Synchronization ability of coupled cell-cycle oscillators in changing environments

Background

The biochemical oscillator that controls periodic events during the Xenopus embryonic cell cycle is centered on the activity of CDKs, and the cell cycle is driven by a protein circuit that is centered on the cyclin-dependent protein kinase CDK1 and the anaphase-promoting complex (APC). Many studies have been conducted to confirm that the interactions in the cell cycle can produce oscillations and predict behaviors such as synchronization, but much less is known about how the various elaborations and collective behavior of the basic oscillators can affect the robustness of the system. Therefore, in this study, we investigate and model a multi-cell system of the Xenopus embryonic cell cycle oscillators that are coupled through a common complex protein, and then analyze their synchronization ability under four different external stimuli, including a constant input signal, a square-wave periodic signal, a sinusoidal signal and a noise signal.

Results

Through bifurcation analysis and numerical simulations, we obtain synchronization intervals of the sensitive parameters in the individual oscillator and the coupling parameters in the coupled oscillators. Then, we analyze the effects of these parameters on the synchronization period and amplitude, and find interesting phenomena, e.g., there are two synchronization intervals with activation coefficient in the Hill function of the activated CDK1 that activates the Plk1, and different synchronization intervals have distinct influences on the synchronization period and amplitude. To quantify the speediness and robustness of the synchronization, we use two quantities, the synchronization time and the robustness index, to evaluate the synchronization ability. More interestingly, we find that the coupled system has an optimal signal strength that maximizes the synchronization index under different external stimuli. Simulation results also show that the ability and robustness of the synchronization for the square-wave periodic signal of cyclin synthesis is strongest in comparison to the other three different signals.

Conclusions

These results suggest that the reaction process in which the activated cyclin-CDK1 activates the Plk1 has a very important influence on the synchronization ability of the coupled system, and the square-wave periodic signal of cyclin synthesis is more conducive to the synchronization and robustness of the coupled cell-cycle oscillators. Our study provides insight into the internal mechanisms of the cell cycle system and helps to generate hypotheses for further research.

     

Mouse Hair Cycle Expression Dynamics Modeled as Coupled Mesenchymal and Epithelial Oscillators

The hair cycle is a dynamic process where follicles repeatedly move through phases of growth, retraction, and relative quiescence. This process is an example of temporal and spatial biological complexity. Understanding of the hair cycle and its regulation would shed light on many other complex systems relevant to biological and medical research. Currently, a systematic characterization of gene expression and summarization within the context of a mathematical model is not yet available. Given the cyclic nature of the hair cycle, we felt it was important to consider a subset of genes with periodic expression. To this end, we combined several mathematical approaches with high-throughput, whole mouse skin, mRNA expression data to characterize aspects of the dynamics and the possible cell populations corresponding to potentially periodic patterns. In particular two gene clusters, demonstrating properties of out-of-phase synchronized expression, were identified. A mean field, phase coupled oscillator model was shown to quantitatively recapitulate the synchronization observed in the data. Furthermore, we found only one configuration of positive-negative coupling to be dynamically stable, which provided insight on general features of the regulation. Subsequent bifurcation analysis was able to identify and describe alternate states based on perturbation of system parameters. A 2-population mixture model and cell type enrichment was used to associate the two gene clusters to features of background mesenchymal populations and rapidly expanding follicular epithelial cells. Distinct timing and localization of expression was also shown by RNA and protein imaging for representative genes. Taken together, the evidence suggests that synchronization between expanding epithelial and background mesenchymal cells may be maintained, in part, by inhibitory regulation, and potential mediators of this regulation were identified. Furthermore, the model suggests that impairing this negative regulation will drive a bifurcation which may represent transition into a pathological state such as hair miniaturization.     

