Integrative analysis of cell cycle control in budding yeast

Robust, tunable biological oscillations from interlinked positive and negative feedback loops

A simple negative feedback loop of interacting genes or proteins has the potential to generate sustained oscillations. However, many biological oscillators also have a positive feedback loop, raising the question of what advantages the extra loop imparts. Through computational studies, we show that it is generally difficult to adjust a negative feedback oscillator's frequency without compromising its amplitude, whereas with positive-plus-negative feedback, one can achieve a widely tunable frequency and near-constant amplitude. This tunability makes the latter design suitable for biological rhythms like heartbeats and cell cycles that need to provide a constant output over a range of frequencies. Positive-plus-negative oscillators also appear to be more robust and easier to evolve, rationalizing why they are found in contexts where an adjustable frequency is unimportant.

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

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

Protein networking: insights into global functional organization of proteomes

The formulation of network models from global protein studies is essential to understand the functioning of organisms. Network models of the proteome enable the application of Complex Network Analysis, a quantitative framework to investigate large complex networks using techniques from graph theory, statistical physics, dynamical systems and other fields. This approach has provided many insights into the functional organization of the proteome so far and will likely continue to do so. Currently, several network concepts have emerged in the field of proteomics. It is important to highlight the differences between these concepts, since different representations allow different insights into functional organization. One such concept is the protein interaction network, which contains proteins as nodes and undirected edges representing the occurrence of binding in large-scale protein-protein interaction studies. A second concept is the protein-signaling network, in which the nodes correspond to levels of post-translationally modified forms of proteins and directed edges to causal effects through post-translational modification, such as phosphorylation. Several other network concepts were introduced for proteomics. Although all formulated as networks, the concepts represent widely different physical systems. Therefore caution should be taken when applying relevant topological analysis. We review recent literature formulating and analyzing such networks.

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.

Modelling molecular interaction pathways using a two-stage identification algorithm

In systems biology, molecular interactions are typically modelled using white-box methods, usually based on mass action kinetics. Unfortunately, problems with dimensionality can arise when the number of molecular species in the system is very large, which makes the system modelling and behavior simulation extremely difficult or computationally too expensive. As an alternative, this paper investigates the identification of two molecular interaction pathways using a black-box approach. This type of method creates a simple linear-in-the-parameters model using regression of data, where the output of the model at any time is a function of previous system states of interest. One of the main objectives in building black-box models is to produce an optimal sparse nonlinear one to effectively represent the system behavior. In this paper, it is achieved by applying an efficient iterative approach, where the terms in the regression model are selected and refined using a forward and backward subset selection algorithm. The method is applied to model identification for the MAPK signal transduction pathway and the Brusselator using noisy data of different sizes. Simulation results confirm the efficacy of the black-box modelling method which offers an alternative to the computationally expensive conventional approach.

Least dissipation cost as a design principle for robustness and function of cellular networks

From a study of the budding yeast cell cycle, we found that the cellular network evolves to have the least cost for realizing its biological function. We quantify the cost in terms of the dissipation or heat loss characterized through the steady-state properties: the underlying landscape and the associated flux. We found that the dissipation cost is intimately related to the stability and robustness of the network. With the least dissipation cost, the network becomes most stable and robust under mutations and perturbations on the sharpness of the response from input to output as well as self-degradations. The least dissipation cost may provide a general design principle for the cellular network to survive from the evolution and realize the biological function.

Systems biology towards life in silico: mathematics of the control of living cells

Systems Biology is the science that aims to understand how biological function absent from macromolecules in isolation, arises when they are components of their system. Dedicated to the memory of Reinhart Heinrich, this paper discusses the origin and evolution of the new part of systems biology that relates to metabolic and signal-transduction pathways and extends mathematical biology so as to address postgenomic experimental reality. Various approaches to modeling the dynamics generated by metabolic and signal-transduction pathways are compared. The silicon cell approach aims to describe the intracellular network of interest precisely, by numerically integrating the precise rate equations that characterize the ways macromolecules' interact with each other. The non-equilibrium thermodynamic or 'lin-log' approach approximates the enzyme rate equations in terms of linear functions of the logarithms of the concentrations. Biochemical Systems Analysis approximates in terms of power laws. Importantly all these approaches link system behavior to molecular interaction properties. The latter two do this less precisely but enable analytical solutions. By limiting the questions asked, to optimal flux patterns, or to control of fluxes and concentrations around the (patho)physiological state, Flux Balance Analysis and Metabolic/Hierarchical Control Analysis again enable analytical solutions. Both the silicon cell approach and Metabolic/Hierarchical Control Analysis are able to highlight where and how system function derives from molecular interactions. The latter approach has also discovered a set of fundamental principles underlying the control of biological systems. The new law that relates concentration control to control by time is illustrated for an important signal transduction pathway, i.e. nuclear hormone receptor signaling such as relevant to bone formation. It is envisaged that there is much more Mathematical Biology to be discovered in the area between molecules and Life.

