Timing of circadian clock regulatory inputs controls duration of activating and repressing phases in a transcriptional D-box-based model

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Basic Mechanisms of the Mammalian Cell Cycle and Circadian Cell Clock Systems

In this Section we brie y expose some of the basic molecular mechanisms of the mammalian cell cycle and cellular clock oscillators. Because Chapters 2 and 3 are written as articles, further explanation and schemes are provided there.

Cell Cycle Mechanisms

A cell that has entered the cell cycle will go through several di erent phases of growth culmi-nating in mitosis (M phase). M phase is the key phase during which two daughter cells are generated. The previous cell cycle stages form the interphase, composed of: the G1 phase of cellular growth, the S phase of DNA replication and the G2 phase of growth and preparation for mitosis [5]. Cell cycle arrest can occur and the cell exits the cell cycle (G0 phase).
Moreover, a characteristic of the eukaryotic cell cycle is that of checkpoints: thresholds of control in the cycle that assess whether a given sequence of events was performed correctly. These are: the G1 checkpoint, where the cell \commits » to divide, the G2/M checkpoint, where possible DNA damage is repaired, and the mitotic spindle checkpoint, that ensures chromosomes are well aligned at the metaphasic plate before releasing the anaphase promoting complex (APC), that promotes cell cycle progression. After each checkpoint the cell cannot revert to its previous cell cycle phase.
The cell cycle phases are characterized at the molecular level by the sequential elevated expression of a family of proteins called Cyclins, each supporting the activity of speci c cyclin-dependent kinases (cdks). Cyclin D forms a complex with either cdk4 or cdk6 and is the cyclin of the G1 phase, Cyclin E pairs mostly with cdk2 and controls the passage from G1 to S phase, cyclin A also pairs with cdk2 and is active during S phase and G2, and nally cyclin B forms a complex with cdk1 and controls the G2/M transition. The cyclin B-cdk1 complex is also known as the mitosis promoting factor (MPF) and is the necessary and su cient element to carry out the mitotic process [6].
Because MPF is the essential cell cycle component, regulatory loops involving this complex are often considered central to the cell cycle. Important regulators of MPF include the wee1 kinase that inactivates MPF by phosphorylation, the cdc25 phosphatase that activates MPF by dephosphorylation and the APC:cdc20 complex that targets MPF for degradation [7], [8]. MPF in turn also phosphorylates these three components, which leads to activation of cdc25, inactivation of wee1 and allows APC to dimerize with cdc20, forming the APC:cdc20 complex. Therefore, MPF forms positive self-regulatory loops via its action in activating its activator cdc25 and repressing its repressor wee1, and a negative feedback loop by promoting the formation of its repressor APC:cdc20. These regulatory mechanisms are important cell cycle controls and are at the center of a variety of cell cycle models (further discussed on Section 1.4), including ours on Chapter 2 (schemes of these interactions are provided on the same Chapter).

Circadian Clock Mechanisms

The central regulatory clock network of mammalian cells is a transcription/translation feedback loop (TTFL) [9]. Two central elements of this network are the CLOCK:BMAL1 and PER:CRY protein complexes. CLOCK:BMAL1 binds to regions of the genome called E-boxes and promotes transcription of the Per and Cry genes. PER and CRY proteins in turn form the PER:CRY complex that reenters the nucleus and binds to CLOCK:BMAL1, blocking its promoter activity. This forms the core negative feedback loop of the mammalian circadian clock.
Furthermore, RORs and REV-ERBs are families of transcription factors that are also impor-tant for clock regulation. CLOCK:BMAL1 promotes expression of Ror and Rev. In turn, ROR proteins are activators of the Clock and Bmal1 genes, while REV-ERBs are repressors. ROR and REV-ERBs compete for binding at the BMAL1 promoter, thus a positive feedback loop is formed between ROR and CLOCK:BMAL1. The negative feedback loop between CLOCK:BMAL1 and REV-ERB is considered an important and fundamental part of the core clock mechanism [10].
Moreover, post-transcriptional mechanisms, including RNA-based mechanisms, are also con-trols of the mammalian circadian clock. Therefore, processes of mRNA stability, translation and alternative splicing are required for maintaining proper clock function [11]. Furthermore, post-translational mechanisms such as phosphorylations and dephosphorylations allow the rapid incorporation of signals by the clock system and play a role in the generation of the 24 h rhythm by controlling the delay of entrance of clock proteins, such as PER, in the nucleus [12].

