Dynamics of microbial propagation: Models ... - Semantic Scholar

BIOTECHNOLOGY AN11 BIOENGINEERING VOL. IX, PAGES 129-170 (1967)

Dynamics of Microbial Propagation: Models Considering Inhibitors and Variable Cell Composition D. RAMKRISHNA, A. G. FREDRICKSON, and H. 14.TSUCHIYA, Department of Chemical Engineering, University of Minnesota, Icfinneapolis, Minnesota 55455

Summary Mathematical models for microbial growth in batch and continuous cultures are formulated. The models have been referred to as distributed models since the microbial population in a culture is looked upon as protoplasmic mass distributed nniformly throughout the culture. Growth is regarded as the increase in this mass by conversion of medium components into biological mass and metabolic products. Two sets of models have been presented. The first arise from inkoducing additional considerations into the model proposed by Nonod to account for ihe stationary phase and the phase of decline in a batch culture. These have been referred to as unstructured, distributed models since t,hey do not recognize any form of structure in the protoplasmic mass. The models in the second set are referred to as structured, distributed models. Structure is introduced by considering the protoplasmic mass to be composed of two groups of substances which interact wit.h each other and with substances in the environment to produce growth. The structured models account for the dependence of growth on the past history of t,he cells; thus they predict all growth phases observed in batch cultures, whereas the unstructured models do mot predict a lag phase. The full implications of the models for continuous propagation, as determined by the method of stability analysis and transient calculations, arc discussed. The models predict a number of new results and should be confronted with experiments.

INTRODUCTION In a microbial culture the individual cells increase in mass by growth and in number by reproduction when the appropriate nutritional arid environmental requirements have been met. The mathematical approach to the study of microbial populations has generally been based either on the use of population density (number of cells per unit volume) or biomass concentration (dry weight per unit volume) 129

130 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

as the dependent variable. * The models based on population density may be referred to as “segregated models” since they recognize that life is segregated into structural and functional units. The models based on biomass concentration may be referred to as “distributed models” since, in effect, they consider the population to be cellular mass distributed uniformly throughout the culture. The segregated models are sometimes based on stochastic (probabilistic) considerations, whereas the distributed models use a deterministic approach. The mathematical equations of the distributed models are more amenable to solution under many situations than those of the segregated models because of the former’s simplicity. Consequently, the distributed models represent an important tool for the analytical study of microbial populations. This paper will consider the formulation of distributed models for microbial growth in batch and continuous cultures. The distributed models are based on interpreting growth as the result of “chemical reactions” between cellular mass and the environment. The chemical reactions (as distinguished from true chemical reactions) are not molecular processes, although they are indeed an overall representation of molecular processes in the cell. I n this sense the approach herein differs from those of Hinshelwoodl and P e ~ r e t ~ . ~ which are based on molecular processes. Monod4 was probably the first to formulate a distributed model using dry weight as the variable. His model does describe batch and continuous growth of microbial cultures under some circumstance^.^ It is just as much the merits of the Monod model as its inadequacies that demand further attention for distributed models. It is possible to consider the protoplasmic mass as further divided into component masses and to regard growth as a result of interactions between component masses and the environment. Such models may be referred to as “structured, distributed models” since the protoplasmic mass is assigned some structure. In this paper both unstructured (when no subdivision of the protoplasmic mass is considered), distributed models and structured, distributed models have been discussed. There appear t o have been no previous attempts to describe in terms of a mechanistic model, the entire batch-growth curve consisting of

* A more complete classification of mathematical models for microbial population has been suggested by Tsuchiya, Fredrickson, and A r i ~ . ~ ~ BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX,ISSUE 2

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the lag phase, the exponential phase (often erroneously referred to as the logarithmic phase), the stationary phase, and the phase of decline. Moreover, the equations employed by most workers for continuous cultures describe only the exponential phase when used for batch growth. This seems to indicate that the continuous culture has only been regarded as a system in which exponential growth is maintained by a continuous supply of nutrients. Admittedly this assumption is not a bad one at low holding times (high dilution rates) and low substrate concentrations in the feed; but as the holding time and the feed substrate concentrations are progressively increased, there is no reason to believe that exponential growth wiIl persist in the continuous culture. Clearly, phenomena underlying the appearance of the various phases during batch growth will also exert their influence in continuous culture. In other words, a mathematical model should first describe the entire batch-growth curve, which the Monod model does not, before it is extended to continuous cultures. It must be emphasized that the models presented here are not intended to provide a final picture of microbial growth. They are intended to show what are the qualitative, as well as the quantitative consequences of a number of possible explanations for the various phases of batch growth. If a model represents an explanation of batch phenomena, then numerical values can be found for the constants defined in the model such that batch growth under a particular set of environmental conditions can be described. It is therefore clear that fitting the batch-growth curve under a single set of environmental conditions is no test of the model’s validity; rather, it represents a necessary but insufficient requirement. Hence some explanations can be rejected at once because they do not satisfy the necessary condition of describing the batch-growth phases. On the other hand, the hypotheses become more plausible if the model with constants determined from batch data can describe the dynamical behavior of the population in other situations. The other situations involve batch and continuous propagation under various conditions. The entire growth curve cannot be described using the dry weight of cells as the growth variable under all circumstances since the dead cells which do not lyse contribute to the dry weight so that a phase of decline is not yielded. The growth variable is therefore defined to be an “active biomass” or viable mass which may be regarded as the dry weight of cells multiplied by the fraction of viable cells in the cul-

132 I). RAMKRISHNA, A. G. FREDRICKSON, €1. hl. TSUCFIIYA

ture. I n cases where the dead cells lyse readily the dry weight can of course be regarded as the active biomass. An assumption often made for the appearance of the phase of decline (and sometimes also for the stationary phase) is that inhibitory products of metabolism accumulate during growth and their subsequent interaction with the viable cells results in the death of these cells. Experimental evidence for the formation of such inhibitory products during growth has been found by Pratt and Fong6 for the green alga Chlorella and by Pratt’ for fungi. Thus an inhibitor will be included as a reactant participating in the reactions representing growth. Presumably there are various inhibitory products formed during growth: but just as the protoplasmic mass built up of a great variety of constituents has been represented by a single entity, the diff went inhibitory products may also be represented by a single entity; this will be denoted simply as inhibitor. The interaction between the inhibitor and the active biomass besides rendering the latter nonviable may result in a number of possibilities as follows: 1. The inhibitor may be permanently tied up with the nonviable mass; i.e., once formed the inhibitor is consumed in the process. 2. The inhibitor may not be consumed in the process; i.e., once formed the inhibitor will exert its influence for all times. 3. More of the inhibitor may be released into the medium; in other words, the inhibitor is produced autocatalytically from its interaction with the viable mass. 4. There may be a lag between the formation of nonviable mass and the release of additional inhibitor into the medium. Other possibilities could also be considered. I n this paper, the above possibilities have been analyzed for the unstructured models. The structured models are, however, based on the third possibility since the analysis of the unstructured models appeared to favor this possibility over the others. It will be clear from the analyses that the models yield predictions characteristic of the assumptions made, thus providing a systematic route to appropriate experiments. The duration of the lag phase is dependent on a number of factors including the previous history of the inoculum, so that a model which aims at describing the batch growth curve including the lag phase needs to include the effect ofpast history on growth rate. It may be assumed that this dependence of growth on past history arises out of the dependence of growth on the current “physiological state” of the cells developed in the course of their past history. Thus it is imBIOTECHNOLOGY AND BIOENGINEERING, VOL. I X , ISSUE 2

