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Conceived and designed the experiments: QT BP ML. Performed the experiments: QT BP PEM FC FYB ML. Analyzed the data: QT BP PEM ML. Wrote the paper: QT BP FYB ML.

The authors have declared that no competing interests exist.

The development of systemic approaches in biology has put emphasis on identifying genetic modules whose behavior can be modeled accurately so as to gain insight into their structure and function. However, most gene circuits in a cell are under control of external signals and thus, quantitative agreement between experimental data and a mathematical model is difficult. Circadian biology has been one notable exception: quantitative models of the internal clock that orchestrates biological processes over the 24-hour diurnal cycle have been constructed for a few organisms, from cyanobacteria to plants and mammals. In most cases, a complex architecture with interlocked feedback loops has been evidenced. Here we present the first modeling results for the circadian clock of the green unicellular alga

Circadian clocks keep time of day in many living organisms, allowing them to anticipate environmental changes induced by day/night alternation. They consist of networks of genes and proteins interacting so as to generate biochemical oscillations with a period close to 24 hours. Circadian clocks synchronize to the day/night cycle through the year principally by sensing ambient light. Depending on the weather, the perceived light intensity can display large fluctuations within the day and from day to day, potentially inducing unwanted resetting of the clock. Furthermore, marine organisms such as microalgae are subjected to dramatic changes in light intensities in the water column due to streams and wind. We showed, using mathematical modelling, that the green unicellular marine alga

Real-time monitoring of gene activity now allow us to unravel the complex dynamical behavior of regulatory networks underlying cell functions

One strategy for obtaining such circuits has been to construct synthetic networks, which are isolated by design

Another strategy is to study natural gene circuits whose function makes them relatively autonomous and stable. The circadian clocks that drive biological processes around the day/night cycle in many living organisms are natural candidates, as these genetic oscillators keep track of the most regular environmental constraint: the alternation of daylight and darkness caused by Earth rotation

Here we report surprisingly good agreement between the mathematical model of a single transcriptional feedback loop and expression profiles of two central clock genes of

Very recently, some light has been shed on the molecular workings of

Interestingly,

We not only found that this two-gene loop model reproduces perfectly transcript profiles of

To characterize the temporal pattern of

Time zero corresponds to dawn. (A) Experimental data points for the

A minimal mathematical model of the two-gene feedback loop comprises four ordinary differential equations (Eq. (2),

Experimental data are recorded under 12∶12 Light/Dark (LD) alternation so that the coupling which synchronizes the clock to the diurnal cycle must be hypothesized. Circadian models usually assume that some parameters depend on light intensity (e.g., a degradation rate is higher in the dark than in the light), and thus take different values at day and night. Parameter space dimension then increases by the number of modulated parameters. Various couplings to light were considered, with 1 to 16 parameters depending on light intensity. We also tested adjustment to model (2) with all parameters constant, which allowed us to quantify the relevance of coupling mechanisms by measuring the difference between best-fitting profiles in the coupled and uncoupled cases.

The free-running period (FRP) of the oscillator in constant day conditions was fixed at 24 hours, which was the mean value observed in experiments

The first result is that an excellent agreement between numerical and experimental profiles is obtained, with a root mean square (RMS) error of a few percent (

The data of

But the more surprising is that a non-coupled model, where all parameters are kept constant, adjusts experimental data (