The Cell Cycle and Circadian Clock Systems { a Brief Discussion

Both clock and cell cycle processes are essential for cellular health in mammals and when un-regulated can result in disease at the organism level. In particular, cancer is characterized by unregulated growth of mutated cells, while disruptions in circadian rhythms have been linked with insulin resistance and in ammation [13]. Due to the tight interconnection between the two oscillators, deregulation in one of them often deregulates the other as well, as evidenced by increased risk of circadian clock disturbances in cancer patients [14].
Furthermore, the circadian clock can impact cancer development [14]. Recently, agonists of REV-ERB (a central clock component) demonstrated e cacy in impairing glioblastoma growth in mice [15]. The mechanism behind these observations may involve REV-ERB-induced inhi-bition of autophagy and de novo lipogenesis, processes that are a part of fat metabolism [15]. This discovery highlights the tight control exerted by the clock oscillator in a variety of cellu-lar internal systems and reveals that pharmacological modulation of circadian components is a promising strategy for cancer therapy.
Besides cancer, circadian rhythms control a variety of cellular processes from energy home-ostasis, insulin secretion, insulin resistance/sensitivity, DNA repair and in ammation. In par-ticular the interplay between the circadian clock and metabolism is of great relevance for un-derstanding a variety of metabolic diseases. For instance, shift workers have a higher incidence of metabolic disorders [16], that are known to be caused by misalignment between the clock of the organism and the external light cycle [17]. In fact, circadian misalignment leads to an increase in markers of insulin resistance and in ammation regardless of sleep loss [13]. The role of the clock in cellular metabolism is currently a subject of vast and ongoing research interest, including recent experimental ([17], [15]) and dynamical modeling ([18]) works alike. In this work, we will explore some ideas about clock connection with metabolism on Chapter 3.
As revealed by genome-wide studies, the majority of drug targets show circadian patterns of control [19]. Moreover, timed drug delivery, or chronotherapy, is an e ective control of drug e cacy, that maximizes the desired drug e ect while simultaneously minimizing undesired side-e ects [19]. Chronotherapy is of importance in the delivery of a variety of treatments, including chemotherapy, though it is not clearly understood exactly what is behind increased e ciency of drugs at certain times. A better understanding of this phenomenon involves studying the circadian clock as well as its possible interactions with the cell cycle.
Concerning the relation between the mammalian circadian clock and cell cycle, rhythms of cell division are observed to be circadian in a variety of organisms [14], which led to an hypothesis of \gating » of the cell cycle by the clock mechanism [20]. This means that the clock mechanism would control the cell cycle so as to only allow mitosis to occur at certain time windows. Under the gating hypothesis the circadian clock would act as a fourth cell cycle checkpoint for the mitotic phase.
Furthermore, several molecular interactions revealing direct action of the clock on the cell cycle have been discovered. Firstly, the CLOCK:BMAL1 protein complex, essential for the circadian clock, induces expression of the wee1 gene [21]. The kinase wee1 phosphorylates and inactivates the cdk1 and cdk2 kinases, thus inhibiting the essential cell cycle complex cyclin B-cdk1, or mitosis promoting factor (MPF). Secondly, the clock components REV-ERB- / and ROR- / regulate the cell cycle inhibitor p21 [22]. Finally, there is also evidence for clock repression of c-Myc, a promoter of cell cycle progression by cyclin E induction [23], that is deregulated in mice de cient in the gene encoding for the core clock protein PER2 [24].
An example of a model of cell cycle gating by the clock is provided by Zamborszky et al., (2007), where critical size control of the mammalian cell cycle was found to be triggered by the clock [25]. By contrast, Gerard and Goldbeter, (2012), simulate entrainment of the cell cycle by the clock, while also suggesting a possible form of gating by the clock at the entry of G1 phase through a mechanism of oscillating growth factor [26].
Moreover, Nagoshi et al., (2004), have analyzed NIH3T3 mouse broblasts in real time and in individual cells and observed autonomous cellular clocks in these cells and that the cell cycle in a population of synchronized cells shows a trimodal frequency distribution of mitosis for speci c circadian clock phases [27]. Up until now, the state of the art included clock control of the cell cycle, exclusively. A breakthrough was made by Feillet et al.,(2014), and Bieler et al., (2014), demonstrating phase-locking between clock and cell cycle with strong evidence for bi-directional coupling [1], [30]. In the same work, the trimodal peak distribution is also obtained for synchronized cells (similarly to Nagoshi et al., (2004) [27]).

