Selection of optimal model for the DNA histogram by analysis of error of estimated parameters.

The ability of four different mathematical models of the DNA histogram to give accurate estimates for the fractions of cells in G1, S, and G2 + M has been investigated. The models studied differ in the form and number of parameters of the function used to represent cells in S-phase. Results obtained from simulated DNA histograms suggest that the standard deviations of the model parameters increase exponentially with the width of the G1 and G2 + M peaks of the histogram. Error analysis is presented as a method to select a model of optimal complexity in relation to the resolution provided by the data in a given set of DNA histograms. Introduction of additional parameters improves the agreement between model and data but may result in a less well-posed model. A model with an optimal number of parameters can therefore be found that will yield parameter estimates with the smallest possible standard deviations.     

Microsatellite mutations during the polymerase chain reaction: mean field approximations and their applications

We develop a novel mathematical model for microsatellite mutations during polymerase chain reaction (PCR). Based on the model, we study the first- and second-order moments of the number of repeat units in a randomly chosen molecule after n PCR cycles and their corresponding mean field approximations. We give upper bounds for the approximation errors and show that the approximation errors are small when the mutation rate is low. Based on the theoretical results, we develop a moment estimation method to estimate the mutation rate per-repeat-unit per PCR cycle and the probability of expansion when mutations occur. Simulation studies show that the moment estimation method can accurately recover the true mutation rate and probability of expansion. Finally, the method is applied to experimental data from single-molecule PCR experiments.     

A pragmatic approach to the analysis of DNA histograms with a definable G1 peak

A method for DNA histogram analysis is described that depends only on the simple assumption that the data are normally distributed and a requirement that a G1 peak is present. A probability density function was derived from the assumption that extracted the S-phase component from the whole histogram. The model was tested with simulated data, and good agreement between predicted and known proportions in G1, S, and G1 + M was found. Good agreement was also found between duplicates of experimentally derived data. Some systematic errors are present in the analysis of certain types of histograms. However, these result in small errors when compared with biological and experimental variation and are less than the average of variation and are less than the average of algorithms in current use. The program required only two queued requests, those of the start and the end channels over which the analysis is to be performed. The algorithms perform rapidly on a microcomputer with only 28K addressable memory. Only two failures occurred in over 350 analyses and the method can be used for drug- and radiation-perturbed populations as well as with unperturbed.     

Multi-type Galton-Watson process as a model for proliferating human tumour cell populations derived from stem cells: estimation of stem cell self-renewal probabilities in human ovarian carcinomas

A mathematical model for proliferation of tumour cell populations is developed. The cell population is assumed to be organized in a hierarchy of decreasing proliferative potential and increasing degree of differentiation. Using some elements of the theory of Multi-type Galton-Watson processes, a method is proposed for the estimation of Psr, the probability of self-renewal of tumour stem cells, from the experimental distribution of clonal unit sizes obtained in cell culture studies. Six data sets from patients with advanced adenocarcinoma of the ovary are used to demonstrate the method. Reasonable estimates are obtained, and the theoretical colony size distributions predicted by the model appear to be in good qualitative agreement with the experimental ones, and lend support to a stem cell model of tumour growth. The possible significance of Psr as a prognostic factor is briefly discussed.     

Evaluation of two unstructured mathematical models for the penicillin G fedbatch fermentation

The mathematical model for the penicillin G fed-batch fermentation proposed by Heijnen et al. (1979) is compared with the model of Bajpai & Reuß (1980). Although the general structure of these models is similar, the difference in metabolic assumptions and specific growth and production kinetics results in a completely different behaviour towards product optimization. A detailed analysis of both models reveals some physical and biochemical shortcomings. It is shown that it is impossible to make a reliable estimation of the model parameters, only using experimental data of simple constant glucose feed rate fermentations with low initial substrate amount. However, it is demonstrated that some model parameters might be key factors in concluding whether or not altering the substrate feeding strategy has an important influence on the final amount of product.It is illustrated that feeding strategy optimization studies can be a tool in designing experiments for parameter estimation purposes.     

