Finishing the cell cycle

Spindle Dynamics Model Explains Chromosome Loss Rates in Yeast Polyploid Cells

Faithful chromosome segregation, driven by the mitotic spindle, is essential for organismal survival. Neopolyploid cells from diverse species exhibit a significant increase in mitotic errors relative to their diploid progenitors, resulting in chromosome nondisjunction. In the model system Saccharomyces cerevisiae, the rate of chromosome loss in haploid and diploid cells is measured to be one thousand times lower than the rate of loss in isogenic tetraploid cells. Currently it is unknown what constrains the number of chromosomes that can be segregated with high fidelity in an organism. Here we developed a simple mathematical model to study how different rates of chromosome loss in cells with different ploidy can arise from changes in (1) spindle dynamics and (2) a maximum duration of mitotic arrest, after which cells enter anaphase. We apply this model to S. cerevisiae to show that this model can explain the observed rates of chromosome loss in S. cerevisiae cells of different ploidy. Our model describes how small increases in spindle assembly time can result in dramatic differences in the rate of chromosomes loss between cells of increasing ploidy and predicts the maximum duration of mitotic arrest.

A stochastic model for the normal tissue complication probability (NTCP) and applicationss

The normal tissue complication probability (NTCP) is a measure for the estimated side effects of a given radiation treatment schedule. Here we use a stochastic logistic birth-death process to define an organ-specific and patient-specific NTCP. We emphasize an asymptotic simplification which relates the NTCP to the solution of a logistic differential equation. This framework is based on simple modelling assumptions and it prepares a framework for the use of the NTCP model in clinical practice. As example, we consider side effects of prostate cancer brachytherapy such as increase in urinal frequency, urinal retention and acute rectal dysfunction.

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

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

ELECTRONIC SUPPLEMENTARY MATERIAL

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

Multi-scale modeling of tissues using CompuCell3D

The study of how cells interact to produce tissue development, homeostasis, or diseases was, until recently, almost purely experimental. Now, multi-cell computer simulation methods, ranging from relatively simple cellular automata to complex immersed-boundary and finite-element mechanistic models, allow in silico study of multi-cell phenomena at the tissue scale based on biologically observed cell behaviors and interactions such as movement, adhesion, growth, death, mitosis, secretion of chemicals, chemotaxis, etc. This tutorial introduces the lattice-based Glazier-Graner-Hogeweg (GGH) Monte Carlo multi-cell modeling and the open-source GGH-based CompuCell3D simulation environment that allows rapid and intuitive modeling and simulation of cellular and multi-cellular behaviors in the context of tissue formation and subsequent dynamics. We also present a walkthrough of four biological models and their associated simulations that demonstrate the capabilities of the GGH and CompuCell3D.

NONLINEAR DYNAMICS OF CELL CYCLES WITH STOCHASTIC MATHEMATICAL MODELS

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

Computational yeast systems biology: a case study for the MAP kinase cascade

Cellular networks and processes can be mathematically described and analyzed in various ways. Here, the case example of a MAP kinase (MAPK) cascade is used to detail steps in the formulation of a system of ordinary differential equations governing the temporal behavior of a signal transduction pathway after stimulation. Different analysis methods for the model are explained and demonstrated, such as stoichiometric analysis, sensitivity analysis, or studying the effect of deletions and protein overexpression. Finally, a perspective on standards concerning modeling in systems biology is given.

From cell population models to tumor control probability: including cell cycle effects

BACKGROUND

Classical expressions for the tumor control probability (TCP) are based on models for the survival fraction of cancer cells after radiation treatment. We focus on the derivation of expressions for TCP from dynamic cell population models. In particular, we derive a TCP formula for a generalized cell population model that includes the cell cycle by considering a compartment of actively proliferating cells and a compartment of quiescent cells, with the quiescent cells being less sensitive to radiation than the actively proliferating cells.

METHODS

We generalize previously derived TCP formulas of Zaider and Minerbo and of Dawson and Hillen to derive a TCP formula from our cell population model. We then use six prostate cancer treatment protocols as a case study to show how our TCP formula works and how the cell cycle affects the tumor treatment.

RESULTS

The TCP formulas of Zaider-Minerbo and of Dawson-Hillen are special cases of the TCP formula presented here. The former one represents the case with no quiescent cells while the latter one assumes that all newly born cells enter a quiescent cell phase before becoming active. From our case study, we observe that inclusion of the cell cycle lowers the TCP.