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portant to develop a concept of physiological state of the cells on which their growth will depend. Moreover, RIBlek concludes from an analysis of the literature that the question of physiological state in continuous cultures is of both theoretical and practical importance.8 The structured models presented in this paper will regard the active biomass as made up of structural component masses. Although the structural complexity of biomass is recognized, it seems worthwhile to consider only two component masses for simplicity. The relative proportions of component masses are postulated to characterize the physiological state of the cells and growth will be assumed to be a function of this physiological state. For a description of the stationary phase and the phase of decline, the same assumptions as those made for the unstructured models are also made for the structured models; i.e., inhibitory products of metabolism are formed during growth which convert the active biomass or the viable mass to nonviable mass. The “autocatalytic step” in the formation of the inhibitor used in the unstructured models will also be retained. The validity of this assumption requires further consideration. The models analyzed in this paper have a fair number of constants. However, it is important to realize that the number of dependent variables is also large. The constants essentially signify the rates of interactions and the quantitative proportions in which the various reactants interact. A growing microbial culture involves a large number of coupled physicochemical processes so that a model of “few” constants would not satisfactorily describe microbial growth.

UNSTRUCTURED, DISTRIBUTED MODELS The models anaIyzed will be referred to as “staling-effect models” since the inhibitory products formed during growth “stale” the culture. Accordingly, staling effect models I, 11, 111, and IV have been worked out. However, it may be noted that models I, 11, and I11 are special cases of a general proposition, whereas model IV includes one additional step.

The Model The reactions representing growth may be denoted symbolically as follows:

134 11. RAMKRISHNA, A. G. FREDRICKSON, H. & TSUCHIYA ‘I.

v + a$

+

+... + (1 + u T ~ ) T+ . . . 2v f aT1’

(1)

V T+N (2) where V = active biomass or viable mass, S = substrate, T = inhibitor, and N = dead protoplasmic mass. The dots represent other products of metabolism. The quantities a,, aT, and aT1 are stoichiometric constants whose significance can be directly inferred from the reactions (1) and ( 2 ) . Thus a , represents the amount of substrate consumed in the process of forming unit mass of active biomass. The constant aT represents the amount of inhibitor formed in the process of forming unit mass of fresh protoplasmic mass. While a , and aT are obviously positive aT1 can be negative, zero, or positive. When aT1 is negative, the inhibitor is consumed in the process of its interaction with active biomass and aT1 would represent the amount of inhibitor consumed in the “deactivation” of unit mass of viable mass [reaction ( 2 ) ] . The model in which aT1 is assumed negative is referred to as “staling effect model I.” When aT1 = 0, no inhibitor is consumed in interacting with the active biomass and the model is analyzed as “staling effect model 11.” When aT1 is positive, more inhibitor is formed from reaction (2) and the model will be called “staling effect model 111”; aT1 would then represent the amount of inhibitor formed in deactivating unit mass of active biomass. The staling effect model will be abbreviated as SEM when convenient.

Batch Culture Studies I n order to write down the differential equations for batch growth it is necessary to assign kinetic expressions for reactions (1) and ( 2 ) . The Monod growth expression will be used for reaction (1) and the interaction between the viable mass and the inhibitor will be taken to be a second order process. The batch culture characteristics can now be analyzed for SEM I, 11, and 111. The differential equations for batch growth are given by:

dCv/dt

=

dC,/dt

=

dCT/dt

=

pCsCv/(Ks

+ CJ - KCTCo

+ C,) aTPCsCo/(K, + Cs) + KaTlCTCv -as~C,Cv/(Ks

(3)

(4 (5)

where C irepresents the concentration of entity i. The constants p and K , are the constants in the Monod model, and K is the rate constant for the deactivation or the death process. It is important to BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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note that a, is not the same as the reciprocal of the “yield coefficient” Y of Herbert et al.599 If there were no deactivation of the viable mass, then Y would be the reciprocal of a,. However, when the deactivation process is considerable, a , and 1 / Y would differ. Thus a , refers to the amount of substrate assimilated in the formation of unit mass of protoplasmic mass without accounting for the deactivation process. The deactivation process would reduce the net gain of protoplasmic mass by growth. I n our opinion the quantity a, is more likely to be constant than Y . The differencebetween 1/Y and a, will be discussed further in the section on continuous cultures. Illustrative calculations for the various models have been presented by assigning realistic values to the constants from their biological significance. Thus the resulting batch-growth curve essentially conformed in trend both qualitatively and quantitatively with data generally obtained on dry weight of cells and substrate concentration. The constants p, K,, and a, can be determined from data on exponential growth where the deactivation process is negligible. The constants K , a,, and aT1,which refer to the inhibitor and its effects cannot be so determined but require further data. For example, data on the stationary phase and the phase of decline are essential. One might wonder how values for these constants can be determined when the identity of the inhibitor is not established and consequently its concentration is not known. The answer is that absolute values of these constants cannot be determined without knowledge of absolute inhibitor concentrations, but relative values of the constants can be determined by fitting equations to the batch-growth curve in its stationary and decline phases. The inhibitor curves that will be presented also have no absolute quantitative significance, but bear a constant ratio to the “actual” inhibitor concentrations. This can be seen from the following: Let C,’ be the “actual” inhibitor concentration (which is not known since the identity of the inhibitor is not established). If the proposed model is an appropriate representation of microbial growth then constants p , K,, a,, K’, aT7,and arl‘ can be found such that the equations

(3‘)

=

+ CJ - K‘CT‘C, -a,pC‘,C,/(K, + C,)

=

aT’pCsCo/(Ks

(5’)

dC,/dt

=

dC,/dt dC,’/dt

pC,C,/(Ks

+ C,) + K’aTl’CT’Co

(4)

136 I). RARIKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

describe growth in batch culture. Since p, K,, a, can be determined without knowledge of the inhibitor, these constants may be assumed to be the same as those in eqs. (3), (4), arid (5), so that calculated values of C, and C, correspond to experimental measurements. It is to be proved that the calculated values of 6, have the relationship claimed with the actual inhibitor concentrations. Equating eqs. (3) and (3’) one finds that CT = (R’/K)CT’, which proves the required result. A few simple manipulations will show that a, = K’a,’/K and uT1 = K‘aTl’/K.