Multi-Type Galton-Watson Process As A Model For Proliferating Human Tumour Cell Populations Derived From Stem Cells: Estimation of Stem Cell Self-Renewal Probabilities In Human Ovarian Carcinomas

Abstract. A mathematical model for proliferation of tumour cell populations is developed. the cell population is assumed to be organized in a hierarchy of decreasing proliferative potential and increasing degree of differentiation. Using some elements of the theory of Multi-type Galton-Watson processes, a method is proposed for the estimation of P_sr, the probability of self-renewal of tumour stem cells, from the experimental distribution of clonal unit sizes obtained in cell culture studies. Six data sets from patients with advanced adenocarcinorna of the ovary are used to demonstrate the method. Reasonable estimates are obtained, and the theoretical colony size distributions predicted by the model appear to be in good qualitative agreement with the experimental ones, and lend support to a stem cell model of tumour growth. the possible significance of P_sr as a prognostic factor is briefly discussed.     

Proof without prejudice revisited: immunofluorescence histogram analysis using cumulative frequency subtraction plus ratio analysis of means

BACKGROUND: Apart from the work of Lampariello and colleagues (Cytometry 15:294-301, 1994; Cytometry 32:241-254, 1998), very little analytical work has been carried out for analysis of immunofluorescence distributions containing an overlapping mixture of labeled and unlabeled cells. The methods developed tend to rely on fitting theoretical distributions to the relevant populations. However, the method described here attempts to produce an analytical solution. METHODS: A new method for immunofluorescence histogram analysis is presented. It uses cumulative frequency distribution subtraction of the test sample from the control to predict the mean of a labeled cell component embedded within a histogram containing unlabeled cells. Ratio analysis of means (RAM) was then carried out to calculate the labeled fraction. The results were submitted to Kolmogorov-Smirnov analysis and Student's t-test for validation at a given level of probability. RESULTS: The method was developed with a data set exhibiting a small "positive" shoulder, which was predicted to contain a labeled fraction comprising 8.0% of the total at the 99% confidence limit. It was then tested with data analyzed and published previously where the Johnson Su family of distributions was used in curve fitting. CONCLUSIONS: There was good agreement between the known and predicted proportions of labeled cells. However, the method is dependent on the symmetry of the distributions. Some minor systematic errors were encountered due, in part, to skewed experimental distributions.     

Bayesian lasso for semiparametric structural equation models

There has been great interest in developing nonlinear structural equation models and associated statistical inference procedures, including estimation and model selection methods. In this paper a general semiparametric structural equation model (SSEM) is developed in which the structural equation is composed of nonparametric functions of exogenous latent variables and fixed covariates on a set of latent endogenous variables. A basis representation is used to approximate these nonparametric functions in the structural equation and the Bayesian Lasso method coupled with a Markov Chain Monte Carlo (MCMC) algorithm is used for simultaneous estimation and model selection. The proposed method is illustrated using a simulation study and data from the Affective Dynamics and Individual Differences (ADID) study. Results demonstrate that our method can accurately estimate the unknown parameters and correctly identify the true underlying model.     

Validation-based model selection for 13C metabolic flux analysis with uncertain measurement errors

Accurate measurements of metabolic fluxes in living cells are central to metabolism research and metabolic engineering. The gold standard method is model-based metabolic flux analysis (MFA), where fluxes are estimated indirectly from mass isotopomer data with the use of a mathematical model of the metabolic network. A critical step in MFA is model selection: choosing what compartments, metabolites, and reactions to include in the metabolic network model. Model selection is often done informally during the modelling process, based on the same data that is used for model fitting (estimation data). This can lead to either overly complex models (overfitting) or too simple ones (underfitting), in both cases resulting in poor flux estimates. Here, we propose a method for model selection based on independent validation data. We demonstrate in simulation studies that this method consistently chooses the correct model in a way that is independent on errors in measurement uncertainty. This independence is beneficial, since estimating the true magnitude of these errors can be difficult. In contrast, commonly used model selection methods based on the χ²-test choose different model structures depending on the believed measurement uncertainty; this can lead to errors in flux estimates, especially when the magnitude of the error is substantially off. We present a new approach for quantification of prediction uncertainty of mass isotopomer distributions in other labelling experiments, to check for problems with too much or too little novelty in the validation data. Finally, in an isotope tracing study on human mammary epithelial cells, the validation-based model selection method identified pyruvate carboxylase as a key model component. Our results argue that validation-based model selection should be an integral part of MFA model development.     

