Modeling of Bacterial Growth under Multiply-Limiting Conditions

588

Biotech” hog. 1994, 10, 588-605

Modeling of Bacterial Growth under Multiply-LimitingConditions. Experiments under Carbon- or/and Nitrogen-Limiting Conditions Jeffrey V. Straight and Doraiswami Ramkrishna’ School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907

As the limiting substrate is altered, a microorganism’s internal structure is also altered by invoking different metabolic pathways for the utilization of the limiting substrate. Transitions between pathways are moderated by the processes of metabolic regulation and are observed under steady-state and transient conditions. Under nitrogen-limited steady-state conditions, the effects of overflow metabolism are observed in the form of polysaccharide synthesis and excess carbon oxidation, two phenomena that are not observed under carbon-limited conditions. Furthermore, during transient periods following dilution rate shifts, metabolic lags are observed t o be a function of the size of the shift a s well a s the limiting substrate. The latter observation indicates the preference of one reaction pathway over another as the status of glucose or NH4+ is altered from limiting to nonlimiting. This paper presents steady-state and transient results from continuous culture experiments using Escherichia coli W. Singly-limiting conditions, when either glucose or ammonia is the limiting substrate, are investigated. Transient conditions are created by quickly increasing or decreasing the dilution rate of the fermentor. By utilizing the control strategies identified by Straight and Ramkrishna (1994) for regulating the processes commonly found within metabolic pathways, a cybernetic model is developed and compared to the steady-state and transient experimental results. Due to the incorporation of metabolic pathways, the development of the model accounts for lumped biosynthetic intermediates in addition to key enzymes that catalyze different cellular processes. The model also accounts for an internal resource that is optimally allocated toward the synthesis of the key enzymes. Furthermore, the model incorporates the effects of maintenance processes and overflow metabolism. Upon incorporating nitrogen utilization, the kinetic aspects of the model do not explicitly reduce to those of previous cybernetic models; however, the regulatory structure is in complete agreement with previous cybernetic models proposed for carbon utilization.

Introduction There are two primary reasons for considering the presence of substrate limitations involving bacterial growth on two or more complementary substrates: One, the maintenance of all substrates but one a t saturating concentrations represents a large pool of underutilized resources and may significantly affect the cost of the overall process, and two, different biological products are produced under limitations that are complementary to carbon limitation. For example, the production of polysaccharides, such as xanthan gums, can occur under nitrogenlimiting conditions, while fermentation products such as butanediol are produced under oxygen-limiting conditions. As engineers continue to enter the field of biotechnology, the large scale production of products other than biomass will require a thorough understanding of which process conditions determine the limiting substrate in order to optimize production of the product of interest. The issue of complementary substrates is not a new one. Previous modeling efforts have been directed specifically toward addressing some of the phenomena observed during microbial growth on limiting complementary substrates. The majority of these efforts, however, has maintained an unstructured approach (Baltzis and Fredrickson, 1988; Mankad and Bungay, 1988; Pavlou and Frederickson, 1989; Tsai and Lee, 1990);therefore,

* Author to whom correspondence should be addressed.

they are incapable of describing the variation in cellular composition that is required under both steady-state and transient conditions when the limiting substrates are complementary. One exception has been the work by Shuler and co-workers (Shuler and Domach, 1983; Domach et al., 1984). Their structured single-cell models, which have been applied to both carbon-limited (Domach et al., 1984) and nitrogen-limited conditions (Shuler and Domach, 1983), continue t o provide insight into understanding microbial physiology and its role in engineering problems. Furthermore, the literature associated with complementary substrates has often been preoccupied with only the appropriate form of the specific growth rate expression (Mankad and Bungay, 1988). This limited perspective offers little or no new understanding of the problems encountered when complementary substrates are limiting. The original cybernetic framework (Kompala et al., 1986)has been expanded recently (Straight and Ramkrishna, 1994) and applied, in part, t o a simple though abstract model system capable of predicting experimentally observed phenomena when microbial growth is limited by complementary substrates. In short, models developed under the tenets of the cybernetic framework are structured in nature and account for the effects of cellular regulatory processes on microbial metabolism. The microorganism is viewed as an optimal system in which the regulatory processes implement optimal strategies (Baloo and Ramkrishna, 1991;Kompala et al., 1986;

8756-7938/94/3010-0588$04.50/00 1994 American Chemical Society and American Institute of Chemical Engineers

Biotechnol. Prog., 1994,Vol. 10, No. 6

589

Table 1. Minimal Medium Salts Solution

component KzHP04.3HzO mzpo4 N&Cl

&so4

concentratiG/l) 18.0 2.86 6.00 0.55

Straight and Ramkrishna, 1994; Turner et al., 1989). The outcome of these strategies is incorporated into the process kinetics by identifying "cybernetic" or control variables. Therefore, a complete cybernetic model includes a kinetic description as well as a n overlying regulatory structure. This paper now explores the incorporation of the cybernetic framework into a structured model developed to describe a n experimental system when glucose or NH4+ may be limiting in continuous cultures. Even though these two substrates are completely complementary, this model system contains both complementary and substitutable processes. The proposed model is also applied to a system in which both glucose and NH4+ may be limiting, i.e., dual-limited conditions. It is important to note that more complex substrates such as amino acids, which are both substitutable for and complementary to glucose, may also be accounted for within the expanded cybernetic framework. However, microbial growth on glucose and NH4+ as limiting substrates adequately demonstrates the applicability of the cybernetic framework toward more complex limiting conditions.

Materials and Methods Microorganism and Inoculum Preparation. Escherichia coli W (ATCC 9637) was obtained from the American Type Culture Collection and used in all experiments. The bacterium was stored on nutrient agar slants a t 4 "C and transferred to a new slant every 4-6 weeks. The slants were initially prepared with DIFCO 1 medium; however, slants prepared with a minimal medium supported higher cell densities. The inoculum was prepared in 250 mL flasks containing glucose and ammonia a t concentrations equivalent or proportional to, i.e., the same ratio of glucose to ammonia, the desired experimental values and grown to stationary phase a t 37 "C in a rotary shaker. After two subsequent subculturings, approximately 15 mL of the second subculture was injected into the fermentor. Medium Preparation. The carbon-free minimal medium is described in Tables 1and 2. The medium was prepared as two solutions: the first contained the potassium and ammonium salts, and the second contained the concentrated trace metals solution, which was added in an amount of 0.5 mu100 mL of salts solution. For nitrogen-limited growth, ammonium chloride was excluded from the salts medium, and the desired amount of ammonium chloride was added separately. Concentrated stock solutions of glucose (200 g/L) and ammonium chloride (20 g/L NH4+, pH 5.5) were also prepared. Fermentor Apparatus and Preparation. A 2 L Virtis fermentor with a magnetic coupled bottom drive was used in all experiments. Prior to continuous operation, the fermentor was operated in the batch mode in order to produce an initial cell culture for continuous operation. The agitation rate was set at 625 rpm, the air flow rate was set a t 1.0 IJmin/L, and the temperature was set a t 37 "C. Previous results (Straight, 1991) have indicated that these rates of air supply and agitation are sufficient to maintain aerobic conditions. The medium (pH 7.25) provided sufficient buffering capacity, and the pH drop in any continuous experiment never exceeded 0.4 unit. The fermentor with its associated feed and exit

Table 2. Trace Metal Stock Solution (200x) component concentration (dl00 mL)

MgSOc7HzO FeS04.7HzO ZnSO4.7HzO MnS04.HzO CaClp2HzO EDTA

concentrated HzSO4

5.0 1.0 0.02 0.02 0.2 1.0 2 mI4100 mL

lines was autoclaved for 45-50 min. The mixing and feed tanks (with an in-line 0.2 pm filter) were autoclaved for 25-30 min. Twenty liters of the salts medium was autoclaved for 60 min. The stock glucose and NH4C1 solutions and the concentrated trace metals solution were autoclaved separately from the salts medium for 25 min. The sterilization temperature was 121 "C. Upon cooling, the desired quantities of glucose, ammonium chloride, and trace metals were added to the salts solution and pumped into the mixing tank. Prior to entering the feed tank, the medium was pumped through a 0.2 pm filter to ensure a sterile feed. From the feed tank the medium was pumped to the fermentor by a Harvard peristaltic pump. The feed rate was calibrated with a n in-line graduated pipet. Before entering the fermentor under the impeller, the feed was passed through two break tubes to prevent contamination of the feed line (Herbert et al., 1965) and mixed with filter-sterilizedhumidified air. The exit line was connected to a n overflow device, which maintained a constant reactor volume of 1350 f 15 mL. Dry Weight Measurements. Cell density was determined from optical density measurements at a wavelength of 540 nm. Optical density was linearly proportional to cell dry weight for absorbances up to 0.30. Absorbance values above 0.30 were sufficiently diluted by a factor of l o x or 5 0 x . Duplicate cell density measurements were made. To correlate optical density measurements with dry weight values a volume of cells was taken from a stationary phase batch culture. A quantity of cells, i.e., 10-20 mg, was chosen in order to minimize filtration time and experimental error. The sample volume was measured, filtered through a 0.2 pm filter (predryed and weighed), washed, and dried at 90 "C for 24 h. The filter plus cell materials was reweighed, and the dry weight of the cell material was determined by difference. Results from batch experiments resulted in the following extinction coefficients: 0.32 g of DW/W OD, for carbon-limited growth, 0.36 g of DWWOD, for nitrogen-limited growth, and 0.34 g of DW/L/OD, when neither glucose nor NH4+ was present in excess relative to the other. Substrate/Carbohydrate Analysis. Glucose was measured by using the glucose oxidase enzymatic assay (Kit No. 510, Sigma Chemicals, St. Louis, MO). Ammonia was also measured enzymatically from the reductive amination of 2-oxoglutarate using glutamate dehydrogenase and reduced nicotinamide adenine dinucleotide (Kit #170-W, Sigma Chemicals, St. Louis, MO). The detection limit of the glucose assay was estimated to be 10-20 mg/L, while the detection limit of the ammonia assay was stated as 0.3 mg/L. The cellular carbohydrate content was determined by the modified anthrone reagent as described by SeiRer et al. (1950).

