Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems.

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as 'multi-synchronization'.

From circadian clock mechanism to sleep disorders and jet lag: Insights from a computational approach.

We present ten insights that can be gained from computational models based on molecular mechanisms for the mammalian circadian clock. These insights range from the conditions in which circadian rhythms occur spontaneously to their entrainment by the light-dark (LD) cycle and to clock-related disorders of the sleep-wake cycle. Endogenous oscillations originate spontaneously from transcription-translation feedback loops involving clock proteins such as PER, CRY, CLOCK and BMAL1. Circadian oscillations occur in a parameter domain bounded by critical values. Outside this domain the circadian network ceases to oscillate and evolves to a stable steady state. This conclusion bears on the nature of arrhythmic behavior of the circadian clock, which may not necessarily be due to mutations in clock genes. Entrainment by the LD cycle occurs in a certain range of parameter values, with a phase that depends on the endogenous period of the circadian clock. A decrease in PER phosphorylation is accompanied by a decrease in endogenous period and a phase advance of the clock; this situation accounts for the familial, advanced sleep phase syndrome (FASPS). The mirror delayed sleep phase syndrome (DSPS) can be accounted for, similarly, by an increase in PER phosphorylation and a rise in autonomous period. Failure of entrainment by the LD cycle in the model corresponds to the non-24 h sleep-wake cycle syndrome, in which the phase of the circadian clock drifts in the course of time. Quasi-periodic oscillations that develop in these conditions sometimes correspond to long-period patterns in which the circadian clock is nearly entrained for long bouts of time before its phase rapidly drifts until a new regime of quasi-entrainment is re-established. In regard to jet lag, the computational approach accounts for the two modes of re-entrainment observed after an advance or delay which correspond, respectively, to an eastward or westward flight: the clock adjusts in a direction similar (orthodromic) or opposite (antidromic) to that of the shift in the LD cycle. Computational modeling predicts that in the vicinity of the switch between orthodromic and antidromic re-entrainment the circadian clock may take a very long time to resynchronize with the LD cycle. Repetitive perturbations of the circadian clock due, for example, to chronic jet lag -a situation somewhat reminiscent of shift work- may lead to quasi-periodic or chaotic oscillations. The latter irregular oscillations can sometimes be observed in normal LD cycles, raising the question of their possible relevance to fragmented sleep patterns observed in narcolepsy. The latter condition, however, appears to originate from disorders in the orexin neural circuit, which promotes wakefulness, rather than from an irregular operation of the circadian clock.

A Computational Model for the Cold Response Pathway in Plants.

Understanding the mechanism by which plants respond to cold stress and strengthen their tolerance to low temperatures is an important and challenging task in plant sciences. Experiments have established that the first step in the perception and transduction of the cold stress signal consists of a transient influx of Ca2+. This Ca2+ influx triggers the activation of a cascade of phosphorylation-dephosphorylation reactions that eventually affects the expression of C-repeat-binding factors (CBFs, notably CBF3), which were shown in many plants to control resistance to cold stress by regulating the expression of cold-regulated (COR) genes. Based on experimental observations mostly made on Arabidopsis thaliana, we build a computational model for the cold response pathway in plants, from the transduction of the cold signal via the transient influx of Ca2+ to the activation of the phosphorylation cascade leading to CBF3 expression. We explore the dynamics of this regulatory network by means of numerical simulations and compare the results with experimental observations on the dynamics of the cold response, both for the wild type and for mutants. The simulations show how, in response to cold stress, a brief Ca2+ influx, which is over in minutes, is transduced along the successive steps of the network to trigger the expression of cold response genes such as CBF3 within hours. Sometimes, instead of a single Ca2+ spike the decrease in temperature brings about a train of high-frequency Ca2+ oscillations. The model is applied to both types of Ca2+ signaling. We determine the dynamics of the network in response to a series of identical cold stresses, to account for the observation of desensitization and resensitization. The analysis of the model predicts the possibility of an oscillatory expression of CBF3 originating from the negative feedback exerted by ZAT12, a factor itself controlled by CBF3. Finally, we extend the model to incorporate the circadian control of CBF3 expression, to account for the gating of the response to cold stress by the plant circadian clock.

Robust synchronization of the cell cycle and the circadian clock through bidirectional coupling.

