Stochastic exit from mitosis in budding yeast: Model predictions and experimental observations

Unlike many mutants that are completely viable or inviable, the CLB2-dbΔ clb5Δ mutant of Saccharomyces cerevisiae is inviable in glucose but partially viable on slower growth media such as raffinose. On raffinose, the mutant cells can bud and divide but in each cycle there is a chance that a cell will fail to divide (telophase arrest), causing it to exit the cell cycle. This effect gives rise to a stochastic phenotype that cannot be explained by a deterministic model. We measure the interbud times of wild-type and mutant cells growing on raffinose and compute statistics and distributions to characterize the mutant''s behavior. We convert a detailed deterministic model of the budding yeast cell cycle to a stochastic model and determine the extent to which it captures the stochastic phenotype of the mutant strain. Predictions of the mathematical model are in reasonable agreement with our experimental data and suggest directions for improving the model. Ultimately, the ability to accurately model stochastic phenotypes may prove critical to understanding disease and therapeutic interventions in higher eukaryotes.Key words: stochastic phenotype, mitotic exit, non-genetic variability, cell cycle modeling, computational biology, stochastic modeling, deterministic modeling     

The dynamical consequences of developmental variability and demographic stochasticity for host-parasitoid interactions

Few age-structured models of species dynamics incorporate variability and uncertainty in population processes. Motivated by laboratory data for an insect and its parasitoid, we investigate whether such assumptions are appropriate when considering the population dynamics of a single species and its interaction with a natural enemy. Specifically, we examine the effects of developmental variability and demographic stochasticity on different types of cyclic dynamics predicted by traditional models. We show that predictions based on the deterministic fixed-development approach are differentially sensitive to variability and noise in key life stages. In particular, we find that the demonstration of half-generation cycles in the single-species model and the multigeneration cycles in the host-parasitoid model are sensitive to the introduction of developmental variability and noise, whereas generation cycles are robust to the intrinsic variability and uncertainty that may be found in nature.     

A simple time delay model for eukaryotic cell cycle

We propose a seven variable model with time delay in one of the variables for the cell cycle in higher eukaryotes. The model consists of four important phosphorylation-dephosphorylation (P-D) cycles that govern the cell cycle, namely Pre-MPF-MPF, Cdc25P-Cdc25, Wee1P-Wee1 and APCP-APC. Other variables are cyclin, free cyclin dependent kinase (Cdk) and mass. The mass acts as a G2/M checkpoint and the checkpoint is represented by a saddle node loop bifurcation. The key feature of the model is that a time lag has been introduced in the activation of anaphase promoting complex (APC) by maturation promoting factor (MPF). This is effected by treating MPF as a time-delayed variable in the activation step of APC. The time lag acts as a spindle checkpoint. Absence of time delay induces a bistability in our model. Time delay also brings about variability in G1 phase timings. The model also reproduces the mutant phenotype experiments on wee1 cells. Stochasticity has been introduced in the model to simulate the dependence of the cycle time on cell birth length. Mutant phenotypes in the stochastic model reproduce the experimental observations better than the deterministic model.     

The distributions of cell size and generation time in a model of the cell cycle incorporating size control and random transitions

A deterministic/probabilistic model of the cell division cycle is analysed mathematically and compared to experimental data and to other models of the cell cycle. The model posits a random-exiting phase of the cell cycle and a minimum-size requirement for entry into the random-exiting phase. By design, the model predicts exponential "beta-curves", which are characteristic of sister cell generation times. We show that the model predicts "alpha-curves" with exponential tails and hyperbolic-sine-like shoulders, and that these curves fit observed generation-time data excellently. We also calculate correlation coefficients for sister cells and for mother-daughter pairs. These correlation coefficients are more negative than is generally observed, which is characteristic of all size-control models and is generally attributed to some unknown positive correlation in growth rates of related cells. Next we compare theoretical size distributions with observed distributions, and we calculate the dependence of average cell mass on specific growth rate and show that this dependence agrees with a well-known relation in bacteria. In the discussion we argue that unequal division is probably not the source of stochastic fluctuations in deterministic size-control models, transition-probability models with no feedback from cell size cannot account for the rapidity with which the new, stable size distribution is established after perturbation, and Kubitschek's rate-normal model is not consistent with exponential beta-curves.     

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.     

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.     

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.     

Formulation and numerical simulations of a continuum model of avascular tumor growth

In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G₀ phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.     

A semi-stochastic model of the transmission of Escherichia coli O157 in a typical UK dairy herd: dynamics, sensitivity analysis and intervention/prevention strategies

When modelling the transmission of infection within small populations, it is necessary to consider the possibility of stochastic fade-out of infection. We present a semi-stochastic model for the transmission of a microparasite, in this case Escherichia coli O157, within a multigroup system, namely a typical UK dairy herd. The model includes birth, death, maturation, the dry/lactating cycle and various types of transmission (i.e. direct, pseudovertical (representing direct faecal-oral transmission between dam and calf within the first 48 h) and indirect (via free-living infectious units in the environment)). We present the results of our simulation study alongside data from empirical studies and also compare simulation results with those for the corresponding deterministic model. We then examine the effects of reducing shedding in the food-producing groups on outbreak size and prevalence of infection. A sensitivity analysis of herd prevalence reveals that, for both the deterministic and the semi-stochastic model, the prevalence within the herd is most sensitive to two parameters relating to the weaned group. This supports our previously reported conclusions for the deterministic model, which were based on an analysis of the next-generation matrix. The sensitivity analysis also indicates that herd prevalence is greatly affected by two other parameters relating to the lactating group. We conclude by discussing the possible efficacy of suggested intervention strategies.     