PERIODIC THERMAL STRESS: OPTIMAL SPACING PREDICTED BY A SIMPLE SET-BACK MODEL OF CELL DIVISION SYNCHRONIZATION

Digital computer simulations have been used to make quantitative predictions based on a simple set-back model of cell division synchronization. According to the model appropriate thermal stress reverses progress within a segment of the division cycle called the set-back interval. In the simulations normally distributed cell-to-cell variations in division cycling rate between periodic thermal shocks were produced with the Monte Carlo method.
The simulations have shown that reasonably good synchronization with the single shock per division strategy requires a relatively long set-back interval and small cell-to-cell variations in rate of progress through the division cycle. The simulations have shown that the degree of synchrony produced by such periodic shocks is highly dependent on the time interval between shocks—with a series of as many as seven shocks inappropriately spaced producing less synchrony than a single shock! The optimal time interval between successive thermal shocks was found approximately equal to the mode division cycle time at synchrony equilibrium multiplied by 1 plus half the fraction of the division cycle occupied by the set-back interval. Position of the set-back interval within the division cycle had little effect on synchrony at the end of the final shock.     

Analysis of cell cycle phases and progression in cultured mammalian cells

Fluorescence activated cell sorting (FACS) analysis has become a standard tool to analyze cell cycle distributions in populations of cells. These methods require relatively large numbers of cells, and do not provide optimal resolution of the transitions between cell cycle phases. In this report we describe in detail complementary methods that utilize the incorporation of nucleotide analogs combined with microscopic examination. While often more time consuming, these protocols typically require far fewer cells, and allow accurate kinetic assessment of cell cycle progression. We also describe the use of a technique for the synchronization of adherent cells in mitosis by simple mechanical agitation (mitotic shake-off) that eliminates physiological perturbation associated with drug treatments.     

Modeling and optimization of alpha-amylase production in a recombinant yeast fed-batch culture taking account of the cell cycle population distribution.

A simple mathematical model describing the cell cycle dependency of rice alpha-amylase production by a recombinant yeast was constructed to investigate the efficiency of cell cycle population control. First, the effects of the glucose concentration and cultivation temperature on the specific growth rate, the specific production rate of rice alpha-amylase, and the distribution of the cell cycle population were studied under balanced growth conditions. On the basis of the results, parameter values for the mathematical model were then estimated. The proposed model was shown to be applicable for unbalanced as well as balanced growth phases. The optimal control strategy in respect of temperature and glucose concentration for maximum rice alpha-amylase production, taking into account the cell cycle population, was determined and the result was compared with that obtained by a simple mathematical model in which cell cycle distribution was not considered. Finally, the effect of the initial population of each cell cycle phase on the final amount of the product under optimal operational conditions was investigated. The simulation and experimental data coincided well with each other, and the model was used to optimize the control strategy for maximum alpha-amylase production.     

Spatio-temporal dynamics of the grey-sided vole in Hokkaido: identifying coupling using state-based Markov-chain modelling

Explaining synchronization of cyclical or fluctuating populations over geographical regions presents ecologists with novel analytical challenges. We have developed a method to measure synchrony within spatial-temporal datasets of population densities applicable to both periodic and irregularly fluctuating populations. The dynamics of each constituent population is represented by a discrete Markov model. The state of a population trajectory at each time-point is classified as one of 'increase', 'decrease', 'peak' or 'trough'. The set of populations at any time-point is characterized by the frequency distribution of these different states, and the time-evolution of this frequency distribution used to test the hypothesis that the dynamics of each population proceeds independently of the others. The analysis identifies years in which population coupling results in synchronous states and onto which states the system converges, and identifies those years in which synchrony remains high but is accounted for by coupling observed in previous years. It also enables identification of which pairs of sites show the highest levels of coupling. Applying these methods to populations of the grey-sided vole on Hokkaido reveals them to be fluctuating in greater synchrony than would be expected from independent dynamics, and that this level of synchrony is maintained through intermittent coupling acting in ca. 1 year in four or five. High synchrony occurs between sites with similar vegetation and of similar altitude indicating that coupling may be mediated through shared environmental stimuli. When coupling is indicated, convergence is equally likely to occur on a peak state as a trough, indicating that synchronization may be brought about by the response of populations to a combination of different stimuli rather than by the action of any single process.     

