Metabolic regulation in bacterial continuous cultures: Idrops/ramkiPapers/100_1991_Baloo... · Metabolic Regulation in Bacterial Continuous Cultures: I S. Baloo and D. Ramkrishna*

Metabolic Regulation in Bacterial Continuous Cultures: I

S. Baloo and D. Ramkrishna* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907

Received November Z 1990/Accepted June 5, 7991

Dilution rate steps in continuous culture experiments with Klebsiella pneumoniae growing on single substrate feeds have brought out interesting features of metabolic regulation not observed in batch cultures. In a step-up experiment, the adjustment of the culture to a new steady state is preceded by an undershoot in cell density. Results of a step-down experiment indicate a corresponding overshoot phenomenon. These observations of the transient behavior of the culture growing on glucose and xylose as well as the steady-state results are interpreted with cybernetic models. The development of the model explicitly accounts for the lumped internal resource, which is optimally allocated toward the synthesis of key enzymes catalyzing different cellular processes. The model also includes a description of the increased maintenance demand observed at low growth rates. It reduces to previous cybernetic models in situations where the cell does not experience a sudden change in its environment and, hence, retains their predictive capability. Key words: Klebsiella pneumoniae - metabolic regulation - continuous cultures, cybernetic model

INTRODUCTION

A description of microbial growth in a changing environment is important from an engineering viewpoint especially because of the use of continuous culture in basic studies of microbial physiology and regulation and as an industrial process. Simple models, like the Monod model that have been developed within the framework of chemical kinetics, represent microbial growth as a single, lumped chemical reaction. They provide a good description of the variation of cell density and substrate concentration with the dilution rate (or the growth rate of the cells) in a chemostat under steady-state conditions. However, if we carry the analogy of microbial growth with classical chemical reaction engineering a step further and apply the Monod model to describe microbial growth in a changing environment, we see that the model fails to describe the transient period (Fig. 1). The main cause of this anomaly is because metabolic regulation, which controls all microbial reactions and functions, has not been incorporated in the description of microbial growth.

Ramkrishna and co-workers’3~23~25~28-30 h ave developed a class of structured growth models, called cybernetic models, that account for the effects of cellular regulatory mechanisms on microbial growth processes.

* To whom all correspondence should be addressed

Biotechnology and Bioengineering, Vol. 38, Pp. 1337-1352 (1991) 0 1991 John Wiley & Sons, Inc.

The cell is viewed as an optimal system with its regulatory processes implementing the optimal strategies. The effects of the regulatory mechanisms of cellular processes are represented by cybernetic variables. A complete representation of a cellular process, therefore, includes the kinetic description of the process as well as the cybernetic variables.

Kompala et al.13,23 introduced lumped key enzymes that catalyze key cellular processes such as growth. Cellular resources required for the synthesis of these enzymes were allocated to the different enzymes systems based on optimality criteria. Metabolic regulation that controls enzyme synthesis through the mechanisms of induction/repression and enzyme activity by activation/repression mechanisms were represented by cybernetic variables. This description provided a very good representation of microbial growth in the presence of multiple substrates in different batch situations in- cluding the ‘diauxie’ phenomenon.

Turner et al.29.30 then applied the cybernetic framework to describe microbial growth in low-growth-rate situations. They addressed the effect of non-growth- associated functions termed maintenance, observed in such situations from a cybernetic perspective. Although the model described low-growth-rate behavior on single and multiple substrates extremely well, it failed to describe the transients observed in continuous cultures. A culture growing at a very slow rate, when subjected to sudden increase in nutrient concentration in the environment, is unable to metabolize all the substrate and immediately grow at a faster rate. The cell first synthe- sizes all the components required to support a higher- growth rate. Further, competition between growth and maintenance functions for the limited resources of the cell restricts cellular response.

The objective of this article is to develop a cybernetic model capable of describing the response of a bacterial culture growing on a single substrate in the transient period following step changes (both increases and decreases) in the dilution rate in a chemostat. The model is compared with experimental data of single substrate growth of Klebsiella pneumoniae on glucose and xylose, respectively, in a chemostat as well as in batch and fed- batch situations. This model forms the basis of a separate article that describes multiple substrate steady-state and transient behavior in a chemostat.

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CYBERNETIC MODEL DEVELOPMENT

Reaction Scheme

Following the formulation of Kompala et aI.l3 the biomass, B, assimilates the limiting nutrient, S, from the environment and converts it into more biomass. This conversion of substrate is catalyzed by a key growth enzyme, E G , and is represented by:

B + S (1 + YG)B + ... The consumption of substrate for maintenance pur- poses, following the development of Turner et aLZ9 is represented by:

B + S E ” - B +

where E M is the key enzyme for the maintenance process. This process is termed basal maintenance to distinguish it from the low-maintenance process introduced later in this development.

Previous cybernetic model formulations assumed that the cells possessed a sufficient quantity of the internal resource required for enzyme synthesis. Hence, although resources were allocated towards the synthesis of different enzyme systems, resource itself was not a limiting factor. This was a good assumption in the cases considered: batch and fed-batch situations under mainly balanced or slowly changing growth conditions. The effect of internal resource limitation was observed in perturbed fed-batch experiment^^^ in which a slow-growing culture was pulsed with a saturating quantity of the growth-limiting substrate. The results that show that the experimental increase in cell density lagged the results of the model simulations point to the existence of a

limiting component other than enzymes and substrate. The extent of the disagreement between the model and experiment depended on the growth rate at the time of the perturbation-the lower the growth rate, the larger the disagreement.

