Sloppy size control of the cell division cycle

In an asynchronous, exponentially proliferating cell culture there is a great deal of variability among individual cells in size at birth, size at division and generation time (= age at division). To account for this variability we assume that individual cells grow according to some given growth law and that, after reaching a minimum size, they divide with a certain probability (per unit time) which increases with increasing cell size. This model is called sloppy size control because cell division is assumed to be a random process with size-dependent probability. We derive general equations for the distribution of cell size at division, the distribution of generation time, and the correlations between generation times of closely related cells. Our theoretical results are compared in detail with experimental results (obtained by Miyata and coworkers) for cell division in fission yeast, Schizosaccharomyces pombe. The agreement between theory and experiment is superior to that found for any other simple models of the coordination of cell growth and division.     

Individual-based modeling of phytoplankton: evaluating approaches for applying the cell quota model

Present phytoplankton models typically use a population-level (lumped) modeling (PLM) approach that assumes average properties of a population within a control volume. For modern biogeochemical models that formulate growth as a nonlinear function of the internal nutrient (e.g. Droop kinetics), this averaging assumption can introduce a significant error. Individual-based (agent-based) modeling (IBM) does not make the assumption of average properties and therefore constitutes a promising alternative for biogeochemical modeling. This paper explores the hypothesis that the cell quota (Droop) model, which predicts the population-average specific growth or cell division rate, based on the population-average nutrient cell quota, can be applied to individual algal cells and produce the same population-level results. Three models that translate the growth rate calculated using the cell quota model into discrete cell division events are evaluated, including a stochastic model based on the probability of cell division, a deterministic model based on the maturation velocity and fraction of the cell cycle completed (maturity fraction), and a deterministic model based on biomass (carbon) growth and cell size. The division models are integrated into an IBM framework (iAlgae), which combines a lumped system representation of a nutrient with an individual representation of algae. The IBM models are evaluated against a conventional PLM (because that is the traditional approach) and data from a number of steady and unsteady continuous (chemostat) and batch culture laboratory experiments. The stochastic IBM model fails the steady chemostat culture test, because it produces excessive numerical randomness. The deterministic cell cycle IBM model fails the batch culture test, because it has an abrupt drop in cell quota at division, which allows the cell quota to fall below the subsistence quota. The deterministic cell size IBM model reproduces the data and PLM results for all experiments and the model parameters (e.g. maximum specific growth rate, subsistence quota) are the same as those for the PLM. In addition, the model-predicted cell age, size (carbon) and volume distributions are consistent with those derived analytically and compare well to observations. The paper discusses and illustrates scenarios where intra-population variability in natural systems leads to differences between the IBM and PLM models.     

Cell cycle modelling

Models able to describe the events of cellular growth and division and the dynamics of cell populations are useful for the understanding of functional control mechanisms and for the theoretical support for automated analysis of flow cytometric data and of cell volume distributions. This paper reports on models that we have developed with this aim for different kinds of cells. The models are composed by two subsystems: one describes the growth dynamics of RNA and protein, and the second accounts for DNA replication and cell division, and describe in a rather unitary frame the cell cycle of eukaryotic cells, like mammalian cells and yeast, and of prokaryotic cells. The model is also used to study the effects of various sources of variability on the statistical properties of cell populations, and we find that in microbial cells the main source of variability appears to be an inaccuracy of the molecular mechanism that monitors cell size. In normal mammalian cells another source of variability, that depends upon the interaction with growth factors which give competence, is apparent. An extended version of the model, which comprises also this additional variability, is presented and used to describe the properties of mammalian cell growth.     

