A stochastic, molecular model of the fission yeast cell cycle: role of the nucleocytoplasmic ratio in cycle time regulation

A hybrid stochastic model of the budding yeast cell cycle

The growth and division of eukaryotic cells are regulated by complex, multi-scale networks. In this process, the mechanism of controlling cell-cycle progression has to be robust against inherent noise in the system. In this paper, a hybrid stochastic model is developed to study the effects of noise on the control mechanism of the budding yeast cell cycle. The modeling approach leverages, in a single multi-scale model, the advantages of two regimes: (1) the computational efficiency of a deterministic approach, and (2) the accuracy of stochastic simulations. Our results show that this hybrid stochastic model achieves high computational efficiency while generating simulation results that match very well with published experimental measurements.

Cellular and metabolic engineering of oleaginous yeast Yarrowia lipolytica for bioconversion of hydrophobic substrates into high-value products

The non-conventional oleaginous yeast Yarrowia lipolytica is able to utilize both hydrophilic and hydrophobic carbon sources as substrates and convert them into value-added bioproducts such as organic acids, extracellular proteins, wax esters, long-chain diacids, fatty acid ethyl esters, carotenoids and omega-3 fatty acids. Metabolic pathway analysis and previous research results show that hydrophobic substrates are potentially more preferred by Y. lipolytica than hydrophilic substrates to make high-value products at higher productivity, titer, rate, and yield. Hence, Y. lipolytica is becoming an efficient and promising biomanufacturing platform due to its capabilities in biosynthesis of extracellular lipases and directly converting the extracellular triacylglycerol oils and fats into high-value products. It is believed that the cell size and morphology of the Y. lipolytica is related to the cell growth, nutrient uptake, and product formation. Dimorphic Y. lipolytica demonstrates the yeast-to-hypha transition in response to the extracellular environments and genetic background. Yeast-to-hyphal transition regulating genes, such as YlBEM1, YlMHY1 and YlZNC1 and so forth, have been identified to involve as major transcriptional factors that control morphology transition in Y. lipolytica. The connection of the cell polarization including cell cycle and the dimorphic transition with the cell size and morphology in Y. lipolytica adapting to new growth are reviewed and discussed. This review also summarizes the general and advanced genetic tools that are used to build a Y. lipolytica biomanufacturing platform.

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.

Systems Level Modeling of the Cell Cycle Using Budding Yeast

Proteins involved in the regulation of the cell cycle are highly conserved across all eukaryotes, and so a relatively simple eukaryote such as yeast can provide insight into a variety of cell cycle perturbations including those that occur in human cancer. To date, the budding yeast Saccharomyces cerevisiae has provided the largest amount of experimental and modeling data on the progression of the cell cycle, making it a logical choice for in-depth studies of this process. Moreover, the advent of methods for collection of high-throughput genome, transcriptome, and proteome data has provided a means to collect and precisely quantify simultaneous cell cycle gene transcript and protein levels, permitting modeling of the cell cycle on the systems level. With the appropriate mathematical framework and sufficient and accurate data on cell cycle components, it should be possible to create a model of the cell cycle that not only effectively describes its operation, but can also predict responses to perturbations such as variation in protein levels and responses to external stimuli including targeted inhibition by drugs. In this review, we summarize existing data on the yeast cell cycle, proteomics technologies for quantifying cell cycle proteins, and the mathematical frameworks that can integrate this data into representative and effective models. Systems level modeling of the cell cycle will require the integration of high-quality data with the appropriate mathematical framework, which can currently be attained through the combination of dynamic modeling based on proteomics data and using yeast as a model organism.