The step-up experimental results from continuous experiments performed in this work also point to the controlling effect of a cornponenth other than the substrate concentration on the dynamic response. Hence, from a modeling standpoint, this implies that the assumption of the existence of sufficient internal resource in the cell is not always valid, and an explicit balance on the resource needs to be considered.

The resource in the cybernetic framework represents a lumped species involved in the synthesis of the key enzymes. The amount of the resource controls the ability of the cell to rapidly alter its growth rate. Other modeling approaches8222 have adopted a similar strategy and considered the ribosomes as the growth-limiting component based on the variation of ribosome level at different growth rates and the importance of ribosomes in protein synthesis. Although the resource is not spe- cifically associated with any component in the cell, it could be viewed as the entire protein synthesizing system (PSS) or a component of the PSS.

Daigger and Grady ’ found that cellular response can be viewed as RNA-limited under certain conditions. In these situations, the resource can be thought of as the fraction of active ribosomes in the cell. The resource is assumed to be synthesized in the presence of substrate, and its synthesis is catalyzed by a key enzyme E R . The resource is not substrate-specific, and its level in the cell along with the other internal components characterizes the physiological status of the cell. Resource synthesis is therefore represented as:

B ” E , - B - + R Transient results in a chemostat show an increased

substrate consumption for maintenance above that accounted for in previous cybernetic models.29 Further- more, there exists literature pointing to increased maintenance functions at very low growth rates such as higher rates of protein degradation” and the accumula- tion of regulatory nucleotides, ppGpp and pppGpp, which cause stringent r e g ~ l a t i o n . ~ , ~ , ~ ~ , ~ ~ Based on the foregoing observations, the existence of an additional low maintenance system that operates maximally at zero growth rates is postulated. One would expect the low maintenance system that complements the basal maintenance systemz9 to become functional quickly at low growth rates when the maintenance demand is high. This low-maintenance process is again catalyzed by a key enzyme E M L and can be represented by:

B + s a B + ... Key enzymes that catalyze the different cellular processes are assumed to be induced in the presence of the

1338 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 38, NO. 11, DECEMBER 20, 1991

substrate and to require the critical resource for their synthesis. The key enzymes syntheses process can be represented by:

B -% B - + E G + EM + EML + ER

The reactions listed above make up the set of processes considered in the cybernetic model. We have introduced an explicit step for the synthesis of the resource, rather than assume that the resource is always present in sufficient amounts within the cell, and a key resource enzyme associated with the resource synthesis step. The cybernetic models developed previously have now been augmented by the addition of a low- maintenance process that is important at low growth rates and in keeping with the cybernetic framework, this low-maintenance process is catalyzed by a low- maintenance enzyme.

Rate Expressions

Having developed the reaction scheme for microbial growth, we now propose kinetic expressions for these reactions. In keeping with previous model^,'^.^^ the rate expressions for all cellular processes (growth and maintenance) are assumed to follow saturation kinetics. The level of the key enzyme that catalyzes the particular process affects the process rate, and the rate expression is modified to account for this fact. Hence, if we consider the growth process, the rate of growth is represented by:

The above expression reduces to the standard Monod formulation when the growth enzyme level is at the maximum.

Enzyme systems are substrate-specific and mainly synthesized in the presence of the particular substrate. Following the development of Turner and Ramkrishna2* the rate of enzyme synthesis is the sum of contribu- tions from a small constitutive rate (a:,) unaffected by regulation and a large inducible rate (aEG) subject to inductionhepression regulatory mechanisms. Hence, if we consider the growth enzyme, we can represent the actual rate of synthesis by:

Metabolic regulation only affects the inducible portion of enzyme synthesis. Being internal components, enzymes are assumed to degrade via a first order process with a rate constant denoted by P E .

Because the critical resource is required for enzyme synthesis, its level in the cell should affect the rate of synthesis and must be included in the rate expression. Saturation type kinetics is assumed to represent this

interaction. Hence, if we again consider the growth enzyme, the rate of enzyme synthesis would be:

This expression is consistent with previous enzyme synthesis formulations.'3929 In situations studied previously, i.e., batch and fed-batch, resource was present in sufficient amounts and so the term (R/KR + R ) is approximately unity as long as the saturation constant is small. In the transient situations, where resource levels are limiting, this term is less than unity and, therefore, lowers the rate of enzyme synthesis. The above form for enzyme synthesis with the inducible/constitutive parts and the resource level contribution is used to describe all enzyme synthesis rates.

Resource, as pointed out earlier, is induce in the presence of substrate, and its synthesis is catalyzed by the resource enzyme ER. The rate of resource synthesis is therefore:

(3)

The resource like other internal components degrades via a first-order process with a rate constant P R .

At this point we have identified the rate form for the different cellular processes. Applying these forms to the reaction scheme, the rate of growth is given by eq. (1) while the rates of basal and low maintenance processes are:

The rate of synthesis of the growth enzyme is given by eq. (2). The rates of synthesis of the basal maintenance enzyme and the resource enzyme have been assumed to be equal to the rate of synthesis of the growth enzyme. This assumption with regard to the model structure has been made purely to reduce the complexity and decrease the number of constants involved in the model. Al- though EG, E M , and ER are distinct enzymes and can have different activities, they are present intracellularly at equal levels.