In-silico modeling of the mitotic spindle assembly checkpoint

Background

The Mitotic Spindle Assembly Checkpoint (^MSAC) is an evolutionary conserved mechanism that ensures the correct segregation of chromosomes by restraining cell cycle progression from entering anaphase until all chromosomes have made proper bipolar attachments to the mitotic spindle. Its malfunction can lead to cancer.

Principle Findings

We have constructed and validated for the human ^MSAC mechanism an in silico dynamical model, integrating 11 proteins and complexes. The model incorporates the perspectives of three central control pathways, namely Mad1/Mad2 induced Cdc20 sequestering based on the Template Model, MCC formation, and APC inhibition. Originating from the biochemical reactions for the underlying molecular processes, non-linear ordinary differential equations for the concentrations of 11 proteins and complexes of the ^MSAC are derived. Most of the kinetic constants are taken from literature, the remaining four unknown parameters are derived by an evolutionary optimization procedure for an objective function describing the dynamics of the APC:Cdc20 complex. MCC:APC dissociation is described by two alternatives, namely the “Dissociation” and the “Convey” model variants. The attachment of the kinetochore to microtubuli is simulated by a switching parameter silencing those reactions which are stopped by the attachment. For both, the Dissociation and the Convey variants, we compare two different scenarios concerning the microtubule attachment dependent control of the dissociation reaction. Our model is validated by simulation of ten perturbation experiments.

Conclusion

Only in the controlled case, our models show ^MSAC behaviour at meta- to anaphase transition in agreement with experimental observations. Our simulations revealed that for ^MSAC activation, Cdc20 is not fully sequestered; instead APC is inhibited by MCC binding.

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.

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.

A quantitative study of the division cycle of Caulobacter crescentus stalked cells

Progression of a cell through the division cycle is tightly controlled at different steps to ensure the integrity of genome replication and partitioning to daughter cells. From published experimental evidence, we propose a molecular mechanism for control of the cell division cycle in Caulobacter crescentus. The mechanism, which is based on the synthesis and degradation of three “master regulator” proteins (CtrA, GcrA, and DnaA), is converted into a quantitative model, in order to study the temporal dynamics of these and other cell cycle proteins. The model accounts for important details of the physiology, biochemistry, and genetics of cell cycle control in stalked C. crescentus cell. It reproduces protein time courses in wild-type cells, mimics correctly the phenotypes of many mutant strains, and predicts the phenotypes of currently uncharacterized mutants. Since many of the proteins involved in regulating the cell cycle of C. crescentus are conserved among many genera of α-proteobacteria, the proposed mechanism may be applicable to other species of importance in agriculture and medicine.

The cell cycle is the sequence of events by which a growing cell replicates all its components and divides them more or less evenly between two daughter cells. The timing and spatial organization of these events are controlled by gene–protein interaction networks of great complexity. A challenge for computational biology is to build realistic, accurate, predictive mathematical models of these control systems in a variety of organisms, both eukaryotes and prokaryotes. To this end, we present a model of a portion of the molecular network controlling DNA synthesis, cell cycle–related gene expression, DNA methylation, and cell division in stalked cells of the α-proteobacterium Caulobacter crescentus. The model is formulated in terms of nonlinear ordinary differential equations for the major cell cycle regulatory proteins in Caulobacter: CtrA, GcrA, DnaA, CcrM, and DivK. Kinetic rate constants are estimated, and the model is tested against available experimental observations on wild-type and mutant cells. The model is viewed as a starting point for more comprehensive models of the future that will account, in addition, for the spatial asymmetry of Caulobacter reproduction (swarmer cells as well as stalked cells), the correlation of cell growth and division, and cell cycle checkpoints.

Identifying dynamical modules from genetic regulatory systems: applications to the segment polarity network

Background

It is widely accepted that genetic regulatory systems are 'modular', in that the whole system is made up of smaller 'subsystems' corresponding to specific biological functions. Most attempts to identify modules in genetic regulatory systems have relied on the topology of the underlying network. However, it is the temporal activity (dynamics) of genes and proteins that corresponds to biological functions, and hence it is dynamics that we focus on here for identifying subsystems.

Results

Using Boolean network models as an exemplar, we present a new technique to identify subsystems, based on their dynamical properties. The main part of the method depends only on the stable dynamics (attractors) of the system, thus requiring no prior knowledge of the underlying network. However, knowledge of the logical relationships between the network components can be used to describe how each subsystem is regulated. To demonstrate its applicability to genetic regulatory systems, we apply the method to a model of the Drosophila segment polarity network, providing a detailed breakdown of the system.

Conclusion

We have designed a technique for decomposing any set of discrete-state, discrete-time attractors into subsystems. Having a suitable mathematical model also allows us to describe how each subsystem is regulated and how robust each subsystem is against perturbations. However, since the subsystems are found directly from the attractors, a mathematical model or underlying network topology is not necessarily required to identify them, potentially allowing the method to be applied directly to experimental expression data.