Analysis of Noise Mechanisms in Cell-Size Control

At the single-cell level, noise arises from multiple sources, such as inherent stochasticity of biomolecular processes, random partitioning of resources at division, and fluctuations in cellular growth rates. How these diverse noise mechanisms combine to drive variations in cell size within an isoclonal population is not well understood. Here, we investigate the contributions of different noise sources in well-known paradigms of cell-size control, such as adder (division occurs after adding a fixed size from birth), sizer (division occurs after reaching a size threshold), and timer (division occurs after a fixed time from birth). Analysis reveals that variation in cell size is most sensitive to errors in partitioning of volume among daughter cells, and not surprisingly, this process is well regulated among microbes. Moreover, depending on the dominant noise mechanism, different size-control strategies (or a combination of them) provide efficient buffering of size variations. We further explore mixer models of size control, where a timer phase precedes/follows an adder, as has been proposed in Caulobacter crescentus. Although mixing a timer and an adder can sometimes attenuate size variations, it invariably leads to higher-order moments growing unboundedly over time. This results in a power-law distribution for the cell size, with an exponent that depends inversely on the noise in the timer phase. Consistent with theory, we find evidence of power-law statistics in the tail of C. crescentus cell-size distribution, although there is a discrepancy between the observed power-law exponent and that predicted from the noise parameters. The discrepancy, however, is removed after data reveal that the size added by individual newborns in the adder phase itself exhibits power-law statistics. Taken together, this study provides key insights into the role of noise mechanisms in size homeostasis, and suggests an inextricable link between timer-based models of size control and heavy-tailed cell-size distributions.

Role of Computational Modeling in Understanding Cell Cycle Oscillators

The periodic oscillations in the activity of the cell cycle regulatory program, drives the timely activation of key cell cycle events. Interesting dynamical systems, such as oscillators, have been investigated by various theoretical and computational modeling methods. Thanks to the insights achieved by these modeling efforts we have gained considerable insights about the underlying molecular regulatory networks that can drive cell cycle oscillations. Here we review the basic features and characteristics of biological oscillators, discussing from a computational modeling point of view their specific architectures and the current knowledge about the dynamics that the life evolution selected to drive cell cycle oscillations.

Measurement and modeling of transcriptional noise in the cell cycle regulatory network

Fifty years of genetic and molecular experiments have revealed a wealth of molecular interactions involved in the control of cell division. In light of the complexity of this control system, mathematical modeling has proved useful in analyzing biochemical hypotheses that can be tested experimentally. Stochastic modeling has been especially useful in understanding the intrinsic variability of cell cycle events, but stochastic modeling has been hampered by a lack of reliable data on the absolute numbers of mRNA molecules per cell for cell cycle control genes. To fill this void, we used fluorescence in situ hybridization (FISH) to collect single molecule mRNA data for 16 cell cycle regulators in budding yeast, Saccharomyces cerevisiae. From statistical distributions of single-cell mRNA counts, we are able to extract the periodicity, timing, and magnitude of transcript abundance during the cell cycle. We used these parameters to improve a stochastic model of the cell cycle to better reflect the variability of molecular and phenotypic data on cell cycle progression in budding yeast.

Robust cell size checkpoint from spatiotemporal positive feedback loop in fission yeast

Cells must maintain appropriate cell size during proliferation. Size control may be regulated by a size checkpoint that couples cell size to cell division. Biological experimental data suggests that the cell size is coupled to the cell cycle in two ways: the rates of protein synthesis and the cell polarity protein kinase Pom1 provide spatial information that is used to regulate mitosis inhibitor Wee1. Here a mathematical model involving these spatiotemporal regulations was developed and used to explore the mechanisms underlying the size checkpoint in fission yeast. Bifurcation analysis shows that when the spatiotemporal regulation is coupled to the positive feedback loops (active Cdc2 promotes its activator, Cdc25, and suppress its inhibitor, Wee1), the mitosis-promoting factor (MPF) exhibits a bistable steady-state relationship with the cell size. The switch-like response from the positive feedback loops naturally generates the cell size checkpoint. Further analysis indicated that the spatial regulation provided by Pom1 enhances the robustness of the size checkpoint in fission yeast. This was consistent with experimental data.

