majewski opt acetate

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Simple Constrained-Optimization View ofAcetate Overflow in E. cob

R. A. Majewski and M .M. DomachDepartment of Chem ical Engineering, Carnegie Me llon University,Pittsburgh, Pennsylvania 15213Accepted for publication Septem ber 1, 1989

The production of acetate by aerobically growing E. coliis examined. The problem is formulated in terms of aflow network that has a s its objective maximal ATP syn-thesis. It is found that when loads are imposed and fluxconstraints exist either at the level of NADH turnoverrate or the activity of a key Krebs cycle enzyme, switch-ing to acetate overflow is predicted. Moreover, the re-sult found for tbe latter constraint can be shown to beformally equivalent to a correlation experimentally de-termined for the specific rate of acetate production byE. coli K-12.INTRODUCTIONThe aerobic production of acetate by E. coli typically oc-curs at rapid growth rates and commences after a criticalgrowth rate has been exceeded. Acetate production is im-portant to biochemical processing for several reasons.First, it is a symptom of a change in cellular physiologicalstate. Acetate production is an overflow phenomena whereacetylCoA is diverted from the Krebs cycle to first acetyl-phosphate, and then acetate which results in the productionof one substrate-level ATP per mole of acetate. Secondly,acetate overflow in E. coli resembles aerobic ethanol pro-duction by yeast in that the overflow occurs at high growthrates when glucose is metabolized. Therefore, acetate andethanol overflow may both arise due to similar regulatorypolicies and constraints. Finally, the accumulation of ace-tate in the growth medium has been suggested to havedeleterious effects on the expression of products by recom-binant E. coli. Consequently, optimal feeding profiles forfed-batch reactor operation have been sought, as well asmethods for analyzing reactor data so that control strate-gies can be im~le rnented.~- ~In an attempt to progress towards developing a rationalefor why acetate secretion occurs, we have examined over-flow metabolism from the capacitated flow network view-point. This viewpoint focuses on the routing of metabolicflows (fluxes) through a connected network where loadsexist. Capacitation refers to the existence of constraintssuch as each network segment (i.e., reaction process) has afinite capacity or the total flow through the network has an

* To whom all correspondence should be addressed.

Biotechnology and Bioengineering, Vol. 35,Pp.732-738 (1990)0 1990 John Wiley 81Sons, Inc.

upper limit. The term load means that some metabolitesare drained from the cycle at a rate dependent on growthrate. For this problem, the use of this viewpoint is straight-forward and consists of hypothesizing that an objective ex-ists and then determining whether acetate overflow occursand what the overflow pattern is when plausible constraintsare imposed.The introduction of an objective and constraints into thisanalysis constitutes an application of the cybernetic per-spective which has been fruitfully applied to other regula-tion problems (e.g., refs. 6 ,7 ) . In general, formulating aproblem in terms of an objective and constraints provides aphysiological rationale (network objective) and accountsfor the phenomenological impact of the mechanistic details(network constraints). Thus, one can envision that networkanalysis could be a useful tool for screening hypotheses onnetwork regulation and also serve as a first step in the for-mulation of a deterministic model. The following describesthe objective chosen, the constraints examined, and thepredicted ovefflow results.

NETWORK STRUCTURE AND OBJECTIVENetwork StructureThe Krebs cycle reactions, load fluxes from the Krebscycle, and acetate-producing and triose-processing reac-tions used by E. coli are shown in Figure 1. The networkshown is similar to that used by Walsh and Koshland torepresent important fluxes in order to derive the mass bal-ance equations required to analyze the data they obtainedfrom experiments with radio-labelled substrates. In addi-tion to fluxes, sites of substrate-level phosphorylation andthe production of reducing equivalents (NADH and FADH)are shown.

Network ObjectiveFirst, we formulate the objective. We assume that maxi-mizing ATP and GTP production is the global objectivethat must be realized by the network shown in Figure 1.

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Figure 1 . Krebs cycle with ovefflow to acetate. The conversion of triose to phosphoenolpyruvate (PE P),pyruvate (P YR), acetylCoA (AcCoA ), citrate (CIT), iswitrate (ICIT), a-ketoglutarate (a-KG), uccinylCoA(SuccCo A), succinate (S UCC ), furnarate (FUM), malate (MAL), oxaloacetate (OAA), and the load on theAcCoA, a - KG, PYR, and O AA pools are shown. Sites of substrate level phosphoIylation which results ineither ATP or GTP formation are indicated a s are sites of reduction equivalent production (-).