Staling Effect Model I The differential equations for batch growth are the same as eqs. (3), (4),and (5) with the restriction that aTl is negative. Thus the

TIME

(HOURS)

Fig. 1. Batch-growth curve for model I on semilog plot. Initial concentration of inhibitor = 0 g./liter in all batch calculations for all models.

model makes the following propositions: During growth, a n inhibitory product is formed which interacts with the viable mass and converts the latter to nonviable mass. As the substrate is exhausted all growth stops and the accumulated inhibitor converts the existing active bioBIOTECHNOLOGY AND BIOENGINEERING, VOL. I X , ISSUE 2

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TABLE I P

Model

I I1 I11

hr.-1 1.0 1.0 1.0

K*,

K,

g./liter

a,

liter/g.-hr.

UT

aT1

0.2 0.2 0.2

2.0 2.0 4.0

1.0 0.4 1.0

0.06 0.1 0.002

-0.03 0 0.03

mass to nonviable mass. It is clear from these coiisiderations that, the stationary phase cannot be described by such a model and calculations confirm this. Figure 1 shows a plot of theviable mass, substrate, and the inhibitor concentrations in a batch culture predicted by the model. Table I lists the values of the constants used for the calculations. The exponential phase comes to an end when the substraie is exhausted. If the constarit uT is larger than that, piclird, thcri the exponential phase would come to an end ns a result of ac~cumulntedinhibitory products.

X t a h g Effect Model II When aT1 is put equal to zero, the differential eqs. (3), (4), and ( 5 ) represent the batch-growth equations for model 11. According to the model, an inhibitor is formed during growth which converts the viable mass into nonviable mass but is not consumed in the process. There is no stationary phase predicted by the model since growth ceases with the exhaustion of substrate and the viable mass concentration thereon drops exponentially. Model 11 is very nearly the same as Model I but for the fact that in the phase of decline in model I the drop in the viable mass concentration is not exponential; in model I1 the death rate in the phase of decline is purely exponential. This gives rise to a distinction between the models which can be put t o test experimentally by plotting the logarithm of viable mass concentration in the phase of decline. A model developed by Rahnlo for retardation of fermentation rates by the products of fermentation as applied t o the growth of yeast by Kleni" is somewhat similar to the staling effect model 11. The rnodcl used by Klem however employed the number of cells per unit volume as the variable and the kinetic expression used was the second-order expression first employed by McKendrick and Pai.12 The mecha-

138 D. RAMKRISHNA, A. G. FREDRICKSON, H. 111. TSUCHIYA

Fig. 2. Batch-growth curve for model 11 on semilog plot.

nism of inhibitor* formation can be shown to be the same as that in the staling effect model 11. It appears that this interesting idea which Rahn advanced in 1929 has somehow eluded further development. If eq. ( 5 ) is divided by eq. (4) and the result integrated from time t = 0, then one obtains

C,

=

a,(C,i - Cs)/a,

(6)

where C S cis the value of C, at t = 0, and Cr has been taken to be zero at t = 0. Substitution of eq. (6) into eq. (3) yields dCw'dt = L'sC,/(Ks

+ CJ - KaT(c,t - Cs>

(7)

Thus the inhibitor concentration has been eliminated as a variable and one needs to solve only eqs. (4) and (7) simultaneously to obtain the batch-growth curve. During exponential growth, the term Ka,(CSi - C,)/a, will be negligible so that the constants p , K,, and a , can be evaluated from experimental data in the exponential phase. Data in the phase of decline will yield the product of the constants

* Klem considered alcohol to be the inhibitor. BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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K and aT but not the individual constants. However, in order to solve eqs. (7) and (4)one needs to know only the product Ka,. The values of the constants picked for calculations are tabulated in Table I. Figure 2 is a plot of viable mass, substrate, and inhibitor concentrations. As already inferred from the hypotheses of the model, the calculations show no lag or stationary phases.

Staling E$ect Model III I n this model, the constant aT1is positive, which implies that the interaction between the viable mass and the inhibitor releases more inhibitor into the medium besides rendering the viable mass nonviable. The batch-growth differential equations are given by eqs. ( 3 ) , (4), and ( 5 ) with the restriction that a,l is positive. The values of the constants used for calculations on the model are in Table I. Figure 3 shows a plot of viable mass, substrate, and inhibitor concentrations in a batch culture calculated for the model. The model predicts all of the observed growth phases with the exception of the lag phase. It is easy to see how these phases are predicted. During exponential growth, the inhibitor accumulates in small quantities. When the substrate is nearly exhausted, growth ceases but the deactivation process is not significant since the inhibitor has not accumulated in sufficient quantities. This period is the stationary phase. But the gradual autocatalytic production of inhibitor increases the rate of deactivation of the active biomass, thus leading into the phase of decline. The microbiologist would have noticed that the initial concentration of the active biomass of 0.25 g./liter used for the calculation (Fig. 3 ) would correspond to a massive inoculum, and hence no lag phase may be observed in practice. However, it must be recognized that the mode1 will predict no lag irrespective of how small the inoculum size may be. Figure 4 shows the effect of the inoculum size on the growth curve. The final yield is almost the same for varying inoculum sizes for a given initial substrate concentration. This is consistent with many experimental observations (for example, Montank’s data on yeastL3). Figure 5 shows the effect of the initial substrate concentration on the growth curve. With increasing initial substrate concentrations, ( I ) the maximum yield of active biomass is increased, (2) the duration of the stationary phase is decreased, ( 3 ) the death rate in the phase of decline increases considerably since increased amounts of

140 D. RAMKRISHNA, A. G. FREDRICKSON, H. RI. TSUCHIYA

Fig. 3. Batch-growth curve for model I11 on semilog plot.

I".

6.01

4.0

0

2

4

6

8

10

12

TIME (HOURS)

Fig. 4. Effect of inoculum size 011 the batch-growth curve (model 111). Initial substrate concentration, Cai = 10 g./liter. BIOTECHNOLOGY AND BIOENGINEERING, VOL. I X , ISSUE 2

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141

1

O.l0

Fig.

5 . Effect

of

2

4

6

8

TIME

(HOURS)

10

initial substrate concentration growth curve (model 111).

(C,J on the

batch

inhibitors are formed. While experiments show that ( I ) is true,14 data of the effect of the initial substrate concentration on the entire batch-growth curve appear to be scanty.

Staling Eflect Model IV The assumptions of this model are essentially the same as those of model 111. However, the process of death docs not produce the inhibitor instantaneously, but the dead cells release more inhibitor into the medium after a finite period of lag. Thus thc intcractioris representing growth arc as follows: V+a,X+28+a,T+. .. V+T+N+T+ ... N + aT1’T . . .

