Modeling the fission yeast cell cycle: quantized cycle times in wee1- cdc25Delta mutant cells

Live-cell imaging defines a threshold in CDK activity at the G2/M transition

Cyclin-dependent kinase (CDK) determines the temporal ordering of the cell cycle phases. However, despite significant progress in studying regulators of CDK and phosphorylation patterns of CDK substrates at the population level, it remains elusive how CDK regulators coordinately affect CDK activity at the single-cell level and how CDK controls the temporal order of cell cycle events. Here, we elucidate the dynamics of CDK activity in fission yeast and mammalian cells by developing a CDK activity biosensor, Eevee-spCDK. We find that although CDK activity does not necessarily correlate with cyclin levels, it converges to the same level around mitotic onset in several mutant backgrounds, including pom1Δ cells and wee1 or cdc25 overexpressing cells. These data provide direct evidence that cells enter the M phase when CDK activity reaches a high threshold, consistent with the quantitative model of cell cycle progression in fission yeast.

Implications of differential size-scaling of cell-cycle regulators on cell size homeostasis

Accurate timing of division and size homeostasis is crucial for cells. A potential mechanism for cells to decide the timing of division is the differential scaling of regulatory protein copy numbers with cell size. However, it remains unclear whether such a mechanism can lead to robust growth and division, and how the scaling behaviors of regulatory proteins influence the cell size distribution. Here we study a mathematical model combining gene expression and cell growth, in which the cell-cycle activators scale superlinearly with cell size while the inhibitors scale sublinearly. The cell divides once the ratio of their concentrations reaches a threshold value. We find that the cell can robustly grow and divide within a finite range of the threshold value with the cell size proportional to the ploidy. In a stochastic version of the model, the cell size at division is uncorrelated with that at birth. Also, the more differential the cell-size scaling of the cell-cycle regulators is, the narrower the cell-size distribution is. Intriguingly, our model with multiple regulators rationalizes the observation that after the deletion of a single regulator, the coefficient of variation of cell size remains roughly the same though the average cell size changes significantly. Our work reveals that the differential scaling of cell-cycle regulators provides a robust mechanism of cell size control.

Uncoupling of Mitosis and Cytokinesis Upon a Prolonged Arrest in Metaphase Is Influenced by Protein Phosphatases and Mitotic Transcription in Fission Yeast

Depletion of the Anaphase-Promoting Complex/Cyclosome (APC/C) activator Cdc20 arrests cells in metaphase with high levels of the mitotic cyclin (Cyclin B) and the Separase inhibitor Securin. In mammalian cells this arrest has been exploited for the treatment of cancer with drugs that engage the spindle assembly checkpoint and, recently, with chemical inhibitors of the APC/C. While most cells arrested in mitosis for prolonged periods undergo apoptosis, others skip cytokinesis and enter G1 with unsegregated chromosomes. This process, known as mitotic slippage, generates aneuploidy and increases genomic instability in the cancer cell. Here, we analyze the behavior of fission yeast cells arrested in mitosis through the transcriptional silencing of the Cdc20 homolog slp1. While depletion of slp1 readily halts cells in metaphase, this arrest is only transient and a majority of cells eventually undergo cytokinesis and show steady mitotic dephosphorylation. Notably, this occurs in the absence of Cyclin B (Cdc13) degradation. We investigate the involvement of phosphatase activity in these events and demonstrate that PP2A-B55^Pab1 is required to prevent septation and, during the arrest, its CDK-mediated inhibition facilitates the induction of cytokinesis. In contrast, deletion of PP2A-B56^Par1 completely abrogates septation. We show that this effect is partly due to this mutant entering mitosis with reduced CDK activity. Interestingly, both PP2A-B55^Pab1 and PP2A-B56^Par1, as well as Clp1 (the homolog of the budding yeast mitotic phosphatase Cdc14) are required for the dephosphorylation of mitotic substrates during the escape. Finally, we show that the mitotic transcriptional wave controlled by the RFX transcription factor Sak1 facilitates the induction of cytokinesis and also requires the activity of PP2A-B56^Par1 in a mechanism independent of CDK.

Modeling the Control of Meiotic Cell Divisions: Entry, Progression, and Exit

Upon nitrogen starvation, Schizosaccharomyces pombe exit the mitotic cell cycle and become irreversibly committed to the completion of meiosis program. Meiotic cell divisions are coordinated with sporulation events to produce haploid spores. In the last few decades, experiments on fission yeast have revealed different molecular players involved in two meiotic cell divisions, meiosis I (MI) and meiosis II (MII). How the MI entry, MI-to-MII transition, and MII exit occur because of the dynamics of the regulatory network is not well understood. In this work, we developed a comprehensive mathematical model of the network that describes the temporal dynamics of meiotic progression. The model accounts for the phenotypes of several experimental data (single and multiple mutations). We demonstrate the control strategy involving multiple feedback loops to yield two successive division cycles. The differential regulation of anaphase-promoting complex/cyclosome (APC/C) coactivators and its inhibitors is crucial for the dynamics of both MI-to-MII transition and MII exit. This model generates mechanistic insights that help in further experiments and modeling.

