Integrative analysis of cell cycle control in budding yeast

Data integration strategies for whole-cell modeling

Data makes the world go round-and high quality data is a prerequisite for precise models, especially for whole-cell models (WCM). Data for WCM must be reusable, contain information about the exact experimental background, and should-in its entirety-cover all relevant processes in the cell. Here, we review basic requirements to data for WCM and strategies how to combine them. As a species-specific resource, we introduce the Yeast Cell Model Data Base (YCMDB) to illustrate requirements and solutions. We discuss recent standards for data as well as for computational models including the modeling process as data to be reported. We outline strategies for constructions of WCM despite their inherent complexity.

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.

An autonomous mathematical model for the mammalian cell cycle

A mathematical model for the mammalian cell cycle is developed as a system of 13 coupled nonlinear ordinary differential equations. The variables and interactions included in the model are based on detailed consideration of available experimental data. A novel feature of the model is inclusion of cycle tasks such as origin licensing and initiation, nuclear envelope breakdown and kinetochore attachment, and their interactions with controllers (molecular complexes involved in cycle control). Other key features are that the model is autonomous, except for a dependence on external growth factors; the variables are continuous in time, without instantaneous resets at phase boundaries; mechanisms to prevent rereplication are included; and cycle progression is independent of cell size. Eight variables represent cell cycle controllers: the Cyclin D1-Cdk4/6 complex, APC^Cdh1, SCF^βTrCP, Cdc25A, MPF, NuMA, the securin-separase complex, and separase. Five variables represent task completion, with four for the status of origins and one for kinetochore attachment. The model predicts distinct behaviors corresponding to the main phases of the cell cycle, showing that the principal features of the mammalian cell cycle, including restriction point behavior, can be accounted for in a quantitative mechanistic way based on known interactions among cycle controllers and their coupling to tasks. The model is robust to parameter changes, in that cycling is maintained over at least a five-fold range of each parameter when varied individually. The model is suitable for exploring how extracellular factors affect cell cycle progression, including responses to metabolic conditions and to anti-cancer therapies.

Network medicine: an approach to complex kidney disease phenotypes

Scientific reductionism has been the basis of disease classification and understanding for more than a century. However, the reductionist approach of characterizing diseases from a limited set of clinical observations and laboratory evaluations has proven insufficient in the face of an exponential growth in data generated from transcriptomics, proteomics, metabolomics and deep phenotyping. A new systematic method is necessary to organize these datasets and build new definitions of what constitutes a disease that incorporates both biological and environmental factors to more precisely describe the ever-growing complexity of phenotypes and their underlying molecular determinants. Network medicine provides such a conceptual framework to bridge these vast quantities of data while providing an individualized understanding of disease. The modern application of network medicine principles is yielding new insights into the pathobiology of chronic kidney diseases and renovascular disorders by expanding the understanding of pathogenic mediators, novel biomarkers and new options for renal therapeutics. These efforts affirm network medicine as a robust paradigm for elucidating new advances in the diagnosis and treatment of kidney disorders.

Inferring cell cycle phases from a partially temporal network of protein interactions

The temporal organization of biological systems is key for understanding them, but current methods for identifying this organization are often ad hoc and require prior knowledge. We present Phasik, a method that automatically identifies this multiscale organization by combining time series data (protein or gene expression) and interaction data (protein-protein interaction network). Phasik builds a (partially) temporal network and uses clustering to infer temporal phases. We demonstrate the method's effectiveness by recovering well-known phases and sub-phases of the cell cycle of budding yeast and phase arrests of mutants. We also show its general applicability using temporal gene expression data from circadian rhythms in wild-type and mutant mouse models. We systematically test Phasik's robustness and investigate the effect of having only partial temporal information. As time-resolved, multiomics datasets become more common, this method will allow the study of temporal regulation in lesser-known biological contexts, such as development, metabolism, and disease.

A continuous-time stochastic Boolean model provides a quantitative description of the budding yeast cell cycle

The cell division cycle is regulated by a complex network of interacting genes and proteins. The control system has been modeled in many ways, from qualitative Boolean switching-networks to quantitative differential equations and highly detailed stochastic simulations. Here we develop a continuous-time stochastic model using seven Boolean variables to represent the activities of major regulators of the budding yeast cell cycle plus one continuous variable representing cell growth. The Boolean variables are updated asynchronously by logical rules based on known biochemistry of the cell-cycle control system using Gillespie's stochastic simulation algorithm. Time and cell size are updated continuously. By simulating a population of yeast cells, we calculate statistical properties of cell cycle progression that can be compared directly to experimental measurements. Perturbations of the normal sequence of events indicate that the cell cycle is 91% robust to random 'flips' of the Boolean variables, but 9% of the perturbations induce lethal mistakes in cell cycle progression. This simple, hybrid Boolean model gives a good account of the growth and division of budding yeast cells, suggesting that this modeling approach may be as accurate as detailed reaction-kinetic modeling with considerably less demands on estimating rate constants.

Leveraging network structure in nonlinear control

Over the last twenty years, dynamic modeling of biomolecular networks has exploded in popularity. Many of the classical tools for understanding dynamical systems are unwieldy in the highly nonlinear, poorly constrained, high-dimensional systems that often arise from these modeling efforts. Understanding complex biological systems is greatly facilitated by purpose-built methods that leverage common features of such models, such as local monotonicity, interaction graph sparsity, and sigmoidal kinetics. Here, we review methods for controlling the systems of ordinary differential equations used to model biomolecular networks. We focus on methods that make use of the structure of the network of interactions to help inform, which variables to target for control, and highlight the computational and experimental advantages of such approaches. We also discuss the importance of nonperturbative methods in biomedical and experimental molecular biology applications, where finely tuned interventions can be difficult to implement. It is well known that feedback loops, and positive feedback loops in particular, play a major determining role in the dynamics of biomolecular networks. In many of the methods we cover here, control over system trajectories is realized by overriding the behavior of key feedback loops.

