Combining lineage correlations and a small molecule inhibitor to detect circadian control of the cell cycle

Summary

Chronotherapy offers an exciting possibility for improving cancer treatments by leveraging the influence of the circadian clock on the cell cycle. While several molecular interactions coupling the two oscillators have been identified, whether they lead to emergent control of cellular proliferation remains unclear. Using stochastic simulations, we demonstrate that the established gene networks underlying the two oscillators are sufficient to generate lineage correlations in cell cycle times, as observed in single-cell microscopy data. The interactions also create a ‘therapeutic window’ between cancer and normal cell proliferation peaks that can be leveraged for chronotherapy. Surprisingly, our model predicts that KL001, a clock inhibitor, minimally affects population growth but significantly alters lineage correlations. Our results suggest that clock control of the cell cycle may not be detectable by measuring changes in population dynamics, but combining measurements of lineage correlations with KL001 treatment may provide a more sensitive approach to detecting the coupling.

Graphical abstract

Highlights

• •Lineage correlations help detect circadian clock mediated cell cycle control
• •Inhibiting circadian clock minimally affects growth rate but greatly changes correlations
• •Circadian clock control creates therapeutic window between cancer and healthy cells

Small molecule; Cell biology

Introduction

The idea that cell proliferation exhibits rhythmicity dates back to at least the 1940s, when Klein and Geisel reported $\sim$ 24 h oscillations in the mitotic rate of duodenal epithelium of mice and rats.¹ This remarkable observation suggested that the cell cycle in the alimentary tract of rodents was under the control of the circadian clock, leading to peaks and troughs over time in the fraction of cycling cells. A number of subsequent studies arrived at the same conclusion, though demonstrating that there could be a large variation in the amplitude and mean levels of the mitotic oscillations depending on the particular region of the alimentary canal under study.^2,3,4,5 Further studies extended these results to other tissues such as the skin, bone marrow, tongue and cornea, some of which showed persistent rhythms even under conditions of constant darkness, implying that the rhythmicity was being imposed by the endogenous circadian clock and not the external light-dark cycles.^6,7 Decades later, these largely physiological studies received strong support when the first molecular links between the circadian clock and cell cycle were discovered. Transcription of several important genes responsible for cell cycle progression —p53,⁸ c-myc,⁹ and WEE1¹⁰ — were found to be directly under the control of circadian clock proteins, via interactions with E-box regions in their promoters. It was also demonstrated, at least in liver hepatocytes, that the control was unidirectional —the clock regulated cell proliferation and not the other way around.¹⁰ Eventually, a number of other molecular connections between the two cellular oscillators were also discovered, primarily acting on the G1-S and G2-M transitions of the cell cycle.^11,12,13 Taken together these studies provided strong evidence that mitotic oscillations exist, allowing for the exciting prospect of chronotherapy in diseases like cancer, where administering drugs during the peak of fast cycling cancer cells could potentially maximize their eradication.^{14,15,16,17,18,19,20}

However, the idea of oscillations in cell proliferation and timing therapy to hit the mitotic peaks has consistently been tempered by studies that have failed to find rhythms in cell proliferation. Since the early study on albino rats by Leblond and Stevens that reported a constant renewal of the intestinal epithelium,²¹ a number of subsequent studies similarly concluded that rhythms in cell proliferation are absent.^22,23 While this earlier debate was based primarily on measurements of bulk datasets (averages over millions of cells), more recent studies using single cells and time-lapse microscopy have also thrown up contradictory results.¹³ Indeed, one such study using a fluorescent reporter of the circadian clock in mouse fibroblasts (NIH3T3cells) arrived at the conclusion that it is the cell cycle that exerts a strong control over the circadian clock,²⁴ not the other way around as has been conventionally believed^13,25 and also concluded from recent theoretical studies.²⁶ This demonstrates the need for developing more accurate approaches to detect the presence of and resolve the directionality of clock-cell cycle coupling, ideally using methods that are robust to the details of the underlying techniques (for example, endogenous versus transgenic reporter systems, kinetics of fluorophore maturation,²⁷ and strong assumptions underlying mathematical models).

A potentially exciting method to detect oscillatory control of the cell cycle was recently suggested,^28,29 based upon an observation made in a series of recent studies —many cell types exhibit higher correlations in cell cycle times among cousins as compared to mother-daughter cell pairs (the so-called ‘cousin-mother inequality’).^28,29,30 The origin of this puzzling phenomenon was suggested to be an oscillatory control over cell cycle speed,^28,29 and the oscillation time-period that best recapitulated measured correlations and cell cycle time distributions was around 24 h.³⁰ The natural candidate for such an oscillatory driver of the cell cycle was the circadian clock, and this expectation was further strengthened when deletion of the clock in bacteria abrogated the lineage correlation structure.²⁹ Investigating the presence of the cousin-mother inequality was therefore suggested as a potential approach to detecting clock control over the cell cycle.²⁹ However, the mathematical models originally used to explain the correlations were phenomenological in nature;^28,29,30 hence, it remains unclear whether the established gene networks underlying the circadian clock and cell cycle oscillators, along with their known molecular couplings, can be expected to generate the experimentally observed correlation structures. Investigating the emergence of lineage correlations from the gene networks becomes especially relevant since noise in the underlying network dynamics can potentially mask the cousin-mother inequality²⁹ and also affect the correlation structure in general.³¹ Furthermore, recent theoretical studies have shown that oscillatory control over the cell cycle is not the only way to obtain such correlation structures.^26,32 Therefore, simply the presence or absence of the cousin-mother inequality is unlikely to be sufficient to establish the presence of cell cycle control by the circadian clock.