The cell cycle and the circadian clock represent major cellular rhythms, which appear to be coupled. Thus the circadian factor BMAL1 controls the level of cell cycle proteins such as Cyclin E and WEE1, the latter of which inhibits the kinase CDK1 that governs the G2/M transition. In reverse the cell cycle impinges on the circadian clock through direct control by CDK1 of REV-ERBα, which negatively regulates BMAL1. These observations provide evidence for bidirectional coupling of the cell cycle and the circadian clock. By merging detailed models for the two networks in mammalian cells, we previously showed that unidirectional coupling to the circadian clock can entrain the cell cycle to 24 or 48 h, depending on the cell cycle autonomous period, while complex oscillations occur when entrainment fails. Here we show that the reverse unidirectional coupling via phosphorylation of REV-ERBα or via mitotic inhibition of transcription, both controlled by CDK1, can elicit entrainment of the circadian clock by the cell cycle. We then determine the effect of bidirectional coupling of the cell cycle and circadian clock as a function of their relative coupling strengths. In contrast to unidirectional coupling, bidirectional coupling markedly reduces the likelihood of complex oscillations. While the two rhythms oscillate independently as long as both couplings are weak, one rhythm entrains the other if one of the couplings dominates. If the couplings in both directions become stronger and of comparable magnitude, the two rhythms synchronize, generally at an intermediate period within the range defined by the two autonomous periods prior to coupling. More surprisingly, synchronization may also occur at a period slightly below or above this range, while in some conditions the synchronization period can even be much longer. Two or even three modes of synchronization may sometimes coexist, yielding examples of birhythmicity or trirhythmicity. Because synchronization readily occurs in the form of simple periodic oscillations over a wide range of coupling strengths and in the presence of multiple connections between the two oscillatory networks, the results indicate that bidirectional coupling favours the robust synchronization of the cell cycle and the circadian clock.

Multi-rhythmicity generated by coupling two cellular rhythms.

The cell cycle and the circadian clock represent two major cellular rhythms, which are coupled because the circadian clock governs the synthesis of several proteins of the network that drives the mammalian cell cycle. Analysis of a detailed model for these coupled cellular rhythms previously showed that the cell cycle can be entrained at the circadian period of 24 h, or at a period of 48 h, depending on the autonomous period of the cell cycle and on the coupling strength. We show by means of numerical simulations that multiple stable periodic regimes, i.e. multi-rhythmicity, may originate from the coupling of the two cellular rhythms. In these conditions, the cell cycle can evolve to any one of two (birhythmicity) or three stable periodic regimes (trirhythmicity). When applied at the right phase, transient perturbations of appropriate duration and magnitude can induce the switch between the different oscillatory states. Such switching is characterized by final state sensitivity, which originates from the complex structure of the attraction basins. By providing a novel instance of multi-rhythmicity in a realistic model for the coupling of two major cellular rhythms, the results throw light on the conditions in which multiple stable periodic regimes may coexist in biological systems.

Revisiting a skeleton model for the mammalian cell cycle: From bistability to Cdk oscillations and cellular heterogeneity.

A network of cyclin-dependent kinases (Cdks) regulated by multiple negative and positive feedback loops controls progression in the mammalian cell cycle. We previously proposed a detailed computational model for this network, which consists of four coupled Cdk modules. Both this detailed model and a reduced, skeleton version show that the Cdk network is capable of temporal self-organization in the form of sustained Cdk oscillations, which correspond to the orderly progression along the different cell cycle phases G1, S (DNA replication), G2 and M (mitosis). We use the skeleton model to revisit the role of positive feedback (PF) loops on the dynamics of the mammalian cell cycle by showing that the multiplicity of PF loops extends the range of bistability in the isolated Cdk modules controlling the G1/S and G2/M transitions. Resorting to stochastic simulations we show that, through their effect on the range of bistability, multiple PF loops enhance the robustness of Cdk oscillations with respect to molecular noise. The model predicts that a rise in the total level of Cdk1 also enlarges the domain of bistability in the isolated Cdk modules as well as the range of oscillations in the full Cdk network. Surprisingly, stochastic simulations indicate that Cdk1 overexpression reduces the robustness of Cdk oscillations towards molecular noise; this result is due to the increased distance between the two branches of the bistable switch at higher levels of Cdk1. At intermediate levels of growth factor stochastic simulations show that cells may randomly switch between cell cycle arrest and cell proliferation, as a consequence of fluctuations. In the presence of Cdk1 overexpression, these transitions occur even at low levels of growth factor. Extending stochastic simulations from single cells to cell populations suggests that stochastic switches between cell cycle arrest and proliferation may provide a source of heterogeneity in a cell population, as observed in cancer cells characterized by Cdk1 overexpression.

