2), and parameters for per1 mutants and per2 mutant are as in (Fig 5).
Table 4.
Summary of PRC.
Fig 7.
Wild type and mutant entrainment to different LD cycles with and without the gating variable.
(A-C) The model entrains to various external LD cycle without the gating variable and (D-F) with gating variable (Eqs 14 and 15, Fig 6A) for WT. Blue and red lines are the simulated per1 and per2 mRNA, whereas blue and red circles are the experimental per1, per2 mRNA respectively. Since an external cue forcing the oscillation, we consider the x axis as external time rather than circadian time. External time is defined as the middle of the light phase which in the present case is time 12. Model simulations of wild type show a maximal per1expression in the light phase whereas per2 gene peaks close to dusk (off set of the light phase) during the entrainment. In (G-L) the per1 mutant that expresses only per2, peaks following the dusk, and in (M-R), per2 mutant, per1 peak follow the dawn (on set of light phase).Under long photoperiods, per1 mutant with gating variable displays more delay in per2 peaking (L) than in without gating variable (I). Compared with WT, In per2 mutant, per1 peaks advanced both in without gating variable (M-O) and with gating variable (P-R), the magnitude of advance is more in with gating variable. Time series are normalized so that maximum value is 1 and minimum value is 0. L values changed in a square wave manner, during the light phase the value of L is 0.1 and in dark phase the value of L is 0. Experimental data points for LD 12:12 were extracted from [48], and all other data points were extracted from [38]. The dark bar in (M-R) is the dark phase, while the unfilled white bar is the light phase. This bar is also common for the entire figure and for clarity it is shown only in the last row. Parameters for WT are as in (Fig 2), and parameters for per1 mutant and per2 mutant are as in (Fig 5).
Fig 8.
Photoperiodic variation of the PER protein.
Experimental and simulated nuclear PER1 and PER2 protein under LD 8:16 (A), and LD 16:8 (B). Compared to the experimental data, phase difference between simulated PER1 and PER2 protein is higher. Experimental data points are extracted from [21]. Time series are normalized so that the maximum value is 1 and minimum value is 0.
Fig 9.
Peaking time of per1 and per2 mRNA under different photoperiods.
(A-D) Simulation without a gating variable. (A) In WT, under short photoperiod, per1 peaks near the dusk, photo periods increases its peaking time, shift towards the middle of the light phase. (B) per2 peaks near the light offset, follow the dawn. (C) In per2 mutant, per1 always peak near the middle of the light phase. (D) In per1 mutant, per2 follows dusk, as in the WT. (E-H) Simulation with gating variable. (E) In WT, per1 peaking time is shifted from middle of the light phase. (F) per2 peaks exactly follows the dusk. (G) In per2 mutant, per1 peaking time always follows the dawn or late night. (H) In per1 mutant, per2 peaking time follows the dusk. Taking together, it is clear that per1 is the part of M oscillator, follows the dawn and per2 is the part of E oscillator, follows the dusk.
Table 5.
Peaking time of per1 and per2 mRNA under different photoperiods.
Fig 10.
Period length as a function of light intensity.
(A) Simulation results are shown for the period length variation with respect to light intensity under constant light condition for wild type (red dots), per1 mutant (black dots) and per2 mutants (green dots) with normal set of parameters (as in Fig 7). Oscillations are found only for the small range of values of L with the corresponding period variation also being very small. (B)) simulation results of period length variation with respect to light intensity under constant light condition with the modified set of parameters (given in the S2 Table, parameters used for LL condition). In wild type and per1 mutant, period increases with increase in light intensity and in per2 mutant, period decreases with increase in the light intensity, which agrees with the experimental results [38].
Fig 11.
Molecular network of coupled oscillator.
Coupled oscillator is constructed from two single cell dual oscillators. M oscillator is present in VL region that guides morning activities and E oscillator in DM region guides evening activities. These two oscillators respond to light differentially. Light accelerates M oscillator while light decelerates E oscillator. The two oscillators are coupled to each other by the neuropeptides AVP and VIP that acts as coupling agents for M and E oscillators. We assume here that VIP and AVP are controlled by PER1/2 in VL and DM regions respectively. Further, we assume that AVP induces per1 and per2 expression in the M oscillator, and VIP induces per1 and per2 expression in the E oscillator.
Fig 12.
Model fitting of AVP and VIP experimental data.
(A) Under DD condition only AVP shows circadian oscillation and VIP is arrhythmic. Red line indicates AVP simulation and red circle indicates AVP data from experiments [58]. (B) Under LD condition, shown in black dotted line, both VIP and AVP show sustained oscillations. Red and blue lines indicate AVP and VIP simulation respectively, while red and blue circles are the data from experiments AVP from [58] and VIP from [57] respectively. Time series are normalized with maximum and minimum as 1 and 0 respectively. Under DD condition, AVP mRNA that peaks at CT4 is taken as the reference point. Simulation results are obtained by integrating the equations given in the S4 Text (Eqs 1–28) with parameter set given in the S2 Table (ME oscillator).
Fig 13.
Stable split under LL condition.
(A) The time series of PER1 protein in VL region (P1nm) and DM region (P1ne) are shown in blue and red lines respectively under normal condition. Simulation results are obtained by integrating the equations given in the S4 Text (Eqs 1–28). Initially we set VIP and AVP coupling parametersto vcm1 = 0.35 nMh-1, vcm2 = 0.25 nMh-1, and the AVP and VIP production rate to kvs1 = 0.001 h-1, kvs2 = 0.015 h-1 for unsplit condition. Rest of the parameters is given in the S2 Table (ME oscillator).The phase difference between proteins PER1, P1nm of morning oscillator and P1ne of the evening oscillator, is very small (~30 min). After 30 days, we vary the coupling parameters to vcm1 = 0.28 nMh-1, vcm2 = 0.25 nMh-1, and the production rates AVP and VIP to kvs1 = 0.001h-1, kvs2 = 0.009 h-1. When parameters are varied, there is an increase in phase difference between P1nm (blue line) and P1ne (red line). (B) Blue circles are experimentally observed PER1 protein at the core region of SCN and red circles are experimentally observed PER1 protein at the shell region of SCN, at the time of splitting [51]. (C) Simulated actogram of the coupled oscillator model. In simulations, actogram is constructed when the normalized value of the variables P1nm or P1ne are above certain threshold value. The threshold we define here is 1.3 times the mean value of the normalized value of P1nm or P1ne. Blue horizontal lines represent M activities and red horizontal lines the E activities. Arrow on the right indicates the time at which parameter is varied. At the beginning of splitting, oscillator in VL region (M oscillator) has a shorter period than oscillator in DM region (E oscillator). After 30 days when parameters are varied, exchange of phase occurs between M and E components as seen in the experimental actogram (Fig 2C in [13]). Finally both the components maintained a stable phase relationship and oscillate with same period. Time series are normalized so that maximum value is 1 and minimum value is 0. We choose L = 0.02 for the simulation.