Development of a two-dimension manifold to represent high dimension mathematical models of the intracellular Mammalian circadian clock

A new focus for mathematical models of the circadian pacemaker involves the encapsulation within the models of detailed biological processes responsible for generating those circadian rhythms. Representing greater biological detail requires more mathematical equations, which pose a greater challenge for the analysis of such systems. Development of a method that retains the predominant dynamics while still providing biologically detailed information is advantageous. Two high-dimension mathematical models of intracellular mammalian circadian pacemakers, Leloup-Goldbeter and Forger-Peskin, with 19 and 73 differential equations, respectively, have been published. The authors projected each of these high-dimension models onto their respective manifold using proper orthogonal functions (POFs) obtained from the empirical decomposition of the model's phase space to obtain a 2-dimension model. The resulting 2-dimension model, represented by 2 differential equations, predicts most of the salient characteristics of a biological clock including approximately 24-h oscillations, entrainment to an LD cycle, phase response curves, and the amplitude recovery dynamics that emerge following amplitude suppression. The manifold representation simplifies the mathematical analysis, since only 2 variables need to be observed and analyzed to understand the behavior of the biological clock. This reduced model derived from a model based on biological variables can be used for the development and analysis of mathematical models of the coupled mammalian oscillators to understand the dynamics of the integrated circadian pacemaker.     

Review: To be or not to be an identifiable model. Is this a relevant question in animal science modelling?

What is a good (useful) mathematical model in animal science? For models constructed for prediction purposes, the question of model adequacy (usefulness) has been traditionally tackled by statistical analysis applied to observed experimental data relative to model-predicted variables. However, little attention has been paid to analytic tools that exploit the mathematical properties of the model equations. For example, in the context of model calibration, before attempting a numerical estimation of the model parameters, we might want to know if we have any chance of success in estimating a unique best value of the model parameters from available measurements. This question of uniqueness is referred to as structural identifiability; a mathematical property that is defined on the sole basis of the model structure within a hypothetical ideal experiment determined by a setting of model inputs (stimuli) and observable variables (measurements). Structural identifiability analysis applied to dynamic models described by ordinary differential equations (ODEs) is a common practice in control engineering and system identification. This analysis demands mathematical technicalities that are beyond the academic background of animal science, which might explain the lack of pervasiveness of identifiability analysis in animal science modelling. To fill this gap, in this paper we address the analysis of structural identifiability from a practitioner perspective by capitalizing on the use of dedicated software tools. Our objectives are (i) to provide a comprehensive explanation of the structural identifiability notion for the community of animal science modelling, (ii) to assess the relevance of identifiability analysis in animal science modelling and (iii) to motivate the community to use identifiability analysis in the modelling practice (when the identifiability question is relevant). We focus our study on ODE models. By using illustrative examples that include published mathematical models describing lactation in cattle, we show how structural identifiability analysis can contribute to advancing mathematical modelling in animal science towards the production of useful models and, moreover, highly informative experiments via optimal experiment design. Rather than attempting to impose a systematic identifiability analysis to the modelling community during model developments, we wish to open a window towards the discovery of a powerful tool for model construction and experiment design.     

Models of classical conditioning

Classical conditioning is a basic form of associative learning in the animal kingdom. Many paradigmatic features of classical conditioning appear to be conserved throughout species and phyla and are independent of stimulus nature. This paper presents an analysis of trial-based and real-time models of classical conditioning which are mathematical abstractions of the underlying processing principles. Various models are reviewed and in a formal analysis, their capability of simulating and explaining classical conditioning is investigated. Since every existing model fails to simulate some particular conditioning phenomena and since some modelling approaches are not appropriate for detailed mathematical analysis, new model components will be introduced that overcome most of the weaknesses observed in the other models.     

系统生物学中的数学模型方法及其应用

综述了当前在系统生物学研究领域中常用数学模型的建立和研究方法,介绍了通量平衡分析、基元通量模式、生化系统理论,以及代谢控制分析等模型的理论基础和发展背景,讨论了这些模型之间的联系、区别,以及适用范围,并总结了这些模型在分析代谢网络结构、优化代谢途径、指导菌种改进以提高琥珀酸、色氨酸、乙醇等重要化工品生产率和转化率中的实际应用。     

A methodology for performing global uncertainty and sensitivity analysis in systems biology

Accuracy of results from mathematical and computer models of biological systems is often complicated by the presence of uncertainties in experimental data that are used to estimate parameter values. Current mathematical modeling approaches typically use either single-parameter or local sensitivity analyses. However, these methods do not accurately assess uncertainty and sensitivity in the system as, by default, they hold all other parameters fixed at baseline values. Using techniques described within we demonstrate how a multi-dimensional parameter space can be studied globally so all uncertainties can be identified. Further, uncertainty and sensitivity analysis techniques can help to identify and ultimately control uncertainties. In this work we develop methods for applying existing analytical tools to perform analyses on a variety of mathematical and computer models. We compare two specific types of global sensitivity analysis indexes that have proven to be among the most robust and efficient. Through familiar and new examples of mathematical and computer models, we provide a complete methodology for performing these analyses, in both deterministic and stochastic settings, and propose novel techniques to handle problems encountered during these types of analyses.     