+

(1)

(W (8)

Reactions (1) and (2a) are the same as those for SEM 11. Hence, model IV may also be regarded as an extension of model 11; the former includes the additional reaction (8). Reaction (8) may be regarded as a process by which lysis of the accumulated dead cells occurs. It will be assumed that this is a firstorder process,

142 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

The batch-growth differential equations are given by

+ C J - KCTCu dC,/dt = - a S ~ C , C , / ( K s+ C J dCT/dt = a!r~CsCu/(Ks+ CJ arltKtCiv dC,/dt = pCsCo/(Ks

dC,/dt

=

aNKCTCu - K'C,

(3)

(4) (9) (10)

where C N is the concentration of nonviable mass. The constant K' represents the rate constant for reaction (8) and aTItand aN are new stoichiometric constants whose significance are as follows: aTlt is the amount of inhibitor formed from unit mass of nonviable mass and is the amount of nonviable mass from deactivation of unit mass of viable mass. The constants used for the calculations on the model are listed in Table 11. Figure 6 shows the active biomass, substrate, and inhibitor concentrations in a batch culture as calculated for the model. As in model 111, with the exception of the lag phase all other growth phases are predicted by model IV. This is of course because there is not much difference between the premises on which the models have been built. It would appear that model I V wouId be more suitable when the dead cells do not lyse readily in which case the measured dry weight should be equal to 6, CN.

+

SUBSTRATE

VIABLE MASS

0.2O.l0

Fig. 6.

2

4 TIME

6 ~HOURS)

8

1

0

1

2

Batch-growth curve for model I V on semilog plot. BIOTECHNOLOGY AND BIOENGINEERING, VOL. I X , ISSUE 2

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TABLE I1

IV

1.0

0.2

4.0

1.0

0.002

0.3

1.4

1.0

Calculations on the effects of inoculum size and initial substrate concentrations are not presented since they are, qualitatively speaking, the same as those for model 111. It is now of interest to determine the continuous culture predictions of the models I, 11,111,and IV.

Continuous Culture Studies The continuous culture equations are obtained by a trivial extension of the batch-growth equations. For models I, 11, and 111, the differential equations for continuous propagation are given by

dC,/dt

=

dCs/dt

=

+ C,) - KCTC, - Cu/O -as&’sCv/(Ks + Cs) + (Cso- CJ/e

dCT/dt

=

~TPC&’,/(K,-I-

pC,C,/(K,

c,) + KaTlCTCu - c,/e

(11) (12) (13)

where C,, is the concentration of the substrate in the feed and O is the holding time (reciprocal of the dilution rate). The possible steady states are obtained by equating the time derivatives to zero and solving the resulting algebraic equations on the digital computer. Whether the solutions of the differential equ% tions actually lead to any one of the possible steady states may be investigated by using the stability theorem of Liapunov.l6 A detailed description of the method is given by Bilous and Amundson.16

Steady States An obvious solution of the steady-state equations is given by = 0, = C,,, and = 0, where represents the steady-state concentration. This represents the condition of washout, and the steady state may be referred to as the washout steady state. Thus, when the washout steady state is “stable,” the continuous propagator will not grow organisms but will be filled with sterile medium. The minimum holding time (maximum diIution rate) is that below which

c8

c,

cT

c

144 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

washout occurs. This holding time can be evaluated analytically by the method of stability analysis. The algebraic equations predict one more steady state in addition to the washout steady state. This steady state will be referred to as the normal steady state since it represents the condition of normal operation.

Stability of Steady States

A steady state is locally stable if the system (the continuous propagator) returns to the steady state in question after a sufficiently small but otherwise arbitrary perturbation from the steady state. The local stability of any steady state is assured when all the eigenvalues of the stability matrix (obtained by linearizing the differential equations about the pertinent steady state) have negative real parts. As an example, the stability of the washout steady state will be worked out below. Let

cv= c " v + x

c,= c",+y

c T =

c"T+x

where x, y, and x are small perturbations from the steady-state values cu, and respectively. The linearization about the steady state is done by expanding the right-hand sides of the differential equations ( l l ) , (12), and (13) in their Taylor series about the steady state and chopping the series off at the first derivative terms. The procedure leads to the following linear vector diff ercntial equations.

c,,

cTJ

+

+

where all = P C J ( K ~ c",) - Kc";. - l/0, a12 = P K , ~ " , / ( K , CJ2, 0113 = --KC",, a 2 1 = -a,pc",/(K, CJ, a 2 2 -= - ~ S P K X(JK s c",)' - 1/09 f f 2 3 = 0, ff31 = U T P ~ " S / ( K Sf CS) KaTlcT, ff32 =,aTpK,Ca/(K,f 0 3 3 = KaTicu - I/$. For stability of the steady state, the eigenvalues of the stability matrix whose elements are a Z jmust have negative real parts. At the washout steady state, the characteristic equation for the eigenvalues becomes

+

c,)',

+

+

BIOTECHNOLOGP AND BIOENGINEERINQ, VOL. IX, ISSUE 2

DYNAMICS OF MICROBIAL PROPAGATION

(A

+ l/ep[x + l / e -

+C~JI

P~,,,/(~,

=

145

o

Hence A*

=

ha = -l/@

and A3

= PCso/(Ks

+ CSJ - l/@

It is clear that the washout steady state will be stable if and only if is negative since XI and Xz are obviously negative. Thus washout will occur when (and only when)

A3

PCso/(Ks

+ cs,> < 1/e

(14)

By a similar analysis it can be proved17that the normal steady state will be stable if and only if 8

> (Ks+ c,,)/Pcs,

(15)

Of course the linearized equations predict only local stability SO that the transient equations should be solved to confirm stability with respect to any starting point. The minimum holding time, en is clearly given by ern =

+ Cd/PCS,

(Ks

(16)

For 0 < Om washout will occur and for e > em normal steady-state operation can be maintained. The minimum holding time is seen to be the same as that predicted by the RiIonod model.5 For the normal steady state it can be seen from the steady-state equations that

C,/(cSu - Cs) = i/as(KCTe + 1)

(17)

For the Monod model, the left-hand side of eq. (17) is equal to the yield coefficient Y , which is assumed to be a constant. Experimental data show this to be otherwise.9 Data obtained by Herbert for Aerobacter aerogenes and Torula utilis show that the yield coefficient decreases with increasing holding times. The staling effect model, on the other hand, predicts such a result because the right-hand side

146 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCI-IIYA

of eq. (17) decreases with increasing holding times since the inhibitor concentration increases with increasing holding times. Since models I and I1 do not predict a stationary phase in a batch culture no continuous culture calculations are presented for these models.* Model I11 is analyzed in some detail.

Staling E$ect Model III The analytical results derived in the previous section hold for models I, 11, and 111. The essential difference in continuous culture predictions between the models lies in the transient behavior.

HOLDING TIME (HOURS)

Fig. 7 . Normal steady-state concentrations of viable mass, substrate, and inhibitor (model 111)versus holding time. Feed substrate concentration, C,, = 20 g./liter.