Consensus clustering for Bayesian mixture models

BACKGROUND

Cluster analysis is an integral part of precision medicine and systems biology, used to define groups of patients or biomolecules. Consensus clustering is an ensemble approach that is widely used in these areas, which combines the output from multiple runs of a non-deterministic clustering algorithm. Here we consider the application of consensus clustering to a broad class of heuristic clustering algorithms that can be derived from Bayesian mixture models (and extensions thereof) by adopting an early stopping criterion when performing sampling-based inference for these models. While the resulting approach is non-Bayesian, it inherits the usual benefits of consensus clustering, particularly in terms of computational scalability and providing assessments of clustering stability/robustness.

RESULTS

In simulation studies, we show that our approach can successfully uncover the target clustering structure, while also exploring different plausible clusterings of the data. We show that, when a parallel computation environment is available, our approach offers significant reductions in runtime compared to performing sampling-based Bayesian inference for the underlying model, while retaining many of the practical benefits of the Bayesian approach, such as exploring different numbers of clusters. We propose a heuristic to decide upon ensemble size and the early stopping criterion, and then apply consensus clustering to a clustering algorithm derived from a Bayesian integrative clustering method. We use the resulting approach to perform an integrative analysis of three 'omics datasets for budding yeast and find clusters of co-expressed genes with shared regulatory proteins. We validate these clusters using data external to the analysis.

CONCLUSTIONS

Our approach can be used as a wrapper for essentially any existing sampling-based Bayesian clustering implementation, and enables meaningful clustering analyses to be performed using such implementations, even when computational Bayesian inference is not feasible, e.g. due to poor exploration of the target density (often as a result of increasing numbers of features) or a limited computational budget that does not along sufficient samples to drawn from a single chain. This enables researchers to straightforwardly extend the applicability of existing software to much larger datasets, including implementations of sophisticated models such as those that jointly model multiple datasets.

Explaining Redundancy in CDK-Mediated Control of the Cell Cycle: Unifying the Continuum and Quantitative Models

In eukaryotes, cyclin-dependent kinases (CDKs) are required for the onset of DNA replication and mitosis, and distinct CDK–cyclin complexes are activated sequentially throughout the cell cycle. It is widely thought that specific complexes are required to traverse a point of commitment to the cell cycle in G1, and to promote S-phase and mitosis, respectively. Thus, according to a popular model that has dominated the field for decades, the inherent specificity of distinct CDK–cyclin complexes for different substrates at each phase of the cell cycle generates the correct order and timing of events. However, the results from the knockouts of genes encoding cyclins and CDKs do not support this model. An alternative “quantitative” model, validated by much recent work, suggests that it is the overall level of CDK activity (with the opposing input of phosphatases) that determines the timing and order of S-phase and mitosis. We take this model further by suggesting that the subdivision of the cell cycle into discrete phases (G0, G1, S, G2, and M) is outdated and problematic. Instead, we revive the “continuum” model of the cell cycle and propose that a combination with the quantitative model better defines a conceptual framework for understanding cell cycle control.

A yeast cell cycle model integrating stress, signaling, and physiology

The cell division cycle in eukaryotic cells is a series of highly coordinated molecular interactions that ensure that cell growth, duplication of genetic material, and actual cell division are precisely orchestrated to give rise to two viable progeny cells. Moreover, the cell cycle machinery is responsible for incorporating information about external cues or internal processes that the cell must keep track of to ensure a coordinated, timely progression of all related processes. This is most pronounced in multicellular organisms, but also a cardinal feature in model organisms such as baker's yeast. The complex and integrative behavior is difficult to grasp and requires mathematical modeling to fully understand the quantitative interplay of the single components within the entire system. Here, we present a self-oscillating mathematical model of the yeast cell cycle that comprises all major cyclins and their main regulators. Furthermore, it accounts for the regulation of the cell cycle machinery by a series of external stimuli such as mating pheromones and changes in osmotic pressure or nutrient quality. We demonstrate how the external perturbations modify the dynamics of cell cycle components and how the cell cycle resumes after adaptation to or relief from stress.

Symmetry and simplicity spontaneously emerge from the algorithmic nature of evolution

SignificanceWhy does evolution favor symmetric structures when they only represent a minute subset of all possible forms? Just as monkeys randomly typing into a computer language will preferentially produce outputs that can be generated by shorter algorithms, so the coding theorem from algorithmic information theory predicts that random mutations, when decoded by the process of development, preferentially produce phenotypes with shorter algorithmic descriptions. Since symmetric structures need less information to encode, they are much more likely to appear as potential variation. Combined with an arrival-of-the-frequent mechanism, this algorithmic bias predicts a much higher prevalence of low-complexity (high-symmetry) phenotypes than follows from natural selection alone and also explains patterns observed in protein complexes, RNA secondary structures, and a gene regulatory network.

Modelling of glucose repression signalling in yeast Saccharomyces cerevisiae

Saccharomyces cerevisiae has a sophisticated signalling system that plays a crucial role in cellular adaptation to changing environments. The SNF1 pathway regulates energy homeostasis upon glucose derepression; hence, it plays an important role in various processes, such as metabolism, cell cycle and autophagy. To unravel its behaviour, SNF1 signalling has been extensively studied. However, the pathway components are strongly interconnected and inconstant; therefore, elucidating its dynamic behaviour based on experimental data only is challenging. To tackle this complexity, systems biology approaches have been successfully employed. This review summarizes the progress, advantages and disadvantages of the available mathematical modelling frameworks covering Boolean, dynamic kinetic, single-cell models, which have been used to study processes and phenomena ranging from crosstalks to sources of cell-to-cell variability in the context of SNF1 signalling. Based on the lessons from existing models, we further discuss how to develop a consensus dynamic mechanistic model of the entire SNF1 pathway that can provide novel insights into the dynamics of nutrient signalling.