For a given triose flux ( F ) , he combined ATP and GTPflux arising from substrate-level phosphorylation (Y,) isYsL = F + ( F - c) + d + ( F - c - d - a - e - f )

= 3F - 2~ - u - e - f ( 1 )The ATP flux from oxidative level phosphorylation (Y or )can likewise be determined as

= 9 F - 7~ - 5d - 5a - 5e - 7 f (2 )where we habe assumed that the P/O ratio is 2 for NADHreducing equivalents and 1 fo r e lec t rons der ived f romFADH. Thus. the total ATP and GTP flux ( Y ) generatedfor a given F is

(3)= 12F - 9c - 5d - 6 a - 6 e - 8fSimplification of Network ObjectiveIn eq. (3 ) . not all the fluxes are independent. Either formalindependence is the case or biochemical intuition and ex-perimental observations suggest that flux-couplings exist.By interrelating the fluxes that are not independent and re-lating the loads (e.g., fluxes a , e , a n d f ) to growth rate,eq. ( 3 ) an be simplified. For example, in E . coli the ratioof net oxaioacetate flux (b) o net a-ketoglutarate flux (a)varies little from 1.6 based on Holms analysis and Walshand Koshlands work. (A similar value can be found by

using the average amino acid profile of E . coli protein andassuming that the major use of a-ketoglutarate and ox-aloacetate skeletons is for amino acid biosynthesis.) Flux ashould also increase as growth rate increases due to the as-similation rate of ammonium ion (catalyzed by glutamatedehydrogenase which consumes a-ketoglutarate) and theconsumption rate of glutamate for biosynthesis both in-creasing. If one assumes that f lux a i s coupled to thegrowth rate, and cell yield on ATP (Yx ,A p) oes not varygreatly, then based on Holms analysis of metabolic fluxesin E . coli (flux a = 1 mmol/g h for p = 0.94 h- andY,.,,, = 10 g/mol where p is the specific growth rate),0.0106 can be found to be the proportionalitj factor be-tween Y and flux a . The net biosynthetic use of acetylCoAshould also increase as growth rate increases due to it be-ing a p recursor in membrane syn thesis . Again , us ingHolms results, one can estimate that e = 0.0258Y (flux ofAcCoA = 2.48 mmol/h g for p = 0.94 h- ) Similarly,the proportionali ty constant between the pyruvate-load(flux f ) nd Y can be found to be 0.0228 (flux f = 2.18mmol/h for p = 0.9 4 h-).9 Finally, flux c can be elimi-nated from eq. ( 3 ) by noting that the sum of the two loadfluxes, a and b , must equal flux c and using the flux rela-t i o n sh ip s d e sc r ib ed ab o v e . On e o b ta in s c = 0 .0 2 7 6 Ywhich upon substitution in eq. ( 3 ) with the other loadsresults in the simpler form of the objective,

(4 )= 7.28F - 3.03dMAJEWSKI AND DOMACH: ACETATE OVERFLOW IN E. COLI 733


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Inspection of eq. (4) indicates that for a given F, ace-tate formation reduces Y and a maximum of approximately14 ATPs per mole of glucose processed to triose resultswhen d = 0. The value of 14 is lower than the theoreticalamount of ATP derived from the full oxidation of glucosebecause the loss of acetylCoA pyruvate, oxaloacetate, anda-ketog l u tara te to b iosyn the tic re ac t ions has been ac-counted for.

NETWORK CONSTRAINTSPrior Observations of the Behavior of CellularRespiratory SystemEthanol production by aerobic yeast has been hypothesizedto be due to the respiratory system having a limiting cap ac-ity. Consequently, at high growth rates, trioses are di-verted to ethanol instead of being oxidized. Bacteria suchas E . coli likewise appear to have respiratory systems withfinite capacity. For example, E . coli fails to adhere to theclassic acceptor control model developed for respiring mi-tochondria in that only state 3 is manifested by growingcells. Evidence for such failures comes from experimentswhere the respiration rate at high growth rates has not beenobserved to be stimulated by agents that destroy the trans-membrane proton gradient. Only after considerable star-vation can the endogeneous respiration rate of E . col i bestimulated by uncouplers such as dinitrophenol. I Simi-lar results ha ve been reported for chemostat-cultivated P a .denitrificans. I The above results and those from respira-tion studiesI3 hav e led Ingledew and Poole to suggest thatelectron transport through the cytochrome chain might bea limiting rate proce ss. S imilar ly, other experimental resultsalso suggest that the capacity of the respiratory system ofsome bacteria can be fully utilized. Studies on the respira-tion rate of chemostat-cultivated K . aerogenes l5 revealedthat at growth rates less than a critical one, the in situ spe-cific respiration rate was less than the potential respirationrate (corresponding to that exhibited in the presence of ex-cess oxygen and nutrie nts). At grow th rates grea ter than thecritical, the in situ and potential respiration rates converge.Constraints ExaminedWhile evidence exists that suggests that the capacity ofE . colis respiratory system may be a constraint that under-lies acetate overflow, the concept of respiratory system istoo broad to be of immediate use for an analysis at themetabolic network level. This difficulty is due to the respi-ratory system being the joint activity of the electron trans-port chain and oxidative enzymes. Th us, w e have examinedthe objective of maximizing ATP production subject to tworespiratory system constraints: ( 1) electro n transport throughthe cytoch rome chain can be a limiting rate process, and(2) the capacity of the Kre bs cycle can bec ome limiting withthe result that all the triose produced from glucose can notbe processed.

