The Effect of the G 1 - S transition Checkpoint on an Age Structured Cell Cycle Model

Knowledge of how a population of cancerous cells progress through the cell cycle is vital if the population is to be treated effectively, as treatment outcome is dependent on the phase distributions of the population. Estimates on the phase distribution may be obtained experimentally however the errors present in these estimates may effect treatment efficacy and planning. If mathematical models are to be used to make accurate, quantitative predictions concerning treatments, whose efficacy is phase dependent, knowledge of the phase distribution is crucial. In this paper it is shown that two different transition rates at the - checkpoint provide a good fit to a growth curve obtained experimentally. However, the different transition functions predict a different phase distribution for the population, but both lying within the bounds of experimental error. Since treatment outcome is effected by the phase distribution of the population this difference may be critical in treatment planning. Using an age-structured population balance approach the cell cycle is modelled with particular emphasis on the - checkpoint. By considering the probability of cells transitioning at the - checkpoint, different transition functions are obtained. A suitable finite difference scheme for the numerical simulation of the model is derived and shown to be stable. The model is then fitted using the different probability transition functions to experimental data and the effects of the different probability transition functions on the model''s results are discussed.     

A Neurocomputational Model of the Mismatch Negativity

The mismatch negativity (MMN) is an event related potential evoked by violations of regularity. Here, we present a model of the underlying neuronal dynamics based upon the idea that auditory cortex continuously updates a generative model to predict its sensory inputs. The MMN is then modelled as the superposition of the electric fields evoked by neuronal activity reporting prediction errors. The process by which auditory cortex generates predictions and resolves prediction errors was simulated using generalised (Bayesian) filtering – a biologically plausible scheme for probabilistic inference on the hidden states of hierarchical dynamical models. The resulting scheme generates realistic MMN waveforms, explains the qualitative effects of deviant probability and magnitude on the MMN – in terms of latency and amplitude – and makes quantitative predictions about the interactions between deviant probability and magnitude. This work advances a formal understanding of the MMN and – more generally – illustrates the potential for developing computationally informed dynamic causal models of empirical electromagnetic responses.     

The duration of cell cycle phases studied using 5-bromodeoxycytidine

The quantitative method is suggested to estimate cell cycle phase durations and dispersions of progress through the phases for population of cells. The method is based on the analysis of frequency of cells with different staining of sister chromatids by means of 5-bromodeoxycytidine. The process of cell population progress is described by the Gauss probability integral. The durations of the cell cycle phases are determined for cell culture of Chinese hamster.     

Zeeman-Stark modeling of the RF EMF interaction with ligand binding

The influence of radiofrequency electromagnetic exposure on ligand binding to hydrophobic receptor proteins is a plausible early event of the interaction mechanism. A comprehensive quantum Zeeman-Stark model has been developed which takes into account the energy losses of the ligand ion due to its collisions inside the receptor crevice, the attracting nonlinear endogenous force due to the potential energy of the ion in the binding site, the out of equilibrium state of the ligand-receptor system due to the basal cell metabolism, and the thermal noise. The biophysical "output" is the change of the ligand binding probability that, in some instances, may be affected by a suitable low intensity exogenous electromagnetic "input" exposure, e.g., if the depth of the potential energy well of a putative receptor protein matches the energy of the radiofrequency photon. These results point toward both the possibility of the electromagnetic control of biochemical processes and the need for a new database of safety standards.     

Stochastic branching model for hemopoietic progenitor cell differentiation

We present algebraic expressions describing the predictions of a stochastic branching model for differentiation of hemopoietic progenitor cells. The model assumes that there is a fixed probability, p (0 less than or equal to p less than or equal to 1), that commitment to a differentiative event occurs per progenitor cell division for each daughter cell. The model describes properties of in vitro hemopoietic cell differentiation including the population structure at the time the first progenitor cell becomes committed, the number of committed progenitor cells engendered by a single progenitor cell, and the probability of eventual commitment of all daughter cells derived from a single progenitor or stem cell. Application of the model to experimental data obtained from erythroid cultures suggests that the observed data can be explained by the stochastic branching model alone without making the deterministic assumption that there is a differentiative hierarchy in the lineage of the progenitors of erythropoiesis (BFU-E). The qualitative and quantitative aspects of the proposed stochastic model are discussed in conjunction with other analogous stochastic branching models.     