Modeling Progression of Single Cell Populations Through the Cell Cycle as a Sequence of Switches

Cell cycle is a biological process underlying the existence and propagation of life in time and space. It has been an object for mathematical modeling for long, with several alternative mechanistic modeling principles suggested, describing in more or less details the known molecular mechanisms. Recently, cell cycle has been investigated at single cell level in snapshots of unsynchronized cell populations, exploiting the new methods for transcriptomic and proteomic molecular profiling. This raises a need for simplified semi-phenomenological cell cycle models, in order to formalize the processes underlying the cell cycle, at a higher abstracted level. Here we suggest a modeling framework, recapitulating the most important properties of the cell cycle as a limit trajectory of a dynamical process characterized by several internal states with switches between them. In the simplest form, this leads to a limit cycle trajectory, composed by linear segments in logarithmic coordinates describing some extensive (depending on system size) cell properties. We prove a theorem connecting the effective embedding dimensionality of the cell cycle trajectory with the number of its linear segments. We also develop a simplified kinetic model with piecewise-constant kinetic rates describing the dynamics of lumps of genes involved in S-phase and G2/M phases. We show how the developed cell cycle models can be applied to analyze the available single cell datasets and simulate certain properties of the observed cell cycle trajectories. Based on our model, we can predict with good accuracy the cell line doubling time from the length of cell cycle trajectory.

From the Belousov-Zhabotinsky reaction to biochemical clocks, traveling waves and cell cycle regulation

In the last 20 years, a growing army of systems biologists has employed quantitative experimental methods and theoretical tools of data analysis and mathematical modeling to unravel the molecular details of biological control systems with novel studies of biochemical clocks, cellular decision-making, and signaling networks in time and space. Few people know that one of the roots of this new paradigm in cell biology can be traced to a serendipitous discovery by an obscure Russian biochemist, Boris Belousov, who was studying the oxidation of citric acid. The story is told here from an historical perspective, tracing its meandering path through glycolytic oscillations, cAMP signaling, and frog egg development. The connections among these diverse themes are drawn out by simple mathematical models (nonlinear differential equations) that share common structures and properties.

Cyclin/Forkhead-mediated coordination of cyclin waves: an autonomous oscillator rationalizing the quantitative model of Cdk control for budding yeast

Networks of interacting molecules organize topology, amount, and timing of biological functions. Systems biology concepts required to pin down 'network motifs' or 'design principles' for time-dependent processes have been developed for the cell division cycle, through integration of predictive computer modeling with quantitative experimentation. A dynamic coordination of sequential waves of cyclin-dependent kinases (cyclin/Cdk) with the transcription factors network offers insights to investigate how incompatible processes are kept separate in time during the eukaryotic cell cycle. Here this coordination is discussed for the Forkhead transcription factors in light of missing gaps in the current knowledge of cell cycle control in budding yeast. An emergent design principle is proposed where cyclin waves are synchronized by a cyclin/Cdk-mediated feed-forward regulation through the Forkhead as a transcriptional timer. This design is rationalized by the bidirectional interaction between mitotic cyclins and the Forkhead transcriptional timer, resulting in an autonomous oscillator that may be instrumental for a well-timed progression throughout the cell cycle. The regulation centered around the cyclin/Cdk-Forkhead axis can be pivotal to timely coordinate cell cycle dynamics, thereby to actuate the quantitative model of Cdk control.

Crosstalk between Plk1, p53, cell cycle, and G2/M DNA damage checkpoint regulation in cancer: computational modeling and analysis

Different cancer cell lines can have varying responses to the same perturbations or stressful conditions. Cancer cells that have DNA damage checkpoint-related mutations are often more sensitive to gene perturbations including altered Plk1 and p53 activities than cancer cells without these mutations. The perturbations often induce a cell cycle arrest in the former cancer, whereas they only delay the cell cycle progression in the latter cancer. To study crosstalk between Plk1, p53, and G2/M DNA damage checkpoint leading to differential cell cycle regulations, we developed a computational model by extending our recently developed model of mitotic cell cycle and including these key interactions. We have used the model to analyze the cancer cell cycle progression under various gene perturbations including Plk1-depletion conditions. We also analyzed mutations and perturbations in approximately 1800 different cell lines available in the Cancer Dependency Map and grouped lines by genes that are represented in our model. Our model successfully explained phenotypes of various cancer cell lines under different gene perturbations. Several sensitivity analysis approaches were used to identify the range of key parameter values that lead to the cell cycle arrest in cancer cells. Our resulting model can be used to predict the effect of potential treatments targeting key mitotic and DNA damage checkpoint regulators on cell cycle progression of different types of cancer cells.

Efficient gradient-based parameter estimation for dynamic models using qualitative data

MOTIVATION

Unknown parameters of dynamical models are commonly estimated from experimental data. However, while various efficient optimization and uncertainty analysis methods have been proposed for quantitative data, methods for qualitative data are rare and suffer from bad scaling and convergence.

RESULTS

Here, we propose an efficient and reliable framework for estimating the parameters of ordinary differential equation models from qualitative data. In this framework, we derive a semi-analytical algorithm for gradient calculation of the optimal scaling method developed for qualitative data. This enables the use of efficient gradient-based optimization algorithms. We demonstrate that the use of gradient information improves performance of optimization and uncertainty quantification on several application examples. On average, we achieve a speedup of more than one order of magnitude compared to gradient-free optimization. In addition, in some examples, the gradient-based approach yields substantially improved objective function values and quality of the fits. Accordingly, the proposed framework substantially improves the parameterization of models from qualitative data.

