The cell division cycle: a physiologically plausible dynamic model can exhibit chaotic solutions.

A mitotic oscillator with one slowly increasing variable (tau L of the order of hours) and one rapidly increasing variable (tau R of the order of minutes) modulated by a timer (ultradian clock) gives an auto-oscillating solution: cells divide when this relaxation oscillator reaches a critical threshold to initiate a rapid phase of the limit cycle. Increasing values of the velocity constant in the slow equation give quasi-periodic, chaotic and periodic solutions. Thus dispersed and quantized cell cycle times are consequences of a chaotic trajectory and have a purely deterministic basis. This model of the dispersion of cell cycle times contrasts with many previous ones in which cell cycle variability is a consequence of stochastic properties inherent in a sequence of many thousands of reactions or the random nature of a key transition step.     

The Cell Cycle

I review recent advances in our knowledge of the eucaryoticcell cycle: the set of processes by which cells grow and divide.Genetic approaches to the cell cycle of somatic cells identifieda pathway of events where the initiation of each event was dependenton the successful completion of the preceding event, as wellas a single key gene, cdc2, that is required both at the beginningand at the end of the cell cycle. The alternative approach ofstudying the cell cycle biochemically in early embryos providedevidence for a cytoplasmic oscillator which alternated betweenmitosis-inducing and interphase-inducing states and identifiedthe mitosis-inducing component as maturation promoting factor(MPF). These two very different views of the cell cycle initiallyseemed irreconcilable. However, a link between the somatic andembryonic cell cycles was provided by the recent discovery thatthe cdc2 protein is one of the components of MPF. In the embryoniccell cycle the activation of MPF and induction of mitosis istriggered by the accumulation of a protein named cyclin whichbecomes a component of MPF. Somehow, MPF induces the proteolyticdegradation of cyclin, which inturn allows MPF to be inactivatedand allows the cell cycle to pass from mitosis into interphase.The more complex cell cycle of somatic cells is probably derivedfrom the embryonic cyclin-based oscillator by imposing a systemof checks and balances on the accumulation and destruction ofcyclin. I also present some thoughts on the relationships between scienceand society, and comment on the way in which scientists describetheir work to the lay world.     

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.     

Non-weak inhibition and phase resetting at negative values of phase in cells with fast-slow dynamics at hyperpolarized potentials

Phase response is a powerful concept in the analysis of both weakly and non-weakly perturbed oscillators such as regularly spiking neurons, and is applicable if the oscillator returns to its limit cycle trajectory between successive perturbations. When the latter condition is violated, a formal application of the phase return map may yield phase values outside of its definition domain; in particular, strong synaptic inhibition may result in negative values of phase. The effect of a second perturbation arriving close to the first one is undetermined in this case. However, here we show that for a Morris–Lecar model of a spiking cell with strong time scale separation, extending the phase response function definition domain to an additional negative value branch allows to retain the accuracy of the phase response approach in the face of such strong inhibitory coupling. We use the resulting extended phase response function to accurately describe the response of a Morris–Lecar oscillator to consecutive non-weak synaptic inputs. This method is particularly useful when analyzing the dynamics of three or more non-weakly coupled cells, whereby more than one synaptic perturbation arrives per oscillation cycle into each cell. The method of perturbation prediction based on the negative-phase extension of the phase response function may be applicable to other excitable cell models characterized by slow voltage dynamics at hyperpolarized potentials.     

