Monotone dependence of the spectral bound on the transition rates in linear compartment models

For linear compartment models or Leslie-type staged population models with quasi-positive matrix the spectral bound of the matrix (the eigenvalue determining stability) is studied in the situation where particles or individuals leave a compartment or stage with some rate and enter another with the same rate. Then the matrix carries the rate with a positive sign in some off-diagonal entry and with a negative sign in the corresponding diagonal entry. Hence the matrix does not depend on the rate in a monotone way. It is shown, however, that the spectral bound is a monotone function of the rate. It is all the time strictly increasing or strictly decreasing or it is constant. A simple algebraic criterion distinguishes between the three cases. The results can be applied to linear systems and to the stability of stationary states in non-linear systems, in particular to models for the transmission of infectious diseases, and in population dynamics.     

三参数增长模型拟合:以季风常绿阔叶林中两个优势乔木种群为例

Logistic、Mitscherlich、Gompertz方程是一类三参数饱和增长曲线模型,广泛地应用于许多学科领域．本文基于logistic方程饱和值K估计的三点法、四点法,推导出Mitscherlich、Gompertz方程K值的三点法、四点法估计公式,并以南亚热带季风常绿阔叶林中两种优势乔木厚壳桂、黄果厚壳桂种群为例,先用三点法或四点法估计出K值,再通过线性回归与非线性回归相结合的方法,可获得三个增长模型中三个参数的最优无偏估计．实例研究表明,两个优势种群增长数据均符合三个增长模型,但更符合增长曲线呈S形的logistic、Gompertz方程,且以logistic方程最适合于观察;黄果厚壳桂种群增长快于厚壳桂种群．     

The usefulness of sensitivity analysis for predicting the effects of cat predation on the population dynamics of their avian prey

Sensitivity analyses of population projection matrix (PPM) models are often used to identify life-history perturbations that will most influence a population's future dynamics. Sensitivities are linear extrapolations of the relationship between a population's growth rate and perturbations to its demographic parameters. Their effectiveness depends on the validity of the assumption of linearity. Here we assess whether sensitivity analysis is an appropriate tool to investigate the effect of predation by cats on the population growth rates of their avian prey. We assess whether predation by cats leads to non-linear effects on population growth and compare population growth rates predicted by sensitivity analysis with those predicted by a non-linear simulation. For a two-stage, age-classified House Sparrow Passer domesticus PPM slight non-linearity arose when PPM elements were perturbed, but perturbation to the vital rates underlying the matrix elements had a linear impact on population growth rate. We found a similar effect with a slightly larger three-stage, age-classified PPM for a Winter Wren Troglodytes troglodytes population perturbed by cat predation. For some avian species, predation by cats may cause linear or only slightly nonlinear impacts on population growth rates. For these species, sensitivity analysis appears to be a useful conservation tool. However, further work on multiple perturbations to avian prey species with more complicated life histories and higher-dimension PPM models is required.     

Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.     

Genomic prediction based on data from three layer lines using non-linear regression models

Background

Most studies on genomic prediction with reference populations that include multiple lines or breeds have used linear models. Data heterogeneity due to using multiple populations may conflict with model assumptions used in linear regression methods.

Methods

In an attempt to alleviate potential discrepancies between assumptions of linear models and multi-population data, two types of alternative models were used: (1) a multi-trait genomic best linear unbiased prediction (GBLUP) model that modelled trait by line combinations as separate but correlated traits and (2) non-linear models based on kernel learning. These models were compared to conventional linear models for genomic prediction for two lines of brown layer hens (B1 and B2) and one line of white hens (W1). The three lines each had 1004 to 1023 training and 238 to 240 validation animals. Prediction accuracy was evaluated by estimating the correlation between observed phenotypes and predicted breeding values.

Results

When the training dataset included only data from the evaluated line, non-linear models yielded at best a similar accuracy as linear models. In some cases, when adding a distantly related line, the linear models showed a slight decrease in performance, while non-linear models generally showed no change in accuracy. When only information from a closely related line was used for training, linear models and non-linear radial basis function (RBF) kernel models performed similarly. The multi-trait GBLUP model took advantage of the estimated genetic correlations between the lines. Combining linear and non-linear models improved the accuracy of multi-line genomic prediction.

Conclusions

Linear models and non-linear RBF models performed very similarly for genomic prediction, despite the expectation that non-linear models could deal better with the heterogeneous multi-population data. This heterogeneity of the data can be overcome by modelling trait by line combinations as separate but correlated traits, which avoids the occasional occurrence of large negative accuracies when the evaluated line was not included in the training dataset. Furthermore, when using a multi-line training dataset, non-linear models provided information on the genotype data that was complementary to the linear models, which indicates that the underlying data distributions of the three studied lines were indeed heterogeneous.

