Age-dependent cell cycle models.

This paper presents age-dependent cell cycle models i.e., models where cell generation time is a random variable given by some distribution function, and the probability of cell division per unit time is a function only of cell age (not, for example, of cell mass). It is shown that there does not exist a stable mass distribution if the cells grow exponentially. In the case of linear growth, conditions for stability of the mass distribution are derived. To show these, the methods different from those considered up till now in the literature, are used. It is also shown that one can consider the cell mass growth as a linear dynamical system with a stochastic perturbation. The sister cell model as an improvement of the Transition Probability Model is derived. Statistical data are obtained for that model, and comparisons are made with some experimental data. As a verification tool, alpha and beta curves, are used.     

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.     

A biological and computational model of megakaryocyte development as a stochastic branching process

The purpose of this paper is to describe a model of megakaryocytopoiesis as a branching process with stochastic processes regulating critical control points of differentiation along the stem cell megakaryocyte platelet axis. Progress of cells through these critical control points are regulated by transitional probabilities, which in turn are regulated by influences such as growth factors. The critical control points include transition of resting megakaryocytic stem cells (CFU-meg) into proliferating stem cells, the cessation of cytokinesis, and the cessation of DNA synthesis. A computerized computational method has been developed for directly fitting the stochastic branching model to colony growth data. The computational model has allowed transitional probabilities to be derived from colony size data. The model provides a unifying explanation for much of the heterogeneity of stages of maturation within populations of megakaryocytes and is fully compatible with historical data supporting the stochastic nature of hematopoietic stem cell regulation and with modern molecular concepts about control of the cell cycle.     

Analysis of growth kinetics by division tracking

Cell division tracking using fluorescent dyes, such as carboxyfluorescein diacetate succinimidyl ester, provides a unique opportunity for analysis of cell growth kinetics. The present review article presents new methods for enhancing resolution of division tracking data as well as derivation of quantities that characterize growth from time-series data. These include the average time between successive divisions, the proportion of cells that survive and the proliferation per division. The physical significance of these measured quantities is interpreted by formulation of a two-compartment model of cell cycle transit characterized by stochastic and deterministic cell residence times, respectively. The model confirmed that survival is directly related to the proportion of cells that enter the next cell generation. The proportion of time that cells reside in the stochastic compartment is directly related to the proliferation per generation. This form of analysis provides a starting point for more sophisticated physical and biochemical models of cell cycle regulation.     

A dynamical system model of neurofilament transport in axons

We develop a dynamical system model for the transport of neurofilaments in axons, inspired by Brown's "stop-and-go" model for slow axonal transport. We use fast/slow time-scale arguments to lower the number of relevant parameters in our model. Then, we use experimental data of Wang and Brown to estimate all but one parameter. We show that we can choose this last remaining parameter such that the results of our model agree with pulse-labeling experiments from three different nerve cell types, and also agree with stochastic simulation results.     

Inference of dynamic biological networks based on responses to drug perturbations

Drugs that target specific proteins are a major paradigm in cancer research. In this article, we extend a modeling framework for drug sensitivity prediction and combination therapy design based on drug perturbation experiments. The recently proposed target inhibition map approach can infer stationary pathway models from drug perturbation experiments, but the method is limited to a steady-state snapshot of the underlying dynamical model. We consider the inverse problem of possible dynamic models that can generate the static target inhibition map model. From a deterministic viewpoint, we analyze the inference of Boolean networks that can generate the observed binarized sensitivities under different target inhibition scenarios. From a stochastic perspective, we investigate the generation of Markov chain models that satisfy the observed target inhibition sensitivities.     