The probability of G1 cells to enter into S increases with their size while S length decreases with cell enlargement in Allium cepa

The distribution of cell surface area projection (cell size) has been measured at birth and at initiation of DNA synthesis in steady-state populations of Allium cepa root meristems. The conditional probability, P(I/G1), that initiation occurs given that the event of being in G1 also occurs has been estimated from these data. P(I/G1) was found to increase when cells became larger. The distribution of G1 duration has been constructed from indicated cell size distributions. The absolute frequencies of G1 times showed a maximum in the zone of cells with short G1 periods; about 14% of cells appear to enter into S with G1 congruent to 1 h. These results suggest that the increase of P(I/G1) was due to cell enlargement and not to cell aging. By comparing the cell size distribution at initiation of S and at the end of this period, a drastic reduction of cell size variability during DNA replication was observed and both curves were seen as rather similar in shape although they obviously had different modal points. These observations support that there is a negative correlation between the initiation size and the duration of genome duplication, and that cells which initiate DNA synthesis with the same size have a similar replication time. From this hypothesis, a plot of S duration versus cell size at initiation of this period was constructed by comparing the distributions of cell size at start and end of replication; this plot was also consistent with the existence of a negative correlation between cell initiation size and S length.     

Cell elongation and division probability during the Escherichia coli growth cycle.

A new method is presented for determining the growth rate and the probability of cell division (separation) during the cell cycle, using size distributions of cell populations grown under steady-state conditions. The method utilizes the cell life-length distribution, i.e., the probability that a cell will have any specific size during its life history. This method was used to analyze cell length distributions of six cultures of Escherichia coli, for which doubling times varied from 19 to 125 min. The results for each culture are in good agreement with a single model of growth and division kinetics: exponential elongation of cells during growth phase of the cycle, and normal distributions of length at birth and at division. The average value of the coefficient of variation was 13.5% for all strains and growth rates. These results, based upon 5,955 observations, support and extend earlier proposals that growth and division patterns of E. coli are similar at all growth rates and, in addition, identify the general growth pattern of these cells to be exponential.     

Models for yeast prions

The cytoplasmic heritable determinant [PSI+] of the yeast Saccharomyces cerevisiae exhibits prion-like properties. The properties of yeast prions are studied in the hope that this will enhance the understanding of mammalian prions, which cause mad-cow, Creutzfeldt-Jakob, and related neurodegenerative diseases. When host cells divide, the yeast prions distribute themselves without loss over the daughter cells. Experimental data provide information on how the proportion of cells with prions decreases over time when priori replication is inhibited. One feature of scientific interest is the unknown mean number, n0, of prions assumed to be present in the cells at the start of the experiment. We develop several stochastic models and by fitting them to the data, we obtain substantially larger estimates of n0 compared with a previous analysis. An interesting feature of a model with constant cell generation times is that the predicted proportion of cells with prions varies over time as a sequence of linked hyperbolic curves. Avenues for future research are outlined, which relax simplifying assumptions made in the models. We make several recommendations for the design of future experiments.     

AN EXTENDED TRANSITION PROBABILITY MODEL OF THE VARIABILITY OF CELL GENERATION TIMES

The transition probability model of variability of cell generation times is extended so that the rate constant for the transition from the A-state to the B-phase of the cell cycle depends on time which a particular cell has already spent in the A-state. A specific time dependence of this rate constant is introduced. It is determined by the value of one constant which is then an additional parameter of the model. The corresponding cell population kinetics are calculated and compared to existing experimental evidence. The model accounts satisfactorily for the generation time distribution function and for the shortening of the G₁ phase of binucleate cells. The time dependence of the transition probability is related to the cell kinetics of an hypothetical cell constituent. A possible relationship is proposed between the chemical parameters within the cell and the parameters of the cell population kinetics.     