To investigate approaches that can overcome the above-mentioned challenges, here, we develop stochastic models to study the emergence of lineage correlations from the known underlying gene networks of the coupled clock-cell cycle oscillators. Our results demonstrate that in a model calibrated to recapitulate measured noise in HCT116 cells, the ‘forward’ coupling between the clock proteins BMAL1-CLOCK and the cell cycle gene WEE1 is sufficient to generate the cousin-mother inequality. Additionally, by incorporating a ‘reverse’ coupling (i.e., inhibition of the circadian clock gene Rev-erb-$\alpha$ by the CycB/CDK1 cell cycle proteins), we provide the crucial check that clock control by the cell cycle cannot explain the observed lineage correlations. Finally, we model the action of the small molecule inhibitor KL001 which stabilizes the CRY protein, increasing the circadian clock time-period and decreasing the amplitude in a concentration dependent manner.³³ Simulating the effect of KL001 on the clock, we interestingly predict a large percentage decrease in the cousin-mother inequality, as opposed to the proliferation rate which exhibits a much smaller and potentially undetectable decrease. More generally, we demonstrate that even if the molecular links from clock to the cell cycle are not sufficient to significantly impact the proliferation rate, robust phase differences between the proliferation peaks of fast versus slow cycling cells are still induced, generating clear ‘therapeutic windows’. Our results therefore suggest that combining KL001 along with measurements of lineage correlations, as opposed to measuring changes in cell proliferation rates, can be a sensitive approach to detecting circadian clock control over the cell cycle.

Results

Stochastic biochemical models for the coupled circadian clock – Cell cycle gene network in single cells

To explore the possibility of lineage correlations arising from the biochemical networks underlying the circadian clock and the cell cycle, we use three models for the coupled oscillators —Model 1, Model 2, and Model 3— which vary in the number of molecular components involved. In Model 1, the circadian clock network was inspired by a more detailed model,^34,35 incorporating the negative auto-regulatory loops involving the PER and CRY genes, whose protein products upon translation and translocation back to the nucleus inhibit their own transcription. The more complex Model 2 is schematically shown in Figure 1A, which includes an additional secondary loop where the oscillatory dynamics of BMAL1 protein is mediated by REV-ERB$\alpha$. BMAL1-CLOCK heterodimer activates the transcription of Rev-erb$\alpha$ mRNA, whose protein product is responsible for inhibiting.

Figure 1.

An overview of the coupled circadian clock - cell cycle gene network and the single cell lineage simulation

(A) The coupled circadian clock - cell cycle gene network (Model 2) includes the transcriptional-translational feedback loops that generate the circadian oscillations. The circadian clock gene network shown in the right panel comprises of the auto-regulatory PER-CRY loop along with the REV-ERV $\alpha$ mediated oscillatory dynamics of BMAL1. In order to depict the nuclear-cytoplasmic translocation processes we have explicitly separated the nucleus and the cytoplasm in the schematic. The cell cycle as shown in the left panel is divided into three phases- G1, S/G2 and M. Mitosis Promoting Factor (MPF) promotes progression through the phases. Forward and Reverse coupling between the two oscillators are via BMAL1-CLOCK control over WEE1 and MPF mediated inhibition of REV-ERB $\alpha$ respectively.

(B) The workflow of our lineage simulation and its output. Left: Algorithm. Middle: A lineage constructed by the simulation where each cell is named as a combination of the lineage it belongs to and the cell number within that lineage at the time of its birth. Right: Time evolution of different proteins for a lineage, recapitulating oscillations with correct phase relationships. The cells belonging to a lineage that were tracked to generate these oscillatory trajectories are labeled at the time of their birth on the graph.

Bmal1 transcription thereby decreasing active BMAL1-CLOCK concentration as shown in Figure 1B. When the inhibition is lifted due to decreasing concentration of REV-ERB$\alpha$, BMAL1-CLOCK concentration increases thereby restarting the oscillation. Both these models lead to 24-h oscillations of the mRNA and protein concentrations as shown in Figures 1B and S2. All figures in the main text correspond to results from Model 2 simulations.

The cell cycle model used in Models 1 and 2 was previously suggested in the literature.³⁵ The mammalian cell cycle is dependent on the activity of multiple Cyclin and Cyclin-Dependent Kinase(CDK) complexes. However, in this simplified network, the authors considered the progression through the cell division process to be dependent on the activity of only CyclinB-CDK1 complex or the mitosis promoting factor (MPF). The circadian control of the cell cycle or forward coupling was incorporated via the control of inhibitory kinase WEE1 by heterodimer BMAL1-CLOCK. As shown in Figure 1A, WEE1 inhibits MPF, keeping its concentration low for the beginning of the cell cycle, till MPF activity rises above WEE1, at which point cell enters the M-phase and ultimately divides. The effect of the cell cycle on the circadian clock or reverse coupling (only present in Model 2) was incorporated via the CDK1 mediated inhibition of REV-ERB$\alpha$.³⁶ Since we did not explicitly have CDK1 in the cell cycle model, reverse coupling was introduced via REV-ERB$\alpha$ inhibition by MPF or CycB/CDK1 complex. The strength of forward and reverse coupling is mediated by changing $C1$ and $C2$ values, respectively, in the set of ODEs mentioned in SI.