Can ODE gene regulatory models neglect time lag or measurement scaling?

MOTIVATION

Many ordinary differential equation (ODE) models have been introduced to replace linear regression models for inferring gene regulatory relationships from time-course gene expression data. But, since the observed data are usually not direct measurements of the gene products or there is an unknown time lag in gene regulation, it is problematic to directly apply traditional ODE models or linear regression models.

RESULTS

We introduce a lagged ODE model to infer lagged gene regulatory relationships from time-course measurements, which are modeled as linear transformation of the gene products. A time-course microarray dataset from a yeast cell-cycle study is used for simulation assessment of the methods and real data analysis. The results show that our method, by considering both time lag and measurement scaling, performs much better than other linear and ODE models. It indicates the necessity of explicitly modeling the time lag and measurement scaling in ODE gene regulatory models.

AVAILABILITY AND IMPLEMENTATION

R code is available at https://www.sta.cuhk.edu.hk/xfan/share/lagODE.zip.

Sexual Differentiation Is Coordinately Regulated by Cryptococcus neoformans CRK1 and GAT1

The heterothallic basidiomycetous fungus Cryptococcus neoformans has two mating types, MATa and MATα. Morphological progression of bisexual reproduction in C. neoformans is as follows: yeast to hyphal transition, filament extension, basidium formation, meiosis, and sporulation. C. neoformans Cdk-related kinase 1 (CRK1) is a negative regulator of bisexual mating. In this study, we characterized the morphological features of mating structures in the crk1 mutant and determined the genetic interaction of CRK1 in the regulatory networks of sexual differentiation. In the bilateral crk1 mutant cross, despite shorter length of filaments than in the wild-type cross, dikaryotic filaments and other structures still remained intact during bisexual mating, but the timing of basidium formation was approximately 18 h earlier than in the cross between wild type strains. Furthermore, gene expression analyses revealed that CRK1 modulated the expression of genes involved in the progression of hyphal elongation, basidium formation, karyogamy and meiosis. Phenotypic results showed that, although deletion of C. neoformans CRK1 gene increased the efficiency of bisexual mating, filamentation in the crk1 mutant was blocked by MAT2 or ZNF2 mutation. A bioinformatics survey predicted the C. neoformans GATA transcriptional factor Gat1 as a potential substrate of Crk1 kinase. Our genetic and phenotypic findings revealed that C. neoformansGAT1 and CRK1 formed a regulatory circuit to negatively regulate MAT2 to control filamentation progression and transition during bisexual mating.

Quantitative Studies for Cell-Division Cycle Control

The cell-division cycle (CDC) is driven by cyclin-dependent kinases (CDKs). Mathematical models based on molecular networks, as revealed by molecular and genetic studies, have reproduced the oscillatory behavior of CDK activity. Thus, one basic system for representing the CDC is a biochemical oscillator (CDK oscillator). However, genetically clonal cells divide with marked variability in their total duration of a single CDC round, exhibiting non-Gaussian statistical distributions. Therefore, the CDK oscillator model does not account for the statistical nature of cell-cycle control. Herein, we review quantitative studies of the statistical properties of the CDC. Over the past 70 years, studies have shown that the CDC is driven by a cluster of molecular oscillators. The CDK oscillator is coupled to transcriptional and mitochondrial metabolic oscillators, which cause deterministic chaotic dynamics for the CDC. Recent studies in animal embryos have raised the possibility that the dynamics of molecular oscillators underlying CDC control are affected by allometric volume scaling among the cellular compartments. Considering these studies, we discuss the idea that a cluster of molecular oscillators embedded in different cellular compartments coordinates cellular physiology and geometry for successful cell divisions.

Adding phosphorylation events to the core oscillator driving the cell cycle of fission yeast

Much is known about the regulatory elements controlling the cell cycle in fission yeast (Schizosaccharomyces pombe). This regulation is mainly done by the (cyclin-dependent kinase/cyclin) complex (Cdc2/Cdc13) that activates specific target genes and proteins via phosphorylation events during the cell cycle in a time-dependent manner. However, more work is still needed to complement the existing gaps in the current fission yeast gene regulatory network to be able to overcome abnormalities in its growth, repair and development, i.e. explain many phenomena including mitotic catastrophe. In this work we complement the previously presented core oscillator of the cell cycle of fission yeast by selected phosphorylation events and study their effects on the temporal evolution of the core oscillator based Boolean network. Thereby, we attempt to establish a regulatory link between the autonomous cell cycle oscillator and the remainder of the cell. We suggest the unclear yet regulatory effect of phosphorylation on the added components, and discuss many unreported points regarding the temporal evolution of the cell cycle and its components. To better visualize the results regardless of the programming background we developed an Android application that can be used to run the core and extended model of the fission yeast cell cycle step by step.