Model System The enzyme-catalyzed processes incorporated into the model are summarized in Figure 1; complete model details are given in the following text. Uppercase letters indicate process variables, while lower case letters iden-

590

77. (nitrogen intermediate)

(NH;)

SA

(glucose)

(biomass) B

SN

,s,

s,

ECtG'L b

(carbon intermediate)

S, (storage compound)

sGl

+

EML,GI

B

B

~

(maintenance)

sGl

+

B

E X,GI

P B

E

~

.

b

~

B-

~

+

S,

(resource synthesis)

Figure 1. Proposed model system to describe multiply-limited microbial growth when glucose, N H 4 + , or both may be limiting. Only the enzyme-mediated processes are shown.

ti@their respective concentration variables. In addition, all key enzymes and kinetic parameters are denoted with an i j or an i j , k subscript to identify the ith process and t h e j t h substrate associated with the particular enzyme or parameter. When an i j , k subscript is utilized, the k subscript identifies the respective substitutable process. Process Equations and Kinetics. The model system in Figure 1 identifies intermediate processes. Specifically, the syntheses of a nitrogen intermediate, SN, and a carbon intermediate, SC, are incorporated. The intermediate SNis a lumped component which represents the pool of nitrogen containing intermediates, e.g., amino acids, contained within the biomass component B. Nitrogen assimilation is represented as

with yield coefficient YNIG~ and stoichiometric coefficient Ym. In agreement with established observations in the literature (Magasanik, 1982; Tempest et al., 19831, two key enzymes are incorporated for the synthesis of SN: a high-affinity enzyme, E N , A ,and ~ , a low-affinity enzyme, EN,A,~, which are analogous to glutamine synthetase and glutamate dehydrogenase, respectively. Process kinetics for both of these processes are assumed to follow dual Monod kinetics, given by

The intermediate SC is also a lumped component, which represents the pool of carbon intermediates not containing nitrogen. Carbon assimilation is represented as 1

key enzymes E c , G ~and , ~ EC,al,Zrepresent low- and highaffinity processes, respectively (Neijssel and Tempest, 1979; Tempest et al., 1983). During the process of synthesizing the necessary biosynthetic intermediates, the carbon, and energy, source is subsequently oxidized. As a result, energy, in the form of ATP and NADH, is generated and made available for biosynthetic processes. The current model development does not explicitly incorporate the presence of these energy resources, but assumes that the quantity of S Gcatabolized ~ to produce energy during the synthesis of SC and SN is equivalent for both intermediates; therefore, the yield coefficients YN/G~ and YC/Gl are assumed to be equal in this model development. Process kinetics for eq 3 are assumed to follow modified Monod kinetics, given by

(overflow metabolism)

Sc + S, + B

-SGl+ yC/Gl

Biotechnol. frog., 1994,Vol. 10, No. 6

B

-Sc + B EC,Gl,k

(3)

with yield coefficient YC/G1. In analogy with eq 1, multiple mechanisms of carbon assimilation are incorporated. The

The two SN synthesis processes as well as the two SC synthesis processes are therefore substitutable in nature, while the syntheses of SNand SC from s G 1 are complementary processes. Once SC and SNbecome available, biomass B is synthesized according to

Yc,Sc

EG,CN + YNiBSN+ B 2B

with stoichiometric coefficients YC/Band YNB. As in eq 2, a dual Monod kinetic form is assumed for the growth process, where

The biomass yield on glucose is assumed to be determined by the conversion of glucose to carbon and nitrogen intermediates. Therefore, the growth process may also be described as a function of the substrates available in the medium by identifying the yield coefficientsYB/G1 and YWA,which are given in notation appropriate for concentration variables as

and derived from the stoichiometric and yield coefficients identified earlier. The yield values Yb/gl and Yb/a are the maximum values in the absence of maintenance and storage compounds, respectively. The development of the model for the system in Figure 1 has addressed only the synthesis and subsequent utilization of key intermediates for the synthesis of biomass components. The necessary key enzymes must also be synthesized. Since the limiting substrates are now complementary, a more complete description of protein synthesis must be incorporated. In addition to RNA, protein synthesis also requires the presence of amino acids, which provide carbon and nitrogen, and generally an appropriate inducing agent. Therefore, the synthesis of each enzyme must in some way depend on ,~ both glucose and NH4+. In the presence of SA,E N , Aand E N , Aare , ~ both synthesized according to

Biotechnol. Prog,, 1994,Vol. 10, No. 6

591

In the case of the key enzyme E N A ~i.e., , the high-affinity enzyme, multiple mechanisms of synthesis have been suggested (Senior, 1975; Tyler, 1978). The synthesis of EN,A,~ is assumed to respond to either the presence or absence of SAwithin the environment. Synthesis in the presence of S A was identified in eq 8. In the absence of SA,the synthesis of EN,A,Iis described by

The induction of enzyme synthesis by a substrate limitation has been previously observed. One particular example is the cellular response to phosphate limitations (Wanner, 1987). In eq 8, SAis identified as the necessary nitrogen source for enzyme synthesis and serves as the appropriate inducing agent. Likewise, S G is ~ identified as the necessary carbon source for enzyme synthesis. The biomass, excluding the key enzyme, is given by B-, and SR is a n RNA-type key resource. The synthesis and regulation of this resource are summarized later in the model development. Synthesis of the remaining key enzymes is assumed to be analogous to eq 8, i.e., requiring an inducer, a carbon source, a nitrogen source, and the resouxce SR. However, either the carbon or the nitrogen source also serves as the inducing agent in order to reduce the complexity of the synthesis process. Induction kinetics are assumed to follow a multiple saturation form. The total synthesis rate of ENAJ is the sum of the two processes described in eqs 8 and 9, i.e., = r-e,n,l r e , n , l where

+

and

The kinetic form of the rate expression for the synthesis of E N A , ~i.e., , re,,,,2, is equivalent to eq 11; however, as expected, the kinetic constants are physically different. Likewise, the induction of the key enzyme Ec,G~,K, which catalyzes the synthesis of SC,is analogous to eq 8 and described by

where S G l serves as the inducing agent and SNsupplies the necessary carbon and nitrogen for protein synthesis. Induction kinetics are given by

Synthesis of the key enzyme EG,CNoccurs in the presence of SC and SN,and the process is described as

B-

sN

SR

B- + EG,CN

Since SNand SCmust be available before cellular growth occurs, these intermediates will always be present a t

some level regardless of the nature of the substrates available in the medium; therefore, the synthesis of EG,CN is intrinsically constitutive in nature. Process kinetics for the synthesis of EG,CN is given by

The increased allocation of carbon source away from cellular growth and toward maintenance processes a t specific growth rates less than the maximum has been incorporated into several modeling efforts [Baloo and Ramkrishna (1991))Pirt (1982))Tsai and Lee (1990), and Turner et al. (1989), among others]. Maintenance rates have been observed to vary greatly, depending upon the nature of the nutrient limitation (Neijssel and Tempest, 1975, 1976a). Empirical observations such as these suggest a need to incorporate multiple maintenance processes within the current modeling framework. Therefore, as the status of the culture is altered, through variations in the limiting substrate or the specific growth rate, multiple processes that consume glucose to provide additional energy and/or intermediates become evident. Three such processes are incorporated and described by sGl+

B

Ei,G1

B

i=M,ML,X

(16)

The processes catalyzed by EM,G~ and EML,G~ are strictly maintenance processes, which were incorporated into the cybernetic framework by Turner et al. (1989) and Baloo and Ramkrishna (1991). The activities of EM,G~ and E ~ , G I are assumed to be a function of the specific growth rate (Turner et al., 1989). The third process, as catalyzed by ExG~, is active only under conditions of excess energy, i.e., excess carbon. The increased oxidation of glucose under energy-sufficient conditions has been viewed as a maintenance process (Pirt, 1982). However, Neijssel and Tempest (1975) preferred the expression “overflow metabolism” for the increased oxidation of glucose when a culture is energy sufEcient. Overflow metabolism results in the synthesis of additional intermediates and energy that cannot be immediately utilized for additional growth. The presence of high concentrations of intermediates that are associated with the assimilation of the limiting substrate may serve to minimize the effects of the prevailing limitation (Neijssel and Tempest, 1975,1976b). Therefore, overflow metabolism is viewed here as an optimal response, which is regulated so that it occurs not just in the presence of high glucose concentrations but rather under energy-sufficient conditions only. The enzymes E M , Gand ~ Ex,G~ are assumed to be synthesized in a manner equivalent to EG,CN, as described by eq 14; therefore, EM,G~, Ex,G~, and EG,CN are present within the biotic phase a t equal concentrations, which implies

However, as stated above, their activities respond to different regulatory signals. An assumption analogous to eq 17, which represents the process of lumping, is made later in the model development for several other , was key enzymes as well. The synthesis Of E m , ~ lwhich proposed by Baloo and Ramkrishna (1991) to account for increased maintenance functions during transient shift-

Biotechnol. Prog., 1994,Vol. 10, No. 6

592

downs in continuous cultures, is described by

feedback regulation in eq 24 has been derived from a kinetic mechanism. Since both Ec,G~,P and EG~,s operate primarily under conditions of excess of glucose and both utilize a form of glucose, the present model development assumes that these enzymes are synthesized a t equivalent rates; therefore, they will be present a t equal concentrations or

with process kinetics given by

The enzyme EML,G~ is not lumped as in eq 17 because the synthesis of EML,G~ is assumed to respond to different kinetic as well as regulatory signals. The rate expressions for r,,gl, rml,gl,and r,,gl are given by

- P4d maxei,gl Ki,gl+ Sgl -

i = m, ml, x

(20)

Under energy-sufficient conditions, e.g., nitrogen limitation, the synthesis of storage compounds has been observed to increase greatly (Dawes and Senior, 1973; Preiss, 1989; Wilkinson, 1959). The significance of these storage polymers is generally assumed to be associated with maintaining the viability of the organism when the limiting non-carbon substrate, e.g., NH4+, is exhausted. The polysaccharide may then be reconverted to glucose and utilized for maintenance functions, thus extending the survival of the organism (Preiss, 1989). Within this model, the synthesis of a carbon storage compound, SS, is catalyzed by a key enzyme Es,G~ and described as

Ss

+B

-

ae,ssgisnSr

(Eis,gl

+ Sgl)(Ks,n + Sn)(Ks,r+ s,)

(27) Process kinetics for the synthesis of SRis given by

Since the synthesis of ER,CN is assumed to be equivalent to the synthesis of EG,CN, this model development assumes

-

The synthesis of Es,G~ is analogous to eq 18, with process kinetics given by ‘e,,

- ec,g1,2

(26) In addition to increased maintenance functions during transients in continuous cultures, the availability of a n RNA-type resource SR,which significantly affects the response of the cell culture during a n increase in dilution rate, has also been incorporated into the cybernetic framework (Baloo and Ramkrishna, 1991). The resource SRis required for enzyme synthesis as already described; however, in contrast to Baloo and Ramkrishna (19911, the synthesis of SRnow utilizes the nitrogen and carbon intermediates SNand SC rather than sG1. The process equation for the synthesis of ER,CN is equivalent to eq 14, while the synthesis of SRby ER,CN is described by eg1,s

(22)

Because SSis a glucose polymer, it may resupply glucose, which is necessary to synthesize both of the intermediates SNand SC, as well as to support essential maintenance functions. The depolymerization of SS (eq 23) is catalyzed by the key enzyme, EG~,s. (23) Process kinetics for the storage and depolymerization of and SS, respectively, are given by

(29) However, as before, their activities may be regulated in different manners. Metabolic Regulation. To complete the model development, the appropriate metabolic regulation must be accounted for by identifying cybernetic or control variables that modify the synthesis and activity of key enzymes. From the results derived by Straight and Ramkrishna (1994), the regulation of the complementary and substitutable processes in Figure 1 can be quickly identified. Within the current model system, two convergent branch points are present: the assimilation of SAinto SNby E N A and , ~ the assimilation of S G l into SCby Ec,Gl,k. Therefore, the synthesis and activity of ENA,k and E c , G ~are , ~ modified by the cybernetic variables for substitutable processes, where %,cn - eg,cn

sG1

0 IU e , k

r,,,,ku:J,k and (25) Experimental results in the literature (Ingraham et al., 1983; Stephanopoulos and Vallino, 1991) have suggested a maximum concentration of intracellular storage compounds; therefore, the rate expression rs,glin eq 24 has also been modified by the inhibition of s,. The synthesis ~ constitutes a divergent branch of SSand SCfrom S G also point in the model; however, SSis not essential for growth (Preiss, 1989) and therefore does not meet the criteria proposed by Straight and Ramkrishna (1994) for regulating branch point structures in metabolic pathways. As a result, the synthesis of SS and SC is not viewed as a divergent branch point that is regulated according to cybernetic principles (Straight, 1991); therefore, the

raJPu:J,k 0 IUfJ,k

I1where

i, j = c,gl and n,a

I1where

i,j = c,gl a n d n,a

with 2 ’;J,k

= r L J , k / ~ r L J p and p=l

‘t,k

= r~~,k’max(r~~,p) P

i,j = c,gl a n d n,a (30) In addition to the two convergent branch points, the nitrogen assimilation enzymes and the carbon assimilation enzymes, which produce SN and SC, respectively, comprise a single divergent branch point. Therefore, both the synthesis and the activity of EN,A,k and E C , G I ~ are further modified by the cybernetic variables for complementary processes, where r,,,,k~;~,~uF0

