Cycle dynamics and synchronization in a coupled network of peripheral circadian clocks

Abstract

The intercellular interactions between peripheral circadian clocks, located in tissues and organs other than the suprachiasmatic nuclei of the hypothalamus, are still very poorly understood. We propose a theoretical and computational study of the coupling between two or more clocks, using a calibrated, reduced model of the circadian clock to describe some synchronization properties between peripheral cellular clocks. Based on a piecewise linearization of the dynamics of the mutual CLOCK:BMAL1/PER:CRY inactivation term, we suggest a segmentation of the circadian cycle into six stages, to help analyse different types of synchronization between two clocks, including single stage duration, total period and maximal amplitudes. Finally, our model reproduces some recent experimental results on the effects of different regimes of time-restricted feeding in liver circadian clocks of mice.

1. Introduction

Most organisms have evolved a circadian timing system to synchronize their physiology and behaviour with the Earth’s light/dark cycle. The core mechanism underlying circadian rhythms is a molecular clock governed by an oscillatory gene network present in virtually all cells and which translates the external time information into optimally phased molecular and cellular rhythms [1]. Environmental or genetic disruption of the circadian clock leads to multiple metabolic alterations in animal models. In humans, misalignment between body time and social time caused for instance by rotating shiftwork or work at night is also a risk factor for cardio-metabolic, inflammatory and malignant diseases [2]. Reciprocally, diseases perturb the circadian coordination thus potentially leading to a vicious cycle. A large body of experimental and clinical evidence suggests that high amplitude and correctly phased oscillations of core clock and clock-controlled genes are a hallmark of health while disrupted or desynchronized rhythms are associated with pathology [3,4]. Whether and how cells temporally coordinate together at the scale of the tissue and organ is therefore a question of primary importance to later implement preventive or therapeutic strategies aiming at reinforcing circadian rhythms.

It is now well established that neurons of the central clock located in the suprachiasmatic nuclei (SCN) of the hypothalamus oscillate in synchrony so that a single phase is observed at the tissue level. The coupling mechanism within the SCN requires intercellular communication through gap junctions and neurotransmitters [5]. Intercellular coupling is an important determinant of oscillator amplitude, robustness and entrainment by synchronizers. The SCN encompasses a pacemaker clock that hierarchically controls other peripheral circadian clocks [6]. It is a largely accepted idea that peripheral clocks function mostly as ‘slave’ oscillators that damp in the absence of internal systemic synchronizers such as body temperature and glucocorticoid hormones.

However, this view has been recently challenged by in vivo experiments performed in mice performed by Schibler’s laboratory [7]. By using a sophisticated technology allowing longitudinal long-term bioluminescence recordings in free living animals, the authors could demonstrate that oscillations of the core clock gene Rev-erbα persist in the livers of animals lacking a functional SCN or circadian oscillators in all extra-hepatic organs. Therefore, it appears that phase coherence between hepatocyte clocks can be achieved independently of external zeitgebers, suggesting that intercellular coupling may occur in the liver. Another recent work [8] by Kramer’s laboratory using the human osteosarcoma U2OS cell line reinforces this hypothesis by providing evidence that peripheral cellular oscillators couple in vitro via paracrine factors. This study further identified transforming growth factor β as a candidate signalling molecule contributing to the intercellular coupling between circadian oscillators although this remains to be established in vivo.

In this paper, we propose a theoretical and computational study of the coupling between two or more clocks, to investigate the question of intercellular coupling between peripheral circadian oscillators. Our objective is to analyse the dynamical evolution and coordination between the main processes involved in intercellular coupling, to better understand the mechanisms leading to oscillator synchronization. For this study, we will use a reduced mathematical model of the circadian clock previously developed in our group by Almeida et al. [9,10], which recovers important properties such as the interactions with the cell cycle. This model is calibrated with data from mouse fibroblast cells cultured in regular medium [11], so we can assume that it is a faithful model of an autonomous peripheral clock.

Mathematical models have been developed specifically for the SCN clock [12], for other organs such as the liver [13] or pancreatic beta-cells [14], but more often for ‘generic’-type cells, blending the literature data from various sources [10,15,16]. Synchronization of neurons in the SCN is studied through a general model in [17] and further detailed in [5], showing the coupling mechanism requires intercellular communication through gap junctions and neurotransmitters.

Synchronization of clocks in other organs or tissues has been studied essentially as a result of SCN control or by entrainment to external cues such as light or food intake [13,14]. In this study, we suggest a segmentation of the circadian cycle into six stages based on a piecewise linearization of our model [9], to help characterize the effect of coupling on the synchronization mechanism. We find that clock synchronization happens at distinct levels: for instance, clocks with slightly differing periods can synchronize both their amplitudes and periods but maintain distinct duration for each of the six cycle stages. For a cluster of neighbouring cells, our coupling model reproduces some of the recent results on mouse hepatocytes [7], suggesting a relevant role for peripheral clock coupling.

2. Coupling and synchronization of two clock models

The circadian clock model used in this paper was developed in our group [9,10] and focuses on the transcriptional details of the two main feedback loops present in the circadian clock mechanism [1,15,16], both centred on the protein complex CLOCK:BMAL1, next briefly described. In this reduced model, paralogous proteins are not distinguished so that for instance the REV variable accounts for both REV-ERBα and REV-ERBβ. The complex CLOCK:BMAL1 promotes transcription of the genes Per and Cry, whose corresponding proteins PER and CRY dimerize to form another complex, PER:CRY. Subsequently, upon nuclear translocation, PER:CRY interacts with CLOCK:BMAL1 and the resulting complex is exported from the nucleus, thus blocking CLOCK:BMAL1 transcriptional activity and closing the first negative feedback loop. In parallel, the complex CLOCK:BMAL1 also promotes transcription of the genes Dbp and Rev-erbα. The protein DBP binds to the D-box clock controlled element located in the Per and Rev-erbα promoters to activate their transcription. Finally, protein REV-ERBα binds to the R-box in the Bmal1 promoter, thereby blocking synthesis of the protein BMAL1 and closing the second negative feedback loop.

