Modeling of Living Systems E-Book

Alain Pavé

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 777

Veröffentlichungsjahr: 2012

Beschreibung

Modeling is now one of the most efficient methodologies in life sciences. From practice to theory, this book develops this approach illustrated by many examples; general concepts and the current state of the art are also presented and discussed. An historical and general introduction informs the reader how mathematics and formal tools are used to solve biological problems at all levels of the organization of life. The core of this book explains how this is done, based on practical examples coming, for the most part, from the author's personal experience. In most cases, data are included so that the reader can follow the reasoning process and even reproduce calculus. The final chapter is devoted to essential concepts and current developments. The main mathematical tools are presented in an appendix to the book and are written in an adapted language readable by scientists, professionals or students, with a basic knowledge of mathematics.

Leseprobe

Table of Contents

Preface

Introduction

Chapter 1. Methodology of Modeling in Biology and Ecology

1.1. Models and modeling

1.2. Mathematical modeling

1.3. Supplements

1.4. Models and modeling in life sciences

1.5. A brief history of ecology and the importance of models in this discipline

1.6. Systems: a unifying concept

Chapter 2. Functional Representations: Construction and Interpretation of Mathematical Models

2.1. Introduction

2.2. Box and arrow diagrams: compartmental models

2.3. Representations based on Forrester diagrams

2.4. “Chemical-type” representation and multilinear differential models

2.5. Functional representations of models in population dynamics

2.6. General points on functional representations and the interpretation of differential models

2.7. Conclusion

Chapter 3. Growth Models – Population Dynamics and Genetics

3.1. The biological processes of growth

3.2. Experimental data

3.3. Models

3.4. Growth modeling and functional representations

3.5. Growth of organisms: some examples

3.6. Models of population dynamics

3.7. Discrete time elementary demographic models

3.8. Continuous time model of the age structure of a population

3.9. Spatialized dynamics: example of fishing populations and the regulation of sea-fishing

3.10. Evolution of the structure of an autogamous diploid population

Chapter 4. Models of the Interaction Between Populations

4.1. The Volterra-Kostitzin model: an example of use in molecular biology. Dynamics of RNA populations

4.2. Models of competition between populations

4.3. Predator–prey systems

4.4. Modeling the process of nitrification by microbial populations in soil: an example of succession

4.5. Conclusion and other details

Chapter 5. Compartmental Models

5.1. Diagrammatic representations and associated mathematical models

5.2. General autonomous compartmental models

5.3. Estimation of model parameters

5.4. Open systems

5.5. General open compartmental models

5.6. Controllabillity, observability and identifiability of a compartmental system

5.7. Other mathematical models

5.8. Examples and additional information

Chapter 6. Complexity, Scales, Chaos, Chance and Other Oddities

6.1. Complexity

6.2. Nonlinearities, temporal and spatial scales, the concept of equilibrium and its avatars

6.3. The modeling of complexity

6.4. Conclusion

APPENDICES

Appendix 1. Differential Equations

A1.1. Outline of systems for locating a point in the plane: Cartesian coordinates, polar coordinates and parametric coordinates

A1.2. Differential equations in R: first-order

A1.3. Ordinary differential equations belonging to R2, second-order differential equations belonging to R – differential systems

A1.4. Studying autonomous nonlinear systems in R2

A1.5. Numerical analysis of solutions to an equation and to an ordinary differential system

A1.6. Partial differential equations (PDE)

Appendix 2. Recurrence Equations

A2.1. Associations with numerical calculations and differential equations

A2.2. Recurrence equations and modeling

Appendix 3. Fitting a Model to Experimental Results

A3.1. Introduction

A3.2. The least squares criterion

A3.3. Models linearly dependent on parameters

A3.4. Nonlinear models according to parameters

A3.5. From the perspective of a statistician

A3.6. Examples of adjustments and types of criteria for the method of least squares, for both the linear model and also for some nonlinear models

Appendix 4. Introduction to Stochastic Processes

A4.1. Non-Markovian processes

A4.2. Introduction to Markov processes

A4.3. Ramification processes (a brief and simple introduction)

Bibliography

Index

First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE LtdJohn Wiley & Sons, Inc.27-37 St George’s Road111 River StreetLondon SW19 4EUHoboken, NJ 07030UKUSAwww.iste.co.ukwww.wiley.com© ISTE Ltd 2012

The rights of Alain Pavé to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2012946442

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-423-1

Preface

At the beginning of the 1970s, I started down the road of mathematical modeling, following in the footsteps of Jean-Marie Legay. He is cited many times in this book: not by way of a posthumous tribute, because the citations were inserted long before his sad demise, but simply because he was one of the founders – the founder, even – of the method, and because the way in which he oversaw my work on my doctoral thesis lent itself perfectly to what I was and what I wanted to do. There were only a few of us in the biometrics laboratory, which he had recently set up, and I recall close collaborations – both scientific and amicable – with the whole team; the first article, penned with Jean-Dominique Lebreton; the first book, written with Jean-Luc Chassé; and the hours spent alongside Jacques Estève preparing mathematical teaching materials for the biology students who were inspired by this bold venture – the attempt to connect two domains which were, at the time, very far removed from one another. At the time, we had to convince both mathematicians and biologists, not only with skilled speechmaking and decorative discourse, but with real results. Today, I believe the battle has been won.