A Markov formulation of the repair-misrepair model of cell survival

Tobias' repair-misrepair (RMR) model of cell survival is formulated as a Markov process, a sequence of discrete repair steps occurring at random times, and the probability of a sequence of viable repairs is calculated. The Markov formulation describes the time evolution of the probability distribution for the number of lesions in a cell. The probability of cell survival is calculated from the distribution of the initial number of lesions and the probabilities of the repair events. The production of lesions is formulated in accordance with the principles of microdosimetry, and the distribution of the initial number of lesions is obtained as an approximation for high and low linear energy transfer cases. The Markov formulation of the RMR model uses the same biological hypotheses as the original version with two statistical approximations deleted. These approximations are the neglect of the effect of statistical fluctuations in calculating the average rate of repair of lesions and the assumption that the final number of unrepaired and lethally misrepaired lesions has a Poisson distribution. The quantitative effect of these approximations is calculated, and a basis is provided for an alternative approach to calculating survival probabilities.     

Stochastic E2F Activation and Reconciliation of Phenomenological Cell-Cycle Models

The transition of the mammalian cell from quiescence to proliferation is a highly variable process. Over the last four decades, two lines of apparently contradictory, phenomenological models have been proposed to account for such temporal variability. These include various forms of the transition probability (TP) model and the growth control (GC) model, which lack mechanistic details. The GC model was further proposed as an alternative explanation for the concept of the restriction point, which we recently demonstrated as being controlled by a bistable Rb-E2F switch. Here, through a combination of modeling and experiments, we show that these different lines of models in essence reflect different aspects of stochastic dynamics in cell cycle entry. In particular, we show that the variable activation of E2F can be described by stochastic activation of the bistable Rb-E2F switch, which in turn may account for the temporal variability in cell cycle entry. Moreover, we show that temporal dynamics of E2F activation can be recast into the frameworks of both the TP model and the GC model via parameter mapping. This mapping suggests that the two lines of phenomenological models can be reconciled through the stochastic dynamics of the Rb-E2F switch. It also suggests a potential utility of the TP or GC models in defining concise, quantitative phenotypes of cell physiology. This may have implications in classifying cell types or states.     

Electrosensibility and electromagnetic hypersensitivity

Electromagnetic sensibility, the ability to perceive electric and electromagnetic exposure, and electromagnetic hypersensitivity (EHS), developing health symptoms due to exposure to environmental electromagnetic fields, need to be distinguished. Increased electrosensibility is a necessary, however, not a sufficient condition for electromagnetic hypersensitivity. At an extended sample of the general population of 708 adults, including 349 men and 359 women aged between 17 and 60 years, electrosensibility was investigated and characterized by perception threshold and its standard deviation. By analyzing the probability distributions of the perception threshold of electric 50 Hz currents, evidence could be found for the existence of a subgroup of people with significantly increased electrosensibility (hypersensibility) who as a group could be differentiated from the general population. The presented data show that the variation of the electrosensibility among the general population is significantly larger than has yet been estimated by nonionizing radiation protection bodies, but much smaller than claimed by hypersensitivity self-aid groups. These quantitative results should contribute to a less emotional discussion of this problem. The investigation method presented, is capable of exclusion diagnostics for persons suffering from the hypersensitivity syndrome.     

Efficiency of organelle capture by microtubules as a function of centrosome nucleation capacity: general theory and the special case of polyspermia

Transport of organelles along microtubules is essential for the cell metabolism and morphogenesis. The presented analysis derives the probability that an organelle of a given size comes in contact with the microtubule aster. The question is asked how this measure of functionality of the microtubule aster is controlled by the centrosome. A quantitative model is developed to address this question. It is shown that for the given set of cellular parameters, such as size and total tubulin content, a centrosome nucleation capacity exists that maximizes the probability of the organelle capture. The developed general model is then applied to the capture of the female pronucleus by microtubules assembled on the sperm centrosome, following physiologically polyspermic fertilization. This application highlights an unintuitive reflection of nonlinearity of the nucleated polymerization of the cellular pool of tubulin. The prediction that the sperm centrosome should lower its nucleation capacity in the face of the competition from the other sperm is a stark illustration of the new optimality principle. Overall, the model calls attention to the capabilities of the centrosomal pathway of regulation of the transport-related functionality of the microtubule cytoskeleton. It establishes a quantitative and conceptual framework that can guide experiment design and interpretation.     