Figure 7 shows a plot of the normal steady-state concentrations of the active biomass, substrate, and the inhibitor. It is worth noting that the substrate concentration has leveled off at a relatively high concentration. This is because the feed-substrate concentration has been deliberately picked to be large to show the effect of the inhibitor. For smaller values of the feed-substrate concentration, the substrate curve will level off at a lower value of the concentration. The steady-state growth and death rates are plotted against holding time in Figure 8. For short holding times above B,, the growth rate * These have been presented elsewhere.'? HIOTECHNOLOGY AND BIOENGINEERING, VOL. I X , ISSUE 2

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0

1

2

HOLDING

3 TIME

4

5

6

147

7

(hrs)

Fig. 8. Normal steady-state growth and death rates versus holding time (model 111). C,, = 20 g./liter.

is considerably higher than the death rate, but as the holding time increases the growth rate approaches the death rate closely. Thus, as Herbertg points out, the cells at low holding times will be like exponential phase cells in a batch culture and as the holding increases they will be more like cells in the stationary phase or the phase of decline. As the holding time increases, the average age of the culture in the propagator also increases. Since the dependence of growth rate on the age of the culture has been ignored (as reflected by the failure of the model to predict a lag phase in batch growth) the results of the model at long holding times should be open to some question. Of course these considerations are also true of the Monod model or any unstructured model. Nature of Transients. The stability analysis for the normal steady state shows that when 8 > 8,, the steady state is either a stable node or a stable focus.* For short holding times, the steady state is a stable node and for long holding times it is a stable focus. When the steady state is a stable focus, the transients are damped oscillations about the steady-state values. As the holding time is increased the rate of damping is lower till the oscillations are practically undamped.

* For definitions of stable node and stable focus see Bilous and Amundson.16

148 1). RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

The equations were solved in the analog computer and the plot of the oscillations on an automatic recorder is shown in Figure 9. Figure 10 shows oscillatory behavior represented on the phase plane of the viable mass and the inhibitor. (This plot was obtained by using an X-Y plotter.) This model might well lead to an understanding of cyclic oscillai tions observed by Finn and Wilson'8 and independently by Maxon19 2.0-

:; LO^ ln

yo

-

m

w

o

(from Lhe analog computer).

L

2 2.0.-

-& \

u)

1.5-

ln

a

a

m 1.0-

w

2 I-

2

.5-

0

Fig. 10. Damped oscillations in continuous culture represented on the phase plane of active biomass and inhibitor (from the analog computer): C , = 20 g./ liter; 0 = 2.5 hr. (model 111). BIOTECHNOLOGY AND BIOENGINEERING. VOL. XX, ISSUE 2

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k W _I

149

TIME (HOURS)

z

Fig. 11. Continuous cri1t)ure tra.iisierits for different, feed substrate concentratioiis (model 111). Note that there are no oscillatjiorrsfor low substrate coiiceiitrat,ions: e = 2.0 hr. (Plots from the analog computer.)

in continuous cultures. * (However, Matcles and Goldtmhwaitez0 have reported that they could not observe oscillatory behavior.) It appears from our analyses that investigation for oscillatory behavior should be made over a wide range of flow rates and feed substrate concentrations. Thus, reference to Figure 11 shows that no oscillations are observed for a feed substrate concentration (CJ of 10 g./liter (at % = 2.5 hr.) whereas on increasing C,, to 20 g./liter, damped oscillations are obtained. These oscillations become less damped when the holding time is increased to 5 hr. (see Fig. 9).

Staling Eflect Aloclel IV The differential equations for continuous propagation are given by (11) and (12), and

+

+ UTI'K'C,

dCvT/dt = U ~ / . ~ C ~ C ~ / C,) (K, dC,/dt

=

aNKCTCli- K'C, - C,/%

- CT/%

(18) (19)

The minimum holding time 8, can again be proved to be given by (16). The normal steady-state concentrations of viable mass, substrate,

* It can be readily proved that the Monod model canriot account for oscillatory behavior.17

150 D. RAMKRISHNA, A. G. FREDRICKSON, H. 31. TSUCHIYA 4.0 _..___._

- 20

3.6-

- I8

3.2 -

-6

2.8-

z

7 E S 2.4-

c

(r

g 2.0r m

z

1.6

-

2-= 1.2-

m Y

a

5 0.004 -

0

0

1

2 3 4 5 HOLDING TIME (HOURS)

6

0 7

Fig. 12. Normal steady-state concentrations of viable mass, substrate, arid inhibitor (model I V ) versus holding time, C,, = 20 g./liter.

and inhibitor are shown in Figure 12. Of particular interest is the behavior of the substrate concentration which after decreasing at first with increasing holding times begins to increase again gradually. The reason for this behavior is obvious. At low holding times, the dead cells are washed out before they lyse and relcasc inhibitor into the medium so that all the substrate is nearly consumed. But as tho holding time is increased the viable mass is converted to nonviable mass, leaving increasing amounts of substrate unconsumed. Such behavior does not appear to have been observed by some, but Jannasch2I has recently provided data of a similar kind. The transient behavior is not much different from that of model 111, especially at large holding times, and is not presented here. The unstructured models do not describe the lag phase in batch growth. Since the duration of the lag phase is dependent on the origin of the inoculum (among other factors) a model should be able to incorporate the effect of past history of the cells on the growth rate BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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in order to predict a lag phase. This cannot be achieved by an unstructured, distributed model (maintaining the autonomous nature of the equations). However, by considering the active biomass to be made up of structural component masses and by regarding growth as the result of interactionq of these masses bctwccri each other and the environment, the effect of past history can bc incorporated into the models. I n other words, structured, distributed models are needed for describing the growth curve including the lag phase. Such models are discussed below.

STRUCTURED, DISTRIBUTED MODELS The Model The active biomass may be divided into two components which we shall denote by G-mass and D-mass. G-mass refers to the nucleic acids and D-mass refers to the rest of the active biomass consisting mainly of proteins. I n a remote sense, this idea is somewhat similar to that which WeissZ2applied to the growth of higher organisms. Growth thus consists of the increase in G and D and occurs as a result of the combined interaction of G and D with substrates. A single substrate for the synthesis of both G and D is assumed, although the inclusion of additional substrates does not overly complicate the problem. The reactions representing growth may be written symbolically as

where S = substrate, T = inhibitor, N c = inactive G, and N , = inactive D. The interrogation marks are to denote various mechanisms of inhibitor formation. The dots represent other products of metnholisin. Thc quantities u s ,us’, u,, uT’,aT,, and uTl’ are stoichiomctric const)ants whose significance will be referred to later. Reaction (20) denotes the synthesis of G-mass, and the reaction will