Symbol | Description | FC (day) | FC (night) | FR |

Minimal |
0.0017 | 0.0016 | 0.0065 | |

CCA1-dependent |
0.93 | 0.29 | 0.67 | |

CCA1 level at |
1.47 | 0.00 | 1.04 | |

Cooperativity of CCA1 | 2 | 2 | 2 | |

mTOC1 half-life (min) | 13.8 | 22.0 | 5.08 | |

mTOC1 degradation saturation threshold (nM) | 8.85 | 18.3 | 1.25 | |

TOC1 translation rate (1/min) | 0.013 | 0.023 | 0.016 | |

TOC1 half-life (min) | 29.9 | 29.0 | 3.58 | |

TOC1 degradation saturation threshold (nM) | 3.85 | 9.78 | 0.76 | |

Minimal |
0.0075 | 0.017 | 0.052 | |

TOC1-dependent |
0.12 | 0.047 | 0.060 | |

TOC1 level at |
100.4 | 1.49 | 44.1 | |

Cooperativity of CCA1 | 2 | 2 | 2 | |

mCCA1 half-life (min) | 13.3 | 52.2 | 0.82 | |

mCCA1 degradation saturation threshold (nM) | 0.56 | 3.76 | 0.063 | |

CCA1 translation rate (1/min) | 0.056 | 0.046 | 0.075 | |

CCA1 half-life (min) | 55.5 | 92.3 | 54.7 | |

CCA1 degradation saturation threshold (nM) | 32.4 | 36.0 | 46.0 |

On the other hand the uncoupled model is equally unrealistic because it cannot be entrained to the day/night cycle, whereas it is observed experimentally that upon a phase shift of the light/dark cycle, CCA1 and TOC1 expression peaks quickly recover their original timings in the cycle. To verify that adjustment by a free-running oscillator model does not depend on the target profile used, we generated a large number of synthetic profiles whose samples where randomly chosen inside the interval of variation observed in biological triplicates, and adjusted a free-running oscillator model to them. In each case, we found that although RMS error slightly degraded compared our target profile (where mCCA1 and mTOC1 samples for a given time always come from the same microarray), it remained on average near 10%, with visually excellent adjustment (

Thus the paradoxical result that data points fall almost perfectly on the temporal profiles of a free-running oscillator is counterintuitive but must nevertheless be viewed as a signature of the clock architecture. As we will see, this in fact does not imply that the oscillator is uncoupled but only that within the class of models considered so far, where parameters of the TOC1–CCA1 loop take day and night values, the uncoupled model is the one approaching experimental data best. Nothing precludes that there are more general coupling schemes that adjust data equally well.

Before unveiling such models, we discuss now whether the simple negative feedback loop described by model (2) is a plausible autonomous gene oscillator. With two transcriptional regulations, it is a simpler circuit than the Repressilator, where three genes repress themselves circularly

To address this issue, we checked robustness of adjustment with respect to parameter variations. We found that the experimental profiles can be reproduced in a wide region of parameter space around the optimum, which is quite remarkable given the simplicity of the model (

The role of post-translational interactions in gene oscillators and circadian clocks has been recently emphasized (see, e.g.,

Circadian models are usually coupled to diurnal cycle by changing some parameter values between day and night

In our case, light/dark alternation has no detectable signature in the dynamics of

For simplicity, we restrict ourselves to models in which some parameters of the TOC1–CCA1 feedback loop are modified between two times of the day, measured relatively to dawn (ZT0). The start and end times of coupling windows are then model parameters instead of being fixed at light/dark transitions. This assumes that the input pathway tracks diurnal cycle instantaneously, without loss of generality for understanding behavior in entrainment conditions. In this scheme, resetting of the two-gene oscillator can be studied by simply shifting the oscillator phase relatively to the coupling windows. The results so obtained will be sufficient to show that there exist coupling schemes which leave no signature on mRNA profiles, and to study their properties.

What makes our approach original is not the gated coupling to diurnal cycle, which can be found in other models, but the fact that we do not try to model the actors of the input pathway, which can be complex. This is because we focus here on the TOC1–CCA1 feedback loop, which mostly behaves as an autonomous oscillator. Thus we only need to specify the action of the unknown mediators on TOC1 or CCA1, the details of their dynamics being irrelevant.

We systematically scanned the coupling window start and end times, adjusting model for each pair. This revealed that many coupling schemes are compatible with experimental data. For example, TOC1 degradation rate

Numerical solutions of model (2) without coupling (dashed lines, same parameter values as in

We also found a family of time windows of different lengths centered around ZT13.33, inside which the CCA1 degradation rate

In these examples, adjustment is sensitive to the timing of these coupling windows: when the start time is modified slightly, the end time must be changed simultaneously so as to recover good adjustment. On the other hand, we found that adjustment error depends little on the coupling strength (measured by the ratio between degradation rates outside and inside the window), especially for short coupling windows.

