
Application of Simple Structured Models in Bioengineering
A. H a r d e r
G i s t  B r o c a d e s R e s e a r c h a n d D e v e l o p m
e n t , P .O. Box 1, 2600 H A Delf t , N e t h e r l a n d s
J. A. R o e l s
v a k g r o e p A l g e m e n e en T e c h n i s c h e Biologic
, J u l i a n a l a a n 67, 2628 B C Del f t ,
N e t h e r l a n d s
t Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 56 2 Brief Survey of Microbiological and
Biochemical Principles Relevant to the Construction
of Structured Models . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 59 3 Corpuscular Description and its Relation to the
Continuum Approach . . . . . . . . . . . . . . . . . 61 4
Construction of Structured Continuum Models . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 65 5 Relaxation
Times and their Relevance to the Construction of Structured Models
. . . . . . . . . 68
5.1 The Concept of Relaxation Times . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 A Model Describing the Dynamics of Product Formation Based on
the Relaxation
Time Concept . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 71 6 Models of Primary Metabolism in Microorganisms . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1 TwoCompartment Models . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75 6.2 A ThreeCompartment Model of Biomass Growth . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 83
7 Models for the Synthesis of Enzymes Subject to Genetic Control
. . . . . . . . . . . . . . . . . . . . . . . 88 7.1 Introduction .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.2
Repressor/Inducer Control . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.3 Catabolite Repression . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 91 7.4 Enzyme Synthesis Mediated by mRNA . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.5 A
Model for Diauxic Growth . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 99 9 Symbols . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 100
10 References . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 103
Mathematical models are an important tool to any engineering
discipline. The mathematical treatment of the processes encountered
in bioengineering is complicated by special problems caused by the
complexity of living systems and the segregated nature of microbial
life. It is especially this last mentioned feature which can result
in errors if the continuum approach commonly used in engineering is
adopted.
The present paper reviews and updates the theory of the
construction of structured continuum models, which become
significant in applications where the common unstructured approach,
e.g. Monod's model, fails. This particularly applies to transient
situations in batch, fed batch or continuous culture.
Emphasize is placed on the need for structured models, which are
as simple as possible. A guide to judging the necessary degree of
complexity is provided using the time constant concept, which is
based on judging the time scales on which the various regulatory
mechanisms are operative.
The significance of structured models to the description of
primary metabolism is described with special reference to growth
energetics.
As a second important range of applications, the dynamics of
extracellular and intracellular enzyme synthesis, is discussed,
both from the viewpoint of product formation and diauxy in growth
on mixed substrates.
The need for experimental verification and the potentialities of
continuous culture, especially in transient situations, in that
respect are indicated to be the main subjects in which research
effort needs to be invested.

56 A. Harder, J, A. ?~oels
1 Introduction
An important aspect of the methodology of presentday physics is
the construction of mathematical models of aspects of the behavior
of a real system. Scientific progress is made possible by testing
the implications of the models, for example by carefully planned
experiments. This generally results in a cyclic process in which
the old model is rejected and a new one postulated (Fig. 1).
It should be emphasized that a model can only represent some of
the properties of a system under consideration. Little would be
gained if the behavior of the system was modelled in all its
intricacies, as this would result in a model scarcely more easy to
handle and understand than the real system it represents.
The assessment of which aspects of the system require
consideration is essential in the construction of a workable model
and should be guided by the application one has in mind and the
possibility of experimental verification of the model.
A model will hence always be based on assumptions concerning the
principles of the system's behavior. These should be clearly
stipulated because their range of validity will have important
consequences for the validity of the conclusions derived from the
model. A more thorough discussion of aspects of the philosophy of
modelling can be found in literature 141
,
TRANSLATION INTO "'] A MATHEMATICAL MODEL
L SOLVING THE t EQUATIONS
DETERMINATION OF' ,,,'1 PARAMETER SRNSIVITY
TESTING THE MODEL I
NO YES
Fig. 1. Flow diagram of the construction of a methematical
model

Application of Simple Structured Models in Bioengineering 57
Two broad groups of approaches to the description of systems
exist. These are the continuum approach and the corpuscular method
5). In a corpuscular model, it is recognized that, probably for all
real systems, typical behavior is caused by the concer ted actions
of objects. The system is inhomogeneous (discontinuous) if length
scales typical of the objects' sizes are considered. For example,
the smallest amount of a chemical substance still having the
properties of that substance is a molecule; likewise, a single
bacterium is the smallest quantity having the properties of a
bacterial :species. In view of our presentday image of matter, the
corpuscular approach must be considered the most realistic method
of description. Nevertheless, in the engineering sciences most
problems commonly encountered are treated in terms of the afore
mentioned continuum approach. In a continuum description, the
corpuscular nature of reality is ignored and the system is
considered to be continuous in space. Variables characteristic of
this approach are temperature, pressure and concentrations of
substances and organisms. The continuum approach is preferred
because these models are conceptually simpler and can more readily
be treated mathematically.
Classical microbiology considers the basic unit of all
functioning organisms to be the cell. Hence, for the description of
microbial systems the corpuscular approach seems to be particularly
suitable. This approach has been pursued by Ramkrishna 6) and by
Fredrickson et al. 7~ who apply the term segregated models to this
class of approaches. Despite the fact that organisms are
corpuscular in nature, the continuum approach, earlier termed
distributive 4~, is most commonly encountered in the description of
bioengineering systems. It can be shown 3,4.7) that in some
instances, continuum models can be formally derived from a
corpuscular description, and both treatments become equivalent in
these cases. In other situations, direct equivalence cannot be
shown and the continuum treatment must be handled with care. A
basic understanding of the corpuscular method is therefore
worthwhile and a brief outline will be given in Sect. 3. After this
short excursion into the corpuscular approach, the continuum
approach will be exclusively used.
A second point relevant to models of bioengineering systems is
the distinction between the deterministic and the probablistic
approaches 3,4~. The difference between both approaches rests in
the nature of the predictions about the future behavior of the
system that the model allows. In a deterministic approach, the
knowledge of the state vector of the system 3~ (a vector composed
of all variables necessary to specify the state of the system at a
given moment in time) allows an exact prediction of the future
behavior during an arbitrary time interval. With the probablistic
approach, it is only possible to specify a probability that the
state vector will be in a given region of state space (state space
is a coordinate system of the dimensionality of the state vector.
Each point in state space corresponds to a single value of the
state vector). The predictions generally become less and less
accurate with increasing time. The probablistic "behavior" of a
system is caused less by the nature of the system than by the
nature of the observer and his observations. A probablistic
approach is often used if the observer is unable to obtain
sufficient information about the state of an object and its
subsequent behavior to allow deterministic predictions (for
example, if not all mechanisms are known, or certain important
state variables cannot be measured). Experience shows that the
necessity of a probablistic approach increases if the number of
individual objects in the system is low. Thus, the behavior of a
large number of organisms growing in a bioreactor can be adequately
described" by a deter

58 A. Harder, J. A. Roels
ministic model, but the behavior of small numbers of organisms
(e.g. in the last stages of sterilization) calls for a probablistic
approach 8~.
In engineering studies deterministic models are almost
exclusively used. This preference is due to the nature of the
predictions and the simpler mathematical structure of these models.
Hence, the principles of the probablistic approach will only be
indicated briefly (Sect. 3), attention will be focussed on
deterministic models.
A further relevant distinction is the classification into
structured and unstructured models 4~. In unstructured models the
state of the organisms in the culture is assumed to be sufficiently
specified by the total number of organisms or the dry weight of
biomass present. However, in a structured model the organism is
described in greater detail, and for example the concentrations of
DNA, RNA and protein per unit dry matter are also specified.
Unstructured models are mathematically more tractable and more
easily verified experimentally. Thus, they are therefore to be
preferred in all applications where their accuracy of description
of a system is suited to the desired application. The Monod
equation 9) for the substrate limited growth of micro organisms is
an example of a successful unstructured model. Originally, it was
empiri cally derived from results on the batch culture of
microorganisms. Herbert ~0) intro duced a term accounting for
endogeneous metabolism, extending it to apply to growth in the
chemostat. More recent work 3.1~) has investigated its application
to fedbatch culture. In general, unstructured models can be
considered a good approximation in two distinct cases. These cases
arise when the composition of the organisms is not relevant to the
aspects of the system the model describes, or when it is
independent of time, i.e. in balanced growth 41
Both conditions are fulfilled in chemostat theory, where the
outcome of the model ling exercise can be shown to be insensitive
to the details of the kinetic assumptions used. Furthermore, at
steady state, the composition of the organisms does not change. The
unstructured approach also assumes composition to be equal at
differing dilution rates, but this is not validated by experimental
evidence ~2,~3)
In short, although unstructured models can often be
advantageously applied to the description of a system's behavior
there are a large number of applications in which these models fail
to be adequate. This applies when the biomass composition changes
drastically, like in some stages of fedbatch processes and the
early stages of batch growth (lag phenomena), and in situations
where a specific constituent (e.g. protein and RNA content in SCP
production) must be modelled. In those cases a structured approach
is necessary. A large number of compositional variables can be
attributed to biomass. If this is performed to the extreme, very
complex models result TM~5). However, these models contain so many
parameters that they become too unwieldy for useful applications in
bioengineering. A class of potentially useful models are formed by
a simple extension of the unstructured approach, in which the
amount and the properties of the biomass are specified by two or
three variables. These are the socalled two or threecompartment
models. They combine a better description of the system's behavior
with moderate mathematical complexity and a sufficiently low number
of parameters to permit experimental verification. Examples of such
models are appearing more frequently in the literature. However,
some conceptual difficulties are inherent to the formulation of
such models. These, if not carefully considered, may lead to models
which are structurally wrong 16).
The objective of this review is to show some applications of
simple structured models

