The Chemostat E-Book

Jerome Harmand

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 325

Veröffentlichungsjahr: 2017

Beschreibung

Invented by J. Monod, and independently by A. Novick and L. Szilard, in 1950, the chemostat is both a micro-organism culturing device and an abstracted ecosystem managed by a controlled nutrient flow. This book studies mathematical models of single species growth as well as competition models of multiple species by integrating recent work in theoretical ecology and population dynamics. Through a modeling approach, the hypotheses and conclusions drawn from the main mathematical results are analyzed and interpreted from a critical perspective. A large emphasis is placed on numerical simulations of which prudent use is advocated. The Chemostat is aimed at readers possessing degree-level mathematical knowledge and includes a detailed appendix of differential equations relating to specific notions and results used throughout this book.

Leseprobe

Table of Contents

Cover

Title

Copyright

Introduction

1 Bioreactors

1.1. Introduction

1.2. Modeling of biological reactions

1.3. Toward “a little more” realism

2 The Growth of a Single Species

2.1. Mathematical properties of the “minimal model”

2.2. Simulations

2.3. Some extensions of the minimal model

2.4. Bibliographic notes

3 Competitive Exclusion

3.1. The case of monotonic growth functions

3.2. Competitive exclusion at steady-state

3.3. Global stability

3.4. The case of non-monotonic growth functions

3.5. Bibliographic notes

4 Competition: the Density-Dependent Model

4.1. Chapter orientation

4.2. Two-species competition

4.3. N-species competition: exclusive intraspecific competition

4.4. N-species competition: the general case

4.5. Bibliographic notes

5 More Complex Models

5.1. Introduction

5.2. Models with aggregated biomass

5.3. The “predator-prey” relationship in the chemostat

5.4. Bibliographic notes

Appendices

Appendix 1: Differential Equations

A1.1. Definitions, notations and fundamental theorems

A1.2. Theory of stability

A1.3. Limit sets

A1.4. Supplements

A1.5. Bibliographic notes

Appendix 2: Indications for the Exercises

A2.1. Chapter 2 exercises

A2.2. Chapter 3 exercises

A2.3. Chapter 4 exercises

A2.4. Chapter 5 exercises

A2.5. Appendix exercises

Bibliography

Index

End User License Agreement

List of Tables

1 Bioreactors

Table 1.1.

Example of a Gujer matrix for the biological system whose reaction scheme is given by [1.3]–[1.4]

Table 1.2.

Example of a Gujer matrix for the biological system whose reaction scheme is given by [1.3]–[1.4] with closure of matter in the growth term

2 The Growth of a Single Species

Table 2.1.

Local stability of the equilibria of [2.2] for “Monod type” μ

Table 2.2.

Local stability of the equilibria of [2.2] for “Monod type” μ

Table 2.3.

Equilibriums of 2.2 for “Haldane type” μ

3 Competitive Exclusion

Table 3.1.

Summary of the various possible situations according to the respective position of the parameter S

in

with respect to break-even concentration λ

i

(

D

)

Table 3.2.

The five possible outcomes of the competition at steady-state between two species whose growth curves are not necessarily monotonic. For a color version of this table, see www.iste.co.uk/harmand/chemostat.zip

Guide

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Table of Contents

Begin Reading

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G1

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e1

Chemostat and Bioprocesses Set

coordinated by

Claude Lobry

Volume 1

The Chemostat

Mathematical Theory of Microorganism Cultures

Jérôme Harmand

Claude Lobry

Alain Rapaport

Tewfik Sari

First published 2017 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 Ltd27-37 St George’s RoadLondon SW19 4EUUK

www.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA

www.wiley.com

© ISTE Ltd 2017

The rights of Jérôme Harmand, Claude Lobry, Alain Rapaport and Tewfik Sari to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2017938650

British Library Cataloguing-in-Publication Data

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

ISBN 978-1-78630-043-0

Introduction

The chemostat is an experimental device invented in the 1950s, almost simultaneously, by Jacques Monod [MON 50] on the one hand, and by Aaron Novick and Leo Szilard on the other hand [NOV 50]. In his seminal article, Monod presented both chemostat equations and an example of an experimental device that operates continuously with the aim of controlling microbial growth by interacting with the inflow rate. Novick and Szilard, for their part, proposed a simpler experimental device, one of the technical difficulties at the time being to design a system capable of delivering a constant supply to a small volume reactor. Originally named “bactogène” by Monod, Novick and Szilard are the ones who propose the name chemostat for chemical [environment] is static. It is used to study microorganisms and especially their growth characteristics on a so-called “limiting” substrate. The other resources essential to their development and reproduction are assumed to be present in excess inside the reactor. It comprises an enclosure containing the reaction volume, an inlet that enables resources to be fed into the system and an outlet through which all components are withdrawn. This device presents two main characteristics: its content is assumed to be perfectly homogeneous and its volume is kept constant by the use of appropriate technical devices making it possible to maintain continuous and identical in and outflow rates. Its reputation is mainly due to the fact that it is capable of fixing the growth rate of the microorganisms that it contains at equilibrium by means of manipulating the inflow supply. First used by microbiologists to study the growth of a given species of microorganisms (referred to as “pure culture”), its usage greatly diversified over time. In the 1960s, it became a standard tool for microbiologists to study relationships between growth and environment parameters. In the 1970–1980s, it would become the focus of a strong interest in mathematical ecology even though it was somewhat neglected by microbiologists. This was mainly because at the time, the attention of the latter was attracted by the development of molecular biology approaches for the monitoring and understanding of microbial ecosystems. Studies on the competition of microorganisms rekindled interest among researchers for the chemostat in the 1980s, especially in the field of microbial ecology. It is not until the 2000s and the advent of the postgenomic era, which requires knowledge and fine control of reaction media, that a renewed interest was really observed for this device among microbiologists. It is used nowadays in scientific areas related to the acquisition of knowledge that is both fundamental, such as ecology or evolutionary biology, and applied such as water treatment, biomass energy recovery and biotechnologies in a broader sense.

