How to Achieve Fast Entrainment? The Timescale to Synchronization

Entrainment, where oscillators synchronize to an external signal, is ubiquitous in nature. The transient time leading to entrainment plays a major role in many biological processes. Our goal is to unveil the specific dynamics that leads to fast entrainment. By studying a generic model, we characterize the transient time to entrainment and show how it is governed by two basic properties of an oscillator: the radial relaxation time and the phase velocity distribution around the limit cycle. Those two basic properties are inherent in every oscillator. This concept can be applied to many biological systems to predict the average transient time to entrainment or to infer properties of the underlying oscillator from the observed transients. We found that both a sinusoidal oscillator with fast radial relaxation and a spike-like oscillator with slow radial relaxation give rise to fast entrainment. As an example, we discuss the jet-lag experiments in the mammalian circadian pacemaker.     

Two redundant oscillatory mechanisms in the yeast cell cycle

The cell division cycle requires oscillations in activity of B-type cyclin (Clb)-Cdk1 kinases. Oscillations are due to periodic cyclin degradation by the anaphase-promoting complex (APC) activated by Cdc20 or Cdh1, and to cyclical accumulation of the Sic1 inhibitor. The results presented here suggest that the regulatory machinery controlling Clb kinase levels embeds two distinct oscillatory mechanisms. One, a "relaxation oscillator," involves alternation between two meta-stable states: Clb high/inhibitors (Sic1/APC-Cdh1) low, and Clb low/inhibitors high. The other, a "negative feedback oscillator," involves Clb kinase activation of APC-Cdc20, leading to Clb degradation. Genetic analysis suggests that these two mechanisms can function independently, and inactivation of both mechanisms is required to prevent mitosis. Computational modeling confirms that two such mechanisms can be linked to yield a robust cell cycle control system.     

Changes in Oscillatory Dynamics in the Cell Cycle of Early Xenopus laevis Embryos

During the early development of Xenopus laevis embryos, the first mitotic cell cycle is long (∼85 min) and the subsequent 11 cycles are short (∼30 min) and clock-like. Here we address the question of how the Cdk1 cell cycle oscillator changes between these two modes of operation. We found that the change can be attributed to an alteration in the balance between Wee1/Myt1 and Cdc25. The change in balance converts a circuit that acts like a positive-plus-negative feedback oscillator, with spikes of Cdk1 activation, to one that acts like a negative-feedback-only oscillator, with a shorter period and smoothly varying Cdk1 activity. Shortening the first cycle, by treating embryos with the Wee1A/Myt1 inhibitor PD0166285, resulted in a dramatic reduction in embryo viability, and restoring the length of the first cycle in inhibitor-treated embryos with low doses of cycloheximide partially rescued viability. Computations with an experimentally parameterized mathematical model show that modest changes in the Wee1/Cdc25 ratio can account for the observed qualitative changes in the cell cycle. The high ratio in the first cycle allows the period to be long and tunable, and decreasing the ratio in the subsequent cycles allows the oscillator to run at a maximal speed. Thus, the embryo rewires its feedback regulation to meet two different developmental requirements during early development.     

Interaction of mitotic oscillators as a source of variability of cell cycle duration

Interaction between membrane mitotic oscillators at the expense of exchange with the molecules of lipids (slow variable) and antioxidants (fast variable) was considered. Parameters of all the oscillators are equal, excluding a small noise added to the equation for lipids. These parameters are chosen in such a way that the oscillators are not far from the transition to the stable stationary state. The numerical modeling has shown that the exchange with lipids brings about the appearance of an additional limit cycle whose period is significantly greater than that of an autonomous oscillator. The addition of noise averages the behaviour of oscillators, and distribution according to cycle duration becomes broad and bimodal. Thus the exchange of the slow variable increases the dispersion of distribution of cell generation times. This conclusion seems to be true for any oscillator with similar dynamic properties.     

The ups and downs of biological timers

Background  

The need to execute a sequence of events in an orderly and timely manner is central to many biological processes, including cell cycle progression and cell differentiation. For self-perpetuating systems, such as the cell cycle oscillator, delay times between events are defined by the network of interacting proteins that propagates the system. However, protein levels inside cells are subject to genetic and environmental fluctuations, raising the question of how reliable timing is maintained.     

Genome streamlining results in loss of robustness of the circadian clock in the marine cyanobacterium Prochlorococcus marinus PCC 9511

The core oscillator of the circadian clock in cyanobacteria consists of 3 proteins, KaiA, KaiB, and KaiC. All 3 have previously been shown to be essential for clock function. Accordingly, most cyanobacteria possess at least 1 copy of each kai gene. One exception is the marine genus Prochlorococcus, which we suggest here has suffered a stepwise deletion of the kaiA gene, together with significant genome streamlining. Nevertheless, natural Prochlorococcus populations and laboratory cultures are strongly synchronized by the alternation of day and night, displaying 24-h rhythms in DNA replication, with a temporal succession of G1, S, and G2-like cell cycle phases. Using quantitative real-time PCR, we show here that in Prochlorococcus marinus PCC 9511, the mRNA levels of the clock genes kaiB and kaiC, as well as a few other selected genes including psbA, also displayed marked diel variations when cultures were kept under a light-dark rhythm. However, both cell cycle and psbA gene expression rhythms damped very rapidly under continuous light. In the closely related Synechococcus sp. WH8102, which possesses all 3 kai genes, cell cycle rhythms persisted over several days, in agreement with established cyanobacterial models. These data indicate a correlation between the loss of kaiA and a loss of robustness in the endogenous oscillator of Prochlorococcus and raise questions about how a basic KaiBC system may function and through which mechanism the daily "lights-on" and "lights-off" signal could be mediated.     

