Mathematical model for optimal control in wastewater discharges: the global performance

In this work we show how mathematical models and optimal control techniques can help us to solve some problems of environmental engineering, more precisely, water pollution problems arising from wastewater discharges into coastal areas or rivers. We deal with a complete two-dimensional mathematical model for the evolution of pollutant concentration in a shallow water domain. By integrating this model we obtain a zero-dimensional one and we use it to study the global performance of the system in a realistic situation. Finally, by using the two-dimensional model, we recall two optimal control problems related to the wastewater disposal problem.     

Modeling tumor growth.

The meaning and limitations of certain mathematical models of tumor growth are discussed, and some new derivations of the existing models are given. A theoretical justification for Gompertz's law of growth for tumors is presented. An age-dependent Von Bertalanffy's equation and diffusion models are introduced, and existence and uniqueness problems are addressed.     

Dose response problems in carcinogenesis.

The estimation of risks from exposure to carcinogens is an important problem from the viewpoint of protection of human health. It also poses some very difficult dose-response problems. Two dose-response models may fit experimental data about equally well and yet predict responses that differ by many orders of magnitude at low doses. Mechanisms of carcinogenesis are not sufficiently understood so that the shape of the dose-response curve at low doses can be satisfactorily predicted. Mathematical theories of carcinogenesis and statistical procedures can be of use with dose-reponse problems such as this and, in addition, can lead to a better understanding of the mechanisms of carcinogenesis. In this paper, mathematical dose-response models of carcinogenesis are considered as well as various proposed dose-response procedures for estimating carcinogenic risks at low doses. Areas are suggested in which further work may be useful. These areas include experimental design problems, statistical procedures for use with time-to-occurrence data, and mathematical models that incorporate such biological features as pharmacokinetics of carcinogens, synergistic effects, DNA repair, susceptible subpopulations, and immune reactions.     

On the syntactic structure of protein sequences and the concept of grammar complexity

It is shown that the concepts of grammar complexity and syntactic structure provide a useful mathematical framework for the investigation of some current problems in protein structure. Grammar complexity gives a measure of the degree of aperiodicity of a sequence and also an optimization criterion for evaluating amino acid categorizations. Three systems of amino acid categorization are compared in relation to their value for describing molecular architecture.     

Permanence and the dynamics of biological systems.

A basic question in mathematical biology concerns the long-term survival of each component, which might typically be a population in an ecological context, of a system of interacting components. Many criteria have been used to define the notion of long-term survival. We consider here the subject of permanence, i.e., the study of the long-term survival of each species in a set of populations. These situations may often be modeled successfully by dynamical systems and have led to the development of some interesting mathematical techniques and results. Our intention here is to describe these and to consider their application to several of the most frequently used models occurring in mathematical biology. We particularly wish to include and cover those models leading to problems that are essentially infinite dimensional, for example reaction-diffusion equations, and to make the discussion accessible to a wide audience, we include a chapter outlining the fundamental theory of these.     

Biological applications of the theory of birth-and-death processes

In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. We conclude that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.     

Mathematical model for cyclic scheduling with work-in-process minimization

This paper is part of an original approach of mathematical modeling for solving cyclic scheduling problems. More precisely, we consider the cyclic job shop. This kind of manufacturing systems is well fitted to medium and large production demands. Many methods have been proposed to solve the cyclic scheduling problem. Among them, we chose the exact techniques, and we focus on the mathematical programming approach. We proposed, in an earlier study, a mathematical programming model for cyclic scheduling with Work-In-Process minimization. We propose here several cutting techniques to improve the practical performances of the model resolution. Some numerical experiments are used to assess the relevance of our propositions. We made a comparison between the original mathematical model and the one endowed by the proposed cuts. This comparison is based on a set of benchmarks generated for this reason. In addition, we make another comparison based on some examples from the literature.     

Does efficiency sensing unify diffusion and quorum sensing?

Quorum sensing faces evolutionary problems from non-producing or over-producing cheaters. Such problems are circumvented in diffusion sensing, an alternative explanation for quorum sensing. However, both explanations face the problems of signalling in complex environments such as the rhizosphere where, for example, the spatial distribution of cells can be more important for sensing than cell density, which we show by mathematical modelling. We argue that these conflicting concepts can be unified by a new hypothesis, efficiency sensing, and that some of the problems associated with signalling in complex environments, as well as the problem of maintaining honesty in signalling, can be avoided when the signalling cells grow in microcolonies.     

Interactions Between the Immune System and Cancer: A Brief Review of Non-spatial Mathematical Models

We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.     

Survey of research in the theory of homogeneous structures and their applications

This work presents some of the author's reflections, the purpose of which is to give an orientation on the present stage and recent developments in the investigations of the theory of homogeneous structures and their applications. This survey appears to be desirable at this time because a number of research workers in mathematical and theoretical biology are becoming interested in making contributions in the theory of homogeneous structures and their applications. It is hoped that this survey will provide a stimulus and some introduction to the subject. The choice of topics is dictated largely by my predilections, and in no way shall I attempt to illustrate the vast and rapidly increasing field of applications of mathematical techniques or even, more generally, the mathematical mode of thinking in the enormous variety of problems in the theory of homogeneous structures and its applications, including biology. Indeed, to present anything like a comprehensive picture would require bulky and manifold volumes of very heterogeneous contributions. However, I shall list a number of biological and engineering fields where techniques of homogeneous structures have been employed and give the reader a short indication of selected bibliographic aids.     