(A) In gray, RMS error when

To gain better insight into the effect of a coupling window, we must take into account the fact that the induced variation in the entrained oscillations can be decomposed as a displacement along the limit cycle (resulting in a phase shift) and a displacement transversely to the limit cycle (resulting in a deformation of the limit cycle). To this end, we apply a variable phase shift to the entrained time profile and optimize this phase shift so as to minimize the adjustment error. We define the waveform error as the minimal value of the latter, and the phase error the value of the phase shift for which it is obtained. A small waveform error indicates that we are following the same limit cycle as in the free-running case, possibly with a different phase than is observed experimentally. Waveform and phase errors for the three windows of CCA1 protein stabilization considered in

Besides the two specific examples shown in

(A) Schematic representation of how window center and duration

As with the examples considered in the previous section, some coupling mechanisms have robust adjustment properties in that a good adjustment is obtained at the two different coupling strengths for the same timings, which coincide with the timings computed in the weak coupling limit. In these cases, adjustment is robust to variations in the coupling strength, which suggests that for these coupling mechanisms, the weak coupling approximation remains valid up to large coupling strengths. For instance, light coupling mechanisms that temporarily increase TOC protein degradation (

Color-coded adjustment RMS error as a function of window duration and modulation ratio (ratio of degradation rates inside and outside the coupling window). (A) Modulation of CCA1 protein degradation rate; (B) Modulation of TOC1 protein degradation rate.

Our analysis shows that several coupling mechanisms are compatible with the experimental data and that discriminating them requires more experimental data. In particular, monitoring gene expression in transient conditions will probably be crucial since the coupling mechanism leaves apparently no signature in the experimental data in entrainement conditions. For simplicity, we restrict ourselves in the following to models in which half-lives of TOC1 or CCA1 proteins are modified during a specific time interval that is determined in

One may wonder about the purpose of coupling schemes with almost no effect on the oscillator. The key point is that our data have been recorded when the clock was entrained by the diurnal cycle and phase-locked to it. A natural question then is: how do such couplings behave when clock is out of phase and resetting is needed? We found that while the two mechanisms shown in

TOC1 (resp. CCA1) degradation rate is multiplied by 2.1 (resp., by 0.6) from ZT0 to ZT6.5 (resp., from ZT12.8 to ZT13.95). After phase-shifting the day/night cycle by 12 hours, (A) mRNA and (B) protein time profiles (logarithmic scale) of numerical solutions (solid lines) converge rapidly to the nominal profile (dashed lines). (C) Residual phase shift one day (black) and five days (blue) after a phase shift ranging from −12 to 12 hours has been applied.

To design this coupling, we utilized the fact that modifying coupling strengths inside windows hardly affects adjustment. We could therefore choose their values so as to minimize the maximal residual phase shift after three days for all possible initial lags (

Why would it be beneficial for a circadian oscillator to be minimally affected by light/dark alternation in normal operation? A tempting hypothesis is that while daylight is essential for synchronizing the clock, its fluctuations can be detrimental to time keeping and that it is important to shield the oscillator from them. If the entrained temporal profile remains close to that of an uncoupled oscillator at different values of the coupling parameter, then it will be naturally insensitive to fluctuations in this parameter. To gain insight into this fundamental question, we subjected the fully coupled and occasionally coupled clock models to fluctuating daylight.