152 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

occur only if G and D are both present in addition to the substrate. Since G-mass is the nucleic acids in the active biomass and D includes enzymatic proteins, reaction (20) may be taken to be an overall representation of the biochemical steps in the synthesis of the nucleic acids. Reaction (21) denotes the synthesis of D-mass, and the interaction will occur only in the presence of both G and D in addition to the substrate. Although reaction (21) can be seen to derive its basis from the biochemical steps leading to the synthesis of proteins, the interaction should be interpreted to include the synthesis of all the constituents in the D-mass. Reactions (22) and (23) together constitute the death process. Individually (22) and (23) denote the deactivation of the G- and Dmasses. Having defined the various interactions, appropriate kinetic expressions must be established for the rates of these interactions. It will be assumed that eqs. (22) and (23) are second-order processes since this would be the simplest representation for reactions involving two reactants. I n order to represent interactions (20) and (21), a double substrate Rlichaelis-Menten type of kinetic expression (see LaidleP) is proposed. Thus the rates of syntheses of G-mass and D-mass denoted by RG and R,, respectively, may be represented by the following expressions:

RG RD

=

pCsC&,/(Ks

+ C,)(KG + CG) + CJ(KG’ + C,)

(24)

M’C,C&,/(KS’ (25) where CG,C, and C, represent concentrations of G-mass, D-mass, and the substrate, respectively. The quantities p and p’ are specific rate constants while K,, K,’, K,, and K,’ are Michaelis constants. Interpretations of the Rilichaelis constants similar to that of K s in the Monod model4 may be derived, I n general, however, they may be roughly regarded as rate-limiting values of the pertinent concentrations. The choice of the kinetic expressions (24) and (25) is a natural one considering the fact that cnzyme-catalyzed reactions form the basis of cellular metabolism. The use of eqs. (24) and (25) bears the same analogy to an enzyme reaction system involving two substrates as does the Monod model to an enzyme reaction system involving a single substrate. Furthermore, it can be shown that observations made by CasperssonZ4and by Gale and Folkes25on the rates of syn=

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theses of nucleic acids and proteins, respectively, are in accord with results on the rates of syntheses of G- and D-masses from the kinetic expressions (24) and (25). Based on two different mechanisms of inhibitor formation, two models, denoted 1 and 2, have been analyzed.

Model 1

G+aSSfD+2G+D.. G

+ a,'X + D

4

2D

.

(20')

+ G + aT'T + . . .

G+T+NG+(l+aT,)T+. D f T+N,+.

.

. .

.

(21)

(22)

(23')

Model 2

G+asS$-D+2G+D+aTT+. G$-a,'S+D+2D+G+. G+T-+N,+.. L ) $-

T + N , f (1

.

.

. .

. aT1')T

(20) (all)

+...

(229 (23)

Thus in model 1the inhibitor is formed from the process of forming D and more of the inhibitor is formed from deactivation of G. The reverse takes place in model 2. Other mechanisms are of course possible, but models 1 and 2 bring out the essential features of the hypotheses involved. The differential equations for batch growth from model 1 are written as: (26)

dCs - _ - -a,&

- as'RD

dt

where CT stands for the concentration of the inhibitor.

154 D. RAi’vIKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

Equations (26), (27), and (28) also hold for model 2, while eq. (29) is replaced by

Of coursc, analytical solutions of these systems of nonlinear equations for various initial conditions cannot be obtained, so one resorts to the digital computer and numerical integration techniques.

Results for Batch Cultures The constants a,, as’, aT, aT‘, a,,, and aT,‘ may be categorized as stoichiometric constants and g, p‘, K , and K’ as rate constants. The former represent the quantitative amounts of reactants involved in the reactions assumed in the models and the latter denote the rate of reaction. A complete description of each constant is appended. Approximate values of these constants for a specific organism could bc determined experimentally by analyzing batch data over specific sections of the batch-growth curve. More accurate values may be obtained by using these values as initial approximation for mathematical iterative Of course such iterative techniques are not to bc confused with ‘‘mere curve fitting’’ procedures which are often a part of bookkeeping operations involving large volumes of empirical data. The need for use of iteration techniques under the current situation simply means that the complicated nonlinear equations in the model do not, permit a direct and straightforward evaluation of the constants defined. By virtue of the biological significarwe of the various constants, realistic. values could be assigned to them. I n other words, the resulting batch-growth curve essentially TABLE I11 Constants

Model

K s ’,

hr.-1

g./liter

KsI, g./liter

2.5

0.2

0.1

P’,

as

as’

6

2

K, liters/ g./hr.

K’, liters/ g./hr.

lA, IB 150

’70

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156 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

Fig. 13. Batch-growth ciirve for model 1A. Initial concentrat,ioris: substrate, C,; = 10.0 g./liter; inhibitsor, C T (for ~ all batch calculations) = 0 g./liter; (a)calculations with ari = 0.03; ( b ) calcidations with mi = 0. Assuming that 1 g. dry weight/liter would contain nearly 10'0 cells/ml., log (cone. of biomass X 10'0) should resemble the log (population density) curve.

conformed in trend both qualitatively and quantitatively with data for cellular mass, substrate concentrations, etc. Tables I11 and IV show the constants used for models 1 and 2. Each one is further classified as A and B, the distinction between A and B arising out of a difference in the choice of nunierical values for the constants. Figure 13 shows a plot of the active biomass against time in a batch culture from calculations based on model 1A. Model 1B shows the same trend for the active biomass. The models can be seen to describe all the phases in a batch growth curve. Figure 13 also shows the batch growth curve for the case where inhibitor formation is not autocatalytic, i.e., where aT, = 0 (all other constants are the same as in model 1A). One sees that no detectable phase of decline appears. If the value of aT' is increased, then there will be a phase of decline but no stationary phase. The fraction of G-mass in the culture given by

R

=

+ C,)

C~'(CG

may be looked upon as the fraction of nucleic acids in the mass of viable cells. This fraction is likely to show trends similar to those BIOTECHNOLOGY AND BIOENGINEERING, VOL. I X , ISSUE 2

DYNAMICS OF MICROBIAL PROPAGATION

157

TIME WOURs)

Fig. 14. T7ariations of G-mass, D-mass, fraction G-mass, and inhibitor in batch culture (model IA): C, = 10.0 g./liter.

TIME

(HOURS)

Fig. 15. Variation of active biomass and fraction G-mass in batch culture (model 1B): Cai = 10.0 $./liter.