CONCLUSION

The cell cycle can be understood as the sequestration of cells in the quiescent compartment, where they are less sensitive to radiation. We suggest that our model can be used in combination with synchronization methods to optimize treatment timing.

Multicell simulations of development and disease using the CompuCell3D simulation environment

Mathematical modeling and computer simulation have become crucial to biological fields from genomics to ecology. However, multicell, tissue-level simulations of development and disease have lagged behind other areas because they are mathematically more complex and lack easy-to-use software tools that allow building and running in silico experiments without requiring in-depth knowledge of programming. This tutorial introduces Glazier-Graner-Hogeweg (GGH) multicell simulations and CompuCell3D, a simulation framework that allows users to build, test, and run GGH simulations.

Preventing aneuploidy: the contribution of mitotic checkpoint proteins

Aneuploidy, an abnormal number of chromosomes, is a trait shared by most solid tumors. Chromosomal instability (CIN) manifested as aneuploidy might promote tumorigenesis and cause increased resistance to anti-cancer therapies. The mitotic checkpoint or spindle assembly checkpoint is a major signaling pathway involved in the prevention of CIN. We review current knowledge on the contribution of misregulation of mitotic checkpoint proteins to tumor formation and will address to what extent this contribution is due to chromosome segregation errors directly. We propose that both checkpoint and non-checkpoint functions of these proteins contribute to the wide array of oncogenic phenotypes seen upon their misregulation.

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 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.

Mitotic checkpoint slippage in humans occurs via cyclin B destruction in the presence of an active checkpoint

In the presence of unattached/weakly attached kinetochores, the spindle assembly checkpoint (SAC) delays exit from mitosis by preventing the anaphase-promoting complex (APC)-mediated proteolysis of cyclin B, a regulatory subunit of cyclin-dependent kinase 1 (Cdk1). Like all checkpoints, the SAC does not arrest cells permanently, and escape from mitosis in the presence of an unsatisfied SAC requires that cyclin B/Cdk1 activity be inhibited. In yeast , and likely Drosophila, this occurs through an "adaptation" process involving an inhibitory phosphorylation on Cdk1 and/or activation of a cyclin-dependent kinase inhibitor (Cdki). The mechanism that allows vertebrate cells to escape mitosis when the SAC cannot be satisfied is unknown. To explore this issue, we conducted fluorescence microscopy studies on rat kangaroo (PtK) and human (RPE1) cells dividing in the presence of nocodazole. We find that in the absence of microtubules (MTs), escape from mitosis occurs in the presence of an active SAC and requires cyclin B destruction. We also find that cyclin B is progressively destroyed during the block by a proteasome-dependent mechanism. Thus, vertebrate cells do not adapt to the SAC. Rather, our data suggest that in normal cells, the SAC cannot prevent a slow but continuous degradation of cyclin B that ultimately drives the cell out of mitosis.

Stuck in division or passing through: what happens when cells cannot satisfy the spindle assembly checkpoint

Cells that cannot satisfy the spindle assembly checkpoint (SAC) are delayed in mitosis (D-mitosis), a fact that has useful clinical ramifications. However, this delay is seldom permanent, and in the presence of an active SAC most cells ultimately escape mitosis and enter the next G1 as tetraploid cells. This review defines and discusses the various factors that determine how long a cell remains in mitosis when it cannot satisfy the SAC and also discusses the cell's subsequent fate.

A model for restriction point control of the mammalian cell cycle

Inhibition of protein synthesis by cycloheximide blocks subsequent division of a mammalian cell, but only if the cell is exposed to the drug before the "restriction point" (i.e. within the first several hours after birth). If exposed to cycloheximide after the restriction point, a cell proceeds with DNA synthesis, mitosis and cell division and halts in the next cell cycle. If cycloheximide is later removed from the culture medium, treated cells will return to the division cycle, showing a complex pattern of division times post-treatment, as first measured by Zetterberg and colleagues. We simulate these physiological responses of mammalian cells to transient inhibition of growth, using a set of nonlinear differential equations based on a realistic model of the molecular events underlying progression through the cell cycle. The model relies on our earlier work on the regulation of cyclin-dependent protein kinases during the cell division cycle of yeast. The yeast model is supplemented with equations describing the effects of retinoblastoma protein on cell growth and the synthesis of cyclins A and E, and with a primitive representation of the signaling pathway that controls synthesis of cyclin D.