Modelling perspectives on aging: can mathematics help us stay young?

We survey several types of mathematical models that keep track of age distributions in a population, or follow some aspects of aging, such as loss of replicative potential of stem cells. The properties of a class of linear models of this type are discussed and compared. We illustrate the applicability of such models with a simple example based on hypothetical stem cell dynamics developed to address age-related telomere loss in the human granulocyte pool. We then describe the contrasting behaviour of nonlinear systems. Examples are drawn from the class of "dynamical diseases" to illustrate some of the aspects of nonlinear systems. Applications of these, and other models to the problems of aging and replicative aging are discussed.     

生态学数学模型的研究进展

当前许多生态学的研究工作使用了数学的方法和手段。因为它可以对生态现象进行量化的研究,而且理论上的分析往往还能使研究结果得以深化。数学模型是数学的思想和方法应用于生态学研究的桥梁。自从60年代数学模型的研究在生态学上再次被人们重视以来,其发     

A one tube flow problem arising in physiology

A mathematical analysis, including existence and uniqueness, is given for some boundary value problems which model the flow of a fluid-solute mixture in a tube which is placed in an interstitium. The model permits an interchange of fluid and solute across the tube walls.     

Mathematical modeling and parameters estimation of anaerobic fermentation processes

In the paper some problems of the mathematical modeling of anaerobic (methane) fermentation of animal waste in stirred tank bioreactors are considered. Laboratory experiments are carried out with highly concentrated organic pollutants and transient step responses of the control output for continuous methane fermentation are obtained. The dynamic behavior of this process is described by sets of deterministic nonlinear differential equations from 2nd order with different structures. Static characteristics are obtained with these models analytically. Investigations by computer simulation of the methane fermentation are performed with the aim to chose appropriate models for automatic control system design of this process.     

Evaluation and propagation of confidence intervals in nonlinear, asymmetrical variance spaces. Analysis of ligand-binding data

Two problems that are often overlooked in studies employing nonlinear least-squares techniques for parameter estimation are confidence-interval estimation and propagation. When the parameters are correlated, the variance space and consequently the confidence intervals are nonlinear and asymmetrical. The presented mathematical method for the evaluation of confidence intervals and error propagation addresses these problems. The examples employed to demonstrate these methods include linear least-squares and the nonlinear least-squares analysis of ligand-binding problems, such as hormone receptor interactions and oxygen binding to human hemoglobin. The mathematical procedures have proven very useful for analyzing the molecular mechanism of cooperativity in human hemoglobin (Johnson, M. L., and G. K. Ackers, 1982. Biochemistry 21:201-211).     

Deterministic chaоs and the problem of predictability in population dynamics

The studies of the processes that can significantly influence the predictability in population dynamics are reviewed and the results of mathematical simulations of population dynamics are compared to the time series obtained in field observations. Considerable attention is given to the chaotic changes in population abundance. Some methods of numerical analysis of chaoticity and predictability of the time series are considered. The importance of comparing the results of mathematical simulation and observation data is tightly linked to problems in detecting chaos in the dynamics of natural populations and estimating the prevalence of chaotic regimes in nature. Insight into these problems can allow identification of the functional role of chaotic regimes in population dynamics.     

A mechanistic approach for modeling temperature-dependent consumer-resource dynamics

Paramount to our ability to manage and protect biological communities from impending changes in the environment is an understanding of how communities will respond. General mathematical models of community dynamics are often too simplistic to accurately describe this response, partly to retain mathematical tractability and partly for the lack of biologically pleasing functions representing the model/environment interface. We address these problems of tractability and plausibility in community/environment models by incorporating the Boltzmann factor (temperature dependence) in a bioenergetic consumer-resource framework. Our analysis leads to three predictions for the response of consumer-resource systems to increasing mean temperature (warming). First, mathematical extinctions do not occur with warming; however, stable systems may transition into an unstable (cycling) state. Second, there is a decrease in the biomass density of resources with warming. The biomass density of consumers may increase or decrease depending on their proximity to the feasibility (extinction) boundary. Third, consumer biomass density is more sensitive to warming than resource biomass density (with some exceptions). These predictions are in line with many current observations and experiments. The model presented and analyzed here provides an advancement in the testing framework for global change scenarios and hypotheses of latitudinal and elevational species distributions.     

分子进化研究中的一些问题

本文在强调生物系统的层次结构和非线性特征的基础上,就分子进化研究中存在的某些认识上的和方法学上的问题,提出了作者的观点和简要分析。     

On the application of compartmental models to radioactive tracer kinetic studies of in vivo protein turnover in animals

A mathematical framework is presented for unifying and extending the various compartmental models and formulae used to calculate fractional protein synthesis and degradation rates in animals from data obtained by infusing labelled amino acids. It is shown how the various schemes can be derived as special cases of the product-precursor model or some three-pool variant. Three-compartment representations, which circumvent the need to measure the specific radioactivity of the precursor pool, are proposed. The mathematical solutions are generally presented in a form that is amenable to parameter estimation by non-linear least squares. The problems of measuring the true precursor pool for protein synthesis are addressed, and theoretical consideration is given to assaying aminoacyl-tRNA.