A toolbox for discrete modelling of cell signalling dynamics

We present a library of network motifs for the development of complex and realistic biological network models using the BioModelAnalyzer, and demonstrate their wider value by using them to construct a model of the cell cycle.

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.

From START to FINISH: computational analysis of cell cycle control in budding yeast

In the cell division cycle of budding yeast, START refers to a set of tightly linked events that prepare a cell for budding and DNA replication, and FINISH denotes the interrelated events by which the cell exits from mitosis and divides into mother and daughter cells. On the basis of recent progress made by molecular biologists in characterizing the genes and proteins that control START and FINISH, we crafted a new mathematical model of cell cycle progression in yeast. Our model exploits a natural separation of time scales in the cell cycle control network to construct a system of differential-algebraic equations for protein synthesis and degradation, post-translational modifications, and rapid formation and dissociation of multimeric complexes. The model provides a unified account of the observed phenotypes of 257 mutant yeast strains (98% of the 263 strains in the data set used to constrain the model). We then use the model to predict the phenotypes of 30 novel combinations of mutant alleles. Our comprehensive model of the molecular events controlling cell cycle progression in budding yeast has both explanatory and predictive power. Future experimental tests of the model’s predictions will be useful to refine the underlying molecular mechanism, to constrain the adjustable parameters of the model, and to provide new insights into how the cell division cycle is regulated in budding yeast.

Time-lapse electrical impedance spectroscopy for monitoring the cell cycle of single immobilized S. pombe cells

As a complement and alternative to optical methods, wide-band electrical impedance spectroscopy (EIS) enables multi-parameter, label-free and real-time detection of cellular and subcellular features. We report on a microfluidics-based system designed to reliably capture single rod-shaped Schizosaccharomyces pombe cells by applying suction through orifices in a channel wall. The system enables subsequent culturing of immobilized cells in an upright position, while dynamic changes in cell-cycle state and morphology were continuously monitored through EIS over a broad frequency range. Besides measuring cell growth, clear impedance signals for nuclear division have been obtained. The EIS system has been characterized with respect to sensitivity and detection limits. The spatial resolution in measuring cell length was 0.25 μm, which corresponds to approximately a 5-min interval of cell growth under standard conditions. The comprehensive impedance data sets were also used to determine the occurrence of nuclear division and cytokinesis. The obtained results have been validated through concurrent confocal imaging and plausibilized through comparison with finite-element modeling data. The possibility to monitor cellular and intracellular features of single S. pombe cells during the cell cycle at high spatiotemporal resolution renders the presented microfluidics-based EIS system a suitable tool for dynamic single-cell investigations.

Validation of signalling pathways: Case study of the p16-mediated pathway

p16 is recognized as a tumor suppressor gene due to the prevalence of its genetic inactivation in all types of human cancers. Additionally, p16 gene plays a critical role in controlling aging, regulating cellular senescence, detection and maintenance of DNA damage. The molecular mechanism behind these events involves p16-mediated signaling pathway (or p16- Rb pathway), the focus of our study. Understanding functional dependence between dynamic behavior of biological components involved in the p16-mediated pathway and aforesaid molecular-level events might suggest possible implications in the diagnosis, prognosis and treatment of human cancer. In the present work, we employ reverse-engineering approach to construct the most detailed computational model of p16-mediated pathway in higher eukaryotes. We implement experimental data from the literature to validate the model, and under various assumptions predict the dynamic behavior of p16 and other biological components by interpreting the simulation results. The quantitative model of p16-mediated pathway is created in a systematic manner in terms of Petri net technologies.

Neutral space analysis for a Boolean network model of the fission yeast cell cycle network

Background

Interactions between genes and their products give rise to complex circuits known as gene regulatory networks (GRN) that enable cells to process information and respond to external stimuli. Several important processes for life, depend of an accurate and context-specific regulation of gene expression, such as the cell cycle, which can be analyzed through its GRN, where deregulation can lead to cancer in animals or a directed regulation could be applied for biotechnological processes using yeast. An approach to study the robustness of GRN is through the neutral space. In this paper, we explore the neutral space of a Schizosaccharomyces pombe (fission yeast) cell cycle network through an evolution strategy to generate a neutral graph, composed of Boolean regulatory networks that share the same state sequences of the fission yeast cell cycle.

Results

Through simulations it was found that in the generated neutral graph, the functional networks that are not in the wildtype connected component have in general a Hamming distance more than 3 with the wildtype, and more than 10 between the other disconnected functional networks. Significant differences were found between the functional networks in the connected component of the wildtype network and the rest of the network, not only at a topological level, but also at the state space level, where significant differences in the distribution of the basin of attraction for the G₁ fixed point was found for deterministic updating schemes.

Conclusions

In general, functional networks in the wildtype network connected component, can mutate up to no more than 3 times, then they reach a point of no return where the networks leave the connected component of the wildtype. The proposed method to construct a neutral graph is general and can be used to explore the neutral space of other biologically interesting networks, and also formulate new biological hypotheses studying the functional networks in the wildtype network connected component.