158 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

7

-

.32

TIME

(HOURS)

Fig. 16. Variations of fraction G-mass and fraction D-mass in batch culture (model 1B): Cdi = 10.0 g./liter.

shown by the amount of nucleic acids per cell ahs measured by Malmgren and Hed&1,~7 Rabotnova, Zaitseva, and Mineeva,28 and many others. The results of calculations for Cot C,, CT,and R are shown in Figure 14 for Model 1A. I n Figure 15 the active biomass and R are plotted against time for model 1B. The trend of R in Figure 14 is similar to the results of Rabotnova et aL28on the variation of nucleic acid composition of cells of Pseudornonas jiuoresceus in batch growth. However, most observations of the amount of nucleic acids per cell show trends similar to that of R in Figure 15 (for example, Malmgren and H e d ~ 5 n ;Rabotnova ~~ et aLZ8). Figure 16 which represents a plot of fraction G-mass and fraction D-mass against time shows striking qualitative agreement with the data on nucleic acids and proteins during the lag phase in a culture of Torula utilis obtained by Rabotnova et a1.28 Clearly, the concentration of G-mass in the culture affects the rate of growth for very small values of C,. As the value of C , increases, the rates of formation of G and D become very nearly independent of BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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159

11.0-

R = .01 10.0-

TIME

(HOURS)

Fig. 17. Effect of inoculum size on batch growth (model IA): CSi = 10.0 g./ liter.

2

4 TIME (HOURS)

6

Fig. 18. Effect of inocrtliim age on batch growth (model IA): Cad= 10.0 g./ liter.

Cc. I n model lA, the fraction of G-mass in the culture increases through the lag phase, levels off in the exponential phase, and drops off rapidly as the phase of decline is in progress. I n model lB, the fraction R increases during the lag phase but falls off during the phase of decline. In either case the culture will show a low value of R during the phase of decline. Most observations on the amount of nucleic acids per cell in batch cultures show the same qualitative trend as does R in model 1B. It appears

160 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

therefore that the values of constants in model 1B are more appropriate than those in model 1A. Thus, if a batch culture is inoculated with a seeding from the phase of decline, a lag phase will be predicted by the models. The fraction of G-mass thus appears to be an index of the physiological age of the culture. As a matter of fact, one may define the reciprocal of R as the physiological age. A young culture will have a relatively low value of 1/R whereas old cultures like those in the phase of decline will have high values of 1/R. One may also note that the nearly constant value of R during the exponential phase in both models 1A and 1B (also true of models 2A and 2B) is consistent with the concept of balanced growth in the exponential phase.29

Fig. 19. Effect of initial substrate concentration, CSi on the growth curve (model 1A).

While the following observations have been made with model IA, they are also true of models l B , 2A, and 2B. However, there are significant differences between models 1 and 2 for continuous operation, which will assist in the assessment of their merits. Figure 17 shows calculations made for model 1A for the effect of the size of the inoculum of a given age on the period of lag. This is consistent with observations in the laboratory. 30,31 I n Figure 18 is shown the effect of the age of the inoculum of a given size on the period of lag. This too conforms to experimental observations. 3Os3I BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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The effect of increased initial substrate concentration, up to a limit, is to increase the yield of the active biomass a t the end of the exponential phase and to decrease the duration of the stationary phase. These resuIts are shown in Figure 19. Experimental data on the effect of the initial substrate concentration on the entire batch-growth curve appear to be scanty, although increased yields of cells have been shown at higher substrate concentration^.'^ Models 1 and 2 predict that the exponential phase comes to an end when the substrate is exhausted in the medium for low initial amounts of the substrate. At higher initial substrate concentrations, the inhibitory products of metabolism halt exponential growth. The batch-growth characteristics of models 2A and 2B are almost the same as those for models 1A and lB, and hence will not be discussed.

Continuous Culture Studies The continuous culture equations are obtained by a trivial extension of the batch equations. For model 1 (A and B) the differential equations become :

where C,, is the concentration of the substrate in the feed stream and 8 is the holding time. The possible steady states are obtained by equating the time derivatives to zero and solving the resulting algebraic equations on the digital computer. As pointed out earlier, whether the solutions of the differential equations actually lead to any one of the possible steady states may be investigated by using the stability theorem of Liapunov.l5

162 D. RAMKRISI-TNA, A. G. FREDRICKSON, H. M. TSUCHIYA

Steady States

It is easily seen that a solution of the algebraic equations obtained by putting the time derivatives equal to zero is given by

CG = c”,

=

CT = 0

Cs =

c,,

This represents the condition of washout; i.e., what we have called the washout steady state. The minimum holding time cannot be obtained analytically in the models presented here; numerically it is found to be slightly greater than 1,’~’. The algebraic equations predict two more steady states in addition to the washout steady state. One of these is close t o the washout steady state and the other is the normal steady state.

Stability of Steady States : Transients The unstructured models discussed earlier predict, as does the Monod model, that no washout will occur for holding times above a certain minimum value. This means that as long as the holding time is above this minimum, regardless of the age and concentration of the cells a t start, a steady-state culture will always develop. This appears to be a rather unrealistic situation. Models 1 and 2, however, predict that the washout steady state can be stable for all holding times. This means that the possibility exists that washout could occur for any holding time depending on the initial condition of the culture. Therefore the minimum holding time in these models would be the holding time for which washout will occur regardless of the initial condition of the culture. Thus washout will occur inevitably if the holding time is below the minimum value; on the other hand, for holding times above the minimum, washout may occur only under certain conditions. The steady state close to the washout steady state was found to be unstable for both models 1 and 2. For niodel 1, the normal steady state was found to be stable for the range of holding times and inlet substrate concentrations considered. The transients are for the most part rapidly damped fluctuations, thus predicting an overshoot phenomenon in the concentration of the activc biomass. Figure 20 shows a plot of the active biomass and the substrate concentrations a t the normal steady state against holding time for model BIOTECHNOLOGY AND BIOENGINEERING. VOL. I X , ISSUE 2

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HOLDING TIME

163

(HOURS)

Fig. 20. Normal steady-state concentrations of active biomass and substrate versus holding time (model 1A). Feed substrate concentration: C,, = 20.0 g./ liter.

1A. The abrupt end of the left-hand side of the curves arises from the fact that the normal steady-state biomass concentration appears to tend to a nonzero limit (as shown by the computer calculations) as the holding time tends to the minimum value. It may be recalled that the normal steady state in the Monod model tends to the washout steady state as the holding time tends to its minimum value. A constant fraction R givcn by CG/’(CG

+ C,>

=

R

(35)

represents a straight line passing through the origin and of slope Rl(1 - R ) in the CG - C, plane. Thus, along the line given by eq. (35), different concentrations of the active biomass of the same physiological age are represented. A constant active biomass concentration given by

cc + c, = c

(36)

164 D. RAMKRISHNA, A. G. FREDRICKSON, H. M. TSUCHIYA

yields a straight line of slope -1 on the Cc - C D plane along which identical concentrations of the active biomass of various physiological ages are represented. Figure 21 shows transients for continuous operation starting with identical concentrations of various physiological ages. It may be noticed that €or some critical value of R either a washout or a steadystate culture can result from small changes in the physiological age of the initial biomass.

Fig. 21. Continuous culture transients on the CC - CD plane; effect of initial age on subsequent growth (model 1A): C,, = 20.0 g./liter; holding time, B = 0.5

hr.