Noise-induced neural impulses

The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.     

Modelling Cell Migration and Adhesion During Development

Cell?Ccell adhesion is essential for biological development: cells migrate to their target sites, where cell?Ccell adhesion enables them to aggregate and form tissues. Here, we extend analysis of the model of cell migration proposed by Anguige and Schmeiser (J. Math. Biol. 58(3):395?C427, 2009) that incorporates both cell?Ccell adhesion and volume filling. The stochastic space-jump model is compared to two deterministic counterparts (a system of stochastic mean equations and a non-linear partial differential equation), and it is shown that the results of the deterministic systems are, in general, qualitatively similar to the mean behaviour of multiple stochastic simulations. However, individual stochastic simulations can give rise to behaviour that varies significantly from that of the mean. In particular, individual simulations might admit cell clustering when the mean behaviour does not. We also investigate the potential of this model to display behaviour predicted by the differential adhesion hypothesis by incorporating a second cell species, and present a novel approach for implementing models of cell migration on a growing domain.     

Non-linear growth mechanics—I. Volterrahamilton systems

In this, the first of a series of papers on stochastic and deterministic non-linear allometric growth models, a deterministic model is proposed which generalizes the widely applicable classical linear model of Huxley and Needham. There aren types of producers, each type depositing a product which accumulates monotonically in the environment. Producers interact via a mass action law satisfying an optimality condition. Coefficients may be interpreted as competition between the various producer types in the usual Volterra sense. An ideal coral reef is studied in which then species of coral polyps lay down aragonite calcium carbonate in building the reef framework. This deterministic model predicts that younger reefs are strongly unstable relative to initial species abundance, while older reefs grow in the classical sense of Huxley and Needham, asymptotically, as time goes to infinity. This research has been partially supported by NSERC-A-7667     

In silico simulation of the effect of hypoxia on MCF-7 cell cycle kinetics under fractionated radiotherapy

The treatment outcome of a given fractionated radiotherapy scheme is affected by oxygen tension and cell cycle kinetics of the tumor population. Numerous experimental studies have supported the variability of radiosensitivity with cell cycle phase. Oxygen modulates the radiosensitivity through hypoxia-inducible factor (HIF) stabilization and oxygen fixation hypothesis (OFH) mechanism. In this study, an existing mathematical model describing cell cycle kinetics was modified to include the oxygen-dependent G1/S transition rate and radiation inactivation rate. The radiation inactivation rate used was derived from the linear-quadratic (LQ) model with dependence on oxygen enhancement ratio (OER), while the oxygen-dependent correction for the G1/S phase transition was obtained from numerically solving the ODE system of cyclin D-HIF dynamics at different oxygen tensions. The corresponding cell cycle phase fractions of aerated MCF-7 tumor population, and the resulting growth curve obtained from numerically solving the developed mathematical model were found to be comparable to experimental data. Two breast radiotherapy fractionation schemes were investigated using the mathematical model. Results show that hypoxia causes the tumor to be more predominated by the tumor subpopulation in the G1 phase and decrease the fractional contribution of the more radioresistant tumor cells in the S phase. However, the advantage provided by hypoxia in terms of cell cycle phase distribution is largely offset by the radioresistance developed through OFH. The delayed proliferation caused by severe hypoxia slightly improves the radiotherapy efficacy compared to that with mild hypoxia for a high overall treatment duration as demonstrated in the 40-Gy fractionation scheme.     

Using a mammalian cell cycle simulation to interpret differential kinase inhibition in anti-tumour pharmaceutical development

Systems biology needs to show practical relevance to commercial biological challenges such as those of pharmaceutical development. The aim of this work is to design and validate some applications in anti-cancer therapeutic development. The test system was a group of novel cyclin-dependent kinase (CDK) inhibitors synthesised by Cyclacel Ltd. The measured in vitro IC50s of each compound were used as input data to a proprietary cell cycle model developed by Physiomics plc. The model was able to predict over three orders of magnitude the cytotoxicity of each compound without model adaptation to specific cancer cell types. This pattern matched the experimentally determined data. One class of compounds was predicted to cause an increase of the cell cycle length with a non-linear dose-response curve. Further work will use apoptosis and DNA replication simulations to look at overall cell effects.     