Refining network reconstruction based on functional reliability

Reliable functioning is crucial for the survival and development of the genetic regulatory networks in living cells and organisms. This functional reliability is an important feature of the networks and reflects the structural features that have been embedded in the regulatory networks by evolution. In this paper, we integrate this reliability into network reconstruction. We introduce the concept of dependency probability to measure the dependency of functional reliability on network edges. We also propose a method to estimate the dependency probability and select edges with high contributions to functional reliability. We use two real examples, the regulatory network of the cell cycle of the budding yeast and that of the fission yeast, to demonstrate that the proposed method improves network reconstruction. In addition, the dependency probability is robust in calculation and can be easily implemented in practice.

A data-driven, mathematical model of mammalian cell cycle regulation

Few of >150 published cell cycle modeling efforts use significant levels of data for tuning and validation. This reflects the difficultly to generate correlated quantitative data, and it points out a critical uncertainty in modeling efforts. To develop a data-driven model of cell cycle regulation, we used contiguous, dynamic measurements over two time scales (minutes and hours) calculated from static multiparametric cytometry data. The approach provided expression profiles of cyclin A2, cyclin B1, and phospho-S10-histone H3. The model was built by integrating and modifying two previously published models such that the model outputs for cyclins A and B fit cyclin expression measurements and the activation of B cyclin/Cdk1 coincided with phosphorylation of histone H3. The model depends on Cdh1-regulated cyclin degradation during G1, regulation of B cyclin/Cdk1 activity by cyclin A/Cdk via Wee1, and transcriptional control of the mitotic cyclins that reflects some of the current literature. We introduced autocatalytic transcription of E2F, E2F regulated transcription of cyclin B, Cdc20/Cdh1 mediated E2F degradation, enhanced transcription of mitotic cyclins during late S/early G2 phase, and the sustained synthesis of cyclin B during mitosis. These features produced a model with good correlation between state variable output and real measurements. Since the method of data generation is extensible, this model can be continually modified based on new correlated, quantitative data.

Boolean network model predicts knockout mutant phenotypes of fission yeast

Boolean networks (or: networks of switches) are extremely simple mathematical models of biochemical signaling networks. Under certain circumstances, Boolean networks, despite their simplicity, are capable of predicting dynamical activation patterns of gene regulatory networks in living cells. For example, the temporal sequence of cell cycle activation patterns in yeasts S. pombe and S. cerevisiae are faithfully reproduced by Boolean network models. An interesting question is whether this simple model class could also predict a more complex cellular phenomenology as, for example, the cell cycle dynamics under various knockout mutants instead of the wild type dynamics, only. Here we show that a Boolean network model for the cell cycle control network of yeast S. pombe correctly predicts viability of a large number of known mutants. So far this had been left to the more detailed differential equation models of the biochemical kinetics of the yeast cell cycle network and was commonly thought to be out of reach for models as simplistic as Boolean networks. The new results support our vision that Boolean networks may complement other mathematical models in systems biology to a larger extent than expected so far, and may fill a gap where simplicity of the model and a preference for an overall dynamical blueprint of cellular regulation, instead of biochemical details, are in the focus.

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.

Model discrimination in dynamic molecular systems: application to parotid de-differentiation network

In modern systems biology the modeling of longitudinal data, such as changes in mRNA concentrations, is often of interest. Fully parametric, ordinary differential equations (ODE)-based models are typically developed for the purpose, but their lack of fit in some examples indicates that more flexible Bayesian models may be beneficial, particularly when there are relatively few data points available. However, under such sparse data scenarios it is often difficult to identify the most suitable model. The process of falsifying inappropriate candidate models is called model discrimination. We propose here a formal method of discrimination between competing Bayesian mixture-type longitudinal models that is both sensitive and sufficiently flexible to account for the complex variability of the longitudinal molecular data. The ideas from the field of Bayesian analysis of computer model validation are applied, along with modern Markov Chain Monte Carlo (MCMC) algorithms, in order to derive an appropriate Bayes discriminant rule. We restrict attention to the two-model comparison problem and present the application of the proposed rule to the mRNA data in the de-differentiation network of three mRNA concentrations in mammalian salivary glands as well as to a large synthetic dataset derived from the model used in the recent DREAM6 competition.

Mathematical modeling of fission yeast Schizosaccharomyces pombe cell cycle: exploring the role of multiple phosphatases

Cell cycle is the central process that regulates growth and division in all eukaryotes. Based on the environmental condition sensed, the cell lies in a resting phase G0 or proceeds through the cyclic cell division process (G1→S→G2→M). These series of events and phase transitions are governed mainly by the highly conserved Cyclin dependent kinases (Cdks) and its positive and negative regulators. The cell cycle regulation of fission yeast Schizosaccharomyces pombe is modeled in this study. The study exploits a detailed molecular interaction map compiled based on the published model and experimental data. There are accumulating evidences about the prominent regulatory role of specific phosphatases in cell cycle regulations. The current study emphasizes the possible role of multiple phosphatases that governs the cell cycle regulation in fission yeast S. pombe. The ability of the model to reproduce the reported regulatory profile for the wild-type and various mutants was verified though simulations.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1007/s11693-011-9090-7) contains supplementary material, which is available to authorized users.