Synchronized dynamics and non-equilibrium steady states in a stochastic yeast cell-cycle network

Applying the mathematical circulation theory of Markov chains, we investigate the synchronized stochastic dynamics of a discrete network model of yeast cell-cycle regulation where stochasticity has been kept rather than being averaged out. By comparing the network dynamics of the stochastic model with its corresponding deterministic network counterpart, we show that the synchronized dynamics can be soundly characterized by a dominant circulation in the stochastic model, which is the natural generalization of the deterministic limit cycle in the deterministic system. Moreover, the period of the main peak in the power spectrum, which is in common use to characterize the synchronized dynamics, perfectly corresponds to the number of states in the main cycle with dominant circulation. Such a large separation in the magnitude of the circulations, between a dominant, main cycle and the rest, gives rise to the stochastic synchronization phenomenon.

Inferring gene regulatory networks by integrating static and dynamic data

OBJECTIVES

The purpose of the paper is to propose a methodology for learning gene regulatory networks from DNA microarray data based on the integration of different data and knowledge sources. We applied our method to Saccharomyces cerevisiae experiments, focusing our attention on cell cycle regulatory mechanisms. We exploited data from deletion mutant experiments (static data), gene expression time series (dynamic data) and the knowledge encoded in the Gene Ontology.

METHODS

The proposed method is based on four phases. An initial gene network was derived from static data by means of a simple statistical approach. Then, the genes classified in the Gene Ontology as being involved in the cell cycle were selected. As a third step, the network structure was used to initialize a linear dynamic model of gene expression profiles. Finally, a genetic algorithm was applied to update the gene network exploiting data coming from an experiment on the yeast cell cycle.

RESULTS

We compared the network models provided by our approach with those obtained with a fully data-driven approach, by looking at their AIC scores and at the percentage of preserved connections in the best solutions. The results show that several nearly equivalent solutions, in terms of AIC scores, can be found. This problem is greatly mitigated by following our approach, which is able to find more robust models by fixing a portion of the network structure on the basis of prior knowledge. The best network structure was biologically evaluated on a set of 22 known cell cycle genes against independent knowledge sources.

CONCLUSIONS

An approach able to integrate several sources of information is needed to infer gene regulatory networks, as a fully data-driven search is in general prone to overfitting and to unidentifiability problems. The learned networks encode hypotheses on regulatory relationships that need to be verified by means of wet-lab experiments.

Stable stochastic dynamics in yeast cell cycle

Chemical reactions in cells are subject to intense stochastic fluctuations. An important question is how the fundamental physiological behavior of the cell is kept stable against those noisy perturbations. In this study, a stochastic model of the cell cycle of budding yeast was constructed to analyze the effects of noise on the cell-cycle oscillation. The model predicts intense noise in levels of mRNAs and proteins, and the simulated protein levels explain the observed statistical tendency of noise in populations of synchronous and asynchronous cells. Despite intense noise in levels of proteins and mRNAs, the cell cycle is stable enough to bring the largely perturbed cells back to the physiological cyclic oscillation. The model shows that consecutively appearing fixed points are the origin of this stability of the cell cycle.

Universally sloppy parameter sensitivities in systems biology models

Quantitative computational models play an increasingly important role in modern biology. Such models typically involve many free parameters, and assigning their values is often a substantial obstacle to model development. Directly measuring in vivo biochemical parameters is difficult, and collectively fitting them to other experimental data often yields large parameter uncertainties. Nevertheless, in earlier work we showed in a growth-factor-signaling model that collective fitting could yield well-constrained predictions, even when it left individual parameters very poorly constrained. We also showed that the model had a "sloppy" spectrum of parameter sensitivities, with eigenvalues roughly evenly distributed over many decades. Here we use a collection of models from the literature to test whether such sloppy spectra are common in systems biology. Strikingly, we find that every model we examine has a sloppy spectrum of sensitivities. We also test several consequences of this sloppiness for building predictive models. In particular, sloppiness suggests that collective fits to even large amounts of ideal time-series data will often leave many parameters poorly constrained. Tests over our model collection are consistent with this suggestion. This difficulty with collective fits may seem to argue for direct parameter measurements, but sloppiness also implies that such measurements must be formidably precise and complete to usefully constrain many model predictions. We confirm this implication in our growth-factor-signaling model. Our results suggest that sloppy sensitivity spectra are universal in systems biology models. The prevalence of sloppiness highlights the power of collective fits and suggests that modelers should focus on predictions rather than on 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.

Understanding diseases by mouse click: the promise and potential of computational approaches in Systems Biology

Computational modelling approaches can nowadays build large-scale simulations of cellular behaviour based on data describing detailed molecular level interactions, thus performing the space- and time-scale integrations that would be impossible just by intuition. Recent progress in the development of both experimental methods and computational tools has provided the means to generate the necessary quantitative data and has made computational methods accessible even to non-theorists, thereby removing a major hurdle that has in the past made many experimentalists hesitate to invest serious effort in formulating quantitative models. We describe how computational biology differs from classical bioinformatics, how it emerged from mathematical biology and elucidate the role it plays for the integration of traditionally separated areas of biomedical research within the larger framework of Systems Biology.