A Short-Term Advantage for Syngamy in the Origin of Eukaryotic Sex: Effects of Cell Fusion on Cell Cycle Duration and Other Effects Related to the Duration of the Cell Cycle-Relationship between Cell Growth Curve and the Optimal Size of the Species, and Circadian Cell Cycle in Photosynthetic Unicellular Organisms

The origin of sex is becoming a vexatious issue for Evolutionary Biology. Numerous hypotheses have been proposed, based on the genetic effects of sex, on trophic effects or on the formation of cysts and syncytia. Our approach addresses the change in cell cycle duration which would cause cell fusion. Several results are obtained through graphical and mathematical analysis and computer simulations. (1) In poor environments, cell fusion would be an advantageous strategy, as fusion between cells of different size shortens the cycle of the smaller cell (relative to the asexual cycle), and the majority of mergers would occur between cells of different sizes. (2) The easiest-to-evolve regulation of cell proliferation (sexual/asexual) would be by modifying the checkpoints of the cell cycle. (3) A regulation of this kind would have required the existence of the G2 phase, and sex could thus be the cause of the appearance of this phase. Regarding cell cycle, (4) the exponential curve is the only cell growth curve that has no effect on the optimal cell size in unicellular species; (5) the existence of a plateau with no growth at the end of the cell cycle explains the circadian cell cycle observed in unicellular algae.

Cell size control in yeast

Cell size is an important adaptive trait that influences nearly all aspects of cellular physiology. Despite extensive characterization of the cell-cycle regulatory network, the molecular mechanisms coupling cell growth to division, and thereby controlling cell size, have remained elusive. Recent work in yeast has reinvigorated the size control field and suggested provocative mechanisms for the distinct functions of setting and sensing cell size. Further examination of size-sensing models based on spatial gradients and molecular titration, coupled with elucidation of the pathways responsible for nutrient-modulated target size, may reveal the fundamental principles of eukaryotic cell size control.

NONLINEAR DYNAMICS OF CELL CYCLES WITH STOCHASTIC MATHEMATICAL MODELS

Cell cycles are fundamental components of all living organisms and their systematic studies extend our knowledge about the interconnection between regulatory, metabolic, and signaling networks, and therefore open new opportunities for our ultimate efficient control of cellular processes for disease treatments, as well as for a wide variety of biomedical and biotechnological applications. In the study of cell cycles, nonlinear phenomena play a paramount role, in particular in those cases where the cellular dynamics is in the focus of attention. Quantification of this dynamics is a challenging task due to a wide range of parameters that require estimations and the presence of many stochastic effects. Based on the originally deterministic model, in this paper we develop a hierarchy of models that allow us to describe the nonlinear dynamics accounting for special events of cell cycles. First, we develop a model that takes into account fluctuations of relative concentrations of proteins during special events of cell cycles. Such fluctuations are induced by varying rates of relative concentrations of proteins and/or by relative concentrations of proteins themselves. As such fluctuations may be responsible for qualitative changes in the cell, we develop a new model that accounts for the effect of cellular dynamics on the cell cycle. Finally, we analyze numerically nonlinear effects in the cell cycle by constructing phase portraits based on the newly developed model and carry out a parametric sensitivity analysis in order to identify parameters for an efficient cell cycle control. The results of computational experiments demonstrate that the metabolic events in gene regulatory networks can qualitatively influence the dynamics of the cell cycle.

Rule-based multi-level modeling of cell biological systems

BACKGROUND

Proteins, individual cells, and cell populations denote different levels of an organizational hierarchy, each of which with its own dynamics. Multi-level modeling is concerned with describing a system at these different levels and relating their dynamics. Rule-based modeling has increasingly attracted attention due to enabling a concise and compact description of biochemical systems. In addition, it allows different methods for model analysis, since more than one semantics can be defined for the same syntax.

RESULTS

Multi-level modeling implies the hierarchical nesting of model entities and explicit support for downward and upward causation between different levels. Concepts to support multi-level modeling in a rule-based language are identified. To those belong rule schemata, hierarchical nesting of species, assigning attributes and solutions to species at each level and preserving content of nested species while applying rules. Further necessities are the ability to apply rules and flexibly define reaction rate kinetics and constraints on nested species as well as species that are nested within others. An example model is presented that analyses the interplay of an intracellular control circuit with states at cell level, its relation to cell division, and connections to intercellular communication within a population of cells. The example is described in ML-Rules - a rule-based multi-level approach that has been realized within the plug-in-based modeling and simulation framework JAMES II.