The chemostat has not only been the subject of numerous publications, but also of several books essential in the field of mathematics. A question can be legitimately raised about what additional work can be done concerning a device that ultimately is very simple in principle. To this question, we can provide the several following answers.

The main source of uncertainty when a biological process is being modeled lies in modeling the growth rate of microorganisms. Bearing in mind that practitioners’ concerns must be addressed, it is based on this fundamental question and from an applied point of view that we have built this book. We do realize that the analytic expression of a growth rate is only an approximation thereof and that the properties of the model should not depend on this expression; that is the reason why we will introduce general models involving different growth models which we will subsequently specify. In particular, we study the influence of the type of growth function being considered on the outcome of a competition between several species. Adopting an increasingly complex approach, and after a general introduction which is covered in the first chapter, the second chapter naturally focuses on the growth of a single species of microorganisms on a resource. The properties of this model are analyzed for the three most important classes of growth rate encountered in biotechnology, namely, those limited and/or inhibited by substrate and those known as density-dependent, in which the growth rate does not depend on the resource only, but also on the density of the existing microorganisms. This first situation becomes more complex in Chapter 3 where we address the case of several species competing on a single resource, when growth rates are resource-dependent only. In particular, the competitive exclusion principle at equilibrium is therein exposed, of which demonstrations are given relating to the existence and the local as well as global stability of the equilibria of the system. The fourth chapter specifically addresses the case of several species competing and coexisting on a resource when growth rates are density-dependent. With the study of this type of model being significantly more complex from the mathematical perspective, it emerges that resorting to numerical simulation tools is a means to bring forward the diversity of the situations encountered. Finally, the final chapter addresses models enabling the consideration of the spatial structuring of microorganisms into several classes, here both in planktonic form and flocs or in biofilms.

We restrict our observations to situations in which a single limiting resource is considered. In a real situation, it is obvious that this will not be the case. Nonetheless, the reasoning needed for the study of these more complex situations will always be carried out based on the tools that we are introducing in this book. This publication is written to allow a linear reading in the order of the chapters that constitute it. However, according to the degree of detail, we propose several times that the reader overlooks certain passages, bearing in mind the possibility of returning to them later on, without confusing the progression during reading.

This book is above all dedicated to engineering students and PhD students wishing to study the techniques for the analysis of dynamic systems related to biological systems used in biotechnology and, in particular, to the chemostat (homogeneous system continuously operating). The primary concern is to address the challenge of studying the qualitative properties of a model already available and dedicated to formalizing a situation of interest. The question of confronting this model to data falls outside the scope of this book. We hope that the educational efforts achieved can make its reading accessible to the greatest number of people, including biotechnology students and not only mathematicians. In particular, important techniques are specifically detailed, whereas the elements requiring more significant developments or secondary significance are proposed as exercises. Furthermore, their solutions are included at the end of the book.

A rather significant appendix (Appendix 1) is dedicated to the theory of differential equations. Strictly speaking, this is not a course but a refresher of the principal notions and results to which we will refer. The reader equipped with the knowledge of a preparatory scientific class or a Bachelor of Science should be able to follow. The book contains a very large number of figures that most often will benefit the student when viewed enlarged (when reading the electronic version).

1Bioreactors

1.1. Introduction

1.1.1.What is a bioreactor?