Yeast systems biology: modelling the winemaker's art

Yeast research represents an important nexus between fundamental and applied research. Just as fundamental yeast research transitioned from classical, reductionist strategies to whole-genome techniques, whole-genome studies are advancing to the next level of biological research, referred to as systems biology. Industries that rely on high-performing yeast, such as the wine industry, are therefore poised to reap the many benefits that systems biology can provide. This includes the promise of strain development at speeds and costs which are unobtainable using current techniques. This article reviews the current state of whole-genome techniques available to yeast researchers and outlines how these processes can be used to obtain 'systems-level' information to provide insights into winemaking.

Mitotic exit in two dimensions

Metaphase of mitosis is brought about in all eukaryotes by activation of cylin-dependent kinase (Cdk1), whereas exit from mitosis requires down-regulation of Cdk1 activity and dephosphorylation of its target proteins. In budding yeast, the completion of mitotic exit requires the release and activation of the Cdc14 protein-phosphatase, which is kept inactive in the nucleolus during most of the cell cycle. Activation of Cdc14 is controlled by two regulatory networks called FEAR (Cdc fourteen early anaphase release) and MEN (mitotic exit network). We have shown recently that the anaphase promoting protease (separase) is essential for Cdc14 activation, thereby it makes mitotic exit dependent on execution of anaphase. Based on this finding, we have proposed a new model for mitotic exit in budding yeast. Here we explain the essence of the model by phaseplane analysis, which reveals two underlying bistable switches in the regulatory network. One bistable switch is caused by mutual activation (positive feedback) between Cdc14 activating MEN and Cdc14 itself. The mitosis-inducing Cdk1 activity inhibits the activation of this positive feedback loop and thereby controlling this switch. The other irreversible switch is generated by a double-negative feedback (mutual antagonism) between mitosis inducing Cdk1 activity and its degradation machinery (APC(Cdh1)). The Cdc14 phosphatase helps turning this switch in favor of APC(Cdh1) side. Both of these bistable switches have characteristic thresholds, the first one for Cdk1 activity, while the second for Cdc14 activity. We show that the physiological behaviors of certain cell cycle mutants are suggestive for those Cdk1 and Cdc14 thresholds. The two bistable switches turn on in a well-defined order. In this paper, we explain how the activation of Cdc20 (which causes the activation of separase and a decrease of Cdk1 kinase activity) provides an initial trigger for the activation of the MEN-Cdc14 positive feedback loops, which in turn, flips the second irreversible Cdk-APC(Cdh1) switch on the APC(Cdh1) side).

Hypothesis testing via integrated computer modeling and digital fluorescence microscopy

Computational modeling has the potential to add an entirely new approach to hypothesis testing in yeast cell biology. Here, we present a method for seamless integration of computational modeling with quantitative digital fluorescence microscopy. This integration is accomplished by developing computational models based on hypotheses for underlying cellular processes that may give rise to experimentally observed fluorescent protein localization patterns. Simulated fluorescence images are generated from the computational models of underlying cellular processes via a "model-convolution" process. These simulated images can then be directly compared to experimental fluorescence images in order to test the model. This method provides a framework for rigorous hypothesis testing in yeast cell biology via integrated mathematical modeling and digital fluorescence microscopy.

Mathematical modeling as a tool for investigating cell cycle control networks

Although not a traditional experimental "method," mathematical modeling can provide a powerful approach for investigating complex cell signaling networks, such as those that regulate the eukaryotic cell division cycle. We describe here one modeling approach based on expressing the rates of biochemical reactions in terms of nonlinear ordinary differential equations. We discuss the steps and challenges in assigning numerical values to model parameters and the importance of experimental testing of a mathematical model. We illustrate this approach throughout with the simple and well-characterized example of mitotic cell cycles in frog egg extracts. To facilitate new modeling efforts, we describe several publicly available modeling environments, each with a collection of integrated programs for mathematical modeling. This review is intended to justify the place of mathematical modeling as a standard method for studying molecular regulatory networks and to guide the non-expert to initiate modeling projects in order to gain a systems-level perspective for complex control systems.

Quantifying robustness and dissipation cost of yeast cell cycle network: the funneled energy landscape perspectives

We study the origin of robustness of yeast cell cycle cellular network through uncovering its underlying energy landscape. This is realized from the information of the steady-state probabilities by solving a discrete set of kinetic master equations for the network. We discovered that the potential landscape of yeast cell cycle network is funneled toward the global minimum, G1 state. The ratio of the energy gap between G1 and average versus roughness of the landscape termed as robustness ratio (RR) becomes a quantitative measure of the robustness and stability for the network. The funneled landscape is quite robust against random perturbations from the inherent wiring or connections of the network. There exists a global phase transition between the more sensitive response or less self-degradation phase leading to underlying funneled global landscape with large RR, and insensitive response or more self-degradation phase leading to shallower underlying landscape of the network with small RR. Furthermore, we show that the more robust landscape also leads to less dissipation cost of the network. Least dissipation and robust landscape might be a realization of Darwinian principle of natural selection at cellular network level. It may provide an optimal criterion for network wiring connections and design.