Calculating sugar yields in high solids hydrolysis of biomass

Calculation of true sugar yields in high solids enzymatic hydrolysis of biomass is challenging due to the varying liquid density and liquid volume resulting from solid solubilization. Ignoring these changes in yield calculations can lead to significant errors. In this paper, a mathematical method was developed for the estimation of liquid volume change and thereafter the sugar yield. The information needed in the calculations include the compositions of the substrate, initial solids loading, initial liquid density, and sugar concentrations before and after hydrolysis. All of these variables are measurable with conventional laboratory procedures. This method was validated experimentally for enzymatic hydrolysis of dilute sulfuric acid pretreated corn stover at solid loadings up to 23% (w/w). The maximum relative error of predicted glucose yield from the true value was less than 4%. Compared to other methods reported in the literature, this method is relatively easy to use and provides good accuracy.     

Growth and survival of Pseudomonas putida strains degrading naphthalene in soil model systems with different moisture levels

The effect of different moisture levels (from 20 to 70%) on the growth and survival of Pseudomonas putida strains G7 and BS3701 degrading naphthalene was studied in soil model systems. P. putida G7 contains plasmid NAH7 and P. putida BS3701 harbours plasmids pBS1141 and pBS1142. A mathematical model is proposed to describe the observed dynamics of the number of viable bacterial cells. Naphthalene and soil organic matter were considered as substrates available to bacteria. Data fitting allowed the estimation of model parameters characterizing microbial growth rate, utilization rate of substrates, specific maintenance rate and yield coefficient. Both the maximum bacterial concentration and the highest yield coefficient were observed at a soil moisture level of 40%. This optimal moisture level is close to but less than the water capacity (48%) of the soil used.     

Experimental evaluation of a model for cometabolism: Prediction of simultaneous degradation of trichloroethylene and methane by a methanotrophic mixed culture

A model for cometabolism is verified experimentally for a defined methanotrophic mixed culture. The model includes the effects of cell growth, endogenous cell decay, product toxicity, and competitive inhibition with the assumption that cometabolic transformation rates are enhanced by reducing power obtained from oxidation of growth substrates. A theoretical transformation yield is used to quantify the enhancement resulting from growth substrate oxidation. A systematic method for evaluating model parameters independently is described. The applicability of the model is evaluated by comparing experimental data for methanotrophic cometabolism of TCE with model predictions from independently measured model parameters. Propagation of errors is used to quantify errors in parameter estimates and in the final prediction. The model successfully predicts TCE transformation and methane utilization for a wide range of concentrations of TCE (0.5 to 9 mg/L) and methane (0.05 to 6 mg/L). (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 492-501, 1997.     

Transition between two excitabilities in mesencephalic V neurons

Neurons can make different responses to identical inputs. According to the emerging frequency of repetitive firing, neurons are classified into two types: type 1 and type 2 excitability. Though in mathematical simulations, minor modifications of parameters describing ionic currents can result in transitions between these two excitabilities, empirical evidence to support these theoretical possibilities is scarce. Here we report a joint theoretical and experimental study to test the hypothesis that changes in parameters describing ionic currents cause predictable transitions between the two excitabilities in mesencephalic V (Mes V) neurons. We developed a simple mathematical model of Mes V neurons. Using bifurcation analysis and model simulation, we then predicted that changes in conductance of two low-threshold currents would result in transitions between type 1 and type 2. Finally, by applying specific channel blockers, we observed the transition between two excitabilities forecast by the mathematical model.     

Microfluorometry analysis

A mathematical model for decomposing an FMF histogram into its G₁S and G₂ + M components is developed. Under certain restrictions, the model applies to both asynchronous and synchronous populations. Two numerical techniques for estimating the percentage of cells in each component are outlined. Using the assumption of exponential growth, theoretical expressions for the percentage of cells in each state and for the S density are derived. This leads to a rapid method for determining the mean time spent in each state by a cell.     