Mathematical model for aerobic culture of a recombinant yeast

A mathematical model was formulated to simulate cell growth, plasmid loss and recombinant protein production during the aerobic culture of a recombinant yeast S. cerevisiae. Model development was based on three simplified metabolic events in the yeast: glucose fermentation, glucose oxidation and ethanol oxidation. Cell growth was expressed as a composite of these metabolic events. Their contributions to the total specific growth rate depended on the activities of the pacemaker enzyme pools of the individual pathways. The pacemaker enzyme pools were regulated by the specific glucose uptake rate. The effect of substrate concentrations on the specific growth rate was described by a modified Monod equation. It was assumed that recombinant protein formation is only associated with oxidative pathways. Plasmid loss kinetics was formulated based on segregational instability during cell division by assuming constant probability of plasmid loss. Experiments on batch fermentation of recombinant S. cerevisiae C468/pGAC9 (ATCC 20690), which expresses Aspergillus awamori glucoamylase gene and secretes glucoamylase into the extracellular medium, were carried out in an airlift bioreactor in order to evaluate the proposed model. The model successfully predicted the dynamics of cell growth, glucose consumption, ethanol metabolism, glucoamylase production and plasmid instability. Excellent agreement between model simulations and our experimental data was achieved. Using published experimental data, model agreement was also found for other recombinant yeast strains. In general, the proposed model appears to be useful for the design, scale-up, control and optimization of recombinant yeast bioprocesses.     

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.     

Modeling brewers' yeast flocculation

Flocculation of yeast cells occurs during the fermentation of beer. Partway through the fermentation the cells become flocculent and start to form flocs. If the environmental conditions, such as medium composition and fluid velocities in the tank, are optimal, the flocs will grow in size large enough to settle. After settling of the main part of the yeast the green beer is left, containing only a small amount of yeast necessary for rest conversions during the next process step, the lagering. The physical process of flocculation is a dynamic equilibrium of floc formation and floc breakup resulting in a bimodal size distribution containing single cells and flocs. The floc size distribution and the single cell amount were measured under the different conditions that occur during full scale fermentation. Influences on flocculation such as floc strength, specific power input, and total number of yeast cells in suspension were studied. A flocculation model was developed, and the measured data used for validation. Yeast floc formation can be described with the collision theory assuming a constant collision efficiency. The breakup of flocs appears to occur mainly via two mechanisms, the splitting of flocs and the erosion of yeast cells from the floc surface. The splitting rate determines the average floc size and the erosion rate determines the number of single cells. Regarding the size of the flocs with respect to the scale of turbulence, only the viscous subrange needs to be considered. With the model, the floc size distribution and the number of single cells can be predicted at a certain point during the fermentation. For this, the bond strength between the cells, the fractal dimension of the yeast, the specific power input in the tank and the number of yeast cells that are in suspension in the tank have to be known. Copyright 1998 John Wiley & Sons, Inc.     

The probability of G1 cells to enter into S increases with their size while S length decreases with cell enlargement in Allium cepa

The distribution of cell surface area projection (cell size) has been measured at birth and at initiation of DNA synthesis in steady-state populations of Allium cepa root meristems. The conditional probability, P(I/G₁), that initiation occurs given that the event of being in g₁ also occurs has been estimated from these data. p(I/G₁) was found to increase when cells became larger. The distribution of G₁ duration has been constructed from indicated cell size distributions. The absolute frequencies of G₁ times showed a maximum in the zone of cells with short G₁ periods; about 14% of cells appear to enter into S with G₁ - 1 h. These results suggest that the increase of p(I/G₁) was due to cell enlargement and not to cell aging. By comparing the cell size distribution at initiation of S and at the end of this period, a drastic reduction of cell size variability during DNA replication was observed and both curves were seen as rather similar in shape although they obviously had different modal points. These observations support that there is a negative correlation between the initiation size and the duration of genome duplication, and that cells which initiate DNA synthesis with the same size have a similar replication time. From this hypothesis, a plot of S duration versus cell size at initiation of this period was constructed by comparing the distributions of cell size at start and end of replication; this plot was also consistent with the existence of a negative correlation between cell initiation size and S length.     