Iu F5

1where i, j = c,gl and n,a

Biotechnol. Pfog., 1994,Vol. 10, No. 6

593

and

riMufjPu; 0 Ivi I1where i , j = c,gl and n,a For the system in Figure 1, the complete forms of the cybernetic variables for the complementary processes are given a s

on Sc and pn is the potential specific growth rate on SN. Both pc and pn are determined from the material balances on SCand SN,i.e., eq 48, after v: and u i are set to unity. The resulting expression for pi is given by

where i , j = c,gl and n,a (36) and

rc,g1,lJsc

u: =

“(rn,a,dSn,

where

r,-,,a,T and ?‘c,gl,T

(32)

rc,gl,dsc)

2

(33)

k=l

The regulation derived for substitutable and complementary processes (Straight and Ramkrishna, 1994) addresses the synthesis and subsequent utilization of intermediate products that are ultimately used for the production of biomass. However, in the case of maintenance processes, the available energy source is oxidized for the purpose of providing the energy required to perform necessary maintenance functions. To account for the increased quantity of glucose consumed for maintenance processes a t reduced specific growth rates, the activities of EM,G~ and EML,G~ are modified by the cybernetic variable um, where (rm,gl

+ rml,gl)vm

0 Ium I1

and

um=l--

P g cn max k c n

(34)

Turner et al. (1989) derived the expression for um in detail; therefore, the derivation is not reproduced here except in its final form. One distinction becomes necessary when considering carbon- and nitrogen-limited growth. Since storage compounds may represent a significant fraction of the total biomass, the synthesis of polysaccharides will necessarily contribute to the overall specific growth rate p. However, only the polysaccharidefree fraction of the total biomass, which is synthesized according to eq 5 , is assumed to reflect the culture status; therefore, u, is a function of pg,cnrather than p. Within this model system, the potential presence of two excess substrates is considered: glucose or NH4+. As proposed by Straight and Ramkrishna (19941,the degree of excess is determined by the difference between the potential specific growth rate supported by a characteristic intermediate and the observed specific growth rate. Within this model development the characteristic intermediate for So1 utilization is SC,while SNis the characteristic intermediate for SAutilization. Therefore, the , given ~ by variables ux,gl and u ~are Ux,gl

=

l u C

-P)

max(pj- ,4 .i

a n d ux,a =

lun

and ‘x,glUx,gl In contrast to excess glucose, the storage of excess nitrogen has not been observed within E. coli W. However, this is not true of all microorganisms (Preiss, 1989). Therefore, while the variable Ux,a may not modify any process within the current system of interest, the utilization of excess non-carbon nutrients still remains a significant problem to which the regulation of excess substrate utilization generally can be applied. As proposed by Straight and Ramkrishna (1994), the presence of alternatives that compete for a common species necessitates the presence of metabolic regulation in order to most efficiently utilize limited resources. Substitutable processes compete to synthesize a common product, while complementary processes compete to utilize a common substrate. Likewise, Baloo and Ramkrishna (1991) identified a competition for limited cellular resources between growth and maintenance enzyme systems and introduced the need to distribute resources between these enzyme systems in a manner that maximized the substrate consumption rate. Therefore, the rate expressions re,g,re,ml,and re,sare modified as rs,glux,gl

are given by

rij,T= xrij,ku:j,k i , j = c,gl a n d n,a

Equation 36 is applicable to steady-state or transient conditions. In the presence of excess glucose, i.e., nitrogenlimiting conditions, two processes are modified by the synthesis of storage compounds and overflow metabolism. Therefore, the activities of E s , ~and l EX,GI are modified by modifying their respective rate expressions, rs,gland rx,gl,as

-P)

max(pj - P ) j

j = c a n d n (35) respectively, where p, is the potential specific growth rate

re,gug, re,mium? and re,suB

where Pg

=

rg,cn/yblgl rpb/gl

+ ‘m,gl,T

rm,gLT

and p m =

rg,cn/yWgl

+ rm,gl,T (37)

and rm,gl,T

= (rm,gl + rml,gl)Um

(38)

Since ultimately also supports maintenance functions a t low to zero growth rates by supplying glucose from stored polysaccharides, the need for the additional allocation of resources is required. Therefore, the synthesis of ES,GIis modified by us. As reviewed by Preiss (19891,the synthesis of enzymes for the storage of carbon appears to respond primarily to the specific growth rate rather than the presence of excess carbon. This conclusion is supported by observations in nitrogen-limited batch cultures that synthesize glycogen primarily near or during the stationary phase, i.e., p 0, even though excess glucose is present throughout the exponential growth phase (Sigal et al., 1964). Furthermore, storage compound synthesis in nitrogen-limited continuous cultures is maximal a t low dilution rates (Holme, 1957; is Preiss, 1989; Straight, 1991). Since the need for EML,G~ also maximal at low specific growth rates, the returns or Es,G~ are assumed received from the synthesis of EML,G~

-

Biotecbnol. Prog., 1994, Vol. 10, No. 6

594

to be given by the total maintenance rate. Therefore, us = um (39) and the enzymes EML,G~ and ES,GIare viewed as being synthesized from the same operon, with the same fraction of resources available to both enzymes. The final aspect of regulation within this model development concerns the synthesis of the key resource SR. In order to address observed metabolic lag periods following an increase in dilution rate, Baloo and Ramkrishna (1991) incorporated a n RNA-type resource into the cybernetic framework. That resource was identified earlier, while its regulation is reproduced here as

rr,cnu, 0 5 u, 5 1 where Therefore, a s pg,cnapproaches p i g , the maximum level of resource is produced to match its maximum need. Control of the synthesis of SRa t a rate that reflects its usage agrees with a n economic perspective of microbial growth, since any resource produced a t a level greater than that required to support the prevailing growth rate would not be efficiently utilized. Material Balances. By combining the kinetic rate expressions and the cybernetic variables, complete material balances can be written in rate form for continuous culture:

ec,gLk

den,a,k --

dt

(

re,n,kui,a,kuk

T

dt dbT + D + + a;,n,k

k = 1,2 (42)

-

(45)

i, j = c,gl a n d n,a (48) (49)

+

where bT = b ss. The glucose and NH4+ feed concentrations are given by sg1,~ and s,,F, respectively, and D is the dilution rate. Equations 41-51 are applicable to steady-state as well as transient conditions. In addition to the process kinetics previously discussed, the model equations account for the following processes: first-order decay of the key enzymes, SR,and SS, dilution of biotic phase components due to expansion of the biomass during growth (Fredrickson, 19761, and small constitutive synthesis rates for the key enzymes and the key resource SR. The maximum key enzyme levels were determined either directly from the model equations or from pseudosteady-state simulations in which the appropriate substrates and intermediates were present in excess quantities. Parameter Estimation. Parameters were estimated from the following sources: literature studies, continuous culture experiments, previous cybernetic modeling efforts, and order of magnitude estimations. Batch experiments were used to estimate the cellular yield on glucose under carbon-limiting conditions and the maximum specific growth rate. A value of 0.49 g of DW/g for was obtained from carbon-limited batch cultures; therefore, Ydg1= Ydgl = 0.49. Since SNis viewed as the nitrogen intermediate pool, Y,,,I,was chosen to reflect the protein content (Daigger and Grady, 1982) of biomass, and Yh = 1 - YA. Furthermore, was determined from the average of the minimum doubling times reported for E. coli in a rich medium and a minimal medium, i.e., 20 and 45 min, respectively (Ingraham et al., 1983; Mandelstam and McQuillen, 1973). The two saturation constants associated with the synthesis of biomass B were not estimated from the experimental data. Rather, K,,g.and Kn,gwere chosen arbitrarily to reflect the size of intracellular intermediate pools, Le., 0.25-5.0% of the biomass dry weight (Cooney et al., 1976; Mandelstam, 1958). The carbon-limited steady-state results in Figure 2 were used to estimate several parameters. The average yield on NH4+ under the conditions of Figure 2 set the value of Ydn once Y,,,I, had been determined. At the highest dilution rate, the decline in cell density due to washout identified the parameters p:g2 and p:y2. Both Kn,a,nand Kc,gl,pwere set a t values equivalent to the glucose saturation constant utilized by Baloo and Ramkrishna (1991). At the opposite end of the steadystate curve, the maintenance parameters pzzl and Kml,gl were determined from the decrease in cell density. Since the enzyme system EM,G~ is a low-affinity system, the parameters pzy and Km,glwere determined from the decline in cell aensity following the overshoot af'ter a shift-down under carbon-limiting conditions. The slow decline in cell density implied that ed,glincreased slowly, and the standard enzyme synthesis rate was utilized for ~ , ~ The 1 . net rate of increase in eml,glis determined by its decay; therefore, Pe,ml was also estimated from the carbon-limited shift-down data. As a high-affinity system that is maximal a t low specific growth rates, the constants Kml,*and Kml,rwere arbitrarily set a t values that would maintain a saturated rate of enzyme synthesis a t low and high growth rates. The carbon-limited shift-up experiments indicated that SRis limiting a t the time of the increase in dilution rate. and Kg,,were Therefore, the parameters p:', K,,,, K,,,,,

qGl

Biotechnol. Prog., 1994, Vol. 10, No. 6

3

\

595

1

I

-3 0.4 -

h

-2

4

.i

5 Ei

+ *

m z

0.2

-1

3

P)

u

s-

n

v

1

l’ l

10

0.0 0.0

0.2

0.4

0.8

0.8

1.0

D i l u t i o n R a t e (Hr-I)