The cell cycle in Models 1 and 2 is driven by a single cyclin/Cdk complex, but it is well established that cell cycle regulation is more intricate. To simulate a more realistic scenario, we also incorporated a more complex cell cycle network in Model 3. Here, there is a combination of four Cyclin/Cdk complexes that regulate the progression of cells through the cell cycle based on their concentrations. This enhanced model allowed us to explore additional molecular interactions between the circadian clock and the cell cycle. Therefore, in addition to BMAL1-CLOCK activation of WEE1 kinase, we also investigated REV-ERB $\alpha$-mediated inhibition of the Cdk inhibitor p21. While the WEE1-mediated forward coupling delays cell cycle progression, REV-ERB$\alpha$-mediated control of p21 accelerates it.

A flowchart of the model simulations is presented in Figure S1 and detailed equations and parameter values are provided in STAR Methods and Sections S1–S3. We converted the above reaction schemes into stochastic differential equations and incorporated them into single cell lineage simulations that mimic the cell division process as demonstrated in Figure 1B (see Sections S1–S5 for details of the equations and the numerical integration method). In the simulations, the cell cycle was divided into three phases - G1, S/G2 and M. Transition through the various phases was based on the concentration of MPF, and the ones that ultimately reached the M-phase divided to give two daughter cells. Cellular lineages were represented using directed graphs, where the relationship between the cells was preserved, to enable inference of lineage-relationships between cell pairs for downstream analysis (details in Section S7). As shown in Figure 1B, the concentrations over time of different proteins for an extracted lineage recapitulated 24 h oscillations, where the anti-phase expression of BMAL1/CLOCK complex and the PER/CRY complex could be observed as expected from previous experiments.³⁷

Circadian clock mediated forward coupling recapitulates experimentally observed correlation structures in cellular lineages

With the models for the two oscillators and correct phase relationships between the trajectories of MPF, BMAL-CLOCK, and WEE1 established, we next asked whether our models exhibit entrainment, as expected from the theory of coupled oscillators. Keeping the circadian clock period ($\sim$ 24 h) and the autonomous cell cycle period ($\sim$ 20 h) fixed, we either increased the forward or the reverse coupling strength. We simulated the time evolution of the gene network and took Fourier transforms to identify the frequency of oscillations of both oscillators (see Section S6 for details). As shown in Figure 2A, the frequency of oscillation of the cell cycle protein MPF changes from 1/20 ${\text{hour}}^{-1}$ when coupling is 0–1/24 ${\text{hour}}^{-1}$ in presence of high forward coupling, suggesting entrainment of the cell cycle by the circadian clock. Similarly, in Figure 2B, the frequency of BMAL1/CLOCK oscillation shifts to 1/20 from 1/24 ${\text{hour}}^{-1}$ with the increase in reverse coupling strength, reflecting entrainment of the circadian clock by the cell cycle.

Figure 2.

Circadian clock control over the cell cycle via BMAL1-CLOCK interaction with WEE1 is sufficient to generate the cousin-mother inequality

(A) Entrainment of the cell cycle by the circadian clock. With increasing forward coupling strength, the frequency of oscillation of the cell cycle gene MPF changes from 1/20 to 1/24 ${\text{hour}}^{-1}$.

(B) Entrainment of circadian clock by the cell cycle. With increasing reverse coupling strength, the circadian clock heterodimer BMAL1/CLOCK oscillation frequency changes from 1/24 to 1/20 ${\text{hour}}^{-1}$. Circadian clock period (TCR) = 24 h, Autonomous cell cycle period (TCC) = 20 h.

(C) Comparison of IMT distribution from simulation and experiment. The simulated IMT distribution (with coupling strength 0.55) approximately matches the experimentally observed IMT distribution (black dashed line).

(D) Increasing forward coupling strength generates high sister correlations and also gives rise to the cousin-mother inequality. In contrast, reverse coupled system fails to recapitulate the cousin-mother inequality. Boxplots were generated from 100 runs of the simulations.

(E) Percentage change in the difference between median values of cousins and mother-daughter correlation in comparison to the uncoupled system, shown for the different coupling strengths. Numbers in the plot refer to the coupling strength either for forward (red) or reverse (blue) coupling.

(F) Schematic to demonstrate the lineage relationship considered with respect to the reference cell marked in red.

(G) Comparison of the IMT correlation down a vertical generation for the different forward coupling strengths. We observe the negative grandmother correlation (green) as seen in the experiments on HCT116 cells.

(H) Effect of different autonomous cell cycle periods on the cousin-mother inequality. Our model predicts WEE1 mediated forward coupling gives rise to cousin mother inequality when the average cell cycle period is $\sim$ 12 to 24 h.