State vector description of the proliferation of mammalian cells in tissue culture. I. Exponential growth

Proliferation of mammalian cells, even under conditions of unlimited growth, presents a complex problem because of the interaction of deterministic and stochastic processes. Division of the cell cycle into a finite number of parts establishes a multidimensional vector space. In this space an arbitrary culture can be represented by a vector called the state vector. The culture's subsequent growth is represented mathematically as a series of transformations of the state vector. The operators effecting these transformations are presented in matrix form and their relationship to the distribution of cell generation times is described. As an application of the model, the growth of an initially synchronized culture is considered and an unambiguous measure of the degree of synchrony is proposed. Results of a computer simulation of such a culture show the behavior with time of the degree of synchrony, the total cell number, and the mitotic index. In particular the importance of the magnitude of the coefficient of variation of the generation time distribution is illustrated.     

Elongation of rod-shaped bacteria.

Three models relating cell length to generation time are considered for rod-shaped bacteria growing under steady-state conditions; all three presuppose linear elongation. The first model assumes that the rate of elongation is proportional to the instantaneous number of chromosome replication forks per cell; the others, that it is inversely related to the generation time and doubles a fixed time prior to cell division. One of these (model 2) treats this relationship as continuous, with the doubling occurring during the last division cycle (at chromosome termination), while the other is a discrete model in which the doubling in rate takes place at chromosome initiation. Expressions are derived for mean cell length and length at birth in each case.Comparison with experimental data on E. coli B/r using non-linear least-squares techniques results in an excellent fit for model 2 and unsatisfactory ones for the others, the best estimate for the time at which the rate doubles being 15·3 min prior to cell division and for the minimum length at birth (i.e., as the growth rate of the culture tends to zero), 1·47 μm.The functional relationship between cell radius and generation time implied by model 2 is also presented. This model again produces a good fit to the experimental data and provides, for the first time, a direct estimate of the volume/origin ratio at initiation of chromosome replication 0·35 ± 0·05 μm³ (s.e.).The results obtained here are compared with various qualitative observations reported in the literature and with such numerical data as are available.     

Evaluation of Multitype Mathematical Models for CFSE-Labeling Experiment Data

Carboxy-fluorescein diacetate succinimidyl ester (CFSE) labeling is an important experimental tool for measuring cell responses to extracellular signals in biomedical research. However, changes of the cell cycle (e.g., time to division) corresponding to different stimulations cannot be directly characterized from data collected in CFSE-labeling experiments. A number of independent studies have developed mathematical models as well as parameter estimation methods to better understand cell cycle kinetics based on CFSE data. However, when applying different models to the same data set, notable discrepancies in parameter estimates based on different models has become an issue of great concern. It is therefore important to compare existing models and make recommendations for practical use. For this purpose, we derived the analytic form of an age-dependent multitype branching process model. We then compared the performance of different models, namely branching process, cyton, Smith–Martin, and a linear birth–death ordinary differential equation (ODE) model via simulation studies. For fairness of model comparison, simulated data sets were generated using an agent-based simulation tool which is independent of the four models that are compared. The simulation study results suggest that the branching process model significantly outperforms the other three models over a wide range of parameter values. This model was then employed to understand the proliferation pattern of CD4+ and CD8+ T cells under polyclonal stimulation.     

Bilinear dynamical systems

In this paper, we propose the use of bilinear dynamical systems (BDS)s for model-based deconvolution of fMRI time-series. The importance of this work lies in being able to deconvolve haemodynamic time-series, in an informed way, to disclose the underlying neuronal activity. Being able to estimate neuronal responses in a particular brain region is fundamental for many models of functional integration and connectivity in the brain. BDSs comprise a stochastic bilinear neurodynamical model specified in discrete time, and a set of linear convolution kernels for the haemodynamics. We derive an expectation-maximization (EM) algorithm for parameter estimation, in which fMRI time-series are deconvolved in an E-step and model parameters are updated in an M-Step. We report preliminary results that focus on the assumed stochastic nature of the neurodynamic model and compare the method to Wiener deconvolution.     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Stochastic simulations on a model of circadian rhythm generation