Figure 2. Carbon-limited steady-state data versus model simulations. Profiles of cell density (0)versus the model simulation (-1 and NH4+ (0)versus the model simulation (- - -) are shown. Average feed concentrations are 0.98 g/L glucose and 1.85 g/L N H 4 + . estimated from carbon-limited shift-up data. The value for pr was taken directly from Baloo and Ramkrishna (1991), while was set a t approximately 1%of p::. To simplifjr the parameter estimation, the resource constants associated with enzyme synthesis were set such that Kc,,,l = Kc,r,2= Kn,r,z= Kg,r. Since glutamine synthetase is synthesized a t high rates even in the presence of low the value of Kn,r,lwas set 2 orders concentrations of of magnitude below that of Kg,r. The saturation constant for the high-affinity NH4+ system is known to be quite low. Therefore, Kn,a,lwas estimated from the residual NH4+ data and the detectability limit of the N H 4 + assay. The undetectable levels of NH4+ and the flatness of the cell density curve a t increasing dilution rates (Figure 4 ) suggested that Kn,a,l should be set 1 order of magnitude less than 0.3 mgL. Since E N A , ~is a low-affinity system with respect to glucose, Kn,gl,lwas set a t a value somewhat larger than that of the glucose saturation constant reported by Baloo and Ramkrishna (1991). The lower limit for Kn,gl,lwas set by carbon-limited shift-up results. The enzyme E N A ~ was assumed to have the highest nitrogen affinity. Once Kn,a,lwas identified, Kc,n,l and Kc,,,,:!were set a t approximately 1 and 2 orders of magnitude above Kn,a,l, respectively. The concentration of glutamine synthetase can amount to several percent of the cellular protein (Tyler, 1978); therefore, ai,,,1was set at a correspondingly high value. Since the concentration of glutamine synthetase has been observed to increase rapidly, the decay rate of E N A , was ~ assumed to be lower (Mandelstam, 1963; Pine, 1972) than the standard value of 0.05 h-l. In contrast to glutamine synthetase, the affinity of glutamate dehydrogenase has been consistently reported within the literature; therefore, Kn,a,2was taken from the results reported by Miller and Stadtman (1972). The kinetics for synthesizing E N A requires ~ both a n inhibition and a saturation constant. The saturation constant was set equal to Kn,a,lwhile the inhibition constant was set equal to Kn,a,Z,since it is reasonable to conclude that the inhibition of ENAJ would become significant as the NH4+ concentration exceeded the NH4+half-saturation constant for the synthesis of EN”:!: In addition to the residual N&+ levels, the decline in cell density a t the highest dilution rate was used to estimate and Kc,gl,l. Since E c , G ~is, ~ a low-affinity system, both Kc,gl,land p:zl are expected to be larger than the values for their corresponding parameters Kc,gl,pand $E2. Once the

4

m+,

maintenance parameters had been identified from the carbon-limited data, the overflow metabolism parameters &;’ and Kx,glwere determined from the residual glucose levels under nitrogen-limiting conditions. In Figure 4 the decline in sa reflects the decline in es,gl; therefore, G,, was determined from the nitrogen-limited steady-state results. As proposed by Sigal et al. (19641, Pe,, was set at a high value equivalent to the decay rate utilized by Baloo and Ramkrishna (1991) for the degradation of the low-maintenance enzyme system. The were estimated from the parameters p t z , Ks,gland KSkgl rate of increase in cell density, while paand Ka,swere estimated from the overshoot in cell density following a shift-down under nitrogen-limiting conditions. The remaining parameters associated with the synthesis of ES,GI,i.e., Ks,,, and K,,r,were assumed to be small, but not so small that the synthesis rate would always be saturated to sn and sr, respectively; therefore, K,,,, and K,,r were set to the lowest values utilized for the key enzymes for growth and intermediate synthesis. The typical value for a: I is 0.1% of the inducible rate, However, results indicated that approximately 1% of the inducible rate was appropriate for az,n,2, a:,ml, and Also, akn,lwas set 1 order of magnitude less than the standard synthesis rate, Le., 0,001 h-l, since the synthesis of ENAJ in the absence of NH4+ is assumed to occur a t a much lower rate than that supported by the process with saturation kinetics. The remaining parameters were set to values utilized in earlier cybernetic models.

Results and Discussion Steady-StateResults. The steady-state growth characteristics of Escherichia coli W under carbon-limiting conditions are compared to model simulations in Figure 2, which includes data from five experiments with feed concentrations of 0.98 f 0.05 g L glucose and 1.85 f 0.03 gL Both the experimental biomass concentration, expressed as grams of DWAiter (DW, dry weight), and the residual N&+ concentration are plotted as a function of the dilution rate. Residual glucose measurements are not included since they were below assay detectability limits. The model quantitatively describes the expected decrease in biomass concentration due to increased maintenance functions a t low dilution rates, as well as the residual NH4+ levels. The model predicts that washout of the culture should occur as the dilution rate increases beyond 0.87 h-l. Unfortunately, wall growth prevented the experimental realization of washout. However, a critical dilution rate of 0.87 h-’ agrees with the maximum specific growth rate of 0.88 h-l, which was measured under carbon-limited batch conditions. To demonstrate that the model is consistent with the standard expression for determining the specific growth rate when complementary substrates are limiting (Baltzis and Fredrickson, 1988),Figure 3 indicates that the model preferentially activates the pathway, i.e., = 1 and u i < 1, which supports the minimum specific growth rate, i.e., pn > p,. Likewise, under steady-state nitrogenlimiting conditions, p, > p,, and uk = 1while u‘, < 1. To test the model’s ability to describe the utilization of complementary substrates, it must be capable of predicting microbial growth when limited by nitrogen alone or potentially limited by carbon and nitrogen, two conditions that the model derived by Baloo and Ramkrishna (1991) was incapable of describing. Steady-state nitrogen-limited data are presented in Figure 4 and compared to appropriate model simulations. Data taken from three experiments with feed concentrations of 5.0

m+.

UX

Biofechnol. Prog., 1994, Vol. 10, No. 6

t /o'6

i o

o'2LL-----

........................

0.1 0.0

3

0.2

0.4

0.8

0.6

Dilution Rate

1.0

( ~ r - 1 )

Figure 3. Steady-state model profiles vz (- - -) and vf, (-1 as well as the potential specific growth rates supported by the intermediates SCand SNunder carbon-limited conditions. The potential specific growth rates on SC and SN are denoted as ,uc (- - -1 and ,un(-1, respectively. Feed conditions are equivalent to those in Figure 2. 0.75

1.00

4

\

M

k

0.75

\M v

0.50

e,

v

h

0.60

5

5 n

B

CI

h

$

G

0.26

0.25

h

-u" 6

d

rl r(

1

u

U 0.0

I

C

)

0

0.0

0.4

Dilution

0.B

. 0.8

0

0.0

1.0

R a t e (Hr-')

Figure 4. Nitrogen-limited steady-state data versus model simulations. Profiles of cell density (0) versus the model simulation (-1, residual glucose ( 0 )versus the model simulation (- - -), and cellular glucose (A) versus the model simulation (. -1 are shown. Average feed concentrations are 5.0 giT.. glucose and 66 m g L NH4+. 6

f 0.1 g/L glucose and 66 f 4 mg/L NH4+ are included in Figure 4. Comparison of the model and experimental profiles in Figure 4 indicates that the model quantitatively describes both the cell density and the residual glucose profiles. Residual NH4+ was below the detectability limits of the assay. Washout of the culture is predicted to occur a t approximately OB0 h-l. Senior (1975) reported similar results utilizing E. coli W. Comparison of Figures 2 and 4 indicates a significant distinction between carbon- and nitrogen-limited microbial growth. Under nitrogen-limiting conditions, the decrease in cell density due to increased maintenance processes a t low dilution rates is not observed. On the contrary, the cell density increases greatly at low dilution rates due to the synthesis of storage compounds, a phenomenon observed experimentally and plotted in Figure 4 for one of the experimental data sets. Since ,uc > p,, under steady-state nitrogen-limiting conditions, ux,gl = 1 while ux,* = 0; therefore, the key enzyme for synthesizing SSis fully active for the conditions in Figure 4. Likewise, since pn > pc under steady-state carbonlimiting conditions, the model predicts that storage compounds are not produced under the conditions of Figure 2, and no indication is given that they are present

0.2

0.4

0.6

0.a

i

D i l u t i o n R a t e (Hr-') Figure 6. Steady-state yields on glucose versus model simulations-two limiting conditions. Profiles for Yb&l (0, 0 ) are compared to model simulations under carbon-limitingconditions (-1 and nitrogen-limiting conditions (- - -1.

in measurable quantities. While the model and the cellular glucose data show good agreement between dilution rates of 0.2 and 0.7 h-l, the two profiles deviate substantially at the two lowest dilution rates, i.e., 0.1 and 0.15 h-l. The model continues to predict an increasing cellular glucose concentration, which exceeds 40% of the total biomass concentration, while the experimental data level off a t a value of about 22% of the dry weight concentration. Since only one data set was obtained, it is not clear whether the data obtained a t the lowest dilution rates represent a reproducible phenomenon. Holme (1957) observed a maximum glycogen concentration of 22.5% of the dry weight. However, the lowest dilution rate investigated by Holme (1957) was 0.13 h-l in steady-state continuous cultures of Escherichia coli B. One aspect of continuing discussion when limiting substrates are complementary is the variation in cell yield values as a function of the limiting substrate. Therefore, cell yields on glucose under carbon- and nitrogen-limiting conditions are compared in Figure 5, while cell yields on N H 4 + under carbon- and nitrogenlimiting conditions are compared in Figure 6. Under carbon-limiting conditions, cell yield on glucose varies between 0.38 and 0.46 g of DW/g. The absence of reduced biomass levels a t low dilution rates under nitrogenlimiting conditions does not imply that glucose is not consumed for maintenance. On the contrary, carbon oxidation is generally much higher under nitrogenlimiting conditions due to the presence of overflow metabolism. Figure 5 indicates that the cell yield on glucose under nitrogen-limiting conditions vanes between 0.175 and 0.375 g of DW/g. Under both limiting conditions, the cell yield on glucose increases with increasing dilution rate due to lower maintenance rates a t higher dilution rates. Batch experiments have indicated a maximum yield on glucose of 0.49 g of DW/g under carbon-limiting conditions. The model predicts that the cellular yield on NH4+ under carbon-limiting conditions should decrease from 5.0 to 4.8 with increasing dilution rate. The average cell yield on NH4+ over all dilution rates is 4.9 g of DW/g, which is in agreement with the model predictions. Since the quantity of NH*+ consumed for biomass synthesis is quite small, a significant amount of scatter in the NH4+ cell yield data is observed. Rutgers et al. (1990) have also reported a cell yield of approximately 4.8 g of DW/g on NH4+ in cultures of Klebsiella pneumoniae growing

Biotechnol. Prog., 1994,Vol. 10, No. 6

597 0.0 n

II

CI

75

0.4

h

v

8 -

h

- =O

3

on

Id

v

p

, ,

B

I

8

T

loo

c)

\

s

n

E

w

m

$

a

40

0

0.2

!I

0

3

I

0

V 0.0

01 0.0

< 0.2

0.4

0.8

0.8

1

D i l u t i o n R a t e (Hr-l)

Figure 6. Steady-state yields on NH4+ versus model simulations-two limiting conditions. Profiles for Ywa(m, 0) are compared to model simulationsunder carbon-limiting conditions (- - -) and nitrogen-limiting conditions (-).

at 0.4h-' under steady-state carbon-limiting conditions. In contrast to carbon-limiting conditions, the cell yield on under nitrogen-limiting conditions varies between 8.4 and 5.2 g of DW/g. The cell yield on NH4+ decreases with increasing dilution rate due to reduced levels of storage compounds at higher dilution rates. In the absence of storage compounds, the model predicts a maximum yield on N&+ of 5.05 g of DW/g. Next, we present the steady-state results when neither glucose nor N&+ is present in the feed at a n excess concentration relative to the other. The reader is again referred to Straight and Ramkrishna (1994)for details concerning how the cybernetic model encompasses both interactive and noninteractive formulations. The maintenance of all substrates but one a t concentrations sufficient to maintain a single limiting substrate may not be economical for large scale processes. Therefore, when more than one complementary substrate is not present in excess, process economics may be improved; however, conditions may occur in which the nutrient supplied by the limiting substrate may vary or there may be simultaneous limitations, e.g., dual limitations. Steady-state results under potentially dual-limited conditions are presented in Figure 7,which compares cell density data and residual NH4+ concentrations to their respective model simulations. Both residual glucose and N&+ levels were measured; however, the residual glucose levels were below the detectability limit of the assay (Straight, 1991). The model predicts that washout of the culture should occur as the dilution rate increases beyond 0.78 h-l. Unfortunately, wall growth prevented the experimental realization of washout. After each experiment, which lasted between 5 and 7 days, a significant film of cells was observed on the glass walls and stainless steel internals of the fermentor, even after they were coated with a silicone surface-treating agent. The effects of wall growth appeared to be most significant when the culture was subjected to dual-limiting conditions. Data from four experiments, with average feed concentrations of 0.98 f 0.03 g/L glucose and 89 f 4 mgfL N&+, are included in Figure 7. If we assume cell yields of 0.49 and 5.0 g of DW/g on glucose and NH4+, respectively, a n upper limit on the NH4+ feed concentration, which is required for nitrogen suficiency, can be calculated to be approximately 95-100 mg/L, assuming a feed concentration of approximately 0.98g/L glucose. Therefore, under the conditions of Figure 7, either glucose,

m+

I

D

0.2

0.4

0.8

Dilution Rate

0.8

-

1.0

(Hr-l)

Figure 7. Dual-limited steady-statedata versus model simulations. Profiles of cell density (0) versus the model simulation (-) and NH4+(0) versus the model simulation (- - -) are shown. Average feed concentrations are 0.98 g/L glucose and 89 mg/L

NH4+.