The assumption that the rates of growth and basal maintenance enzyme synthesis are equal was made earlier.29 This essentially implies that there is a certain amount of maintenance associated with growth, and this is referred to here as basal maintenance. The additional assumption of the rate of resource enzyme synthesis being equal to that of the growth enzyme is made purely as a first approximation. It is, however, a logical assumption as high-growth enzyme levels require a high-resource level, which in turn requires a large re-

BALOO AND RAMKRISHNA: REGULATION IN CONTINUOUS CULTURES: I 1339

source synthesis rate. From the assumption of equal synthesis rates for the growth, basal maintenance and resource enzymes, we write:

r E G = rE.w = r E R (6)

The maximum enzyme level eEax is calculated from a mass balance on the enzyme and assuming that the cells possess a maximum amount of growth enzyme when growing at their maximum rate.13 The maximum enzyme level is therefore:

(7)

At low growth rates, the resource level within the cell is very low. However, it is at these growth rates that the low-maintenance enzyme process is important. It is therefore assumed that the resource level does not affect the rate of low-maintenance enzyme synthesis. Another perspective is that instead of choosing a very low value of KR for low-maintenance enzyme synthesis, the constant has been arbitrarily set to zero. The rate of low enzyme synthesis is therefore:

The enzyme level of the low-maintenance enzyme should be high at low growth rates when the low- maintenance process is important. It is, therefore, assumed that the maximum enzyme level is attained at zero growth rate. This is borne out by Matin,I6 who com- piled an exhaustive list of enzyme activity as a function of the growth rate and found that several catabolic enzymes showed maximum activity at the minimum growth rate. Hence, the maximum maintenance enzyme level can be calculated from a simple balance by setting rc = 0 and assuming that there is still sufficient substrate present in the environment. Thus,

(9)

where PEML is the degradation rate constant of the enzyme.

Basic Postulates and Formulation of Cybernetic Variables

The cybernetic model is complete only when the regulation represented by the cybernetic variables is included in the kinetic description. The cybernetic variables representing control of enzyme activity (denoted by v) modify the rates of the different processes (growth, maintenance, and resource synthesis), while the cybernetic variables representing control of enzyme synthesis (denoted by u) modify the distinct enzyme synthesis rates (growth enzyme and low-maintenance enzyme). Before the specific forms of the cybernetic variables are developed, the assumptions made about metabolism are

stated in the form of postulates. These postulates have been formulated based on our experience to date with regulation. While we do not imply them to be “univer- sal” postulates, for biological diversity is not so permis- sive, they are nevertheless useful as clear statements of our present viewpoint. They must be regarded as a nec- essarily evolving set with a hopefully increasing range of applicability.

Basic Postulates

1. Growth is of paramount importance to the micro- organism.

2. Maintenance metabolism manifests itself at growth rates less than the maximum growth rate and becomes more active at lower growth rates.

3. The processes of growth and maintenance compete for the limited cellular resources with the competition becoming increasingly more important at very low growth rates. These processes are regulated such that the substrate consumption rate is maximized.

4. The activity of the maintenance metabolism is such that the total rate of substrate consumption is maximized.

5. The cellular demand for resource increases with the growth rate. The maximum demand for resource is realized at the maximum growth rate.

Postulates 1, 2, and 4 were enunciated p r e v i ~ u s l y * ~ ~ ~ ~ and are restated here as they are also applicable to this modeling effort.

Postulate 1 essentially states that the organism grows at the fastest rate possible under the given environmental and cellular conditions. The cell will therefore maintain the activity of the growth enzyme at its maximum. Even though maintenance competes with the growth process for cellular resources (postulate 3), the cell does not reduce the activity of the growth enzyme already synthesized. Thus, the growth process which is of “paramount” importance is always activated to the fullest extent.

Postulate 2 states that maintenance requirements are automatically met when cells are growing at their maximum growth rate. It is only at submaximal growth rates that maintenance metabolism is activated and the activity of this process increases as the growth rate is lowered.

Postulate 3 states that severe substrate limitation, i.e., very low growth rates, brings out the competition between growth and maintenance for the limited cellular resources. Because basal maintenance is associated with growth (EG = E M ) , only the low maintenance process, which is catalyzed by a distinct key enzyme (EM,) from that of the growth enzyme, competes with the growth process. Under these severe conditions, the cell has large maintenance requirements and therefore pro- motes the low maintenance process over growth by allocating the majority of its resource toward low- maintenance enzyme synthesis. However, under normal


conditions when the maintenance demand is not high, the cell diverts most of its resource toward synthesis of the growth enzyme and allocates only a small fraction toward the synthesis of the low-maintenance enzyme. This results in lower low-maintenance enzyme levels in the cell and correspondingly low-maintenance rates.

From postulate 4 it would seem that the best way to maximize substrate uptake would be to activate both growth and maintenance processes to the maximum levels, unity, because they are the only processes that con- sume the substrate. The activity of the growth process has already been set to unity by postulate 1. We deem it, however, inefficient for the cell to promote the activity of maintenance metabolism when maintenance requirements are not high. Because maintenance requirements increase at the lower growth rates (postulate 2), we assume that maintenance activity is proportional to the deficiency in growth rate, i.e., p Y - rG. Hence, at the maximum growth rate, maintenance requirements are zero, while zero growth rates fully activate the maintenance functions (consistent with postulate 2).

Postulate 5 essentially indentifies the requirements for resource. Resource is required by the cell for enzyme synthesis, and these requirements increase at high growth rates when growth enzyme levels are high. There is a large demand for the resource only when the cell is growing at a fast rate. Therefore, it would be wasteful for the cell to synthesize large quantities of resource at low growth rates. Because the demand for the resource increases with growth rate, we assume that the activity of the resource enzyme is proportional to the growth rate. This assumption is consistent with postulate 5 because the resource synthesis rate will be high at high growth rates when the demand is high and vice versa.

