Modeling of Living Systems: From Cell to Ecosystem

By Alain Pavé

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.

• Language: English
• Publisher: Wiley
• Release date: Dec 27, 2012
• ISBN: 9781118569696

Book preview

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:

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

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British Library Cataloguing-in-Publication Data

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

ISBN: 978-1-84821-423-1

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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)

Figure I.1.gif

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.

Chapter 1

Methodology of Modeling in Biology and Ecology

1.1. Models and modeling

The notion of models in the field of biology emerged over the course of the 1960s and 1970s when it became apparent that more precision was required between the real-world subject of study and its representation through a mathematical object: the mathematical model. For living systems, one of the first major syntheses was created by J.-M. Legay [LEG 73]¹. Since then, many groups and individuals have contributed to a more precise definition of this notion and, especially, the definition of a model construction and use strategy, i.e. a modeling approach. However, the development of this method has been a source of debate, with certain parties neglecting or denying its usefulness in biology and ecology, and other experts struggling with the idea that the mathematical and physical objects in question may appear simpler than the simplest virus or the most elementary macromolecule. Moreover, the difficulty of producing general laws written in a mathematical form is not always fully appreciated. This difficulty has two implications: firstly, it limits the applicable range of models, leading to a diversification of approaches, and secondly, it leads to a questioning of the approach itself. Finally, the difficulty in obtaining precise and sufficiently numerous measurements does not facilitate our task. This being said, we may no longer doubt the fact that modeling is recognized as an effective tool, particularly when integrated into a rigorous experimental approach.

Finally, while modeling is not necessarily a form of theorization, the construction of a model may prove to be a determining element; however, we must remember that a formula or mathematical object must, first and foremost, be operational, i.e. respond to the aims of the modeling activity, and, as far as possible, be interpretable in biological or ecological terms. This formula must be able to be translated into simple terms, accessible to all, to avoid esotericism in the language used: esotericism has a tendency to cover ignorance and leads to obscurantism.

1.1.1. Models

A model can be described as a symbolic representation of certain aspects of a real-world object or phenomenon, i.e. an expression or formula written following the rules of the symbolic system from which this representation stems. This may seem somewhat obscure. Firstly, we should illustrate the notion of a symbolic system. Let us take the example of natural integers. In everyday life, we use a set of ten symbols to represent the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9: the figures in the decimal system. The numbers we use to count, for example, the number of people in a room, are made up of a figure (from zero individuals, 0, to nine people, written as 9) or two or more figures in conjunction (for more than nine people). Moreover, we can perform defined operations using these numbers, for example addition, which allows us to use two numbers to obtain a third. These rules were not defined by chance, but correspond to a concrete reality. For example, if we know that a room contains 21 people and eleven more people enter the room, the addition 11 + 21 allows us to calculate the final number of people without needing to recount: 11 + 21 = 32. This demonstrates the operation of any symbolic system: we have a set of basic symbols which we can associate and transform using a set of rules specific to this system.

In practice, the symbolic systems used in modeling today are rather more complex, but are still based on the same principle. We encounter a wide variety of systems; in addition to everyday language, significant examples include:

– mathematical language, in which case we talk of mathematical models;

– computerized modes of representation (programming languages, database formalisms, knowledge-base formalisms, multi-agent formalisms, etc.;

– geometric representations: curves, surfaces, maps, etc.;

– and many others.

Figure 1.1. gives an idea of the most widespread symbolic representation systems. See the following for an example.

The Gompertz (mathematical) model

or

This model represents the growth of certain morphological variables, including height and body mass, in higher organisms (for example, the development of body mass in the muskrat presented in Chapter 3, section 3.5.2).

The Monod model

x represents the size of a population. This model provides a good representation of the growth of bacterial populations; we shall return to this example later.

Topographical maps and choremes

These provide users with a geometrical vision of the physical environment: for example, vegetation maps give a representation of vegetal coverage. Other symbolic systems may be used by geographers, for example, the choreme system proposed by Roger Brunet (see Brunet et al., [BRU 92], for example) to represent spatial organization and dynamics.

Functional representation

This diagram represents the growth of a biomass x in the presence of a growth factor f for which the dynamics are independent of the biomass. This diagram may be associated with the Gompertz model. It assists in the interpretation of the growth phenomena described by this model in functional terms.

One of the first problems we encounter is the choice of representation: we must represent well in order to solve well [MIN 88]. The integrated form of the Gompertz model, for example, is operational from a perspective of numerical calculation; the differential form, on the other hand, is better suited to interpretation. This expression is used to develop a block diagram, leading to the interpretation of the model in biological terms (see Chapter 2 and 3).