Having demonstrated that our coupled oscillator model generates the expected entrainment behavior, we next asked if the forward and reverse coupling can generate lineage related intermitotic time (IMT) correlation structures observed in previous experiments.³⁰ In particular, we and others have shown that cousin cells often exhibit stronger IMT correlations than mother-daughter pairs (the cousin-mother inequality), and that this surprising phenomenon may arise from circadian gating of the cell cycle.^28,29,30 To study the origin of the cousin-mother inequality, we utilized our previously discussed lineage simulation that creates ancestor cells’ lineages based on the stochastic gene network provided as input. Since noise in the cell cycle times can affect the correlations,³⁰ we determined the optimum noise level and coupling strength for our model by comparing the IMT distribution generated by our simulation to that observed experimentally in the HCT116 cell line.³⁰ As shown in the blue distribution in Figure 2C and in Figure S3, we were able to approximately reproduce the experimentally observed IMT distribution both for a forward and reverse coupled gene network respectively. We then extracted related cell pairs from our simulations, and compared their cell cycle time correlations for both forward and reverse coupled cases separately. We observed the cousin-mother inequality emerge upon increasing the forward coupling strength, showing a significant increase in the cousin correlations at the coupling strength of 0.55 where the IMT distribution matched experimental results (Figures 2D and 2E). In contrast, in the reverse coupled gene network where the cell cycle influences the circadian clock, the cousin-mother inequality was not observed even for a high reverse coupling strength (Figures 2D and 2E). Additionally, to check whether having different levels of noise for the clock and cell cycle networks affects the observation of the cousin-mother inequality, we ran a set of simulations with varying degrees of noise strength for the circadian clock components. We found that the inequality was always observed (Figure S3). Since these simulations were performed with constant noise, we also used state-dependent noise in the lineage simulations for both Model 2 and Model 3. As shown in Figures S3 and S4, consistent with our previous results, the cousin-mother inequality appeared only in the forward-coupled scenario.

We also extracted the vertical generation (grandmother, great-grandmother, etc) correlations in the cell cycle times (Figures 2F and 2G). As was observed in the experiment,^26,30 the grandmother correlation was negative, thereby suggesting that our model results are consistent with the overall lineage correlation structure measured using time lapse imaging of HCT116 cells.

Finally, we also explored the effect of changing the autonomous cell cycle time on the lineage correlations. Since mammalian cells can have both shorter ($\sim 12$ hours for stem cells) as well as larger ($\sim 24$ hours for fibroblasts) average cell cycle times, we varied the mean of the IMT distribution within $12-30$ hours. The results of these simulations (Figure 2H) predict that the cousin-mother inequality will be stronger at lower IMT values, and is lost closer to average cell cycle times of $\sim 24-30$ hours. Interestingly, this is seemingly consistent with measurements on mouse fibroblasts (average division time $\sim 24$ hours) where the inequality was not observed,³⁸ and HCT116, L1210 cells (average division time of both cell types $\sim 16$ hours) where a strong cousin-mother inequality was observed.^28,30 Note, however, that whether a forward coupling exists in NIH3T3cells, is currently debated and further reports have shown that the circadian clock strength is also weaker in NIH3T3.^24,39,40

Therefore, in summary, our results in this section suggest that circadian clock control over the cell cycle (via BMAL1-CLOCK interaction with Wee1 or REV-ERB$\alpha$ interaction with p21) might give rise to the cousin-mother inequality, but only when the average autonomous cell cycle time is approximately within $12-24$ hours.

KL001 mediated inhibition of the circadian clock leads to a large decrease in the cousin-mother inequality

Though our results from the last section demonstrated that the molecular coupling between BMAL1-CLOCK and WEE1 can generate the cousin-mother inequality, observation of this inequality in a dataset does not necessarily prove the existence of circadian clock – cell cycle coupling. Recent studies suggested that imposing a cell size dependent regulation of cell cycle speed, can also give rise to similar IMT correlations.^26,32 Therefore, we next asked if perturbations to the circadian clock, and hence the lineage correlations, could potentially be a way to probe the existence of forward coupling between the clock and the cell cycle. Selectively modulating the circadian clock can be achieved experimentally using a multitude of small molecules, such as KL001 and SR9009. KL001, for example, stabilizes the CRY protein by preventing its ubiquitin mediated degradation leading to damped oscillations with decreased amplitude and increased time period.³³ A particularly attractive feature of KL001 is that its effects are concentration dependent.³³ We, therefore, modeled KL001 in our lineage simulations, to study its effects on the correlation structure in cell cycle times.