CONCLUSIONS

Rule-based languages are a suitable starting point for developing a concise and compact language for multi-level modeling of cell biological systems. The combination of nesting species, assigning attributes, and constraining reactions according to these attributes is crucial in achieving the desired expressiveness. Rule schemata allow a concise and compact description of complex models. As a result, the presented approach facilitates developing and maintaining multi-level models that, for instance, interrelate intracellular and intercellular dynamics.

A model of yeast cell-cycle regulation based on multisite phosphorylation

In order for the cell's genome to be passed intact from one generation to the next, the events of the cell cycle (DNA replication, mitosis, cell division) must be executed in the correct order, despite the considerable molecular noise inherent in any protein-based regulatory system residing in the small confines of a eukaryotic cell. To assess the effects of molecular fluctuations on cell-cycle progression in budding yeast cells, we have constructed a new model of the regulation of Cln- and Clb-dependent kinases, based on multisite phosphorylation of their target proteins and on positive and negative feedback loops involving the kinases themselves. To account for the significant role of noise in the transcription and translation steps of gene expression, the model includes mRNAs as well as proteins. The model equations are simulated deterministically and stochastically to reveal the bistable switching behavior on which proper cell-cycle progression depends and to show that this behavior is robust to the level of molecular noise expected in yeast-sized cells (approximately 50 fL volume). The model gives a quantitatively accurate account of the variability observed in the G1-S transition in budding yeast, which is governed by an underlying sizer+timer control system.

Investigating the two-moment characterisation of subcellular biochemical networks

While ordinary differential equations (ODEs) form the conceptual framework for modelling many cellular processes, specific situations demand stochastic models to capture the influence of noise. The most common formulation of stochastic models for biochemical networks is the chemical master equation (CME). While stochastic simulations are a practical way to realise the CME, analytical approximations offer more insight into the influence of noise. Towards that end, the two-moment approximation (2MA) is a promising addition to the established analytical approaches including the chemical Langevin equation (CLE) and the related linear noise approximation (LNA). The 2MA approach directly tracks the mean and (co)variance which are coupled in general. This coupling is not obvious in CME and CLE and ignored by LNA and conventional ODE models. We extend previous derivations of 2MA by allowing (a) non-elementary reactions and (b) relative concentrations. Often, several elementary reactions are approximated by a single step. Furthermore, practical situations often require the use of relative concentrations. We investigate the applicability of the 2MA approach to the well-established fission yeast cell cycle model. Our analytical model reproduces the clustering of cycle times observed in experiments. This is explained through multiple resettings of M-phase promoting factor (MPF), caused by the coupling between mean and (co)variance, near the G2/M transition.

Exploring the roles of noise in the eukaryotic cell cycle

The DNA replication-division cycle of eukaryotic cells is controlled by a complex network of regulatory proteins, called cyclin-dependent kinases, and their activators and inhibitors. Although comprehensive and accurate deterministic models of the control system are available for yeast cells, reliable stochastic simulations have not been carried out because the full reaction network has yet to be expressed in terms of elementary reaction steps. As a first step in this direction, we present a simplified version of the control system that is suitable for exact stochastic simulation of intrinsic noise caused by molecular fluctuations and extrinsic noise because of unequal division. The model is consistent with many characteristic features of noisy cell cycle progression in yeast populations, including the observation that mRNAs are present in very low abundance (approximately 1 mRNA molecule per cell for each expressed gene). For the control system to operate reliably at such low mRNA levels, some specific mRNAs in our model must have very short half-lives (<1 min). If these mRNA molecules are longer-lived (perhaps 2 min), then the intrinsic noise in our simulations is too large, and there must be some additional noise suppression mechanisms at work in cells.

Computational systems biology of the cell cycle

One of the early success stories of computational systems biology was the work done on cell-cycle regulation. The earliest mathematical descriptions of cell-cycle control evolved into very complex, detailed computational models that describe the regulation of cell division in many different cell types. On the way these models predicted several dynamical properties and unknown components of the system that were later experimentally verified/identified. Still, research on this field is far from over. We need to understand how the core cell-cycle machinery is controlled by internal and external signals, also in yeast cells and in the more complex regulatory networks of higher eukaryotes. Furthermore, there are many computational challenges what we face as new types of data appear thanks to continuing advances in experimental techniques. We have to deal with cell-to-cell variations, revealed by single cell measurements, as well as the tremendous amount of data flowing from high throughput machines. We need new computational concepts and tools to handle these data and develop more detailed, more precise models of cell-cycle regulation in various organisms. Here we review past and present of computational modeling of cell-cycle regulation, and discuss possible future directions of the field.