The reduced form of the model studied in [9] describes the dynamics of the protein complexes CLOCK:BMAL1 (denoted B), DBP (denoted D), REV (denoted R) and PER:CRY (denoted P). The simplified interaction diagram is depicted in figure 1 and the ordinary differential equations model from [9] can be written as

$$\begin{array}{rl}\dot{B}& =V_R{h}^{-}(R)-{\gamma }_{B}BP,\\ \dot{D}& =V_{B}B-{\gamma }_{D}D,\\ \dot{R}& =V_{D}D-{\gamma }_{R}R\\ \mathrm{and}\dot{P}& =V_{D}D-{\gamma }_{B}BP,\end{array}\}$$ 2.1

where $h^{-}(R)=k_R^4/(k_R^4+R^4)$ is a decreasing function. The model was calibrated with REV-ERBα experimental data from [11] and its concentration units are in percentages of the REV-ERBα data mean value, which was used to normalize the data (see [10]). The set of estimated parameters can be found in table 2 of [9] and is used throughout this paper (unless stated otherwise):

$$\begin{array}{rl}& V_R=44.4\text{\%}\hspace{0.17em}{\mathrm{h}}^{-1},\hspace{0.17em}{k}_R=80.1\text{\%},\hspace{0.17em}{V}_B=0.142\text{\%}\hspace{0.17em}{\mathrm{h}}^{-1},\hspace{0.17em}{V}_D=19\text{\%}\hspace{0.17em}{\mathrm{h}}^{-1}\\ \mathrm{and}& {\gamma }_R=0.241\hspace{0.17em}{\mathrm{h}}^{-1},\hspace{0.17em}{\gamma }_D=0.156\hspace{0.17em}{\mathrm{h}}^{-1},\hspace{0.17em}{\gamma }_B=2.58\hspace{0.17em}{\mathrm{h}}^{-1},\end{array}\}$$ 2.2

with a period of τ₁ = 24.5 h.

Figure 1.

Diffusive coupling between two circadian clocks through the complex PER:CRY.

2.1. Coupling between two peripheral clocks

To study the coupling between two or more oscillators, we will include two new observations into our model (2.1): the variation of the oscillator’s period depending on expression level of gene Cry2 [18] and the role of gene Per2 in intercellular clock communication [8]. Although the proteins CRY and PER are not specifically represented in system (2.1), their complex PER:CRY is described by variable P. Therefore, both new observations will be modelled by modifying equation $\dot{P}$ in two different ways, as follows.

To represent circadian period variability among a population of cells, we will follow experimental results showing that Cry2 knockout or knockdown cells have a longer period than wild type cells [18,19]. To represent this in our model, we add a factor ɛ in the equation of PER:CRY, $\dot{P}=\epsilon V_{D}D-{\gamma }_{B}BP$, where ɛ < 1 describes a lower availability of protein CRY2 and thus a lower rate (ɛV_D) of complex PER:CRY formation (table 2 in appendix A shows the correspondence between ɛ values and periods).

Several earlier studies have shown that the Per2 gene is responsive to multiple signalling pathways both in the SCN and in peripheral organs [20,21]. Interestingly, recent experimental data [8] show that paracrine signalling downregulates Dbp and upregulates Per2 expression, suggesting that gene Per2 is an early responder to intercellular clock communication. In this paper, we will assume that this communication is represented by a direct diffusive action on the complex PER:CRY variable P. Although a highly simplified representation, it avoids the introduction of new uncalibrated variables into the model, and allows us to continue our analytical study. The resulting coupled model is

$$\begin{array}{rl}{\dot{B}}_i& =V_R{h}^{-}(R_i)-{\gamma }_B{B}_i{P}_i,\\ {\dot{D}}_i& =V_B{B}_i-{\gamma }_D{D}_i,\\ {\dot{R}}_i& =V_D{D}_i-{\gamma }_R{R}_i\\ \mathrm{and}{\dot{P}}_i& ={\epsilon }_i{V}_D{D}_i-{\gamma }_B{B}_i{P}_i+k(P_j-P_i).\hspace{0.17em}\end{array}\}$$ 2.3

In the next sections, we will analyse the effect of ɛ and k in the coupled network, assuming a weak coupling rate [22] and a wider interval for the circadian periods ɛ_min < ɛ < ɛ_max.

2.2. Synchronization of two systems

In a first step, we will study the synchronization of two systems (using ɛ₁ = 1 and ɛ₂ = ɛ), as a function of the factor ɛ and the coupling strength k. To do this, we compute the total difference between the variables, $|r|=\sqrt{r_B^2+r_D^2+r_R^2+r_P^2}$, with r_B = B₁ − B₂, r_D = D₁ − D₂, r_R = R₁ − R₂, r_P = P₁ − P₂.

The total difference $|r|$ will increase with the perturbation (1 − ɛ) (mutation ‘strength’) and decrease with the coupling strength k. Figure 2 illustrates the opposite effects of period differences and coupling strength on the error. In the case of equal periods (i.e. 1 − ɛ = 0), the error converges to zero and the two clocks synchronize exactly, while the coupling strength cannot guarantee exact synchronization.