Application of Simple Structured Models in Bioengineering 59
in biotechnology. The theory of the construction of structured
continuum models will be treated to clearly point out the
difficulties and to show how these can be avoided.
2 Brief Survey of Microbiological and Biochemical Principles
Relevant to the Construction of Structured Models
In microorganisms, a great variety of chemical reactions take
place between a limited number of precursor molecules. This
reaction pattern results in a complex macro molecular machinery of
great structural diversity. In order to grow optimally under
varying external conditions, organisms must be able to adapt their
activities to changing environmental conditions. A number of
mechanisms operative in influencing the reaction pattern inside an
organism can be distinguished:
a) Direct massaction law regulation Changes in the
concentration of one or more of the intermediates or substrates of
a reaction pattern causes changes in the rates of the reactions
constituting the pattern. These changes, however, are generally not
beneficial to the organism. One of the possible theories behind the
Monod equation is an example of a deduction based on massaction
law considerations 1~. In general, the time constants of these
changes are small (i.e., the action is quickly established). b)
Regulation of the activity of enzymes Enzymes are macromolecutes
with complex secondary, tertiary and quaternary structures.
Interactions of these molecules with small molecules, effectors,
may cause changes in the enzyme's conformation and hence in its
catalytic action. Controls have been demonstrated for the main
energy supplying pathways 18) and in anabolic bio synthetic
sequences 19, 20)
It is now accepted that the mechanisms, known as the allosteric
controls, are vital to the integration of microbial metabolism.
General and useful mathematical models for a single regulatory
enzyme have been proposed 21,2z). A remarkable general approach to
the study of sequences of enzymes with regulatory characteristics
has been described by Savageau 23) The time constants of these
controls are generally larger than those of the mechanism described
under a). c) Regulation of the macromolecutar composition of the
cell The concentrations of the various macromolecules of the cell
are adapted to changing environmental conditions by altering their
rates of synthesis. The changes in the steadystate concentrations
of RNA, proteins, DNA and carbohydrates in response to dilution
rates in continuous culture are well established ~2,13k The RNA
concen tration, especially, is known to increase markedly with
increasing dilution rate.
In Fig. 2, the results of various investigators 12.24.28) are
summarized. As can be ~een, the relationship between the RNA
content and the specific growth rate in t steady state appears to
be independent of the nature of the organism and of :he means of
the growth limitation employed. This is the basis of the "constant
:fficiency hypothesis" for protein synthesis at the ribosomes, i.e.
each ribosome )roduces protein at a constant rate, independent of
environmental factors 12~. This ~ypothesis was later refined. It
was shown that, especially at low specific growth

6O
RNA in d r y moss (%)
4 0 
A. Harder, J. A. Roels
3 0 
20
10
o tx : + 0 +
(1 I
0 0 .5 1.0 1,5 Ix ( h 1 )
x A z o t o b a c t e r c h r o o c o c c u m o B a c i l l u s
m e g a t e r l u m
Aerobacter aerogenes + Canc l ida u t ] l i s S a l m o n e l l
a t y p h l m u r i u m ~ Esche r [ ch io col i
Fig. 2. Compilation of data of RNA % as a function of dilution
rate
rates, more RNA is present than is required by the constant
efficiency hypothesis ~3) This unused protein synthesis capacity
was shown to be mobilized quickly in transient states following a
sudden increase of the specific growth rate 29)
More drastic changes in the cellular composition are known to
follow alterations in the type of the nutrient supplied. The
amounts of the various enzymes produced by the cell are regulated
to meet requirements. The operon model postulated by Jacob and
Monod 3o) explains these phenomena from the existence of controls
concerning the rate of transcription of the codons present on the
genetic material. The rate of transcription of a codon onto
messenger gNA is controlled by regulatory genes. The cell produces
a repressor protein which, in the active form, binds to the
operator and blocks transcription. An effector, often derived from
the substrate of the enzymic sequence the operon codes for,
interacts allosterically with the repressor protein, either binding
to or releasing the operator, depending on whether the effector is
an antiinducer or an inducer, respectively. Thus, this mechanism
allows the organism to change its enzymic constitution to suit the
demands posed by nutritional changes in the environment.
In recent years 3~), it has been recognized that a second
important control of the transcription of codons exists. Efficient
transcription to mRNA is postulated to only take place if a
complex of cAMP (cyclic AMP) and CAP (catabolite activator
protein) is bound to a promotor gene on the DNA. Certain
catabotites, such as glucose, apparently reduce the cAMP
concentration and inhibit the expression of the codon (positive
control, catabolite repression).
The genetic control mechanisms mentioned are relevant to the
description of lag phase phenomena, diauxy and product formation
(intracellular and extracellular enzymes). The time constants of
these mechanisms are larger than those mentioned under b).

Application of Simple Structured Models in Bioengineering 61
d) Selection within a population of a species Natural selection
offers a further possible mode of adaption. Genetic variation
within a species may lead to the selection of an individual having
properties which confer an advantage in the environment under
consideration. This causes a shift in the mean properties of the
population, and is particularly relevant to continuous culture
techniques which generally favor fast growing organisms. This can
cause problems in industrial processes where organisms with a lower
productivity may have a selectio nal advantage over the industrial
strain as they may direct more energy to growth and less to product
formation. The population thus gradually becomes less productive.
This has been shown to happen in an Qtamylase producing strain
32~. These selectional processes are characterized by time
constants larger than those of the adaptational processes. e)
Changes in the composition of a mixed species population In a
number of important applications, for example in waste water
treatment, the biotic phase is made up of a mixture of organisms
rather than of a single species. Changes in environmental
conditions may induce changes in the fractions of the different
species 33). m model for waste water treatment must allow for these
phenomena in order to describe dynamical situations with some
accuracy. The time constant for such changes may be very large.
3 Corpuscular Description and its Relation to the Continuum
Approach
A continuum model of a population of microorganisms assumes the
organisms to be homogeneously distributed throughout the culture
fluid, the cellular nature of organisms being considered to be
irrelevant. This approach leads to loss of realism, but it is
easier mathematically. In some instances continuum models can be
formally derived from a corpuscular treatment by the use of
suitable averaging techniques over all objects present in the
culture. An important aspect of such a procedure is that it leads
to a better understanding of the correct formulation of kinetic
equations in the continuum representation 3, 7.34)
In this context some aspects of corpuscular theory will be
briefly reviewed. More complete accounts can be found in the
literature 6,7,35)
A collection of objects is considered (e.g. a number of
microorganisms). The state of each of the organisms is
characterized by a state vector ~, containing variables which, for
example, describe the composition of the organism in terms of the
macromolecules DNA, RNA, protein and carbohydrates at a given
moment of time. A multidimensional probabilitydensity function,
W(o), is now defined, giving the probabilitydensity for the state
vector to have a value in a certain region of state space (i.e. a
probablistic approach). This probabilitydensity function is
defined by the following equation:
dN(o~) = NtW(o) iIi/dco i (1)

62 A. Harder, J. A. Roels
Equation (1) shows the relationship between the number of
organisms, dN(o~), having a state vector in the state space volume
element, 1~ dei, the number or organisms per
i
unit volume, N,, and the probability~ensity function. The
moments of the multidimensional probabilitydensity function are
important
quantities. For simplicity, these will be demonstrated for the
case of a unidimensionai probabilitydensity function, i.e. the
case in which the state vector contains only one variable (i.e. i =
1, fJ)i = .0).
The first moment of the probabilitydensity function is defined
by
co
(to) = ~ oW(~0) do~ (2) 0
The first moment is the average value of o for all organisms
present in the culture.
Another important quantity is the second moment, (co2), of the
probabilitydensity function:
( 62 ) = ~o flqJ(c) do (3) 0
The meaning of ((02 ) is best illustrated by comparing it to
variance, 0 2 , as used in statistics 36~:
eo
o 2 = J" ( o  ( o ) ) 2 ~ ( o ) d o C41 0
The following relationship is easily shown tO hold:
o 2 =  2 (5)
Now a function of the property o, fro), is considered, its
average value for all objects in the culture is given by:
(f(o)) = ~ f(o) q~(o) &0 (61 0
To illustrate the application of Eq. (6) the following example
is considered: A culture or organisms performs an enzymatic
reaction due to the action of an enzyme E. The amount of enzyme per
organism is e. The probabilitydensity function for e is W(e). The
number of organisms is assumed to be sufficiently large and a
MichaelisMenten type Eq. (37) is assumed to apply to each cell.
Then the rate of enzymatic reaction
per cell, RE, can be written as:
RE _ keC~ (7~ K M + C~
where C s = concentration of substrate, K M = Michaelis
constant.