Detection of mutual determinism between a pair of spike trains

 A method for detecting mutual deterministic dependence between a pair of spike trains is proposed. When it is assumed that a cell assembly, which is a subgroup of neurons processing a common task, is constituted as a dynamical system, then the mutual determinism between constituent neurons may be directly reflected in functional connectivity in the assembly. The deterministic dependence between two spike trains can be measured with statistical significance using a method of nonlinear prediction. Some examples of simulations are demonstrated in both deterministic and stochastic cases. Received: 8 August 1999 / Accepted in revised form: 29 March 2001     

Cluster analysis of dynamic parameters of gene expression

Cluster analysis has proven to be a valuable statistical method for analyzing whole genome expression data. Although clustering methods have great utility, they do represent a lower level statistical analysis that is not directly tied to a specific model. To extend such methods and to allow for more sophisticated lines of inference, we use cluster analysis in conjunction with a specific model of gene expression dynamics. This model provides phenomenological dynamic parameters on both linear and non-linear responses of the system. This analysis determines the parameters of two different transition matrices (linear and nonlinear) that describe the influence of one gene expression level on another. Using yeast cell cycle microarray data as test set, we calculated the transition matrices and used these dynamic parameters as a metric for cluster analysis. Hierarchical cluster analysis of this transition matrix reveals how a set of genes influence the expression of other genes activated during different cell cycle phases. Most strikingly, genes in different stages of cell cycle preferentially activate or inactivate genes in other stages of cell cycle, and this relationship can be readily visualized in a two-way clustering image. The observation is prior to any knowledge of the chronological characteristics of the cell cycle process. This method shows the utility of using model parameters as a metric in cluster analysis.     

Evidence for an intraclonal random shift between two types of cell cycle times in an embryonal carcinoma cell line

In a recent paper we reported the discovery of an intraclonal bimodal-like cell cycle time variation within the multipotent embryonal carcinoma (EC) PCC3 N/1 line growing in the exponential phase in the undifferentiated state. The variability was found to be localized in the G1 period. Furthermore, an inverse relation between cell size and cell generation time was found in the cell system analysed. It was suggested that the bimodal-like intraclonal time variability previously reported was attributable to an intraclonal shift between two types of cell-growth-rate cycles and that the cell-growth cycle has a supramitotic character, being dissociated from the DNA-division cycle. The growth rate heterogeneity in the cell population was found to need three cell cycles to reach full dispersion in time. This was assumed to be due to a decreased inheritance from sister cell pairs to second cousin cell pairs. Thus, the interesting feature is that in one and the same multipotent cell line there was evidence for an intraclonal instability with a random shift between two types of cell cycle differing in the duration of their G1 period.     

Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep

Under selected conditions, nonlinear dynamical systems, which can be described by deterministic models, are able to generate so-called deterministic chaos. In this case the dynamics show a sensitive dependence on initial conditions, which means that different states of a system, being arbitrarily close initially, will become macroscopically separated for sufficiently long times. In this sense, the unpredictability of the EEG might be a basic phenomenon of its chaotic character. Recent investigations of the dimensionality of EEG attractors in phase space have led to the assumption that the EEG can be regarded as a deterministic process which should not be mistaken for simple noise. The calculation of dimensionality estimates the degrees of freedom of a signal. Nevertheless, it is difficult to decide from this kind of analysis whether a process is quasiperiodic or chaotic. Therefore, we performed a new analysis by calculating the first positive Lyapunov exponent L ₁ from sleep EEG data. Lyapunov exponents measure the mean exponential expansion or contraction of a flow in phase space. L ₁ is zero for periodic as well as quasiperiodic processes, but positive in the case of chaotic processes expressing the sensitive dependence on initial conditions. We calculated L ₁ for sleep EEG segments of 15 healthy men corresponding to the sleep stages I, II, III, IV, and REM (according to Rechtschaffen and Kales). Our investigations support the assumption that EEG signals are neither quasiperiodic waves nor a simple noise. Moreover, we found statistically significant differences between the values of L ₁ for different sleep stages. All together, this kind of analysis yields a useful extension of the characterization of EEG signals in terms of nonlinear dynamical system theory.     

Role of the autonomic nervous system in generating non-linear dynamics in short-term heart period variability.

We evaluated the role played by the autonomic nervous system in producing non-linear dynamics in short heart period variability (HPV) series recorded in healthy young humans. Non-linear dynamics are detected using an index of predictability based on a local non-linear predictor and a surrogate data approach. Different types of surrogates are utilized: (i) phase-randomized Fourier-transform based (FT) data; (ii) amplitude-adjusted FT (AAFT) data; and (iii) iteratively refined AAFT (IAAFT) data of two types (IAAFT-1 and IAAFT-2). The approach was applied to experimental protocols activating or blocking the sympathetic or parasympathetic branches of the autonomic nervous system or periodically perturbing cardiovascular control via paced respiration at different breathing rates. We found that short-term HPV was mostly linear at rest. Experimental protocols activating the sympathetic or parasympathetic nervous system did not produce non-linear dynamics. In contrast, paced respiration, especially at slow breathing rates, elicited significantly non-linear dynamics. Therefore, in short-term HPV ( approximately 300 beats) the use of non-linear models is not supported by the data, except under conditions whereby the subject is constrained to a slow respiratory rate.