A mathematical model of mitotic exit in budding yeast: the role of Polo kinase

Cell cycle progression in eukaryotes is regulated by periodic activation and inactivation of a family of cyclin-dependent kinases (Cdk's). Entry into mitosis requires phosphorylation of many proteins targeted by mitotic Cdk, and exit from mitosis requires proteolysis of mitotic cyclins and dephosphorylation of their targeted proteins. Mitotic exit in budding yeast is known to involve the interplay of mitotic kinases (Cdk and Polo kinases) and phosphatases (Cdc55/PP2A and Cdc14), as well as the action of the anaphase promoting complex (APC) in degrading specific proteins in anaphase and telophase. To understand the intricacies of this mechanism, we propose a mathematical model for the molecular events during mitotic exit in budding yeast. The model captures the dynamics of this network in wild-type yeast cells and 110 mutant strains. The model clarifies the roles of Polo-like kinase (Cdc5) in the Cdc14 early anaphase release pathway and in the G-protein regulated mitotic exit network.

Overexpression limits of fission yeast cell-cycle regulators in vivo and in silico

Cellular systems are generally robust against fluctuations of intracellular parameters such as gene expression level. However, little is known about expression limits of genes required to halt cellular systems. In this study, using the fission yeast Schizosaccharomyces pombe, we developed a genetic 'tug-of-war' (gTOW) method to assess the overexpression limit of certain genes. Using gTOW, we determined copy number limits for 31 cell-cycle regulators; the limits varied from 1 to >100. Comparison with orthologs of the budding yeast Saccharomyces cerevisiae suggested the presence of a conserved fragile core in the eukaryotic cell cycle. Robustness profiles of networks regulating cytokinesis in both yeasts (septation-initiation network (SIN) and mitotic exit network (MEN)) were quite different, probably reflecting differences in their physiologic functions. Fragility in the regulation of GTPase spg1 was due to dosage imbalance against GTPase-activating protein (GAP) byr4. Using the gTOW data, we modified a mathematical model and successfully reproduced the robustness of the S. pombe cell cycle with the model.

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.

Dynamic modeling and analysis of cancer cellular network motifs

With the advent of high-throughput biology, we now routinely scan cells and organisms at practically all levels, from genome to protein, metabolism, signaling and other cellular functions. This methodology allowed biological studies to move from a reductionist approach, such as isolation of specific pathways and mechanisms, to a more integrative approach, where biological systems are seen as a network of interconnected components that provide specific outputs and functions in response to stimuli. Recent literature on biological networks demonstrates two important concepts that we will consider in this review: (i) cellular pathways are highly interconnected and should not be studied separately, but as a network; (ii) simple, recurrent feedback motifs within the network can produce very specific functions that favor their modular use. The first theme differs from the traditional approach in biology because it provides a framework (i.e., the network view) in which large datasets are analyzed with an unbiased view. The second theme (feedback motifs) shows the importance of locally analyzing the dynamic properties of biological networks in order to better understand their functionality. We will review these themes with examples from cell signaling networks, gene regulatory networks and metabolic pathways. The deregulation of cellular networks (metabolism, signaling etc.) is involved in cancer, but the size of the networks and resulting non-linear behavior do not allow for intuitive reasoning. In that context, we argue that the qualitative classification of the 'building blocs' of biological networks (i.e. the motifs) in terms of dynamics and functionality will be critical to improve our understanding of cancer biology and rationalize the wealth of information from high-throughput experiments. From the examples highlighted in this review, it is clear that dynamic feedback motifs can be used to provide a unified view of various cellular processes involved in cancer and this will be critical for future research on personalized and predictive cancer therapies.

Optimizing periodicity and polymodality in noise-induced genetic oscillators

Many cellular functions are based on the rhythmic organization of biological processes into self-repeating cascades of events. Some of these periodic processes, such as the cell cycles of several species, exhibit conspicuous irregularities in the form of period skippings, which lead to polymodal distributions of cycle lengths. A recently proposed mechanism that accounts for this quantized behavior is the stabilization of a Hopf-unstable state by molecular noise. Here we investigate the effect of varying noise in a model system, namely an excitable activator-repressor genetic circuit, that displays this noise-induced stabilization effect. Our results show that an optimal noise level enhances the regularity (coherence) of the cycles, in a form of coherence resonance. Similar noise levels also optimize the multimodal nature of the cycle lengths. Together, these results illustrate how molecular noise within a minimal gene regulatory motif confers robust generation of polymodal patterns of periodicity.