The theory of biological robustness and its implication in cancer

One of the essential issues in systems biology is to identify fundamental principles that govern living organisms at the system level. In this chapter, I argue that robustness is a fundamental feature of living systems where its relationship with evolution-trade-offs among robustness, fragility, resource demands, and performance-provides a possible framework for how biological systems have evolved and been organized. In addition, diseases can be con- sidered as a manifestation of fragility of the system. In some cases, such as cancer, the disease state establishes its own robustness against therapeutic interventions. Understanding robustness and its intrinsic properties will provide us with a more profound understanding of biological systems, their anomalies, and countermeasures.

Bayesian approaches to reverse engineer cellular systems: a simulation study on nonlinear Gaussian networks

BACKGROUND

Reverse engineering cellular networks is currently one of the most challenging problems in systems biology. Dynamic Bayesian networks (DBNs) seem to be particularly suitable for inferring relationships between cellular variables from the analysis of time series measurements of mRNA or protein concentrations. As evaluating inference results on a real dataset is controversial, the use of simulated data has been proposed. However, DBN approaches that use continuous variables, thus avoiding the information loss associated with discretization, have not yet been extensively assessed, and most of the proposed approaches have dealt with linear Gaussian models.

RESULTS

We propose a generalization of dynamic Gaussian networks to accommodate nonlinear dependencies between variables. As a benchmark dataset to test the new approach, we used data from a mathematical model of cell cycle control in budding yeast that realistically reproduces the complexity of a cellular system. We evaluated the ability of the networks to describe the dynamics of cellular systems and their precision in reconstructing the true underlying causal relationships between variables. We also tested the robustness of the results by analyzing the effect of noise on the data, and the impact of a different sampling time.

CONCLUSION

The results confirmed that DBNs with Gaussian models can be effectively exploited for a first level analysis of data from complex cellular systems. The inferred models are parsimonious and have a satisfying goodness of fit. Furthermore, the networks not only offer a phenomenological description of the dynamics of cellular systems, but are also able to suggest hypotheses concerning the causal interactions between variables. The proposed nonlinear generalization of Gaussian models yielded models characterized by a slightly lower goodness of fit than the linear model, but a better ability to recover the true underlying connections between variables.

Phosphorylation of the Sic1 inhibitor of B-type cyclins in Saccharomyces cerevisiae is not essential but contributes to cell cycle robustness

In budding yeast, B-type cyclin (Clb)-dependent kinase activity is essential for S phase and mitosis. In newborn G(1) cells, Clb kinase accumulation is blocked, in part because of the Sic1 stoichiometric inhibitor. Previous results strongly suggested that G(1) cyclin-dependent Sic1 phosphorylation, and its consequent degradation, is essential for S phase. However, cells containing a precise endogenous gene replacement of SIC1 with SIC1-0P (all nine phosphorylation sites mutated) were fully viable. Unphosphorylatable Sic1 was abundant and nuclear throughout the cell cycle and effectively inhibited Clb kinase in vitro. SIC1-0P cells had a lengthened G(1) and increased G(1) cyclin transcriptional activation and variable delays in the budded part of the cell cycle. SIC1-0P was lethal when combined with deletion of CLB2, CLB3, or CLB5, the major B-type cyclins. Sic1 phosphorylation provides a sharp link between G(1) cyclin activation and Clb kinase activation, but failure of Sic1 phosphorylation and proteolysis imposes a variable cell cycle delay and extreme sensitivity to B-type cyclin dosage, rather than a lethal cell cycle block.

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.

Cell size at S phase initiation: an emergent property of the G1/S network

The eukaryotic cell cycle is the repeated sequence of events that enable the division of a cell into two daughter cells. It is divided into four phases: G₁, S, G₂, and M. Passage through the cell cycle is strictly regulated by a molecular interaction network, which involves the periodic synthesis and destruction of cyclins that bind and activate cyclin-dependent kinases that are present in nonlimiting amounts. Cyclin-dependent kinase inhibitors contribute to cell cycle control. Budding yeast is an established model organism for cell cycle studies, and several mathematical models have been proposed for its cell cycle. An area of major relevance in cell cycle control is the G₁ to S transition. In any given growth condition, it is characterized by the requirement of a specific, critical cell size, P_S, to enter S phase. The molecular basis of this control is still under discussion. The authors report a mathematical model of the G₁ to S network that newly takes into account nucleo/cytoplasmic localization, the role of the cyclin-dependent kinase Sic1 in facilitating nuclear import of its cognate Cdk1-Clb5, Whi5 control, and carbon source regulation of Sic1 and Sic1-containing complexes. The model was implemented by a set of ordinary differential equations that describe the temporal change of the concentration of the involved proteins and protein complexes. The model was tested by simulation in several genetic and nutritional setups and was found to be neatly consistent with experimental data. To estimate P_S, the authors developed a hybrid model including a probabilistic component for firing of DNA replication origins. Sensitivity analysis of P_S provides a novel relevant conclusion: P_S is an emergent property of the G₁ to S network that strongly depends on growth rate.