The formulations of the cybernetic variables can be derived from the above discussion of the postulates in a straightforward manner. Postulate 1 requires that the growth process be always fully active, i.e., V G = 1. Therefore, the actual growth rate r G v G , is simply r G .

Because postulate 2 applies to maintenance metabolism, the activities of both the low-maintenance and basal-maintenance functions are controlled by a single cybernetic variable V M . Therefore, the actual rate of maintenance is

( r M + rML)vM 0 I VM 5 1

Further, the postulate requires that vM = 0 when

The expressions for the cybernetic variables uG and uM controlling the synthesis of growth and low- maintenance enzyme systems can be derived from postulate 3. The total specific uptake rate qs is given by:

r G = p y .

q s = rG + (rML + r M ) v M

Because the growth and maintenance enzymes systems compete for the limited cellular resources, the optimal allocation of resources is achieved by applying the

Matching Law l1 as shown by Kompala et al.13 If we view the return from each enzyme system as the rate of substrate consumption of the process catalyzed by that enzyme system, the return from the growth process is:

r G - Y G

while the return from the maintenance processes is:

rTMvM(rTM = r M + r M L )

Hence, the fractional allocation to the growth process ( u ~ ) such that the substrate consumption rate is maximized is

(10) r C / y G

r G / Y G + rTMvM U G =

and the fractional allocation to the maintenance process (uM) subject to the same constraint is:

These variables modify the rates of synthesis of the growth enzyme and the low-maintenance enzyme.

From postulate 4 and the assumption that V M is proportional to the deficiency in growth rate, we can derive the expression for the cybernetic variable vM. Because the derivation has been presented previ0usly,2~ write only the final expression for vM.

The expression for the cybernetic variable vR can be derived from postulate 5. The assumption that the activity of the resource enzyme is proportional to the growth rate results in

V R =

Because 0 I vR 5 1, the bounds on A are

0 I A I 1/pF

Because the demand for resource is greatest at the maximum growth rate, the resource synthesis rate should be the highest at p Y . Therefore, at rG = p Y , vR = 1 implies

A = l/pY

or

At rG = 0, vR = 0, and so no resource is synthesized as resource is not required by the cell. At r G = 1, V R = 1, and the resource is at its maximum level because the demand is the greatest.

The cybernetic variables have been derived from optimal criteria coupled with our understanding of


metabolic regulation (stated here in the form of basic postulates). This approach is borne out by Tempest and Neijssel," who stated that the regulation of metabolism by the control mechanisms achieve an optimization of substrate utilization and its conversion into biomass. The cybernetic variables along with the concentration variables of the kinetic expressions make up the complete description of microbial growth. The complete actual rate expressions for the different reactions therefore are:

Growth: ~ G V G (VG = 1) (14) rG = Pc"""e- S

emax K , + s

Maintenance: rTM VM (16)

Growth, basal maintenance and resource enzyme:

rEG = rEM = rER = rE (18)

(19) S R

r E = a2 + a€-- KE + s K R + RUG Low-maintenance enzyme:

Resource:

The cybernetic variables uG, u M , vM, and V R are calculated from eq. (10-13), respectively. The internal components, i.e., resource, growth enzyme, and low- maintenance enzyme, degrade via first order processes with rate constants p ~ , BE, and / 3 ~ ~ ~ , respectively.

khtification of Parameters

The cybernetic model for growth on a single growth- limiting substrate has 18 parameters. All these parameters are associated with the structure (kinetic aspects) of the model. The cybernetic variables that are calculated from optimal strategies do not introduce any new parameters. The parameter values are obtained in a sequential manner by performing experiments where only a few of the model elements are important and, therefore, only they govern the observed behavior.

Qpically, a microbial culture in a batch reactor with a saturating initial substrate concentration grows slowly during the initial lag period when all the internal components of the cell are adjusting to the new environment. The culture then grows at the maximum rate until the substrate is exhausted. From the model standpoint, this means that there is an initial buildup of the

growth enzyme, followed by rapid growth. Because the cells are growing at their maximum rate, resource levels are high and, therefore, do not limit the growth process. Further, maintenance functions are also not a factor because the cells are growing maximally. Hence, the important variables in describing batch growth are those associated with growth ( p F x , K,, and YG) and the growth enzyme synthesis (a:, ( Y E , PE , and K E ) .

The results of steady-state continuous culture experiments are used to obtain the parameter values for the basal maintenance system. At intermediate growth rates maintenance starts to become important, and so the basal maintenance system parameters ( p r and KM) govern the exact amount of substrate consumed for maintenance and, hence, the observed cell density. In the steady-state experiments, resource is not limiting, and the low-maintenance system is not yet functional; therefore, these processes do not affect cellular behavior. We would also expect that the saturation constant for the maintenance process (KM) to be a very small value relative to the saturation constant for growth (K,).

The dilution rate step-up experiments point to the inability of the cell to respond quickly to changes in the environmental conditions, thus indicating that the resource is rate-limiting. Therefore, the values of the resource parameters ( a R , KRS, p ~ , and K R ) dictate the observed culture behavior. Because the resource levels vary significantly with the growth rate, the degradation rate of the resource should be fairly high.

The dilution rate step-down experiments bring out the effects of the low-maintenance system. The growth rate drops to very low levels during the period immediately following the shift, and it is under these conditions that the low-maintenance process is promoted by the cell. Therefore, the parameters associated with the low- maintenance process (PE, a;.,,, a E M L , KEML, and BE^^) can be determined from the step-down experiments. We would expect that the saturation constant for the low-maintenance enzyme to be very low relative to the saturation constant for growth (K, ) , because the process is activated only under severe substrate limita- tions. Further, because the maintenance demand can increase very quickly, the synthesis and degradation rate of the low-maintenance enzyme, aEML and / 3 ~ ~ ~ , should be high.