Application of Simple Structured Models in Bioengineering 63
The average rate of reaction per cell for all organisms in the
culture is given by"
f keC~ ( R E ) = K M + C~   W(e) de (8)
Equation (8) can be modified to:
kCs (RE) =   (e) (9)
K M 4 C s
The overall rate of reaction, rE, for all organisms in the
culture is obtained if the right hand side of Eq. (9) is
multiplied by N t
kC, rE  KM + C~ N,(e) (10)
The product Nt(e) is the amount of the enzyme per unit volume of
the culture; it hence is a continuum variable which will be
indicated by C E. Thus, Eq. (10) can be written as:
kCs rE  K M + C~~ CE ( 11 )
Equation (11) is the continuum formulation of the
MichaelisMenten model for the culture. It was shown above to be a
direct consequence of a formal corpuscular treatment. Hence, the
corpuscular and the continuum approach are equivalent in this
case.
The reasoning presented above can be easily generalized to the
case of a multidi mensional probabilitydensity function, a
situation relevant to the construction of structured continuum
models. For example, the rate of a sequence of enzymatic reactions,
R, is considered,' which is a function of the amounts of a number
of compounds present in the cell, expressed by the vector to, and a
number of extra cellular concentrations of chemical
substances.
R = R(to, y) (12)
In Eq. (12) y is the vector of abiotic, extracellular,
concentrations. The average rate of the enzymatic reaction per cell
for all cells present in the
culture, (R) , is now given by:
o co
(R) = ... f R(to, y) W((o) dto (13) O O
For the general case, the integral at the righthand side of Eq.
(13) cannot be simplified further. Straightforward evaluation is
possible if the following conditions hold:

64 A. Harder, J. A. Roels
a) R(to, y) can be factored out with respect to the individual
elements of the organism's state vector:
R(o, y) = k~T(y) I ] O)i (14) i
In Eq. (14) T(y) is a function of the extracellular state
vector, y. b) The properties, %, of the cell are statistically
independent; in this case, the probabilitydensity function can be
written as:
y(~o) = H %() (15) i
If Eqs. (15) and (14) are combined with Eq. (13), also
considering Eq. (2) for the restrictive case to which both
conditions mentioned under a and b apply, it follows that
(R) = k~T(y) 1[ (ml) (16) i
in which the (toi) are the average values of each of the
individual properties of the cells.
Using Eq. (16), the total rate of reaction in the culture is now
given by:
r E = k~T(y) 1~ (%) N, (17) i
If one cell has a mass W, and the mass fractions of the various
compounds in the cell are given by %, it follows:
r E = k~T(y) l~ .  t wiW C X (18) i
In Eq. (18), n is the dimensionality of the state vector to and
C X is the concentration of biomass dry matter.
Equation (18) shows that in the correct approach to structured
continuum models, the extracellular and intracellular
concentrations should be treated differently. Special precautions
are not necessary for the abiotic concentrations (vector y); they
can be expressed as concentrations per unit of culture volume. The
biotic concentrations (i.e. the concentrations of cellular
compounds) are, however, best expressed as mass fractions of the
cellular mass, the socalled intrinsic concentrations 16) Finally,
the rate equation (18) is shown to be first order with respect to
the total biomass concentration, Cx, a feature, which is
intuitively correct 3)
It is also possible to construct a correct rate equation using
biotic concentrations expressed per unit of culture volume, when
the general form of the rate equation
becomes:
,1 1, (19) r E = kCT(y) [ I xiW C~
xi in Eq. (19) is the biotic concentration of compound i
expressed per unit volume.

Application of Simple Structured Models in Bioengineering 65
A kinetic equation of the following type is often proposed:
rE = keT(Y) H xi (20)
The structure of this equation is based on the massaction law
rate equations fundamental to most approaches to chemical kinetics
3s~. It represents, however, an incorrect approach to
bioengineering kinetics when reactions between cellular
constituents are also considered. This is obvious from a comparison
of Eqs. (20) and (19). This difficulty was first pointed out by
Fredrickson ~6~ who dealt with examples of such errors in the
literature 39,40). These errors have however also appeared in the
recent literature 4~
The problems resulting from the use of equations similar to Eq.
(20) were illustrated by Roels and Kossen 3~ by referring to the
model of Williams 39)
There is another problem associated with the use of the
continuum approach which must be discussed. The averaging process
according to Eq. (13) only leads to meaningful deterministic values
if the number of objects considered is sufficiently large. In
general, the order of magnitude of the variance of a sample of N
objects is equal to the ensemble variance divided by N. In view of
this, if the number of organisms considered becomes less then
10210 * 3), the deterministic continuum approach should be
handled with caution.
An important problem involves the application of massaction law
considerations at the level of the bacterial cell where, in many
cases, there are only a few molecules per individual cell. Examples
of such problems have been discussed for MichaelisMenten kinetics
4.2) and for the operon model 43~ These exercises clearly show that
in such cases, a mass action law approach to kinetics may lead to
errors. In the present review, however, these problems will be
ignored.
4 Construction of Structured Continuum Models 34, 44)
In the following a culture of microorganisms will be considered.
The concentration of biomass present in the culture is C x. The
state of the culture is defined by an overall state vector C which
contains the concentrations of the compounds in the biotic and
abiotic phases.
According to the arguments developed in Sect. 3, the overall
state vector is divided into biotic and abiotic parts:
C = {y, x} (21)
The abiotic state vector, y, contains the concentrations of k
compounds which are not part of the intact biomass. The biotic
state vector, x, contains the concentrations of n compounds which
are part of the biomass.
Components present in both the biotic and abiotic phases are
identified by distinct numbers in both state vectors. The compounds
specified by the state vector x, are assumed to account for all
biomass dry matter which, however, does not necessarily imply the
specification of the concentration of each component of the
biomass

66 A. Harder, J. A. Roels
separately. The elements of x may also refer to groups of
compounds. Under this condition, the following relationship
holds:
C~ = L xi (22) i = l
In the preceding section it was shown that a correct approach to
bioengineering kinetics is facilitated by the use of intrinsic
concentrations for the biotic phase. Hence, a vector w of intrinsic
concentrations must be defined.
The elements, w i, of that vector are given by:
w i = xl/C x (23)
In a culture of constant volume, the concentrations of the
various compounds present can be treated as extensive quantities
and their rate of change can be obtained from the general procedure
for the formulation of balance equations funda mental to all
physical theory. This balance principle is stated as in Ref.
3).
ACCUMULATION IN
SYSTEM = CONVERSION + TRANSPORT
Two sources contribute to the accumulation of a compound in a
system. These are transport of the compound to the system and
production of that compound within the system. In vector notation,
the verbal statement can be represented as:
= rA + ~ ( 2 4 )
In this equation r A is the vector of the net rates of the
production of each compound in the reaction pattern in the system.
@ is the vector of the rates of transport of these compounds to the
system. The reaction pattern inside the system is now characterized
by the vector r of the m independent reactions taking place in the
system ,s). The net rate of formation of each compound is now given
by:
r A = rot (25)
Equation (25) defines the stoichiometry matrix, at, an m x p
matrix (p is the dimen sionality of vector C). In this matrix, the
element Qtij gives the amount of compound j produced in the ith
reaction.
Expressions analogous to Eqs. (24) and (25) may now be
formulated for the rates of change of the abiotic and biotic state
vectors:
= r % + @y (26)
= r atx + @x (27 )

Application of Simple Structured Models in Bioengineering 67
In Eqs. (26) and (27), ay and ~ are the stoichiometric matrices
for abiotic and biotic compounds; ~y and O~ are the vectors of
rates of transport for abiotic and biotic compounds.
Equation (26) can be used to describe the dynamics of the
abiotic state vector. The balance equation for the biotic state
vector poses special problems.
Firstly, an equation for ~ must be formulated. As the state
vector x refers to intact cells, transport of compounds to or from
the system can only take place as intact cells. This excludes the
possibility of removal or addition of cells of a composition other
than the population mean. Thus, the following equation holds:
O. = q~w (28)
where ~, is a scalar representing the rate of transport of
biomass to the system, expressed per unit of system volume.
Secondly, as previously stated (Sect. 3), the kinetic equations
are, as far as the biotic compounds are concerned, best expressed
in terms of intrinsic concentrations, i.e. in terms of the vector
w.
A direct formulation of a balance for the biotic state vector
is, however, impaired by the fact that, even if the system's volume
is constant, the intrinsic variables are not extensive quantities.
It is, however, possible to formulate an expression for the
dynamics of the intrinsic state vector starting with Eq. (27).
Inserting Eqs. (23) and (28) it follows:
(w'C~) = r a~ + @~w (29)
By differentiation of the lefthand side of Eq. (29) it
follows:
C;# + w ~ = rat. + ~xw (30)
If the n component equations of the vectorequation (Eq. (30))
are added, it follows:
i = l i = l i= 1 (31)
In this equation 1 is a column vector of dimensionally n
composed of ones. The matrix product r aq # 1 in Eq. (31) is equal
to the net growth rate, r~, of the total
amount of biomass dry matter. As the sum of all n elements of
the vector w equals unity, and the sum of the
time derivatives of the n elements of w equals zero, Eq. (31)
can be written as:
(~x = r~ + @x (32)
If Eq. (32) is substituted into Eq. (30), the following equation
for the dynamics of w is obtained after rearrangement:
= (r=~  wrO/C, (33)