Figure 2.

Maximum of the total difference $|r|=\sqrt{r_B^2+r_D^2+r_R^2+r_P^2}$ as a function of ɛ, for two coupled systems (2.3). The parameters are those listed in (2.2).

To further study the role of coupling in maintaining cell synchronization and the capacity of the cells to synchronize, we performed two types of simulations to calculate the amount of time needed to desynchronize two cells once coupling is disabled or, conversely, to calculate the time needed to synchronize cells that start with some phase difference. (Throughout this section, we compute the oscillator phases relative to the crossing of threshold value k_R by variable R, as it decreases.) Figure 3 shows the number of circadian cycles needed to desynchronize or synchronize two cells, as one of the periods varies in terms of ɛ. Unsurprisingly, the time needed to desynchronize cells increases in an exponential form as the two oscillator periods become closer (black curve). More interestingly, the form of the synchronization curve highly depends on the initial phase difference between the two oscillators. For phase differences up to ≈8 h, the number of cycles to synchronization decreases with the difference between periods. For low initial phase differences (about 5 to 6 h), the time to synchronization is less than six circadian cycles for two cells with periods τ₁ = 24.5 h and 24.8 < τ₂ < 25.5 (that is, 0.85 < ɛ < 0.95). For a higher initial phase difference of about 7.3 h, the two cells need from 20 to 5 cycles to synchronize. Finally, for very high initial phase differences (around 10 h, see lighter blue curve) the trend is inverted, since synchronization times are around five circadian cycles for 25.2 < τ₂ < 25.5 (that is, 0.85 < ɛ < 0.9) and then increase to about seven cycles for τ₂ = 24.8 (or ɛ = 0.95).

Figure 3.

Number of circadian cycles needed for desynchronization and synchronization of two oscillators. Desynchronization (black curve): the two oscillators start from a point where their phase difference is smallest and the coupling is turned off. Synchronization (blue curves): the two oscillators start from a point where their phase difference is either 0.2τ₁ = 4.9 h, 0.25τ₁ = 6.1 h, 0.3τ₁ = 7.3 h or 0.4τ₁ = 9.8 h, with τ₁ = 24.5 h and the coupling is turned to k = 0.04. In both cases, the oscillators are considered synchronized while their phase difference is lower than 0.15τ₁ = 3.6 h.

Our simulations evoke an observation on SCN clocks, for which it has been reported that the role of neuropeptide VIP (a factor implicated in cell communication) in synchronization is dose dependent, with a low VIP leading to synchronization but a high VIP leading to desynchronization [23].

3. Characterization of circadian cycle timing

In order to better understand how the coupling mechanism works, the effect of the period differences (characterized by ɛ) and the coupling strength k on the synchronization, we will divide the circadian cycle into six different stages, based on the dynamics of the CLOCK:BMAL1/PER:CRY mutual inactivation. To do this, the nonlinear terms of model (2.1) are simplified to create a piecewise linear model [24] that partitions the state space into six different regions. The time spent by the orbit in each of these regions can then be explicitly calculated in terms of the system’s parameters and switching conditions, providing estimates for the duration of each stage. A computational study shows the duration of each stage and their reorganization in the coupled network.

3.1. A piecewise linear clock model

There are two nonlinear terms in system (2.1). First, let us consider the decreasing Hill function $f\hspace{0.17em}(R)=V_R(k_R^4/(k_R^4+R^4))$. This class of functions is often simplified by a step function [24]:

$$f\hspace{0.17em}(R)\approx \{\begin{array}{cc}0,& R>k_R\\ V_R,& R<k_R.\end{array}$$ 3.1

The second nonlinear term is the common degradation or inactivation of the complexes CLOCK:BMAL1 and PER:CRY, that is, −γ_BBP. As the oscillator performs one full cycle, this term goes through several well-marked conditions (figure 4). Define B* and P* to be low levels of reference for the variables B and P, respectively. Consider also B_high and P_high to be high levels of reference for those variables. Let B_max and P_max denote their maximal values. These reference values define a family of intervals in B and P ([0, B*], [B*, B_max] and [0, P*], [P*, P_high], [P_high, P_max]) where the CLOCK:BMAL1 and PER:CRY common inactivation term can be simplified by setting either B or P to be at their low or high reference values.

Figure 4.

Protein concentrations over time and γ_BBP and f(R) curves.

Notice that while B is approximately constant at its low level B*, or when P is very large (P > P_high), the two terms γ_BBP and f(R) have similar values, so they add up approximately to 0 in the B equation. This suggests the following simplification:

$${\gamma }_{B}BP\approx \{\begin{array}{ll}0,& R>k_R\text{ and }B<B^{\ast }\\ V_R,& R<k_R\text{ and }P>P_{\mathrm{high}}.\end{array}$$ 3.2

To approximate γ_BBP in the other cases, we adopt the following criterion: for intermediate levels of P, we assume that the inactivation term is dominated by the B dynamics with P constant, while for very low P, the opposite scenario holds. Taking into account the phase opposition between B and P, we expect B to be close to its maximum value when P < P*. We thus propose the following piecewise linear approximation for γ_BBP:

$${\gamma }_{B}BP\approx \{\begin{array}{lll}0,\hspace{0.17em}& R>k_R,B<B^{\ast }\hspace{0.17em}& \text{stage 1}\\ V_R,\hspace{0.17em}& R<k_R,P>P_{\mathrm{high}}\hspace{0.17em}& \text{stage 2}\\ {\gamma }_{B}B{P}^{\ast },\hspace{0.17em}& R<k_R,P^{\ast }<P<P_{\mathrm{high}}& \text{stage 3}\\ {\gamma }_B{B}_{\max }P,& R<k_R,P<P^{\ast }\hspace{0.17em}& \text{stage 4}\\ {\gamma }_B{B}_{\max }P,& R>k_R,B>B^{\ast },P<P^{\ast }& \text{stage 5}\\ {\gamma }_{B}B{P}^{\ast },\hspace{0.17em}& R>k_R,B>B^{\ast },P>P^{\ast }& \text{stage 6}.\end{array}$$ 3.3

These conditions cover all the state space and ensure positivity of the piecewise linear system composed of (2.1)–(3.1)–(3.3). To simulate this system, we need to provide suitable values for the reference levels.