Live cell imaging and systems biology

Much of the experimental data used to construct mathematical models of molecular networks are derived from in vitro measurements. However, there is increasing evidence that in vitro measurements fail to capture both the complexity and the individuality found in single, living cells. These limitations can be overcome by live cell microscopy which is evolving to enable in vivo biochemistry. Here, we survey the current capabilities of live cell microscopy and illustrate how a number of different imaging approaches could be applied to analyze a specific molecular network. We argue that incorporation of such quantitative live-cell imaging methods is critical for the progress of systems biology.

CytoSys: a tool for extracting cell-cycle-related expression dynamics from static data

Computational models of biological processes are important building blocks in Systems Biology studies. Calibration and validation are two important steps for moving a mathematical model to a computational model. While calibration refers to finding numerical value of the coefficients such as rate constants in a mathematical model, validation refers to verifying that the calibrated model behaves the same as the biological system under previously unseen conditions such as environmental changes (e.g., drug treatment) or mutations. In lieu of direct measurements of rate constants, modeling of the molecular mechanisms that govern biological behaviors may be able to use dynamic expression profiles of reactant biomolecules for calibration. For validation, similar data, obtained under new conditions, are probably better than direct measurements of rate constants. In any case, direct measurement of rate constants is almost always impractical and difficult or impossible. Here, we show a computer-assisted methodology to extract embedded dynamic profiles of cell-cycle proteins from statically sampled, multivariate cytometry data guided by heuristics assembled from canonical cell-cycle knowledge. The methodology is implemented using standard "list mode" cytometry data-processing software followed by CytoSys - a software tool with an easy-to-use graphical interface. We demonstrate the use of CytoSys with a case study of exponentially growing, human erythroleukemia cells and extract the dynamic expression profiles of cyclin A for calibrating an existing deterministic mathematical model of the cell cycle.

Process-based network decomposition reveals backbone motif structure

A central challenge in systems biology today is to understand the network of interactions among biomolecules and, especially, the organizing principles underlying such networks. Recent analysis of known networks has identified small motifs that occur ubiquitously, suggesting that larger networks might be constructed in the manner of electronic circuits by assembling groups of these smaller modules. Using a unique process-based approach to analyzing such networks, we show for two cell-cycle networks that each of these networks contains a giant backbone motif spanning all the network nodes that provides the main functional response. The backbone is in fact the smallest network capable of providing the desired functionality. Furthermore, the remaining edges in the network form smaller motifs whose role is to confer stability properties rather than provide function. The process-based approach used in the above analysis has additional benefits: It is scalable, analytic (resulting in a single analyzable expression that describes the behavior), and computationally efficient (all possible minimal networks for a biological process can be identified and enumerated).

A mathematical model for cell size control in fission yeast

Experimental investigations of cell size control in fission yeast Schizosaccharomyces pombe have illustrated that the cell cycle features 'sizer' and 'timer' phases which are distinguished by a growth rate changing point. Based on current biological knowledge of fission yeast size control, we propose here a model of ordinary differential equations (ODEs) for a possible explanation of the facts and control mechanism which is coupled with the cell cycle. Simulation results of the ODE model are demonstrated to agree with experimental data for the wild type and the cdc2-33 mutant. We show that the coupling of cell growth to cell division by translational control may account for observed properties of size control in fission yeast. As the translational control in the expression of cycle proteins Cdc13 and Cdc25 constructs positive feedback loops, the dynamical activities of the key components undergoes a rapid rising after a preliminary stage of slow increase. The coupling of this dynamical behavior to the elongation of the cell naturally gives rise to a rate change point and to 'sizer' and 'timer' phases, which characterize the cell cycle of fission yeast.

Timing robustness in the budding and fission yeast cell cycles

Robustness of biological models has emerged as an important principle in systems biology. Many past analyses of Boolean models update all pending changes in signals simultaneously (i.e., synchronously), making it impossible to consider robustness to variations in timing that result from noise and different environmental conditions. We checked previously published mathematical models of the cell cycles of budding and fission yeast for robustness to timing variations by constructing Boolean models and analyzing them using model-checking software for the property of speed independence. Surprisingly, the models are nearly, but not totally, speed-independent. In some cases, examination of timing problems discovered in the analysis exposes apparent inaccuracies in the model. Biologically justified revisions to the model eliminate the timing problems. Furthermore, in silico random mutations in the regulatory interactions of a speed-independent Boolean model are shown to be unlikely to preserve speed independence, even in models that are otherwise functional, providing evidence for selection pressure to maintain timing robustness. Multiple cell cycle models exhibit strong robustness to timing variation, apparently due to evolutionary pressure. Thus, timing robustness can be a basis for generating testable hypotheses and can focus attention on aspects of a model that may need refinement.

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.