68 A. Harder, J. A. Roels
Equations (26), (32) and (33), together with a set of
constitutive equations for the rates of reaction r and constitutive
equations for ~x and ~y, form a complete structured continuum model
in which the biotic compounds are treated in terms of intrinsic
concentrations. Equation (33) shows that in the state equation for
the intrinsic biotic state vector w a term wrx appears. This
accounts for the dilution of the biotic compounds by the increase
in the total amount of biomass. Omission of this term in the
formulation of an equation for the biotic state vector dynamics is
another important source of errors in structured continuum models
(see article of Fredrick s o n 16)).
The approach to structured continuum models developed in this
section will be applied to some examples in the following
sections.
5 Relaxation Times and their Relevance to the Construction of
Structured Models
5.1 The Concept of Relaxation Times
Bioengineering systems are, like all engineering systems, of a
complex nature and a rigorous description of their behavior leads
to large sets of complex mathematical equations containing a large
number of parameters not readily obtainable experi mentally.
Hence, a consistent procedure must be developed to simplify this
description to a realistic model relevant to the desired
application. An interesting approach to the depiction of complex
systems was developed in thermodynamics about 1950 46, 47). It may
be extended to the treatment of bioengineering systems. This is the
theory of socalled incomplete systems which are described using
the concept of "hidden variables".
Thermodynamics concerns the description of systems in terms of a
black box approach, using only macroscopic variables which can be
observed from outside the system. However, processes which cannot
be externally observed and yet still contribute to the behavior of
the system often occur, e.g. when chemical reactions take place
within the system. A representative example is an unstructured
approach to the depiction of continuous culture, where the directly
measured macroscopic variables are the concentration of biomass, C
x, and the concentration of the substrate, C s. The internal
processes of the organisms will adjust immediately after a shift in
dilution rate. These changes, for example in RNA and protein
content, cannot be directly observed but certainly influence the
behavior of the organisms.
In thermodynamics, the theory of incomplete systems introduces
the concept of the natural times or the relaxation times of the
internal processes. The system is described in terms of the
externally observable variables and a number of relaxation times
which characterize the rate of the adaptation of the internal
processes to a change in external conditions. A small relaxation
time characterizes a mechanism which adjusts quickly. This approach
is more or less analogous to the transfer function approach to the
dynamic behavior of systems 48). The application of the latter
approach to bioengineering systems has been investigated 4952~

Application of Simple Structured Models in Bioengineering 69
The time constant concept provides a direct route to the choice
of the degree of complexity required for the description of the
behavior of a system. In principle, the behavior of a culture of
organisms is described by a vast number of relaxation times
resulting from, amongst others, the various regulatory mechanisms
discussed in Sect. 2. These mechanisms generally have largely
different relaxation times, a highly speculative picture of which
is given in Fig. 3. A description of the system can be simplified
by basing an approach on a comparison of the relaxation times of
the internal processes and those characterizing the relevant
changes in external con ditions.
If the changes in environmental conditions are slow compared
with the rate of adaptation of a given mechanism, i.e. if the
relaxation time of the latter is much smaller, the dynamics of that
mechanism may be ignored. In the case mentioned, the organism will
be at steady state compared to that mechanism and external
variables suffice to describe the state of the organism. An
additional relaxation time associated with the dynamics of
adaptation of the given mechanism is not needed. The model can be
simplified by the socalled pseudosteadystate hypothesis with
respect to the mecha nism under consideration. A totally different
situation occurs when the relaxation times of the changes in the
environment are small with respect to those of the cell's
adaptational mechanism, i.e. if the internal state adjusts very
slowly. The mechanism can then be totally ignored and the state of
the organism with respect to that mechanism will be characterized
by the initial state throughout the process. The description of the
behavior of the system can now be simplified by deleting that
mechanism. In order to clarify the nature of both types of
simplification vital to the construction of workable models, some
examples will now be dealt with. a) In the kinetic description of
enzymatic reactions, the MichealisMenten equation is often used 3
7 ) :
k W E f s rs =   C~ (34)
K M k C s
The state of the organism is described by the mass fraction of
the enzyme in the biomass, w E.
10 6 i 0 5 10 t~ 103 10 2 10 1 101 102 103 104 105 10
6
I I I I I I I o I I I I I I ~ s s ACTION LAW 10
ALLOSTERI C CONTROLS ,I, m ,I
mRNA CONTROL
R E L A X A T I O N T I M E
( S E C O N D S )
CHANGES IN ENZYMIC
CONCENTRATIONS
SELECTION WITHIN A
POPULATION OF ONE OR MORE SPECIES
EVOLUTIONARy
CHANGES
Fig. 3. Various internal mechanisms and order of magnitude of
their relaxation times

70 A. Harder, J. A. Rods
The derivation of Eq. (34) is based on the following kinetic
scheme:
E + S ~ E S ~ E + P (35)
The enzyme is assumed to associate with the substrate to form an
intermediate, ES; this intermediate subsequently dissociates to
yield free enzyme and the product. A detailed solution of the
dynamics according to Eq. 35 would require a description in terms
of w E and WEs, the mass fraction of enzyme and enzymesubstrate
complex. Equation (34) is, however, obtained if the relaxation time
of the adjustment of the ES concentration is very small, compared
with the other time constants 53, 54) b) A general approach to the
bioenergetics of microbial growth has recently been developed. This
is based upon the pseudosteady state hypothesis with respect to
the energy metabolism intermediates, ATP and NADH. These have very
small time constants for the adaptation of their concentrations 55,
56). c) The steadystate behavior of a continuous culture can be
adequately described by the unstructured Monod model lO~. When a
continuous culture reaches a steady state, the relaxation times of
the changes in environmental conditions have become "infinite" and
the pseudosteadystate hypothesis is justified with respect to all
adaptational mechanisms. It may, however, take a long time
(approximately 3 times the largest relaxation time) for all
processes to reach their steadystate values. The phenomenon of
selection in continuous culture is an example. This may cause
changes in the steady state of a continuous culture on a time scale
which is large compared with that of other mechanisms. This is a
known problem in continuous culture 32~ as well as an effective
tool in the selection of organisms with desirable properties from a
mixed culture of organisms 57, 58~. The Monod equation cannot be as
successfully applied to the description of the transient behavior
of pure and mixed cultures 49, s9,6o~ Alter a transient shift, the
relaxation times for the changes in experimental conditions are
smaller. The pseudosteadystate hypothesis is then valid with
respect to a more restricted class of internal processes, i.e.
those having a relaxation time smaller by a factor 3 than that of
the smallest environmental relaxation time.
Although the application of the relaxation time concept to the
simplification of the description of a system could be further
discussed, we will, however, limit ourselves to indicating its
application in a number of examples to be treated in the
next section. As already pointed out, the transfer function
approach, roughly an analogue of the
treatment in terms of relaxation times, has been advocated for
the application to bioengineering systems 495~. It is our opinion
that such an approach provides a valuable tool in the
identification of the number and the order of magnitude of the
relaxation times necessary for an adequate description of a system.
It should, however, be borne in mind that the transfer function
approach basically only applies to linear systems. In other words,
it only holds in the region around a given initial state where the
system can be sufficiently well described by a linearized set of
differential equations. This severely limits any application to
bioengineering where the~ systems are strongly nonlinear. Although
the same holds, in principle, for the time constant concept, it may
be more easily understood in terms of mechanisms and more readily
adapted to provide a realistic depiction of bioengineering systems.
An attempt to show this will be undertaken in Sect. 5.2. Both
approaches are basically "black

Application of Simple Structured Models in Bioengineering 71
box" approaches. The realism of the model can be evaluated by
attempting to translate time constants or transfer functions into a
model based on known regulatory properties. At least, one example
of such a procedure for the transfer function approach has been
published 59)
Finally, an important conclusion may be formulated. It is
improbable that, in a given situation, more than two or three
adaptational mechanisms have relaxation times of the order of
magnitude of those of the changes in external conditions. Hence,
all remaining relaxation times can be eliminated from the
description of the dynamics of the system either by a
pseudosteadystate hypothesis, for the small relaxation times, or,
by ignoring the mechanism, for the large ones. On the basis of this
reasoning, it can be stated that a two or threecompartment model
will generally suffice to describe the system's dynamics. In Sect.
6 an example of a twocompartment model, containing one internal
mechanism, will be treated.
5.2 A Model Describing the Dynamics of Product Formation Based
on the Relaxation Time Concept
In order to show the effect of the relaxation times of internal
adaption processes on the dynamics of microorganisms, a general
model describing product formation processes will be developed.
Continuous culture is a powerful tool for the study of microbial
product formation. Apart from problems arising in connection with
strain degeneration, which may typically occur with the high
yielding strains used in industrial practice, an organism may be
studied under steadystate conditions. This allows to establish the
steadystate relationship between the specific rate of product
formation, qp, and the specific growth rate (the continuous culture
dilution rate) in cases where such a relationship exists 61)
Apart from trivial cases in which the product formation rate is
directly related to energy generation 3, 62,63), the relationship
between continuous culture results and the more dynamic batch and
fedbatch systems is not readily apparent. A number of theoretical
and empirical studies on this problem have been reported 6468).
The activity function as proposed by Powell 69~ for the dynamics of
growth can provide a basis for the development of a description of
the dynamics of product formation.
In the development of the model, the following assumptions are
adopted:  The total rate of substrate uptake depends on the
concentration of the limiting
(and energy supplying) substrate according to a Monod
relationship:
qs, maxCs rs  Ks + Cx (36)
where q . . . . . = saturation value of the specific rate of
conversion of substrate. The rate of product formation depends on
the rate of substrate uptake (and hence energy generation)
according to:
rp = Qr~ (37)