Phase-locking of the Mammalian Circadian Clock and Cell Cycle$

The work of Feillet et al., (2014), changed the previous state of the art concerning the interconnection between the clock and cell cycle systems in that it showed substantial evidence for a control of the cell cycle on the clock [1]. This is evidenced by observing the periods of the clock and cell cycle systems in NHI3T3 mouse broblasts under di erent growth conditions. The authors are able to measure the phases of both cell cycle and clock in NHI3T3 mouse broblasts at each point in time, using two independent reporter systems: a single-live-cell imaging of a REV-ERB ::VENUS fusion protein, as a clock reporter, and two uorescent cell cycle reporters, Cdt1 and Geminin, based on uorescent ubiquitination of the cell cycle (FUCCI).
The cell cycle oscillator is known to be period-responsive to the concentration of growth factors in the medium { these are expressed as % of FBS (Fetal Bovine Serum) and comprised of a variety of nutrients and growth factors, such as insulin-like growth factor 1 (IGF-1), that stimulate the cell cycle in a variety of mammalian cells. The rate of cell division increases with FBS concentration.
Feillet et al., (2014), observe that increasing the concentration of growth factor in the medium results not only in the expected increased frequency of the cell cycle, but also in an equal trend of increase in clock frequency [1], such that the two oscillators always remain synchronized for a variety of values of % of FBS.
Furthermore, they verify that cell cycle division occurs at a speci c clock phase for all cells. This means their observations are consistent with a model of oscillators that are phase-locked. Phase locking is characterized by convergence of the combined phase of oscillation (t) = ( 1(t); 2(t)) to a closed curve { an attractor. The phase-locking is distinct from the gating model, as phase-locked oscillators are synchronized through the entire cycle { knowing the phase of one oscillator determines the phase of the other, in ideal noise-free systems. By contrast, in the gating hypothesis only the mitotic phase would have to align with speci c clock phases.
Fig. 1.1 shows the 1:1 phase-locking results of Feillet et al., (2014), [1], for 10 % FBS and 15 % FBS. For cells grown in 10 % FBS the mean clock period is 21.9 1.1 h and the mean cell-cycle period is 21.3 1.3 h, while for cells grown in 15 % FBS the mean clock period is 19.4 0.5 h and the mean cell-cycle period is 18.6 0.6 h. Furthermore, the peak in REV-ERB ::VENUS expression is phase-locked with the mitotic phase: the mean delay of REV-ERB ::VENUS after mitosis is of 8,6 h for 10 % FBS and of 7,1 h for 15 % FBS (see also Traynard et al., (2016), [28]).
Figure 1.1: Result from Feillet et al., (2014) for 1:1 clock/cell cycle phase-locking [1]. The increase in growth factor concentration in the culture medium of NHI3T3 mouse broblasts increases the frequency of the cell cycle, which results in an increase in clock frequency as well. The oscillators are in 1:1 phase-locking. For 10 % FBS mean clock period is 21.9 1.1 h and mean cell-cycle period is 21.3 1.3 h and REV-ERB ::VENUS peaks on average 8,6 h after mitosis. For 15 % FBS mean clock period is 19.4 0.5 h and mean cell-cycle period is 18.6 0.6 h and REV-ERB ::VENUS peaks on average 7,1 h after mitosis Moreover Fig. 1.2 shows an histogram of cell density versus clock phase from Feillet et al., (2014), [1]. For 15 % FBS, increases in cell density, due to cellular division, occur at a preferential clock phase.
