Diffusion models for chemotaxis: a statistical analysis of noninteractive unicellular movement.

A program is developed for applying stochastic differential equations to models for chemotaxis. First a few of the experimental and theoretical models for chemotaxis both for swimming bacteria and for cells migrating along a substrate are reviewed. In physical and biological models of deterministic systems, finite difference equations are often replaced by a limiting differential equation in order to take advantage of the ease in the use of calculus. A similar but more intricate methodology is developed here for stochastic models for chemotaxis. This exposition is possible because recent work in probability theory gives ease in the use of the stochastic calculus for diffusions and broad applicability in the convergence of stochastic difference equations to a stochastic differential equation. Stochastic differential equations suggest useful data for the model and provide statistical tests. We begin with phenomenological considerations as we analyze a one-dimensional model proposed by Boyarsky, Noble, and Peterson in their study of human granulocytes. In this context, a theoretical model consists in identifying which diffusion best approximates a model for cell movement based upon theoretical considerations of cell physiology. Such a diffusion approximation theorem is presented along with discussion of the relationship between autocovariance and persistence. Both the stochastic calculus and the diffusion approximation theorem are described in one dimension. Finally, these tools are extended to multidimensional models and applied to a three-dimensional experimental setup of spherical symmetry.     

Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.     

Maximum Likelihood Estimation of Long‐Term HIV Dynamic Models and Antiviral Response

Summary HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short‐term dynamic models, recent works considered more complex long‐term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation‐maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long‐term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2–ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between‐patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.     

Reduced Models of Algae Growth

The simulation of biological systems is often plagued by a high level of noise in the data, as well as by models containing a large number of correlated parameters. As a result, the parameters are poorly identified by the data, and the reliability of the model predictions may be questionable. Bayesian sampling methods provide an avenue for proper statistical analysis in such situations. Nevertheless, simulations should employ models that, on the one hand, are reduced as much as possible, and, on the other hand, are still able to capture the essential features of the phenomena studied. Here, in the case of algae growth modeling, we show how a systematic model reduction can be done. The simplified model is analyzed from both theoretical and statistical points of view.     

Mathematical correlation between biomaterial and cellular parameters--critical reflection of statistics

For mathematical modelling of the biomaterial-cell contact, it is necessary to find both parameters characterizing physical and chemical properties of the material surface and also such describing the reaction of the adhering cells. Only those material and cell parameters that correlate with each other are applicable to model this contact mathematically. Only few papers are dealing with this special problem. The aim of this paper is to present results of physical/chemical and biological investigations made on differently modified rough titanium implant surfaces in order to find out only the correlating parameters. Furthermore we discuss several ways to apply statistical methods to the correlation problem. Only few ones of all investigated parameters both on material and on cellular side were applicable for correlation. For example we found in our studies that fractal structure parameter topothesy has influence on the spreading behaviour of the osteoblastic cells. However the value of the correlation coefficient and its statistical significance heavily depend on the method of averaging the available data. Especially the biological data (spreading area) were afflicted with relatively high error up to 30%. Averaging of this data masks the true facts. That is why the correlation coefficient considerably decreases if the biological parameters are not averaged. On the other hand, the statistical reliability increases due to the higher number of investigated cases. Critical error discussion is necessary in statistical correlation between material and biological parameters. Often the results are heavily influenced by the statistical handling of data, especially if only few data are available. May be that new unconventional methods like bootstrap method can show a way out of this dilemma.     

Hyperparameter estimation using stochastic approximation with application to population pharmacokinetics

A stochastic approximation algorithm is proposed for recursive estimation of the hyperparameters characterizing, in a population, the probability density function of the parameters of a statistical model. For a given population model defined by a parametric model of a biological process, an error model, and a class of densities on the set of the individual parameters, this algorithm provides a sequence of estimates from a sequence of individuals' observation vectors. Convergence conditions are verified for a class of population models including usual pharmacokinetic applications. This method is implemented for estimation of pharmacokinetic population parameters from drug multiple-dosing data. Its estimation capabilities are evaluated and compared to a classical method in population pharmacokinetics, the first-order method (NONMEM), on simulated data.     