Figure 1.1. Principal categories of formal operational models: traditional models can only represent a small part of a complex system. This is the case, for example, of living systems or systems with biological components (e.g. ecosystems). Current modes of representation offered by computer sciences, for example multi-agent models, allow us to extend these representations by encircling them and, moreover, offering new possibilities in terms of manipulation and reasoning

ch1-image1.1.gif

1.1.2. Modeling

Clearly, modeling is the approach which leads to the creation of a model. The process takes into account:

– the object and/or phenomenon being represented;

– the formal system selected;

– objectives, i.e. the use for which the model is intended;

– data (in relation to variables) and information (concerning the relationship between variables) already available or accessible through experimentation or observation.

Figure 1.2. Example of formal representations of a biological system. Here, we have highlighted formalisms based on an object-centered computer representation used in certain IT systems. Note the (possible) links with other representations (mathematical, graphical or otherwise)

ch1-image1.2.gif

The tasks which need to be accomplished clearly depend on the biological situation and formal system selected. However, in all cases, we must:

– carry out formalization activities in correspondence with the writing of the model;

– manipulate this model within the formal system to render it more useable (for example to obtain an integrated expression from a differential equation) and to study its properties;

– establish relationships with other representations (for example, the graph of a function, or the computer program which will allow users to calculate numerical values);

– interpret and compare the different representations obtained in the formal world with the biological reality (this reality is generally seen through experimental data).

1.2. Mathematical modeling

As we have seen, modeling may be based on formal systems other than mathematics. However, mathematical modeling is the best known, explored and developed system (having been in use for over 2000 years), both in terms of its internal operations and in terms of the relationships between mathematical objects and real objects or other representations (for example, geometric representations). The construction of mathematical representations follows the same type of schema laid out in Figure 1.2 (see Figure 1.3).

However, we should remember that most current knowledge in mathematics was obtained through problems in the domain of physics. The capacity of mathematics to solve certain problems in this discipline is astonishing: objects and concepts have emerged, the existence of which was suggested by the logic inherent in mathematics, permitting subtle physical interpretations. To take a current example, we might refer to gauge theory, or to the consequences of Lagrangian invariance for certain transformations known as symmetries. However, these extraordinary successes should not distract us from the fact that the picture is less rosy as soon as we move away from fundamental physics, even to the domain of everyday physics, particularly in terms of the physics of complex systems (for example, the correct treatment of the painter’s ladder problem, including friction and the oscillations produced by the painter climbing the ladder, which is no easy matter). Tackling biological systems is harder still. We do not know if nature is essentially mathematical², but one thing we can say, without excessive positivism, is that mathematics developed essentially around certain physical problems, and while it is encouraging and astounding that it responds so well to these questions, this seems to be greater proof of the excellence of the human mind rather than of the presence of a profound mathematical essence in the world.

With our current knowledge, we are able to give a more detailed vision of modeling using a global diagram where the elementary steps are laid out (see Figure 1.4). Our discussion will be based on this diagram. In Figures 1.2 and 1.3 and, to a lesser extent, in Figure 1.4., we see that we refer back to the real-world and to experiments on numerous occasions during the construction of a model. The aim of modeling is not only to describe, using a static object, but also, and especially, to generate a dynamic of thought in and around the scientific act.

Figure 1.3. Mathematical modeling consists of proposing the representation of a real-world object or phenomenon, for example, the growth of an organism or a population, using the formal system of mathematics. This figure is similar to Figure 1.2, but the earlier figure was more general

ch1-image1.3.gif

1.2.1. Analysis of the biological situation and problem

The first step consists of establishing a synthetic view of the biological situation, including existing knowledge and available or accessible data, notably experimental data, while remaining focused on the aims of the modeling process. This last point is important as, contrary to popular belief, the same formalism is not always chosen for the same situation, and the choice may be different for different problems, or even in relation to the available or accessible data. Let us take the following example.

EXAMPLE.– Model and method of dosage

The day-to-day operations of a biology laboratory often involve the use of dosages. Firstly, we develop the method and study performances to define, for example, admissible domains of use, after which calibration is carried out.