Furthermore, Feillet et al., (2014), observe synchronization of cells under the application a of Dexamethasone pulse (during 2 hours) in the medium [1]. Dexamethasone is a corticosteroid drug, known to synchronize clocks in populations of mammalian cells by inducing PER expres-sion in all cells. Corticosteroids induce expression of circadian clock PER proteins via activation of transcriptional activator glucocorticoid receptor GR [29]. Feillet et al., (2014), verify that application of a Dexamethasone pulse results in di erent synchronization ratios depending on the concentration of growth factor [1].
For cells grown in 20% FBS, the population of cells segregates into two groups, one with cells and synchronizing in 1:1 phase-locking (just as without Dexamethasone application) and the other group of cells showing a slower clock than cell cycle. From several analysis of cells grown in 10% FBS and 20% FBS the period locked ratios are determined to be roughly 5:4 for 10% FBS and 3:2 for 20% FBS (second group); this is further predicted by mathematical modeling [1]. The synchronization dynamics of the second group for 20% FBS after Dexamethasone application is similar to that observed by Nagoshi et al., (2004), under a similar protocol [27].
Fig. 1.3 A) shows the clustering of cells in one simulation for 20% FBS after synchronization by a pulse of Dexamethasone. In this particular case the clock to cell cycle period ratios are 1,8 and 1,09. Fig. 1.3 B) shows the distribution of cell density with clock phase for the two groups of cells.
From these results as well as mathematical modeling, the authors conclude the existence of multiple attractors for clock and cell cycle phase-locked behavior [1], i.e. that the input of Dexamethasone may be shifting the oscillators from one limit-cycle to another.
While the three peak distribution of cell density on itself doesn’t exclude the \gating » hypoth-esis, the observations of 1:1 period-lock are supportive of the phase-locked coupled oscillators hypothesis and suggests coupling from the cell cycle to the circadian clock in mammalian cells. Thus, there is likely bidirectional coupling between the oscillators. Our work aims at gaining insight on dynamical mechanisms that may be behind the observations of Feillet et al., (2014),
[1] presented in this Section, in particular the di erent synchronization ratios, and explore uni-and bi- directional forms of coupling.
A) Scatter plot showing segregation of two groups of cells in terms of their clock and cell cycle period ratios. In the blue cluster, the median clock period is 29 h 1,05 h and the median cell cycle period is 16,5 h 0,48 h; in the red cluster, the median clock period is 21,25 h 0,36 h and the median cell cycle period is 19,5 h 0,42 h. In this experiment the mean period ratios are 1,8 for the blue group and 1,09 for the red group. B) Distribution of cell densities with clock phase for the two cluster groups. On the left plot, the rst group shows a three peak distribution, where the middle, left and right peaks correspond respectively to the rst, second, and third divisions (hence the increase in cell density). On the right plot, the second group phase-locks in 1:1 ratio, similarly to the system without Dexamethasone (see Fig. 1.2).
of clock and cell cycle with the increase of growth factor [30], thus further corroborating this hypothesis. A recent study by Traynard et al., (2016) has presented model-based investigations of possible bidirectional mechanisms of clock and cell cycle coupling, that didn’t result in recovering the rational period-lock ratios [28].

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Models of the Mammalian Cell Cycle and Circadian Clock Oscillators

In this Section, we present a brief review of reference models for the cell cycle and the circadian cellular clock systems.

