Simplified normal mode analysis of conformational transitions in DNA-dependent polymerases: the elastic network model

The Elastic Network Model is used to investigate the open/closed transition in all DNA-dependent polymerases whose structure is known in both forms. For each structure the model accounts well for experimental crystallographic B-factors. It is found in all cases that the transition can be well described with just a handful of the normal modes. Usually, only the lowest and/or the second lowest frequency normal modes deduced from the open form give rise to calculated displacement vectors that have a correlation coefficient larger than 0.50 with the observed difference vectors between the two forms. This is true for every structural class of DNA-dependent polymerases where a direct comparison with experimental structural data is available. In cases where only one form has been observed by X-ray crystallography, it is possible to make predictions concerning the possible existence of another form in solution by carefully examining the vector displacements predicted for the lowest frequency normal modes. This simple model, which has the advantage to be computationally inexpensive, could be used to design novel kind of drugs directed against polymerases, namely drugs preventing the open/closed transition from occurring in bacterial or viral DNA-dependent polymerases.     

Predicting the QRS complex and detecting small changes using principal component analysis.

In this paper, a new method for QRS complex analysis and estimation based on principal component analysis (PCA) and polynomial fitting techniques is presented. Multi-channel ECG signals were recorded and QRS complexes were obtained from every channel and aligned perfectly in matrices. For every channel, the covariance matrix was calculated from the QRS complex data matrix of many heartbeats. Then the corresponding eigenvectors and eigenvalues were calculated and reconstruction parameter vectors were computed by expansion of every beat in terms of the principal eigenvectors. These parameter vectors show short-term fluctuations that have to be discriminated from abrupt changes or long-term trends that might indicate diseases. For this purpose, first-order poly-fit methods were applied to the elements of the reconstruction parameter vectors. In healthy volunteers, subsequent QRS complexes were estimated by calculating the corresponding reconstruction parameter vectors derived from these functions. The similarity, absolute error and RMS error between the original and predicted QRS complexes were measured. Based on this work, thresholds can be defined for changes in the parameter vectors that indicate diseases.     

Modelling the flow of cytometric data obtained from unperturbed human tumour cell lines: Parameter fitting and comparison

In this paper we firstly present three alternative formulations of a mathematical model for human tumour cell lines unperturbed by cancer therapy. The model counts the number density of cells in each phase of the cell cycle over time where cells are differentiated by their DNA content. Data are available from the Auckland Cancer Society Research Centre, Auckland, New Zealand, in the form of DNA histograms or profiles from 11 different human tumour cell lines (i.e. in vitro) unperturbed by cancer therapy. We then apply one (computationally fast) formulation of the model and discover that although in general different combinations of parameter values give rise to very different DNA profiles it is possible that different combinations of parameter values give rise to virtually identical profiles. Experimental estimates of the rate of transition from the G ₁-phase (growth) to the S-phase (DNA synthesis) enable us to uniquely determine other model parameters of interest that give the least square error between the model and data. We finally apply our model to each of the 11 different cell lines and compare cell cycle phase transit times. Although the DNA histograms of each of the cell lines have similar shapes these cell lines have different combinations of transit times to each other, which could explain why they often react very differently when exposed to anti-cancer therapies during laboratory experiments. An understanding of the in vitro situation may give an insight into why some human cancer patients do not respond to cancer therapy. An erratum to this article is available at .     

An evolutionary computing approach for parameter estimation investigation of a model for cholera

We consider the problem of using time-series data to inform a corresponding deterministic model and introduce the concept of genetic algorithms (GA) as a tool for parameter estimation, providing instructions for an implementation of the method that does not require access to special toolboxes or software. We give as an example a model for cholera, a disease for which there is much mechanistic uncertainty in the literature. We use GA to find parameter sets using available time-series data from the introduction of cholera in Haiti and we discuss the value of comparing multiple parameter sets with similar performances in describing the data.     