A deterministic model of the cell cycle

The variability of the duration of the cell cycle is explained by the phenomenon of sensitive dependence upon initial conditions; as may occur in deterministic non-linear systems. Chaotic dynamics of a system is the result of this sensitive dependence. First a deterministic system is formulated that is equivalent to the Smith-Martin transition probability model of the cell cycle. Next the model is extended to a dynamic process that ranges over the cell generations. A deterministic non-linear relationship between the cycle time of the mother and daughter cell is established. It clarifies the variability of mother-daughter correlation for the different cell types. The model is fitted to two different cell cultures; it shows that the graph of the non-linear relation has the same shape for different cell types.     

Conditional probability analysis for a domain model of Ca(2+)-inactivation of Ca2+ channels.

The domain model of Ca2+ inactivation of Ca2+ channels, which has been used to explain rapid inactivation of whole cell Ca2+ currents in pancreatic beta cells, is applied to single-time and conditional open probability measurements on guinea pig ventricular myocyte Ca2+ channels. These two measurements greatly constrain the choice of kinetic constants in the model. Calculations with the model provide a simple quantitative explanation of recent experimental results, including a slow increase in the inactivation rate.     

关于质体随细胞分裂传递的数学模型

本文根据质体是具有遗传功能的细胞器这一事实出发,对质体传递规律首次进行了定量研究,建立了突变质体在细胞分裂中传递的全概率和条件概率公式．并对此公式的应用作了进一步的探讨．本文所提供的方法将为质体遗传开辟一条定量研究的新路．     

Commitment to erythroid differentiation by friend erythroleukemia cells: a stochastic analysis

A method for the clonal analysis of murine erythroleukemia cells has been developed which allows the precise characterization of the number of progeny produced by each cell and the degree of differentiation of each progeny cell. The potential of almost every cell in the culture can be monitored because a plating efficiency close to 100% has been achieved. The effects of treatment with an inducer of differentiation (DMSO) on the proliferative capacity of the treated cells have been studied with this technique. Cells from a mass culture treated with inducer give rise to colonies of differentiated progeny when subsequently cloned in the absence of inducer. Colonies exhibiting this phenotype represent the progeny of cells committed to the differentiation pathway by treatment with inducer. We observe that the commitment decision limits the subsequent proliferative capacity of the cell to four additional cell divisions. A quantitative analysis suggests that the commitment decision for each cell is made in a stochastic manner. Irreversible commitment to the expression of differentiated functions occurs with discrete probability per cell generation for many cell generations. The value for this probability is a function of the concentration of inducer (DMSO). A correlative biochemical study suggests that an irreversible commitment decision by a significant proportion of the population precedes or accompanies increases in cytoplasmic globin mRNA levels, one of the earliest detectable biochemical markers for erythroid differentiation in this system.A specific kinetic model based on these considerations has been developed to predict clonal phenotypes as a function of time and probability of commitment. Quantitative predictions based on this model are in excellent agreement with experimental observations. The effectiveness of a stochastic model in predicting the behavior of this system is discussed in relation to the stochastic behavior of normal hematopoiesis and the biochemical mechanisms which control these differentiation programs.     

Can promotion of initiated cells be explained by excess replacement of radiation-inactivated neighbor cells?

Recently, the observed promotion in the clonal expansion of a two-stage cancer model was attributed to a small excess replacement probability for the initiated cells. The proposed mechanism of excess replacement was evaluated for single intermediate cells surrounded by normal cells. This paper investigates this mechanism further using the same biological parameters. If the formation of clones of intermediate cells is taken into account in a quantitative analysis of the proposed mechanism, it turns out that (1) for the initial strong increase of the promotional effect with exposure, a much larger and unlikely excess replacement probability is needed, and (2) the leveling of the promotional effect for high exposures cannot be explained by multiple normal neighbors of an intermediate cell being inactivated within one cell cycle, as it had been suggested. Perhaps these discrepancies could be partly resolved by a re-scaling of the original parameters, but this should be investigated further.     