Stochastic Petri Net extension of a yeast cell cycle model

This paper presents the definition, solution and validation of a stochastic model of the budding yeast cell cycle, based on Stochastic Petri Nets (SPN). A specific family of SPNs is selected for building a stochastic version of a well-established deterministic model. We describe the procedure followed in defining the SPN model from the deterministic ODE model, a procedure that can be largely automated. The validation of the SPN model is conducted with respect to both the results provided by the deterministic one and the experimental results available from literature. The SPN model catches the behavior of the wild type budding yeast cells and a variety of mutants. We show that the stochastic model matches some characteristics of budding yeast cells that cannot be found with the deterministic model. The SPN model fine-tunes the simulation results, enriching the breadth and the quality of its outcome.

Modelling dynamic processes in yeast

Yeast molecular and cell biology has accumulated large amounts of qualitative and quantitative data of diverse cellular processes. The results are often summarized as verbal or graphical descriptions. Moreover, a series of mathematical models has been developed that should help to interpret such data, to integrate them into a coherent picture and to allow for an understanding of the underlying processes. Dynamic modelling of regulatory processes in yeast focuses on central carbon metabolism, on a number of selected signalling pathways and on cell cycle regulation. These models can explain questions of general relevance, such as whether the dynamics of a network can be understood from the combination of in vitro kinetics of its individual reactions. They help to elucidate complicated dynamic features, such as glycolytic oscillations, effects of feedback regulation or the optimal regulation of gene expression. The availability of comprehensive qualitative information, such as protein interactions or pathway composition, and sets of quantitative data make yeast a perfect model organism. Therefore, yeast-related data are often used to develop and examine computational approaches and modelling methods.

The role of modelling in identifying drug targets for diseases of the cell cycle

The cell cycle is implicated in diseases that are the leading cause of mortality and morbidity in the developed world. Until recently, the search for drug targets has focused on relatively small parts of the regulatory network under the assumption that key events can be controlled by targeting single pathways. This is valid provided the impact of couplings to the wider scale context of the network can be ignored. The resulting depth of study has revealed many new insights; however, these have been won at the expense of breadth and a proper understanding of the consequences of links between the different parts of the network. Since it is now becoming clear that these early assumptions may not hold and successful treatments are likely to employ drugs that simultaneously target a number of different sites in the regulatory network, it is timely to redress this imbalance. However, the substantial increase in complexity presents new challenges and necessitates parallel theoretical and experimental approaches. We review the current status of theoretical models for the cell cycle in light of these new challenges. Many of the existing approaches are not sufficiently comprehensive to simultaneously incorporate the required extent of couplings. Where more appropriate levels of complexity are incorporated, the models are difficult to link directly to currently available data. Further progress requires a better integration of experiment and theory. New kinds of data are required that are quantitative, have a higher temporal resolution and that allow simultaneous quantitative comparison of the concentration of larger numbers of different proteins. More comprehensive models are required and must accommodate not only substantial uncertainties in the structure and kinetic parameters of the networks, but also high levels of ignorance. The most recent results relating network complexity to robustness of the dynamics provide clues that suggest progress is possible.

A simple time delay model for eukaryotic cell cycle

We propose a seven variable model with time delay in one of the variables for the cell cycle in higher eukaryotes. The model consists of four important phosphorylation-dephosphorylation (P-D) cycles that govern the cell cycle, namely Pre-MPF-MPF, Cdc25P-Cdc25, Wee1P-Wee1 and APCP-APC. Other variables are cyclin, free cyclin dependent kinase (Cdk) and mass. The mass acts as a G2/M checkpoint and the checkpoint is represented by a saddle node loop bifurcation. The key feature of the model is that a time lag has been introduced in the activation of anaphase promoting complex (APC) by maturation promoting factor (MPF). This is effected by treating MPF as a time-delayed variable in the activation step of APC. The time lag acts as a spindle checkpoint. Absence of time delay induces a bistability in our model. Time delay also brings about variability in G1 phase timings. The model also reproduces the mutant phenotype experiments on wee1 cells. Stochasticity has been introduced in the model to simulate the dependence of the cycle time on cell birth length. Mutant phenotypes in the stochastic model reproduce the experimental observations better than the deterministic model.