An ensemble of models of the acute inflammatory response to bacterial lipopolysaccharide in rats: results from parameter space reduction

In previous work, we developed an 8-state nonlinear dynamic model of the acute inflammatory response, including activated phagocytic cells, pro- and anti-inflammatory cytokines, and tissue damage, and calibrated it to data on cytokines from endotoxemic rats. In the interest of parsimony, the present work employed parametric sensitivity and local identifiability analysis to establish a core set of parameters predominantly responsible for variability in model solutions. Parameter optimization, facilitated by varying only those parameters belonging to this core set, was used to identify an ensemble of parameter vectors, each representing an acceptable local optimum in terms of fit to experimental data. Individual models within this ensemble, characterized by their different parameter values, showed similar cytokine but diverse tissue damage behavior. A cluster analysis of the ensemble of models showed the existence of a continuum of acceptable models, characterized by compensatory mechanisms and parameter changes. We calculated the direct correlations between the core set of model parameters and identified three mechanisms responsible for the conversion of the diverse damage time courses to similar cytokine behavior in these models. Given that tissue damage level could be an indicator of the likelihood of mortality, our findings suggest that similar cytokine dynamics could be associated with very different mortality outcomes, depending on the balance of certain inflammatory elements.     

Transmission dynamics of lymphatic filariasis: density-dependence in the uptake of Wuchereria bancrofti microfilariae by vector mosquitoes

Gaining a better understanding of parasite infection dynamics in the vector mosquito (Diptera: Culicidae) population is central to improving knowledge regarding the transmission, persistence and hence control of lymphatic filariasis. Here, we use data on mosquito feeding experiments collated from the published literature to examine the available evidence regarding the functional form of the first component of this parasite-vector relationship for Wuchereria bancrofti (Filarioidea: Onchocercidae) causing Bancroftian filariasis, i.e. the rate of microfilariae (mf) uptake from the blood of infected humans by the feeding mosquito vector. Using a simple logarithmic regression model for describing the observed relationships between the mean numbers of mf ingested per mosquito and parasite load in humans in each study, and a linear mixed-effects meta-analytical framework for synthesizing the observed regressions across studies, we show here for the first time clear evidence for the existence of density-dependence in this process for all the three major filariasis transmitting mosquito vectors. An important finding of this study is that this regulation of mf uptake also varies significantly between the vector genera, being weakest in Culex, comparatively stronger in Aedes and most severe and occurring at significantly lower human mf loads in Anopheles mosquitoes. The analysis of the corresponding mf uptake prevalence data has further highlighted how density-dependence in mf uptake may influence the observed distributions of mf in vector populations. These results show that whereas strong regulation of mf uptake, especially when it leads to saturation in uptake at low human parasite intensities, can lead to static distributions of mf per mosquito with host parasite intensity, a weaker regulation of mf ingestion can give rise to changes in both mean mf loads and in the frequency distribution of parasites/mosquito with increasing human parasite intensity. These findings highlight the importance of considering local vector infection dynamics when attempting to predict the impacts of community-based filariasis control. They also emphasize the value of developing and applying robust meta-analytic methods for estimating functional relationships regarding parasitic infection from population ecological data.     

The dead space in a compartmental lung model

A compartmental lung model with any number of synchronously filling and emptying functional chambers and a common dead space or conducting region is considered. It is shown that the model gives rise to an output, in an open circuit washout determination, which is a weighted sum of exponentials. From estimates of these weights and exponential components, estimates of the model parameters can be recovered. Relations giving the unique correspondence between the output parameters and the model parameters are derived and the existence and uniqueness of solutions established.     

Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant tau is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. The computational methods described here are a efficient alternative to other well known estimation techniques for leaky integrate-and-fire models. Moreover, it could serve as a template for performing parameter inference on more complex integrate-and-fire neuronal models.     