Cell cycle dynamics inferred from the static properties of cells in balanced growth

The duration of a morphological phase of the cell cycle is reflected in the steady state distribution of the sizes of cells in that phase. Relationships presented here provide a method for estimating the timing and variability of any cell cycle phase. It is shown that the mean size of cells initiating and finishing any phase can be estimated from (1) the frequency of cells exhibiting the distinguishing morphological or autoradiographic features of the phase; (2) the mean size of cells in the phase; and (3) their coefficient of variation. The calculations are based on a submodel of the Koch-Schaechter Growth Controlled Model which assumes that (i) the distribution of division sizes is Gaussian; (ii) there is no correlation in division sizes between successive generations; and (iii) every cell division gives rise to two daughter cells of equal size. The calculations should be useful for a wider range of models, however, because the extrapolation factors are not sensitive to the chosen model. Criteria are proposed to allow the user to check the method's applicability for any experimental case. The method also provides a more efficient test of the dependence of growth on cell size than does the Collins-Richmond method. This is because the method uses the mean and coefficient of variation of the size of the total population, in conjunction with those of the cells in a final phase of the cell cycle, to test potential growth laws. For Escherichia coli populations studied by electron microscopy, an exponential growth model provided much better agreement than did a linear growth model. The computer simulations were used to generate rules for three types of cell phases: those that end at cell division, those that start at cell division, and those totally contained within a single cell cycle. For the last type, additional criteria are proposed to establish if the phase is well enough contained for the formulae and graphs to be used. The most useful rule emerging from these computer studies is that the fraction of the cell cycle time occupied by a phase is the product of the frequency of the phase and the ratio of the mean size of cells in that phase to the mean size of all cells in the population. A further advantage of the techniques presented here is that they use the 'extant' distributions that were actually measured, and not hypothesized distributions nor the special distributions needed for Collins-Richmond method that can only be calculated from the observed distributions of dividing or newborn cells on the basis of an assumed growth law.     

Incorporating phenotype-dependent growth rates into the color-shift model for preneoplastic hepatocellular lesions

Multi-stage models occupy a central position in modeling the carcinogenesis process. These models formalize the hypothesis that cells have to undergo several transformations on their way to malignancy. This hypothesis assumes that a preneoplastic cell of a later stage arises through a mutational event of a single cell of a previous stage and that preneoplastic cells proliferate clonally. However, there is some evidence that multi-stage models cannot adequately describe the formation and the progression of preneoplastic lesions at least in certain organs [Math. Biosci. 168 (2000) 167]. An alternative model assuming that all cells in a colony of altered hepatocytes change their phenotype more or less simultaneously rather than by mutation of single cells has already been introduced [Math. Biosci. 148 (1998) 181] and is called color-shift model (CSM). This model assumed deterministic phenotype-independent growth for the foci once they are generated. An expansion of the CSM allowing for variability between deterministic growth behaviour of phenotypically different colonies is presented (modCSM) and the model is applied to focal lesion data from a rat hepatocarcinogenesis experiment. The fit of the originally proposed and the modCSM are compared with respect to their ability to predict numbers and radii of preneoplastic cell foci.     

Plasmid stability in budding yeast populations: Steady-state growth with selection pressure

Plasmid gene product accumulation in a cell population depends on the fraction of plasmid-containing cells and the distribution of single-cell plasmid content. These important population properties have been related to plasmid replication regulation and kinetics and to plasmid segregation rules at the single-cell level using population balance mathematical models. Budding yeast populations are considered in detail because of the practical potential of yeast host-vector systems and because of the model complications introduced by the asymmetric division pattern observed for Saccharomyces cerevisiae at all but the largest growth rates. Solutions are presented for several different reasonable models of plasmid replication and segregation. The results offer potential for identification of important qualitative features of yeast plasmid replication and of model parameter values from average and segregated experimental data on yeast populations.     

Towards understanding of the complex structure of growing yeast populations

In a growing Saccharomyces cerevisiae population, cell size is finely modulated according to both the chronological and genealogical ages. This generates the complex heterogeneous structure typical of budding yeast populations. In recent years, there has been a growing interest in developing mathematical models capable of faithfully describing population dynamics at the single cell level. A multistaged morphologically structured model has been lately proposed based on the population balance theory. The model was able to describe the dynamics of the generation of a heterogeneous growing yeast population starting from a sub-population of daughter unbudded cells. In this work, which aims at validating the model, the simulated experiment was performed by following the release of a homogeneous population of daughter unbudded cells. A biparametric flow cytometric approach allowed us to analyse the time course joint distribution of DNA and protein contents at the single cell level; this gave insights into the coupling between growth and cell cycle progression that generated the final population structure. The comparison between experimental and simulated size distributions revealed a strong agreement for some unexpected features as well. Therefore, the model can be considered as validated and extendable to more complex situations.     