To estimate B_max, consider the steady state obtained when transcription is activated and the degradation term is fixed at γ_BBP*:

$$\dot{B}\approx V_R-{\gamma }_{B}B{P}^{\ast }\Rightarrow V_R-{\gamma }_B{B}_{\max }{P}^{\ast }=0\iff B_{\max }=\frac{V_R}{{\gamma }_B{P}^{\ast }}.$$

To estimate the low level P*, consider the cycle’s phase opposition property and assume that both B and D reached their maximum values, and P is produced at a rate ɛV_DD_max:

$$\dot{D}\approx V_B{B}_{\max }-{\gamma }_{D}D\Rightarrow V_B{B}_{\max }-{\gamma }_D{D}_{\max }=0\iff D_{\max }=\frac{V_B{B}_{\max }}{{\gamma }_D}$$

and

$$\begin{array}{c}\dot{P}\approx \epsilon V_D{D}_{\max }-{\gamma }_D{B}_{\max }P\Rightarrow \epsilon V_D{D}_{\max }-{\gamma }_D{B}_{\max }\overline{P}=0\iff \\ \overline{P}={\displaystyle \frac{\epsilon V_D{D}_{\max }}{{\gamma }_B{B}_{\max }}={\displaystyle \frac{\epsilon V_D{V}_B}{{\gamma }_B{\gamma }_D}.}}\end{array}$$

For a safer estimate we consider $P^{\ast }=\overline{P}/4$, so that

$$P^{\ast }=\frac{\epsilon V_D{V}_B}{4{\gamma }_B{\gamma }_D}\mathrm{and}{B}_{\max }=\frac{4V_R{\gamma }_D}{\epsilon V_D{V}_B}.$$

By contrast, there is no obvious way to estimate a suitable non-zero value for B*, so we choose a fixed low value, B* = 0.01, for the simulations in this paper. Set B_high = B_max since a distinction is not required between them. Finally, P_max is not used explicitly but an intermediate value for P must be fixed. In the current simulations, we use an appropriately high value within the system’s parameters and set P_high = V_R.

Figure 5 shows oscillations of the four proteins of the piecewise linear model (2.1)–(3.1)–(3.3), as well as the duration of each stage (black straight lines), with parameter set (2.2). We observe that, depending on the parameter set, the system may not go through all six stages.

Figure 5.

Protein concentrations over time and stage durations, for the piecewise linear model. The stages are represented by the black straight lines, corresponding to the ordinates 50, 100, 150, 175, 200 and 225, respectively.

3.2. A partition of the circadian cycle

Through the simplification of the continuous model (2.1) into a piecewise affine model, we analyse the circadian cycle as a sequence of six main stages, based on the repression of CLOCK:BMAL1 by REV and the dynamics of the mutual degradation/sequestration between CLOCK:BMAL1 and PER:CRY. Our stages can be compared to the common circadian time (CT) description [1], which is based on the circadian transcriptional landscape in the mouse liver. This means that night hours (from CT12 to CT0/24) correspond to the mouse activity and feeding phase while the daylight hours (from CT0 to CT12) correspond to the resting and fasting phase. In this context, we find that the circadian night is divided into two stages, both characterized by a decreasing or low concentration of CLOCK:BMAL1 and the common degradation term either modulated by CLOCK:BMAL1 dynamics (stage 6) or at a very low level (stage 1).

The circadian daylight interval is divided into approximately three stages. An early day stage (2) corresponds to a very high PER:CRY level, which imposes a high but saturated common degradation term, and CLOCK:BMAL1 de-repressed. The beginning of this stage would approximately correspond to the beginning of the circadian time count CT0. The second daylight stage (3) corresponds to CLOCK:BMAL1 augmentation as PER:CRY decreases with their degradation modulated by the dynamics of CLOCK:BMAL1. In the third daylight stage (4), PER:CRY is now at a low level while CLOCK:BMAL1 is still increasing, but their common degradation term is now modulated by PER:CRY dynamics. The end of this stage would correspond to middle circadian time (CT12). Finally, there is a day-to-night transition (stage 5), where REV has reached its switching value and repression of CLOCK:BMAL1 is turned on.

3.3. Duration of each cycle stage in terms of the system’s parameters

The advantage of the piecewise linear approximation is to allow the explicit computation of a solution for four of the six cycle stages and hence also an expression for these four stages’ duration in terms of the parameters. We can express the duration of stages 6, 1, 2 and 3 in terms of the parameters, but expressions for stages 4 and 5 are more complex to derive. However, in all our examples, stages 4 and 5 have very short duration (lower than about 2 h).