Since we did not explicitly have CRY proteins in our model, we mimicked the effect of KL001 by decreasing the degradation rates of the PER-CRY complex, which leads to damped circadian oscillations as shown in Figure 3A, similar to ones experimentally observed in.³³ We divided the degradation rates of the PER-CRY complex by the different numbers indicated in Figure 3D to simulate addition of increasing concentration of KL001. Thus, the uninhibited or control system is depicted when KL001 value is equal to 1 (see details in Section S8). To carefully compare and contrast the experimental results with our simulations, we digitally extracted KL001 time-series datasets from³³ (Figures 3B and 3C), and computed the time periods and amplitudes of the oscillations as a function of KL001 concentration (Figures 3B and 3C; see details in Section S9). The experimental data showed an abrupt jump in the time period on increasing KL001 concentration, which was well captured by our model (Figures 3B and 3D). The amplitude on the other hand showed a more continuous decrease, which was also recapitulated in our simulations (Figures 3C and 3E). However, two aspects of the experimental data did not precisely match our simulations: (1) a remnant oscillation even at very high KL001 concentrations, resulting in a non-monotonic behavior of the measured time period (Figure 3B), and (2) in our simulations, the amplitude of the clock oscillations decays earlier than the increase in time period upon KL001 treatment, while this is not the case in the experiments. Discrepancy (1) is unlikely to affect our results, since at the high concentrations of KL001 the amplitude of oscillations was essentially zero (Figures 3C and 3E). For discrepancy (2), the main result of the earlier decay of amplitude (with relatively fixed time period) in our simulation is the non-monotonic decrease of the coefficient of variation and lineage correlations (Figures 3G–3I). Because the cousin-mother inequality is strongly determined by the oscillatory nature of the clock-cell cycle coupling, an initial decrease in clock amplitude without changes to the time period results in no decrease in the CMI (Figure 3I). Once again, however, the high KL001 concentration results do not get affected. Overall, our KL001 model therefore largely captures the observed effects of the inhibitor on clock oscillations.

Figure 3.

Circadian clock perturbation with KL001 decreases IMT correlations in related cell pairs

(A) Damped circadian clock oscillations as observed with increasing KL001 inhibitor concentration in our Model 2 simulations.

(B and C) Oscillation time period and amplitude respectively as functions of increasing KL001 concentration. Time series datasets were digitally extracted from experimental results in³³ and analyzed as described in STAR Methods and Section S9.

(D and E) Oscillation time period and amplitude respectively as functions of increasing KL001 concentration from our Model 2 simulations.

(F) Change in inter-mitotic time distribution as observed with increasing KL001 concentration. The model predicts a decrease in variance of the distribution in comparison to the experimental observation (black dashed line).

(G) Decrease in the coefficient of variation of Inter-mitotic time distribution.

(H) Increasing concentration of circadian clock inhibitor KL001 diminishes the cousin-mother inequality.

(I) Percentage change in median value of cousin-mother inequality in comparison to the control case, shown for the different concentrations of KL001. (Boxplot and median value calculated for 100 runs of simulation. Here, C1 = 0.55, C2 = 0, TCC = 15 hrs). All plots were generated using simulations of Model 2.

With the model for KL001 established, we next asked how increasing KL001 concentration (by reducing the degradation rate of PER-CRY complex) would affect the lineage correlations. Simulating lineages with circadian clock inhibition by KL001 showed a decrease in variability of distribution of cell-cycle times (Figure 3F), which was further quantified by comparing the coefficient of variation (COV) of the IMT distribution for 100 runs of the simulation as shown in Figure 3G. Similar observations were previously reported in Cyanobacteria, where time lapse experiments performed on wild-type and clock deleted mutants revealed decreased cell cycle duration variability in case of the mutant.²⁹ This reduction in variance was attributed to the loss of clock oscillations in the mutant, which otherwise generates larger variability in cell cycle times. Further, we observe that simulating lineages keeping the forward coupling strength high, with increasing circadian inhibition causes the related cell correlation to decrease at highest inhibitor concentration. When comparing the percentage difference in the median value of cousins and the mother-daughter correlations for the inhibited systems (KL001 = 2 to 1000 represents increasing concentration) in comparison to the control system (KL001 = 1), the cousin-mother inequality diminishes by $\sim$ 50% (Figures 3H and 3I).

In summary, our results suggest that the effect of the small molecule clock inhibitor KL001 on clock oscillations can be recapitulated using a relatively simple network model of the circadian clock. Furthermore, addition of KL001 at high concentrations is expected to significantly reduce the cousin-mother inequality in cell cycle time correlations, that should be detectable using live cell microscopy techniques.

Population growth rates are relatively unaffected by KL001-induced clock inhibition

A natural, and in principle simpler, alternative to measuring KL001-induced changes in the lineage correlations would be to measure cell proliferation rate changes. Inhibiting the clock oscillations, and thereby the coupling to the cell cycle, would be expected to also change the proliferation rate.³⁵ As we demonstrated earlier in Figures 2A and 2B, entrainment causes the average cell cycle time to shift closer to the clock period, hence, one might expect KL001 treatment to revert the clock-driven proliferation rate back to the autonomous (no coupling) scenario. If this were to be the case, single-cell lineage tracking experiments would not be required, and simply measuring population dynamics in bulk assays would provide an easier approach to detecting clock control of the cell cycle.