A critical review of mathematical models and data used in diabetology

The literature dealing with mathematical modelling for diabetes is abundant. During the last decades, a variety of models have been devoted to different aspects of diabetes, including glucose and insulin dynamics, management and complications prevention, cost and cost-effectiveness of strategies and epidemiology of diabetes in general. Several reviews are published regularly on mathematical models used for specific aspects of diabetes. In the present paper we propose a global overview of mathematical models dealing with many aspects of diabetes and using various tools. The review includes, side by side, models which are simple and/or comprehensive; deterministic and/or stochastic; continuous and/or discrete; using ordinary differential equations, partial differential equations, optimal control theory, integral equations, matrix analysis and computer algorithms.     

Empirical and mechanistic mathematical models of temporal evolution of milk production in ruminants.

In the various sectors of animal science there has been little exploration of the theoretical mathematical aspects of data analysis and modelling. The dominant statistical methods used for the analysis of experimental data are rarely valuable for developing a deeper understanding of the problem. In addition they do not take account of the evolution over time of those variables of major interest to be studied. Only recently have more sophisticated methods of mathematical modelling begun to be used. Nonetheless attention tends to be focused exclusively on empirical models. Mathematical models with greater explanatory power, in particular those which use differential equations, are as yet little used. This work develops a mathematical approach to a problem that is of great interest in animal science: the development over time of milk production in economically important ruminant species.     

Theoretical analysis of reverse-time correlation for idealized orientation tuning dynamics

A theoretical analysis is presented of a reverse-time correlation method used in experimentally investigating orientation tuning dynamics of neurons in the primary visual cortex. An exact mathematical characterization of the method is developed, and its connection with the Volterra–Wiener nonlinear systems theory is described. Various mathematical consequences and possible physiological implications of this analysis are illustrated using exactly solvable idealized models of orientation tuning.     

A rumor transmission model with various contact interactions

We consider a rumor transmission model with various contact interactions and explore what effect such interactions have on the spread of a rumor, in particular whether they can explain the rumor recursion. Through mathematical analysis and computer simulations, we conjecture that rumor recursion remains a major challenge to mathematical models of rumors beyond our model proposed here.     

On dependency properties of the ISIs generated by a two-compartmental neuronal model

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.     

Perspectives on Mathematical Modeling for Receptor-Mediated Processes

Mathematical modeling is a potent in silico tool that can help investigate, interpret, and predict the behavior of biological systems. The first step is to develop a working hypothesis of the biology. Then by “translating” the biological phenomena into equations, models can harness the power of mathematical analysis techniques to explore the dynamics and interactions of the biological components. Models can be used together with traditional experimental models to help design new experiments, test hypotheses, identify mechanisms, and predict outcomes. This article reviews the process of building, calibrating, and using mathematical models in the context of the kinetics of receptor and signal transduction biology. An example model related to the androgen receptor-mediated regulation of the prostate is presented to illustrate the steps in the modeling process and to highlight the potential for mathematical modeling in this area.     

Perspectives on mathematical modeling for receptor-mediated processes

Mathematical modeling is a potent in silico tool that can help investigate, interpret, and predict the behavior of biological systems. The first step is to develop a working hypothesis of the biology. Then by "translating" the biological phenomena into equations, models can harness the power of mathematical analysis techniques to explore the dynamics and interactions of the biological components. Models can be used together with traditional experimental models to help design new experiments, test hypotheses, identify mechanisms, and predict outcomes. This article reviews the process of building, calibrating, and using mathematical models in the context of the kinetics of receptor and signal transduction biology. An example model related to the androgen receptor-mediated regulation of the prostate is presented to illustrate the steps in the modeling process and to highlight the potential for mathematical modeling in this area.     

Geographic models and control of cattle liver flukes in the Southern USA

John Malone and Stanley Zukowski describe climate forecast and soil hydrology-based geographic information system models of snail habitat extent that provide an environmental context for site-specific mathematical models of fascioliasis. These have potential for the development of broad-scale control models that include a cost-benefit analysis component.     

农业废弃物养分循环利用技术模式评估模型的研究进展

农业废弃物的养分循环利用技术模式是实现农业循环经济的重要手段,其评估模型为优化养分循环利用技术提供了重要支撑。本文总结了农业废弃物养分循环技术模式评估框架、评估模型及评价指标、模型的数据源及其不确定性分析,以及模型应用尺度的研究进展。当前,常用于评估养分流动的模型主要是过程数学模型和产业生态学模型。过程数学模型和产业生态学模型在评估结果的可靠性和模拟尺度上存在较大差异,前者主要集中在实验室或中试规模,精度较高;后者可以实现从微观到宏观的多尺度模拟,数据的获取方式导致其具有较高的不确定性。最后,本文对农业废弃物养分循环利用技术评估模型的研究进行展望,提出为了在区域尺度上实现对农业生产系统废弃物资源化利用技术的准确评估,可以将过程数学模型与工业生态学模型相结合,建立可靠的模型框架和数据库,同时,在工厂、农场、村落、乡镇、区域等地理尺度进行模型拓展研究。     

Mathematical simulation and analysis of cellular metabolism and regulation.

