Cell migration and organization in the intestinal crypt using a lattice-free model

We present a novel class of spatial models of cell movement and arrangement applied to the two-dimensional cellular organization of the intestinal crypt. The model differs from earlier approaches in using a dynamic movement on a lattice-free cylindrical surface. Cell movement is a consequence of mitotic activity. Cells interact by viscoelastic forces. Voronoi tessellation permits simulations of individual cell boundaries. Simulations can be compared with experimental data obtained from cell scoring in sections. Simulation studies show that the model is consistent with the experimental results for the spatial distribution of labelling indices, mitotic indices and other observed phenomena using a fixed number of stem cells and a fixed number of transit cell divisions.     

On the application of active learning and Gaussian processes in postcryopreservation cell membrane integrity experiments

Biological cell cryopreservation permits storage of specimens for future use. Stem cell cryostorage in particular is fast becoming a broadly spread practice due to their potential for use in regenerative medicine. For the optimal cryopreservation process, ultralow temperatures are needed. However, elevated temperatures are often unavoidable in a typical sample handling cycle which in turn negatively affects the postcryopreservation integrity of cells. In this paper, we present an application of active learning using an underlying Gaussian Process (GP) model in an experimental study on postcryopreservation membrane integrity response to a range of elevated temperature conditions. We tailored this technique for the current investigation and developed an algorithm which enabled identification of the sampling locations for the experiments in order to obtain the highest information return about the process from a limited size sample set. We applied this algorithm in the experimental study investigating the effects of severe temperature elevation (ranging from -40 to 20 °C) over a short term event (48 hours) on the postcryopreservation membrane integrity of Mesenchymal Stem Cells (MSCs) derived from human bone marrow. The algorithm showed excellent performance by selecting the locations which maximized the reduction of variance of the process response estimate. An approximating GP model developed from this experimental data shows that the elevated temperatures during cryopreservation have an imminent detrimental effect on cell integrity.     

The systems biology of signaling pathways

Signaling networks play the central role in the regulation of processes in a single cell and in the entire body. A recent breakthrough in technologies for systems biology, which combine experimental and mathematical methods, permits scientists to model signaling pathways in an individual cell and in cell populations. This approach provides new information on mechanisms that regulate a variety of biological processes. Here we discuss the mathematical formalisms that are applied to signaling pathway modeling and relevant experimental methods.     

Aggregation behavior of red blood cells in shear flow. A theoretical interpretation of simultaneous rheo-optical and viscometric measurements

A theoretical interpretation of simultaneous viscosity measurements and light backscattering experiments is carried out in the framework of the structural model for concentrated dispersions proposed previously by one of the authors. The work is mainly focused on erythrocyte aggregation, hence spherical as well as linear aggregates (rouleaux) were considered in the modeling. A connection between the structural parameters provided by each technique is established, in particular the characteristic shear rates for break up of aggregates. Theoretical predictions were then applied to experimental data of human blood collected from patients with different diseases in a hospital data bank. Finally, we conclude that the structural modeling proposed permits a reasonably good correlation between experimental data of viscometry and light backscattering from blood samples, leading to new perspectives in the analysis of the red blood cell aggregation phenomena.     

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.     

Glucose transport in a model of the rat proximal tubule epithelium

A mathematical model of the rat proximal tubule epithelium has been extended to include terms for glucose-sodium cotransport, as well as the passive permeability properties of urea. Except for a metabolically driven Na⁺-K⁺ exchanger at the cell basolateral membrane, all membrane transport is represented by the relations of linear nonequilibrium thermodynamics. Use of this formalism permits the explicit calculation of the intracellular depolarization immediately following the luminal application of glucose, and shows the magnitude of this potential deflection proportional to the glucose chemical-potential change. The steady-state glucose transport by this model epithelium, like experimental data, is fitted remarkably well by a three-parameter pump-leak model of transport. In view of the nonsaturability of the cotransporter of the model epithelium, the goodness of fit to the three-parameter model is surprising and underscores the uncertainty in extracting individual membrane properties from whole epithelial data. Experiments are simulated in which hypertonic glucose placed in the bath induces cell swelling and K⁺ uptake; a hypertonic impermeant induces cell shrinkage and K⁺ loss. Although this parallels the observations in vivo, the large K⁺ shifts predicted by the model suggest the absence of important volume-regulatory mechanisms from the model scheme.     