NH4+, or both may be limiting in a continuous culture. Model feed conditions for the steady-state and transient simulation results are 0.98f 0.07 gfL and 89 f 4 mg/L respectively. Therefore, the variafor glucose and tions in the model glucose and NH4+ feed concentrations are 7% and 4.5%, respectively. The NH4+ feed variation for the model simulations is equivalent to the measured variation in the NH4+feed concentration. The measured variation in glucose feed concentration under duallimiting conditions is approximately 3%; however, for comparative purposes, the variation in the glucose feed concentration for the model simulations was increased to approximately 7%. This increase in glucose feed concentration for the model simulations alters the absolute cell density levels without significantly altering the dynamic response of the model. The data in Figure 7 suggest that the culture is carbonlimited below a dilution rate of 0.4 h-l and nitrogenlimited as the dilution rate exceeds 0.4 h-l. Several observations support these conclusions. First, the cell density a t the lowest dilution rates is lower than the cell density a t the highest dilution rates, indicating the effects of maintenance processes that have been observed when the culture is carbon-limited (Baloo and Ramkrishna, 1991). Second, the high residual NH4+concentration and a n undetectable residual glucose level between D = 0.1 h-l and D = 0.4 h-l also indicate that the culture is carbon-limited in this region. The residual NH4+ concentration decreases to approximately 2-4 mg/L as the dilution rate exceeds 0.4 h-l, which suggests that the culture becomes nitrogen-limited past D = 0.4 h-l. Furthermore, the cell density data are lower than would be expected with a n equivalent glucose feed concentration under carbon-limiting conditions a t the higher dilution rates. The concentration of intracellular glucose, i.e., glycogen, was not measured under the conditions of Figure 7 since glycogen was not expected to be present in significant quantities due to the lack of a large excess of glucose in the medium. Results in the literature (Lee et al.,1984)have indicated that a region of dual-limiting conditions should exist; however, in the absence of good residual glucose data, the results in Figure 7 cannot confirm this observation. The model description of the cell density data in Figure 7 is semiquantitative. While good agreement between the model and the experimental NH4+ data is apparent, the model consistently overpredicts the cell density data

m+,

Biotechnol. Prog,, 1994,Vol. 10, No. 6

598

0' o'e l'=

t8

0.4-

cl

4

0.3

I

2 W

I.

0.2-

i 0.0 0.0

0

0.2

0.4

0.6

0.8

1.0

D i l u t i o n R a t e (Hr-I)

. 0.0

0

1

0.2

0

. 0.4

0.8

Dilution Rate

0.8

1.0

(Hr-I)

Figure 8. Steady-state model profiles of b~ (-), u: (- - -), and u: (.* -1 under dual-limiting conditions. Feed conditions are equivalent t o those in Figure 1.

Figure 9. Steady-stateyield values versus model simulations. Profiles for y b . r i g l ( 0 ) versus the model simulations (-1 and Y b & (0) versus the model simulation (- - -) are shown.

except a t low and high dilution rates. The maximum difference between the model and cell density profiles is approximately 10%. The consistency of the data, however, does indicate that experimental error is not responsible for the model overpredictions, but points to a more fundamental issue, i.e., a variable extinction coefficient. Therefore, the measured extinction coefficient (Straight, 1991) of 0.34 g of DW/OD, appears to be a reasonable representation of the biomass concentration when the culture is sufficiently away from intermediate growth rates, i.e., between D = 0.2 and 0.55 h-' in Figure 7. Within this range of dilution rates it appears that culture conditions are too far removed from the batch conditions under which the extinction coefficient was determined (Straight, 1991). Since the exponential growth rate in batch cultures is high (Straight, 19911, better agreement between the model and the experimental data a t high dilution rates is expected. Although the model does not track the cell density data over the entire range of dilution rates, it does predict quite well the occurrence of a transition at an intermediate dilution rate. With respect to the model cell density profile, a dip in the cell density is predicted to occur a t the point of transition, i.e., D = 0.4 h-l. Also, the NH4+ model profile decreases significantly as the dilution rate is increased above D = 0.4 h-l, before rising again prior to the predicted culture washout. While the data in Figure 7 do not confirm a region of simultaneous limitation by carbon and nitrogen, the proposed transition point appears to fall within the transition region reported by Lee et al. (1984). Below D = 0.3 h-l, their data suggested that the culture was carbon-limited. Above D = 0.45 h-l, their data suggested that the culture was nitrogenlimited. Within the transition region, between D = 0.3 and 0.45 h-l, Lee et al. (1984) concluded that the culture was simultaneously limited by both carbon and nitrogen. With regard to the model, the clearest indication of a transition between carbon- and nitrogen-limited growth is evident in Figure 8, which describes the profiles of uz and u:. When D < 0.4 h-l, = 1while v i 1. As D equals and exceeds 0.4 h-l, the two cybernetic variables flip-flop and u: < 1 while u: = 1. The cybernetic variable identifies the intermediate that limits the specific growth rate. From this result, we propose that the results in Figure 8 indicate that the culture is carbonor glucose-limited when D < 0.4 h-l and nitrogen- or NH4+-limited when D t 0.4 h-l.

Comparisons between cell yield values under duallimiting conditions are plotted in Figure 9, which describes cell yields on glucose and NH4+, respectively. In comparison to the results presented under singly-limiting conditions, the cell yield on glucose under the duallimiting conditions of Figure 7 falls between its corresponding values under carbon- or nitrogen-limiting conditions. A minimum value of approximately 0.37 g of DW/g and a maximum value of approximately 0.435 g of DW/g were calculated from the data in Figure 7. The model overpredicts the cell yield in the intermediate dilution rate range. The cell yields on NH4+ are also presented in Figure 9. Under dual-limiting conditions, the experimental yield on NH4+ appears to be slightly lower than its corresponding value under carbon-limiting conditions. However, the model predicts that the cell yield on NH4+ should increase under dual-limiting conditions when compared to carbon-limiting conditions. The differences between the experimental and the model cell yield values are not surprising since the model overpredicts the cell density data. Therefore, many of the calculated cell yields under dual-limiting conditions will be lower than the model predictions. When more than one complementary substrate is present a t a potentially limiting concentration, the model's ability to respond quickly to a variable environment initially may be in question due to the past failures of the interactive approach to describe experimental growth rate data (Lee et al., 1984; Mankad and Bungay, 1988). However, since both substrates are present at low concentrations, both of the preferred systems for assimilating s , ~and sa are the high-affinity enzymes, Ec,G~,P and E N A ~respectively, , as indicated in Figure 10. Therefore, while the maximum rates of substrate assimilation may be reduced (see Table 31, the high affinity for substrate better prepares the microorganism for low substrate environments. Hence, the steady-state results predict that even when interactive kinetics are utilized (Bader, 1978),transients under dual-limiting conditions should be as short as or shorter than transients following equivalent shifts in dilution rate under singly-limiting conditions. Previous investigators (Lee et al., 1984) who have considered dual-limiting conditions have not reported the levels of established enzyme systems that are responsible for carbon and nitrogen assimilation. Carbon-LimitedTransient Results. Two types of transient conditions were imposed: shift-up and shiftdown experiments in which the dilution rate was rapidly increased or decreased, respectively. The shift in each

UE

UF

Biotechnol. Prog., 1994,Vol. 10, No. 6

599 0.5

i l

.

O

O

~

h

c;i

0.754

I’

i ale ”

_--_____-------

I

1

\

0.501 N

e,

r

.

?

E1 0.4 W

h

3

’/I

.r(

v,

$

0.3

P

0.85

oi‘

\

4 rl

- _- - _- - - - - - _ _ _ _ _ _ _ . ’uoi

0.2 0

1

,

,

,

,

2

3

4

5

,

6

I

,

7

8

Time (Hrs)

Figure 11. Carbon-limited transient data versus model simulations. hofiles of cell density data (0,0 ) versus model simulations after a shift-up in dilution rate from 0.095 to 0.70 h-l (-) and from 0.405 to 0.74 h-I (- - -) at t = 0 h. Feed concentrations are equivalent to those in Figure 2.

than the dilution rate. Once this occurs, the biomass concentration begins to rise while sgl,sc, and snbegin to decrease due to their increased rates of consumption at higher biomass concentrations. As sr exceeds its steadystate value, the specific growth rate exceeds the dilution rate and drives sgl below its steady-state value. The culture cannot support a high growth rate at this lower glucose level and the specific growth rate declines, producing the observed overshoot in cell density prior to the new steady state. As the magnitude of the shift decreases, the length of the lag period also decreases since the magnitude of the resource limitation decreases. The cybernetic model also provides a good description of transient culture behavior following a shift-down in dilution rate under carbon-limiting conditions. Figure 14 compares cell density data to model simulations after a shift-down in the dilution rate. Immediately following the shift-down, the growth rate is greater than the parameter

value 0.001 g/g of DW*h

%CJ

%c,2 ae,n,l

ae.n.1 &,n,z

Qed &,s

ay ay>1 ap2 ay,1

ay,2

a p ae,s

ar Be,g

Be,c,l Be,c,2 Be,n,l

Be,n,2 Beml Be,a

0.001 g/g of DW-h 0.001 g/g of DW-h 0.0001 g/g of DW*h 0.05 g/g of DW*h 0.001 g/g of DW*h 0.001 d g of DW*h 0.015 g/g of DW*h 1x g/g of DW*h 1x g/g of DW-h 1x g/g of DW-h 1 x loT6g/g of DW*h 1 x 10-5g/gofDW*h 1x g/g of DW-h 0.001 g/g of DW-h 0.001 g/g of DW-h 0.005 h-1 0.05 h-1 0.005 h-1 0.02 h-l 0.05 h-l 0.50 h-1 6.0 h-l

parameter B T

Bs fi% max fin,e,2 1;:if max fic,gl,2 max P,,,

fi:: max k g l

&GI

1 : ;