AVAILABILITY AND IMPLEMENTATION

The proposed approach is implemented in the open-source Python Parameter EStimation TOolbox (pyPESTO). pyPESTO is available at https://github.com/ICB-DCM/pyPESTO. All application examples and code to reproduce this study are available at https://doi.org/10.5281/zenodo.4507613.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Cluster Gauss-Newton and CellNOpt Parameter Estimation in a Small Protein Signaling Network of Vorinostat and Bortezomib Pharmacodynamics

Ordinary differential equation (ODE)-based models of signal transduction pathways often contain parameters that are unidentifiable or unmeasurable by experimental data, and calibrating such models to data remains challenging. Here, two efficient parameter estimation methods, cluster Gauss-Newton (CGN) and CellNOpt (CNO), were applied to fit a signaling network model of U266 multiple myeloma cells to the activity dynamics of key proteins in response to vorinostat and/or bortezomib. A logic-based network model was constructed and transformed to 17 ODEs with 79 parameters estimated within broad ranges of biologically plausible values. The top 10% best-fit parameters by both methods had high uncertainties with CV > 50% for the majority of parameters. The root mean square and prediction errors were comparable without statistically significant differences between the two methods. Despite uncertain parameter estimation, protein dynamics after the sequential combination of bortezomib and vorinostat was predicted with reasonable accuracy and precision. Global sensitivity analyses of partial rank correlation coefficients and Sobol sensitivity demonstrated that apoptosis induction was most sensitive to parameters governing the activity of the proteasome-JNK-caspase-8 axis. Simulations revealed that the greatest magnitude of pharmacodynamic drug interactions between bortezomib and vorinostat occurred at caspase-9, AKT, and Bcl-2. Two sequential combinations were explored in silico, and the outcome matched qualitatively with an empirical evaluation of the pharmacodynamic interaction based on cell viability. Overall, the CGN and CNO algorithms performed similarly for this ODE-based network model calibration, and the calibrated model provided meaningful insights into cellular signaling mechanisms in response to pharmacological perturbations.

Boolean factor graph model for biological systems: the yeast cell-cycle network

Background

The desire to understand genomic functions and the behavior of complex gene regulatory networks has recently been a major research focus in systems biology. As a result, a plethora of computational and modeling tools have been proposed to identify and infer interactions among biological entities. Here, we consider the general question of the effect of perturbation on the global dynamical network behavior as well as error propagation in biological networks to incite research pertaining to intervention strategies.

Results

This paper introduces a computational framework that combines the formulation of Boolean networks and factor graphs to explore the global dynamical features of biological systems. A message-passing algorithm is proposed for this formalism to evolve network states as messages in the graph. In addition, the mathematical formulation allows us to describe the dynamics and behavior of error propagation in gene regulatory networks by conducting a density evolution (DE) analysis. The model is applied to assess the network state progression and the impact of gene deletion in the budding yeast cell cycle. Simulation results show that our model predictions match published experimental data. Also, our findings reveal that the sample yeast cell-cycle network is not only robust but also consistent with real high-throughput expression data. Finally, our DE analysis serves as a tool to find the optimal values of network parameters for resilience against perturbations, especially in the inference of genetic graphs.

Conclusion

Our computational framework provides a useful graphical model and analytical tools to study biological networks. It can be a powerful tool to predict the consequences of gene deletions before conducting wet bench experiments because it proves to be a quick route to predicting biologically relevant dynamic properties without tunable kinetic parameters.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04361-8.

Efficient gradient-based parameter estimation for dynamic models using qualitative data

Motivation

Unknown parameters of dynamical models are commonly estimated from experimental data. However, while various efficient optimization and uncertainty analysis methods have been proposed for quantitative data, methods for qualitative data are rare and suffer from bad scaling and convergence.

Results

Here, we propose an efficient and reliable framework for estimating the parameters of ordinary differential equation models from qualitative data. In this framework, we derive a semi-analytical algorithm for gradient calculation of the optimal scaling method developed for qualitative data. This enables the use of efficient gradient-based optimization algorithms. We demonstrate that the use of gradient information improves performance of optimization and uncertainty quantification on several application examples. On average, we achieve a speedup of more than one order of magnitude compared to gradient-free optimization. In addition, in some examples, the gradient-based approach yields substantially improved objective function values and quality of the fits. Accordingly, the proposed framework substantially improves the parameterization of models from qualitative data.

Availability and implementation

The proposed approach is implemented in the open-source Python Parameter EStimation TOolbox (pyPESTO). pyPESTO is available at https://github.com/ICB-DCM/pyPESTO. All application examples and code to reproduce this study are available at https://doi.org/10.5281/zenodo.4507613.

Supplementary information

Supplementary data are available at Bioinformatics online.

Dynamical Modeling of Optogenetic Circuits in Yeast for Metabolic Engineering Applications

Dynamic control of engineered microbes using light via optogenetics has been demonstrated as an effective strategy for improving the yield of biofuels, chemicals, and other products. An advantage of using light to manipulate microbial metabolism is the relative simplicity of interfacing biological and computer systems, thereby enabling in silico control of the microbe. Using this strategy for control and optimization of product yield requires an understanding of how the microbe responds in real-time to the light inputs. Toward this end, we present mechanistic models of a set of yeast optogenetic circuits. We show how these models can predict short- and long-time response to varying light inputs and how they are amenable to use with model predictive control (the industry standard among advanced control algorithms). These models reveal dynamics characterized by time-scale separation of different circuit components that affect the steady and transient levels of the protein under control of the circuit. Ultimately, this work will help enable real-time control and optimization tools for improving yield and consistency in the production of biofuels and chemicals using microbial fermentations.

Unifying the mechanism of mitotic exit control in a spatiotemporal logical model

The transition from mitosis into the first gap phase of the cell cycle in budding yeast is controlled by the Mitotic Exit Network (MEN). The network interprets spatiotemporal cues about the progression of mitosis and ensures that release of Cdc14 phosphatase occurs only after completion of key mitotic events. The MEN has been studied intensively; however, a unified understanding of how localisation and protein activity function together as a system is lacking. In this paper, we present a compartmental, logical model of the MEN that is capable of representing spatial aspects of regulation in parallel to control of enzymatic activity. We show that our model is capable of correctly predicting the phenotype of the majority of mutants we tested, including mutants that cause proteins to mislocalise. We use a continuous time implementation of the model to demonstrate that Cdc14 Early Anaphase Release (FEAR) ensures robust timing of anaphase, and we verify our findings in living cells. Furthermore, we show that our model can represent measured cell-cell variation in Spindle Position Checkpoint (SPoC) mutants. This work suggests a general approach to incorporate spatial effects into logical models. We anticipate that the model itself will be an important resource to experimental researchers, providing a rigorous platform to test hypotheses about regulation of mitotic exit.