This victory is also due in part to the project run by Greco (Groupement de recherche coordonné – Coordinated Research Group – at CNRS), “Analyse des systèmes” (Analysis of Systems), where I worked with Arlette Chéruy and our mutual colleagues, pooling her experience in the field of automation and mine in the field of biometrics. Together we solidified the methodological foundations for the modeling of biological systems. Also at that time, the Société Française de Biométrie (French Biometric Society) was beginning to supplement its traditionally statistical approach with forays into mathematical modeling, particularly under the guidance of Richard Tomassone and – of course – Jean-Marie Legay. In 1983, the Groupe, later to become the Club Edora (Equations Différentielles Ordinaires et Récurrentes Appliquées – Applied Ordinary and Recurrent Differential Equations), was created within Inria, and lived a stimulating life for a decade. There were several of us at the root of this pleasant and effective association: Pierre Bernhard, Jacques Demongeot, Claude Lobry, François Rechenmann and myself, along with a group of (somewhat younger!) researchers, including Jean-Luc Gouzé. For a good modeling approach in life sciences, it is necessary to be firmly within the biological and ecological reality; thus, biologists, ecologists, agronomists and doctors actively participated in our cogitation, such as Paul Nival and Antoine Sciandra, biologists from the marine environment, or Jean-Pierre Flandrois and Gérard Carret, and other doctors, chemists and researchers. This club contributed greatly to the emergence of mathematical modeling in life sciences. In 1989, the thinktank “interactions des mathématiques” (mathematical interactions) from the CNRS published a crucial contribution in its periodic report. I remember the scintillating debates that took place in this group, impelled by Jean-Pierre Kahane. In the wake of these reflections, in 1990 within the CNRS’ environmental program run by Alain Ruellan, we set up the topical program “méthodes, modèles et théories” (methods, models and theories). Beyond mathematical modeling, Alain was convinced of the usefulness and even the necessity of it. By his side, I learnt to design and drive large-scale scientific operations. In 1996, the CNRS’ Programme Environnement, Vie et Sociétés (Environment, Life and Societies Program), the successor to the environmental program, organized a conference about “new trends in mathematical modeling for the environment”, and managed, thanks to the quality of the debates and the written publications, to demonstrate the pertinence of the approach in this vast domain.

This dynamic was also developing in parallel in other communities; in some others it had long been established. It contributed greatly to the progress of modeling and its extension to most scientific disciplines. In order to foster this dynamic and promote better links between these diverse communities, in 1997 the CNRS created the interdisciplinary program “modélisation et simulation numérique” (modeling and digital simulation). It was conceived and driven forward by Claudine Schmidt-Lainé. I contributed a little to it, and my collaboration with Claudine continued for a number of years, when she took up the post of Scientific Director of Cemagref, which became Irstea (Institut national de recherche en sciences et technologies pour l’environnement et l’agriculture – National Institute for Research in Science and Technology for the Environment and Agriculture). During that time, we published several articles together, showing – in particular – how mathematical modeling facilitates the practice of interdisciplinarity.

Since then, work has continued, and although I am in charge of a project situated a very long way from Metropolitan France, I have continued to take an interest in mathematical modeling and to promote it. It occupies a significant place in the work done in French Guiana since 2002, of which a narrative can be found in the book cowritten with Gaëlle Fornet, Amazonie, une aventure scientifique et humaine du CNRS (Amazonia: a scientific and human venture by the CNRS).

In hindsight, we can give an outline. To begin with, all this bears the hallmarks of a social activity, with many personal relationships. We draw strength and inspiration from our extensive reading, our multitude of discussions, individual or collective dreams, and also from our friendships. I have learnt a great deal from my colleagues, friends, the people with whom I have worked, and from the efficiency of Mrs Piéri, to whom I owe much of the composition in my earliest published works. No less can be said of my little family – not by a long stretch. Marie-José supported me for many years, which were cut short too soon. Marc is fulfilling all the hopes we had for him. As an historian, his knowledge extends to numerous sectors, and in addition, he is always a wise and critical reader of my written work.