A major property of living cells is their ability to maintain mass homeostasis throughout cell divisions. It has been proposed that in order to achieve such homeostasis, some critical event(s) in the cell cycle will take place only when the cell has grown beyond a critical cell size. In the budding yeast Saccharomyces cerevisiae, a widely used model for the study of the eukaryotic cell cycle, a large body of evidence indicates that cells have to reach a critical size before they start to replicate their DNA and to form bud, which will give rise to the daughter cell. This critical cell size is modulated by growth rate, hence by nutritional conditions and the multiplicity of genetic material (i.e., ploidy). The authors present a mathematical model of the regulatory molecular network acting at the G₁ to S transition. The major novel features of this model compared with previous models of this process are (1) the accounting for cell growth (i.e., the increase in cell volume); (2) the explicit consideration of the fact that cells have a nucleus and a cytoplasm, and that key cell cycle regulatory molecules must move between these different compartments and can only react or regulate each other if they are in the same compartment; and (3) the requirement of sequential overcoming of two molecular thresholds given by a cyclin-dependent kinase/cyclin and a cyclin-dependent kinase inhibitor. The model was tested by simulating the processes during G₁ to S transition for different growth conditions or for different mutants and by comparing the results with experimental data. A parameter sensitivity analysis (i.e., testing the model predictions when parameters are varied), newly indicates that the critical cell size is an emergent property of the G₁ to S network. The model leads to a unified interpretation of seemingly disparate experimental observations and makes predictions to be experimentally verified.

Reliable cell disruption in yeast

We report the results of an optical assay to determine the degree of cell wall disruption in yeast. The results indicate that cell wall disruption with glass beads yields reproducible results that can be modelled with an integral measure of time to failure that implies a decreasing failure rate. It is shown that a standard protocol results in only 60% disruption, with a relatively large coefficient of variation. The data show that the yield of total RNA harvested is proportional to the degree of cellular disruption, and that there is no loss of RNA quality with > 90% disruption. The data also show that cell disruption of a synchronous culture varies with the cell cycle. We speculate that the decreasing failure rate is related to the cell cycle phase-dependent disruptability.

Biological robustness in complex host-pathogen systems

Infectious diseases are still the number one killer of human beings. Even in developed countries, infectious diseases continue to be a major health threat. This article explores a conceptual framework for understanding infectious diseases in the context of the complex dynamics between microbe and host, and explores theoretical strategies for anti-infectives. The central pillar of this conceptual framework is that biological robustness is a fundamental property of systems that is closely interlinked with the evolution of symbiotic host-pathogen systems. There are specific architectural features of such robust yet evolvable systems and interpretable trade-offs between robustness, fragility, resource demands, and performance. This concept applies equally to both microbes and host. Pathogens have evolved to exploit the host using various strategies as well as effective escape mechanisms. Modular pathogenicity islands (PAI) derived from horizontal gene transfer, highly variable surface molecules, and a range of other countermeasures enhance the robustness of a pathogen against attacks from the host immune system. The host has likewise evolved complex defensive mechanisms to protect itself against pathogenic threats, but the host immune system includes several trade-offs that can be exploited by pathogens and induces undesirable inflammatory reactions. Due to the complexity of the dynamics emerging from the interactions of multiple microbes and a host, effective counter-measures require an in-depth understanding of system dynamics as well as detailed molecular mechanisms of the processes that are involved.

Whole-cell modeling framework in which biochemical dynamics impact aspects of cellular geometry

A mathematical framework for modeling biological cells from a physicochemical perspective is described. Cells modeled within this framework consist of at least two regions, including a cytosolic volume encapsulated by a membrane surface. The cytosol is viewed as a well-stirred chemical reactor capable of changing volume while the membrane is assumed to be an oriented 2-D surface capable of changing surface area. Two physical properties of the cell, namely volume and surface area, are determined by (and determine) the reaction dynamics generated from a set of chemical reactions designed to be occurring in the cell. This framework allows the modeling of complex cellular behaviors, including self-replication. This capability is illustrated by constructing two self-replicating prototypical whole-cell models. One protocell was designed to be of minimal complexity; the other to incorporate a previously reported well-known mechanism of the eukaryotic cell cycle. In both cases, self-replicative behavior was achieved by seeking stable physically possible oscillations in concentrations and surface-to-volume ratio, and by synchronizing the period of such oscillations to the doubling of cytosolic volume and membrane surface area. Rather than being enforced externally or artificially, growth and division occur naturally as a consequence of the assumed chemical mechanism operating within the framework.

Modelling Cell Growth and its Modulation of the G1/S Transition

We present a model for the regulation of the G(1)/S transition by cell growth in budding yeast. The model includes a description of cell size, the extracellular nutrient concentration and a simplified model of the G(1)/S transition as originally reported by Chen et al. [Mol. Biol. Cell 11:369-391, 2000]. By considering cell growth proportional to cell size we show that the cell grows exponentially. In the case where cell growth is considered proportional to the concentration of a sizer protein within the cell, our model exhibits both exponential and linear cell growth for varying parameter values. The effects of varying nutrient concentration and initial cell size are considered in the context of whether progression through the cell-size checkpoint occurs. We consider our results in relation to recent experimental evidence and discuss possible experiments for testing our theoretical predictions.