Somite-derived cells replace ventral aortic hemangioblasts and provide aortic smooth muscle cells of the trunk

We have previously shown that endothelial cells of the aortic floor give rise to hematopoietic cells, revealing the existence of an aortic hemangioblast. It has been proposed that the restriction of hematopoiesis to the aortic floor is based on the existence of two different and complementary endothelial lineages that form the vessel: one originating from the somite would contribute to the roof and sides, another from the splanchnopleura would contribute to the floor. Using quail/chick orthotopic transplantations of paraxial mesoderm, we have traced the distribution of somite-derived endothelial cells during aortic hematopoiesis. We show that the aortic endothelium undergoes two successive waves of remodeling by somitic cells: one when the aortae are still paired, during which the initial roof and sides of the vessels are renewed; and a second, associated to aortic hematopoiesis, in which the hemogenic floor is replaced by somite endothelial cells. This floor thus appears as a temporary structure, spent out and replaced. In addition, the somite contributes to smooth muscle cells of the aorta. In vivo lineage tracing experiments with non-replicative retroviral vectors showed that endothelial cells do not give rise to smooth muscle cells. However, in vitro, purified endothelial cells acquire smooth muscle cells characteristics. Taken together, these data point to the crucial role of the somite in shaping the aorta and also give an explanation for the short life of aortic hematopoiesis.     

A model of tumor growth based on cell cycle kinetics

This work describes a mathematical model of growth based on the kinetics of the cell cycle. A traditional model of the cell cycle has been used, with the addition of a resting (G₀) state from which cells could reenter the reproductive cycle. The model assumes that a growth regulatory substance regulates the transition of cells to and from the resting state. Other transitions between the phases of the cycle were modeled as a first order process. Cell loss is an important feature of growth kinetics, and has been represented by a general but tractable mathematical form. The resulting model forms a system of ordinary nonlinear differential equations. Analytic methods are employed first in the study of this system. Simplifying assumptions regarding cell loss give rise to special cases for which equilibrium solutions can be found. One special case, which assumes first order loss from all cell cycle phases at equal rates, is presented here. For small time values, approximations corresponding to exponential growth were developed. The equations describing an intrinsic growth rate were derived. Simulation methods were used to further characterize the behavior of this model. Parameter values were chosen based on animal tumor cell cycle kinetic data, resulting in a set of 45 model simulations. Several tumor treatment protocols were simulated which illustrated the importance of the intrinsic growth rate and cell loss concepts. Although the qualitative behavior regarding absolute and relative growth is reasonable, this model awaits data for model fitting, parameter estimation, or revision of the equations.     

Minimal output sets for identifiability

Ordinary differential equation models in biology often contain a large number of parameters that must be determined from measurements by parameter estimation. For a parameter estimation procedure to be successful, there must be a unique set of parameters that can have produced the measured data. This is not the case if a model is not uniquely structurally identifiable with the given set of outputs selected as measurements. In designing an experiment for the purpose of parameter estimation, given a set of feasible but resource-consuming measurements, it is useful to know which ones must be included in order to obtain an identifiable system, or whether the system is unidentifiable from the feasible measurement set. We have developed an algorithm that, from a user-provided set of variables and parameters or functions of them assumed to be measurable or known, determines all subsets that when used as outputs give a locally structurally identifiable system and are such that any output set for which the system is structurally identifiable must contain at least one of the calculated subsets. The algorithm has been implemented in Mathematica and shown to be feasible and efficient. We have successfully applied it in the analysis of large signalling pathway models from the literature.     

Application of MUTIC to the exploration of gene expression data in prostate cancer

We show here an example of the application of a novel method, MUTIC (model utilization-based clustering), used for identifying complex interactions between genes or gene categories based on gene expression data. The method deals with binary categorical data which consist of a set of gene expression profiles divided into two biologically meaningful categories. It does not require data from multiple time points. Gene expression profiles are represented by feature vectors whose component features are either gene expression values, or averaged expression values corresponding to gene ontology or protein information resource categories. A supervised learning algorithm (genetic programming) is used to learn an ensemble of classification models distinguishing the two categories based on the feature vectors corresponding to their members. Each feature is associated with a "model utilization vector", which has an entry for each high-quality classification model found, indicating whether or not the feature was used in that model. These utilization vectors are then clustered using a variant of hierarchical clustering called Omniclust. The result is a set of model utilization-based clusters, in which features are gathered together if they are often considered together by classification models - which may be because they are co-expressed, or may be for subtler reasons involving multi-gene interactions. The MUTIC method is illustrated here by applying it to a dataset regarding gene expression in prostate cancer and control samples. Compared to traditional expression-based clustering, MUTIC yields clusters that have higher mathematical quality (in the sense of homogeneity and separation) and that also yield novel insights into the underlying biological processes.     