Linking cell division to cell growth in a spatiotemporal model of the cell cycle

Cell division must be tightly coupled to cell growth in order to maintain cell size, yet the mechanisms linking these two processes are unclear. It is known that almost all proteins involved in cell division shuttle between cytoplasm and nucleus during the cell cycle; however, the implications of this process for cell cycle dynamics and its coupling to cell growth remains to be elucidated. We developed mathematical models of the cell cycle which incorporate protein translocation between cytoplasm and nucleus. We show that protein translocation between cytoplasm and nucleus not only modulates temporal cell cycle dynamics, but also provides a natural mechanism coupling cell division to cell growth. This coupling is mediated by the effect of cytoplasmic-to-nuclear size ratio on the activation threshold of critical cell cycle proteins, leading to the size-sensing checkpoint (sizer) and the size-independent clock (timer) observed in many cell cycle experiments.

Coherence and Timing of Cell Cycle Start Examined at Single-Cell Resolution

Cell cycle "Start" in budding yeast involves induction of a large battery of G1/S-regulated genes, coordinated with bud morphogenesis. It is unknown how intra-Start coherence of these events and inter-Start timing regularity are achieved. We developed quantitative time-lapse fluorescence microscopy on a multicell-cycle timescale, for following expression of unstable GFP under control of the G1 cyclin CLN2 promoter. Swi4, a major activator of the G1/S regulon, was required for a robustly coherent Start, as swi4 cells exhibited highly variable loss of cooccurrence of regular levels of CLN2pr-GFP expression with budding. In contrast, other known Start regulators Mbp1 and Cln3 are not needed for coherence but ensure regular timing of Start onset. The interval of nuclear retention of Whi5, a Swi4 repressor, largely accounts for wild-type mother-daughter asymmetry and for variable Start timing in cln3 mbp1 cells. Thus, multiple pathways may independently suppress qualitatively different kinds of noise at Start.

A comprehensive continuous-time model for the appearance of CGH signal due to chromosomal missegregations during mitosis

Aneuploidy, the gain or loss of large regions of the genome, is a common feature in cancer cells. Irregularities in chromosomal copy number caused by missegregations of chromosomes during mitosis can be visualized by cytogenetic techniques including fluorescence in situ hybridization (FISH), spectral karyotyping (SKY) and comparative genomic hybridization (CGH). In the current work, we consider the propagation of irregular copy numbers throughout a cell population as the individual cells progress through ordinary mitotic cell cycles. We use an algebraic model to track the different copy numbers as states in a stochastic process, based on the model of chromosome instability of Gusev, Kagansky, and Dooley, and consider the average copy number of a particular chromosome within a cell population as a function of the cell division rate. We review a number of mathematical models for determining the length of the cell cycle, including the Smith-Martin transition probability model and the 'sloppy size' model of Wheals, Tyson and Diekmann. The program MITOSIM simulates the growth of a population of cells using the aforementioned models of the cell cycle. MITOSIM allows the cell population to grow, with occasional resampling, until the average copy number of a given chromosome in the population reaches a preset threshold signifying a positive copy number alteration in this region. MITOSIM calculates the relationship between the missegregation rate and the growth rate of the cell population. This allows the user to test hypotheses regarding the effect chromosomal aberrations have upon the cell cycle, cell growth rates, and time to population dominance.

Bifurcation analysis of a model of the budding yeast cell cycle

We study the bifurcations of a set of nine nonlinear ordinary differential equations that describe regulation of the cyclin-dependent kinase that triggers DNA synthesis and mitosis in the budding yeast, Saccharomyces cerevisiae. We show that Clb2-dependent kinase exhibits bistability (stable steady states of high or low kinase activity). The transition from low to high Clb2-dependent kinase activity is driven by transient activation of Cln2-dependent kinase, and the reverse transition is driven by transient activation of the Clb2 degradation machinery. We show that a four-variable model retains the main features of the nine-variable model. In a three-variable model exhibiting birhythmicity (two stable oscillatory states), we explore possible effects of extrinsic fluctuations on cell cycle progression.