Cell Cycle Models

In 1991, Tyson rst proposed a cell cycle modeling approach centered on MPF, the essential cell cycle component, by modeling the interactions of cdc2 (cdk1) with cyclin B and showed the existence of three modes of stability: a steady state with high MPF activity, a spontaneous oscillator and an excitable switch [31]. Following this work, Novak and Tyson, (1993), were the rst to select the regulatory loops between MPF and cdc25/wee1, as well as the negative feedback loop where MPF stimulates its own degradation by activation of the ubiquitination pathway, as essential mechanisms that in themselves form the central cell cycle network of eukaryotic cells [32]. This work resulted in a model for the cell cycle in Xenopus oocyte extracts.
A decade later, Pomerening et al., (2003), studied the negative feedback-loop between MPF and APC:cdc20, by developing a model that includes auto-regulatory positive loops [33]. This model generates relaxation oscillations. Furthermore, the authors verify experimentally that the activation response of cdc2 to non-degradable cyclin B is consistent with a bistable dynamical behavior [33]. Moreover, Qu et al., (2003), presented a generic mathematical model of the eukaryotic cell cycle that allows simulation of both the G1/S and G2/M transitions [34]. In this model, the cell cycle checkpoint is a Hopf bifurcation point. Later, Pomerening et al., (2005), highlighted the importance of positive feedback loops in maintaining sustained cell cycle oscillations and veri ed experimentally that the cdc2/APC system in Xenopus egg extracts behaves like a relaxation oscillator [35].
In more recent years, Gerard and Goldbeter, (2009), proposed a detailed, 39-variable model of the mammalian cell cycle, containing four cdk modules regulated by reversible phosphoryla-tion, cdk inhibitors, and protein synthesis or degradation [36]. This model extensively describes the network of cyclin-dependent kinases and rst includes the role of growth factors in inducing the system’s transition from a quiescent state to an oscillatory cdk network. Later, the authors reduced this model ([36]) to a skeleton model of 5 variables (see Gerard and Goldbeter, (2011)), where the growth factor role in stability control is maintained and progression along the G1, S, G2 and M phase is still veri ed via sequential activation of the cyclin/cdk complexes [37]. Moreover, Gerard et al., (2012), extended this skeleton model via the incorporation of phos-phorylation/dephosphorylation cdk regulation as well as the positive feedback loops between MPF and cdc25/wee1 and veri ed that these controls promote the occurrence of bistability and increase the amplitude of oscillations in the various cyclin/cdk complexes [38]. Furthermore, including these regulatory mechanisms improves robustness of the cdk oscillations with respect to molecular noise, as shown by stochastic modeling [38].
Finally, Gerard et al., (2015), built and analyzed a mathematical model of the molecular interactions controlling the G1/S and G2/M transitions in yeast cells with a minimal cdk network consisting of a single cyclin-cdk fusion protein [39].