Involvement of a cell size control mechanism in the induction and maintenance of oscillations in continuous cultures of budding yeast

Spontaneous oscillations occur in glucose-limited continuous cultures of Saccharomyces cerevisiae under aerobic conditions. The oscillatory behavior is detectable as a periodic change of many bioparameters such as dissolved oxygen, ethanol production, biomass concentration, as well as cellular content of storage carbohydrates and is associated to a marked synchronization of the yeast population. These oscillations may be related to a periodic accumulation of ethanol produced by yeast in the culture medium.The addition of ethanol to oscillating yeast cultures supports this hypothesis: indeed, no effect was observed if ethanol was added when already present in the medium, while a marked phase oscillation shift was obtained when ethanol was added at any other time. Moreover, the addition of ethanol to a nonoscillating culture triggers new oscillations. An accurate analysis performed at the level of nonoscillating yeast populations perturbed by addition of ethanol showed that both the growth rate and the protein content required for cell division increased in the presence of mixed substrate (i.e., ethanol plus limiting glucose). A marked synchronization of the yeast population occurred when the added ethanol was exhausted and the culture resumed growth only on limiting glucose. A decrease of protein content required for cell division was also apparent. These experimental findings support a new model for spontaneous oscillations in yeast cultures in which the alternative growth on limiting glucose and limiting glucose plus ethanol modifies the critical protein content required for cell division.     

Optimization of the exposure of normal and transformed cell populations to phase-specific agents

Dynamic behavior of stem cells population of the "critical" tissue (normal population) and tumor cell population under periodic treatment with a phase-specific cytotoxic agent was considered. The results were used for optimization of anticancer chemotherapy. The schedules of treatment were found which provide a maximum rate of tumor-cell elimination for any given rate of the normal population size decrease. If the mean generation times of normal and tumor populations differ (which was stated for many tumors), usage of the optimal period markedly increases the selectivity of therapy, while application of other periods can result in selective elimination of the normal population. Problems concerned with practical realization of the proposed regimes are discussed.     

A simple model for cell volume and developmental compartments in nutrient limited cyclostat cultures of algae

Daily light-dark cycles can entrain cell growth and division cycles in populations of algae growing in nutrient limited continuous cultures, or cyclostats. In this study a simple model for the flux of cells between discrete developmental stages is formulated for periodic cyclostat cultures of algae. Cell growth, in terms of volume, was set as being constant within a given developmental compartment, but variable between compartments. Growth within a given compartment or transition between compartments was restricted to specific intervals of the subjective day. The model was calibrated to phosphate limited cyclostat growth of Euglena gracilis, with the intervals for transition between compartments fixed at the times relative to the subjective dawn corresponding to critical transition points in the phased cell cycle of this organism. The model output for mean population volume per cell agreed well with experimental data. Although greatly simplified, the periodic behavior of the model volume frequency distributions for the discrete compartments provide reasonable approximation of experimentally determined distributions.     

The influence of ion concentrations on the dynamic behavior of the Hodgkin–Huxley model-based cortical network

Action potentials (APs) in the form of very short pulses arise when the cell is excited by any internal or external stimulus exceeding the critical threshold of the membrane. During AP generation, the membrane potential completes its natural cycle through typical phases that can be formatted by ion channels, gates and ion concentrations, as well as the synaptic excitation rate. On the basis of the Hodgkin–Huxley cell model, a cortical network consistent with the real anatomic structure is realized with randomly interrelated small population of neurons to simulate a cerebral cortex segment. Using this model, we investigated the effects of Na⁺ and K⁺ ion concentrations on the outcome of this network in terms of regularity, phase locking, and synchronization. The results suggested that Na⁺ concentration does slightly affect the amplitude but not considerably affects the other parameters specified by depolarization and repolarization. K⁺ concentration significantly influences the form, regularity, and synchrony of the network-generated APs. No previous study dealing directly with the effects of both Na⁺ and K⁺ ion concentrations on regularity and synchronization of the simulated cortical network-generated APs, allowing for the comparison of results obtained using our methods, was encountered in the literature. The results, however, were consistent with those obtained through studies concerning resonance and synchronization from another perspective and with the information revealed through physiological and pharmacological experiments concerning changing ion concentrations or blocking ion channels. Our results demonstrated that the regularity and reliability of brain functions have a strong relationship with cellular ion concentrations, and suggested the management of the dynamic behavior of the cellular network with ion concentrations.     

Cell cycle-dependent protein secretion by Saccharomyces cerevisiae.