72 A. Harder, J. A. Roels
In Eq. (37), Q is the product formation activity function. The
rational behind Eq. (37) is the assumption that part of the energy
flux through the organism is directed toward product formation, the
fraction of the total flux being determined by the activity
function Q.
It is assumed that the fraction of energy directed to product
formation remains small, compared with the total rate of substrate
uptake. (This assumption is easily avoided, but it results in less
complicated equations which adequately represent the general
features of the more complex case). The specific growth rate, p,
can, for this case, be calculated from the Herbert/Pirt 10, 70~
equation:
tl = Y~x q . . . . . C~ K~ + C~ m~Y~x (38)
where Y~x = yield factor for substrates on biomass x, m~ =
maintenance requirements of substrate.
From Eqs. (38) and (37) it follows:
qp la = Y~x ~  m.,Ysx (39)
Equation (39) provides a relationship between the specific rate
of product formation, %, the activity function, Q, and the specific
growth rate la.
In the equation developed above, it is assumed that the
substrate is only used as an energy source. The carbon requirement
for growth is assumed to come from presupplied monomers. Again,
this assumption can be easily avoided by involving slightly more
complex mathematics. An interesting direct application of Eq. (39)
is obtained if the product formation is directly related to energy
generation (for example, in the anaerobic formation of alcohol,
lactic acid, etc.). In this case Q is a constant which is directly
obtained from stoichiometric considerations, and the familiar
Luede kingPiret 63) equation results from a rearrangement of Eq.
39:
Q (40) % = ~~ la+Qm~
In this simple case, Eq. (40) suffices to describe continuous
cultures as well as fed batch or batch cultures, provided that
relaxation times of primary metabolism adap tion are small
compared with those of the changes in external conditions. However,
there are cases in which Q is regulated in response to
environmental changes in a manner not directly related to the
specific growth rate. In these instances, a definite relationship
between Q and Ix in a steady state can still be assumed to
exist:
Q* = f(~t) (41)
where Q* is the value of Q in a steady state, for example after
a sufficiently long period of continuous culture growth. It is
assumed to be an arbitrary function of p, f(p). From Eqs. (41) and
(39) it follows:
Y~q* (42) Q~ p + msY~x

Application of Simple Structured Models in Bioengineering 73
where q* is the steadystate value of the specific rate of
product formation at a specific growth rate g. Eqs. (42) and (41)
allow the determination of Q*(g) from continuous culture
experiments. In order to extend the theory to dynamical situations
an equation is needed for the rate of adaptation of Q to changes in
environmental conditions. Such an equation can be obtained using
the following reasoning: a) The activity function is assumed to be
equal or proportional to an identifiable substance in the cell,
i.e. the dynamics of Q can be described by the intrinsic balance
equation derived in Sect. 4:
1 O = ~ (rQ  r,Q) (43)
In Eq. (43), rQ is the rate of synthesis of Q. In a steady
state, the specific rate of Q synthesis, q~, is given by:
q~ = gQ* (44)
b) When not in a steady state, control mechanisms, which adapt
the specific rate of Q synthesis, are assumed to operate. The
difference between the actual rate of Q synthe sis and the
steadystate rate is assumed to depend on the difference between Q
and the value of Q* consistent with a steadystate at the
environmental conditions existing at the moment considered:
qQ = q~ + g ( Q  Q*) (45)
Now the function g(Q  Q*) is approximated by a Taylor series
expansion 71) around Q*, truncated as a first approximation after
the second term:
g(Q  Q*) = g(O) + (Q  Q*) (46) O =Q*
From the definition of the function g(Q  Q*) it is clear that
g(O) equals zero. Furthermore, from the condition that the steady
state must be stable it follows:
Q=Q, =
If Eqs. (46) and (47) are combined it follows:
q Q = q ~  K ( Q  Q * ) (48)
where K is a positive constant given by:
~g K =  (~Q)Q=Q, (49)

74 A. Harder, J, A. Roels
When introducing constant K, the further assumption that the
first derivative of g with respect to Q, evaluated at Q = Q*, does
not depend on Q* is made.
Combining Eqs. (43), (44) and (48) for the rate of the change of
Q it results:
( ) =   ( K + p ) ( Q  Q * ) (50)
Equation (50) can now be applied, for example, to a shift in
continuous culture, showing the relaxation time for the adaptation
to a new steady state to be equal to 1/(p t K). If K is large, a
new steady state will be reached almost instantaneously. Then, the
organism will not show any lag in its adaptation to a new steady
state. Alternatively, if K is small, the time constant for
adaptation will be equal to l/p, i.e. dilution through growth will
control the adaptational process.
The model presented above has been numerically simulated for a
situation where decreases exponentially. The organism was
considered to be fully adapted to the initial growth rate. The
value of K was chosen to vary between 10 3 times and 100 times the
specific growth rate decay constant. The steadystate relationship
for Q* and la was assumed to be:
Q* = 0.5p (51)
In Figure 4 the apparent relationship between Q and la in the
dynamic situation is compared with the steadystate relationship
according to Eq. (51) for various values of K. As can be seen, when
K is large the steadystate relationship is obtained. For very low
values of K the function Q is higher than the value according to
the steady state relationship, and in the extreme case changes are
only due to dilution by growth. This simple model may be used in a
first effort to explain the behavior of product formation systems
having a largely varying rate of adaptation to environmental
changes. The constants and relationships of the model can, in
principle, be easily determined experimentally. First, the
steadystate relationship between Q* and ta
c 0.5 0
0.4
0.2 t )
~, o.1
~ ~ ystate / / relationship change of J~
I I I I 0.1 0.2 0.3 0.4 0.5 0.6 03 0.8 o.g 1.0
Specific growth rate
Fig. 4. Steadystate and dynamic re lationship between specific
rate of product formation and specific growth rate for various
rates of the exponential decrease of the specific growth rate

Application of Simple Structured Models in Bioengineering 75
can be determined using continuous culture. The constant of
adaptation, K, can be ascertained using, for example, shiftdown or
shiftup experiments in continuous culture. It must be emphasized
that the model is a first approximation and can be refined by
allowing K to depend on Q* and by the introduction of higher order
terms of the Taylor series expansion.
6 Models of Primary Metabolism in Microorganisms
6.1 TwoCompartment Models
In unstructured models, biomass is considered as a black box,
and regulatory processes inside the black box are ignored. As was
discussed in the preceding section, relaxation phenomena inside the
black box may cause the system to behave as if it had a memory of
its preceding state. These phenomena may formally be treated by the
introduction of "hidden variables", by the transfer function
approach or, alternatively, by specifying the process causing the
delayed response, i.e. by a mechanistic approach. The difference
between these approaches becomes significant if the processes
occurring inside the system are known. Molecular biology has
revealed much of the internal functioning of microorganisms.
Knowledge seems to have advanced sufficiently to investigate its
exploitation in bioengineering kinetics.
Early attempts to include structure in the description of the
biomass were based upon a distinction of two sections in the
biomass. One approac h distinguished between a section, responsible
for the synthesis of cellular macromolecules and a structural
section containing the macromolecules necessary for the functioning
of the cellular machinery. The first compartment was often
considered to consist of RNA and precursor molecules and the second
of protein and DNA. The models of Williams 39,721 and Ramkrishna et
al. 401 are based on this distinction. The model of Verhoff et al.
73~ distinguishes an assimilating compartment which takes up
nutrients and transforms them to energy carriers and biomass
precursors, and a synthetic compartment which produces new biomass.
On the latter distinction, the models of Bijkerk and Hall ~4~ and
Pamment et al. 7~ for the growth of yeast are based. A slightly
different approach to the introduction of structure into the
biomass is the socalled IerusalimskyPowell bottleneck model
4"A'~A''60'69'76'77). In this model, a "bottleneck" in metabolism
is specified. This is a measure of the maximum specific rate at
which the organism is able to convert substrate to biomass. Models
which are basically analogous to this approach have been published
39, 788oj. In some instances, the bottleneck is specified to be
the RNAconcentration because RNA plays a central role in the
synthesis of protein (constant efficiency hypothesis for the
synthesis of protein at the ribosomes, see Sect. 2). The time
constant for the adaptation of the RNA concentration seems to be of
an order of magnitude relevant to most applications, namely about
0.11 h 1 in bacteria.
Table 1 summarizes some features of a number of models which
have appeared in the literature.
The development of a typical twocompartment model will now be
shown in greater detail to familiarize the reader with the basic
formalism. The model is