The positive circadian regulators CLOCK and BMAL1 control G2/M cell cycle transition through Cyclin B1.

We previously identified a tight bidirectional phase coupling between the circadian clock and the cell cycle. To understand the role of the CLOCK/BMAL1 complex, representing the main positive regulator of the circadian oscillator, we knocked down Bmal1 or Clock in NIH3T33C mouse fibroblasts (carrying fluorescent reporters for clock and cell cycle phase) and analyzed timing of cell division in individual cells and cell populations. Inactivation of Bmal1 resulted in a loss of circadian rhythmicity and a lengthening of the cell cycle, originating from delayed G2/M transition. Subsequent molecular analysis revealed reduced levels of Cyclin B1, an important G2/M regulator, upon suppression of Bmal1 gene expression. In complete agreement with these experimental observations, simulation of Bmal1 knockdown in a computational model for coupled mammalian circadian clock and cell cycle oscillators (now incorporating Cyclin B1 induction by BMAL1) revealed a lengthening of the cell cycle. Similar data were obtained upon knockdown of Clock gene expression. In conclusion, the CLOCK/BMAL1 complex controls cell cycle progression at the level of G2/M transition through regulation of Cyclin B1 expression.

Dissipative structures in biological systems: bistability, oscillations, spatial patterns and waves.

The goal of this review article is to assess how relevant is the concept of dissipative structure for understanding the dynamical bases of non-equilibrium self-organization in biological systems, and to see where it has been applied in the five decades since it was initially proposed by Ilya Prigogine. Dissipative structures can be classified into four types, which will be considered, in turn, and illustrated by biological examples: (i) multistability, in the form of bistability and tristability, which involve the coexistence of two or three stable steady states, or in the form of birhythmicity, which involves the coexistence between two stable rhythms; (ii) temporal dissipative structures in the form of sustained oscillations, illustrated by biological rhythms; (iii) spatial dissipative structures, known as Turing patterns; and (iv) spatio-temporal structures in the form of propagating waves. Rhythms occur with widely different periods at all levels of biological organization, from neural, cardiac and metabolic oscillations to circadian clocks and the cell cycle; they play key roles in physiology and in many disorders. New rhythms are being uncovered while artificial ones are produced by synthetic biology. Rhythms provide the richest source of examples of dissipative structures in biological systems. Bistability has been observed experimentally, but has primarily been investigated in theoretical models in an increasingly wide range of biological contexts, from the genetic to the cell and animal population levels, both in physiological conditions and in disease. Bistable transitions have been implicated in the progression between the different phases of the cell cycle and, more generally, in the process of cell fate specification in the developing embryo. Turing patterns are exemplified by the formation of some periodic structures in the course of development and by skin stripe patterns in animals. Spatio-temporal patterns in the form of propagating waves are observed within cells as well as in intercellular communication. This review illustrates how dissipative structures of all sorts abound in biological systems.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems.

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

Dissipative structures and biological rhythms.

Sustained oscillations abound in biological systems. They occur at all levels of biological organization over a wide range of periods, from a fraction of a second to years, and with a variety of underlying mechanisms. They control major physiological functions, and their dysfunction is associated with a variety of physiological disorders. The goal of this review is (i) to give an overview of the main rhythms observed at the cellular and supracellular levels, (ii) to briefly describe how the study of biological rhythms unfolded in the course of time, in parallel with studies on chemical oscillations, (iii) to present the major roles of biological rhythms in the control of physiological functions, and (iv) the pathologies associated with the alteration, disappearance, or spurious occurrence of biological rhythms. Two tables present the main examples of cellular and supracellular rhythms ordered according to their period, and their role in physiology and pathophysiology. Among the rhythms discussed are neural and cardiac rhythms, metabolic oscillations such as those occurring in glycolysis in yeast, intracellular Ca++ oscillations, cyclic AMP oscillations in Dictyostelium amoebae, the segmentation clock that controls somitogenesis, pulsatile hormone secretion, circadian rhythms which occur in all eukaryotes and some bacteria with a period close to 24 h, the oscillatory dynamics of the enzymatic network driving the cell cycle, and oscillations in transcription factors such as NF-ΚB and tumor suppressors such as p53. Ilya Prigogine's concept of dissipative structures applies to temporal oscillations and allows us to unify within a common framework the various rhythms observed at different levels of biological organization, regardless of their period and underlying mechanism.