Investigation into the experimental kinetic support of the two-state model of the cell cycle

Five previously published cell generation-time distribution functions have been examined in an effort to elucidate the parameters of the two-state model of the cell cycle. These parameters are the fractional number of cells that bypass the G0 state, the probability of exit from G0, and the distribution of traversal times through the active state. To explain observed beta-curve behavior of cell populations, it is necessary to define the parameters in terms of pairwise behavior of newborn sister cells. From the beta-curve, we demonstrate that at least 50% of the cells must pass through the G0 state. The alpha-curve is consistent with any positive fraction of newborn cells passing through the G0 state, and provides no further information. We explore a possible method for resolving the remaining indeterminacy regarding the number of cells bypassing the G0 state, namely, examination of the generation-time distribution functions of fast sister cells only. Such an approach, although theoretically attractive, presents formidable experimental difficulties, however. If it should turn out that indeed only 50% of the cells are apparently passing through a random-exiting phase of the cell cycle, then an alterative plausible biological mechanism for the observed variability in generation times is supplied by Prescott's hypothesis: variability is a consequence of the inequality in the metabolic content of sister cells at birth.     

Error analysis in flow microfluorometry

Sources of error in a typical algorithm for the analysis of single flow-microfluorometric histograms are identified. A new statistical model for such data is presented, by means of which the error sources are quantitatively investigated. These theoretical investigations lead to three practical observations: A more detailed characterization of the fluorescence dispersion process is needed for a more refined algorithm. Levels of dispersion typically experienced are such that from a single histogram the distribution of cells within S-phase cannot be finely resolved; but the crude distribution of cells among the three phases G1, S, and G2-M may be accurately estimated. If currently typical levels of dispersion can be halved, then the S-phase distribution can be finely resolved.     

A Bayesian approach to functional-based multilevel modeling of longitudinal data: applications to environmental epidemiology

Flexible multilevel models are proposed to allow for cluster-specific smooth estimation of growth curves in a mixed-effects modeling format that includes subject-specific random effects on the growth parameters. Attention is then focused on models that examine between-cluster comparisons of the effects of an ecologic covariate of interest (e.g. air pollution) on nonlinear functionals of growth curves (e.g. maximum rate of growth). A Gibbs sampling approach is used to get posterior mean estimates of nonlinear functionals along with their uncertainty estimates. A second-stage ecologic random-effects model is used to examine the association between a covariate of interest (e.g. air pollution) and the nonlinear functionals. A unified estimation procedure is presented along with its computational and theoretical details. The models are motivated by, and illustrated with, lung function and air pollution data from the Southern California Children's Health Study.     

Application of Statistical Classification Technique to the Forecasting of Outcomes of Diseases

A probabilistic statistical model based on statistical classification is proposed for disease outcome forecasting. In particular, cardiac infarction outcome is forecasted and a table of the forecasting results is given. A conclusion of theoretical character is made about the preferable use of a non-uniform band width in the construction of a general histogram for unknown distribution density estimation.     

Cell-cycle analysis of perturbed cell populations: computer simulation of sequential DNA distributions.

A mathematical model is presented that permits simulation of a time sequence of DNA distributions with a single set of cell-cycle parameters. The method is particularly suited to the quantitative analysis of sets of sequential DNA distributions from perturbed cell populations. The model permits determination of the durations and associated dispersions of the phases of the cell cycle as well as the point in the cell cycle at which the perturbing agent exerts its effect. The mathematical details of the simulation technique are presented, and the technique is applied to the analysis of DNA distributions from perturbed cell populations. Three cell populations are modeled: CHO-line cells released from a block at the interface of the G1-and S-phases, 3T3 cells released from a G1-phase block produced by serum starvation, and S49 mouse lymphoma cells responding to a block in the G1-phage produced by N6,02'-dibutyryl adenosine 3':5'-cyclic monophosphate (Bt2cAMP).