4; Fc;:

Kn,arl Kn,gl,1 Kn,r,l Kll,a,2 Kw12 Kn,r,z Kc,gl,l Ken,! Kc,r,l KC,,l,Z

value

parameter

value

2.5h-’ 0.50 h-l 0.70 g/g of DW.h 1.15 g/g of DW-h 0.63 g/g of DW*h 0.60 g/g of DW-h 0.08 g/g of DW-h 1.30 g/g of DW*h 0.20 g/g of DW*h 0.15 g/g of DW*h 0.70 g/g of DW-h 1.0 gg of DW-h 0.05 g/g of DW-h 3 x 10-5g~ 0.005 g/L 2x g/g of DW 0.05 gL 0.003 g L 0.002 g/g of DW

Kc,n,z Kc,r,2 K,,C Kr,n Kg,c Km Kg,r Km,gl Km1,gl Kd,r Km1,n Kx,gl Ks,gl Ksal KSJ Ks,n Ks,r KS,S

0.003 g/g of DW 0.002 g/g of DW 0.005 g/g of DW 0.005 g/g of DW 0.003 g/g of DW 0.005 gig of DW 0.002 g/g of DW 0.001 g/L 1 x 10-6 g/L 1x g/g of DW 1x g/g of DW 0.10 g/L 0.50 g/L

Ydgl

0.30 g/L 0.0002 g/g of DW 0.002 g/g of DW 0.003 gL

Ydgl

0.49 0.49

Ydn

0.33

0.005 g/L 2x g/g of DW

0.0002 g/g of DW 2x g/g of DW

1.0 g/L

Y&

0.40

YIlAl

0.60

Biotechnol. Prog., 1994, Vol. 10, No. 6

600

\ a

0'5A0 n

I4

\ $

n

0.45W

-0.3 0

m

h

0

s

-0.1

2v1

G

$

0.35-

c1 0.2I: 0

, 1

,

0

2

3

.- - - - -

, c y - ,

4

---------. ,

;I

5

6

7

w .-I

0.0

P)

u

8

Time ( H r s )

0.25

Figure 12. Carbon-limited experimental data versus the model simulations after a shift-up in dilution rate from 0.10 to 0.61 h-' at t = 0 h. Profiles of cell density (M) versus the model versus the model simulation (- - -) simulation (--) and glucose (0) are shown. Feed concentrations are equivalent to those in Figure 2.

(Hrs)

5.0

mu "

v

\

E

4.0

el

0.5 v

3.0

h

$

0.3

2.0 0.4

1

2

3

4

5

6

7

s

s

2 E

W

0 5

;j

E'

b!

I

! 8 1

0

1 .o

I d

I / I I

10

v

m

P

0.2

2 0

5

c

' ,

15

el

i I.

2

20

15

h

lo-'

0.4

10

Figure 14. Carbon-limited cell density data (0)versus the model simulation (-1 after a shift-down in dilution rate from 0.72 to 0.26 h-l at t = 0 h. Feed concentrationsare equivalent to those in Figure 2. 0.6

t rl

5

Time

loo

0.5

e

0

U 10-3

Time (Hrs)

Figure 13. Transient model profiles of b~ (-1, s,l (- -1, sr (- - -1, sc (. *.), and sn(- - -) after a shift-upin dilution rate from 0.095 to 0.70 h-l under carbon-limiting conditions. Feed conditions are equivalent to those in Figure 2.

dilution rate and the culture rapidly consumes the residual glucose. Subsequently, the cell density rises. A reduced resource level, since ur decreases with decreasing p , coupled to the low glucose level forces the specific growth rate and the cell density to decrease, producing the observed overshoot and a slow decline in cell density. At the lower specific growth rate, resources are distributed to the low-maintenance enzyme system, Le., u m and um increase with decreasing p. Since maintenance rates are relatively low, sgl does not exhibit a significant undershoot; therefore, no undershoot in cell density is predicted or observed as the final steady state is attained. Nitrogen-LimitedTransient Results. As for carbonlimited cultures, a series of transient experiments was also investigated under nitrogen-limiting conditions. Under carbon-limiting conditions, the observed growth dynamics following a shift-up in dilution rate was primarily attributed to the unavailability of resources, which prevented the culture from immediately growing a t a rate equivalent to the postshift dilution rate. Under nitrogen-limiting conditions, the resource SRis not the primary factor determining growth dynamics after a shift-up in dilution rate. Figure 15 compares experimental data to model simulations after a shift-up in dilution rate under nitrogen-limiting conditions. Straight (1991)

0.3

0.0

0

Time (Hrs)

Figure 15. Nitrogen-limitedexperimental data versus model simulations after a shift-up in dilution rate from 0.11 to 0.68 h-1 at t = 0 h. Profiles of cell density (m)versus the model versus the model simulation (- - -), simulation (-1, glucose (0) and N€&+(A) versus the model simulation * .) are shown. Feed concentrations are equivalent to those in Figure 4.

provides more such simulations to compare favorably with experimental data or shiftups in dilution rate. Profiles for cell density, residual glucose, and residual NH4+ are plotted. The respective model simulations quantitatively describe all three profiles when available. The significance of the high-affinity system E N , Ais ,~ immediately apparent in Figure 16. Since is high even a t a dilution rate of 0.11 h-l, the synthesis of SNis maintained a t a high rate and sn rises quickly immediately following the shift. Likewise, since the lowaffinity system, E c , G ~has , ~ , a rate of glucose conversion to SCthat is higher than that of the high-affinity system, s, is higher under nitrogen-limiting conditions. The maintenance of high intermediate levels is further promoted by the actions of the cybernetic variables uf and ur. Since these variables strive to maintain balanced levels of SC and SN, a high level of one intermediate can push the synthesis of the other upward as well, even if existing levels are already high. Therefore, additional synthesis of SC is maintained during the transient by high levels of SN. The presence of high intermediate concentrations maintains a constant increase in sruntil it accumulates to a level that supports an increasing

Biotechnol, Prog., 1994,Vol. 10, No. 6

601 0.3

0.55

... ...

n

d

\

I I

mc

.....,

cl

. ..

k

t

'\\

..

'_.-.-,-,-,-.-.-.-.-.-.-r..-.-.-,.

0.45

.

.. .

n

el

0.2

v

A

$

-

0.4

m

0.35

0.1

2

,

,

3

4

Time

,

5

, 6

,

7

,

8

/

1o - ~

s

g

+.

20

'd'

,

,

,

,

3

1

2

3

'a.-

4

5

---------L8

7

8

0.0

0

9

Time (Hrs)

(Hrs)

-

Figure 16. Transient model profiles of bT (-1, sr(- - -), sc (. -1, and s, (- - -) after a shift-up in dilution rate from 0.11 to 0.68

h-l under nitrogen-limiting conditions. Feed conditions are equivalent to those in Figure 4. 6.0

0.6

.:..............

2

3

aE W

0

0.25

,

40

0

P

,' , 0 1

M

0

d

0.31

\ W

4 .* m

v

80

Figure 18. Dual-limited experimental data versus model

simulations after a shift-up in dilution rate from 0.105 to 0.66 h-l at t = 0 h. Profiles of cell density (m) versus the model simulation (-), glucose ( 0 )versus the model simulation (. *), and NH4+ (A) versus the model simulation (- - -) are shown. Average feed concentrations are 0.98 g/L glucose and 89 mg/L NH4+.

-

n

el

\ $

h

fl

4

bo 0 . 5

4.0

\

u

W

v

h

Q,

3

II)

.d

0

v)

$

0.4

fl w -f, 0,

0

i

2.0

1

z

V

0.0

0.3 0

4

8

12

16

2C

Time (Hrs)

Figure 17. Nitrogen-limitedexperimental data versus model

simulationsaRer a shift-downin dilution rate from 0.68 to 0.145 h-l at t = 0 h. Profiles of cell density (W) versus the model simulation (-) and glucose (0)versus the model simulation(- - -1 are shown. Feed conditions are equivalent to those in Figure 4.

growth rate. Subsequently, srand the key enzymes reach their peak values more quickly when compared to results under carbon-limiting conditions. Therefore, in the presence of EN,A,~, E c , G ~and , ~ , coupling by metabolic regulation, transient periods are reduced when compared to transients of equivalent shift-ups under carbon-limiting conditions. During transients following a shift-down in dilution rate, growth dynamics under nitrogen-limiting conditions is primarily determined by the synthesis of storage compounds in much the same way that maintenance processes domainate the cellular response after a shiftdown under carbon-limiting conditions. Figure 17 compares cell density and glucose profiles to their respective model simulations after a shift-down in dilution rate under nitrogen-limiting conditions. Quantitative agreement between the experimental data and the model simulations is evident from the figures. Straight (1991) presents more such demonstrations. Residual NH4+ levels were undetectable during the transient periods. As under carbon-limiting conditions, resources are allocated to the synthesis of enzymes for maintenance and the synthesis of storage compounds. However, since ux,gl = 1under the conditions of Figure 17, storage compounds are produced rapidly due to a fully active enzyme system.

Therefore, the cell density rises rather than declines. Following the shift-down, sa decreases rapidly to an extremely low level due to its consumption by growth processes as well as its reduced availability at the lower dilution rate. Subsequently, sn decreases rapidly. The culture cannot maintain a high specific growth rate when sn is low, and the growth rate declines allowing snto begin to rise. As sn increases, the resource level rises slowly and the specific growth begins to increase to its final steady-state. Subsequently, u m declines producing an overshoot in es,gland ss. Straight (1991) also presents the dynamics of the cybernetic variables up and v i during the shift-down process, providing a complete explanation of transient behavior. Carbon-and Nitrogen-LimitedTransient Results. To complete the present investigation, transient experiments under dual-limiting conditions were also performed. Figure 18 compares experimental data to model simulations after a shift from 0.105 to 0.66 h-l under dual-limiting conditions. Average feed conditions are equivalent to those in Figure 20, Le., 0.98 giL glucose and 89 mg/L NH4+. Profiles for cell density, residual glucose, and residual N&+ are plotted in Figure 18. The model quantitatively describes both the cell density and the residual NH4+ profiles. However, comparison of the experimental and model profiles for residual glucose indicates that the model overpredicts the experimental data in the same manner as under carbon-limiting conditions. The magnitude of the overprediction, however, is a little larger under dual-limiting conditions. Comparison of the experimental results in Figure 18 to results following similar increases in dilution rate under carbon-limiting conditions indicates that the response time following a shift-up under dual-limiting conditions is as fast as or faster than the response time under carbon-limiting conditions for similar size shifts, which agrees with the proposal made on the basis of the steadystate results above. The dynamic behavior following a shift-up under duallimiting conditions is explained through Figures 19-21, which plot profiles for several of the model components. As when the culture is carbon-limited, the resource and enzyme levels a t the time of the shift are not sufficient to support a specific growth rate equivalent to the postshift dilution rate; therefore, the culture initially washes out of the fermentor. However, under duallimiting conditions, Figure 10 indicates that both of the

Biotechnol. Prog., 1994, Vol. 10, No. 6

602

of resource synthesis, as compared to carbon-limiting conditions. 5 The biomass level begins to rise once again """1 0 ' 5 after the resource concentration reaches a level that can /--. i

.."'.\ .

I

\. .

r

\'-. .............