Bifurcation and oscillatory dynamics of delayed CDK1-APC feedback loop

Extensive experimental evidence has been demonstrated that the dynamics of CDK1-APC feedback loop play crucial roles in regulating cell cycle processes, but the dynamical mechanisms underlying the regulation of this loop are still not completely understood. Here, the authors systematically investigated the stability and bifurcation criteria for a delayed CDK1-APC feedback loop. They showed that the maximum reaction rate of CDK1 inactivation by APC can drive sustained oscillations of CDK1 activity ([inline-formula removed]) and APC activity ([inline-formula removed]), and the amplitude of these oscillations is increasing with the increase of the reaction rate over a wide range; a certain range of the self-activation rate for CDK1 is also significant for generating these oscillations, for too high or too low rates the oscillations cannot be generated. Moreover, they derived the sufficient conditions to determine the stability and Hopf bifurcations, and found that the sum of time delays required for activating CDK1 and APC can induce [inline-formula removed] and [inline-formula removed] to be oscillatory, even when the [inline-formula removed] and [inline-formula removed] settle in a definite stable steady state. Furthermore, they presented an explicit algorithm for the properties of periodic oscillations. Finally, numerical simulations have been presented to justify the validity of theoretical analysis.

A systems biology approach to discovering pathway signaling dysregulation in metastasis

Total metastatic burden is the primary cause of death for many cancer patients. While the process of metastasis has been studied widely, much remains to be understood. Moreover, few agents have been developed that specifically target the major steps of the metastatic cascade. Many individual genes and pathways have been implicated in metastasis but a holistic view of how these interact and cooperate to regulate and execute the process remains somewhat rudimentary. It is unclear whether all of the signaling features that regulate and execute metastasis are yet fully understood. Novel features of a complex system such as metastasis can often be discovered by taking a systems-based approach. We introduce the concepts of systems modeling and define some of the central challenges facing the application of a multidisciplinary systems-based approach to understanding metastasis and finding actionable targets therein. These challenges include appreciating the unique properties of the high-dimensional omics data often used for modeling, limitations in knowledge of the system (metastasis), tumor heterogeneity and sampling bias, and some of the issues key to understanding critical features of molecular signaling in the context of metastasis. We also provide a brief introduction to integrative modeling that focuses on both the nodes and edges of molecular signaling networks. Finally, we offer some observations on future directions as they relate to developing a systems-based model of the metastatic cascade.

Conformational state switching and pathways of chromosome dynamics in cell cycle

The cell cycle is a process and function of a cell with different phases essential for cell growth, proliferation, and replication. It depends on the structure and dynamics of the underlying DNA molecule, which underpins the genome function. A microscopic structural-level understanding of how a genome or its functional module chromosome performs the cell cycle in terms of large-scale conformational transformation between different phases, such as the interphase and the mitotic phase, is still challenging. Here, we develop a non-equilibrium, excitation-relaxation energy landscape-switching model to quantify the underlying chromosome conformational transitions through (de-)condensation for a complete microscopic understanding of the cell cycle. We show that the chromosome conformational transition mechanism from the interphase to the mitotic phase follows a two-stage scenario, in good agreement with the experiments. In contrast, the mitotic exit pathways show the existence of an over-expanded chromosome that recapitulates the chromosome in the experimentally identified intermediate state at the telophase. We find the conformational pathways are heterogeneous and irreversible as a result of the non-equilibrium dynamics of the cell cycle from both structural and kinetic perspectives. We suggest that the irreversibility is mainly due to the distinct participation of the ATP-dependent structural maintenance of chromosomal protein complexes during the cell cycle. Our findings provide crucial insights into the microscopic molecular structural and dynamical physical mechanism for the cell cycle beyond the previous more macroscopic descriptions. Our non-equilibrium landscape framework is general and applicable to study diverse non-equilibrium physical and biological processes such as active matter, differentiation/development, and cancer.

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.

Random Parametric Perturbations of Gene Regulatory Circuit Uncover State Transitions in Cell Cycle

Many biological processes involve precise cellular state transitions controlled by complex gene regulation. Here, we use budding yeast cell cycle as a model system and explore how a gene regulatory circuit encodes essential information of state transitions. We present a generalized random circuit perturbation method for circuits containing heterogeneous regulation types and its usage to analyze both steady and oscillatory states from an ensemble of circuit models with random kinetic parameters. The stable steady states form robust clusters with a circular structure that are associated with cell cycle phases. This circular structure in the clusters is consistent with single-cell RNA sequencing data. The oscillatory states specify the irreversible state transitions along cell cycle progression. Furthermore, we identify possible mechanisms to understand the irreversible state transitions from the steady states. We expect this approach to be robust and generally applicable to unbiasedly predict dynamical transitions of a gene regulatory circuit.

The palette of techniques for cell cycle analysis

The cell division cycle is the generational period of cellular growth and propagation. Cell cycle progression needs to be highly regulated to preserve genomic fidelity while increasing cell number. In multicellular organisms, the cell cycle must also coordinate with cell fate specification during development and tissue homeostasis. Altered cell cycle dynamics play a central role also in a number of pathophysiological processes. Thus, extensive effort has been made to define the biochemical machineries that execute the cell cycle and their regulation, as well as implementing more sensitive and accurate cell cycle measurements. Here, we review the available techniques for cell cycle analysis, revisiting the assumptions behind conventional population-based measurements and discussing new tools to better address cell cycle heterogeneity in the single-cell era. We weigh the strengths, weaknesses, and trade-offs of methods designed to measure temporal aspects of the cell cycle. Finally, we discuss emerging techniques for capturing cell cycle speed at single-cell resolution in live animals.