Robustness of the latent variable model for correlated binary data

The marginal regression model offers a useful alternative to conditional approaches to analyzing binary data (Liang, Zeger, and Qaqish, 1992, Journal of the Royal Statistical Society, Series B 54, 3-40). Instead of modelling the binary data directly as do Liang and Zeger (1986, Biometrika 73, 13-22), the parametric marginal regression model developed by Qu et al. (1992, Biometrics 48, 1095-1102) assumes that there is an underlying multivariate normal vector that gives rise to the observed correlated binary outcomes. Although this parametric approach provides a flexible way to model different within-cluster correlation structures and does not restrict the parameter space, it is of interest to know how robust the parameter estimates are with respect to choices of the latent distribution. We first extend the latent modelling to include multivariate t-distributed latent vectors and assess the robustness in this class of distributions. Then we show through a simulation that the parameter estimates are robust with respect to the latent distribution even if latent distribution is skewed. In addtion to this empirical evidence for robustness, we show through the iterative algorithm that the robustness of the regression coefficents with respect to misspecifications of covariance structure in Liang and Zeger's model in fact indicates robustness with respect to underlying distributional assumptions of the latent vector in the latent variable model.     

On the Population Dynamics of the Malaria Vector

A deterministic differential equation model for the population dynamics of the human malaria vector is derived and studied. Conditions for the existence and stability of a non-zero steady state vector population density are derived. These reveal that a threshold parameter, the vectorial basic reproduction number, exist and the vector can established itself in the community if and only if this parameter exceeds unity. When a non-zero steady state population density exists, it can be stable but it can also be driven to instability via a Hopf Bifurcation to periodic solutions, as a parameter is varied in parameter space. By considering a special case, an asymptotic perturbation analysis is used to derive the amplitude of the oscillating solutions for the full non-linear system. The present modelling exercise and results show that it is possible to study the population dynamics of disease vectors, and hence oscillatory behaviour as it is often observed in most indirectly transmitted infectious diseases of humans, without recourse to external seasonal forcing.     

On Genetic Map Functions

Various genetic map functions have been proposed to infer the unobservable genetic distance between two loci from the observable recombination fraction between them. Some map functions were found to fit data better than others. When there are more than three markers, multilocus recombination probabilities cannot be uniquely determined by the defining property of map functions, and different methods have been proposed to permit the use of map functions to analyze multilocus data. If for a given map function, there is a probability model for recombination that can give rise to it, then joint recombination probabilities can be deduced from this model. This provides another way to use map functions in multilocus analysis. In this paper we show that stationary renewal processes give rise to most of the map functions in the literature. Furthermore, we show that the interevent distributions of these renewal processes can all be approximated quite well by gamma distributions.     