Deconstructing the core dynamics from a complex time-lagged regulatory biological circuit

Complex regulatory dynamics is ubiquitous in molecular networks composed of genes and proteins. Recent progress in computational biology and its application to molecular data generate a growing number of complex networks. Yet, it has been difficult to understand the governing principles of these networks beyond graphical analysis or extensive numerical simulations. Here the authors exploit several simplifying biological circumstances which thereby enable to directly detect the underlying dynamical regularities driving periodic oscillations in a dynamical nonlinear computational model of a protein-protein network. System analysis is performed using the cell cycle, a mathematically well-described complex regulatory circuit driven by external signals. By introducing an explicit time delay and using a 'tearing-and-zooming' approach the authors reduce the system to a piecewise linear system with two variables that capture the dynamics of this complex network. A key step in the analysis is the identification of functional subsystems by identifying the relations between state-variables within the model. These functional subsystems are referred to as dynamical modules operating as sensitive switches in the original complex model. By using reduced mathematical representations of the subsystems the authors derive explicit conditions on how the cell cycle dynamics depends on system parameters, and can, for the first time, analyse and prove global conditions for system stability. The approach which includes utilising biological simplifying conditions, identification of dynamical modules and mathematical reduction of the model complexity may be applicable to other well-characterised biological regulatory circuits. [Includes supplementary material].

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.

A genetic timer through noise-induced stabilization of an unstable state

Stochastic fluctuations affect the dynamics of biological systems. Typically, such noise causes perturbations that can permit genetic circuits to escape stable states, triggering, for example, phenotypic switching. In contrast, studies have shown that noise can surprisingly also generate new states, which exist solely in the presence of fluctuations. In those instances noise is supplied externally to the dynamical system. Here, we present a mechanism in which noise intrinsic to a simple genetic circuit effectively stabilizes a deterministically unstable state. Furthermore, this noise-induced stabilization represents a unique mechanism for a genetic timer. Specifically, we analyzed the effect of noise intrinsic to a prototypical two-component gene-circuit architecture composed of interacting positive and negative feedback loops. Genetic circuits with this topology are common in biology and typically regulate cell cycles and circadian clocks. These systems can undergo a variety of bifurcations in response to parameter changes. Simulations show that near one such bifurcation, noise induces oscillations around an unstable spiral point and thus effectively stabilizes this unstable fixed point. Because of the periodicity of these oscillations, the lifetime of the noise-dependent stabilization exhibits a polymodal distribution with multiple, well defined, and regularly spaced peaks. Therefore, the noise-induced stabilization presented here constitutes a minimal mechanism for a genetic circuit to function as a timer that could be used in the engineering of synthetic circuits.

The transition from differential equations to Boolean networks: a case study in simplifying a regulatory network model

Methods for modeling cellular regulatory networks as diverse as differential equations and Boolean networks co-exist, however, without much closer correspondence to each other. With the example system of the fission yeast cell cycle control network, we here discuss these two approaches with respect to each other. We find that a Boolean network model can be formulated as a specific coarse-grained limit of the more detailed differential equations model for this system. This demonstrates the mathematical foundation on which Boolean networks can be applied to biological regulatory networks in a controlled way.

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.

Boolean network model predicts cell cycle sequence of fission yeast

A Boolean network model of the cell-cycle regulatory network of fission yeast (Schizosaccharomyces Pombe) is constructed solely on the basis of the known biochemical interaction topology. Simulating the model in the computer faithfully reproduces the known activity sequence of regulatory proteins along the cell cycle of the living cell. Contrary to existing differential equation models, no parameters enter the model except the structure of the regulatory circuitry. The dynamical properties of the model indicate that the biological dynamical sequence is robustly implemented in the regulatory network, with the biological stationary state G1 corresponding to the dominant attractor in state space, and with the biological regulatory sequence being a strongly attractive trajectory. Comparing the fission yeast cell-cycle model to a similar model of the corresponding network in S. cerevisiae, a remarkable difference in circuitry, as well as dynamics is observed. While the latter operates in a strongly damped mode, driven by external excitation, the S. pombe network represents an auto-excited system with external damping.

Computational analysis of mammalian cell division gated by a circadian clock: quantized cell cycles and cell size control

Cell cycle and circadian rhythms are conserved from cyanobacteria to humans with robust cyclic features. Recently, molecular links between these two cyclic processes have been discovered. Core clock transcription factors, Bmal1 and Clock (Clk), directly regulate Wee1 kinase, which inhibits entry into the mitosis. We investigate the effect of this connection on the timing of mammalian cell cycle processes with computational modeling tools. We connect a minimal model of circadian rhythms, which consists of transcription-translation feedback loops, with a modified mammalian cell cycle model from Novak and Tyson (2004). As we vary the mass doubling time (MDT) of the cell cycle, stochastic simulations reveal quantized cell cycles when the activity of Wee1 is influenced by clock components. The quantized cell cycles disappear in the absence of coupling or when the strength of this link is reduced. More intriguingly, our simulations indicate that the circadian clock triggers critical size control in the mammalian cell cycle. A periodic brake on the cell cycle progress via Wee1 enforces size control when the MDT is quite different from the circadian period. No size control is observed in the absence of coupling. The issue of size control in the mammalian system is debatable, whereas it is well established in yeast. It is possible that the size control is more readily observed in cell lines that contain circadian rhythms, since not all cell types have a circadian clock. This would be analogous to an ultradian clock intertwined with quantized cell cycles (and possibly cell size control) in yeast. We present the first coupled model between the mammalian cell cycle and circadian rhythms that reveals quantized cell cycles and cell size control influenced by the clock.