Cell Fate Specification Based on Tristability in the Inner Cell Mass of Mouse Blastocysts.

During development, interactions between transcription factors control the specification of different cell fates. The regulatory networks of genetic interactions often exhibit multiple stable steady states; such multistability provides a common dynamical basis for differentiation. During early murine embryogenesis, cells from the inner cell mass (ICM) can be specified in epiblast (Epi) or primitive endoderm (PrE). Besides the intracellular gene regulatory network, specification is also controlled by intercellular interactions involving Erk signaling through extracellular Fgf4. We previously proposed a model that describes the gene regulatory network and its interaction with Erk signaling in ICM cells. The model displays tristability in a range of Fgf4 concentrations and accounts for the self-organized specification process observed in vivo. Here, we further investigate the origin of tristability in the model and analyze in more detail the specification process by resorting to a simplified two-cell model. We also carry out simulations of a population of 25 cells under various experimental conditions to compare their outcome with that of mutant embryos or of embryos submitted to exogenous treatments that interfere with Fgf signaling. The results are analyzed by means of bifurcation diagrams. Finally, the model predicts that heterogeneities in extracellular Fgf4 concentration play a primary role in the spatial arrangement of the Epi/PrE cells in a salt-and-pepper pattern. If, instead of heterogeneities in extracellular Fgf4 concentration, internal fluctuations in the levels of expression of the transcription factors are considered as a source of randomness, simulations predict the occurrence of unrealistic switches between the Epi and the PrE cell fates, as well as the evolution of some cells toward one of these states without passing through the previous ICM state, in contrast to what is observed in vivo.

Dynamics of the mammalian cell cycle in physiological and pathological conditions.

A network of cyclin-dependent kinases (Cdks) controls progression along the successive phases G1, S, G2, and M of the mammalian cell cycle. Deregulations in the expression of molecular components in this network often lead to abusive cell proliferation and cancer. Given the complex nature of the Cdk network, it is fruitful to resort to computational models to grasp its dynamical properties. Investigated by means of bifurcation diagrams, a detailed computational model for the Cdk network shows how the balance between quiescence and proliferation is affected by activators (oncogenes) and inhibitors (tumor suppressors) of cell cycle progression, as well as by growth factors and other external factors such as the extracellular matrix (ECM) and cell contact inhibition. Suprathreshold changes in all these factors can trigger a switch in the dynamical behavior of the network corresponding to a bifurcation between a stable steady state, associated with cell cycle arrest, and sustained oscillations of the various cyclin/Cdk complexes, corresponding to cell proliferation. The model for the Cdk network accounts for the dependence or independence of cell proliferation on serum and/or cell anchorage to the ECM. Such computational approach provides an integrated view of the control of cell proliferation in physiological or pathological conditions. Whether the balance is tilted toward cell cycle arrest or cell proliferation depends on the direction in which the threshold associated with the bifurcation is passed once the cell integrates the multiple signals, internal or external to the Cdk network, that promote or impede progression in the cell cycle.

Gata6, Nanog and Erk signaling control cell fate in the inner cell mass through a tristable regulatory network.

During blastocyst formation, inner cell mass (ICM) cells differentiate into either epiblast (Epi) or primitive endoderm (PrE) cells, labeled by Nanog and Gata6, respectively, and organized in a salt-and-pepper pattern. Previous work in the mouse has shown that, in absence of Nanog, all ICM cells adopt a PrE identity. Moreover, the activation or the blockade of the Fgf/RTK pathway biases cell fate specification towards either PrE or Epi, respectively. We show that, in absence of Gata6, all ICM cells adopt an Epi identity. Furthermore, the analysis of Gata6(+/-) embryos reveals a dose-sensitive phenotype, with fewer PrE-specified cells. These results and previous findings have enabled the development of a mathematical model for the dynamics of the regulatory network that controls ICM differentiation into Epi or PrE cells. The model describes the temporal dynamics of Erk signaling and of the concentrations of Nanog, Gata6, secreted Fgf4 and Fgf receptor 2. The model is able to recapitulate most of the cell behaviors observed in different experimental conditions and provides a unifying mechanism for the dynamics of these developmental transitions. The mechanism relies on the co-existence between three stable steady states (tristability), which correspond to ICM, Epi and PrE cells, respectively. Altogether, modeling and experimental results uncover novel features of ICM cell fate specification such as the role of the initial induction of a subset of cells into Epi in the initiation of the salt-and-pepper pattern, or the precocious Epi specification in Gata6(+/-) embryos.