Synchronized Saccharomyces cerevisiae cell populations were used to examine secretion rates of a heterologous protein as a function of cell cycle position. The synchronization procedure had a profound effect on the type and quality of data obtained. When cell synchrony was induced by cell cycle-arresting drugs, a significant physiological perturbation of cells was observed that obscured representative secretion data. In contrast, synchronization with centrifugal elutriation resulted in synchronized first-generation daughter cells with undetectable perturbation of the physiological state. The synchronized cells did not secrete significant amounts of protein until they reached cell division, suggesting that the secretion process in these cells is strongly cell cycle dependent. However, the maximum secretion rate of the synchronized culture (7-14 molecules/cell/second) was significantly lower than that of an asynchronous culture (29-51 molecules/cell/second). This result indicates that young daughter cells isolated in the synchronization process exhibit different protein secretion behavior than older mother cells that are absent in the synchronized cell population but present in the asynchronous culture.     

Tumor Growth Rate Determines the Timing of Optimal Chronomodulated Treatment Schedules

In host and cancer tissues, drug metabolism and susceptibility to drugs vary in a circadian (24 h) manner. In particular, the efficacy of a cell cycle specific (CCS) cytotoxic agent is affected by the daily modulation of cell cycle activity in the target tissues. Anti-cancer chronotherapy, in which treatments are administered at a particular time each day, aims at exploiting these biological rhythms to reduce toxicity and improve efficacy of the treatment. The circadian status, which is the timing of physiological and behavioral activity relative to daily environmental cues, largely determines the best timing of treatments. However, the influence of variations in tumor kinetics has not been considered in determining appropriate treatment schedules. We used a simple model for cell populations under chronomodulated treatment to identify which biological parameters are important for the successful design of a chronotherapy strategy. We show that the duration of the phase of the cell cycle targeted by the treatment and the cell proliferation rate are crucial in determining the best times to administer CCS drugs. Thus, optimal treatment times depend not only on the circadian status of the patient but also on the cell cycle kinetics of the tumor. Then, we developed a theoretical analysis of treatment outcome (TATO) to relate the circadian status and cell cycle kinetic parameters to the treatment outcomes. We show that the best and the worst CCS drug administration schedules are those with 24 h intervals, implying that 24 h chronomodulated treatments can be ineffective or even harmful if administered at wrong circadian times. We show that for certain tumors, administration times at intervals different from 24 h may reduce these risks without compromising overall efficacy.     

Set-back model of division-synchronization: optimal spacing of thermal shocks that accelerate cell cycling.

Monte Carlo simulations have been used to predict the dependence of synchrony on the timing of periodic thermal shocks that synchronize division by cell cycle set-backs. In many of the simulations each set-back augmented the subsequent rate of progression of individual cells through the division cycle. In this study a subtle error in previous synchronization simulations was corrected. The simulations show that whether or not set-backs affect subsequent cell-cycling rates the degree of synchrony attained is acutely dependent on the spacing of thermal shocks administered once per division. Set-back-dependent increases in division-cycling rates usually decrease the difference between maximum and minimum synchrony. According to the simulations the more cell cycle rates between shocks are augmented by set-back the shorter the optimum time span between shocks. Whether or not set-backs affect subsequent division-cycling rates the intershock time span providing maximum synchrony allows cell number to precisely double.     

Identification of proteins whose synthesis is modulated during the cell cycle of Saccharomyces cerevisiae

We examined the synthesis and turnover of individual proteins in the Saccharomyces cerevisiae cell cycle. Proteins were pulse-labeled with radioactive isotope (35S or 14C) in cells at discrete cycle stages and then resolved on two-dimensional gels and analyzed by a semiautomatic procedure for quantitating gel electropherogram-autoradiographs. The cells were obtained by one of three methods: (i) isolation of synchronous subpopulations of growing cells by zonal centrifugation.; (ii) fractionation of pulse-labeled steady-state cultures according to cell age; and (iii) synchronization of cells with the mating pheromone, alpha-factor. In confirmation of previous studies, we found that the histones H4, H2A, and H2B were synthesized almost exclusively in the late G1 and early S phases. In addition, we identified eight proteins whose rates of synthesis were modulated in the cell cycle, and nine proteins (of which five, which may well be related, were unstable, with half-lives of 10 to 15 min) that might be regulated in the cell cycle by periodic synthesis, modification, or degradation. Based on the time of maximal labeling in the cell cycle and on experiments with alpha-factor and hydroxyurea, we assigned the cell cycle proteins to two classes: proteins in class I were labeled principally in early G1 phase and at a late stage of the cycle, whereas those in class II were primarily synthesized at times ranging from late G1 to mid S phase. At least one major control point for the cell cycle proteins occurred between "start" and early S phase. A set of stress-responsive proteins was also identified and analyzed. The rates of synthesis of these proteins were affected by certain perturbations that resulted during selection of synchronous cell populations and by heat shock.     