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

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

The early impact of genetics on our understanding of cell cycle regulation in Aspergillus nidulans

The application of genetic analysis was crucial to the rapid progress that has been made in cell cycle research. Ron Morris, one of the first to apply genetics to cell cycle research, developed Aspergillus nidulans into an important model system for the analysis of many aspects of cell biology. Within the area of cell cycle research, Ron's laboratory is noted for development of novel cell biological and molecular genetic approaches as well as seminal insights regarding the regulation of mitosis, checkpoint regulation of the cell cycle, and the role of microtubule-based motors in chromosome segregation. In this special edition of FGB dedicated to Ron Morris, and in light of the recent progress in fungal genomics, we review the outstanding contributions his work made to our understanding of mitotic regulation. Indeed, his efforts have provided many mutants and experimental tools along with the conceptual framework for current and future studies of mitosis in A. nidulans.

Regulation of the eukaryotic cell cycle: molecular antagonism, hysteresis, and irreversible transitions

In recent years, molecular biologists have uncovered a wealth of information about the proteins controlling cell growth and division in eukaryotes. The regulatory system is so complex that it defies understanding by verbal arguments alone. Quantitative tools are necessary to probe reliably into the details of cell cycle control. To this end, we convert hypothetical molecular mechanisms into sets of nonlinear ordinary differential equations and use standard analytical and numerical methods to study their solutions. First, we present a simple model of the antagonistic interactions between cyclin-dependent kinases and the anaphase promoting complex, which shows how progress through the cell cycle can be thought of as irreversible transitions (Start and Finish) between two stable states (G1 and S-G2-M) of the regulatory system. Then we add new pieces to the "puzzle" until we obtain reasonable models of the control systems in yeast cells, frog eggs, and cultured mammalian cells.

Biological timing and the clock metaphor: oscillatory and hourglass mechanisms

Living organisms have developed a multitude of timing mechanisms--"biological clocks." Their mechanisms are based on either oscillations (oscillatory clocks) or unidirectional processes (hourglass clocks). Oscillatory clocks comprise circatidal, circalunidian, circadian, circalunar, and circannual oscillations--which keep time with environmental periodicities--as well as ultradian oscillations, ovarian cycles, and oscillations in development and in the brain, which keep time with biological timescales. These clocks mainly determine time points at specific phases of their oscillations. Hourglass clocks are predominantly found in development and aging and also in the brain. They determine time intervals (duration). More complex timing systems combine oscillatory and hourglass mechanisms, such as the case for cell cycle, sleep initiation, or brain clocks, whereas others combine external and internal periodicities (photoperiodism, seasonal reproduction). A definition of a biological clock may be derived from its control of functions external to its own processes and its use in determining temporal order (sequences of events) or durations. Biological and chemical oscillators are characterized by positive and negative feedback (or feedforward) mechanisms. During evolution, living organisms made use of the many existing oscillations for signal transmission, movement, and pump mechanisms, as well as for clocks. Some clocks, such as the circadian clock, that time with environmental periodicities are usually compensated (stabilized) against temperature, whereas other clocks, such as the cell cycle, that keep time with an organismic timescale are not compensated. This difference may be related to the predominance of negative feedback in the first class of clocks and a predominance of positive feedback (autocatalytic amplification) in the second class. The present knowledge of a compensated clock (the circadian oscillator) and an uncompensated clock (the cell cycle), as well as relevant models, are briefly re viewed. Hourglass clocks are based on linear or exponential unidirectional processes that trigger events mainly in the course of development and aging. An important hourglass mechanism within the aging process is the limitation of cell division capacity by the length of telomeres. The mechanism of this clock is briefly reviewed. In all clock mechanisms, thresholds at which "dependent variables" are triggered play an important role.

In silico predictions of Escherichia coli metabolic capabilities are consistent with experimental data

A significant goal in the post-genome era is to relate the annotated genome sequence to the physiological functions of a cell. Working from the annotated genome sequence, as well as biochemical and physiological information, it is possible to reconstruct complete metabolic networks. Furthermore, computational methods have been developed to interpret and predict the optimal performance of a metabolic network under a range of growth conditions. We have tested the hypothesis that Escherichia coli uses its metabolism to grow at a maximal rate using the E. coli MG1655 metabolic reconstruction. Based on this hypothesis, we formulated experiments that describe the quantitative relationship between a primary carbon source (acetate or succinate) uptake rate, oxygen uptake rate, and maximal cellular growth rate. We found that the experimental data were consistent with the stated hypothesis, namely that the E. coli metabolic network is optimized to maximize growth under the experimental conditions considered. This study thus demonstrates how the combination of in silico and experimental biology can be used to obtain a quantitative genotype-phenotype relationship for metabolism in bacterial cells.