Modeling brain resonance phenomena using a neural mass model

Stimulation with rhythmic light flicker (photic driving) plays an important role in the diagnosis of schizophrenia, mood disorder, migraine, and epilepsy. In particular, the adjustment of spontaneous brain rhythms to the stimulus frequency (entrainment) is used to assess the functional flexibility of the brain. We aim to gain deeper understanding of the mechanisms underlying this technique and to predict the effects of stimulus frequency and intensity. For this purpose, a modified Jansen and Rit neural mass model (NMM) of a cortical circuit is used. This mean field model has been designed to strike a balance between mathematical simplicity and biological plausibility. We reproduced the entrainment phenomenon observed in EEG during a photic driving experiment. More generally, we demonstrate that such a single area model can already yield very complex dynamics, including chaos, for biologically plausible parameter ranges. We chart the entire parameter space by means of characteristic Lyapunov spectra and Kaplan-Yorke dimension as well as time series and power spectra. Rhythmic and chaotic brain states were found virtually next to each other, such that small parameter changes can give rise to switching from one to another. Strikingly, this characteristic pattern of unpredictability generated by the model was matched to the experimental data with reasonable accuracy. These findings confirm that the NMM is a useful model of brain dynamics during photic driving. In this context, it can be used to study the mechanisms of, for example, perception and epileptic seizure generation. In particular, it enabled us to make predictions regarding the stimulus amplitude in further experiments for improving the entrainment effect.     

Identifying predator-prey processes from time-series

The functional response is a key element in predator-prey models as well as in food chains and food webs. Classical models consider it as a function of prey abundance only. However, many mechanisms can lead to predator dependence, and there is increasing evidence for the importance of this dependence. Identification of the mathematical form of the functional response from real data is therefore a challenging task. In this paper we apply model-fitting to test if typical ecological predator-prey time series data, which contain both observation error and process error, can give some information about the form of the functional response. Working with artificial data (for which the functional response is known) we will show that with moderate noise levels, identification of the model that generated the data is possible. However, the noise levels prevailing in real ecological time-series can give rise to wrong identifications. We will also discuss the quality of parameter estimation by fitting differential equations to such time-series.     

Dynamic Bayesian sensitivity analysis of a myocardial metabolic model

Dynamic compartmentalized metabolic models are identified by a large number of parameters, several of which are either non-physical or extremely difficult to measure. Typically, the available data and prior information is insufficient to fully identify the system. Since the models are used to predict the behavior of unobserved quantities, it is important to understand how sensitive the output of the system is to perturbations in the poorly identifiable parameters. Classically, it is the goal of sensitivity analysis to asses how much the output changes as a function of the parameters. In the case of dynamic models, the output is a function of time and therefore its sensitivity is a time dependent function. If the output is a differentiable function of the parameters, the sensitivity at one time instance can be computed from its partial derivatives with respect to the parameters. The time course of these partial derivatives describes how the sensitivity varies in time.When the model is not uniquely identifiable, or if the solution of the parameter identification problem is known only approximately, we may have not one, but a distribution of possible parameter values. This is always the case when the parameter identification problem is solved in a statistical framework. In that setting, the proper way to perform sensitivity analysis is to not rely on the values of the sensitivity functions corresponding to a single model, but to consider the distributed nature of the sensitivity functions, inherited from the distribution of the vector of the model parameters.In this paper we propose a methodology for analyzing the sensitivity of dynamic metabolic models which takes into account the variability of the sensitivity over time and across a sample. More specifically, we draw a representative sample from the posterior density of the vector of model parameters, viewed as a random variable. To interpret the output of this doubly varying sensitivity analysis, we propose visualization modalities particularly effective at displaying simultaneously variations over time and across a sample. We perform an analysis of the sensitivity of the concentrations of lactate and glycogen in cytosol, and of ATP, ADP, NAD⁺ and NADH in cytosol and mitochondria, to the parameters identifying a three compartment model for myocardial metabolism during ischemia.     

Melanoblast proliferation dynamics during mouse embryonic development. Modeling and validation

In this paper, we are looking for mathematical modeling of mouse embryonic melanoblast proliferation dynamics, taking into account, the expression level of β‐catenin. This protein plays an important role into the whole signal pathway process. Different assumptions on some unobservable features lead to different candidate models. From real data measured, from biological experiments and from a priori biological knowledge, it was able to validate or invalidate some of the candidate models. Data assimilation and parameter identification allowed us to derive a mathematical model that is in very good agreement with biological data. As a result, the produced model can give tracks for biologists into their biological investigations and experimental evidence. Another interest is the use of this model for robust hidden parameter identification like double times or number of founder melanoblasts.