Rapid cycling and precocious termination of G1 phase in cells expressing CDK1AF

In Xenopus embryos, the cell cycle is driven by an autonomous biochemical oscillator that controls the periodic activation and inactivation of cyclin B1-CDK1. The oscillator circuit includes a system of three interlinked positive and double-negative feedback loops (CDK1 -> Cdc25 -> CDK1; CDK1 -/ Wee1 -/ CDK1; and CDK1 -/ Myt1 -/ CDK1) that collectively function as a bistable trigger. Previous work established that this bistable trigger is essential for CDK1 oscillations in the early embryonic cell cycle. Here, we assess the importance of the trigger in the somatic cell cycle, where checkpoints and additional regulatory mechanisms could render it dispensable. Our approach was to express the phosphorylation site mutant CDK1AF, which short-circuits the feedback loops, in HeLa cells, and to monitor cell cycle progression by live cell fluorescence microscopy. We found that CDK1AF-expressing cells carry out a relatively normal first mitosis, but then undergo rapid cycles of cyclin B1 accumulation and destruction at intervals of 3-6 h. During these cycles, the cells enter and exit M phase-like states without carrying out cytokinesis or karyokinesis. Phenotypically similar rapid cycles were seen in Wee1 knockdown cells. These findings show that the interplay between CDK1, Wee1/Myt1, and Cdc25 is required for the establishment of G1 phase, for the normal approximately 20-h cell cycle period, and for the switch-like oscillations in cyclin B1 abundance characteristic of the somatic cell cycle. We propose that the HeLa cell cycle is built upon an unreliable negative feedback oscillator and that the normal high reliability, slow pace and switch-like character of the cycle is imposed by a bistable CDK1/Wee1/Myt1/Cdc25 system.     

Dynamics from a time series: can we extract the phase resetting curve from a time series?

Recordings of the membrane potential from a bursting neuron were used to reconstruct the phase curve for that neuron for a limited set of perturbations. These perturbations were inhibitory synaptic conductance pulses able to shift the membrane potential below the most hyperpolarized level attained in the free running mode. The extraction of the phase resetting curve from such a one-dimensional time series requires reconstruction of the periodic activity in the form of a limit cycle attractor. Resetting was found to have two components. In the first component, if the pulse was applied during a burst, the burst was truncated, and the time until the next burst was shortened in a manner predicted by movement normal to the limit cycle. By movement normal to the limit cycle, we mean a switch between two well-defined solution branches of a relaxation-like oscillator in a hysteretic manner enabled by the existence of a singular dominant slow process (variable). In the second component, the onset of the burst was delayed until the end of the hyperpolarizing pulse. Thus, for the pulse amplitudes we studied, resetting was independent of amplitude but increased linearly with pulse duration. The predicted and the experimental phase resetting curves for a pyloric dilator neuron show satisfactory agreement. The method was applied to only one pulse per cycle, but our results suggest it could easily be generalized to accommodate multiple inputs.     

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.     

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.     

An autonomous cell-cycle oscillator involved in the coordination of G1 events

In early embryonic development, the cell cycle is paced by a biochemical oscillator involving cyclins and cyclin-dependent kinases (cdks). Essentially the same machinery operates in all eukaryotic cells, although after the first few divisions various braking mechanisms (the so-called checkpoints) become significant. Haase and Reed have recently shown that yeast cells have a second, independent oscillator which coordinates some of the events of the G1 phase of the cell cycle.(1) Although the biochemical nature of this oscillator is not known,it seems unlikely to be a redundant cyclin/cdk system. BioEssays 22:3-5, 2000.     

Apparent discontinuities in the phase-resetting response of cardiac pacemakers

Injection of a brief stimulus pulse resets the spontaneous periodic activity of a sinoatrial node cell: a stimulus delivered early in the cycle generally delays the time of occurrence of the next action potential, while the same stimulus delivered later causes an advance. We investigate resetting in two models, one with a slow upstroke velocity and the other with a fast upstroke velocity, representing central and peripheral nodal cells, respectively. We first formulate each of these models as a classic Hodgkin-Huxley type of model and then as a model representing a population of single channels. In the Hodgkin-Huxley-type model of the slow-upstroke cell the transition from delay to advance is steep but continuous. In the corresponding single-channel model, due to the channel noise then present, repeated resetting runs at a fixed stimulus timing within the transitional range of coupling intervals lead to responses that span a range of advances and delays. In contrast, in the fast-upstroke model the transition from advance to delay is very abrupt in both classes of model, as it is in experiments on some cardiac preparations ("all-or-none" depolarization). We reduce the fast-upstroke model from the original seven-dimensional system to a three-dimensional system. The abrupt transition occurs in this reduced model when a stimulus transports the state point to one side or the other of the stable manifold of the trajectory corresponding to the eigendirection associated with the smaller of two positive eigenvalues. This stable manifold is close to the slow manifold, and so canard trajectories are seen. Our results demonstrate that the resetting response is fundamentally continuous, but extremely delicate, and thus suggest one way in which one can account for experimental discontinuities in the resetting response of a nonlinear oscillator.     