The balance between cell cycle arrest and cell proliferation: control by the extracellular matrix and by contact inhibition.

To understand the dynamics of the cell cycle, we need to characterize the balance between cell cycle arrest and cell proliferation, which is often deregulated in cancers. We address this issue by means of a detailed computational model for the network of cyclin-dependent kinases (Cdks) driving the mammalian cell cycle. Previous analysis of the model focused on how this balance is controlled by growth factors (GFs) or the levels of activators (oncogenes) and inhibitors (tumour suppressors) of cell cycle progression. Supra-threshold changes in the level of any of these factors can trigger a switch in the dynamical behaviour of the Cdk network corresponding to a bifurcation between a stable steady state, associated with cell cycle arrest, and sustained oscillations of the various cyclin/Cdk complexes, corresponding to cell proliferation. Here, we focus on the regulation of cell proliferation by cellular environmental factors external to the Cdk network, such as the extracellular matrix (ECM), and contact inhibition, which increases with cell density. We extend the model for the Cdk network by including the phenomenological effect of both the ECM, which controls the activation of the focal adhesion kinase (FAK) that promotes cell cycle progression, and cell density, which inhibits cell proliferation via the Hippo/YAP pathway. The model shows that GFs and FAK activation are capable of triggering in a similar dynamical manner the transition to cell proliferation, while the Hippo/YAP pathway can arrest proliferation once cell density passes a critical threshold. The results account for the dependence or independence of cell proliferation on serum and/or cell anchorage to ECM. Whether the balance in the Cdk network is tilted towards cell cycle arrest or proliferation depends on the direction in which the threshold associated with the bifurcation is passed once the cell integrates the multiple, internal or external signals that promote or impede progression in the cell cycle.

Oscillatory enzyme reactions and Michaelis-Menten kinetics.

Oscillations occur in a number of enzymatic systems as a result of feedback regulation. How Michaelis-Menten kinetics influences oscillatory behavior in enzyme systems is investigated in models for oscillations in the activity of phosphofructokinase (PFK) in glycolysis and of cyclin-dependent kinases in the cell cycle. The model for the PFK reaction is based on a product-activated allosteric enzyme reaction coupled to enzymatic degradation of the reaction product. The Michaelian nature of the product decay term markedly influences the period, amplitude and waveform of the oscillations. Likewise, a model for oscillations of Cdc2 kinase in embryonic cell cycles based on Michaelis-Menten phosphorylation-dephosphorylation kinetics shows that the occurrence and amplitude of the oscillations strongly depend on the ultrasensitivity of the enzymatic cascade that controls the activity of the cyclin-dependent kinase.

Critical phase shifts slow down circadian clock recovery: implications for jet lag.

Advancing or delaying the light-dark (LD) cycle perturbs the circadian clock, which eventually recovers its original phase with respect to the new LD cycle. Readjustment of the clock occurs by shifting its phase in the same (orthodromic re-entrainment) or opposite direction (antidromic re-entrainment) as the shift in the LD cycle. To investigate circadian clock recovery after phase shifts of the LD cycle we use a detailed computational model previously proposed for the cellular regulatory network underlying the mammalian circadian clock. The model predicts the existence of a sharp threshold separating orthodromic from antidromic re-entrainment. In the vicinity of this threshold, resynchronization of the clock after a phase shift markedly slows down. The type of re-entrainment, the position of the threshold and the time required for resynchronization depend on multiple factors such as the autonomous period of the clock, the direction and magnitude of the phase shift, the clock biochemical kinetic parameters, and light intensity. Partitioning the phase shift into a series of smaller phases shifts decreases the impact on the recovery of the circadian clock. We use the phase response curve to predict the location of the threshold separating orthodromic and antidromic re-entrainment after advanced or delayed phase shifts of the LD cycle. The marked increase in recovery times predicted near the threshold could be responsible for the most severe disturbances of the human circadian clock associated with jet lag.

From quiescence to proliferation: Cdk oscillations drive the mammalian cell cycle.