Biological phenomena are often modeled by differential equations, where states of a model system are described by continuous real values. When we consider concentrations of molecules as dynamical variables for a set of biochemical reactions, we implicitly assume that numbers of the molecules are large enough so that their changes can be regarded as continuous and they are described deterministically. However, for a system with small numbers of molecules, changes in their numbers are apparently discrete and molecular noises become significant. In such cases, models with deterministic differential equations may be inappropriate, and the reactions must be described by stochastic equations. In this study, we focus a clock gene expression for a circadian rhythm generation, which is known as a system involving small numbers of molecules. Thus it is appropriate for the system to be modeled by stochastic equations and analyzed by methodologies of stochastic simulations. The interlocked feedback model proposed by Ueda et al. as a set of deterministic ordinary differential equations provides a basis of our analyses. We apply two stochastic simulation methods, namely Gillespie's direct method and the stochastic differential equation method also by Gillespie, to the interlocked feedback model. To this end, we first reformulated the original differential equations back to elementary chemical reactions. With those reactions, we simulate and analyze the dynamics of the model using two methods in order to compare them with the dynamics obtained from the original deterministic model and to characterize dynamics how they depend on the simulation methodologies.     

Deconstructing the role of myosin contractility in force fluctuations within focal adhesions

Force fluctuations exhibited in focal adhesions that connect a cell to its extracellular environment point to the complex role of the underlying machinery that controls cell migration. To elucidate the explicit role of myosin motors in the temporal traction force oscillations, we vary the contractility of these motors in a dynamical model based on the molecular clutch hypothesis. As the contractility is lowered, effected both by changing the motor velocity and the rate of attachment/detachment, we show analytically in an experimentally relevant parameter space, that the system goes from decaying oscillations to stable limit cycle oscillations through a supercritical Hopf bifurcation. As a function of the motor activity and the number of clutches, the system exhibits a rich array of dynamical states. We corroborate our analytical results with stochastic simulations of the motor-clutch system. We obtain limit cycle oscillations in the parameter regime as predicted by our model. The frequency range of oscillations in the average clutch and motor deformation compares well with experimental results.     

Transient responses of budding yeast populations

The unequal-division model for budding yeast is used to formulate a population-balance model for the transient behavior of populations of these organisms. The model consists of linear partial differential equations coupled through algebraic equations. It is shown how the solution of this system of equations can be obtained in a systematic stepwise fashion. The special case of a population subjected to a step change in growth rate is described in some detail, and solutions for two special cases are determined for transients following an age-distribution perturbation. It is shown how experimental data on transient behavior of a cell population can yield information on single-cell mass-synthesis kinetics and on the manner in which individual cells control certain critical parameters in the cell cycle.     

Quasi-exponential generation time distributions from a limit cycle oscillator

In spite of the apparently random behaviour and the often exponential distribution of generation times expressed in cell populations, there is evidence for rather precise timekeeping in the cell cycle. In experiments using time-lapse video-tape microscopy, we have noted that cell generation times are often not distributed smoothly but in many cases seem to cluster at roughly 4 hr intervals. Phase shift responses following application of heat shock, ionizing radiation or serum pulses in each case show a pattern which is repeated twice in cells with an 8-9 hr modal generation time. We describe here a cell cycle model with an independent cellular clock controlling cell cycle events which accounts for the phase response data, while also reconciling the stochastic and periodic behaviour characteristic of animal cells.     

Quasi-Exponential Generation Time Distributions From A Limit Cycle Oscillator

In spite of the apparently random behaviour and the often exponential distribution of generation times expressed in cell populations, there is evidence for rather precise timekeeping in the cell cycle. In experiments using time-lapse video-tape microscopy, we have noted that cell generation times are often not distributed smoothly but in many cases seem to cluster at roughly 4 hr intervals. Phase shift responses following application of heat shock, ionizing radiation or serum pulses in each case show a pattern which is repeated twice in cells with an 8–9 hr modal generation time. We describe here a cell cycle model with an independent cellular clock controlling cell cycle events which accounts for the phase response data, while also reconciling the stochastic and periodic behaviour characteristic of animal cells.