For stage 6, we have the following equation for B: $\dot{B}=$ $-{\gamma }_{B}B{P}^{\ast }$ whose analytical solution is $B(t)={\mathrm{e}}^{-{\gamma }_B{P}^{\ast }(t-t_0)}B(t_0)$. Letting (t₃, t₆) denote the times at which the cycle end stage 3 and 6, respectively, we assume that B(t₃) = B_max = V_R/(γ_BP*) and B(t₆) = B*. So, recalling P* = ɛV_DV_B/(4γ_Bγ_D), we obtain

$$\begin{array}{c}B(t_6)={\text{e}}^{-{\gamma }_B{P}^{\ast }(t_6-t_3)}B(t_3)\Rightarrow t_6-t_3\\ =-\frac{4{\gamma }_D}{\epsilon V_D{V}_B}\ln \left(\frac{\epsilon B^{\ast }{V}_D{V}_B}{4V_R{\gamma }_D}\right).\end{array}$$ 3.4

Since ɛ represents Cry2 knockout, as ɛ decreases the period of the system increases. This expression shows that the duration of stage 6 also increases with the effectiveness of Cry2 knockout.

In a similar way, we can compute the duration of stages 1, 2 and 3:

$$t_1-t_6=-\frac{1}{{\gamma }_D}\ln \left(\frac{D(t_1)-(V_B{B}^{\ast }/{\gamma }_D)}{D(t_6)-(V_B{B}^{\ast }/{\gamma }_D)}\right),$$ 3.5

$$t_2-t_1=-\frac{1}{{\gamma }_D}\ln \left(\frac{D(t_2)-(V_B{B}^{\ast }/{\gamma }_D)}{D(t_1)-(V_B{B}^{\ast }/{\gamma }_D)}\right)$$ 3.6

$$\mathrm{and}{t}_3-t_2=-\frac{1}{{\gamma }_B{P}^{\ast }}\ln \left(\frac{B(t_3)-(V_R/{\gamma }_B{P}^{\ast })}{B^{\ast }-(V_R/{\gamma }_B{P}^{\ast })}\right).$$ 3.7

Duration of stage 6 depends only on the parameters, whereas duration of the other stages (k = 1, …, 5) depends also on the values reached by D and B at the beginning (D(t_k−1), B(t_k−1)) or at the end (D(t_k), B(t_k)) of stage k, which we do not necessarily know how to estimate.

Notice that stages 6, 1 and 2 follow each other in this order. Since the equation of D is the same for all stages, during stages 1 and 2, its solution is

$$D(t)={\mathrm{e}}^{-{\gamma }_D(t-t_0)}(D(t_0)-\frac{V_B{B}^{\ast }}{{\gamma }_D})+\frac{V_B{B}^{\ast }}{{\gamma }_D}.$$ 3.8

The lower level D* of D is reached when B = B* so we can estimate it:

$$\begin{array}{r}\dot{D}\approx V_B{B}^{\ast }-{\gamma }_{D}D\Rightarrow V_B{B}^{\ast }-{\gamma }_D{D}^{\ast }=0\iff D^{\ast }=\frac{V_B{B}^{\ast }}{{\gamma }_D}.\end{array}$$

So, as D(t₀) > V_BB*/γ_D, D(t) is decreasing during stages 1 and 2, so that D(t₆) > D(t₁) > D(t₂).

Figure 6 shows the duration of each stage as a function of ɛ. For parameter set (2.2), the piecewise linear system does not go through stages 4 and 5. With respect to ɛ, stage 6 decreases strongly then stabilizes, stage 3 increases then stabilizes, stage 1 increases for very low ɛ then decreases. Stage 2 seems to increase close to linear. Let us focus now on the duration of the period: because of the strong decrease of stage 6, it decreases, then because of the increase of stage 2 and the stabilization of stages 3 and 5, it increases. More specifically, it is interesting to compare the model behaviour for ɛ < 1 with Cry2 knockout cells: lower ɛ values represent more effective knockouts, with less available protein CRY2. Cry2 knockouts have higher periods and amplitudes [22], and our model shows that the period increases with the effectiveness of the knockout. Moreover, our model shows that the longer period is due to a longer duration of stage 6 corresponding to a longer end of the day, in agreement with the property that Cry2 deficient mice represent models of ‘eveningness’ [25] (showing activity patterns later in the day).

Figure 6.

Duration of each stage as a function of ɛ.

4. Circadian stages and period duration in a mixed-cell population

To study the coupling dynamics in a cell population with mixed periods, we now consider the continuous system (2.3) with ɛ₁ = 1 and ɛ₂ = ɛ, to reproduce two coupled oscillators with different periods. To further analyse the dynamics, we measure the duration of the six stages for each oscillator, according to the conditions summarized in table 1. Figure 7 shows the stage and period durations of the coupled system, as a function of ɛ, for two values of the coupling strength k.

Table 1.

Characterization of the six circadian time stages based on the mutual CLOCK:BMAL1 and PER:CRY inactivation dynamics.

stage	1	2	3	4	5	6
circadian	night	early day	day	day	day/night	night
time		CT0$\to$		$\to$ CT12	transition
CLOCK:BMAL1	low	low	increasing	high	high	decreasing
(B)	B < B*	B > 0	B > 0	B > 0	B > B*	B > B*
PER:CRY	increasing	high	decreasing	low	low	increasing
(P)	P > 0	P > Phigh	P* < P < Phigh	P < P*	P < P*	P > P*
common	negligible	saturated	modulated	modulated	modulated	modulated
degradation		by P	by B	by P	by P	by B
repression of	on	off	off	off	on	on
CLOCK:BMAL1	R > KR	R < KR	R < KR	R < KR	R > KR	R > KR

Figure 7.