Effects of stochasticity in models of the cell cycle: from quantized cycle times to noise-induced oscillations

Noise and fluctuations are ubiquitous in living systems. Still, the interaction between complex biochemical regulatory systems and the inherent fluctuations ('noise') is only poorly understood. As a paradigmatic example, we study the implications of noise on a recently proposed model of the eukaryotic cell cycle, representing a complex network of interactions between several genes and proteins. The purpose of this work is twofold: First, we show that the inclusion of noise into the description of the system accounts for several recent experimental findings, as e.g. the existence of quantized cycle times in wee1- cdc25delta double-mutant cells of fission yeast. In the main part, we then focus on more general aspects of the interplay between noise and the dynamics of the system. In particular, we demonstrate that a stochastic description leads to qualitative changes in the dynamics, such as the emergence of noise-induced oscillations. These findings will be discussed in the light of an ongoing debate on models of cell division as limit-cycle oscillators versus checkpoint mechanisms.

Growth during the cell cycle

During the cell cycle, major bulk parameters such as volume, dry mass, total protein, and total RNA double and such growth is a fundamental property of the cell cycle. The patterns of growth in volume and total protein or RNA provide an "envelope" that contains and may restrict the gear wheels. The main parameters of cell cycle growth were established in the earlier work when people moved from this field to the reductionist approaches of molecular biology, but very little is known on the patterns of metabolism. Most of the bulk properties of cells show a continuous increase during the cell cycle, although the exact pattern of this increase may vary. Since the earliest days, there have been two popular models, based on an exponential increase and linear increase. In the first, there is no sharp change in the rate of increase through the cycle but a smooth increase by a factor of two. In the second, the rate of increase stays constant through much of the cycle but it doubles sharply at a rate change point (RCP). It is thought that the exponential increase is caused by the steady growth of ribosome numbers and the linear pattern is caused by a doubling of the structural genes during the S period giving an RCP--a "gene dosage" effect. In budding yeast, there are experiments fitting both models but on balance slightly favoring "gene dosage." In fission yeast, there is no good evidence of exponential increase. All the bulk properties, except O2 consumption, appear to follow linear patterns with an RCP during the short S period. In addition, there is in wild-type cells a minor RCP in G2 where the rate increases by 70%. In mammalian cells, there is good but not extensive evidence of exponential increase. In Escherichia coli, exponential increase appears to be the pattern. There are two important points: First, some proteins do not show peaks of periodic synthesis. If they show patterns of exponential increase both they and the total protein pattern will not be cell cycle regulated. However, if the total protein pattern is not exponential, then a majority of the individual proteins will be so regulated. If this majority pattern is linear, then it can be detected from rate measurements on total protein. However, it would be much harder at the level of individual proteins where the methods are at present not sensitive enough to detect a rate change by a factor of two. At a simple level, it is only the exponential increase that is not cell cycle regulated in a synchronous culture. The existence of a "size control" is well known and the control has been studied for a long time, but it has been remarkably resistant to molecular analysis. The attainment of a critical size triggers the periodic events of the cycle such as the S period and mitosis. This control acts as a homeostatic effector that maintains a constant "average" cell size at division through successive cycles in a growing culture. It is a vital link coordinating cell growth with periodic events of the cycle. A size control is present in all the systems and appears to operate near the start of S or of mitosis when the cell has reached a critical size, but the molecular mechanism by which size is measured remains both obscure and a challenge. A simple version might be for the cell to detect a critical concentration of a gene product.