Critical slowing down and attractive manifold: A mechanism for dynamic robustness in the yeast cell-cycle process

Biological processes that execute complex multiple functions, such as the cell cycle, must ensure the order of sequential events and maintain dynamic robustness against various fluctuations. Here, we examine the mechanisms and fundamental structure that achieve these properties in the cell cycle of the budding yeast Saccharomyces cerevisiae. We show that this process behaves like an excitable system containing three well-decoupled saddle-node bifurcations to execute DNA replication and mitosis events. The yeast cell-cycle regulatory network can be divided into three modules-the G1/S phase, early M phase, and late M phase-wherein both positive feedback loops in each module and interactions among modules play important roles. Specifically, when the cell-cycle process operates near the critical points of the saddle-node bifurcations, a critical slowing down effect takes place. Such interregnum then allows for an attractive manifold and sufficient duration for cell-cycle events, within which to assess the completion of DNA replication and mitosis, e.g., spindle assembly. Moreover, such arrangement ensures that any fluctuation in an early module or event will not transmit to a later module or event. Thus, our results suggest a possible dynamical mechanism of the cell-cycle process to ensure event order and dynamic robustness and give insight into the evolution of eukaryotic cell-cycle processes.

Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems

We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.

Genetic interactions derived from high-throughput phenotyping of 6589 yeast cell cycle mutants

Over the last 30 years, computational biologists have developed increasingly realistic mathematical models of the regulatory networks controlling the division of eukaryotic cells. These models capture data resulting from two complementary experimental approaches: low-throughput experiments aimed at extensively characterizing the functions of small numbers of genes, and large-scale genetic interaction screens that provide a systems-level perspective on the cell division process. The former is insufficient to capture the interconnectivity of the genetic control network, while the latter is fraught with irreproducibility issues. Here, we describe a hybrid approach in which the 630 genetic interactions between 36 cell-cycle genes are quantitatively estimated by high-throughput phenotyping with an unprecedented number of biological replicates. Using this approach, we identify a subset of high-confidence genetic interactions, which we use to refine a previously published mathematical model of the cell cycle. We also present a quantitative dataset of the growth rate of these mutants under six different media conditions in order to inform future cell cycle models.

Bayesian inference using qualitative observations of underlying continuous variables

MOTIVATION

Recent work has demonstrated the feasibility of using non-numerical, qualitative data to parameterize mathematical models. However, uncertainty quantification (UQ) of such parameterized models has remained challenging because of a lack of a statistical interpretation of the objective functions used in optimization.

RESULTS

We formulated likelihood functions suitable for performing Bayesian UQ using qualitative observations of underlying continuous variables or a combination of qualitative and quantitative data. To demonstrate the resulting UQ capabilities, we analyzed a published model for immunoglobulin E (IgE) receptor signaling using synthetic qualitative and quantitative datasets. Remarkably, estimates of parameter values derived from the qualitative data were nearly as consistent with the assumed ground-truth parameter values as estimates derived from the lower throughput quantitative data. These results provide further motivation for leveraging qualitative data in biological modeling.

AVAILABILITY AND IMPLEMENTATION

The likelihood functions presented here are implemented in a new release of PyBioNetFit, an open-source application for analyzing Systems Biology Markup Language- and BioNetGen Language-formatted models, available online at www.github.com/lanl/PyBNF.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Differential Scaling of Gene Expression with Cell Size May Explain Size Control in Budding Yeast

Yeast cells must grow to a critical size before committing to division. It is unknown how size is measured. We find that as cells grow, mRNAs for some cell-cycle activators scale faster than size, increasing in concentration, while mRNAs for some inhibitors scale slower than size, decreasing in concentration. Size-scaled gene expression could cause an increasing ratio of activators to inhibitors with size, triggering cell-cycle entry. Consistent with this, expression of the CLN2 activator from the promoter of the WHI5 inhibitor, or vice versa, interfered with cell size homeostasis, yielding a broader distribution of cell sizes. We suggest that size homeostasis comes from differential scaling of gene expression with size. Differential regulation of gene expression as a function of cell size could affect many cellular processes.

Clb3-centered regulations are recurrent across distinct parameter regions in minimal autonomous cell cycle oscillator designs

Some biological networks exhibit oscillations in their components to convert stimuli to time-dependent responses. The eukaryotic cell cycle is such a case, being governed by waves of cyclin-dependent kinase (cyclin/Cdk) activities that rise and fall with specific timing and guarantee its timely occurrence. Disruption of cyclin/Cdk oscillations could result in dysfunction through reduced cell division. Therefore, it is of interest to capture properties of network designs that exhibit robust oscillations. Here we show that a minimal yeast cell cycle network is able to oscillate autonomously, and that cyclin/Cdk-mediated positive feedback loops (PFLs) and Clb3-centered regulations sustain cyclin/Cdk oscillations, in known and hypothetical network designs. We propose that Clb3-mediated coordination of cyclin/Cdk waves reconciles checkpoint and oscillatory cell cycle models. Considering the evolutionary conservation of the cyclin/Cdk network across eukaryotes, we hypothesize that functional ("healthy") phenotypes require the capacity to oscillate autonomously whereas dysfunctional (potentially "diseased") phenotypes may lack this capacity.

A hybrid stochastic model of the budding yeast cell cycle

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

Parameter Estimation and Uncertainty Quantification for Systems Biology Models

Mathematical models can provide quantitative insights into immunoreceptor signaling, and other biological processes, but require parameterization and uncertainty quantification before reliable predictions become possible. We review currently available methods and software tools to address these problems. We consider gradient-based and gradient-free methods for point estimation of parameter values, and methods of profile likelihood, bootstrapping, and Bayesian inference for uncertainty quantification. We consider recent and potential future applications of these methods to systems-level modeling of immune-related phenomena.

Mathematical Model Explaining the Role of CDC6 in the Diauxic Growth of CDK1 Activity during the M-Phase of the Cell Cycle

In this paper we propose a role for the CDC6 protein in the entry of cells into mitosis. This has not been considered in the literature so far. Recent experiments suggest that CDC6, upon entry into mitosis, inhibits the appearance of active CDK1 and cyclin B complexes. This paper proposes a mathematical model which incorporates the dynamics of kinase CDK1, its regulatory protein cyclin B, the regulatory phosphatase CDC25 and the inhibitor CDC6 known to be involved in the regulation of active CDK1 and cyclin B complexes. The experimental data lead us to formulate a new hypothesis that CDC6 slows down the activation of inactive complexes of CDK1 and cyclin B upon mitotic entry. Our mathematical model, based on mass action kinetics, provides a possible explanation for the experimental data. We claim that the dynamics of active complexes CDK1 and cyclin B have a similar nature to diauxic dynamics introduced by Monod in 1949. In mathematical terms we state it as the existence of more than one inflection point of the curve defining the dynamics of the complexes.