76 A. Harder , J. A. Roe l s
.u
._=
8
"t= 0
'8 ,=
a
W.
= .. .= ._ = "=
=
=.= ~_~ ~ ~.
< < < Z Z Z
0
~g
0
..0
=
=
S

Application of Simple Structured Models in Bioengineering 77
[..
..~ ~ ~ o ~
~ ~  ~
~ ~ ~ ~ ~ ~ ' ~ ~ "" ~:.~ ~ ~ ' " ~ ~ ~  ' ~ ~ ~ .~ ~ ~ ~ ~ ~ o
~ ~ ~'~,,
0 0 o
~ . ~
. , ~ ~o~
< , ~
I=
e,I) I~
"0 0
8
o ~ ~ . ~ ~ o
" ' ~ o.~
~ . ~ r.. o~ ~ " ~ ~= ~ o ~ Z
0 0
~ s : ~ "~
0 0
~ o

78 A. Harder, J. A. Roels
essentially based on that of Williams 39,721, and on its
modification and extension by Roels and Kossen 3), Roels s4) and
Harder 44).
A culture in which an organism is growing on one single source
of carbon and energy is considered. The organism is assumed to
consist of two compartments (Fig. 5). The G compartment is thought
to contain the enzymes which convert the substrate to the various
building blocks for the macromolecules of the cell. The remainder
of the biomass, the K compartment, is storage material, genetic
material, RNA and the pools of the various precursors. The G
compartment is assumed to be synthetized from the K compartment
under the catalytic action of the K compartment. The latter
compartment is thought to be synthetized from substrate under the
catalytic action of the G compartment. This is a great
simplification of the complexity of the cellular processes, but is
significantly closer to reality than the unstructured approach.
For a description of the state of the culture as a function of
time, the approach advocated in Sect. 4 can be used without
problems when the kinetics and the stoichiometry of the processes
involved are defined:
a) The conversion of the substrate to the K compartment. The
rate will be defined by rsK. The stoichiometry is defined by the
yield factor Y~K, the amount of K compart ment material
synthetized per unit of substrate used. Bearing in mind that this
synthesis occurs from the substrate under the catalytic action of
the G compartment, the following generalized relationship is
proposed:
rsK = fl(fs) f2(WG) C x (52)
in which w o is the weight fraction of G compartment in the
biomass. In the formulation of this equation, the notions derived
from a corpuscular
treatment are taken into account, i.e. rates are assumed to be a
function of the fraction of the G compartment in the biomass and to
be first order in the total biomass concentration (Sect. 3).
In practice, Eq. (52) must be specified. For example, it may be
formulated as
qs, maxCs WG r s K    C~ (53)
K~+C~ K o + w ~
rSK
Kcompart~
rKG .~compartme,n t
L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . t
Fig. 5. Schematic representation of a twocom partment model
(r~k = rate of conversion of substrate to Kcompartment, r~G rate
of con version of K compartment to G compartment, roK rate of
depolymerization of G compartment to K compartment)

Application of Simple Structured Models in Bioengineering 79
Eq. (53) assumes a Monodtype relationship for the dependence
ofrsK on the substrate concentration, C~, and the weight fraction
of the G compartment in the biomass. b) The transformation of the K
compartment into the G compartment. The rate is defined as rKo and
the stoichiometry by the yield constant Y~o, i.e. the amount of G
compartment produced per unit K compartment consumed. The following
relationship is proposed for the rate of transformation of the K to
the G compartment:
rKo = f3(w~) Cx (54)
Equation (54) expresses the assumption that the transformation
of the K to the G compartment is governed only by composition of
the biomass (f.e. wo) and is not a direct function of the substrate
concentration. c) Turnover of the compartments of the biomass. In
the present example, the G compartment will be assumed to be
subject to turnover. The turnover process is assumed to be modelled
by
roK = mGw~C x (55)
m o = specific turnover rate of compartment G. It is assumed to
be a depolymerization process and is first order in the total
amount of G compartment (w~Cx). The specific rate of
depolymerization is m o. The yield constant for the formation of
the K compartment from the G compartment is assumed to be unity,
i.e. no mass is lost during the depolymerization of the G
compartment to precursors.
The balance equations for the rate of change of the substrate
concentrations, the biomass concentration and the fraction of the G
compartment are now obtained by the application of the formalism
treated in Sect. 4. Table 2 summarizes the resulting equations.
An interesting feature of the equations in Table 2 is the fact
that the intrinsic balance equation is independent of the mode of
operation, i.e. batch, fed batch or continuous culture.
The problem of the expression fs(wo) will now be discussed. It
is wellknown that the composition of the biomass in a steadystate
continuous culture changes with dilution rate. Usually, the amount
of RNA present increases with rising dilution rate (Fig. 2). In our
terminology, this might mean a decreasing amount of the G
compartment with increasing growth rate. A simple and apparently
reliable approximation to the relationship is linear. Thus, the
continuous culture steadystate fraction of the G compartment can
be modelled by:
w* = W~o + [3ola (56)
w* indicates the steadystate fraction of the G compartment;
we~,o is the fraction of the G compartment present in a steady
state when the growth rate is extrapolated to zero.
As was first pointed out by Koch is), the existence of a given
linear relationship between an intrinsic concentration and the
dilution rate in a continuous culture steady

80 A. Harder, J. A. Roels
Table 2, State equations of the twocompartment model. The
following Equations are obtained by substitution of the rates and
flows into Eqs. (26), (32) and (33)
dEs qs, maxCs WG
dt K, + C~ K c + w G C,+@~ 2.1
dC~ qs, m~C~ wG   = YsK Cx + (Y~G  1) f3(WG) C x + qb x 2.2
dt K~+C~ K o + w G
dWG YsK q~" maxfs WG   = w G + f~(wG) {wo + Ylca(1  wa) } 
mGw ~ dt K~+C~ K o + w a
2.3
The Equations contain the transport contributions

Application of Simple Structured Models in Bioengineering 81
Now the function f3(w6) introduced in Eq. (54) is identified for
a steady state as being proportional to q*c.
The extrapolation which is now made is based on an assumption
implicit in each of the two compartment models mentioned earlier.
It is assumed that the fraction of the biomass composed of G or K
compartments is sufficient to specify the activity of the biomass,
i.e. the two values C x and w G provide sufficient information to
rigorously define the amount and activity of the biomass. This
clarifies the relationship with the unstructured approach. In an
unstructured model, the biomass concentration alone is considered
sufficient to specify the activities of the biomass. The next
alternative, a twocompartment model, specifies one compositional
variable.
Under the assumptions presented above, Eq. (60) can be
generalized to hold even if the system is not in a steady
state:
1 ( qKc = YKGh(w~) = ~ (wc) 2 + m~   ~  c ] wc (61)
The reasoning presented above has resulted in a model of the two
compartment type having minimal complexity. Although highly
simplified, it may provide a useful alternative to unstructured
models in situations where these models fail.
The procedure outlined in Sect. 5.2 could also have been
applied, resulting in a different approach in which the adaptation
rate is also assumed to be directly influenced by environmental
conditions. This is a more complex, but more flexible approach.
The necessity for a model of such simplicity, while sufficient
knowledge is available for the construction of a model of much
greater realism may not be obvious.
Two factors should be considered: a) A number of regulatory
mechanisms at the level of energy generation and consumption
operate with such small relaxation times that a pseudosteadystate
hypothesis with respect to these mechanisms is justified. Hence,
the introduction of these details seems to be unnecessary. b) A
minimum of complexity is desirable because a complex model often
proves very difficult to verify and may fit experimental results
without having any relationship with the behavior of the organism
a). Only after obtaining experimental evidence that the simple
model should be rejected because of unsufficient fit of the data or
unrealistic parameter values, should additional complexity be
introduced. An additional "hidden" variable must be specified.
Careful study however, of the biochemistry of RNA and protein
synthesis 13,8184), may result in model structures of greater
realism and a complexity similar to that treated in this
section.
To give an impression of the typical features of the model
presented above, an analysis of the continuous culture growth using
the twocompartment model presented above was performed. The model
exhibits all features of the classical chemostat theory 26) but
also allows for the description of alterations in biomass
composition with growth rate changes (Fig. 6). For a steadystate
culture, the advantages of the twocompartment model over the
unstructured approach are not readily apparent. The differences
become clearer in transient situations such as washout from
continuous culture. Figure 7 shows the results of a simulation of
washout for cells pregrown at two different dilution rates. The
present model

