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 Two-Compartment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6.2 A Three-Compartment 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 present-day 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 1-41

,

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 present-day 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 fed-batch 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 fed-batch 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 so-called two- or three-compartment 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 mass-action 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 mass-action 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 steady-state 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 anti-inducer 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 m-RNA is postulated to only take place if a complex of c-AMP (cyclic AMP) and CAP (catabolite activator protein) is bound to a promotor gene on the DNA. Certain catabotites, such as glucose, apparently reduce the c-AMP 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 Qt-amylase 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 probability-density function, W(o), is now defined, giving the probability-density for the state vector to have a value in a certain region of state space (i.e. a probablistic approach). This probability-density function is defined by the following equation:

dN(o~) = NtW(o) iI-i/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 probability-density function are important

quantities. For simplicity, these will be demonstrated for the case of a unidimensionai probability-density 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 probability-density 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 probability-density 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 probability-density function for e is W(e). The number of organisms is assumed to be sufficiently large and a Michaelis-Menten 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 Michaelis-Menten 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 probability-density 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 a-biotic, 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 right-hand 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 probability-density 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 a-biotic 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 so-called 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 mass-action 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 102--10 * 3), the deterministic continuum approach should be handled with caution.

An important problem involves the application of mass-action 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 Michaelis-Menten 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 a-biotic phases.

According to the arguments developed in Sect. 3, the overall state vector is divided into biotic and a-biotic parts:

C = {y, x} (21)

The a-biotic 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 a-biotic 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 i-th reaction.

Expressions analogous to Eqs. (24) and (25) may now be formulated for the rates of change of the a-biotic 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 a-biotic and biotic compounds; ~y and O~ are the vectors of rates of transport for a-biotic and biotic compounds.

Equation (26) can be used to describe the dynamics of the a-biotic 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 left-hand side of Eq. (29) it follows:

C;# + w ~ = rat. + ~xw (30)

If the n component equations of the vector-equation (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 so-called 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 49-52~

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 so-called pseudo-steady-state 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 Michealis-Menten 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~ 10-3 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

m-RNA 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 enzyme-substrate 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 pseudo-steady 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 steady-state 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 pseudo-steady-state 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 steady-state 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 pseudo-steady-state 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 49-5~. 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 non-linear. 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 pseudo-steady-state 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 three-compartment model will generally suffice to describe the system's dynamics. In Sect. 6 an example of a two-compartment 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 micro-organisms, 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 steady-state conditions. This allows to establish the steady-state 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 fed-batch systems is not readily apparent. A number of theoretical and empirical studies on this problem have been reported 64-68). 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 pre-supplied 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- king-Piret 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 steady-state 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 steady-state rate is assumed to depend on the difference between Q and the value of Q* consistent with a steady-state 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 steady-state 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 steady-state relationship according to Eq. (51) for various values of K. As can be seen, when K is large the steady-state 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 steady-state relationship between Q* and ta

c 0.5 0

0.4

0.2 t )

~, o.1

~ ~ y-state / / 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. Steady-state 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, shift-down or shift-up 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 Two-Compartment 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 so-called Ierusalimsky-Powell 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, 78-8oj. In some instances, the bottleneck is specified to be the RNA-concentration 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 0.1--1 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 two-compartment 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

K-compart~

rKG .~compartme,n t

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t

Fig. 5. Schematic representation of a two-com- partment model (r~k = rate of conversion of substrate to K-compartment, 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 Monod-type 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 well-known that the composition of the biomass in a steady-state 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 steady-state fraction of the G compartment can be modelled by:

w* = W~o + [3ola (56)

w* indicates the steady-state 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 two-compartment 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 two-compartment 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 pseudo-steady-state 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 13,81-84), 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 two-compartment 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 steady-state culture, the advantages of the two-compartment model over the unstructured approach are not readily apparent. The differences become clearer in transient situations such as wash-out from continuous culture. Figure 7 shows the results of a simulation of wash-out for cells pre-grown 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 two-compartment model (arbitrary parameter values)

reveals that the cells pre-grown at the lowest growth rate exhibit a more rapid wash-out.

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 steady-state 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 two-compartment 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 wash-out 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 ss-ss)

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 Three-Compartment 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 three-compartment 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 three-compartment approach over the two-compartment 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 three-compartment model

One of the interesting features of the model is an argument which is independent of the details of the kinetic treatment. The three-compartments 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 pseudo-steady-state 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 pseudo-steady-state 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 YATp-COncept (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 YATp-Values are the ATP-yields (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 ATP-consumption 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 three-compartment 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. G-compartment turnover) does not exceed 3 %/h 96), while in non-growing 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 three-compartment 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 steady-state relationships for the rates of synthesis of these compartments, which follow directly from the steady-state 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 99-101) He proposes a n-th 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 wild-type 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 13-galactosidase 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 13-galactosidase increases by a factor 1.000-- 10.000 102. lo3.zo4~. The research con- cerning the lactose-inducible IB-galactosidase 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 so-called 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 co-ordinated 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 lac-operon 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 L-arabinose 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 m-RNA (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 = anti-inducer). These regulatory phenomena are termed negative controls. Based on the research of Gilbert and Mfiller-Hill 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 ligand-repressor-operator 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 anti-inducer, 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 10-20 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 pseudo-steady-state 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 i-th 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 pseudo-steady state. In wild-type E. coli cells, there are 10-20 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 anti-inducer, 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 CAP-cyclic AMP complex stimulates the fl-galactosidase synthesis by binding at the promoter site (P) of the lac-operon and initiating the transcription by RNA potymerase (see Fig. 9)138,139) This antagonistic effect of catabo