Circadian Clock Model

The rst circadian clock oscillatory model was proposed by Goodwin, (1975), and is based on a negative feedback loop between a protein and its own gene [40]. In later years, Leloup and Goldbeter, (2003), developed a detailed model of the mammalian circadian clock (of 16 to 19 variables) and observe sustained versus damped oscillations as well as entrainment of the system by light/dark cycles [41]. Furthermore, the authors verify a sensitivity of the oscillator’s phase in relation to changes of parameters that potentially relates to syndromes of advanced or delayed sleep phase observed in humans [41]. In the same year, Forger and Peskin, (2003), propose a di erent yet also detailed model of the mammalian circadian clock, using mass action kinetics, that is calibrated to data and accurately reproduces characteristics of oscillatory clock proteins and mRNAs, such as the shape and amplitude of oscillation and the phase of entrainment to the 24 h light/dark cycle [42].
Moreover, Leloup and Goldbeter, (2004), further extend studies on the Leloup and Gold-beter, (2003) [41], model and observe that the oscillatory behavior and period of the system are most sensitive to parameters involved in the synthesis or in the degradation of Bmal1 mRNA and BMAL1 protein, and that these regulatory mechanisms may be su cient for generating sustained oscillations [43]. Furthermore, in the same study the authors veri ed that the phase of oscillations upon entrainment to the light/dark cycle strongly depends on the parameters that govern the level of CRY protein [43]. On the same year, Becker-Weimann et al., (2004), propose a model using a reduced number of species, that is able to reproduce the rescue of cir-cadian oscillations in P er2Brdm1=Cry2 = double-mutant mice [44]. Di erently, Mirsky et al., (2009), propose a model minimizing post-translation modi ed species that is evaluated against experimental knockout phenotypes in what concerns retention of rhythmicity and changes in expression levels of clock species [45].
Relogio et al., (2011), developed a circadian clock model based on data for the amplitude and phase of the core clock components that highlights the role of the ROR/BMAL1/REV-ERB loop as important to the circadian clock [46]. Moreover, Comet et al., (2012), identi ed mechanisms common to circadian clocks across species, using di erential equations as well as discrete models [47]. The authors simpli ed as much as possible in order to obtain minimal networks of essential interactions and reduced the model of Leloup and Goldbeter, (2004) [43], to eight and four variables [47].
A di erent type of model is proposed by Korenci et al., (2012), describing a six-variable gene regulatory network of the liver core clock, whose negative feedback includes time-delayed variables [48]. This model is able to reproduce time pro les, amplitudes and phases of clock genes and shows that intrinsic combinatorial gene regulation governs the liver circadian clock
[48]. Moreover, Jolley et al., (2014), propose a simpli ed clock model that highlights the role of the clock controlled genomic binding region D-box and reproduces predictions on the dual regulation of Cry1 by D-box and REV-ERB /ROR [49].

Table of contents :

1 Introduction
1.1 Basic Mechanisms of the Mammalian Cell Cycle and Circadian Cell Clock Systems
1.1.1 Cell Cycle Mechanisms
1.1.2 Circadian Clock Mechanisms
1.2 The Cell Cycle and Circadian Clock Systems { a Brief Discussion
1.3 Phase-locking of the Mammalian Circadian Clock and Cell Cycle
1.4 Models of the Mammalian Cell Cycle and Circadian Clock Oscillators
1.4.1 Cell Cycle Models
1.4.2 Circadian Clock Models
1.5 Principles, Methods and Goals
1.6 Work Overview
2 Modeling the Mammalian Cell Cycle 
2.1 A comprehensive reduced model of the mammalian cell cycle
2.1.1 Abstract
2.1.2 Introduction
2.1.3 A 7D intermediate model
2.1.4 Model Reduction and Calibration
2.1.5 Mathematical Analysis
2.1.6 Conclusion
2.1.7 Aknowledgments
2.2 Function Approximation in Cell Cycle Modeling
2.2.1 Graphical Function Approximation
2.2.2 Piecewise Quadratic Approximation
3 Modeling the Mammalian Circadian Clock
3.1 Timing of circadian clock regulatory inputs controls duration of activating and repressing phases in a transcriptional D-box-based model
3.1.1 Introduction
3.1.2 Model Design, Calibration and Robustness
3.1.3 Results and Discussion
3.1.4 Model Reduction
4 Coupling the Mammalian Cell Cycle and Circadian Clock Oscillators 
4.1 Coupling via MPF-induced phosphorylation of REV-ERB
4.2 Coupling via BMAL1-mediated wee1 activation (indirect repression of MPF)
4.2.1 Cell Cycle Period Control via the Clock
4.3 Bidirectional coupling
4.4 Coupling via GF-induced inhibition of R-box
4.5 Final Discussion
5 Conclusions and Perspectives 
5.1 Conclusions
5.2 Perspectives and Future Work
5.2.1 Design of Synthetic Oscillators
5.2.2 Clock and Cell Cycle Modeling
5.2.3 Experimental Studies

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