The above discussion brings out the sequential manner in which the parameters can be estimated from experimental data. The parameters identify both the substrate and the organism and are listed in Table I.

MATERIALS AND METHODS

Organism

Klebsiella pneumoniae (ATCC 13882) obtained from the American Type Culture Collection (Rockville, MD) was used in all experiments. The bacteria were stored


Table I. Model parameter values for growth on glucose and xylose.

Parameter Units Glucose Xylose

1.223 0.17 0.39 0.5 1 0.003 0.003 0.003 0.003 1 x 1 x 10-6 8 x lo-' 0.001 8 x 0.05 0.125 0.05 6.0 2.5

0.56 0.20 0.40 0.39 0.013 0.013 0.013 0.003 1 x 1 x 5 x 1 0 - ~ 0.001 2 x 1 0 - ~ 0.05 0.125 0.05 2.5 2.5

in small vials in a freezer at -60°C. The organisms were suspended in a glycerol-rich (20 g/L) double- strength minimal salts medium so as to prevent shock while freezing.

Medium and lnoculum Preparation

The preparation of the growth medium and the inoculum for each experiment has been described previo~sly.'~

Fermentor Description

A 2-L New Brunswick fermentor with a magnetic coupled drive was used in all experiments. The fermentor was modified for use in continuous culture experiments. Before each experiment, the fermentor internals and the glass vessel and all glass connectors in contact with the fermentor broth were immersed in a silicone surface-treating solution (1% solution of Prosil-28, PCR Inc., Gainesville, FL) for 30 s, rinsed, and then dried in the 90°C oven for 1 h to reduce bacterial attachment to the walls. The fermentor filled with 1 L of the carbon- free salts medium was autoclaved with dissolved oxygen and pH probes in place. The entire assembly was autoclaved at 121°C for 50 min.

The fermentor was then placed in a 37°C constant- temperature water bath. Required amounts of trace metals and the sugar solution were added to the fermentor to make up the correct concentrations of all the components. The medium (pH 7.1) used in the fer- mentations provided sufficient buffering capacity and the pH drop in any experiment was never greater than 0.3 units. Filtered air was bubbled through the fermentor at a rate of 1.0 L/min, and the agitation rate was

All lines and components in the continuous culture system were autoclaved for 50 min at 121°C and connected in a sterile fashion. Twenty liters of medium mixed in a reserve tank was pumped over to the feed tank through two filters. Initially, 1 L of the medium was wasted to flush out the filters before unclamping the lines leading to the feed tank. Between each refilling of the feed tank, the filters were cleaned with dilute sulfu- ric acid and distilled water. The medium was pumped to the fermentor by a Harvard peristaltic pump. A gradu- ated pipette connected to the feed line was used to measure the flow rate of medium into the fermentor. Before entering the fermentor, the feed stream was passed through two break tubes" to prevent any back growth, mixed with the filtered air, and discharged under the impeller. The discharge line was connected to an overflow device that maintained a constant volume in the fermentor. Samples were withdrawn as desired through a sample port with a syringe.

A high flow rate recycle line circulated the medium through a bubble trap and a spectrophotometer. The spectrophotometer (Bausch and Lomb) was adapted with a modified flow ce11I4 to monitor continuously the absorbance of the fermentation broth. The recycle loop was flushed with sterilized 2 N potassium hydroxide solution and sterilized distilled water every day to remove any bacterial attachment at the connector junctions and to re-zero the spectrophotometer. The spectrophotometer was connected through an amplifier board to an Apple I1 plus computer for continuous data acquisition. Absorbance readings were averaged over 2 min and the average was stored in a file and also printed to the screen. The continuous absorbance readings were correlated with off-line absorbance readings (from the Perkin-Elmer spectrophotometer) and then converted to cell density values. The correlation accounted for the baseline drift of the on-line spectrophotometer.

Growth Measurements

Cell dry weight was estimated from absorbance readings of the culture broth with a spectrophotometer (Perkin- Elmer) set at a wavelength of 540 nm. An absorbance unit was found to be equal to 0.35 g dry weight of biomass/L. Dense cultures (absorbance readings above 0.3 units) were diluted to ensure that the absorbance readings were within the correlation range. Readings from the on-line spectrophotometer were correlated to the absorbance readings from the Perkin-Elmer spectrophotometer. Periodic direct dry weight measurements were also made to check the correlation.

Sugar Measurements

maintained at 600 rpm. These two parameters ensured that growth was always aerobic, as verified by dissolved oxygen measurements.

The supernatant obtained as a result of centrifuging the fermentation broth to separate the cells was used in the sugar assays. Glucose was measured by using the glu-


cose oxidase enzymatic assay (Kit #510, Sigma Chemi- cals, St. Louis, MO). The technique was modified by changing the proportion of the assay components so as to measure concentrations as low as 10 mg/L. Xylose was measured by using the orcinol assay.' The assay results were very sensitive to the heating time (90°C for 20 min), and, therefore, a set of standards was pro- cessed with each batch of 20 samples. The detection limit of xylose assay was 3 mg/L.