A Novel Method for the Topographic Analysis of Neural Activity Reveals Formation and Dissolution of ‘Dynamic Cell Assemblies’

The study of synchronous oscillations in neural systems is a very active area of research. However, cognitive function may depend more crucially upon a dynamic alternation between synchronous and desynchronous activity rather than synchronous behaviour per se. The principle aim of this study is to develop and validate a novel method of quantifying this complex process. The method permits a direct mapping of phase synchronous dynamics and desynchronizing bursts in the spatial and temporal domains. Two data sets are analyzed: Numeric data from a model of a sparsely coupled neural cell assembly and experimental data consisting of scalp-recorded EEG from 40 human subjects. In the numeric data, the approach enables the demonstration of complex relationships between cluster size and temporal duration that cannot be detected with other methods. Dynamic patterns of phase-clustering and desynchronization are also demonstrated in the experimental data. It is further shown that in a significant proportion of the recordings, the pattern of dynamics exhibits nonlinear structure. We argue that this procedure provides a 'natural partitioning' of ongoing brain dynamics into topographically distinct synchronous epochs which may be integral to the brain's adaptive function. In particular, the character of transitions between consecutive synchronous epochs may reflect important aspects of information processing and cognitive flexibility.     

A scenario-based approach to modeling development: a prototype model of C. elegans vulval fate specification

Studies of developmental biology are often facilitated by diagram “models” that summarize the current understanding of underlying mechanisms. The increasing complexity of our understanding of development necessitates computational models that can extend these representations to include their dynamic behavior. Here we present a prototype model of Caenorhabditis elegans vulval precursor cell fate specification that represents many processes crucial for this developmental event but that are hard to integrate using other modeling methodologies. We demonstrate the integrative capabilities of our methodology by comprehensively incorporating the contents of three seminal papers, showing that this methodology can lead to comprehensive models of developmental biology. The prototype computational model was built and is run using a language (Live Sequence Charts) and tool (the Play-Engine) that facilitate the same conceptual processes biologists use to construct and probe diagram-type models. We demonstrate that this modeling approach permits rigorous tests of mutual consistency between experimental data and mechanistic hypotheses and can identify specific conflicting results, providing a useful approach to probe developmental systems.     

An on-lattice agent-based Monte Carlo model simulating the growth kinetics of multicellular tumor spheroids

PurposeTo develop an on-lattice agent-based model describing the growth of multicellular tumor spheroids using simple Monte Carlo tools.MethodsCells are situated on the vertices of a cubic grid. Different cell states (proliferative, hypoxic or dead) and cell evolution rules, driven by 10 parameters, and the effects of the culture medium are included. About twenty spheroids of MCF-7 human breast cancer were cultivated and the experimental data were used for tuning the model parameters.ResultsSimulated spheroids showed adequate sizes of the necrotic nuclei and of the hypoxic and proliferative cell phases as a function of the growth time, mimicking the overall characteristics of the experimental spheroids. The relation between the radii of the necrotic nucleus and the whole spheroid obtained in the simulations was similar to the experimental one and the number of cells, as a function of the spheroid volume, was well reproduced. The statistical variability of the Monte Carlo model described the whole volume range observed for the experimental spheroids. Assuming that the model parameters vary within Gaussian distributions it was obtained a sample of spheroids that reproduced much better the experimental findings.ConclusionsThe model developed allows describing the growth of in vitro multicellular spheroids and the experimental variability can be well reproduced. Its flexibility permits to vary both the agents involved and the rules that govern the spheroid growth. More general situations, such as, e. g., tumor vascularization, radiotherapy effects on solid tumors, or the validity of the tumor growth mathematical models can be studied.     