Electronic supplementary material

The online version of this article (doi:10.1186/s12711-014-0075-3) contains supplementary material, which is available to authorized users.     

Kernel-imbedded Gaussian processes for disease classification using microarray gene expression data

Background  

Designing appropriate machine learning methods for identifying genes that have a significant discriminating power for disease outcomes has become more and more important for our understanding of diseases at genomic level. Although many machine learning methods have been developed and applied to the area of microarray gene expression data analysis, the majority of them are based on linear models, which however are not necessarily appropriate for the underlying connection between the target disease and its associated explanatory genes. Linear model based methods usually also bring in false positive significant features more easily. Furthermore, linear model based algorithms often involve calculating the inverse of a matrix that is possibly singular when the number of potentially important genes is relatively large. This leads to problems of numerical instability. To overcome these limitations, a few non-linear methods have recently been introduced to the area. Many of the existing non-linear methods have a couple of critical problems, the model selection problem and the model parameter tuning problem, that remain unsolved or even untouched. In general, a unified framework that allows model parameters of both linear and non-linear models to be easily tuned is always preferred in real-world applications. Kernel-induced learning methods form a class of approaches that show promising potentials to achieve this goal.     

Nonlinear systems in medicine

Many achievements in medicine have come from applying linear theory to problems. Most current methods of data analysis use linear models, which are based on proportionality between two variables and/or relationships described by linear differential equations. However, nonlinear behavior commonly occurs within human systems due to their complex dynamic nature; this cannot be described adequately by linear models. Nonlinear thinking has grown among physiologists and physicians over the past century, and non-linear system theories are beginning to be applied to assist in interpreting, explaining, and predicting biological phenomena. Chaos theory describes elements manifesting behavior that is extremely sensitive to initial conditions, does not repeat itself and yet is deterministic. Complexity theory goes one step beyond chaos and is attempting to explain complex behavior that emerges within dynamic nonlinear systems. Nonlinear modeling still has not been able to explain all of the complexity present in human systems, and further models still need to be refined and developed. However, nonlinear modeling is helping to explain some system behaviors that linear systems cannot and thus will augment our understanding of the nature of complex dynamic systems within the human body in health and in disease states.     

The impact of exclusion processes on angiogenesis models

Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.

     

Protein Hypersaline Adaptation: Insight from Amino Acids with Machine Learning Algorithms

Traditional bioinformatics methods performed systematic comparison between the halophilic proteins and their non-halophilic homologues, to investigate the features related to hypersaline adaptation. Therefore, proposing some quantitative models to explain the sequence-characteristic relationship of halophilic proteins might shed new light on haloadaptation and help to design new biocatalysts adapt to high salt concentration. Five machine learning algorithm, including three linear and two non-linear methods were used to discriminate halophilic and their non-halophilic counterparts and the prediction accuracy was encouraging. The best prediction reliability for halophilic proteins was achieved by artificial neural network and support vector machine and reached 80 %, for non-halophilic proteins, it was achieved by linear regression and reached 100 %. Besides, the linear models have captured some clues for protein halo-stability. Among them, lower frequency of Ser in halophilic protein has not been report before.     

On the numerical integration of non-local terms for age-structured population models

We formulate explicit second-order finite difference schemes for the numerical integration of non-linear age-dependent population models. These methods have been designed by means of a representation formula for the theoretical solution of the integro-differential equation joint with open quadrature formulae for the numerical approximation of non-local terms. The schemes are analyzed and some numerical experiments are also reported in order to show numerically their accuracy.     

A comparison of methods for estimating the kinetic parameters of two simple types of transport process

Sets of experimental data, with known characteristics and error structures, have been simulated for the Michaelis-Menten equation plus a second term, either for linear transport or for competitive inhibition. The Michaelis-Menten equation plus linear term was fitted by several methods and the accuracy and the precision of the parameter estimates from the several methods were compared. The model-fitting methods were: three for least-squares non-linear regression, computer versions of two graphical methods and of two non-parametric methods. The most precise and accurate method was that of D.W. Marquardt (J. Soc. Ind. Appl. Math. 11 (1963) 431–441). The Michaelis-Menten equation with competitive inhibition was also fitted by several methods, viz., two for least-squared non-linear regression, a non-parametric method and four variants of the Preston-Schaeffer-Curran plot (Preston, R.L. et al. (1974) J. Gen. Physiol. 64, 443–467). The most precise and accurate of these was the non-linear regression method of W.W. Cleland (Adv. Enzymol. 29 (1967) 1–32). For both these models, the various graphical methods and non-parametric methods gave poor results and are not recommended.     