0.0

RESULTS AND DISCUSSION

I I I I

Batch Culture

The results of batch growth of K. pneurnoniae on 2.5 g/L of glucose are plotted in Figure 2(a) along with simulation results of the cybernetic model presented earlier. The maximum specific growth rate (pEaX) was calculated from the experiment to be 1.223 h-', while the yield coefficient of 0.51 gdw/gm for growth on glucose was obtained from a dry-weight determination at the end of the experiment. The batch simulation results were relatively insensitive to changes in the value of the saturation constant for glucose (K) . The value of 0.003 g/L used in all simulations was obtained from the steady-state continuous culture experiments. The values of 0.001 h-' for the inducible rate of growth enzyme synthesis (aE) and 0.05 h-' for growth enzyme decay ( B E ) from previous model^'^,*^ were retained in the present simulations. The growth enzyme saturation constant (&) and the resource synthesis saturation constant (&) were chosen to be the same as the saturation constant for growth (K5). The constitutive rates of growth enzyme and low maintenance enzyme synthesis (a: and a&,L) were fixed at 8.0 x and 8 x lo-' h-', respectively. These values correspond to the constitutive machinery contributing 1% to the total enzyme synthesized: a: = 0.01 emax and a,&L = 0.01 e i y .

The rates of synthesis and decay of resource and low- maintenance enzyme are high because we expect the processes to be activated rapidly. Further, because maintenance processes are important at low-growth rates, the saturation constant for maintenance ( K M ) and the saturation constant for the synthesis of the low- maintenance enzyme (&,) were set at 1.0 x 1 O P g/L so as not to be inhibited by relatively low substrate levels in continuous culture. The initial conditions for the cell density and substrate concentration were given by experimental conditions. The initial growth enzyme level was chosen to be 85% of the maximum enzyme level because the inoculum was precultured on glucose.13 The simulation results were insensitive to the initial low-maintenance enzyme and resource levels. This is to be expected because the culture essentially grows at the maximum rate after the initial lag period. The initial growth enzyme level characterizes the length of the lag period. The simulation results also show mini-

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Figure 2. Batch growth: Experimental and cybernetic model simulation results. (a) Growth on 2.5 g/L glucose, (b) growth on 0.25 g/L xylose.

ma1 substrate consumption for maintenance as expected in batch cultures.

Similar batch growth experiments on xylose were also performed. The experimental data and model simulation results for growth on 1 g/L of xylose are plotted in Figure 2(b). The maximum growth rate and saturation constant for xylose were found to be 0.56 h-' and 0.013 g/L, respectively. As with the case of growth on glucose, we have set the saturation constant for the synthesis of the xylose growth enzyme and the saturation constant for synthesis of resource from xylose equal to the saturation constant for growth, i.e., K E = KRS =

I

i


Ks. The rates of inducible xylose enzyme synthesis and decay have been chosen to be the same as the glucose enzymes29 (aE = 0.001 h-' and P E = 0.05 h-'). The constitutive rates of syntheses of the xylose growth enzyme and low-maintenance enzyme ((YE* and a&) were fixed at 5 x h-' and correspond to the constitutive machinery contributing 3% and 1% to total enzyme synthesis: a: = 0.03 emax and a iL = 0.01 e E .

The values of all the parameters used in the xylose growth simulations are also listed in Table I. The batch simulations were sensitive only to the initial level of the xylose growth enzyme because the level determined the duration of the lag period, which was taken to be 85% of the maximum enzyme level because the inoculum was precultured on xylose.

The results of batch growth show that the cybernetic model developed describes microbial growth on glucose and xylose in a batch reactor. The present model essentially reduces to the cybernetic model developed by Kompala et al.13 because resource and maintenance are not important in a batch situation.

and 2 x

Perturbed Fed-Batch Culture

The cybernetic model can be used to describe the perturbed fed-batch experiments of Turner et al.29 The behavior of Klebsiella oxytoca during one such experiment performed by Turner has been plotted in Figure 3 along with simulation results of the cybernetic model developed by Turner and the present cybernetic model. The culture that is growing very slowly under a glucose limitation suddenly finds the growth limitation relieved by the addition of a pulse of glucose. The organisms re-

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spond to the nutrient-rich environment by regulating their metabolism to exploit the additional glucose in environment. Because they have been growing on glucose prior to the pulse, they do not need to synthesize a com- pletely different set of enzymes but just augment the set already available. However, the time taken to accom- plish this depends on the availability of the resources required for the increased enzyme requirement. Fur- ther, because the growth limitation has been lifted, the cell now distributes resources to preferentially synthesize the enzymes required for growth and lowers the maintenance activity.

The present model, which accounts for the resource level in the cell and the allocation of resources between growth and maintenance enzymes through the cybernetic variables uG and uM, shows an improvement over the previous cybernetic formulation in describing the response of the culture to the glucose pulse. This improvement is more significant in situations where the growth rate at the time of addition of the pulse is lower, thus implying a lower resource level. Both models describe the response of the culture to the second pulse satisfactorily. This is because resource that was synthesized in response to the first pulse is still present in the cell when the second pulse is introduced.

Continuous Culture

Microbial growth in a continuous culture system brings out important features of physiology and regulation that are expressed only in a transient manner in batch systems. Continuous culture experiments therefore provide an excellent test of the cybernetic model developed. The balance equations for the different cellular components of the cybernetic model in a chemostat are:

dc dt - = (rc - D)c

where the rates of growth, maintenance, enzyme, and resource syntheses are given by eqs. (14-21), respectively, the cybernetic variables are given by eqs. (10-13) and D, the dilution rate or the inverse of the mean resi- dence time, is given by F / K The steady state balance equations are obtained by setting the time derivatives in eqs. (22-26) equal to zero.

The results of the steady state growth of K . pneumoniae in a continuous culture plotted on the same figure.