A bioreactor is an enclosure containing a nutrient medium consisting of a cocktail of various molecules − referred to as “substrates” − upon which one or more populations of microorganisms grow, and as such the set of these microorganisms is called “biomass”. Bioreactors are used to perform operations for transforming matter through biological pathways, most often accompanied, but not systematically, by the increase of biomass in the reaction medium. Microbiology teaches us that only soluble substrates − that have created chemical bonds with water molecules − are available for the growth of living cells. Within the context of this book, from a formal point of view, a biological reaction will therefore describe the transformation of elements existing in the medium in soluble form into a solid form, biomass and possibly into a certain number of metabolites and/or gas. However, it may happen that a number of resources are present in solid form. A so-called “hydrolysis” step is necessary to transform this solid substrate into a soluble form assimilable by microorganisms. The conception that claims that a microorganism “grabs” molecules passing nearby is somehow a figment of the imagination. For example, it is accepted in soils that most microorganisms, whether they be mobile or not, excrete enzymes around them and recover the nutrients that reach them by diffusion. It is therefore essential to properly distinguish between the different processes involved before initiating the modeling of phenomena as complex as the degradation of a set of substrates by microorganisms. This is the subject of this chapter that describes the most important processes involved in this type of matter transformation and the systematic approach widely adopted in process engineering to model them. In what follows, we will only cover microbial ecosystems that are implemented in most bioprocesses.

1.1.2.Classification of biological reactors

In process engineering, bioreactors are first classified according to their mode of operation, in other words the way in which they are supplied with matter, and depending on whether microorganisms are free in the medium (so-called “planktonic” organisms) or fixed on a support; the latter could itself be fixed or mobile.

As a result, it is possible to distinguish continuously-fed systems, systems whose supply is semi-continuous and those operating in closed mode. In continuous reactors, the reaction volume remains constant, in- and outflow rates being identical. It is the most commonly used operating mode in industries aiming to process a large amount of material arriving continuously, as it is the case, for example, in the treatment of water by biological means. We will often refer to this mode in this chapter insofar as this is one of the most significant industries in terms of quantities of processed materials. Semi-continuous (or fedbatch) reactors are systems whose inflow rate is not zero but whose outflow rate is zero. In such a system, the reaction volume is thus increasing over time from a minimal to a maximal value. This type of system is particularly suitable for the production of biomass as the amount of substrate can be supplied according to the specific needs of microorganisms. It is also used when the risk of inhibition due to the substrate accumulation or a metabolic intermediary in the medium is present. Depending on the physiological state of microorganisms, it is then possible to decrease, or on the contrary, to increase the amount of resource fed into the reactor. Finally, batch mode − or reactor − designates a closed system in the sense where there is neither supply nor withdrawal of the system: substrates (the different nutrients necessary for the growth of microorganisms) as well as the inoculum (biomass) are introduced at the initial time. Therefore, the reaction volume of the system is constant over time (if possible liquid-gas exchanges are neglected) and the reaction takes place up to the moment when it is measured (or considered) that it has completed. This operating mode is widely used in agri-food, pharmaceutical and chemical industries, notably for the production of molecules with high-added value, and more generally in cultures in which the risk of contamination through the feed is high.

Since biomass is the catalyst for reaction, the effectiveness of a biological system will be all the more significant when the substrate necessary for its growth is in an appropriate form (this is referred to as biodegradability) and accessible (so-called accessibility). The homogeneity of the medium as well as biomass and resource densities will consequently play essential roles in the operation of these systems. In order to maximize the concentration of the existing biomass − but mainly to facilitate the separation of the biomass from the residual reaction medium (in other words, to facilitate the separation of the liquid and solid phases of the medium) − it is possible to resort to using a support upon which biomass will tend to settle in the form of “biofilms”1. In laboratories, even today, many engineers are testing the effectiveness of all kinds of fixed or mobile supports and are studying the properties of associated processes. In fact, it is essentially based on these considerations − relating to the feeding modes of reactors and to the manner in which biomass is retained within the system − that different technologies of reactors have been proposed. Finally, the last major element for the classification of bioprocesses is linked to the same biological processes that condition bacterial growth. It designates the set of conditions that must prevail in the medium to enable the growth of microorganisms (this is often referred to as “environmental conditions”). They are essentially ecosystem-dependent. In the next section, we review a certain number of concepts that are necessary to understand the formalization of the model of the chemostat, which we will next present in several ways in the book.

1.1.3.A brief reminder of microbiology

To grow and multiply, a (micro-) organism needs a multitude of elements, including some at trace level. A natural manner to classify organisms is to refer to the mechanisms that they implement to capture the matter necessary for their growth and to produce their energy. When we concentrate on the major factors influencing growth, a source carbon and an energy source are in effect essentially needed. Carbon is found in two basic forms in nature: organic or inorganic. For simplicity, assume here that organic carbon is the carbon produced by living entities, inorganic carbon designating the CO2 in its different chemical forms (carbonic acid, bicarbonate, dissolved CO2; we will later on return to these elements to talk about the mutual influence of biological reactions and chemical balance of the medium, which are fundamental factors affecting the growth of microorganisms).

There are essentially two sources of energy: light and chemical energy. Organisms that extract their energy from light are called phototrophs, those who take it from chemistry being called chemotrophs. Regarding sources of carbon, organisms being able to utilize organic matter are called heterotrophs, those using CO2 are known as autotrophic. By combining the carbon and the energy source being utilized, four major classes of microorganisms can then be defined:

– chemoautotrophs utilize

CO

2

as a carbon source and derive their energy from the consumption of inorganic substrates;