Both cyclin B levels and DNA-replication checkpoint control the early embryonic mitoses in Drosophila

The earliest embryonic mitoses in Drosophila, as in other animals except mammals, are viewed as synchronous and of equal duration. However, we observed that total cell-cycle length steadily increases after cycle 7, solely owing to the extension of interphase. Between cycle 7 and cycle 10, this extension is DNA-replication checkpoint independent, but correlates with the onset of Cyclin B oscillation. In addition, nuclei in the middle of embryos have longer metaphase and shorter anaphase than nuclei at the two polar regions. Interestingly, sister chromatids move faster in anaphase in the middle than the posterior region. These regional differences correlate with local differences in Cyclin B concentration. After cycle 10, interphase and total cycle duration of nuclei in the middle of the embryo are longer than at the poles. Because interphase also extends in checkpoint mutant (grapes) embryo after cycle 10, although less dramatic than wild-type embryos, interphase extension after cycle 10 is probably controlled by both Cyclin B limitation and the DNA-replication checkpoint.

Estimation of nuclear volume as an indicator of maturation of glial precursor cells in the developing rat spinal cord: a stereological approach

Studies on nuclear volume have shown that it is an indication of the state of differentiation of cells. This study provides evidence indicating increasing nuclear volume during cell maturation. Using unbiased stereological techniques, nuclear volume of both proliferating and non-proliferating glial cells was analysed in the developing spinal cord. Proliferating glial precursor cells were identified using a 5-bromo-2'-deoxyuridine (BrdU) incorporation assay. The nuclear volume of BrdU-labelled cells and unlabelled cells was determined in both periventricular regions and the white matter of the cord at different embryonic ages. In the periventricular region BrdU-labelled nuclei were smaller than unlabelled nuclei at all ages examined. These labelled cells represent dividing undifferentiated progenitors. The unlabelled neighbouring cells with larger nuclei represent a more differentiated population. In the white matter BrdU-labelled nuclei were of similar volume to the unlabelled nuclei. Both of these groups represent glial precursor cells that have migrated from deeper regions and are at similar stages of differentiation, perhaps with different proliferative potential. These findings indicate that the nuclear volume of early glial cells increases as these cells migrate and differentiate.

Regularities and irregularities in the cell cycle of the fission yeast, Schizosaccharomyces pombe (a review)

In an exponentially growing wild-type fission yeast culture a size control mechanism ensures that mitosis is executed only if the cells have reached a critical size. However, there is some scattering both in cell length at birth (BL) and in cycle time (CT). By computational simulations we show here that this scattering cannot be explained solely by asymmetric cell division, therefore we assume that nuclear division is a stochastic, asymmetric process as well. We introduce an appropriate stochastic variable into a mathematical model and prove that this assumption is suitable to describe the CT vs. BL graph in a wild-type fission yeast population. In a double mutant of fission yeast (namely wee1-50 cdc25 delta) this CT vs. BL plot is even more curious: cycle time splits into three different values resulting in three clusters in this coordinate system. We show here that it is possible to describe these quantized cycles by choosing the appropriate values of the key parameters of mitotic entry and exit and even more the clustered behavior may be simulated by applying a further stochastic parameter.

The dynamics of cell cycle regulation

Major events of the cell cycle--DNA synthesis, mitosis and cell division-are regulated by a complex network of protein interactions that control the activities of cyclin-dependent kinases. The network can be modeled by a set of nonlinear differential equations and its behavior predicted by numerical simulation. Computer simulations are necessary for detailed quantitative comparisons between theory and experiment, but they give little insight into the qualitative dynamics of the control system and how molecular interactions determine the fundamental physiological properties of cell replication. To that end, bifurcation diagrams are a useful analytical tool, providing new views of the dynamical organization of the cell cycle, the role of checkpoints in assuring the integrity of the genome, and the abnormal regulation of cell cycle events in mutants. These claims are demonstrated by an analysis of cell cycle regulation in fission yeast.

Checking cell size in yeast

To remain viable, cells have to coordinate cell growth with cell division. In yeast, this occurs at two control points: the boundaries between G1 and S phases, also known as Start, and between G2 and M phases. Theoretically, coordination can be achieved by independent regulation of growth and division, or by participation of surveillance mechanisms in which cell size feeds back into cell-cycle control. This article discusses recent advances in the identification of sizing mechanisms in budding and in fission yeast, and how these mechanisms integrate with environmental stimuli. A comparison of the G1-S and G2-M size-control modules in the two species reveals a degree of conservation higher than previously thought. This reinforces the notion that internal sizing could be a conserved feature of cell-cycle control throughout eukaryotes.