A stochastic model of cell replicative senescence based on telomere shortening, oxidative stress, and somatic mutations in nuclear and mitochondrial DNA.

Human diploid fibroblast cells can divide for only a limited number of times in vitro, a phenomenon known as replicative senescence or the Hayflick limit. Variability in doubling potential is observed within a clone of cells, and between two sister cells arising from a single mitotic division. This strongly suggests that the process by which cells become senescent is intrinsically stochastic. Among the various biochemical mechanisms that have been proposed to explain replicative senescence, particular interest has been focussed on the role of telomere reduction. In the absence of telomerase--an enzyme switched off in normal diploid fibro-blasts-cells lose telomeric DNA at each cell division. According to the telomere hypothesis of cell senescence, cells eventually reach a critically short telomere length and cell cycle arrest follows. In support of this concept, forced expression of telomerase in normal fibroblasts appears to prevent cell senescence. Nevertheless, the telomere hypothesis in its basic form has some difficulty in explaining the marked stochastic variations seen in the replicative lifespans of individual cells within a culture, and there is strong empirical and theoretical support for the concept that other kinds of damage may contribute to cellular ageing. We describe a stochastic network model of cell senescence in which a primary role is played by telomere reduction but in which other mechanisms (oxidative stress linked particularly to mitochondrial damage, and nuclear somatic mutations) also contribute. The model gives simulation results that are in good agreement with published data on intra-clonal variability in cell doubling potential and permits an analysis of how the various elements of the stochastic network interact. Such integrative models may aid in developing new experimental approaches aimed at unravelling the intrinsic complexity of the mechanisms contributing to human cell ageing.     

Analysis of unstable recombinant Saccharomyces cerevisiae population growth in selective medium

Widely applied selection strategies for plasmid-containing cells in unstable recombinant populations are based upon synthesis in those cells of an essential, selection gene product. Regular partitioning of this gene product combined with asymmetric plasmid segregation produces plasmid-free cells which retain for some time the ability to grow in selective medium. This theory is elaborated here in terms of a segregated model for an unstable recombinant population which predicts population growth characteristics and composition based upon experimental data for stable strain growth kinetics, plasmid content, and selection gene product stability. Analytical solutions from this model are compared with an unsegregated phenomenological model to evaluate the effective specific growth rate of plasmid-free cells in selective medium. Model predictions have been validated using experimental growth kinetics and flow cytometry data for Saccharomyces cerevisiae D603 populations containing one of the plasmids YCpG1ARS1, YCpG1DeltaR8, YCpG1DeltaR88, YCpG1DeltaH103, YCpG1DeltaH200, pLGARS1, and pLGSD5. The recombinant strains investigated encompass a broad range of plasmid content (from one to 18 plasmids per cell) and probability alpha of plasmid loss at division (0.05 相似文献   

A stochastic model for cancer risk

We propose a simple stochastic model based on the two successive mutations hypothesis to compute cancer risks. Assume that only stem cells are susceptible to the first mutation and that there are a total of D stem cell divisions over the lifetime of the tissue with a first mutation probability mu(1) per division. Our model predicts that cancer risk will be low if m = mu(1)D is low even in the case of very advantageous mutations. Moreover, if mu(1)D is low the mutation probability of the second mutation is practically irrelevant to the cancer risk. These results are in contrast with existing models but in agreement with a conjecture of Cairns. In the case where m is large our model predicts that the cancer risk depends crucially on whether the first mutation is advantageous or not. A disadvantageous or neutral mutation makes the risk of cancer drop dramatically.