With the light input pathway unknown, we must allow for the fact that light fluctuations may be strongly attenuated upon reaching the

Although the two types of model adjust experimental data equally well when subjected to a regular alternation, they have completely different responses to daylight fluctuations. In

(A) Light intensity varying randomly from day to day. The time evolution of TOC1 concentration is shown for: (B), (C) the permanently coupled clock model of

In contrast to this, the two occasionally coupled oscillators of

We also studied the effect of fluctuations at shorter time scales. When light intensity was varied randomly each hour, but with the same mean intensity each day, the permanently coupled model was still affected but much less than in

The results described above may seem to rely on the FRP being equal to 24 hours. When the FRP is smaller or larger, coupling is required to achieve frequency locking and pull the oscillation period to 24 hours. To investigate this more general case, we scaled kinetic constants of the free-running model used in

Gated coupling can also synchronize free-running clock models with a FRP of 23.5h or 25h without leaving any signature in mRNA profiles. Top left, (A)–(C): numerical solutions of model (2) for a FRP of 23.5h, subjected to coupling windows shown as shaded areas. TOC1 (resp. CCA1) protein degradation rate is multiplied (resp. divided) by three from ZT3 to ZT5.5 (resp. ZT15 to ZT17.5). Top right, (D)–(F): numerical solutions of model (2) for a FRP of 25h, subjected to coupling windows shown as shaded areas. TOC1 (resp. CCA1) protein degradation rate is multiplied (resp. divided) by three from ZT22.75 to ZT24 (resp. ZT11.75 to ZT12). (A), (D) RNA in log scale; crosses (resp. circles) indicate

Gating of light input by rectangular profiles does not reflect the fact that the concentration of the mediators modulating the oscillator typically vary in a gradual way. The existence of nested coupling windows such that models with shorter windows can adjust data with larger parameter modulation (see

The behavior of the model using the optimized modulation profiles (

(A) Temporal profile of the CCA1 protein stability modulation coefficient

Our findings illustrate how mathematical modeling can give insight into the architecture of a genetic module. Not only can expression profiles of two

Why would a circadian oscillator decouple from the day/night cycle when in phase with it so as to generate quasi-autonomous oscillations? A natural hypothesis is that this protects the clock against daylight fluctuations, which can be important in natural conditions

In nature, the daylight intensity sensed by an organism depends not only on time of day but also on various factors such as sky cover or, for marine organisms such as

A clock permanently coupled to light is also permanently subjected to its fluctuations. Depending on the coupling scheme, keeping time may become a challenge when fluctuations induce phase resettings and continuously drive the clock away from its desired state. Indeed, we found that a mathematical model with properly timed coupling windows was insensitive to strong light intensity fluctuations while a permanently coupled model became erratic even for very small coupling strengths. For simplicity, we only tested the robustness of a model with modulated TOC1 and CCA1 protein degradation. However, it should be stressed that all other light-coupling mechanisms that were found to be robust with respect to adjustment (see

These results lead to enquire whether similar designs exist in other circadian clocks. Although the importance of this problem was noted some time ago

We thus conjecture that a circadian clock must be built so as to be insensitive to daylight intensity fluctuations when entrained by the day/night cycle, just as it is insentitive to molecular or temperature fluctuations, and that this can be achieved by keeping the oscillator as close to the free-running limit cycle as possible, scheduling coupling at a time when the oscillator is not responsive. An important consequence of this principle is that it allows us to discriminate between different possible coupling mechanisms for a given model, as our analysis revealed dramatic differences in the ability of different parametric modulations to buffer fluctuations. It also allows us to determine the preferred timing for a given coupling mechanism, which may prove very helpful when trying to identify the molecular actors which mediate the light information to the clock.

When the FRP is close to 24 hours, as in much of our analysis, it is easy to understand why robustness to daylight fluctuations requires that the forced oscillation shadows the free-running solution. Robustness manifests itself in the time profile remaining constant when subjected to random sequences of daylight intensity. This includes strongly fluctuating sequences as well as sequences of constant daylight intensity at different levels. Thus, the oscillator response should be the same at high and low daylight intensities, which implies that the solution must remain close to the free-running one as forcing is increased from zero. Note that this only holds in entrainment conditions, where coupling is not needed. When the clock is out of phase, strong responses to forcing are expected, with resetting being faster as forcing is stronger.

When the natural and external periods are significantly different, the problem may seem more complex as coupling is required to correct the period mismatch. There is a minimal coupling strength under which the oscillator is not frequency-locked and entrainment cannot occur. Nevertheless, we showed that timing the coupling windows properly is as effective for oscillators with FRP of 23.5 and 25 hours as for the 24-hour example we had considered. Again, the forced solution remains close to the free-running limit cycle even if proceeding at a different speed to correct the period mismatch. This also shows that FRP is not a critical parameter for adjustment of the experimental data used here.