Advanced Modeling of Cellular Proliferation: Toward a Multi-scale Framework Coupling Cell Cycle to Metabolism by Integrating Logical and Constraint-Based Models

Biological functions require a coherent cross talk among multiple layers of regulation within the cell. Computational efforts that aim to understand how these layers are integrated across spatial, temporal, and functional scales represent a challenge in Systems Biology. We have developed a computational, multi-scale framework that couples cell cycle and metabolism networks in the budding yeast cell. Here we describe the methodology at the basis of this framework, which integrates on off-the-shelf logical (Boolean) models of a minimal yeast cell cycle with a constraint-based model of metabolism (i.e., the Yeast 7 metabolic network reconstruction). Models are implemented in Python code using the BooleanNet and COBRApy packages, respectively, and are connected through the Boolean logic. The methodology allows for incorporation of interaction data, and validation through -omics data. Furthermore, evolutionary strategies may be incorporated to explore regulatory structures underlying coherent cross talks among regulatory layers.

PyBioNetFit and the Biological Property Specification Language

In systems biology modeling, important steps include model parameterization, uncertainty quantification, and evaluation of agreement with experimental observations. To help modelers perform these steps, we developed the software PyBioNetFit, which in addition supports checking models against known system properties and solving design problems. PyBioNetFit introduces Biological Property Specification Language (BPSL) for the formal declaration of system properties. BPSL allows qualitative data to be used alone or in combination with quantitative data. PyBioNetFit performs parameterization with parallelized metaheuristic optimization algorithms that work directly with existing model definition standards: BioNetGen Language (BNGL) and Systems Biology Markup Language (SBML). We demonstrate PyBioNetFit's capabilities by solving various example problems, including the challenging problem of parameterizing a 153-parameter model of cell cycle control in yeast based on both quantitative and qualitative data. We demonstrate the model checking and design applications of PyBioNetFit and BPSL by analyzing a model of targeted drug interventions in autophagy signaling.

A stochastic model of size control in the budding yeast cell cycle

BACKGROUND

Cell size is a key characteristic that significantly affects many aspects of cellular physiology. There are specific control mechanisms during cell cycle that maintain the cell size within a range from generation to generation. Such control mechanisms introduce substantial variabilities to important properties of the cell cycle such as growth and division. To quantitatively study the effect of such variability in progression through cell cycle, detailed stochastic models are required.

RESULTS

In this paper, a new hybrid stochastic model is proposed to study the effect of molecular noise and size control mechanism on the variabilities in cell cycle of the budding yeast Saccharomyces cerevisiae. The proposed model provides an accurate, yet computationally efficient approach for simulation of an intricate system by integrating the deterministic and stochastic simulation schemes. The developed hybrid stochastic model can successfully capture several key features of the cell cycle observed in experimental data. In particular, the proposed model: 1) confirms that the majority of noise in size control stems from low copy numbers of transcripts in the G1 phase, 2) identifies the size and time regulation modules in the size control mechanism, and 3) conforms with phenotypes of early G1 mutants in exquisite detail.

CONCLUSIONS

Hybrid stochastic modeling approach can be used to provide quantitative descriptions for stochastic properties of the cell cycle within a computationally efficient framework.

Reverse engineering genetic networks using nonlinear saturation kinetics

A gene regulatory network (GRN) represents a set of genes along with their regulatory interactions. Cellular behavior is driven by genetic level interactions. Dynamics of such systems show nonlinear saturation kinetics which can be best modeled by Michaelis-Menten (MM) and Hill equations. Although MM equation is being widely used for modeling biochemical processes, it has been applied rarely for reverse engineering GRNs. In this paper, we develop a complete framework for a novel model for GRN inference using MM kinetics. A set of coupled equations is first proposed for modeling GRNs. In the coupled model, Michaelis-Menten constant associated with regulation by a gene is made invariant irrespective of the gene being regulated. The parameter estimation of the proposed model is carried out using an evolutionary optimization method, namely, trigonometric differential evolution (TDE). Subsequently, the model is further improved and the regulations of different genes by a given gene are made distinct by allowing varying values of Michaelis-Menten constants for each regulation. Apart from making the model more relevant biologically, the improvement results in a decoupled GRN model with fast estimation of model parameters. Further, to enhance exploitation of the search, we propose a local search algorithm based on hill climbing heuristics. A novel mutation operation is also proposed to avoid population stagnation and premature convergence. Real life benchmark data sets generated in vivo are used for validating the proposed model. Further, we also analyze realistic in silico datasets generated using GeneNetweaver. The comparison of the performance of proposed model with other existing methods shows the potential of the proposed model.

Dissecting the mechanisms of cell division

Cell division is a highly regulated and carefully orchestrated process. Understanding the mechanisms that promote proper cell division is an important step toward unraveling important questions in cell biology and human health. Early studies seeking to dissect the mechanisms of cell division used classical genetics approaches to identify genes involved in mitosis and deployed biochemical approaches to isolate and identify proteins critical for cell division. These studies underscored that post-translational modifications and cyclin-kinase complexes play roles at the heart of the cell division program. Modern approaches for examining the mechanisms of cell division, including the use of high-throughput methods to study the effects of RNAi, cDNA, and chemical libraries, have evolved to encompass a larger biological and chemical space. Here, we outline some of the classical studies that established a foundation for the field and provide an overview of recent approaches that have advanced the study of cell division.