Ultimately, the effectiveness of the method has been proven, and a genuine scientific community has sprung up. The laboratory of a dozen people in 1966 has become a research unit that is home to over 200 scientists, and has developed in numerous directions: biometrics, mathematical modeling, biocomputing, molecular evolution, ecology and evolution biology. In 2012, we are celebrating its 50th anniversary. In my book “La course de la gazelle”, I go into further detail about this rich history. Regarding the method, this book attempts to communicate the essential bases of it with the complicity and spectacular efficiency of the publisher who agreed to take it on.

Introduction

Our aim in writing this book is to provide methodological elements for approaching modeling, by means of a general overview and through the presentation of specific examples. We shall examine the process leading to the creation of a model, i.e. a formal representation of a real-world object or phenomenon, in this case from the domains of biology and ecology. The best-known part of modeling is based on mathematics, more specifically on models using numerical variables and parameters. While this category covers the bulk of the examples given here, we should remember that other approaches are possible, for the following reasons:

– Not everything can be measured; it is therefore not always reasonable to associate an observation of the real-world with a “real” number to give a “physical” meaning to elementary arithmetical operations. Coding implemented with the aim of using tools based on arithmetic, algebra and analysis with R using the set of real numbers1 is not without its dangers.

– In certain cases, symbolic approaches may be preferable to classic numerical approaches, or may advantageously be used to complement the classic method. In cases where a decision process is being modeled, representations from the field of artificial intelligence can prove highly effective.

Nevertheless, taking a broad view of models and modeling, covering all representations using formal systems, we see that the basic concepts observed in numerical modeling may be transposed into other contexts (for example, concepts of identification or validation). In Chapter 1, we shall attempt to give a general overview of these concepts.

In this book, we shall both discuss general methods used to assist in modeling and present a number of detailed examples. In most cases, the data given in the text is real data, and is therefore suitable for use by the reader.

As we are mostly dealing with mathematical modeling, a technical reminder of the main mathematical objects and of the bulk of the methods used is included as an appendix to this book. Our focus is mainly on “specific” models established in relation to real situations, often particular experiments. However, Chapters 4 and 6 of this work clearly demonstrate the interest of “paradigmatic” models as “ideal types”, where they are used to generate ideas, explore virtual realms or speculate on possibilities. This can lead to interesting conjectures concerning real-world possibilities. From this perspective, the case of deterministic chaos is an ideal example (see [HAK 90, LET 06]).

While the model is the main focus of our attention, we should not forget that models form part of a general approach within the context of systems analysis, and are strongly connected to experimentation and observation.

Figure I.1.Allegorical figure of mathematics, the mother of modeling: Pierre de Fermat and his “muse”, imagined by the sculptor (Théophile Barrau, 1848-1913). Pierre de Fermat was one of the most famous mathematicians in history, known for his celebrated theorem which resisted all attempts to solve it for over three centuries. He was also, alongside Blaise Pascal, one of the inventors of probability theory. Salle des Illustres, Mairie de Toulouse (photo credit: Muriel Preux-Pavé)

This book has been written in such a way as to enable nonlinear reading, allowing the reader to pick and choose sections according to personal taste or requirements. For this reason, certain sections somewhat overlap.

The scientific status of modeling

As in the case of systems analysis, we may state that modeling is a methodology which transcends specific scientific disciplines; similar concepts, identical techniques and a shared language can be found in domains as different as biometrics, automatic control and econometrics. This work contains elements of this common language. However, our methodology cannot be developed independently of the underlying scientific context. Firstly, each domain of use has its particularities, and secondly, methodological development should be articulated around questions which are discipline-specific. For a biometrician, for example, a biological or ecological problem leads to the development of a methodology, and not vice-versa. It is important to avoid situations where we “produce guns to hunt dinosaurs, then spend our lives looking for the dinosaurs in order to use the guns”. We should also be careful to avoid confusing modeling with theorization; we may theorize without modeling and model without theorizing. However, modeling is a valuable tool for use in theoretical approaches; the fact that modeling may contribute to the development of these theories themselves renders it even more effective.

In concrete terms, modeling plays a part in three main functions of scientific research: (i) the detection and expression of questions, (ii) problematization and the acquisition of data and information, and (iii) the definition of actions and the study of their consequences.

The status of the modeler: is there a place for this specialism?