Synchronization by irregular inactivation

Many natural and technological systems have on/off switches. For instance, mitosis can be halted by biochemical switches which act through the phosphorylation state of a complex called mitosis promoting factor. If switching between the on and off states is periodic, chaos is observed over a substantial portion of the on/off time parameter plane. However, we have discovered that the chaotic state is fragile with respect to random fluctuations in the on time. In the presence of such fluctuations, two uncoupled copies of the system (e.g., two cells) controlled by the same switch rapidly synchronize.

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

A detailed mathematical model for the fission yeast mitotic cycle is developed based on positive and negative feedback loops by which Cdc13/Cdc2 kinase activates and inactivates itself. Positive feedbacks are created by Cdc13/Cdc2-dependent phosphorylation of specific substrates: inactivating its negative regulators (Rum1, Ste9 and Wee1/Mik1) and activating its positive regulator (Cdc25). A slow negative feedback loop is turned on during mitosis by activation of Slp1/anaphase-promoting complex (APC), which indirectly re-activates the negative regulators, leading to a drop in Cdc13/Cdc2 activity and exit from mitosis. The model explains how fission yeast cells can exit mitosis in the absence of Ste9 (Cdc13 degradation) and Rum1 (an inhibitor of Cdc13/Cdc2). We also show that, if the positive feedback loops accelerating the G(2)/M transition (through Wee1 and Cdc25) are weak, then cells can reset back to G(2) from early stages of mitosis by premature activation of the negative feedback loop. This resetting can happen more than once, resulting in a quantized distribution of cycle times, as observed experimentally in wee1(-) cdc25Delta mutant cells. Our quantitative description of these quantized cycles demonstrates the utility of mathematical modeling, because these cycles cannot be understood by intuitive arguments alone.

Kinetic analysis of a molecular model of the budding yeast cell cycle

The molecular machinery of cell cycle control is known in more detail for budding yeast, Saccharomyces cerevisiae, than for any other eukaryotic organism. In recent years, many elegant experiments on budding yeast have dissected the roles of cyclin molecules (Cln1-3 and Clb1-6) in coordinating the events of DNA synthesis, bud emergence, spindle formation, nuclear division, and cell separation. These experimental clues suggest a mechanism for the principal molecular interactions controlling cyclin synthesis and degradation. Using standard techniques of biochemical kinetics, we convert the mechanism into a set of differential equations, which describe the time courses of three major classes of cyclin-dependent kinase activities. Model in hand, we examine the molecular events controlling "Start" (the commitment step to a new round of chromosome replication, bud formation, and mitosis) and "Finish" (the transition from metaphase to anaphase, when sister chromatids are pulled apart and the bud separates from the mother cell) in wild-type cells and 50 mutants. The model accounts for many details of the physiology, biochemistry, and genetics of cell cycle control in budding yeast.

APC(Cdc20) promotes exit from mitosis by destroying the anaphase inhibitor Pds1 and cyclin Clb5

Ubiquitin-mediated proteolysis due to the anaphase-promoting complex/cyclosome (APC/C) is essential for separation of sister chromatids, requiring degradation of the anaphase inhibitor Pds1, and for exit from mitosis, requiring inactivation of cyclin B Cdk1 kinases. Exit from mitosis in yeast involves accumulation of the cyclin kinase inhibitor Sic1 as well as cyclin proteolysis mediated by APC/C bound by the activating subunit Cdh1/Hct1 (APC(Cdh1)). Both processes require the Cdc14 phosphatase, whose release from the nucleolus during anaphase causes dephosphorylation and thereby activation of Cdh1 and accumulation of another protein, Sic1 (refs 4-7). We do not know what determines the release of Cdc14 and enables it to promote Cdk1 inactivation, but it is known to be dependent on APC/C bound by Cdc20 (APC(Cdc20)) (ref. 4). Here we show that APC(Cdc20) allows activation of Cdc14 and promotes exit from mitosis by mediating proteolysis of Pds1 and the S phase cyclin Clb5 in the yeast Saccharomyces cerevisiae. Degradation of Pds1 is necessary for release of Cdc14 from the nucleolus, whereas degradation of Clb5 is crucial if Cdc14 is to overwhelm Cdk1 and activate its foes (Cdh1 and Sic1). Remarkably, cells lacking both Pds1 and Clb5 can proliferate in the complete absence of Cdc20.