The same experimental results are also plotted against simulation results of the Pirt model” and the cybernetic model developed by Turner et al.” [Fig. 4(b)]. All models provide a good description of the steady-state behavior and describe the decrease in cell density at the lower growth rates because of substrate consumption for maintenance functions.

Similar steady-state results with the culture growing on xylose with a feed concentration of 0.49 g/L are plotted in Figure 4(c). The simulation results using the xylose parameters from Table I are also plotted on the same figure. The model again predicts the decreased cell density at low dilution rates. The maximum rates of basal maintenance and the low maintenance processes

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(P;~’ and p l y ) during growth on either glucose and xylose are comparable. This is to be expected because the maintenance processes of the cell are the same regard- less of the substrate on which it is growing.

The true test of the present cybernetic model is its ability to describe transient behavior in a continuous culture because previous cybernetic modelsz9 and the Pirt model,*’ which described microbial growth observed thus far in batch, fed-batch, and steady-state continuous culture systems satisfactorily, fail to describe the transient behavior in a chemostat (Fig. 1). The results of one such transient are plotted in Figure 5(a). In this experiment the dilution rate of a steady-state continuous culture growing on glucose with a feed concen-

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Figure 4. Steady-state growth of Klebsiellu pneumoniae in a continuous culture: (a) Present cybernetic model simulations of growth on glucose (sf = 0.48 g/L); (b) simulations of the Pirt model’’ and the cybernetic model of Turner et al.4 of growth on glucose (sf = 0.48 g/L); (c) present cybernetic model simulations of growth on xylose (s, = 0.49 g/L).


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Figure 5. Transient period of a continuous culture growing on glucose following a step-up in dilution rate from 0.22 to 0.91 h-'. (a) Experimental results and cybernetic model simulations; (b) profiles of cell density, resource, glucose, and growth rate from simulations; (c) profiles of cell density, growth enzyme level, cybernetic variables U G and V R from simulations.

tration of 0.49 g/L was suddenly increased from a value of 0.22 h-' to 0.91 h-'. The model simulation results that are plotted on the same figure show good agreement with the experimental behavior. The model provides an accurate description of the lag period during which the cell is adjusting to its new environment and a qualitative description of the overshoot in cell density just before the new steady state is reached. The experimental results are similar to those observed by other researcher^.^^^^'^,^^

The working of the model during this shift-up experiment is understood by following the profiles of the individual components and the cybernetic variables [Figs. 5(b) and 5(c)]. Following the increase in dilution rate there is an immediate rise in the substrate concentration in the reactor. The growth rate also increases sharply to 0.5 h-' as the substrate limitation is lifted.

However, the resource level in the cell increases more slowly because it first has to be synthesized by the cell. There is a washout of the culture during this period because the dilution rate is greater than the growth rate. When the resource level builds up to a point where it can support a high growth, the cells begin to accumu- late in the fermentor. The increased biomass and the high growth rate cause a rapid depletion of the substrate, and this in turn lowers the growth rate.

A small overshoot in cell density is predicted by the model because of the interaction between the high resource level and the low substrate concentration. The resource level being higher than that required to maintain a growth rate of 0.91 h-' permits the cell to grow faster, and this causes an undershoot in the substrate concentration. The profiles of the growth enzyme level and the cybernetic variables uG and vR plotted in Fig- ure 5(c) provide a clear picture of the regulation during the step-up experiment. The value of u G , the cybernetic variable controlling the amount of resource allocated to growth enzyme synthesis, increases sharply from 0.75 to 0.95. The rate of growth enzyme synthesis, therefore, increases, and this results in an increased level of enzyme within the cell. Further, the value of vR, the cybernetic variable controlling the activity of the resource synthesis process, increases during the transient period, and results in a higher resource concentration at the higher growth rates.

The cybernetic model also provides a reasonable description of the transient behavior of a culture following a step-down in dilution rate from 0.91 to 0.24 h-' [Fig. 6(a)]. The model predicts all features observed during the transient: the initial overshoot in cell density, the subsequent sharp but sustained drop in cell density, and the final undershoot in biomass concentration before the new steady-state value is reached. The profiles of the model components and the cybernetic variables during the transient are plotted in Figures 6(b) and 6(c). In the step-down experiments, the effect of maintenance is more pronounced than in the step-up experiments,

and, therefore, we have plotted the profiles of .EM and ~ 1 , ~ instead of EG and U G in Figure 6(c). The cell density rises immediately after the step-down in dilution rate because the growth rate of the culture is higher than the dilution rate. The increased biomass level and the slower feed rate of substrate cause the glucose concentration to drop precipitously in the fermentor. This in turn causes the growth rate of the culture to drop to 0.09 h-I. At this low growth rate, maintenance functions dominate.

The cell allocates most of the available resource to the low-maintenance enzyme synthesis as indicated by an increase in the value of p*.M from approximately zero to 0.7 [Fig. 6(c)]. The level of low-maintenance enzyme rises rapidly (because the system has a high rate of synthesis) and the incoming substrate is mainly consumed for maintenance. The cells wash out of the culture. Be- cause they are essentially not growing, resource is no longer synthesized (given by the low v M values) resulting in a drop in resource level. The presence of fewer cells in the reactor allows a buildup of glucose, thus permit- ting a slow increase in the growth rate of the culture. The rise in growth rate causes the cell to reallocate ex- isting resources toward growth enzyme synthesis represented by the drop in the value of pM. Consequently, the low-maintenance enzyme level decreases rapidly (because of the high rate of decay). The final undershoot in cell density is caused by the interaction between the high residual glucose concentration and low resource level.