While mathematics has long been used to represent observed phenomena, notably in the physical sciences, modeling as a specialism has emerged only recently (within the last 30 years, at most) when it was noted that, in other disciplines, such as life sciences or engineering, the construction and use of formulae necessitated the methodical assembly of techniques from other disciplines (e.g. mathematics, statistics and computer science). Currently, scientists are generally in favor of this label. Modeling constitutes a movement which participates in the dynamics of the sciences; the existence of modelers is a result of the emergence of a specific approach and particular techniques. However, we feel that the activities of a modeler cannot be dissociated from a particular scientific domain; these individuals require a strong background in their own specific field. The modeler also needs to master a wide variety of techniques and methods. Specialists of this kind are a rare breed, if indeed they exist at all. As the modeler cannot be omniscient, he or she must have a specific area of expertise, be it statistics and probability or analysis or computer science.

Essentially, the modeler must specialize in a strategy, or, in other words, know how to model effectively in the specific discipline to which his or her skills are to be applied.

The role of the modeler in a scientific project

Clearly, the role of the modeler is an important one: modeling leads to a form of synthesis, and the modeler acquires a global, often critical, view of the project with which he or she is involved. He or she may be the only individual to benefit from this unique vision.

As knowledge so often equals power, the modeler is in a privileged position. However, generally speaking, there is no reason to recognize modelers as holding such an important position. It would even be risky to adopt this as a general rule, implicit or otherwise; leadership of a project requires a number of other qualities. Another point should also be made: where the modeling function involves the manipulation of formal objects and leads to a “pencil – paper – computer” way of working, the modeler needs to have had real contact and even practical on-the-ground experience, using the equipment and experimental techniques concerned. The modeler should be well aware of the real techniques involved in measurement and observation. A modeler working on the dynamics of macromolecules must understand the workings of the measurement apparatus; an individual modeling the evolution of inter-tropical forestry systems would do well to acquire on-the-ground experience. On occasion, the modeler’s perspective as a “naïve expert” may lead to the representation of observed entities in a way which is distinct from that used by specialists in the specific domain, who are influenced by the dominant concepts of their disciplines. To maintain the “freshness” of their viewpoint, the modeler needs to immerse him/herself in the knowledge of the biological aspects involved in specific research activity while, at the same time, maintaining variety in his/her objects and subjects of study. In this way, the modeler will avoid becoming locked into the use of dominant representations and concepts. These dominant ideas have their place, being selected on the basis of suitability for specific disciplines, but diversity of perspective must be maintained for these disciplines to evolve. In this way, the modeler may play a critical and constructive role.

In all cases, it should be clear that the model does not constitute an end in itself, but is simply a tool in the scientist’s toolbox. It forms part of the model-experiment dialectic encountered in scientific discourse and practice. Finally, we must note that data is usually proven to be right, but this is not always true; in certain cases, a model may validate or invalidate data. In this way, a model may be used as a monitoring tool.

How far can these skills be adopted, in whole or in part, by the scientific community as a whole?

A few years ago, we attempted to respond to this question based on the use of modern computing tools. The tools in question have undergone considerable development since then, making them increasingly useful both to modelers and to laypeople. However, there is no reason for despair; we must vary our approaches and not rely uniquely on the miracles of computing (even if it is “artificially intelligent”), but continue down the time-honored routes of teaching and training (including the introduction of modeling into university syllabi and distribution of this information through schools, publications, etc.). This is the context in which the present work was written.

The contents of this volume are intentionally partial, some might say biased; this is not an encyclopedia. Only certain problems have been covered, and, while the author takes a broad view of modeling, precise and operational developments are based on the real-life examples we have encountered, which were themselves specific and dealt with in some detail. The approach is essentially quantitative, giving us access to a wide range of effective tools. Nevertheless, as we mentioned earlier, it is important to remember that not everything is quantifiable or measureable. These characteristics do not, however, disqualify something from being the object of modeling. On this point, we must be wary of a fundamentalist or even reactionary viewpoint, held by supporters of a particular technique, which promotes a very narrow view of modeling. We should also be careful to avoid taking a purely commercial standpoint at the expense of the ethical obligations proper to researchers and engineers: the model should not be the peremptory argument for a decision which has already been made, or a tool used to support an ideology (i.e. to promote a particular worldview, where the model is made to fit this idea). Modeling may, however, be used as an instrument in defining a technical or political decision. While we are aware of the importance of an open and investigative spirit, we also consider it essential to promote honesty and rigor. For this reason, we would support the creation of a deontology of modeling (we insist upon this point in reaction to certain practices we have observed which have filled us with horror).

This last phrase concluded the foreword to the 1994 edition. Twenty years on, considerable progress has been made in the field of modeling, but the ethical and deontological considerations involved in the domain are still shaky, and everyday life continues to demonstrate the limitations of these techniques, for example in economics where the use of modeling may, on occasion, prove somewhat questionable.

1 The term is misleading when we consider the degree of sophistication of the formal construction of this set of numbers.