:

I I

lo-'

cl

a

.

Y

"

lo-*

r lo-=

h

a el

Y

- _ _ - - _ _ . .......................

.I-

t 0

1

2

3

4

5

6

7

8

mu

9

Time (Hrs)

Figure 20. Transient model profiles of b~ (-), sc (- - -1, S, (- - -), and en,a,l(- -1 after a shift-up in dilution rate from 0.105 to 0.66 h-l under dual-limitingconditions. Feed conditions are equivalent to those in Figure 5. ec,gl,2(.*.),

0.55

0.45

I I

k

1.5

v_ _ _ - ,,,.... --

v

...............

1.0

---------________

'' 0.35

*

'. 0.25

7

0

uc

.

oo

>

.

. ,

1

1

I

,

2

3

4

,

,

,

,

,

5

6

7

8

8

0.5

Time ( H r s )

Figure 21. Transient model profiles of b~ (-1, uz (- - -1, and uf, (..*) after a shift-up in dilution rate from 0.105 to 0.66 h-l under dual-limiting conditions. Feed conditions are equivalent to those in Figure 5.

preferred mechanisms for synthesizing SC and SN are predicted to be the high-afEnity systems. This preference by the microorganism results in a high transient level of sn,which produces a high rate of resource synthesis for the first 2 h after the shift-up. Therefore, even though there is potentially a lower capacity for resource synthesis under dual-limiting conditions, the outcome of regulatory processes ultimately results in a higher transient rate

support a specific growth rate higher than the postshift dilution rate. Model profiles for vz and v i indicate that a change in the limiting pathway occurs (Figure 21) during the transient following the shift-up in dilution rate, i.e., v: = 1. Since sa is much decreased from its corresponding value under carbon-limiting conditions, the culture cannot maintain the high transient s, level. Subsequently, snpeaks early and decreases to a steadystate value that is lower than the corresponding value under carbon-limiting conditions. Due to the decreased sn level, the magnitude of the overshoot in sr is also reduced, and the specific growth rate begins to decrease much earlier than it does when the culture is carbonlimited. The decreased overshoot in specific growth rate greatly reduces the undershoot in glucose; therefore, little or no overshoot in cell density is predicted to occur prior to the final steady state. However, some overshoot is evident from the cell density data in Figure 18. Perhaps the most striking difference between the shiftup results under carbon-limiting and dual-limiting conditions is the rate of increase in cell density following the initial undershoot in cell density. Since the preferred SG~ and SA assimilation processes are the high-affinity processes, the rate of increase in cell density under duallimiting conditions is slower than that observed under carbon-limiting conditions due to the lower maximum rates supported by the highdaffnity systems. Therefore, the overshoots in cell density that precede the final steady state are less sharp under dual-limiting conditions. A reduced rate of increase in biomass concentration is evident in both the experimental and model profiles (Figure 18). Even tllough the rate of increase in cell density is decreased under dual-limiting conditions, the final steady-state value is also lower in magnitude compared to carbon-limiting conditions; therefore, the transient period after a shift-up under dual-limiting conditions is comparable to or shorter than the transient period after an equivalent shift-up under carbon-limiting conditions. Straight (1991) compared several additional shift-up experiments t o model simulation in his doctoral thesis. In addition to biomass, both residual glucose and NH4+ concentrations have been presented for several of the experiments. The agreement between the model and the experimental results remains semiquantitative. In general, as the size of the shift-up decreased, the ability of the culture to respond faster is evident from the cell density data. As under carbon-limiting conditions, the additional requirement for resource at the postshift dilution rate decreases as the size of the shift decreases; therefore, the culture exhibits little or no washout. However, the model continues to predict some washout of the culture following the increase in dilution rate. Comparison of the dynamics exhibited by a carbonlimited and a dual-limited culture after a shift-down in dilution rate reveals many similarities as well as some differences. Figure 22 compares experimental cell density and NH4+ profiles to their respective model simulations after a shift in dilution rate from 0.655 to 0.145 h-l. Average feed conditions are quivalent to those in , sa, sr,and p are plotted Figure 7. Model profiles for b ~sgl, in Figure 23, while the profiles for ec,g1,2,en,a,i,sc, and Sn are plotted in Figure 24. The model results indicate that, as in carbon-limited cultures, maintenance plays a significant role following the shift-down in dilution rate, i.e., ug decreases and u m increases, resulting in a reallocation of resources from eg,cnto e,,,l,gl; therefore, eg,cn

Biotechnol. Prog., 1994,Vol. 10, No. 6

603

to carbon-limited conditions during the shift-down tran-

n cl

sient (results not shown). 0'55j4 Since the model predicts a nitrogen limitation a t D = O'-

n

cl

0.451

3

v

E

h

4J

-20 w

I*

+ *

I

0.25

, 5

0

15

10

-

10

, I

0

x z

20

Time (Hrs)

Figure 22. Dual-limited experimental data versus model simulations after a shift-down in dilution rate from 0.655 to 0.145h-' at t = 0 h. Profiles of cell density (W) versus the model

simulation (-) and NH4+ (A) versus the model simulation (- - -1 are shown. Average feed concentrations are 0.98 g/Lglucose and 89 mg/L NH4+. lo-'

l'ool

0.75

"

0'59

II

L

h

lo-a

\Mcl Y

0.50

a'

0'45

0.35

lo-=

4

0'251

\M

............................

W

0.25

0.0

"

105

0

15

10

2

Time ( H r s ) sgl (- - -), sa sr (- - -), and p (. after a shifl-downin dilution rate from 0.655to 0.145h-l under dual-limiting conditions. Feed conditions are equivalent to those in Figure 18.

Figure 23. Transient model profiles of b~ (-), (- -),

.e)

--

0.55

1 oo

I ....... .....................

A

0.45

I

i

i I

01-

-

10-l

I

~-0 1)5 2.0

j

I

I

!'

0

5

10

15

20

Time (Hrs) Figure 24. Transient model profiles of bT (-), sc (- - -), sn (- -1, ec,gl,z(.* and en,a,l(- -) after a shift-downin dilution rate from 0.655to 0.145h-' under dual-limiting conditions. Feed conditions are equivalent to those in Figure 18.

-

e),

decreases while eml,glincreases. Likewise, u m increases and ur decreases a s the specific growth rate declines below the postshift dilution rate. In contrast to Figure 21, the model predicts a transition from nitrogen-limited

0.655 h-l, i.e., uz < 1 and u i = 1, sc is higher than its corresponding value under carbon-limiting conditions; therefore, sgl is also higher as glucose utilization is restricted by u:. Furthermore, the preference for the high-affinity system en,a,lmaintains sna t a level equivalent to its corresponding value under carbon-limiting conditions when sa is much higher. Since sc and sn are higher than or equal to carbon-limited values, the reduction in specific growth rate after the shift-down is less than predicted for equivalent carbon-limited conditions. Therefore, the period of declining cell density following the initial overshoot in cell density is predicted to occur a t a slower rate under dual-limiting conditions, with no undershoot in cell density occurring prior to attaining the final steady state. Comparison of the model and cell density profiles in Figure 22, however, indicates that the predicted decline in cell density is slower than the observed rate. A slower rate of decline should indicate a longer transient period; however, due to the effects of overflow metabolism, i.e., ux,gl = 1at D = 0.655 h-l, under nitrogen-limiting conditions the total change in cell density after the shift-down is less under dual-limiting conditions. Therefore, the transient period is approximately equal to the transient period following a n equivalent shift-down under carbon-limitingconditions. Straight (1991) has presented two additional shift-down experiments and compared them to model simulations of biomass levels and the residual NH4+ concentration. Residual glucose levels were undetectable during the shift-downs. The agreement between the experimental results and the model simulations were comparable to the agreement exhibited in Figure 22.

Conclusions In contrast to the standard unstructured approach, the cybernetic framework has now resolved several of the problems associated with describing the utilization of complementary substrates in microbial systems under steady-state and transient conditions. The chosen perspective, one of metabolic pathways, identifies biosynthetic intermediates as the common elements of both complementary and substitutable substrates and permits the derivation of appropriate regulatory strategies for complementary and substitutable processes. The earlier work on the regulation of substitutable substrates (Baloo and Ramkrishna, 1991; Kompala et al., 1986; Turner et al., 1989) is a n important aspect to consider. However, an ability to predict microbial growth when limiting substrates are complementary in nature is important in all systems, regardless of whether they are microbial in nature or of a more complex nature, such as plant and animal systems. Comparison between the model and experimental results exhibits quantitative agreement in the majority of cases. The largest discrepancy between the model and experimental profiles is apparent in the results for the residual glucose profile during a carbon-limited shift-up transient. The most probable cause of this discrepancy can be attributed to uncoupled oxidation of the carbod energy source during the sampling procedure (Rutgers et al., 1990). The experimental results presented within this paper are only a representative sample. Parameter estimation for these results and others was accomplished, in part, by choosing a data set, e.g., a carbon-limited shiftup experiment, to determine a small subset of critical parameters. The remaining data sets, e.g., the remaining

Biotechnol. Prog., 1994, Vol. 10, No. 6

604

carbon-limited shift-up results, were used to check the quality of the resulting parameter values; therefore, the values given in Table 3 are not a result of a global optimization scheme, but rather, they are identified from specific steady-state or transient results in a systematic manner. Each new system of interest requires identification of the appropriate kinetic mechanisms in order to describe the observed experimental behavior. Since the range of conditions examined within this paper is very diverse, the number of significant processes is correspondingly large. However, the proposed regulatory structure is independent of the system and derived from the pertinent process kinetics and concentration variables (Straight and Ramkrishna, 1994). Therefore, while the kinetic detail may be significant, the simplicity of the regulatory structure greatly reduces the overall complexity of the model in comparison to alternative structured models proposed to describe similar conditions (Shuler and Domach, 1983; Domach et al., 1984). A good example of the simple representation of optimal control is found in the actions of the control variables for complementary processes. The variables and uF act to maintain both SNand SC a t high levels during transient periods by switching the preferred pathway, i.e., = 1, back and forth between the two intermediates, a s demonstrated in Figure 20. Therefore, under the conditions of the transient described by Figure 17, the presence of excess SN results in the activation of EC,GI,~ to produce Sc, a s well as the inhibition of E N A , ~This . phenomenon has been referred to as transactivation by Stephanopoulos and Vallino (1991). Incorporation of feedback regulation from a purely kinetic mechanism would produce a slower and noninteractive response. Furthermore, the rate of intermediate synthesis only decreases as the product concentration rises when feedback regulation is derived from a kinetic mechanism. In contrast, the cybernetic formulation can promote higher rates of intermediate synthesis, even when both SC and SNare present in high concentrations. The present results therefore establish the cybernetic framework again as a viable alternative to purely kinetic models and suggests alternative process strategies for maximizing product synthesis under complementary limitations. Culturing microorganisms under dual-limited conditions offers the possibility of improving process economics by minimizing the amount of unused substrates within the medium. The presence of two substrates a t nonsaturating concentrations also does not increase the length of transient periods following shifts in dilution rate. The availability of multiple mechanisms for substrate assimilation and their regulation enables the culture and the model, even when it utilizes interactive kinetic forms, to respond a s fast a s or faster than when the fermentor is operated under singly-limiting conditions. Furthermore, the potential improvement of product output by utilizing operating strategies that exploit the advantages of different limitations involving complementary substrates is also very much apparent. Present and previous model results demonstrate that both of the above applications can now be addressed by models that account for the processes of metabolic regulation according to the principles of the cybernetic framework. Even though the agreement between the experimental results and the model simulations is quantitative in the majority of cases, two experimental problems must be addressed in future endeavors to investigate bacterial growth on limiting complementary substrates. First, the overprediction of the cell density data by the