Mathematical modeling of intracellular signaling pathways

Dynamic modeling and simulation of signal transduction pathways is an important topic in systems biology and is obtaining growing attention from researchers with experimental or theoretical background. Here we review attempts to analyze and model specific signaling systems. We review the structure of recurrent building blocks of signaling pathways and their integration into more comprehensive models, which enables the understanding of complex cellular processes. The variety of mechanisms found and modeling techniques used are illustrated with models of different signaling pathways. Focusing on the close interplay between experimental investigation of pathways and the mathematical representations of cellular dynamics, we discuss challenges and perspectives that emerge in studies of signaling systems.

The modular systems biology approach to investigate the control of apoptosis in Alzheimer's disease neurodegeneration

Apoptosis is a programmed cell death that plays a critical role during the development of the nervous system and in many chronic neurodegenerative diseases, including Alzheimer's disease (AD). This pathology, characterized by a progressive degeneration of cholinergic function resulting in a remarkable cognitive decline, is the most common form of dementia with high social and economic impact. Current therapies of AD are only symptomatic, therefore the need to elucidate the mechanisms underlying the onset and progression of the disease is surely needed in order to develop effective pharmacological therapies. Because of its pivotal role in neuronal cell death, apoptosis has been considered one of the most appealing therapeutic targets, however, due to the complexity of the molecular mechanisms involving the various triggering events and the many signaling cascades leading to cell death, a comprehensive understanding of this process is still lacking.

Modular systems biology is a very effective strategy in organizing information about complex biological processes and deriving modular and mathematical models that greatly simplify the identification of key steps of a given process.

This review aims at describing the main steps underlying the strategy of modular systems biology and briefly summarizes how this approach has been successfully applied for cell cycle studies. Moreover, after giving an overview of the many molecular mechanisms underlying apoptosis in AD, we present both a modular and a molecular model of neuronal apoptosis that suggest new insights on neuroprotection for this disease.

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.

Quantitative modeling in cell biology: what is it good for?

Recently, there has been a surge in the number of pioneering studies combining experiments with quantitative modeling to explain both relatively simple modules of molecular machinery of the cell and to achieve system-level understanding of cellular networks. Here we discuss the utility and methods of modeling and review several current models of cell signaling, cytoskeletal self-organization, nuclear transport, and the cell cycle. We discuss successes of and barriers to modeling in cell biology and its future directions, and we argue, using the field of bacterial chemotaxis as an example, that the closer the complete systematic understanding of cell behavior is, the more important modeling becomes and the more experiment and theory merge.

Funneled landscape leads to robustness of cell networks: yeast cell cycle

We uncovered the underlying energy landscape for a cellular network. We discovered that the energy landscape of the yeast cell-cycle network is funneled towards the global minimum (G0/G1 phase) from the experimentally measured or inferred inherent chemical reaction rates. The funneled landscape is quite robust against random perturbations. This naturally explains robustness from a physical point of view. The ratio of slope versus roughness of the landscape becomes a quantitative measure of robustness of the network. The funneled landscape can be seen as a possible realization of the Darwinian principle of natural selection at the cellular network level. It provides an optimal criterion for network connections and design. Our approach is general and can be applied to other cellular networks.

Cellular networks are in general quite robust and perform their biological functions against environmental perturbations. There are so far very few studies of why networks should be robust and perform biological functions from the physical point of view. In this work, Wang, Huang, Xia, and Sun studied the global properties of the network from physical perspectives. The aim of this paper is to provide a conceptual framework and a tool to study the global nature of the cellular network. The main conclusion is that by uncovering the underlying potential landscape of the budding yeast cell cycle the authors show that it is funneled and robust against the perturbation from kinetic rates and environmental disturbances through noise. This provides the physical explanation of the robustness and stability of the network for performing biological functions. They believe the energy landscape is useful in exploring global properties of protein–protein interaction networks. They also believe the funneled landscape may provide a possible quantitative realization of the Darwinian principle of natural selection at the cellular network level. Finally, Wang et al. derived a quantitative criterion for robustness of the network function. This criterion may provide a novel algorithm for optimizing the network connections to improve the design of synthetic networks.

BiologicalNetworks: visualization and analysis tool for systems biology

Systems level investigation of genomic scale information requires the development of truly integrated databases dealing with heterogeneous data, which can be queried for simple properties of genes or other database objects as well as for complex network level properties, for the analysis and modelling of complex biological processes. Towards that goal, we recently constructed PathSys, a data integration platform for systems biology, which provides dynamic integration over a diverse set of databases [Baitaluk et al. (2006) BMC Bioinformatics7, 55]. Here we describe a server, BiologicalNetworks, which provides visualization, analysis services and an information management framework over PathSys. The server allows easy retrieval, construction and visualization of complex biological networks, including genome-scale integrated networks of protein–protein, protein–DNA and genetic interactions. Most importantly, BiologicalNetworks addresses the need for systematic presentation and analysis of high-throughput expression data by mapping and analysis of expression profiles of genes or proteins simultaneously on to regulatory, metabolic and cellular networks. BiologicalNetworks Server is available at .