Using our lineage simulation, we generated trajectories of cell population as a function of time as shown in Figure 4A. We fit a linear model to the log(cell number) versus time data, to estimate the proliferation rate of the population. First, we checked our model results for increasing forward coupling strength, and found that the proliferation rate decreased as expected due to entrainment (Figure 4B). However, under circadian clock inhibition with KL001, the proliferation rate interestingly showed minimal increase of $\sim$ 2% (Figure 4C). This result can be explained by observing that the mean of the cell cycle times distribution does not change with increasing KL001 concentration (Figure 3F). This happens because even after circadian oscillations are dampened, a non-zero coupling exists, maintaining the same mean proliferation rate across various doses of KL001 as shown in Figure 4D. Coupling persists even after clock inhibition by KL001 because the concentrations of clock proteins do not drop to zero but instead stabilize at specific non-zero levels. This behavior was observed in the experiment and recapitulated in our simulations (Figure 3).

Figure 4.

Cell proliferation rates are predicted to remain relatively unaffected by circadian clock inhibition using KL001

(A) Representative exponential trajectories of number of cells in a population as a function of time for different runs of the simulation.

(B) Decreasing proliferation rate observed for increasing forward coupling strength owing to entrainment of the cell cycle by the circadian clock. We plot the percentage change in the median proliferation rate relative to the uncoupled system i.e., $C1$ = 0.

(C) The cell proliferation rate remains unaffected with increasing KL001 mediated circadian clock inhibition. Here, the proliferation rate is plotted as percentage change relative to the uninhibited system depicted by KL001 = 1. (Boxplot and median value calculated for 100 runs of simulation. Here, C1 = 0.55, C2 = 0, TCC = 15 h).

(D) The circadian clock remains coupled to the cell cycle across various KL001 dose levels, hence maintaining an almost unchanged mean cell cycle time and hence proliferation rate. All plots were generated using simulations of Model 2.

Our results in this section predict that the proliferation rate increase upon adding KL001 to proliferating cells will be minimal, and likely undetectable within experimental resolution. Changes in the lineage correlations are much larger as we found in the previous section, and hence will likely be a more powerful approach to detecting circadian control over the cell cycle.

The circadian clock induces phase shifts between proliferation peaks of cells with different autonomous cell cycle periods

Given the relatively small impact predicted for the clock’s oscillatory control on cell proliferation rates at the population level, we then asked what other effects the coupling could have on cell proliferation that might be harnessed for chronotherapy. Previous work using phenomenological models have suggested that clock coupling to the cell cycle of cancer versus normal cells can induce a ‘therapeutic window’ in proliferation peaks.⁴¹ To investigate whether our molecular models can also generate such therapeutic windows even in the absence of significant changes to the population dynamics, we examined the distribution of cells in various cell cycle phases in the presence and absence of circadian clock coupling.

Using our lineage simulation of Model 2, we generated deep lineages for 100 asynchronous ancestral cells and tracked the proportion of cells in different stages of the cell cycle. As shown in Figure 5A, only when coupled to the circadian clock do the proportions of cells in the M phase exhibit 24-h oscillations (all other phases exhibit the same behavior upon coupling as well). In the absence of circadian clock coupling, the distribution of cells across cell cycle phases can be predicted by ergodic rate analysis, which posits that the fraction of cells in each phase is proportional to the duration of that phase, with an enrichment of G0 cells due to the production of two daughter cells of age 0 from a single mother.⁴² As shown in Figure 5A, in the uncoupled system the predicted and simulated proportions align exactly (red dotted line), while in the coupled system, the oscillations occur around the predicted mean (blue dotted line). We further tested whether the ergodic principle holds across different cell cycle phases by dividing the cell cycle into G1 and S/G2/M phases. For both cases, we observed that the oscillations centered around the predicted mean proportion, as shown in Figure 5B (details in Section S10).

Figure 5.

Circadian clock mediated forward coupling induces phase shifts between proliferation peaks of cells with different autonomous cell cycle periods

(A) Emergence of 24-h oscillations in the proportion of cells in M phase of the cell cycle when circadian clock mediated forward coupling is present. The dotted lines (blue and red) denote the proportion of cells in M phase as predicted by ergodic theory.

(B) Oscillatory behavior in the proportion of cells in G1 and S/G2/M phases of the cell cycle in the forward coupled scenario ($C1$ = 0.55). These oscillations are centered around the mean proportion as predicted by ergodic principle (red and blue dotted lines).

(C) Phase shifted oscillations in the M-phase proportion when different autonomous cell cycle periods were considered. Interestingly, the time period of oscillation for all three curves is the same, set by the circadian clock time period. All curves were generated with Model 2 and in each run the forward coupling strength $C1$ was maintained at 0.55.

To explore the effect of changing the underlying autonomous cell cycle period, we ran simulations with different autonomous periods: 12 h, 20 h, and 24 h, as shown in Figure 5C. All cell cycle phases exhibit similar oscillations, but only the M phase is shown in Figure 5C. Interestingly, in all three cases, the time period of oscillation of the phase proportion was the same – 24 h, as set by the circadian clock time period. This was true even for the 12-h autonomous cell cycle case, where the coupling was not strong enough to shift the average cell cycle time to 24 h. However, there were clear phase differences established between the peaks of the oscillations, creating a ‘therapeutic window’ for preferentially targeting one cell type over another. For instance, if the faster and slower cycling cells are associated with cancer and normal cells respectively, the clear separation of the times at which their proliferations peak (Figure 5C) opens up the possibility of maximizing cancer cell elimination while minimizing toxicity due to normal cell death.