UF

UF

model a t intermediate growth rates appears to be due to a highly variable cell morphology and connected to the use of optical density measurements, which are utilized to estimate biomass concentrations. Lee et al. (1984) have confirmed this phenomenon by measuring the variation in cell volume as a function of dilution rate in steady-state continuous cultures of Escherichia coli B/r. Under singly-limiting conditions, i.e., either carbon- or nitrogen-limited, the cell volume increased by 25-40% as the dilution rate was increased from 0.2 to 0.8 h-l. In comparison to these results, they reported variations in cell volume, which increased by greater than 200% over the same range of dilution rates under dual-limiting conditions. Such a large variation in cell size under duallimiting conditions suggests that the cell density cannot be quantitatively described by a single constant correlation coefficient over the entire range of dilution rates (Straight, 1991). Clearly, cell morphology should be a factor to consider when future work is performed with multiply-limited microbial cultures involving complementary substrates. One potential solution is to directly measure a macromolecular component of the biomass, e.g., protein, rather than to measure the biomass concentration indirectly, e.g., optical methods. Second, the absence of measurable residual glucose levels under steady-state conditions in Figure 7 and the overprediction of the glucose profile by the model under carbon-limiting transient conditions in Figure 18 suggest that the culture sampling technique needs improvement. The time spent during culture sampling under carbonor dual-limiting conditions is critical for good residual glucose data. Rutgers et al. (1990) investigated singlyand dual-limited cultures of Klebsiella pneumoniae that were limited by glucose and/or NH4+. Their results indicate that culture sampling times, which include stopping cellular metabolism, should be reduced to times on the order of 0.1 s. Otherwise, carbon oxidation, which is uncoupled from growth processes, can significantly reduce the residual glucose level. Typical sampling times for the data presented within this paper, which included culture removal, centrifugation, and freezing, were completed within 4-5 min (Straight, 1991); therefore, it is not surprising that residual glucose levels were undetectable or overpredicted under the conditions we concluded to be carbon- or dual-limited. Regardless of these two experimental problems, the utilization of limiting complementary substrates by bacterial cultures can now be described or predicted within a single unified approach when the culture is subjected to either singly- or dual-limiting conditions involving complementary substrates. Furthermore, steady-state and transient continuous cultures are within the scope of the cybernetic framework whether the limiting substrates are complementary or substitutable (Daigger and Grady, 1982) in nature. Multiple models are not required to describe microbial growth, which is subject to multiply-limiting conditions. Because the models in this paper are able to account for transient effects, they are important in the control of bioreactors. They also have potential applications in the modeling of animal cell cultures because media for their growth are expensive and it is not possible to arrange for a single limiting nutrient. Other applications include modeling of microbial processes such as bioremediation. The evolution of effective biodegradation strategies must seek the quantitative guidance of models that account for such effects as those of growth intermediates and other cometabolites on the uptake of pollutants. The models presented here are attractive candidates for the same.

Biotechnol. Prog., 1994,Vol. 10, No. 6

Acknowledgment Support from the National Science Foundation through Grants CBT-8609293 and CBT-8813170 is gratefully acknowledged. We also thank Professor Alan Konopka for all of his help and suggestions in the completion of this work.

Literature Cited Bader, F. G. Analysis of Double-Substrate Limited Growth. Biotechnol. Bioeng. 1978,20, 183-202. Baloo, S.; Ramkrishna, D. Metabolic Regulation in Bacterial Continuous Cultures-I. Biotechnol.Bweng. 1991,38,13371352. Baltzis, B. C.; Fredrickson, A. G. Limitation of Growth Rate by Two Complementary Nutrients: Some Elementary but Neglected Considerations. Biotechnol. Bioeng. 1988, 31, 7586. Cooney, C. L.; Wang, D. I. C.; Mateles, R. I. Growth of Enterobacter aerogenes in a Chemostat with Double Nutrient Limitations. Appl. Environ. Microbiol. 1976, 31, 91-98. Daigger, G. T.; Grady, C. P. L. An Assessment of the Role of Physiological Adaptation in the Transient Response of Bacter i d Cultures. Biotechnol. Bioeng. 1982,24, 1427-1444. Dawes, E. A,; Senior, P. J. Energy Reserve Polymers in Microorganisms. Adu. Microbiol. Physiol. 1973,10,135-266. Domach, M. M.; Leung, S. K.; Cahn, R. E.; Cocks, G. G.; Shuler, M. L. Computer Model for Glucose-Limited Growth of a Single Cell of Escherichia coli B/r-1. Biotechnol. Bioeng. 1984,26, 203-216. Fredrickson, A. G. Formulation of Structured Growth Models. Biotechnol. Bioeng. 1976, 18, 1481-1486. Herbert, D.; Phipps, J. P.; Tempest, D. W. The Chemostat: Design and Instrumentation. Lab. Pract. 1965,14, 1150. Holme, T. Continuous Culture Studies on Glycogen Synthesis in Escherichia coli B. Acta Chem. Scand. 1957, 11, 763775. Ingraham, J. L.; Maaloe, 0.;Neidhardt, F. C. In In Growth of the Bacterial Cell; Sinauer Associates, Inc.: Sunderland, MA, 1983. Kompala, D. S.; Ramkrishna, D.; Jansen, N. B.; Tsao, G. T. Investigation of Bacterial Growth on Mixed Substrates: Experimental Evaluation of Cybernetic Models. Biotechnol. Bioeng. 1986,28, 1044-1055. Lee, A. L.; Ataai, M. M.; Shuler, M. L. Double-Substrate-Limited Growth of Escherichia coli. Biotechnol. Bioeng. 1984, 26, 1398-1401. Magasanik, B. Genetic Control of Nitrogen Assimilation in Bacteria. Annu. Rev. Genet. 1982, 16, 135-168. Mandelstam, J. The Free Amino Acids in Growing and NonGrowing Populations of Escherichia coli. Biochem. J . 1958, 69, 103-110. Mandelstam, J. Protein Turnover and Its Function in the Economy of the Cell. Ann. N.Y. Acad. Sci. 1963,102, 621636. Mandelstam, J.; McQuillen, K. In Biochemistry of Bacterial Growth; John Wiley & Sons: New York, 1993. Mankad, T.; Bungay, H. R. Model for Microbial Growth with More than One Limiting Nutrient. J. Biotechnol. 1988, 7, 161-166. Miller, R. E.; Stadtman, E. R. Glutamate Synthase from Escherichia coli. J. Biol. Chem. 1972,247, 7407-7419. Neijssel, 0.M.; Tempest, D. W. The Regulation of Carbohydrate Metabolism in the Growth ofKlebsiella aerogenes NCTC 418 Organisms, Growing in Chemostat Culture. Arch. Microbiol. 1975,106, 251-258. Neijssel, 0. M.; Tempest, D. W. The Role of Energy-Spilling Reactions in the Growth of Klebsiella aerogenes NCTC 418 in Aerobic Chemostat Culture. Arch. Microbiol. 1976a,110, 305-311.

605 Neijssel, 0. M.; Tempest, D. W. Bioenergetic Aspects of Aerobic Growth of Klebsiella aerogenes NCTC 418 in Carbon-Limited and Carbon-Sacient Chemostat Culture. Arch. Microbiol. 1976b, 110, 305-311. Neijssel, 0. M.; Tempest, D. W. The Physiology of Metabolite Over-Production. SOC. Gen. Microbiol. 1979,29, 53-81. Pavlou, S.; Fredrickson, A. G. Growth of Microbial Populations in Nonminimal Media: Some Considerations for Modeling. Biotechnol. Bioeng. 1989, 34, 971-989. Pine, M. J. Turnover of Intracellular Proteins. Arch. Microbiol. 1972,26,103-126. Pirt, S. J. Maintenance Energy: A General Model for EnergyLimited and Energy-Sufficient Growth. Arch. Microbiol. 1982,133, 300-302. Preiss, J. Chemistry and Metabolism of Intracellular Reserves. In Bacteria in Nature; Leadbetter, E. R., Ed.; Plenum Press: New York, 1989; Vol. 3, pp 189-258. Rutgers, M.; Balk, P. A.; van Dam, K. Qualification of MultipleSubstrate Controlled Growth-Simultaneous Ammonium and Glucose Limitation in Chemostat Cultures of Klebsiella pneumoniae. Arch. Microbiol. 1990, 153, 478-484. Seifter, S.; Dayton, S.;Novic, B.; Muntwyler, E. The Estimation of Glycogen with the Anthrone Reagent. Arch. Biochem. 1950,25,191-200. Senior, P. J. Regulation of Nitrogen Metabolism in Escherichia coli and Klebsiella aerogenes: Studies with the ContinuousCulture Technique. J. Bacteriol. 1975, 123, 407. Shuler, M. L.; Domach, M. M. Mathematical Models of the Growth of Individual Cells. In Foundations of Biochemical Engineering: Kinetics and Thermodynamics in Biological Systems; Stephanopoulos, G., Ed.; American Chemical Society: Washington, D.C., 1983; pp 93-133. Sigal, N.; Cattaneo, J.; Segel, I. H. Glycogen Accumulation by Wild-Type and Uridine Diphosphate Glucose F’yrophosphorylase-Negative Strains of Escherichia coli. Arch. Biochem. Biophys. 1964,108, 440-451. Stephanopoulos, G.; Vallino, J. J. Network Rigidity and Metabolic Engineering in Metabolite Overproduction. Science 1991,252, 1675-1681. Straight, J. V. Microbial Growth in Continuous Cultures Subject t o Single and Multiple Limitations Involving Carbon and/or Nitrogen. Ph.D. Thesis, Purdue University, West Lafayette, IN, 1991. Straight, J. V.; Ramkrishna, D. Cybernetic Modeling and Regulation of Metabolic Pathways. Growth on Complementary Nutrients. Biotechnol. Prog. 1994, 10, 574-587. Tempest, D. W.; Neijssel, 0. M.; Zevenboom, W. Properties and Performance of Microorganisms in Laboratory Culture; Their Relevance to Growth in Natural Ecosystems. SOC.Gen. Microbiol. 1983,34, 119-152. Tsai, S. P.; Lee, Y. H. A Model for Energy-Sufficient Culture Growth. Biotechnol. Bioeng. 1990,35, 138-145. Turner, B. G.; Ramkrishna, D.; Jansen, N. B. Cybernetic Modeling of Bacterial Cultures at Low Growth Rates: SingleSubstrate Systems. Biotechnol. Bioeng. 1989,34, 252-261. Tyler, B. Regulation of the Assimilation of Nitrogen Compounds. Annu. Rev. Biochem. 1978,47, 1127-1162. Wanner, B. L. Phosphate Metabolism and Cellular Regulation in Microorganisms; Yagil, E., Ed.; American Society for Microbiology: Washington, D.C., 1987; p 12. Wilkinson, J. F. The Problem of Energy-Storage Storage Compounds in Bacteria. Exp. Cell Res., Suppl. 1959,7,111-130. Accepted July 6, 1994.@ Abstract published in Advance ACS Abstracts, August 15, 1994. @