We recently proposed a detailed model describing the dynamics of the network of cyclin-dependent kinases (Cdks) driving the mammalian cell cycle (Gérard and Goldbeter, 2009). The model contains four modules, each centered around one cyclin/Cdk complex. Cyclin D/Cdk4-6 and cyclin E/Cdk2 promote progression in G1 and elicit the G1/S transition, respectively; cyclin A/Cdk2 ensures progression in S and the transition S/G2, while the activity of cyclin B/Cdk1 brings about the G2/M transition. This model shows that in the presence of sufficient amounts of growth factor the Cdk network is capable of temporal self-organization in the form of sustained oscillations, which correspond to the ordered, sequential activation of the various cyclin/Cdk complexes that control the successive phases of the cell cycle. The results suggest that the switch from cellular quiescence to cell proliferation corresponds to the transition from a stable steady state to sustained oscillations in the Cdk network. The transition depends on a finely tuned balance between factors that promote or hinder progression in the cell cycle. We show that the transition from quiescence to proliferation can occur in multiple ways that alter this balance. By resorting to bifurcation diagrams, we analyze the mechanism of oscillations in the Cdk network. Finally, we show that the complexity of the detailed model can be greatly reduced, without losing its key dynamical properties, by considering a skeleton model for the Cdk network. Using such a skeleton model for the mammalian cell cycle we show that positive feedback (PF) loops enhance the amplitude and the robustness of Cdk oscillations with respect to molecular noise. We compare the relative merits of the detailed and skeleton versions of the model for the Cdk network driving the mammalian cell cycle.

Entrainment of the mammalian cell cycle by the circadian clock: modeling two coupled cellular rhythms.

The cell division cycle and the circadian clock represent two major cellular rhythms. These two periodic processes are coupled in multiple ways, given that several molecular components of the cell cycle network are controlled in a circadian manner. For example, in the network of cyclin-dependent kinases (Cdks) that governs progression along the successive phases of the cell cycle, the synthesis of the kinase Wee1, which inhibits the G2/M transition, is enhanced by the complex CLOCK-BMAL1 that plays a central role in the circadian clock network. Another component of the latter network, REV-ERBα, inhibits the synthesis of the Cdk inhibitor p21. Moreover, the synthesis of the oncogene c-Myc, which promotes G1 cyclin synthesis, is repressed by CLOCK-BMAL1. Using detailed computational models for the two networks we investigate the conditions in which the mammalian cell cycle can be entrained by the circadian clock. We show that the cell cycle can be brought to oscillate at a period of 24 h or 48 h when its autonomous period prior to coupling is in an appropriate range. The model indicates that the combination of multiple modes of coupling does not necessarily facilitate entrainment of the cell cycle by the circadian clock. Entrainment can also occur as a result of circadian variations in the level of a growth factor controlling entry into G1. Outside the range of entrainment, the coupling to the circadian clock may lead to disconnected oscillations in the cell cycle and the circadian system, or to complex oscillatory dynamics of the cell cycle in the form of endoreplication, complex periodic oscillations or chaos. The model predicts that the transition from entrainment to 24 h or 48 h might occur when the strength of coupling to the circadian clock or the level of growth factor decrease below critical values.

Effect of positive feedback loops on the robustness of oscillations in the network of cyclin-dependent kinases driving the mammalian cell cycle.

The transitions between the G(1), S, G(2) and M phases of the mammalian cell cycle are driven by a network of cyclin-dependent kinases (Cdks), whose sequential activation is regulated by intertwined negative and positive feedback loops. We previously proposed a detailed computational model for the Cdk network, and showed that this network is capable of temporal self-organization in the form of sustained oscillations, which govern ordered progression through the successive phases of the cell cycle [Gérard and Goldbeter (2009) Proc Natl Acad Sci USA 106, 21643-21648]. We subsequently proposed a skeleton model for the cell cycle that retains the core regulatory mechanisms of the detailed model [Gérard and Goldbeter (2011) Interface Focus 1, 24-35]. Here we extend this skeleton model by incorporating Cdk regulation through phosphorylation/dephosphorylation and by including the positive feedback loops that underlie the dynamics of the G(1)/S and G(2)/M transitions via phosphatase Cdc25 and via phosphatase Cdc25 and kinase Wee1, respectively. We determine the effects of these positive feedback loops and ultrasensitivity in phosphorylation/dephosphorylation on the dynamics of the Cdk network. The multiplicity of positive feedback loops as well as the existence of ultrasensitivity promote the occurrence of bistability and increase the amplitude of the oscillations in the various cyclin/Cdk complexes. By resorting to stochastic simulations, we further show that the presence of multiple, redundant positive feedback loops in the G(2)/M transition of the cell cycle markedly enhances the robustness of the Cdk oscillations with respect to molecular noise.