Several dilution rate step-up and step-down experiments with the culture growing on glucose were performed; in all cases good agreement with model predictions was observed. The magnitude of the change in dilution rate determines the exact transient behavior observed. The undershoot in cell density observed following a large decrease in dilution rate from 1.0 to 0.22 h-' [Fig. 7(a)] is not observed in an experiment where the dilution rate was decreased from 0.804 to 0.302 h-' [Fig. 7(b)]. Similarly, the overshoot in biomass concentration observed following a large step-up [Fig. 5(a)] is not observed in an experiment where the change in dilution rate is small [Fig. 7(c)]. The large changes in dilution rate cause a severe change in the environmental conditions; this translates into dramatic changes in the physiological makeup of the cells and an increased importance of metabolic regulation. The comparison of experimental and simulation results shows that the model is able to track this varied behavior of the culture by accounting for the key cellular components and metabolic regulation in the form of cybernetic variables.

Similar continuous culture dilution rate step-up and step-down experiments were performed with the organism growing on xylose (sF = 0.5 g/L) as the only growth-limiting nutrient. Simulations of the cybernetic model with the xylose parameters (Table I) provide a


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Figure 7. Experimental data and cybernetic model simulations of the transient period of a continuous culture growing on glucose: (a) following a step-down in dilution rate from 1.0 to 0.22 h-'; (b) following a step-down in dilution rate from 0.804 to 0.302 h-l; (c) following a step-up in dilution rate from 0.367 to 0.56 h-'.

reasonable descriptidn of the observed transient behavior. In the step-up experiment where the dilution rate is increased from 0.205 to 0.480 h-' [Fig. 8(a)], the cell density decreases for a period of 3 h before increasing slowly over the next 5 h to the final steady-state value. Due to the high saturation constant for growth on xylose ( K s = 0.013 g/L), the cell density at 0.48 h-' is lower than that at 0.205 h-' even though maintenance functions are more important at the lower dilution rate. The high saturation constant also results in a high residual substrate concentration (0.1 g/L) at the high dilution rate. This effect is pronounced in a step-down experiment.

In the transient experiment where the dilution rate is decreased from 0.48 to 0.259 h-' [Fig. 8(b)], the cell density increases from 0.14 to 0.185 gdw/L over a period of 2 h until all the residual xylose is consumed. The lower dilution rate cannot, however, support this cell density, so there is a gradual wash-out of the culture over the next 3 h until the final steady state is attained. The workings of the model during the step-up and step- down experiments with xylose can be qualitatively explained in the same fashion as those of the glucose transients. However, because of the very different values of the saturation constant and the maximum growth rate of the organism on glucose and xylose, the glucose


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Figure 8. Experimental data and cybernetic model simulations of the transient period of a continuous culture growing on xylose: (a) following a step-up in dilution rate from 0.205 to 0.48 h-'; (b) following a step-down in dilution rate from 0.48 to 0.259 h-'; (c) following a step-down in dilution rate from 0.45 to 0.17 h-'.

and xylose transients are very different from a quanti- tative standpoint. The model predictions of both the cell density and residual xylose concentration are reasonable in all experiments. The effect of a large magnitude change in dilution rate is seen even in the xylose transients.

Following the initial rise in cell density, there is an undershoot just before the final steady-state value is reached in the 0.45 to 0.17 h-' step-down experiment [Fig. 8(c)]. The transient results of the 0.48 to 0.259 h-' step-down experiment [Fig. 8(b)] do not exhibit this undershoot. The results of other step-up and step-down experiments5 are also described by the single substrate model. The transients following step-up and step-down of dilution rates show different trends, and it is significant that the model presented here shows good quanti- tative agreement in all the cases observed.

CONCLUSIONS

The transient continuous culture experiments shown here were performed by changing the dilution rate once steady state was reached and then following the culture behavior until a new steady state corresponding to the new dilution rate was achieved. A sequence of five or six transients was recorded before an experiment that typically lasted 6 days was terminated. Toward the end of the experiment, a thin bacterial film on the walls and internals of the fermentor was observed even though all the internals were coated with a silicone surface-treating agent. Wall growth was particularly noticeable in the continuous culture xylose experiments where the culture did not wash out even when the dilution rate was increased past the maximum growth rate observed in the batch experiments. This probably


caused the discrepancy between the observed and simu- lated sugar concentrations in the xylose experiments.

In this article we have addressed the transient behavior of a microbial culture from a cybernetic perspective. Previous models were found inadequate in describing the complex dynamics arising from sudden steps (both increases and decreases) in dilution rate in a continuous culture. The cybernetic model developed here explicitly accounts for cellular resource that becomes limiting especially during the long metabolic lag period in the step-up experiments. The model also accounts for the increased demand for the limited amount of substrate present and interactions between growth and maintenance for scarce cellular resources in low growth rate situations following a step-down in dilution rate. Rea- sonable agreement was observed between the model simulations and the experimental data for continuous culture growth on glucose and xylose. The model essentially reduced to previous cybernetic formulations in the case of batch and steady-state continuous culture situations because the effects of resource level and increased maintenance functions are insignificant.

The findings in this article clearly show the importance of metabolic regulation in describing continuous culture transients. In the past, it has been of interest to investigate the effect of forced feeds, sinusoidally or ~therwise,”-’~ on improving the performance of a continuous culture. The use of models not incorporat- ing regulatory features for simulating such situations is ill-advised.1726

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Metabolic regulation in bacterial continuous cultures: Idrops/ramkiPapers/100_1991_Baloo... · Metabolic Regulation in Bacterial Continuous Cultures: I S. Baloo and D. Ramkrishna*