Quasi-exponential generation time distributions from a limit cycle oscillator

In spite of the apparently random behaviour and the often exponential distribution of generation times expressed in cell populations, there is evidence for rather precise timekeeping in the cell cycle. In experiments using time-lapse video-tape microscopy, we have noted that cell generation times are often not distributed smoothly but in many cases seem to cluster at roughly 4 hr intervals. Phase shift responses following application of heat shock, ionizing radiation or serum pulses in each case show a pattern which is repeated twice in cells with an 8-9 hr modal generation time. We describe here a cell cycle model with an independent cellular clock controlling cell cycle events which accounts for the phase response data, while also reconciling the stochastic and periodic behaviour characteristic of animal cells.     

Quasi-Exponential Generation Time Distributions From A Limit Cycle Oscillator

In spite of the apparently random behaviour and the often exponential distribution of generation times expressed in cell populations, there is evidence for rather precise timekeeping in the cell cycle. In experiments using time-lapse video-tape microscopy, we have noted that cell generation times are often not distributed smoothly but in many cases seem to cluster at roughly 4 hr intervals. Phase shift responses following application of heat shock, ionizing radiation or serum pulses in each case show a pattern which is repeated twice in cells with an 8–9 hr modal generation time. We describe here a cell cycle model with an independent cellular clock controlling cell cycle events which accounts for the phase response data, while also reconciling the stochastic and periodic behaviour characteristic of animal cells.     

Intermittent erythropoiesis in anemic newts

A group of 88 newts, Triturus cristatus carnifex (Laurenti), was rendered totally anemic by administering acetylphenylhydrazine (APH) in the breeding water for 48 h at a concentration of 25 mg/liter. The course of erythron restoration was followed for 5 months, sacrificing four specimens per week and analyzing the blood and spleen hemopoietic tissue. The return to the normal values of the red blood cell count occurred through marked increases in concentration at fairly regular intervals, which is best explained by a discontinuous, rhythmic erythropoiesis. This fact is strictly correlated with the intermittent mitotic activity observed in the spleen and with the periodic appearance of large quantities of immature elements in the blood smears. The APH-induced synchronization of newt erythropoietic activity revealed the approximate length of each erythropoietic cycle to be 4 to 5 weeks and the erythropoietic life span to be 50 to 60 days.     

Cell synchrony techniques. I. A comparison of methods

Selected cell synchrony techniques, as applied to asynchronous populations of Chinese hamster ovary (CHO) cells, have been compared. Aliquots from the same culture of exponentially growing cells were synchronized using mitotic selection, mitotic selection and hydroxyurea block, centrifugal elutriation, or an EPICS V cell sorter. Sorting of cells was achieved after staining cells with Hoechst 33258. After synchronization by the various methods the relative distribution of cells in G1, S, or G2 + M phases of the cell cycle was determined by flow cytometry. Fractions of synchronized cells obtained from each method were replated and allowed to progress through a second cell cycle. Mitotic selection gave rise to relatively pure and unperturbed early G1 phase cells. While cell synchrony rapidly dispersed with time, cells progressed through the cell cycle in 12 hr. Sorting with the EPICS V on the modal G1 peak yielded a relatively pure but heterogeneous G1 population (i.e. early to late G1). Again, synchrony dispersed with time, but cell-cycle progression required 14 hr. With centrifugal elutriation, several different cell populations synchronized throughout the cell cycle could be rapidly obtained with a purity comparable to mitotic selection and cell sorting. It was concluded that, either alone or in combination with blocking agents such as hydroxyurea, elutriation and mitotic selection were both excellent methods for synchronizing CHO cells. Cell sorting exhibited limitations in sample size and time required for synchronizing CHO cells. Its major advantage would be its ability to isolate cell populations unique with respect to selected cellular parameters.