201
1.5
10 / 10
0 '  0
Wk qSK
1.0
0.5
i I I  
82
c: 20
0.5
I L I_  
0.5 1.0 1.5 0 0.5 1.0 1.5 Dilution rate D (h q} Dilution rate D
(h 1)
A. Harder, J. A. Rods
Fig. 6. The steady state values of substrate concentration, C~,,
biomass concentration, C~*, mass fraction of K compartment in the
biomass, wk*, and the specific rate of conversion of substrata to K
compartment, q,*~ according to simulations with the twocompartment
model (arbitrary parameter values)
reveals that the cells pregrown at the lowest growth rate
exhibit a more rapid washout.
Although the model is primarily designed to apply to pure
cultures, its application to mixed cultures certainly merits
investigation. As was pointed out in Sect. 2 (see Fig. 2), there
exists a tendency for organisms of different origin to have the
same steadystate RNA content at the same specific growth rate.
Hence, a compartmental approach describing a mixed culture in terms
of average RNA content could be an interesting approach to the
study of the dynamics of mixed cultures 44).
The twocompartment model exhibits many features observed in
batch and continuous culture experiments..The method is certainly
promising as an approach to the modelling of microbial growth in
situations where the relaxation times of the changes in
environmental conditions are of the order of magnitude of the
relaxation time of one of the internal adaptational mechanisms,
e.g. in batch or fed batch growth or during transients in
continuous culture. The particular model presented, however, must
be considered as a preliminary proposal because many of the kinetic
assumptions do not rest on solid biochemical facts about the
internal regulation of the ccU. Furthermore, there are difficulties
in identifying the compositional nature of the K and G compartments
in terms of structural compounds of the cell. It is clear that a
more thorough study of known regulation phenomena and an empirical
study of transient situations, for example in continuous culture,
is needed. In this respect, the chemostat is a valuable research
tool. It is gratifying to note that the study

Application of Simple Structured Models in Bioengineering 83
Cx/Col Shift D= 0.1 ~1 .5 (h "t) ] Shift D= 0.9~1.5 (h 4)
1.0'
0.5
0 t 2 3 l. 5 Time Time efter shift (h)
of shift
Fig. 7. Simulation of the washout curves for bio mass from
continuous culture. Plot of the biomass concentration relative to
the initial biomass con centration, Cx/C~0 against time for a
shift of the dilution rate from 0.1 to 1.5 (solid line) and a shift
from a dilution rate of 0.9 to 1.5 (dotted line)
of transient phenomena is apparently attracting the attention of
a growing number of investigators ssss)
In judging the succes of a modelling excercise, it is important
to remember that appropriate fit to an empirical biomass
concentration or an oxygen uptake profile provides little evidence
of an adequate model structure if the model parameters have been
obtained by a least squares optimization. Independent evidence
concerning the degree of realism of parameters and mechanisms
postulated, or a good fit to a response obtained under conditions
different from those under which the parameter estimation was
performed, are needed to critically evaluate the validity of a
model. This holds for the models proposed in the literature as well
as for the example treated in this section.
6.2 A ThreeCompartment Model of Biomass Growth
There exist a number of situations in which organisms are known
to produce large amounts of intracellular storage compounds (e.g.
macromolecules of glucose such as glycogen 89, 90)
Recently, a threecompartment model in which this phenomenon is
considered has been developed by Harder 44) and applied to the
description of the dynamics of activated sludge. In this model,
three groups of constituents, termed R, K and G compartments, are
distinguished (Fig. 8). The K compartment is the microbial RNA, the
G compartment consists of protein, and the R compartment is the
remainder of the biomass, mainly consisting of carbohydrates and
precursor molecules such as nucleic acids and amino acids. The
clear advantage of the threecompartment approach over the
twocompartment model treated in the preceding section is in the
easier identification of the compartments in terms of actual
constituents of the biomass. In this review, only the main aspects
of the model are discussed. The reader is referred to the original
literature for a more detailed treatment.

84 A. Harder, J. A. Roeis
TURN OVER
I
,I TURN OVER
Fig. 8. Schematic representation of a threecompartment
model
One of the interesting features of the model is an argument
which is independent of the details of the kinetic treatment. The
threecompartments are assumed to be synthetized from an external
substrate and, as there is only one carbon source, this wilt be
utilized for both ATP generation and precursor synthesis. In the
model a pseudosteadystate hypothesis for ATP is implicit as it is
assumed that ATP consumption always matches ATP production. This
assumption is generally correct because the relaxation time for
adaptation of the ATP concentration is quite small. Furthermore, it
is assumed that both K and G compartments are subject to turnover,
a phenomenon which may be interpreted as maintenance 9,~.
The R compartment is not subject to turnover, and the following
equation for substrate consumption due to the R compartment
synthesis can thus be formulated:
1 r~ = .7 rR (62)
where r R is the rate of the R compartment synthesis, Y*R the
yield factor for the R compartment with respect to the substrate
(kg R compartment per kg substrate) and rsR the rate of substrate
consumption for the R compartment synthesis. For the K and G
compartments, the situation is more complicated. It is best
described by the assumption that pools of precursors for both
compartments are contained in the R compartment. These pools are
supplied by the transformation of substrate to K and G compartment
precursors and by depolymerization of macromolecules to their
precursors. Precursors are drawn from the pools for the synthesis
of K and G compartment. If a pseudosteadystate hypothesis is
applied to the pools (i.e. if the relaxation times for their
adaptation are small), the rates of substrate consumption due to K
and G compartment synthesis are given by:
1    r K ( 6 3 )
r s K  YsK
l r~G = . 7  r C (64)
YsG

Application of Simple Structured Models in Bioengineering 85
where rsK and rso are the rates of substrate consumption due to
the synthesis of the compartments K and G; YsK and Yso are their
yield factors and r~: and r o the net rates of synthesis of K and G
compartments. The latter rates are obtained from the total rates
(rK) t and (ro) t through correction for turnover:
r K = (rK) t   mKwKC x (65)
r o = (ro) ,   mowoC ~ (66)
The constants m~ and m G a r e the specific rates of turnover of
the compartments. The last factor contributing to the rate of
substrate consumption is the amount
of substrate required for the production of ATP necessary for
the synthesis of macromolecules from precursors. These contribution
are calculated by the intro duction of a modification of the
YATpCOncept (Bauchop and Elsden 92)). The rates of ATP consumption
due to R, K and G compartment synthesis are assumed to be:
1 r A T P , R   Y A T P R rR (67)
1 FATP, K   YATP,~ (rK + mKwKC~) (68)
1 rATP, G   YATP.C (r + mwCx) (69)
In these equations, the YATpValues are the ATPyields (kg per
mol ATP) for the various compartments. As can be seen, a turnover
contribution appears in the rates of ATP consumption for the
synthesis of K and G compartments. Finally, the contributions in
Eqs. (67)(69) are converted into substrate consumption by the
introduction of the stoichiometry constant cz c. This is the amount
of ATP (moles) produced per kg of substrate catabolized. The total
rate of substrate consumption for energy generation is as
follows:
rG + r s =   rR + y~Tp K rK + y~xp G 0~ P, R , ,
+ 1 1 mGWGCx I (70) YATP, K mKWKCx + TAre, G
The total rate of substrate consumption is now obtained by the
adding up Eqs. (62)(64) and (70). The result is given in Table 3,
together with the equation for the total rate of ATPconsumption
obtained by adding up Eqs. (67)(69). The equations are derived
for a steady state with constant proportions or R, G and K
compartments in the biomass. Equations (1) and (2) of Table 3
provide a structured

86 A. Harder, J. A. Roels
Table 3. Equations for the rate ofsubstrate and ATP consumption
on the basis of a threecompartment approach
Rate of substrate consumption:
, wry, , t ' 'tl I +   + ~ ~  I +~cYATe. R LY,T,,~  E c
< ,
Cx mGWG t
Rate of ATP consumption:
 ( Y ATP. R YATP, R [) WG FATP = ~ '.YATP. K \YATP. G
f mgw g mGWG ) +C~   +
~YATP, K ~ /
3.1
3.2
extension of the equations proposed by Herbert 1o) and Pirt 7o)
and Stouthamer and Bettenhausen 93), respectively, for the
unstructured approach:
1 r~ = .z7 r~ + msC~ (71)
1 _ _ + mArpCx (72)
rATe = (YArP)max
In which (YATP)max is the maximum yield of biomass on ATP. These
equations are only analogous to those in Table 3 when the
biomass
composition is constant. The values of the parameters in the
equations of Table 3 can be obtained in principle from the work oi"
Forrest and Walker 94) and Stou thamer 95). These authors have
theoretically evaluated the values of YATP, R' YATP, K and YATP, G"
The values of Ysa' Ysg and Y~6 are obtained from stoichiometric
considerations. The value of 0~c can be obtained from metabolic
pathways whereas for aerobic growth the P/O ratio must be known. As
this parameter is still open to dispute, two values are assumed.
The estimation of m K and m G remains a problem on which little
detailed information is available. For rapidly growing bacteria and
fungi the rate of protein turnover (i.e. Gcompartment turnover)
does not exceed 3 %/h 96), while in nongrowing organisms, the
turnover may be as high as 5 %/h 97). For the purpose of the
present exercise a value of 3 % per h is assumed. Because even less
is known about RNA turnover, a value of 3 ~o per h is arbitrarily
adopted. Table 4 summarizes the parameter values for growth on
glucose it> it >ub>lt'atc..\~ can bc seen from Table 5,
the use of the model results in reasonable yields for a P: O ratio
of about 1. An important tendency is obvious: The yield increases
with rising