A consequence of the small deviation of the limit cycle from the free-running one when coupling strength is varied is that oscillations should vary little upon a transition from LD to LL or DD conditions (see, e.g.,

The

An important problem is how a clock with occasional coupling can adjust to different photoperiods so as to anticipate daily events all along the year. We can only touch briefly this question here as it requires understanding how the temporal profile of the coupling windows changes with photoperiod and thus a detailed description of the unknown light input pathways and additional feedback loops that control the timing of these windows. The key point is that the phase of the entrained oscillations is controlled by the position of the coupling windows. Thus the role of light input pathways and additional feedback loops, whose internal dynamics will typically be affected by input from photoreceptors and feedback from the TOC1–CCA1 oscillator, is to time the coupling windows as needed for each photoperiod so that the correct oscillation timing is generated

Our results also bring some insight into the recent observation that a circadian clock may require multiple feedback loops to maintain proper timing of expression peaks in response to noisy light input across the year

Finally, robustness to intensity fluctuations may explain why it is important to have a self-sustained oscillator at the core of the clock, as a forced damped oscillator permanently needs forcing to maintain its amplitude, and is thereby vulnerable to amplitude fluctuations. Confining the dynamics near the free-running limit cycle allows to have a pure phase dynamics for the core oscillator, uncoupled from intensity fluctuations. Understanding how to construct it will require taking into account the sensitivity of the free-running oscillator to perturbations across its cycle

A simple organism as

A minimal mathematical model of the transcriptional loop where

Model (2) has 16 free continuously varying parameters besides the cooperativities

Adjustment was carried out by using a large number of random parameter sets as starting points for an optimization procedure based on a Modified Levenberg–Marquardt algorithm (routine LMDIF of the MINPACK software suite

To study the effect of daylight fluctuations, parameters were modulated as follows.

The parameters of the Gaussian-shaped modulation profiles were determined by optimizing resetting. For all possible variable initial time lag ranging from −12 to 12 hours, the effect of the coupling scheme based on the two profiles modulating TOC1 degradation and CCA1 degradation was characterized as follows. The time lag was applied to the free-running cycle adjusting experimental data. Then, the coupling scheme was applied for one or 5 days. Finally, the coupling was switched off and the residual phase error was measured after two days. The set of six parameters defining modulation profiles were obtained as those which minimize RMS residual phase error across the 24-hour interval.

Transition from light/dark alternation(LD) to constant light (LL) and constant darkness (DD) for the fully coupled model. Time evolution of mRNA concentrations for the fully coupled model shown in

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Influence of experimental errors on adjustement of a free running oscillator model to data. Alternate target profiles with samples randomly chosen inside the interval of variation observed are generated and adjusted. Each random target corresponds to a slightly different parameter set and to a different adjustment RMS error (A) RMS error distribution; (B) The five target profiles most distant from each other have been selected and are associated with different colors. Crosses (resp. circles) indicate the

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Probability distribution for parameter values in parameter sets with adjustment RMS error below 10%. Parameters are determined as explained in _{10}. The probability distributions of parameter values for the model with all parameters modulated are shown in red and blue for the day and night values, respectively. The probability distribution of parameter values for the model with all parameters constant is shown in black.

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Characterization of coupling schemes. (A) iPRC characterizing the phase change induced by an infinitesimal perturbation of parameters _{X}_{X}_{X}_{m}_{0}_{0}_{0}_{0}_{0}

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Resetting of the clock model of

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Response of the fully coupled and occasionally coupled clock models to fluctuations in daylight intensity occurring on a time scale of one hour. The figure is otherwise similar to

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Response of the two occasionally coupled clock models of

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Characterization of gated coupling mechanisms in the weak modulation limit.

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We thank Bernard Vandenbunder for his helpful guidance in the early stages of this work and Constant Vandermoere for assistance with data analysis.