Cell size and morphological properties of yeast Saccharomyces cerevisiae in relation to growth temperature

Cell volume is an important parameter for modelling cellular processes. Temperature-induced variability of cellular size, volume, intracellular granularity, a fraction of budding cells of yeast Saccharomyces cerevisiae CEN.PK 113-7D (in anaerobic glucose unlimited batch cultures) were measured by flow cytometry and matched with the performance of the biomass growth (maximal specific growth rate (μmax), specific rate of glucose consumption, the rate of maintenance, biomass yield on glucose). The critical diameter of single cells was 7.94 μm and it is invariant at growth temperatures above 18.5°C. Below 18.5°C, it exponentially increases up to 10.2 μm. The size of the bud linearly depends on μmax, and it is between 50% at 5°C and 90% at 31°C of the averaged single cell. The intracellular granularity (side scatter channel (SSC)-index) negatively depends on μmax. There are two temperature regions (5-31°C vs. 33-40°C) where the relationship between SSC-index and various cellular parameters differ significantly. In supraoptimal temperature range (33-40°C), cells are less granulated perhaps due to a higher rate of the maintenance. There is temperature dependent passage through the checkpoints in the cell cycle which influences the μmax. The results point to the existence of two different morphological states of yeasts in these different temperature regions.

Efficiently Encoding Complex Biochemical Models with the Multistate Model Builder (MSMB)

Biologists seek to create increasingly complex molecular regulatory network models. Writing such a model is a creative effort that requires flexible analysis tools and better modeling languages than offered by many of today's biochemical model editors. Our Multistate Model Builder (MSMB) supports multistate models created using different modeling styles that suit the modeler rather than the software. MSMB defines a simple but powerful syntax to describe multistate species. Our syntax reduces the number of reactions needed to encode the model, thereby reducing the cognitive load involved with model creation. MSMB gives extensive feedback during all stages of model creation. Users can activate error notifications, and use these notifications as a guide toward a consistent, syntactically correct model. Any consistent model can be exported to SBML or COPASI formats. We show the effectiveness of MSMB's multistate syntax through realistic models of cell cycle regulation and mRNA transcription. MSMB is an open-source project implemented in Java and it uses the COPASI API. Complete information and the installation package can be found at http://copasi.org/Projects/ .

A transcriptome-wide analysis deciphers distinct roles of G1 cyclins in temporal organization of the yeast cell cycle

Oscillating gene expression is crucial for correct timing and progression through cell cycle. In Saccharomyces cerevisiae, G1 cyclins Cln1-3 are essential drivers of the cell cycle and have an important role for temporal fine-tuning. We measured time-resolved transcriptome-wide gene expression for wild type and cyclin single and double knockouts over cell cycle with and without osmotic stress. Clustering of expression profiles, peak time detection of oscillating genes, integration with transcription factor network dynamics, and assignment to cell cycle phases allowed us to quantify the effect of genetic or stress perturbations on the duration of cell cycle phases. Cln1 and Cln2 showed functional differences, especially affecting later phases. Deletion of Cln3 led to a delay of START followed by normal progression through later phases. Our data and network analysis suggest mutual effects of cyclins with the transcriptional regulators SBF and MBF.

The cell-cycle transcriptional network generates and transmits a pulse of transcription once each cell cycle

Multiple studies have suggested the critical roles of cyclin-dependent kinases (CDKs) as well as a transcription factor (TF) network in generating the robust cell-cycle transcriptional program. However, the precise mechanisms by which these components function together in the gene regulatory network remain unclear. Here we show that the TF network can generate and transmit a "pulse" of transcription independently of CDK oscillations. The premature firing of the transcriptional pulse is prevented by early G1 inhibitors, including transcriptional corepressors and the E3 ubiquitin ligase complex APC^Cdh1. We demonstrate that G1 cyclin-CDKs facilitate the activation and accumulation of TF proteins in S/G2/M phases through inhibiting G1 transcriptional corepressors (Whi5 and Stb1) and APC^Cdh1, thereby promoting the initiation and propagation of the pulse by the TF network. These findings suggest a unique oscillatory mechanism in which global phase-specific transcription emerges from a pulse-generating network that fires once-and-only-once at the start of the cycle.

Control of Eukaryotic DNA Replication Initiation-Mechanisms to Ensure Smooth Transitions

DNA replication differs from most other processes in biology in that any error will irreversibly change the nature of the cellular progeny. DNA replication initiation, therefore, is exquisitely controlled. Deregulation of this control can result in over-replication characterized by repeated initiation events at the same replication origin. Over-replication induces DNA damage and causes genomic instability. The principal mechanism counteracting over-replication in eukaryotes is a division of replication initiation into two steps—licensing and firing—which are temporally separated and occur at distinct cell cycle phases. Here, we review this temporal replication control with a specific focus on mechanisms ensuring the faultless transition between licensing and firing phases.

Flavin-based metabolic cycles are integral features of growth and division in single yeast cells

The yeast metabolic cycle (YMC) is a fascinating example of biological organization, in which cells constrain the function of specific genetic, protein and metabolic networks to precise temporal windows as they grow and divide. However, understanding the intracellular origins of the YMC remains a challenging goal, as measuring the oxygen oscillations traditionally associated with it requires the use of synchronized cultures growing in nutrient-limited chemostat environments. To address these limitations, we used custom-built microfluidic devices and time-lapse fluorescence microscopy to search for metabolic cycling in the form of endogenous flavin fluorescence in unsynchronized single yeast cells. We uncovered robust and pervasive metabolic cycles that were synchronized with the cell division cycle (CDC) and oscillated across four different nutrient conditions. We then studied the response of these metabolic cycles to chemical and genetic perturbations, showing that their phase synchronization with the CDC can be altered through treatment with rapamycin, and that metabolic cycles continue even in respiratory deficient strains. These results provide a foundation for future studies of the physiological importance of metabolic cycles in processes such as CDC control, metabolic regulation and cell aging.

Modeling the dynamic behavior of biochemical regulatory networks

Strategies for modeling the complex dynamical behavior of gene/protein regulatory networks have evolved over the last 50 years as both the knowledge of these molecular control systems and the power of computing resources have increased. Here, we review a number of common modeling approaches, including Boolean (logical) models, systems of piecewise-linear or fully non-linear ordinary differential equations, and stochastic models (including hybrid deterministic/stochastic approaches). We discuss the pro's and con's of each approach, to help novice modelers choose a modeling strategy suitable to their problem, based on the type and bounty of available experimental information. We illustrate different modeling strategies in terms of some abstract network motifs, and in the specific context of cell cycle regulation.