Moment fitting for parameter inference in repeatedly and partially observed stochastic biological models

The inference of reaction rate parameters in biochemical network models from time series concentration data is a central task in computational systems biology. Under the assumption of well mixed conditions the network dynamics are typically described by the chemical master equation, the Fokker Planck equation, the linear noise approximation or the macroscopic rate equation. The inverse problem of estimating the parameters of the underlying network model can be approached in deterministic and stochastic ways, and available methods often compare individual or mean concentration traces obtained from experiments with theoretical model predictions when maximizing likelihoods, minimizing regularized least squares functionals, approximating posterior distributions or sequentially processing the data. In this article we assume that the biological reaction network can be observed at least partially and repeatedly over time such that sample moments of species molecule numbers for various time points can be calculated from the data. Based on the chemical master equation we furthermore derive closed systems of parameter dependent nonlinear ordinary differential equations that predict the time evolution of the statistical moments. For inferring the reaction rate parameters we suggest to not only compare the sample mean with the theoretical mean prediction but also to take the residual of higher order moments explicitly into account. Cost functions that involve residuals of higher order moments may form landscapes in the parameter space that have more pronounced curvatures at the minimizer and hence may weaken or even overcome parameter sloppiness and uncertainty. As a consequence both deterministic and stochastic parameter inference algorithms may be improved with respect to accuracy and efficiency. We demonstrate the potential of moment fitting for parameter inference by means of illustrative stochastic biological models from the literature and address topics for future research.     

The dynamics of the lynx–hare system: an application of the Lotka–Volterra model

The Lotka–Volterra model of predator–prey dynamics was used for approximation of the wellknown empirical time series on the lynx–hare system in Canada that was collected by the Hudson Bay Company in 1845–1935. The model was assumed to demonstrate satisfactory data approximation if the sets of deviations of the model and empirical data for both time series satisfied a number of statistical criteria (for the selected significance level). The frequency distributions of deviations between the theoretical (model) trajectories and empirical datasets were tested for symmetry (with respect to the Y-axis; the Kolmogorov–Smirnov and Lehmann–Rosenblatt tests) and the presence or absence of serial correlation (the Swed–Eisenhart and “jumps up–jumps down” tests). The numerical calculations show that the set of points of the space of model parameters, when the deviations satisfy the statistical criteria, is not empty and, consequently, the model is suitable for describing empirical data.     

Robustness of the approximate likelihood of the protracted speciation model

The protracted speciation model presents a realistic and parsimonious explanation for the observed slowdown in lineage accumulation through time, by accounting for the fact that speciation takes time. A method to compute the likelihood for this model given a phylogeny is available and allows estimation of its parameters (rate of initiation of speciation, rate of completion of speciation and extinction rate) and statistical comparison of this model to other proposed models of diversification. However, this likelihood computation method makes an approximation of the protracted speciation model to be mathematically tractable: it sometimes counts fewer species than one would do from a biological perspective. This approximation may have large consequences for likelihood‐based inferences: it may render any conclusions based on this method completely irrelevant. Here, we study to what extent this approximation affects parameter estimations. We simulated phylogenies from which we reconstructed the tree of extant species according to the original, biologically meaningful protracted speciation model and according to the approximation. We then compared the resulting parameter estimates. We found that the differences were larger for high values of extinction rates and small values of speciation‐completion rates. Indeed, a long speciation‐completion time and a high extinction rate promote the appearance of cases to which the approximation applies. However, surprisingly, the deviation introduced is largely negligible over the parameter space explored, suggesting that this approximate likelihood can be applied reliably in practice to estimate biologically relevant parameters under the original protracted speciation model.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behaviour of a system, differential equations have proved to be powerful for modelling and unravelling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge–Kutta fourth-order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