The plausibility of the first constraint has already beennoted above while that of the second constraint is supportedby several facts. For example, the results of chemostat ex-per iments ind ica te tha t the crude ex t rac t ac t iv i t ies o fa-ketoglutarate dehydrogenase (KDH), succinate dehydro-genase (SDH), and malate dehydrogenase (MDH) decline,or attain a maxima and thereafter decline as the growthrate on glucose minimal media increases.I6 KD H in par-ticular is a volatile enzyme. It has been found to be totallyrepressed under anaerobic condition^".'^ or barely detect-ab le in exponent ia l ly g rowing , aerob ic E . c o l i K-12.I8Thus. the activity and catabolic role of the E . coli Krebscycle apparently decreases as growth rate increases whichin turn could lead to acetate overflow.

Identification of the enzy me s responsible for constra iningKrebs cycle flux can be attempted by combining observa-tions on the induction patterns with metabolic logic argu-ments. First, it has been observed that in comparison to thelevels of the enzymes present when aerobic growth oncomplex media occurs, derepression of the tricarboxylican d f iv e - ca r b o n d eca r b o x y l i c ac id en zy m es o ccu r s i nconjunction with repression of KDH when growth occurson glucose minimal medi um . Suc h an induction hier-archy has been noted to be logical because the diversion ofa-ketoglutarate to glutamate is the outcom e. Inde ed, if thea-ketoglutarate branchpoint is examined further, one findsthat the Michaelis constant of KDH for a-ketoglutarate

IS approximately 10-fold less than that ofglutamate de hydrogenase (6 x 10-JM). Therefore, it isplausible that at high growth rates, the high saturation ofKDH in combination with the repression of KDH results ina significant redirection of KDH flux to glutamate. Such aredirection is also consistent with Holms calculations. Itcan be found from his results that as the growth rate in-creases from 0 .7 2 to 0.9 4 h -, the ratio of the net flux toglutamate-to-KDH flux increases much more than the ratioof the net flux to glutamate. Thus, because ( 1 ) KDH activ-ity has been observed to small in rapidly growing E . coli;(2) KDH saturation is associated with a substantial gluta-mate dehydrogenase flux; and (3) saturation and repressionof KDH can account for an increased flux to glutamate athigh growth rates. we suggest that KDH is a worthwhilecandidate to explore as a flux constraint.Another potential candidate for a flux constraint is CS.If the crude extract, biosynthetic activities (i.e., not the ac-tivities of KDH and SDH which are typically measuredwith artificial electron acceptors) found in rapidly growingcel ls a re compared , one f inds tha t CS i s the smal les t .However, CS flux must equal the su m of KDH flux andthe net flux to glutamate. Additionally. CS is inhibited bya-ketoglutarate. Thus, it is difficult to envision how themaximal velocity of CS can be substantially less than KDHwhen KDH is highly satu rated, CS is inhibite d, and the fluxto glutamate is comparable to the KDH flux. Rather, bymaintaining CS and KD H activities sm all in comparison toother cycle enzymes, and coordinately repressing them asgrowth rate increases, the diversion of a-ketoglutarate and

(7 10-?M)19.0 .

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oxaloacetate to biosynthesis would occur as growth rate in-creases. Hou ever, maintaining greater CS activity thanKDH activity would insure that adequate flux to glutamateand succinyICoA is the case.Summary of Problem FormulationBased on the above, the optimization problem and con-straints can be stated as

( 5 4AX[Y] = 7.28F - 3.03dsubject to

or

Equation ( 5a ) represents the objective of maximizingATP flux for a given F. Constraint (5b) corresponds to fi-nite electron transfer capacity (c , ) being present whichcannot be exceeded by the sum of reducing equivalentsproduced at each site (fj,).he alternate constraint (5c)represents the presence of a limiting Krebs cycle enzymeactivity ( P ). which if KDH, can not be exceeded by theflux through KDH ( f K D H ) . In terms of network analysisformalism. constraint (Sb) corresponds to each segmenthaving unbounded capacity but the distribution and magni-tude of the fluxes in each segment can not result in c , be-ing exceeded. Constraint (5 c) is simply one segment has aminimum capacity.