The results of this section thus demonstrate how circadian clock-regulated cell cycle control can create optimal therapeutic windows in cancer treatment. Our model suggests that even if the clock coupling is not strong enough to significantly change population proliferation rates, robust phase differences can be induced in the proliferation peaks of fast versus slow cycling cells by the underlying gene regulatory networks.

Discussion

One of the key ideas behind cancer chronotherapy is administration of treatments at a time of day when cancer cell proliferation is at its peak.⁴³ The important assumption underlying this idea is that circadian clock control over the cell cycle will lead to oscillations over time in the fraction of cells in any cell cycle phase.^18,41 However, even though substantial evidence exists for a molecular level coupling of the clock and the cell cycle, whether there is any emergent control of cell proliferation by the clock remains a heavily debated subject. The question of emergent control is particularly relevant since the circadian clock oscillations are known to be low-amplitude and noisy, and therefore molecular interactions with the cell cycle are not guaranteed to result in strong control of proliferation. The key unanswered questions, therefore, are whether the underlying microscopic dynamics at the level of the proteins and mRNA in single cells is sufficient to generate cell cycle control by the oscillating circadian clock, and how to establish sensitive approaches to demonstrate evidence for such control, if it exists.

Here, we use stochastic, mechanistic models to predict that some of the best studied molecular connections –BMAL1-CLOCK driving of the cell cycle via the G2-M inhibitor WEE1, and REV-ERB$\alpha$ mediated inhibition of Cdk inhibitor protein p21– are likely to be sufficient to exert detectable control over cellular proliferation as measured by lineage correlations in cell cycle times. By carefully calibrating the noise and coupling coefficients, we ensure that our models generate distributions of cell cycle times that are similar to those measured experimentally in HCT116 cells. Given that these distributions can strongly affect the lineage correlations,^29,30,31,38 our results demonstrate that realistic noisy dynamics of mRNA and proteins can still generate the cousin-mother inequality in cell cycle time correlations. Perhaps, more importantly, we show that our simulated KL001 treatment, which recapitulates the experimentally observed abrupt increase in clock time period and more continuous reduction in amplitude, gives rise to a large reduction $\left(>50\%\right)$ in the cousin-mother inequality. Interestingly, such a large change is not observed in the population growth rate of the cells, since even very high KL001 concentrations only flatten out the PER-CRY/BMAL1-CLOCK oscillations without reducing their absolute levels to zero (thereby maintaining a steady coupling to the cell cycle). Our results, therefore, suggest that measuring KL001 mediated reduction in the cousin-mother inequality, but not the population growth rates, could be a sensitive readout for clock control over the cell cycle. Importantly, we also demonstrate that even if the molecular links generate weak control over the cell cycle such that proliferation rates are not strongly affected, robust phase differences in proliferation peaks between fast and slow cycling cells will be established (Figure 5C). Hence, even a weak coupling could potentially be utilized for timing therapy administration to minimize normal cell death.

The advantage of our proposed method is that it relies on two relatively simple experiments requiring just the ability to track cell divisions – (1) measuring lineage correlations of cell cycle times in vitro and (2) repeating the same measurements in the presence of the small molecule inhibitor of the clock, KL001. The method also requires no addition of synchronizing agents such as dexamethasone, but rests on the assumption that the clock phase gets inherited by daughter cells at time of division, thereby maintaining some degree of phase synchrony within lineages.³⁰ However, we anticipate potential limitations that might render this approach applicable only in limited cell types. For conclusively determining clock control of the cell cycle, our approach requires positive results from both the above experiments —a strong cousin-mother inequality to begin with, and subsequently a large decrease in the inequality upon KL001 addition. A negative result in either experiment is likely to be hard to interpret: the absence of the inequality may arise from particular combinations of cell cycle and clock periods (as we show in Figure 2H) or even simply due to masking by noise.²⁹ Maintenance of the inequality even in the presence of KL001 might indicate the presence of other stronger hidden variables (such as cell size^26,32) that couple to the cell cycle and mask the effect of clock-generated correlations.

Limitations of the study

There are a few limitations with our modeling approach —as with any mechanistic model based on biomolecular reactions; the results may be somewhat dependent on the precise network connections chosen. For example, while the BMAL1-CLOCK connection to WEE1 is best characterized, other connections between the clock and cell cycle also exist and have been computationally modeled, which we have ignored.⁴⁴ Additionally, our models cannot recapitulate the absolute values of the lineage correlations as seen in experiments, but rather explain only the qualitative structure of the cousin-mother inequality. This is expected, since it is well understood that a wide variety of inherited components from mother to daughter cells such as mRNA/protein levels, transcription rate, and epigenetic states could all contribute to setting the absolute values of the correlations.^30,31,45,46 However, since our method conceptually rests upon identifying the qualitative inequality in the correlations, our inability to reproduce the absolute correlation values is unlikely to be a major limitation. Finally, as is often done, we also use the Michaelis-Menten form of enzyme kinetics in our stochastic models, which is not entirely accurate since one should ideally use a Gillespie algorithm to simulate the elementary reactions.⁴⁷

Conclusions

Recent discoveries of small molecule inhibitors and activators of the circadian clock provide the exciting possibility of manipulating cellular processes coupled to the clock. The cell cycle is one such important potential target, that might be indirectly controlled via perturbations of the clock for chronotherapeutic purposes. Using stochastic computational models, we suggest combining measurements of cell cycle time correlations on lineages with the clock inhibitor KL001, to detect whether the clock controls cell proliferation – a question that remains heavily debated in the field. Even weak coupling that can be detected via changes in lineage correlations but not proliferation rate, can induce a phase shift between proliferating cancer versus normal cells. Whether our predictions hold and the conclusions are generalizable to a variety of cell types remains to be demonstrated with carefully executed future experiments.