Quantized cycling time in artificial gene networks induced by noise and intercell communication

We propose a mechanism for the quantized cycling time based on the interplay of cell-to-cell communication and stochasticity, by investigating a model of coupled genetic oscillators with known topology. In addition, we discuss how inhomogeneity can be used to enhance such quantizing effects, while the degree of variability obtained can be controlled using the noise intensity or adequate system parameters.

A data integration approach for cell cycle analysis oriented to model simulation in systems biology

Background

The cell cycle is one of the biological processes most frequently investigated in systems biology studies and it involves the knowledge of a large number of genes and networks of protein interactions. A deep knowledge of the molecular aspect of this biological process can contribute to making cancer research more accurate and innovative. In this context the mathematical modelling of the cell cycle has a relevant role to quantify the behaviour of each component of the systems. The mathematical modelling of a biological process such as the cell cycle allows a systemic description that helps to highlight some features such as emergent properties which could be hidden when the analysis is performed only from a reductionism point of view. Moreover, in modelling complex systems, a complete annotation of all the components is equally important to understand the interaction mechanism inside the network: for this reason data integration of the model components has high relevance in systems biology studies.

Description

In this work, we present a resource, the Cell Cycle Database, intended to support systems biology analysis on the Cell Cycle process, based on two organisms, yeast and mammalian. The database integrates information about genes and proteins involved in the cell cycle process, stores complete models of the interaction networks and allows the mathematical simulation over time of the quantitative behaviour of each component. To accomplish this task, we developed, a web interface for browsing information related to cell cycle genes, proteins and mathematical models. In this framework, we have implemented a pipeline which allows users to deal with the mathematical part of the models, in order to solve, using different variables, the ordinary differential equation systems that describe the biological process.

Conclusion

This integrated system is freely available in order to support systems biology research on the cell cycle and it aims to become a useful resource for collecting all the information related to actual and future models of this network. The flexibility of the database allows the addition of mathematical data which are used for simulating the behavior of the cell cycle components in the different models. The resource deals with two relevant problems in systems biology: data integration and mathematical simulation of a crucial biological process related to cancer, such as the cell cycle. In this way the resource is useful both to retrieve information about cell cycle model components and to analyze their dynamical properties. The Cell Cycle Database can be used to find system-level properties, such as stable steady states and oscillations, by coupling structure and dynamical information about models.

Modeling the septation initiation network (SIN) in fission yeast cells

Cytokinesis in fission yeast is controlled by a signal transduction pathway called the Septation Initiation Network (SIN). From a dynamical point of view the most interesting questions about the regulation of fission yeast cytokinesis are: how do wild type cells ensure that septation is initiated only once per cycle? Why does the control system stay in a continuously septating state in some mutant strains? And how is it that the SIN remains active when cytokinesis fails? To answer these questions we construct a simplified mathematical model of the SIN and graft this regulatory module onto our previous model of cyclin-dependent kinase (Cdk) dynamics in fission yeast cells. The SIN is both activated and inhibited by mitotic Cdk/cyclin complexes. As a consequence of this dual regulation, the SIN gets activated only once at the end of mitosis, when Cdk activity drops. The mathematical model describes the timing of septation not only in wild type cells but also in mutants where components of the SIN are knocked out. The model predicts phenotypes of some uncharacterized mutant cells and shows how a cytokinesis checkpoint can stop the cell cycle if septation fails.

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.

Analysis of a generic model of eukaryotic cell-cycle regulation

We propose a protein interaction network for the regulation of DNA synthesis and mitosis that emphasizes the universality of the regulatory system among eukaryotic cells. The idiosyncrasies of cell cycle regulation in particular organisms can be attributed, we claim, to specific settings of rate constants in the dynamic network of chemical reactions. The values of these rate constants are determined ultimately by the genetic makeup of an organism. To support these claims, we convert the reaction mechanism into a set of governing kinetic equations and provide parameter values (specific to budding yeast, fission yeast, frog eggs, and mammalian cells) that account for many curious features of cell cycle regulation in these organisms. Using one-parameter bifurcation diagrams, we show how overall cell growth drives progression through the cell cycle, how cell-size homeostasis can be achieved by two different strategies, and how mutations remodel bifurcation diagrams and create unusual cell-division phenotypes. The relation between gene dosage and phenotype can be summarized compactly in two-parameter bifurcation diagrams. Our approach provides a theoretical framework in which to understand both the universality and particularity of cell cycle regulation, and to construct, in modular fashion, increasingly complex models of the networks controlling cell growth and division.

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.