Computer programs to implement retinal models

To implement retinal models of some complexity a set of computer programs, coordinated by a main program is required. A set of said programs is presented here which permits the design and the experimentation with linear and non-linear retinal layered models. A main program, called RETINA, is the basis for the building of a model, the generation of the input stimuli and the experimentation with the overall system.     

Comparison of non-linear and linear models for estimating haemoglobin adduct stability

According to the kinetic theory for the build-up and elimination of haemoglobin (Hb) adducts, unstable Hb adducts are simultaneously eliminated by zero-order Hb turnover and first-order chemical instability. Thus, the elimination of unstable Hb adducts is non-linear with respect to time. Nonetheless, many studies of Hb adduct stability have characterized the elimination of Hb adducts using linear zero-order or linear first-order models. This paper demonstrates the use of non-linear regression to estimate the first-order rate constant of Hb adduct instability (k) using data on the elimination of Hb adducts in rats dosed with benzene or ortho -toluidine. Results obtained using non-linear regression models are compared with results from the more commonly employed zero- and first-order linear models. It is shown that exposure estimates based on measured levels of unstable Hb adducts can be severely biased if zero-order turnover is assumed. Furthermore, based on published data, estimates of k are subject to estimated relative biases in the range of -4% to 96% when first-order linear models are used to characterize Hb adduct instability.     

Comparison of non-linear and linear models for estimating haemoglobin adduct stability

According to the kinetic theory for the build-up and elimination of haemoglobin (Hb) adducts, unstable Hb adducts are simultaneously eliminated by zero-order Hb turnover and first-order chemical instability. Thus, the elimination of unstable Hb adducts is non-linear with respect to time. Nonetheless, many studies of Hb adduct stability have characterized the elimination of Hb adducts using linear zero-order or linear first-order models. This paper demonstrates the use of non-linear regression to estimate the first-order rate constant of Hb adduct instability (k) using data on the elimination of Hb adducts in rats dosed with benzene or ortho -toluidine. Results obtained using non-linear regression models are compared with results from the more commonly employed zero- and first-order linear models. It is shown that exposure estimates based on measured levels of unstable Hb adducts can be severely biased if zero-order turnover is assumed. Furthermore, based on published data, estimates of k are subject to estimated relative biases in the range of -4% to 96% when first-order linear models are used to characterize Hb adduct instability.     

Compound Regression Models for Ordered Categorical Data

A general class of sequential models for the analysis of ordered categorical variables is developed and discussed. The models apply if the ordinal response may be subdivided into two or more meaningful sets of response categories. The parametrization explicitly makes use of this subdivision. The models furnish a linear alternative to non-linear models which incorporate a scale parameter. They are shown to be special cases of multivariate generalized linear models. Applications are discussed with the use of several examples.     

Merging Spline Approximations with Analysis of Nonlinear Regression Models

The paper outlines a method of analysis of intricically non-linear regression functions. The nonlinear model is first approximated by a cubic spline function. Thereafter, the standard methods of analysis in linear models are applied to obtain estimates and test statistics.     

Computational methods for the screening of pharmacokinetic parameters in metabolism experiments

A computational strategy is presented which is useful for the qualitative and quantitative assessment of reaction-kinetic parameters in a one-compartment open model for the metabolism of substances in a biological system, especially the metabolism of dialkylnitrosamines to the ultimate carcinogen. The mathematical model used is that of first order kinetics and results in Bateman-functions. The strategy allows to decide about stability or instability of a substance occurring in a chain of transformations. In the case of stability quantitative estimates of the half life are provided. The procedure compares parameter estimates from models of different qualitative nature. These estimates are derived by linear or non-linear regression methods.     

The numerical solution of stiff differential equations

This paper first discusses the conditions in which a set of differential equations should give stable solutions, starting with linear systems assuming that these do not differ greatly in this respect from non-linear systems. Methods of investigating the stability of particular systems are briefly discussed. Most real biochemical systems are known from observation to be stable, but little is known of the regions over which stability persists; moreover, models of biochemical systems may not be stable, because of inaccurate choice of parameter values.The separate problem of stability and accuracy in numerical methods of approximating the solution of systems of non-linear equations is then treated. Stress is laid on the consistently unsatisfactory results given by explicit methods for systems containing "stiff" equations, and implicit multistep methods are particularly recommended for this class of problem, which is likely to include many biochemical model systems. Finally, an iteration procedure likely to give convergence both in multistep methods and in the steady-state approach is recommended, and areas in which improvement in methods is likely to occur are outlined.