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Dr. Shaon Chakrabarti (shaon@ncbs.res.in).

Materials availability

This study did not generate any new reagents.

Data and code availability

• •Publicly available dataset analyzed in the paper is listed in key resources table.
• •All codes used in the manuscript are available at: https://github.com/Shaonlab.
• •Any additional information required is available from the lead contact upon request.

Acknowledgments

S.C. acknowledges funding from SERB (Government of India) under project number SPR/2021/000486 as well as intramural funds from National Center for Biological Sciences–Tata Institute of Fundamental Research (NCBS-TIFR).

Author contributions

S.C. conceptualized and designed the study; A.N. developed the models and performed the computations, calculations, and analyses to study the origin of lineage correlation and population growth rates, A.S. performed the computations and calculations to study the emergence of therapeutic window; A.N. and S.C. wrote the paper with inputs from A.S.; and S.C. supervised the work.

Declaration of interests

The authors declare no conflicts of interest.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
Analyzed dataset	Tsuyoshi Hirota et al.	N/A
Software and algorithms
Lineage Simulations	This paper	https://github.com/Shaonlab
Analysis codes	This paper	https://github.com/Shaonlab

Method details

Models 1, 2 and 3 for the circadian clock and cell cycle gene networks

We used three models of the coupled circadian clock - cell cycle networks to investigate the emergence of cell cycle control. Model 1 includes the negative auto-regulatory PER-CRY feedback loop for the circadian clock, a simplified cell cycle model driven by MPF, and a link between the two oscillators via BMAL1-CLOCK activation of WEE1 kinase.³⁵ To investigate effects of a reverse coupling, i.e., the impact of cell cycle control on the circadian clock and lineage correlations, we expanded the circadian clock network in Model 1 by incorporating the REV-ERB$\alpha$ loop, enabling MPF-mediated inhibition of the circadian protein REV-ERB$\alpha$. This expanded framework is referred to as Model 2 in which the forward coupling is via BMAL1-CLOCK activation of WEE1 kinase and the reverse coupling is via MPF (cell cycle protein) inhibition of REV-ERB$\alpha$. Recognizing the complexity of mammalian cell cycle regulation, we also tested our findings using a more realistic cell cycle model, Model 3, driven by four Cyclin/Cdk complexes. In Model 3 we investigated two separate molecular links by which circadian clock controls the cell cycle – (a) BMAL1-CLOCK activation of WEE1 kinase and (b) REV-ERB$\alpha$ mediated inhibition of Cdk inhibitor protein p21. The models are set up using Ordinary Differential Equations(ODEs). Owing to the inherent stochasticity of gene expression in biological systems we also implement the Chemical Langevin Equation (CLE) framework to convert these ODEs into Stochastic Differential equations (SDEs) and integrate them using the Euler-Maruyama scheme.⁴⁸ Details of the models and the parameter values can be found in Sections S1–S3, while the integration schemes are explained in Sections S4 and S5.

Lineage simulations to extract correlations in cell cycle times

To study the effect of coupled gene network on the Inter-Mitotic times of related cells, we developed a lineage simulation framework where the clock-cell cycle networks are simulated for individual cells, cell division is driven by the levels of cell cycle proteins, and finally inheritance of all cellular components is symmetric at time of division (details in Section S7). The IGRAPH package in R was used to represent the lineage as a directed graph.^49,50 We initialise a population of ancestral cells and assign the concentration of the different molecular species to each of them. Numerically integrating the system of SDEs drives the cells through the cell cycle. Once the mother cells reach the M-phase, we introduce two daughter cells that have the same concentrations as their mother, and remove the mother cell from the population. The simulation runs until a user-defined population size is reached. Once we have the lineage graph we create unique related cell pairs and study the IMT correlation between them.

Modeling the effect of KL001 on the circadian clock

We incorporate the effect of a small molecule inhibitor of the circadian clock, KL001, which prevents ubiquitin-mediated inhibition of CRY proteins thereby stabilising their concentration. This leads to loss of oscillations in concentration of circadian clock genes.³³ Since we do not explicitly have CRY proteins in our gene network, we simulate KL001’s effect by decreasing the degradation rate of the PER/CRY complex in our model (see Section S8 for details). To model increasing concentration of KL001, we divide the PER/CRY degradation rate by a numeric factor; the factor is set to 1 for the control case where no KL001 is present.

Quantification and statistical analysis

We computed the Pearson’s correlation coefficients between the Inter-mitotic times for different pairs of related cells – Sisters, Cousins and Mother-daughter pairs.

Published: March 22, 2025