The ups and downs of modeling the cell cycle

We discuss the impact of mathematical modeling on our understanding of the cell cycle. Although existing, detailed models confirm that the known interactions in the cell cycle can produce oscillations and predict behaviors such as hysteresis, they contain many parameters and are poorly constrained by data which are almost all qualitative. Questions about the basic architecture of the oscillator may be more amenable to modeling approaches that ignore molecular details. These include asking how the various elaborations of the basic oscillator affect the robustness of the system and how cells monitor their size and use this information to control the cell cycle.     

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.     

The cell cycle of lymphocytes in Fanconi anemia

Summary BrdU-incorporation techniques were used to study the cell cycle in 18 cases of Fanconi's anemia (FA).By comparison with controls, a significant slowing of the cell cycle of lymphocytes in vitro was observed in all FA patients, and possibly in FA heterozygotes, although to a lesser degree. It is probable that the demonstration of the slowing is dependent on the culture conditions. No slowing was observed in other patients affected by at least one of the symptoms of FA. The slow cell cycle of FA cells is mostly due to a very long G₂-phase. A relationship between slow cell cycle and chromatid anomalies exists, the slower cells being significantly more frequently carriers of radial figures than the faster cells, in the same patient.     

Circadian range of entrainment in the semilunar eclosion rhythm of the marine insect clunio marinus

The semilunar eclosion of the intertidal chironomid Clunio is controlled by a semilunar timing of pupation in combination with a daily timing of emergence. This results in reproductive activities of a laboratory population every 15 days at a distinct time of day (in nature mostly in correlation with the afternoon low water time on days with spring tides). The entrainment of the timing processes has been tested under various periods of the daily light-dark cycle in order to check the circadian organization of the timing mechanisms as suggested for the perception of the semilunar zeitgeber situation (a distinct phase relationship between the 24 h light-dark cycle and the 12.4 h tidal cycle recurring after every 15th light-dark cycle, named semimonthly zeitgeber cycle) as well as for the daily zeitgeber (the 24 h light-dark cycle). With respect to the semilunar timing, a strong entrainment was only possible in semimonthly zeitgeber cycles with light-dark cycle periods close to the 24-h day (light-dark cycles of 10:10 to 14:14). This limited circadian range of entrainment of an endogenous circasemilunar long-term rhythm (syn. oscillator) conforms with the hypothesis for a circadian clock component as an intrinsic part of the semilunar zeitgeber perception.The range of entrainment for the daily timing was obviously wider which may be discussed either in relation to a multioscillatory circadian organization of the midges or in relation to different coupling characteristics of one circadian oscillator during semilunar and daily timing.     

A modified radial isochron clock with slow and fast dynamics as a model of pacemaker neurons

A simple mathematical model of living pacemaker neurons is proposed. The model has a unit circle limit cycle and radial isochrons, and the state point moves slowly in one region and fast in the remaining region; regions can correspond to the subthreshold activity and to the action potentials of pacemaker neurons, respectively. The global bifurcation structure when driven by periodic pulse trains was investigated using one-dimensional maps (PTC), two-dimensional bifurcation diagrams, and skeletons involving stimulus period and intensity. The existence of both the slow and the fast dynamics has a critical influence on the global bifurcation structure of the oscillator when stimulated periodically.Supported by Trent H. Wells Jr. Inc.     

Two distinct states of sugar transport system in cultures of BHK cells

The sugar transport of growing and quiescent cultures of BHK-21 cells is studied by the equilibrium exchange method. Two distinct components of sugar transport can be detected. One component displays fast transport rates and is evident in cells at low cell density. The other displays slow transport properties and is typical of quiescent cells. In the course of increase in cell density or following serum-activation of quiescent cells, these two components are present in the same cell-culture. The two components of transport are interpreted as resulting from the presence of two types of cells, one in a “fast” and the other in a “slow” transport state. The transition in each cell from one state of transport into the other appears to be a discrete and sudden event. The gradual change in the cell population results from a change in the number of cells in each state. Cells in the fast transport state show a saturable and a non saturable component of sugar transport. Cells in the slow transport state display only a non saturable component.