Quantitativein situ hybridization to measure single-cell changes in vasopressin and oxytocin mRNA levels after osmotic stimulation

1. The measurement of cellular mRNA content by quantitative in situ hybridization is a valuable approach to the study of gene expression in brain since this tissue exhibits a high degree of phenotypic heterogeneity. 2. The cellular content of vasopressin and oxytocin mRNA in hypothalamo-neurohypophysial system neurons was altered by maintaining rats for 24 hr on 2% sodium chloride water. 3. Statistical and graphical techniques were then used to analyze cell by cell how mRNA levels were altered as a result of osmotic stimulation. We propose that the negative binomial probability distribution is a suitable model to describe how mRNA content varies across a defined cell population. For both measures of oxytocin and vasopressin mRNA levels, maximum-likelihood estimation indicated that this model adequately described empirical findings obtained from rats drinking tap water or salt water. 4. Both graphical and statistical analyses suggested how the defined neural system responds to osmotic stimulation: mRNA content was altered as a multiplicative function of "initial state." The utility and limitations of the quantitative approach are discussed.     

Stochastic kinetics of intracellular huntingtin aggregate formation

Neurodegeneration in Huntington disease is described by neuronal loss in which the probability of cell death remains constant with time. However, the quantitative connection between the kinetics of cell death and the molecular mechanism initiating neurodegeneration remains unclear. One hypothesis is that nucleation of protein aggregates containing exon I fragments of the mutant huntingtin protein (mhttex1), which contains an expanded polyglutamine region in patients with the disease, is the explanation for the infrequent but steady occurrence of neuronal death, resulting in adult onset of the disease. Recent in vitro evidence suggests that sufficiently long polyglutamine peptides undergo a unimolecular conformational change to form a nucleus that seeds aggregation. Here we use this nucleation mechanism as the basis to derive a stochastic mathematical model describing the probability of aggregate formation in cells as a function of time and mhttex1 protein concentration, and validate the model experimentally. These findings suggest that therapeutic strategies for Huntington disease predicated on reducing the rate of mhttex1 aggregation need only make modest reductions in huntingtin expression level to substantially increase the delay time until aggregate formation.     

Spatiotemporal dynamics of the biological interface between cancer and the microenvironment: a fractal anomalous diffusion model with microenvironment plasticity

ABSTRACT: BACKGROUND: The invasion-metastasis cascade of cancer involves a process of parallel progression. A biological interface (module) in which cells is linked with ECM (extracellular matrix) by CAMs (cell adhesion molecules) has been proposed as a tool for tracing cancer spatiotemporal dynamics. METHODS: A mathematical model was established to simulate cancer cell migration. Human uterine leiomyoma specimens, in vitro cell migration assay, quantitative real-time PCR, western blotting, dynamic viscosity, and an in vivo C57BL6 mouse model were used to verify the predictive findings of our model. RESULTS: The return to origin probability (RTOP) and its related CAM expression ratio in tumors gradually decreased with increased tumor size, and approached the 3D Polya constant (0.340537) in a periodic structure. The biphasic pattern of cancer cell migration revealed that cancer cells initially grew together and subsequently began spreading. A higher viscosity of fillers applied to the cancer surface was associated with a significantly greater inhibitory effect on cancer migration, in accordance with the Stokes-Einstein equation. CONCLUSION: The positional probability and cell-CAM-ECM interface (module) in the fractal framework helped us decipher cancer spatiotemporal dynamics; in addition we modeled the methods of cancer control by manipulating the microenvironment plasticity or inhibiting the CAM expression to the Polya constant.     

Tracking a fluctuating environment: a study of sampling

The problem of how animals keep track of unpredictable changes in the profitability of foraging sites was studied. An optimality model was used to predict the frequency with which a forager should sample a foraging site in which the probability of reward fluctuates randomly between high and low. The alternative foraging site is stable and offers an intermediate probability of reward. The model was tested with pigeons in a shuttle-box the two ends of which represented the two foraging sites. The pigeons succeeded in tracking the changes in the fluctuating site and the payoff attained was close to the optimum. Variations in the frequency of sampling between experimental treatments were in qualitative agreement with the model for some treatments but not others. The quantitative details of sampling behaviour were not as predicted by the optimality model, but many features could be accounted for by a mechanistic model of choice. The pigeons' choice rule, although different from that of the optimality model, is one that produces near-optimal payoffs under the conditions of this experiment.