Application o f Simple Structured Models in Bioengineering
Table 4. Parameters of a threecompartment model
87
0t, Y~a Y~K YsG YATP. a YArV. K YATP, O mK mo
88.9* 0.9 0.78 0.78 81.0x 10 3 26.8 10 3 25.6 x 10  s 0.03
0.03 222**
* P/O = 1 ** P/O = 3
Table 5. Calculated values o f Y~, m,, (YATP)mx and mAT P for
microorganisms of various compositional characteristics
Organism Y~x (YAlrp)max m s x 103 mA.rp
x103 P/O P/O P/O P/O 1 3 1 3
Aerobacter aerogenes (w G = 0.67, wt = 0.31) 0.59 0.69 26.5 13 5
!.I Aerobacter aerogenes (w G = 0.80, w K = 0.16) 0.59 0.69 26.7 12
5 1.1 Candida utilis (w G = 0.31, w t = 0A0) 0.68 0.78 43.4 6 2 0.4
Activated sludge (w G = 0.49, w K = 0.13) 0.65 0.74 35.0 8 3 0.7
Activated sludge (w o = 0.62, w K = 0.10) 0.63 0.73 32.0 9 4
0.8
carbohydrate content of the organism (e.g. yeast). The
maintenance factors reveal the reverse tendency, being lower for a
high carbohydrate content. This phenomenon may also partly account
for the abnormally low maintenance coefficients reported for
activated sludge 44,98~ and axenic cultures 9s). The systematic
increase of the fraction of storage carbohydrate with decreasing
growth rate will result in a low estimate of the maintenance factor
if determined by the conventional double reciprocal plot of yield
factor against growth rate.
In the publication of Harder 4,,j, a modified version of the
model structure presented above is used. The uptake of substrate is
assumed to take place by conversion to the R compartment which is
subsequently converted to K and G compartments. Furthermore, the
technique described in Sect. 6.1 was used to model the rates of
synthesis of K and G compartments, i.e. the continuous culture
steadystate relationships for the rates of synthesis of these
compartments, which follow directly from the steadystate mass
fractions, are assumed to be adequate even when the organism is not
in a steady state.
Furthermore, Harder argues that Monod's equation is unfit to
describe the uptake of substrate by a mixed population, a view
which is supported by some authors 99101) He proposes a nth order
power law equation for the substrate consumption. Harder shows that
the model fits the results of his continuous culture experiments
and is in fair agreement with some preliminary transient
experiments although no attempts

88 A. Harder, J. A. Roels
have been made to adjust the constants of the model such that
they optimally fit the experimental curves. Although the validity
of the various assumptions and the kinetic equations to be used are
still uncertain, the method presented appears to be of future value
as an alternative to unstructured models.
7 Models for the Synthesis of Enzymes Subject to Genetic
Control
7.1 Introduction
In a single wildtype cell of E. coli growing on glucose or
glycerol, the constitutive enzymes of the glycolytic pathway are
always present in 100.000 copies or more per cell. In balanced
growth, these enzymes are formed at constant rates. However, a
variety of enzymes are subject to control mechanisms at the genetic
level. An enzyme like 13galactosidase is present in only 5 copies
per cell if the substrate is glucose or glycerol. On switching over
to a galactoside like lactose, the amount of 13galactosidase
increases by a factor 1.000 10.000 102. lo3.zo4~. The research
con cerning the lactoseinducible IBgalactosidase system in E.
coli was the initiating point for the formulation of two
fundamental physiological concepts of cellular regu lation lo4):
1) the transcription of structural genes can be controlled by other
socalled regulatory genes, 2) this control is carried out by
products, i.e. proteins of the regulatory genes themselves. These
proteins can turn off the transcription of the structural
genes.
A schematic representation of the lactose operon in E. coil is
given in Fig. 9. It is an example of a coordinated unit of
structural and regulatory genes in microorganisms. The quantitative
description of the dynamics of enzyme synthesis is more or less
based on the extensively investigated lacoperon in E. coli.
Variations in the lac operon theme are, for example, the
tryptophan operon of E. coli 1o5~, the hut (histidine utilization)
system in Klebsiella aerogenes lo6) and the Larabinose operon of
E. coli17).
In the following, we will review the quantitative descriptions
of the genetically regulated consumption of substrates and
formulate a model of rather limited complexity which describes the
known phenomenon of diauxic lag 34}. The model can also be used to
depict the synthesis of extracellular enzymes.
7.2 Repressor/Inducer Control
The expression of structural genes of an operon is controlled by
a regulatory gene which produces a protein called cytoplasmic
repressor (R) at the ribosomes via transcription on mRNA (Fig. 9).
This protein controls the regulatory gene on the operator gene (O)
by blocking transcription if bound to that gene. An effector (E),
which commonly interacts allosterically with the repressor, can
decrease the affinity of the repressor for the operator site (i.e.
effector = inducer) or increase it (i.e. effector = antiinducer).
These regulatory phenomena are termed negative controls. Based on
the research of Gilbert and MfillerHill lo8,109) and the kinetic
treatment of Yagii and Yagil 110), which

Application of Simple Structured Models in Bioengineering 89
/ / GLUCOSE
/'71 TlllO GALACTOSIDE TRANSACETYLASE
c~roPL~xxc XE~RANE /41 LACTOSE PEP.KEASE
~  GALACTOSIDASE
~ A D E I ~ L A T ~ CYCLASE ~
POSITIVE
 // m I~A
REI~ESSOR + INDUCER CHRO~,OSC~t,
DNA
Fig. 9. Schematic representation of the lac operon; its negative
and positive control units
is also the base of the operon models of several authors 34., l
l l 114,121), the interaction between the cytoplasmic repressor
(R), the operator (O) and an effector (E) can be formulated in
terms of chemical equilibria.
The ligandrepressoroperator interaction can be described by
the following reaction scheme 115)~
K2 O + R .~OR
+ +
nE nE
":'41" K41l ' ~ 0 + R E . ~ ORE.
The number of binding sites for the effector on the repressor is
given by n. If the effector (E) is an inducer, the RE, complex
results and if it is an antiinducer, the ORE. complex is
formed.
The equilibrium constants K,, K2, K3, and K 4 are given by
w~E, K I  w*t..,*~" (73)
R~VVE!

90 A . H a r d e r , J. A . R o e l s
K2 =   (74) W o W R
W * ORE n K3 w* *w**" (75)
OR 't , El
W* OREn K 4 = ,    ~  (76)
W o W R E n
In Eqs. (73)(76) the intracellular concentrations of O, R, E,
and of the different complexes at equilibrium are given in terms of
moles per unit of cellular dry mass.
So far it must be kept in mind that the following assumptions
have been made: a) A descriptio n in terms of moles per unit dry
mass, i.e. a macroscopic description, has been used, but
microorganisms contain not more than 2 ~ copies of one type of
operator per cell and 1020 copies of the repressor protein ~0s,
rag} For such small enti ties (see also Sect. 3), the meaning of
concentration and of thermodynamic equilibrium is disputable.
However, according to Hill 116) and Berg and Blomberg 43),
thermodynamic reasoning can, in some cases, be applied to small
systems as long as a large ensemble of such systems is present.
Actually, the various concentrations have to be defined as the
probability to find the compound concerned in a given state 11o.
H7} b) The pseudosteadystate hypothesis is applied to the
reactions between the various regulatory compounds. The mechanism
under consideration, i.e. the synthesis of enzymes, involves a
considerable larger relaxation time (see Sect. 5) compared to the
establishment of the equilibria between R, O and E. The dynamics of
the establishment of these equilibria are therefore ignored. c) The
affinity of the repressor protein for the ith effector molecule is
not influenced by the already bound (i   I) effector molecules.
If the binding of one effector molecule at one site entraces or
decreases the birding of subsequent molecules at the other sites,
the theory has to be refined no)
Balances on repressor, operator and effector lead to the
following equations:
(%%=w~+w* + w % + w * RE n ORE n
(w~), = w~ + w * + w* ORE n
(w~), = w~ + nw'~E" + nW*E,
(77)
(78)
(79)
* * W * In the balance equations, (w~) t, (Wo) t and ( E)t are
the total numbers of moles of R, O and E, respectively, in the
pseudosteady state. In wildtype E. coli cells, there are 1020
times more repressor molecules than there are operators los~. In
this case, w* R and w* can be neglected in Eq. (77). A more
complicated theory is obtained
ORE n
if this is not the case ns) The affinity of the repressor
molecules for the operator descreases considerably
if the effector is an inducer and W~REn can be omitted in Eq.
(78). In the case of an antiinducer, w* R instead of W~RE, can be
neglected in this equation.

Application of Simple Structured Models in Bioengineering 91
Equation (79) can be simplified by the assumption that the
intracellular con centration of effector molecules is in
sufficient excess over repressor molecules so that w* + w*

92 A. Harder, J. A. Roels
It has been shown that the CAPcyclic AMP complex stimulates the
flgalactosidase synthesis by binding at the promoter site (P) of
the lacoperon and initiating the transcription by RNA potymerase
(see Fig. 9)138,139) This antagonistic effect of catabo