Quantized cell cycle times: interaction between a relaxation oscillator and ultradian clock pulses

Control of the timing of cell division is considered to result from a relaxation cell cycle oscillator: this has one slow and one rapid component and obeys a system of two ordinary differential equations. Interactions of the slow component with an ultradian oscillator leads to quantization of cell cycle times when the free parameters of the cell cycle oscillator are chosen close to its bifurcation point. This model fits the experimental results previously reported.     

Recycling the cell cycle: cyclins revisited

I discuss advances in the cell cycle in the 21 years since cyclin was discovered. The surprising redundancy amongst the classical cyclins (A, B, and E) and cyclin-dependent kinases (Cdk1 and Cdk2) show that the important differences between these proteins are when and where they are expressed rather than the proteins they phosphorylate. Although the broad principles of the cell cycle oscillator are widely accepted, we are surprisingly ignorant of its detailed mechanism. This is especially true of the anaphase promoting complex (APC), the machine that triggers chromosome segregation and the exit of mitosis by targeting securin and mitotic cyclins for destruction. I discuss how a cyclin/Cdk-based engine could have evolved to assume control of the cell cycle from other, older protein kinases.     

From growth to cell cycle control.

How does a quiescent cell decide to re-enter the cell cycle and start replicating its DNA? What controls cell proliferation? These are fundamental questions that have to be solved in order to understand the mechanisms of oncogenesis. Some recent data have provided clues about how signal transduction pathways may be connected to the cell cycle. A protein kinase cascade starting from the membrane growth factor receptor is thought to be involved in transducing extracellular stimuli to the master switches of the cell cycle control machinery. The recently identified extracellular-signal regulated kinases (ERKs) appear to play an important role in this pathway. Expression of cyclins, which are regulatory subunits of the universal cell cycle oscillator cdc2, may also be controlled through this kinase cascade. The products of tumor suppressor genes Rb and p53 also play an important role in regulating cell proliferation by interfering with the cell cycle pathway. Here, I will review and discuss the importance of these different new results.     

A model for the enhancement of fitness in cyanobacteria based on resonance of a circadian oscillator with the external light-dark cycle

Fitness enhancement based on resonating circadian clocks has recently been demonstrated in cyanobacteria [Ouyang et al. (1998). Proc. Natl Acad. Sci. U.S.A.95, 8660-8664]. Thus, the competition between two cyanobacterial strains differing by the free-running period (FRP) of their circadian oscillations leads to the dominance of one or the other of the two strains, depending on the period of the external light-dark (LD) cycle. The successful strain is generally that which has an FRP closest to the period of the LD cycle. Of key importance for the resonance phenomenon are observations which indicate that the phase angle between the circadian oscillator and the LD cycle depends both on the latter cycle's length and on the FRP. We account for these experimental observations by means of a theoretical model which takes into account (i) cell growth, (ii) secretion of a putative cell growth inhibitor, and (iii) the existence of a cellular, light-sensitive circadian oscillator controlling growth as well as inhibitor secretion. Building on a previous analysis in which the phase angle was considered as a freely adjustable parameter [Roussel et al. (2000). J. theor. Biol.205, 321-340], we incorporate into the model a light-sensitive version of the van der Pol oscillator to represent explicitly the cellular circadian oscillator. In this way, the model automatically generates a phase angle between the circadian oscillator and the LD cycle which depends on the characteristic FRP of the strain and varies continuously with the period of the LD cycle. The model provides an explanation for the results of competition experiments between strains of different FRPs subjected to entrainment by LD cycles of different periods. The model further shows how the dominance of one strain over another in LD cycles can be reconciled with the observation that two strains characterized by different FRPs nevertheless display the same growth kinetics in continuous light or in LD cycles when present alone in the medium. Theoretical predictions are made as to how the outcome of competition depends on the initial proportions and on the FRPs of the different strains. We also determine the effect of the photoperiod and extend the analysis to the case of a competition between three cyanobacterial strains.     

Quantifying the dynamics of coupled networks of switches and oscillators

Complex network dynamics have been analyzed with models of systems of coupled switches or systems of coupled oscillators. However, many complex systems are composed of components with diverse dynamics whose interactions drive the system's evolution. We, therefore, introduce a new modeling framework that describes the dynamics of networks composed of both oscillators and switches. Both oscillator synchronization and switch stability are preserved in these heterogeneous, coupled networks. Furthermore, this model recapitulates the qualitative dynamics for the yeast cell cycle consistent with the hypothesized dynamics resulting from decomposition of the regulatory network into dynamic motifs. Introducing feedback into the cell-cycle network induces qualitative dynamics analogous to limitless replicative potential that is a hallmark of cancer. As a result, the proposed model of switch and oscillator coupling provides the ability to incorporate mechanisms that underlie the synchronized stimulus response ubiquitous in biochemical systems.     

Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2

In the early embryonic cell cycle, Cdc2-cyclin B functions like an autonomous oscillator, whose robust biochemical rhythm continues even when DNA replication or mitosis is blocked. At the core of the oscillator is a negative feedback loop; cyclins accumulate and produce active mitotic Cdc2-cyclin B; Cdc2 activates the anaphase-promoting complex (APC); the APC then promotes cyclin degradation and resets Cdc2 to its inactive, interphase state. Cdc2 regulation also involves positive feedback, with active Cdc2-cyclin B stimulating its activator Cdc25 (refs 5-7) and inactivating its inhibitors Wee1 and Myt1 (refs 8-11). Under the correct circumstances, these positive feedback loops could function as a bistable trigger for mitosis, and oscillators with bistable triggers may be particularly relevant to biological applications such as cell cycle regulation. Therefore, we examined whether Cdc2 activation is bistable. We confirm that the response of Cdc2 to non-degradable cyclin B is temporally abrupt and switch-like, as would be expected if Cdc2 activation were bistable. We also show that Cdc2 activation exhibits hysteresis, a property of bistable systems with particular relevance to biochemical oscillators. These findings help establish the basic systems-level logic of the mitotic oscillator.     

The relationship between the transition probability and oscillator concepts of the cell cycle and the nature of the commitment to replication

The oscillator concept of the cell cycle predicts the existence of threshold conditions within the cell which must be exceeded before replication is initiated. It is shown (a) that if the conditions are subthreshold, random disturbances can trigger a cycle of the oscillation (round of replication) and (b) that the probability of this occurring increases as the state of the system approaches the threshold. It is concluded that the oscillator concept explains the data on which the transition probability model is based. It also accounts for the ability of cells to replicate even when the mitogen is present for a limited period only.     

Evidence toward a Dual Phosphatase Mechanism That Restricts Aurora A (Thr-295) Phosphorylation during the Early Embryonic Cell Cycle

The mitotic kinase Aurora A (AurA) is regulated by a complex network of factors that includes co-activator binding, autophosphorylation, and dephosphorylation. Dephosphorylation of AurA by PP2A (human, Ser-51; Xenopus, Ser-53) destabilizes the protein, whereas mitotic dephosphorylation of its T-loop (human, Thr-288; Xenopus, Thr-295) by PP6 represses AurA activity. However, AurA(Thr-295) phosphorylation is restricted throughout the early embryonic cell cycle, not just during M-phase, and how Thr-295 is kept dephosphorylated during interphase and whether or not this mechanism impacts the cell cycle oscillator were unknown. Titration of okadaic acid (OA) or fostriecin into Xenopus early embryonic extract revealed that phosphatase activity other than PP1 continuously suppresses AurA(Thr-295) phosphorylation during the early embryonic cell cycle. Unexpectedly, we observed that inhibiting a phosphatase activity highly sensitive to OA caused an abnormal increase in AurA(Thr-295) phosphorylation late during interphase that corresponded with delayed cyclin-dependent kinase 1 (CDK1) activation. AurA(Thr-295) phosphorylation indeed influenced this timing, because AurA isoforms retaining an intact Thr-295 residue further delayed M-phase entry. Using mathematical modeling, we determined that one phosphatase would be insufficient to restrict AurA phosphorylation and regulate CDK1 activation, whereas a dual phosphatase topology best recapitulated our experimental observations. We propose that two phosphatases target Thr-295 of AurA to prevent premature AurA activation during interphase and that phosphorylated AurA(Thr-295) acts as a competitor substrate with a CDK1-activating phosphatase in late interphase. These results suggest a novel relationship between AurA and protein phosphatases during progression throughout the early embryonic cell cycle and shed new light on potential defects caused by AurA overexpression.     

ATM/Wip1 activities at chromatin control Plk1 re‐activation to determine G2 checkpoint duration

After DNA damage, the cell cycle is arrested to avoid propagation of mutations. Arrest in G2 phase is initiated by ATM‐/ATR‐dependent signaling that inhibits mitosis‐promoting kinases such as Plk1. At the same time, Plk1 can counteract ATR‐dependent signaling and is required for eventual resumption of the cell cycle. However, what determines when Plk1 activity can resume remains unclear. Here, we use FRET‐based reporters to show that a global spread of ATM activity on chromatin and phosphorylation of ATM targets including KAP1 control Plk1 re‐activation. These phosphorylations are rapidly counteracted by the chromatin‐bound phosphatase Wip1, allowing cell cycle restart despite persistent ATM activity present at DNA lesions. Combining experimental data and mathematical modeling, we propose a model for how the minimal duration of cell cycle arrest is controlled. Our model shows how cell cycle restart can occur before completion of DNA repair and suggests a mechanism for checkpoint adaptation in human cells.