Figure 22 shows transients for continuous operation starting with different initial concentrations of active biomass of the same physiological age. Lower initial concentrations tend to cause washout. Critical ranges similar to those of the previous situation also exist. I n model 2 the normal steady state was found to be unstable when the holding time was increased sufficiently. For lower holding times (but above the minimum value) the normal steady state is a stable BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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focus. The oscillations arising during transient operation damp out so slowly that they are practically undamped. Figure 23 depicts the oscillations during transient operation in the

/

/

I

2.5

2.0 D (CONC. X 1 0 ' )

Fig. 22. Continuous culture transients on the CC - CD plane; effect of initial concentration on subsequelit growth (model 1A): C, and 0 as in Figure 21. HOLDING TIME = 1.2 HOURS

1 I

I

n 1.10

0

4

8 TIME (HOURS)

12

16

20

Fig. 23. Oscillations in continuous culture of concentrations of active biomass G-mass and D-mass: C,, = 20 g./liter (model 2A).

166 TI. RAMXRISITNA, A. G. l~ItEl)RICKSON, II. M. TSUCHIYA

concentrations of G-mass, D-mass, and the total active biomass starting from unsteady-state conditions (for model 2A). The rate of damping becomes lower with increasing holding times until the stable focus switches to an unstable focus. When the normal steady state is an unstable focus, the concentrations fluctuate with increasing amplitudes until the organisms are washed out of the propagator. Figure 24 shows a phase-plane plot of a transient around an unstable focus for model 2A. This washout is independent of the initial condition of the culture. The definition of the minimum holding time is therefore not adequat,e

-L

I

b .\

W

D fgm./liter)

Fig. 24. Oscillations of increasing amplitudes eventually leading to washout in continuous culture (model 2A).

for model 2. Rather, model 2 predicts that in order to prevent washout the holding time should be somewhere between a n upper and a lower limit. The lower limit may be referred to as the minimum holding time. The appearance of the upper limit is easily explainied. Increased holding times provide for increased inhibitor formaton, which in turn drops the active G-mass concentration in the culture, thus lowering the growth rate sufficiently to cause washout. Alternatively, when the holding time is increased without limit, the culture becomes increasingly older and consequently cannot grow fast enough to balance the washout rate. RIOTECIINOLOGY 4 N D BIOENGINEERING, VOL. IX, ISSUE 2

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CONCLUSIONS It is evident that the models analyzed above predict a number of new results which can be subjected to experimental investigation. Thus the effect of the initial substrate concentration on the entire growth curve should provide some clue to the mechanism of inhibitor formation. Conditions under which oscillations appear can also be verified. Unstructured Models Since models I and I1 do not predict a stationary phase, whereas models 111 and IV do, the criterion of choice between the pairs of models is clear. I n cases where no stationary phase appears, models I and IT can be distinguished from the fact that model I1 predicts a purely exponential death in the phase of decline whereas model I predicts a slower than exponential death rate. Between models 111 and IV the difference is noticeably observed in the steady-state concentration of the substrate as a function of holding time. It is clear that the models provide a systematic layout for experiments which can be performed to determine the plausibility of the hypotheses behind the various models.

Structured Models A great variety of situations not predicted by the uristructured models are predicted by these models. A complete verification of the predictions does not appear possible with current information, although certain gross results can be easily verified. The possibility of a model which predicts the various phases of batch growth as well as the transient behavior of continuous cultures rests upon t.he assignment of structure to organisms. T h a t is, organisms are not chemically or spatially homogeneous and an adequate model must reflect this. It is the interaction of the various cell constituentJs with each other and with their environment that produces t.he phases of batch growth and the rhamcteristic changes in cell composition observed therein. Thus, there is no single equation of the growth curve, as implied in some of t,he loiological literature, nor is the inclusion of further equations to describe the state of the environment sufficient. Any attempt to describe the growth phases

168

L).

RARIKRISHNA, A. G. FREDRICKSON, H. 31. TSUCHIYA

with a single equation results in an non-autonomous model which introduces an arbitrary character into the model. Though the models described in this paper do not consider spatial structure, the inclusion of “simple” chemical structure appears sufficient to explain most features of the batch-growth curve. The phenomenon of endogenous metabolism has not been considered in the analyses presented herein. The stationary and the decline phases in a batch culture can also be explained on entirely different premises from those used in this paper. Thus an endogenous substrate formed during growth may maintain viability of the cells during the stationary phase; when the endogenous substrate is exhausted, the phase of decline sets in and the viable cells die off more rapidly. These explanations have also been formulated into mathematical models by us.17 The results obtained are different from those presented here so that experiments can be performed to investigate the relative plausibility of explanations based on the formation of inhibitors, those based on endogenous metabolism and those based on a combination of the two. This work was supported in part by NASA Grants NsG 79-60 and NASA-24005-056.

Nomenclature a,

1. Stoichiometric constant expressing the amount of substrale used up in forming unit mass of viable mass (unstructured models). 2. Stoichiometric constant expressing the amount of substrate used up in forming unit mass of G (structured models).

a,’

Stoichiometric constant expressing the amount of substrate used up in iorming unit mass of D. 1. Stoichiometric constant expressing the amount of inhibitor formed from producing unit mass of viable mass (unstructured models). 2. Stoichiometric constant expressing the amount of inhibitor formed from producing unit mass of G (structured models). Stoichiometric constant expressing the amount of inhibitor formed from producing unit mass of D. 1. Stoichiometric constant expressing the amount of inhibitor formed from the deactivation of unit mass of viable mass (unstructured models). 2 . Stoichiometric constant expressing the amount of inhibitor formed from the deactivation of unit mass of G (structured models). Stoichiometric constant expressing the amount of inhibitor formed from the deactivation of unit mass of D. Concentration of entity i, g./liter.

aT

aT’ aT1

UTl‘

Ct

BIOTECHNOLOGY AND BIOENGINEERING, VOL. IX, ISSUE 2

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ei

169

Steady-state concentration of entity i, g./liter Concentration of substrate in feed to continuous culture, g./liter. D-mass

C,, D G K

G-mass

K'

KG,K s KG', K,' R S

T V a

x P P'

0

1. Rate constant for the deactivation of viable mass by the inhibitor (unstructured models). 2. Rate constant for the deactivation of G by the inhibitor (structured models), liter/g./hr. Rate constant for the deactivation of D by the inhibitor (structured models), liter/g./hr. Michaelis constants for the formation of G, g./liter. (In the nnstructured models K s is the Michaelis constant in the htonod model growth expression.) Michaelis constants for the formation of D, g./liter. Fraction of G-mass in the culture. Substrate. Inhibitor. Viable mass or active biomass. Stability matrix. Eigenvalne. Rate constant for the formation of G, hr.-l. (Also specific growth rate of viable mass in the unstructured models.) Rate constant for the formation of D, hr.-l. Holding time, hr.

References

c.

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Received May 2, 1966

BIOTECHNOLOGY A N D RIOENGINEERING, VOL. IX, ISSUE 2