Duration of stages and periods for system (2.3) as a function of ɛ. (a) k = 0.04; (b) k = 0.09. In both panels, the solid lines represent oscillator 1 (ɛ = 1) and the dashed lines represent oscillator 2 (ɛ ≠ 1). The thin solid and dashed yellow lines represent the periods of the oscillators before coupling.

As expected, a higher coupling constant allows synchronization of the periods for a wider interval around ɛ = 1. For ɛ < 1, this simulation evokes the mixture between Cry2 knockout and wild type cell populations performed for instance in [22] or [8]. The period becomes longer as ɛ < 1 decreases (check thin dashed lines, uncoupled oscillator 2), as is the case in Cry2 knockout cells. The mixture of wild type and Cry2 knockout cells is represented by the coupled oscillators: solid and dashed thick lines, respectively (ɛ < 1). In this case, we see that the longer periods are shortened by up to 2 h (dashed line) while the wild-type cells remain at approximately the same period (solid line) in a small neighbourhood of ɛ = 1, in good agreement with the recent observations by Finger et al. [8, fig. 1H,J].

Although amplitudes and periods of both oscillators remain similar, notice that the duration of each stage may differ substantially from one oscillator to the other (compare same-colour solid and dashed lines in figure 7): the two oscillators spend different amounts of time in different stages, while remaining synchronized on the full period. Figure 8 confirms that, in a close neighborhood of ɛ = 1, the coupled oscillators have essentially the same periods but with a phase difference, in particular the variables’ peaks are not reached simultaneously.

Figure 8.

Phase advance of oscillator 2 (ɛ ≠ 1) over oscillator 1 (ɛ = 1), as an average over about 50 periods. (a) k = 0.04; (b) k = 0.09. Thin lines represent average plus standard deviation.

To further understand the mechanisms leading to the distinct duration of stages 1 and 2 in the two coupled oscillators, we first identified the entry and exit points from each stage and then identified and compared the dominant processes involved in each of the stages. In agreement with (3.3), stage 1 is completely defined by the value of BMAL1, since both entry and exit points from this stage correspond to the crossing of threshold B* by BMAL1, as this protein is either decreasing (entry to stage 1) or increasing (exit to stage 1) (figure 9a). The entry and exit to stage 2 are defined by REV and PER:CRY, respectively. The oscillators enter into stage 2 when REV crosses its activation level k_R while decreasing (figure 9b) and exit stage 2 when PER:CRY crosses its high level P_high while decreasing (figure 9c).

Figure 9.

Dynamics of two coupled oscillators with k = 0.04 and ɛ = 0.82. Oscillator 1 is shown in solid curves, oscillator 2 is shown in dashed curves. The uncoupled system periods are ${\tau }_1^u=24.5\hspace{0.17em}\mathrm{h}$ and ${\tau }_2^u=25.5\hspace{0.17em}\mathrm{h}$, while the coupled periods are ${\tau }_1^c={\tau }_2^c=24.4\hspace{0.17em}\mathrm{h}$. (a–c) Time series of variables B (red), R (green), and P (blue). The solid and dashed black horizontal lines denoted the length of the six stages. The vertical lines denote the beginning and ending of stages 1 and 2. (d,e) Time series of the main processes at play in the equation of the systems: synthesis of B and P (V_Rh⁻¹(R_j) and V_DD_j), their mutual degradation term (γ_BB_jP_j), and the effect of coupling (plotted as V_DD_i − k(P_i − P_j)). The dark grey solid and dashed horizontal lines correspond to the duration of stage 1 while the light grey solid and dashed horizontal lines correspond to the duration of stage 2.

We then compared the strength of the different processes during the two stages (figure 9d,e). It follows that during stage 1, repression of BMAL1 by REV is high and the mutual degradation of BMAL1 and PER:CRY is at a low level. The dominant process during stage 1 is clearly the production term of PER:CRY, V_DD. Comparison of the dark and light blue curves of figure 9e shows that the coupling term is also negligible during this stage. This last observation indicates that the two oscillators are operating in a close to uncoupled mode and therefore the longer period oscillator naturally shows a longer duration of stage 1.

Conversely, during stage 2, the opposite happens, with low repression of BMAL1 by REV and the mutual degradation of BMAL1 and PER:CRY at high level is now the dominant process (figure 9d). Also, during stage 2, the coupling term is at its highest strength, as can be seen by its effect on PER:CRY dynamics: the coupling term increases (respectively, decreases) the production rate of the shorter (respectively, longer) period oscillator. This leads to the longer period oscillator decreasing PER:CRY levels faster and hence reaching the threshold P_high faster than expected relative to the uncoupled oscillator. Inversely, the shorter period oscillator reaches P_high slower than expected. Thus, during stage 2, the coupling term is largely responsible for decreasing the difference between the two periods by imposing a shorter duration of this stage for the longer period oscillator and a longer duration of this stage for the shorter period oscillator.

These conclusions are confirmed by figure 10, which show that the duration of stage 1 is roughly the same in the case of coupled or uncoupled oscillators in a small neighbourhood of ɛ = 1. By contrast, the duration of stage 2 for oscillator 2 decreases significantly from the uncoupled to coupled situation (for ɛ < 1), while it increases slightly for oscillator 1. The synchronization dynamics between the two oscillators along the circadian cycle is summarized in figure 11, which highlights the dominant processes at each stage.

Figure 10.

Comparison of duration of stages 1 and 2 for the coupled and uncoupled oscillators.

Figure 11.

Dominant processes throughout one circadian cycle. During stage 1 (circadian night) the two oscillators evolve in a weakly coupled mode, each oscillator following its individual natural period. Concentrations of PER:CRY and REV increase overall. During stage 2 (circadian early day), coupling is at its strongest level, leading to the adjustment of the two oscillators’ periods, with a shorter stage 2 duration for the longer period oscillator.