Modelling the network of cell cycle transcription factors in the yeast Saccharomyces cerevisiae

BACKGROUND

Reverse-engineering regulatory networks is one of the central challenges for computational biology. Many techniques have been developed to accomplish this by utilizing transcription factor binding data in conjunction with expression data. Of these approaches, several have focused on the reconstruction of the cell cycle regulatory network of Saccharomyces cerevisiae. The emphasis of these studies has been to model the relationships between transcription factors and their target genes. In contrast, here we focus on reverse-engineering the network of relationships among transcription factors that regulate the cell cycle in S. cerevisiae.

RESULTS

We have developed a technique to reverse-engineer networks of the time-dependent activities of transcription factors that regulate the cell cycle in S. cerevisiae. The model utilizes linear regression to first estimate the activities of transcription factors from expression time series and genome-wide transcription factor binding data. We then use least squares to construct a model of the time evolution of the activities. We validate our approach in two ways: by demonstrating that it accurately models expression data and by demonstrating that our reconstructed model is similar to previously-published models of transcriptional regulation of the cell cycle.

CONCLUSION

Our regression-based approach allows us to build a general model of transcriptional regulation of the yeast cell cycle that includes additional factors and couplings not reported in previously-published models. Our model could serve as a starting point for targeted experiments that test the predicted interactions. In the future, we plan to apply our technique to reverse-engineer other systems where both genome-wide time series expression data and transcription factor binding data are available.

Few crucial links assure checkpoint efficiency in the yeast cell-cycle network

MOTIVATION

The ability of cells to complete mitosis with high fidelity relies on elaborate checkpoint mechanisms. We study S- and M-phase checkpoint responses in silico in the budding yeast with a stochastic dynamical model for the cell-cycle. We aim to provide an unbiased functional classification of network interactions that reflect the contribution of each link to checkpoint efficiency in the presence of cellular fluctuations.

RESULTS

We developed an algorithm BNetDyn to compute stochastic dynamical trajectories for an input gene network and its structural perturbations. User specified output measures like the mutual information between trigger and output nodes are then evaluated on the stationary state of the Markov process. Systematic perturbations of the yeast cell-cycle model by Li et al. classify each link according to its effect on checkpoint efficiencies and stabilities of the main cell-cycle phases. This points to the crosstalk in the cascades downstream of the SBF/MBF transcription activator complexes as determinant for checkpoint optimality; a finding that consistently reflects recent experiments. Finally our stochastic analysis emphasizes how dynamical stability in the yeast cell-cycle network crucially relies on backward inhibitory circuits next to forward induction.

AVAILABILITY

C++ source code and network models can be downloaded at http://www.vital-it.ch/Software/

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.

Simulation of microarray data with realistic characteristics

Background

Microarray technologies have become common tools in biological research. As a result, a need for effective computational methods for data analysis has emerged. Numerous different algorithms have been proposed for analyzing the data. However, an objective evaluation of the proposed algorithms is not possible due to the lack of biological ground truth information. To overcome this fundamental problem, the use of simulated microarray data for algorithm validation has been proposed.

Results

We present a microarray simulation model which can be used to validate different kinds of data analysis algorithms. The proposed model is unique in the sense that it includes all the steps that affect the quality of real microarray data. These steps include the simulation of biological ground truth data, applying biological and measurement technology specific error models, and finally simulating the microarray slide manufacturing and hybridization. After all these steps are taken into account, the simulated data has realistic biological and statistical characteristics. The applicability of the proposed model is demonstrated by several examples.

Conclusion

The proposed microarray simulation model is modular and can be used in different kinds of applications. It includes several error models that have been proposed earlier and it can be used with different types of input data. The model can be used to simulate both spotted two-channel and oligonucleotide based single-channel microarrays. All this makes the model a valuable tool for example in validation of data analysis algorithms.

Downregulation of PP2A(Cdc55) phosphatase by separase initiates mitotic exit in budding yeast

After anaphase, the high mitotic cyclin-dependent kinase (Cdk) activity is downregulated to promote exit from mitosis. To this end, in the budding yeast S. cerevisiae, the Cdk counteracting phosphatase Cdc14 is activated. In metaphase, Cdc14 is kept inactive in the nucleolus by its inhibitor Net1. During anaphase, Cdk- and Polo-dependent phosphorylation of Net1 is thought to release active Cdc14. How Net1 is phosphorylated specifically in anaphase, when mitotic kinase activity starts to decline, has remained unexplained. Here, we show that PP2A(Cdc55) phosphatase keeps Net1 underphosphorylated in metaphase. The sister chromatid-separating protease separase, activated at anaphase onset, interacts with and downregulates PP2A(Cdc55), thereby facilitating Cdk-dependent Net1 phosphorylation. PP2A(Cdc55) downregulation also promotes phosphorylation of Bfa1, contributing to activation of the "mitotic exit network" that sustains Cdc14 as Cdk activity declines. These findings allow us to present a new quantitative model for mitotic exit in budding yeast.