MOTIVATION: A better understanding of the biological phenomena observed in cells requires the creation and analysis of mathematical models of cellular metabolism and physiology. The formulation and study of such models must also be simplified as far as possible to cope with the increasing complexity demanded and exponential accumulation of the metabolic reconstructions computed from sequenced genomes. RESULTS: A mathematical simulation workbench, DBsolve, has been developed to simplify the derivation and analysis of mathematical models. It combines: (i) derivation of large-scale mathematical models from metabolic reconstructions and other data sources; (ii) solving and parameter continuation of non-linear algebraic equations (NAEs), including metabolic control analysis; (iii) solving the non-linear stiff systems of ordinary differential equations (ODEs); (iv) bifurcation analysis of ODEs; (v) parameter fitting to experimental data or functional criteria based on constrained optimization. The workbench has been successfully used for dynamic metabolic modeling of some typical biochemical networks (Dolgacheva et al., Biochemistry (Moscow), 6, 1063-1068, 1996; Goldstein and Goryanin, Mol. Biol. (Moscow), 30, 976-983, 1996), including microbial glycolytic pathways, signal transduction pathways and receptor-ligand interactions. AVAILABILITY: DBsolve 5. 00 is freely available from http://websites.ntl.com/ approximately igor.goryanin. CONTACT: gzz78923@ggr.co.uk     

Comparing functional responses in predator-infected eco-epidemics models

The current paper deals with the mathematical models of predator–prey system where a transmissible disease spreads among the predator species only. Four mathematical models are proposed and analysed with several popular predator functional responses in order to show the influence of functional response on eco-epidemic models. The existence, boundedness, uniqueness of solutions of all the models are established. Mathematical analysis including stability and bifurcation are observed. Comparison among the results of these models allows the general conclusion that relevant behaviour of the eco-epidemic predator–prey system, including switching of stability, extinction, persistence and oscillations for any species depends on four important parameters viz. the rate of infection, predator interspecies competition and the attack rate on susceptible predator. The paper ends with a discussion of the biological implications of the analytical and numerical results.     

Leg stiffness in human running: Comparison of estimates derived from previously published models to direct kinematic-kinetic measures

It is not presently clear whether mathematical models used to estimate leg stiffness during human running are valid. Therefore, leg stiffness during the braking phase of ground contact of running was calculated directly using synchronous kinematic (high-speed motion analysis) and kinetic (force platform) analysis, and compared to stiffness calculated using four previously published kinetic models. Nineteen well-trained male middle distance runners (age=21.1±4.1yr; VO(2max)=69.5±7.5mlO(2)kg(-1)min(-1)) completed a series of runs of increasing speed from 2.5 to 6.5ms(-1). Leg stiffness was calculated directly from kinetic-kinematic analysis using both vertical and horizontal forces to obtain the resultant force in the line of leg compression (Model 1). Values were also estimated using four previously published mathematical models where only force platform derived and anthropometric measures were required (Models 2-5; Morin et al., 2005, Morin et al., 2011, Blum et al., 2009, Farley et al., 1993, respectively). The greatest statistical similarity between leg stiffness values occurred with Models 1 and 2. The poorest similarity occurred when values from Model 4 were compared with Model 1. Analyses suggest that the poor correlation between Model 1 other models may have resulted from errors in the estimation in change in leg length during the braking phase. Previously published mathematical models did not provide accurate leg stiffness estimates, although Model 2, used by Morin et al. (2005), provided reasonable estimates that could be further improved by the removal of systematic error using a correction factor (K=1.0496K(Model2)).     

Regression analyses and motile sperm subpopulation structure study as improving tools in boar semen quality analysis

A precise estimation of the fertilizing ability of a boar ejaculate would be very useful to improve pig assisted reproduction results. For this purpose, we tested the mathematical combination of several parameters of the boar semen quality analysis, including the computer-assisted semen motility analysis (CASA), as a predictive fertility tool. The utilized mathematical relations among parameters were logistic and linear regressions. Two mathematical models obtained by logistic regression involving Osmotic Resistance Test (ORT Test), Hyperosmotic Resistance Test (HRT Test) and viability of fresh samples, showed a significant (P<0.05) correlation between semen characteristics and conception rate. However, none of the obtained models produced a significant correlation model between semen characteristics and prolificacy. The CASA analyses show that three separate subpopulations of spermatozoa with different motility characteristics coexist in boar ejaculates. There were significant (P<0.001) differences in the distribution of these subpopulations among boars, but no clear relationship between motile subpopulation structure and fertility was obtained. Our results support the belief that the predictive use of the results obtained in a standard boar semen quality analysis can reasonably be achieved by applying logistic correlation analyses among several function parameters of boar semen quality analysis and in vivo conception rates obtained after artificial insemination (AI).