Figure 1.4. Schematic diagram showing the different steps involved in the mathematical modeling of a biological system. The results (static objects) are shown in rectangles, while procedures (dynamic processes) are shown in rectangles with rounded corners. During modeling, we do not always follow the full pathway set out in this idealized chart, either because technical considerations prevent us from going further, or because the desired results are obtained earlier in the process

ch1-image1.4.gif

We may have two different aims, and these do not contradict each other: (i) to create a mechanistic model of the dosage process itself in order to better comprehend and test its operation and, potentially, its performances; and (ii) to construct a model which is as simple to use as possible. More often than not, we end up with two models expressed in different ways.

Let us begin by clarifying what is meant by dosage: an experimental operation which links a quantity y (the measurement), to an unknown quantity x which represents, for example, the concentration of any given compound in a solution. We therefore need to deduce x from the known value, y. This operation may be presented simply (see Figure 1.5).

Figure 1.5. Schematic diagram of a dosage: a milieu containing an unknown quantity, x, of a substance. An experimental operation allows us to obtain a measurement y which is linked to x (for example an optical density). Our aim is to estimate x based on a known value of y obtained through the experimental procedure and, where possible, to evaluate the precision of this estimation

ch1-image1.5.gif

We first look at the relationship y = f (x), then the method is calibrated. This study consists of observing the relationships between x and y using samples for which the concentration x is known (generally, such samples are created in the laboratory).

Let us suppose that the relationship obtained by experimentation takes a sigmoid form. This is frequently the case, due to the nonlinearity of responses for weak and strong concentrations (Figure 1.6).

We could attempt to represent y as a function of x using a mathematical model which faithfully reproduces this form. This model might come, for example, from a mechanistic modeling of the dosage.

Let us suppose that the logistic model results from an analysis of the phenomenon (generally, the model obtained is more complicated). The form of the response is compatible with this model. However, the model is nonlinear as a function of its parameters (K and r), a fact which poses a certain number of technical problems: the estimation of the parameters and the evaluation of the precision obtained for variable x; y is supposed to be a known quantity.

Figure 1.6. Example of the response given by a dosage method during calibration. The mean curve was obtained by manual smoothing

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In practical terms, we would – understandably – be tempted to simplify the problem. We could, for example, choose a range of concentrations which provides an almost linear response (Figure 1.7) for use in our experiment. This situation would allow us to use statistical techniques from the linear domain. In particular, it would also allow us to not only estimate the parameters a0 and a1 of the model where y = a0 + a1 x simply by linear regression, but also to evaluate the precision of an estimation of x where y is known (Figure 1.8).

Figure 1.7. To avoid technical problems caused by a nonlinear response and especially the nonlinearity of the model as a function of its parameters, we may decide to carry out our experiments in the zone of responses where a linear model would be acceptable

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Figure 1.8. By placing ourselves in the linear response zone, we may estimate the parameters of the model simply using linear regression. We can determine a domain of applicability around the straight line, allowing us to associate an evaluation of precision to a value x0, as long as we establish reasonable hypotheses regarding the distribution of errors

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This example shows that, for reasons connected with the use of the model, we may be led to choose a simpler model than that suggested by the data and by mechanistic analysis.

1.2.2. Characterization and analysis of the system

The second step involves characterizing and analyzing the system, the phenomenon or phenomena or the real object which we wish to represent.

At this point, two strategies are possible. The first, data-guided strategy boils down to looking for a model which successfully describes experimental data. For example, a model with a representative curve y = f(x) which successfully shows the calibration data of a dosage method, as shown above, would fall into this category. Using this strategy, we obtain essentially practical models for descriptive use; generally, models of this type do not give us information on the subjacent processes and mechanisms (however, this is not always the case – we shall return to this point later). These may also be black box models (for example, the contents of the dosage box in Figure 1.5. are not examined, and thus it constitutes a black box). In such cases, we are only interested in modeling the output, y, as a function of the input, x, in such a way as to render it useable in practice.

The other strategy, known as a concept guided strategy, consists of seeking a representation, a model, which describes the operation of the box in the system. These models look inside the box, or at least produce and formalize plausible hypotheses. They are generally more complex than descriptive models, and also harder to work with. It should come as no surprise, therefore, to note once again that the same system may be represented by several different models depending on the intended use. In this case, we must carefully specify the nature of the system, its components and the relationships which are to be represented in the middle. To do this, we use the following strategy:

(i) Specify whether the system is isolated, closed or open. An isolated system maintains no relationship with the outside world (generally, these systems are idealizations of experimental systems). A closed system only exchanges energy with the outside world. An open system exchanges both matter and energy with its environment.