A dynamic model for functional mapping of biological rhythms

Functional mapping is a statistical method for mapping quantitative trait loci (QTLs) that regulate the dynamic pattern of a biological trait. This method integrates mathematical aspects of biological complexity into a mixture model for genetic mapping and tests the genetic effects of QTLs by comparing genotype-specific curve parameters. As a way of quantitatively specifying the dynamic behavior of a system, differential equations have proven to be powerful for modeling and unraveling the biochemical, molecular, and cellular mechanisms of a biological process, such as biological rhythms. The equipment of functional mapping with biologically meaningful differential equations provides new insights into the genetic control of any dynamic processes. We formulate a new functional mapping framework for a dynamic biological rhythm by incorporating a group of ordinary differential equations (ODE). The Runge-Kutta fourth order algorithm was implemented to estimate the parameters that define the system of ODE. The new model will find its implications for understanding the interplay between gene interactions and developmental pathways in complex biological rhythms.     

Estimation of Human Biological Age Based on the Parameters of Heart Rhythmic Activity

A linear regression model has been constructed for estimating human biological age, with the parameters of heart rhythmic activity being used as biological markers. One of the advantages of using these parameters as biological markers of aging is the possibility to measure a number of parameters for an individual subject in a brief (6–7 min) procedure of rhythmogram recording. This makes the collection of data for a statistically reliable sample much easier, increases the accuracy of the model, and permits its use along with other methods for mass examinations of a population and for control of the effects of drugs and food additives. The model may be extended and supplemented with other biological markers in order to improve the approximation of biological age.     

Log-linear Models for Polytomous Data

The present study discusses two variants of linear logistic models for polytomous variables for ?unordered”? and for ?ordered”? categories (polydimensional and one-dimensional model). The ML-estimation equations and the possibilities to test the validity of the model are given for both. A test for goodness-of-fit (external validity) and a test for equality of the parameter estimates for split data (interval validity) are suggested. In addition, statistical tests for the significance of individual parameters on the basis of the information matrix and likelihood ratio tests for one or more parameters are described. The presentation is completed by an empirical example from the area of audiology.     

Statistical analysis of synaptic transmission: model discrimination and confidence limits.

Procedures for discriminating between competing statistical models of synaptic transmission, and for providing confidence limits on the parameters of these models, have been developed. These procedures were tested against simulated data and were used to analyze the fluctuations in synaptic currents evoked in hippocampal neurones. All models were fitted to data using the Expectation-Maximization algorithm and a maximum likelihood criterion. Competing models were evaluated using the log-likelihood ratio (Wilks statistic). When the competing models were not nested, Monte Carlo sampling of the model used as the null hypothesis (H0) provided density functions against which H0 and the alternate model (H1) were tested. The statistic for the log-likelihood ratio was determined from the fit of H0 and H1 to these probability densities. This statistic was used to determine the significance level at which H0 could be rejected for the original data. When the competing models were nested, log-likelihood ratios and the chi 2 statistic were used to determine the confidence level for rejection. Once the model that provided the best statistical fit to the data was identified, many estimates for the model parameters were calculated by resampling the original data. Bootstrap techniques were then used to obtain the confidence limits of these parameters.     

生物协同学,Lorenz模型和种群动力学

由协同学方程出发,可以描述种群的大迁徙,由此又能够得到Lorenz模型,它可以描述两种种群的变化关系．当取绝热近似时,还可以导致种群动力学的不同模型．因此,生物协同学能够深刻揭示不同物种之间,既竞争又协同的复杂的非线性关系．     

Mathematical Models of the Kinetics of the Cultivation of Microorganisms

Various equations of mathematical models for the kinetics of the development of various biological processes were obtained on the basis of the generalized differential equation of biomass growth. Aerobic periodic cultivation of the yeast Saccharomyces cerevisiae was carried out to provide a comparative evaluation of advantages and disadvantages of four types of mathematical models. It is shown that the exponential model is a particular solution to the generalized differential equation. The developed mathematical model can be used to predict the course of biological processes in time and can serve as a tool for a computational experiment in order to clarify the dependence of the rate of a biological process on changes in certain parameters that affect the development of cells.