Breeding and management of BB rats by the aid of the computer program BB RADABA.

In the process of breeding laboratory animals for experimental research a host of data will be produced. Our computer program BB RADABA permits the management of breeding and experimental data as well as planning of experiments and evaluation of experimental results. More than one year of practical work with this program has shown that BB RADABA provides a better survey over the data, moreover it is able to supply objective recommendations of suitable breeding pairs and had the advantage of graphic and statistic data presentation.     

Temperature and the Growth of Plant Cells

In cell elongation, the juvenile cell vacuolates, takes up water, and expands by irreversible extension of the growth-limiting primary walls. This process was elaborated analytically by Lockhart in the mid-1960s. His growth equation does not, however, include the influence of the environmental temperature at which cell growth takes place. In this article we consider a phenomenological model including temperature in the equation of growth. Also, by introducing the possible influence of growth regulators treated here as external perturbations, linear and nonlinear solutions are found. A comparison of experimental and theoretical results permits qualitative and quantitative conclusions concerning change in the magnitude of the cell wall yielding coefficient Φ as a function of both time and temperature (with or without external perturbations), which has acquired reasonable values throughout.     

Hierarchical analysis of large-scale two-dimensional gel electrophoresis experiments

Large-scale two-dimensional gel experiments have the potential to identify proteins that play an important role in elucidating cell mechanisms and in various stages of drug discovery. Such experiments, typically including hundreds or even thousands of related gels, are notoriously difficult to perform, and analysis of the gel images has until recently been virtually impossible. In this paper we describe a scalable computational model that permits the organization and analysis of a large gel collection. The model is implemented in Compugen's Z4000 system. Gels are organized in a hierarchical, multidimensional data structure that allow the user to view a large-scale experiment as a tree of numerous simpler experiments, and carry out the analysis one step at a time. Analyzed sets of gels form processing units that can be combined into higher level units in an iterative framework. The different conditions at the core of the experiment design, termed the dimensions of the experiment, are transformed from a multidimensional structure to a single hierarchy. The higher level comparison is performed with the aid of a synthetic "adaptor" gel image, called a Raw Master Gel (RMG). The RMG allows the inclusion of data from an entire set of gels to be presented as a gel image, thereby enabling the iterative process. Our model includes a flexible experimental design approach that allows the researcher to choose the condition to be analyzed a posteriori. It also enables data reuse, the performing of several different analysis designs on the same experimental data. The stability and reproducibility of a protein can be analyzed by tracking it up or down the hierarchical dimensions of the experiment.     

Convective paracellular solute flux. A source of ion-ion interaction in the epithelial transport equations

An electrolyte model of an epithelium (a cell and a tight junction in parallel, both in series with a lateral interspace basement membrane) is analyzed using the formalism of nonequilibrium thermodynamics. It is shown that if the parallel structures are heteroporous (i.e., reflection coefficients for two ion species differ between the components), then a cross-term will appear in the overall transport equations of the epithelium. Formally, this cross-term represents an ion-ion interaction. With respect to the rat proximal tubule, data indicating epithelial ionic reflection coefficients less than unity, together with the assumption of no transcellular solvent drag, imply the presence of convective paracellular solute flux. This means that a model applicable to a heteroporous structure must be used to represent the tubule, and, in particular, the cross-terms for ion-ion interaction must also be evaluated in permeability determinations. A series of calculations is presented that permits the estimation of the Na-Cl interaction for rat proximal tubule from available experimental data. One consequence of tubule heteroporosity is that an electrical potential may be substantially less effective than an equivalent concentration gradient in driving reabsorptive ion fluxes.     