5. Clock synchronization in response to time-restricted feeding

So far, we have studied the coupling between two oscillators. However, the study of cell periods in a mixture of wild type and Cry2 knockout cells suggests that hepatocytes are coupled in clusters of 5–10 cells [22]. In liver lobules, cells are organized in parallel rows. We tested five distinct topologies (star, all-to-all, ring, rows and chain) to represent different patterns of exchanges between the cells. We report our results on star topology (figure 12).

Figure 12.

Star topology.

In this section, we use our clock model to study the relevance of peripheral clock coupling in the response to different regimes of time-restricted feeding in liver circadian clocks of mice. Recent in vivo experiments by Sinturel et al. [7] show that coherent oscillations are maintained in the liver of mice devoid of a functional SCN or having a clock exclusively in hepatocytes. We designed a computational study to reproduce the experimental protocol in these experiments, consisting of feeding mice (with and without SCN) ad libitum for about 20 days, then alternating periods of fasting and feeding (of 12 h each) for about 20 days before feeding them again ad libitum. Circadian oscillations were monitored longitudinally in free-living animals via PER2::luciferase reporter. The experiments show that during ad libitum intervals, oscillations in mice without SCN are damped. However, a 12 h alternating between eating and fasting intervals permits rhythmic and increasing oscillations in peripheral organs.

After food intake, glycaemia increases and causes the pancreatic β-cells to secrete insulin [26]. Glucose and insulin levels are strongly linked and that is why we model the food intake with insulin (I(t)), added in the CLOCK:BMAL1 equation:

$$\dot{B}=V_R{h}^{-}(R)-{\gamma }_{B}BP+I(t).$$ 5.1

We consider a cluster of 8 cells with initial periods taken randomly between 23.01 h and 26.32 h (0.7 ≤ ɛ ≤ 1.2), coupled in a diffusive way through PER:CRY (k = 0.04) in a star topology. To model ad libitum food intake in SCN lesioned mice (arrhythmic feeding situation), we let insulin be at a constant but low level $I(t)=2\text{\%}\hspace{0.17em}{\mathrm{h}}^{-1}$, throughout the first 20 days. To model time-restricted feeding, we add 2 h pulses of insulin with high amplitude $I(t)=13\text{\%}\hspace{0.17em}{\mathrm{h}}^{-1}$, once every circadian cycle. As the peak of insulin lasts about 2 h and using results of [10], the 2 h pulse is applied from CT5 to CT7 during ten days and then from CT17 to CT19 during the next ten days. This schedule corresponds to the restricted access to the food during either the resting or activity phase, respectively, as in [7].

Figure 13 reports the average concentration of PER:CRY among the 8 cells, $\overline{P}(t)=\sum _{i=1}^8{P}_i(t)/8$, throughout the sequence of 20 + 10 + 10 + 20 days. To check the effect of coupling, we performed exactly the same protocol to a cluster of eight individual cells without any coupling (figure 13a) or with diffusive coupling, in a star topology, with weak coupling strength k = 0.04 (figure 13b). Comparison of figure 13a,b shows that, during periods of time-restricted feeding, oscillations are rhythmic and their amplitude increases strongly, especially in the case of weakly coupled cells. For a more detailed analysis of these signals, we performed fast Fourier transforms (FFT) for the ad libitum intervals and the time-restricted feeding intervals without and with coupling (k = 0.04) (see figure 14, blue and orange curves, respectively).

Figure 13.

Average PER:CRY concentrations over time for ad libitum feeding (black bands) or time-restricted feeding (white bands), over a cluster of eight cells with different periods (τ₁ = 24.11 h, τ₂ = 23.48 h, τ₃ = 23.55 h, τ₄ = 26.29 h, τ₅ = 26.17 h, τ₆ = 26.15 h, τ₇ = 24.95 h and τ₈ = 24.58 h). (a) Without coupling and (b) star topology and k = 0.04.

Figure 14.

FFT analysis of figure 13. Blue curve: without coupling. Orange curve: star topology and k = 0.04. (a) Periods during ad libitum feeding. (b) Periods during time-restricted feeding.

The FFT analysis highlights several aspects. First, in the intervals of time-restricted feeding (figure 14b), the cluster of uncoupled cells shows two small amplitude peaks around 22 h and 25 h. Conversely, the cluster of coupled cells shows only one well defined peak around 23 h, with at least a twofold higher magnitude. This observation indicates that the coupled cluster has the capacity to produce a significant coherent signal under the time-restricted feeding regime, corresponding well to the experimental results of [7]. A clear contrast is presented by the uncoupled cluster, unable to generate coherent oscillations and showing two low amplitude periods, conforming to those present in the eight individual cells.

A second observation is that the main peak representing ad libitum feeding (figure 14a) is centred around 25 h for the uncoupled cluster, while it has a lower period for the coupled cluster.

To complete these observations, we analysed the amplitudes of the three main peaks of the FFT in terms of different values of insulin, for ad libitum feeding periods (values of insulin between 0.5 and 5) and time-restricted feeding period values of insulin between 11 and 15.5) separately (figure 15), with and without coupling.

Figure 15.

Amplitudes of the three main peaks of FFT for k = 0 (solid lines) and k = 0.04 (dashed lines) in terms of insulin values. (a) Ad libitum feeding; (b) time-restricted feeding.