(ii) Organize the variables acting on a biological or ecological system into three major groups:

– state variables, which describe the state of the system (size or density of a population, concentration of a product, etc.);

– action variables which modify the state of the system through external action (e.g. modification of temperature by heating or cooling, injection or removal of a production, controlled immigration or emigration, etc.);

– observation variables or observable variables, which provide information on the state of the system (these observable variables may also be state variables).

This classification is taken from the field of automatics (i.e. the science of automatic control), a science which looks at the operation and command of technological systems. The modeling approach owes a good deal to automaticians (for example, see the work produced by A. Chéruy [CHÉ 88]).

In addition, we may identify input variables in the system (input of matter or energy). This input may be controllable, in which case these variables are action variables, but this is not necessarily the case. In the same way, we are able to identify output (of matter or energy), which should not be confused with observable variables.

Finally, it may be necessary to take into account the space if the hypothesis of homogeneity cannot be applied: a representation may be linear (1D), flat (2D) or in a physical 3D space.

(iii) Define the relationships between variables: these are processes inherent to the system (for example the growth of an individual or population by consumption of environmental resources) or due to experimental or technological operations (commands, measurement, evaluation, etc.).

EXAMPLE.– Dynamics of a Fusarium population

Let us consider the following experimental system: micro-organisms are immersed in a medium containing resources which will allow them to grow. The system (micro-organism population and resource-containing medium) is closed. An example of this would be the growth of bacteria such as Escherichia coli in a liquid medium in a test tube. This medium contains resources needed for the population to grow. We may also cite studies concerning the growth of microscopic soil-dwelling fungi in sterile, reconstituted soil placed in a beaker. The experiment consists of sowing at time t = 0 (beginning of the experiment) then following the development of the population over time (Figure 1.9). Reliable methods of estimating the biomass and/or number of individuals have been established in this case.

All things being equal, we consider that the dynamic of the system is described by a pair of variables (x = measurement of the size of the micro-organism population (biomass), s = quantity of resources available); the interaction between the two translates as consumption and assimilation. The result of this interaction is the production of biomass. This interaction and this result may be shown using a functional relationship (see below and Chapter 2).

Figure 1.9. Schematic representation of the experimental operations allowing us to follow the evolution of a population of micro-organisms in a medium containing resources to allow the growth of this population

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A precise example, which we shall encounter again in Chapter 4, concerns the growth of a microscopic soil-dwelling fungus of the genre Fusarium. The available data is displayed in Table 1.1.

Table 1.1. Available data for the growth of a microscopic soil-dwelling fungus of the genre Fusarium

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To obtain a clearer general picture, it is helpful to create a graph (see Figure 1.10).

1.2.3. Choice or construction of a model

The third step consists of choosing and creating a model. This step has already been covered in the dosage example. Here, we shall look at a more mechanistic aspect: the construction of a model which is no longer simply descriptive, but sets out mechanisms, following an approach based on the strategy laid out above. It would be excessive – and, most likely, impossible – to cite all the possible ways of creating a model here. However, certain tools may be useful (for example, intermediate representations using descriptive languages). Moreover, certain conditions must be respected to ensure coherence between the model and that which it represents (described in Figure 1.4 as functional coherence). Let us take an example.

Figure 1.10. Example of the results of a study of the dynamics of a population of Fusarium (microscopic soil-dwelling fungus) in a laboratory (reconstituted sterile soil). The mean curve gives a global description of the phenomenon. The experimental data results are shown as points on the graph

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EXAMPLE.– The xs model, a new expression of the logistic model

Let us reuse the terms from the example above. We wish to represent the growth of a population of micro-organisms in an isolated environment with limited resources (i.e. no input or output, no immigration or emigration). Let the symbols x and s represent the biomass and resources respectively (where, evidently, x ≥ 0 and s ≥ 0).

The problem is approached from the angle of the speed of growth of the population. For simplicity’s sake, let us suppose that the medium and the distribution of individuals across the medium are homogeneous. Thus, the speed of growth may be expressed using an ordinary differential equation:

Production in terms of growth is considered to be constant, meaning that the same quantity of resources consumed, Δs, gives the same quantity of biomass, Δx, regardless of the stock levels of resources and biomass: Δs and Δx are thus independent of the values of s and x. At any given time t:

This relationship expresses the fact that the mass balance in a closed system is constant.

In other words, the phenomenon may be represented by the system of two ordinary differential equations:

However, the speed of growth of the population must fulfill the following conditions:

– f(x, s) is a positive function in the first quadrant (x ≥ 0, s ≥ 0);

– if x = 0 then f(0, s) = 0 (there is no spontaneous generation!). The simplest model taking into