Mathematical modeling of thrombus growth in mesenteric vessels

Richardson’s phenomenological mathematical model of the thrombi growth in microvessels is extended to describe the realistic features of the phenomenon. The main directions of the generalization of Richardson’s model are: (1) the dependence of platelet activation time on the distance from the injured vessel wall; (2) the non-homogeneity of the platelet distribution in blood flow in the vicinity of the vessel wall; (3) the adequate choice of the phenomenological function describing the dependence of blood velocity on the thrombus size. The generalization of the model corresponds to the main experimental results and theoretical considerations concerning thrombus formation obtained in recent years. The extended model permits to achieve qualitative agreement between model and experimental data.     

A comprehensive model of the crypts of the small intestine of the mouse provides insight into the mechanisms of cell migration and the proliferation hierarchy

A comprehensive model has been formulated for the proliferative behaviour of the crypts of the small intestine based on individual cell to cell relationships rather than on the average effects of all cells. The model accommodates a wide range of cell kinetic data and provides an insight into the mechanisms involved in cell movement within the columnar sheet of cells and into the relationship between the stem cells and their progeny. The model permits the number of stem cells and transit generations to be estimated. The number of stem cells is predicted to be not less than 4 and not more than 16 per crypt with cell cycle times of between 12 and 32 h respectively. Certain conclusions can be drawn concerning the mechanisms involved in the initial cell displacements after cell division. The model also allows an estimation of parameters which cannot be measured directly such as the degree of cell generation disorder and the amount of dispersion of cells within a cell lineage.     

A stochastic two-dimensional model of intercellular Ca2+ wave spread in glia

We describe a two-dimensional stochastic model of intercellular Ca(2+) wave (ICW) spread in glia that includes contributions of external stimuli, ionotropic and metabotropic P2 receptors, exo- and ecto-nucleotidases, second messengers, and gap junctions. In this model, an initial stimulus evokes ATP and UTP release from a single cell. Agonists diffuse and are degraded both in bulk solution and at cell surfaces. Ca(2+) elevation in individual cells is determined by bound agonist concentrations s and by number and features of P2 receptors summed with that generated by IP(3) diffusing through gap junction channels. Variability of ICWs is provided by randomly distributing a predetermined density of cells in a rectangular grid and by randomly selecting within intervals values characterizing the extracellular compartment, individual cells, and interconnections with neighboring cells. Variability intervals were obtained from experiments on astrocytoma cells transfected to express individual P2 receptors and/or the gap junction protein connexin43. The simulation program (available as Supplementary Material) permits individual alteration of ICW components, allowing comparison of simulations with data from cells expressing connexin43 and/or various P2 receptor subtypes. Such modeling is expected to be useful for testing phenomenological hypotheses and in understanding consequences of alteration of system components under experimental or pathological conditions.     

Mathematical models for optimizing tumour radiotherapy. I: A simple two compartment cell-kinetic model for the unperturbed growth of transplantable tumours

Tumour growth kinetics has been analysed on the basis of interactions between two compartments comprising the proliferating and non-proliferating cells. Starting from the differential equations of growth of the cell-populations in the two compartments and assuming the process of intercompartmental cell transfers to be linear, an analytic expression on the variation of growth-fraction with the age of the tumour is obtained. The restricted conditions on the cell-cycle time and cell-loss-rate, under which these differential equations lead to a Gompertzian growth of the tumour, are critically analysed. The formalism permits the estimation of some important cell-kinetic parameters, like growth-fraction or cell-loss-factor, from a knowledge of the tumour-growth curve, cell-cycle-time and a single measurement of the cell-loss-rate of the matured tumour, provided the tumour follows a Gompertzian growth. The validity of the model has been verified with the experimental data on 4 different transplantable murine tumour systems. Usefulness of the model has been demonstrated by making some interesting predictions regarding the radiation response of the tumours from the cell-kinetic parameters.     

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.