In the intervals of time-restricted feeding, the coupled cluster shows a peak with a significantly higher amplitude (see figure 15b, dashed red line) than the other peaks, suggesting that the cells synchronize over the same period. At the opposite, for uncoupled clusters all peaks have comparable amplitudes indicating that several different periods are similarly represented; thus uncoupled cells are weakly or not synchronized. In the intervals of ad libitum feeding, peaks have similar amplitudes, no matter whether there is coupling or not, suggesting weak synchronization.

Comparison between the amplitude of the peaks of the FFT between ad libitum feeding and time-restricted feeding for a coupled system (figure 15) shows that the main peak of time-restricted feeding is significantly higher than the main peak of ad libitum feeding, in agreement with the experimental results of [7, fig. 2F]. There is no such difference between the peaks of ad libitum feeding and time restricted feeding for an uncoupled system, again suggesting a lack of synchronization in uncoupled cells.

Remarkably, our simulations are in agreement with experimental results for mice without SCN [7], suggesting that small clusters of weakly coupled cells are important elements for achieving coherence and synchronization of peripheral clocks.

6. Discussion

This theoretical and computational study was motivated by recent experimental observations suggesting that intercellular coupling between circadian clocks occurs in peripheral organs such as the liver and contributes to maintain coherent oscillations in the absence of the SCN master clock. We constructed a network of coupled peripheral clocks to provide an early analysis of some aspects of the coupling dynamics.

6.1. Using a validated transcriptional model of the clock

Our model of the mammalian circadian clock was developed in previous work, calibrated with data from mouse fibroblasts, and validated on a number of applications (for instance, on the interactions between the mammalian clock and cell cycle). This model, with its validated set of physiological parameters, constituted a solid basis for our coupled network and its analysis, a clear advantage relative to generalist phase models used in many studies. Namely, this model allowed us to represent populations of cells with either longer or shorter periods, by tuning only one parameter (ɛ).

6.2. A partition of circadian cycle into six stages to characterize synchronization dynamics

To more closely analyse the coupling dynamics, we devised a partition of the circadian cycle based on the repression of BMAL1 and the common inactivation between the complexes CLOCK:BMAL1 and PER:CRY. In this way we obtain a circadian cycle with six stages, each representing one part of the circadian day and its length characterized by the system’s parameters. This partition was used to study the coupling process in a mixture of wild type and longer period Cry2^−/− cells. Comparison of the clock periods with or without coupling shows that the longer period Cry2^−/−-type cells shorten their periods in a coupled network, while wild type cells do not greatly change their periods, a result in very good agreement with the recent work of [8].

Our results highlight the fact that two coupled oscillators may be synchronized in period but have quite different duration for each stage (figure 9). Analysis of the main processes (figure 11) shows that, during stage 1 (circadian night), the coupling term is negligible, and the oscillators operate near to uncoupled mode, with the longer period oscillator naturally exhibiting a longer stage 1. Conversely, during stage 2 (circadian early day), the coupling is at its strongest level and succeeds in balancing the two periods by shortening the longer period oscillator.

6.3. Synchronization in a cluster of cells in response to time-restricted feeding

In the liver, communication between hepatocytes spreads to clusters of less than 10 neighbour cells, under weak coupling [22]. We used our model to represent these cell clusters as observed in liver lobules, and simulated both uncoupled and coupled clusters with high period variability (between 23 and 26.5 h), to study the average PER:CRY signal in response to either ad libitum or time-restricted feeding. Our simulations indicate that both timed food intake and intercellular coupling are determinant factors for the PER:CRY amplitude and period of oscillations (figure 13). Moreover, both factors are needed to reproduce the experimental observations, as shown by comparison of results with and without network coupling (figures 14a,b and 15). The model with network coupling reproduces more faithfully than the one without coupling the recent experiments showing that hepatocyte clocks still synchronize in mice devoid of a functional SCN. This comparison of model and data thus suggests that coherence and synchronization of peripheral clocks is possible thanks to coupling between clocks.

Based on a physiological model of peripheral circadian clocks, our theoretical and computational results are among the first to analyse peripheral clock coupling. Our modelling approach provides a dynamical view of the coupling mechanism and illustrates how the coordination between the different stages leads to synchronization of two peripheral oscillators. Our results also show that intercellular coupling of peripheral clocks is a major factor in obtaining a coherent signal in response to feeding schedules.

Appendix A

Table 2.

Equivalences between ɛ values and period of (uncoupled) oscillator 2.

ɛ	0.2	0.4	0.6	0.7	0.8	0.85	0.9	0.95
period (h)	27.2	27.2	26.7	26.3	25.8	25.5	25.2	24.8
ɛ	1.0	1.05	1.1	1.15	1.2	1.4	1.6	1.8
period (h)	24.5	24.1	23.8	23.4	23.0	21.4	19.6	17.9

Data accessibility

This paper uses a mathematical model (developed in a previous paper [9] and described in the text), and extends it to include new terms. Numerical simulations of this model generate the data analysed in the paper. All parameters and conditions are provided to allow for computational reproduction of the data.

Authors' contributions

O.B.: conceptualization, formal analysis, investigation, methodology, writing—original draft, writing—review and editing; M.T.: funding acquisition, investigation, resources, validation, writing—review and editing; F.D.: funding acquisition, investigation, resources, supervision, validation, writing—review and editing; M.C.: conceptualization, formal analysis, funding acquisition, investigation, methodology, resources, supervision, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

This work was supported in part by project UCA^JEDI ANR-15-IDEX-01 and the LABEX SIGNALIFE (ANR-11-LABX-0028-01) from the programme Investments for the Future of the French National Agency for Research and by project ICycle ANR-16-CE33-0016-01.