OPTIMAL SOLUTIONEnzymatic ConstraintThe op t imal so lu t ion of e q . (5a) fo r const ra in t (5c) isstraightforward. First, constraint (Sc) can be rewritten as

e, 2 0.368F - 0.737d ( 6 )by noting that f K D H = F - c - d - a - e - f Fig . 1)and eliminating fluxes, c, a , e , and f with the aforemen-tioned relationships. Since a nonzero value of flux d re-duces Y, he optimal policy for F 5 2.72 e , is

Y / e , = 7 .2 8 ( F /e , ) (7a)d l e , = 0 (7b)

and for F 2 .72 e ,Y/e:= 5 . 7 6 ( F / e , ) + 4.12

dl e , = 0 . 5 0 2 ( F / e r )- 1.36 (7c)( 7 4The interesting features of the solution are acetate over-flow begins at a critical value of F / e , and a linear depen-dence on F i r : exists.

Reducing Equivalent Turnover ConstraintThe optimal solution for constraint (5b ) is quali tativelysimilar to that for constraint (5c). In this case. constraint(5c) can be rewritten as

(8 )T2 2.74F - 2.06dTherefore, for F 5 0 . 3 6 5 ~ ~

Y/cT = 7 . 2 8 ( F / c T ) (9a)d / C T = 0 (9b)

and for F > 0 . 3 6 5 ~ ~Y/cT = 3 . 2 5 ( F / c T )+ 1.47 (9c)

d / c T = 1 . 3 3 ( F / c T )- 0.485 ( 9 4Again, an onset condition exists for acetate overflow tobegin and a monotonic dependence o n a specific acetateflux with respect to a specific triose flux is predicted.DISCUSSIONBoth constraints exam ined yield qualitativel>- similar re-sults and could prove to be applicable to explaining over-flow metabolism and respiration experiment results. Forexample, the onset of acetate formation could be due to abottleneck at KDH and as Poole and Ingledeu suggested,the upperbound constraint on growth rate and associatedflux through glycolysis and the Krebs cycle could be setor matched by the capacity of the electron transport chainto tu rnover NADH and FADH. Such a dual const ra in tsituation would be consi stent with acetate formation begin-ning at a critical growth rate and in situ and potential respi-ration rates converging as the maximal growth rate isapproached. Interestingly, evokin g both cons traints couldalso prove helpful for describing the switching and calcu-lating the energetics associated with the more complicatedoverflow metabolism network of B . subtilis. First, acetateoverflows at low to medium growth rates. Then as growthrate increases, acetate secretion ceases and is replaced bylac ta te ~ ecr e t io n . ~hus, first an oxidized metabolite (ace-tate) overflows whereafter as growth rate increases. a re-duced metabolite (lactate ) overflo ws. Such a hierarchy isconsistent with the capacity of the Krebs cycle being firstexceeded and then at higher growth rates, the Krebs cycleconstraint is replaced by the constraint of inadequate ca-pacity for the disposal of reducing equivalents.

In an attempt to examine the above ideas in a quantita-tive manner, we first tranform the results to correspond toa correlation found to fit the specific rate of acetate pro-duction ( v ) of chemostat-cultivated, . coli. The correla-tion was reported by Bajpa i3 and it contains two constants( v , , p c ) ; is given by

p L & . v = o ( IOa)p 2 pc3 v = v , ( p - k . 1 ( lob)

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where p is the specific growth rate and p, is the thresholdgrowth rate associated with the onset of acetate produc-tion. The values of the constants, v, and p,, were reportedto be 0.81 g acetate/g cell and 0.51 h-, respectively. Itappears that this correlation not only can fit Bajpais data,but it also can fit reasonably well other data we found inthe literature. For example, the specific rate of acetate pro-duction of c hemostat-cultivated E . coli at 30C can be fi tif p c = 0.145 h- and v l = 1.6 g acetate/g cell . Forchemostat-cultivated E . coli-B/r at 37C, p, was found tobe 0.47 h- and v increased with dilution rate to 0.78 h-lwhere v was 0.2 h-I; he nce, v , = 0.645.22The transformation can be accomplished by noting thatthe specific molar rate of glucose consumption has beenobserved to be proportional to growth rate.* and glucoseconsumption is the sum of dissimilatory and assimilatoryfluxes. Therefore, by performing a molar carbon balance,total molar glucose consumption ( Q , ) can be found to be

(11)where E , X, and V, and p are a proportionality constant,cell mass concentration, cultivator volume, and the frac-tion of triose carbon dissimilated, respectively. Addition-ally, the last term in eq. (1 ) represents the incorporationrate of glucose-carbon into cellular-carbon where it is as-sumed that carbon content is 50% of the dry weight andindependent of p . If the e, constraint is assumed to be re-sponsible for acetate overflow, then substituting for F intoeq . (7d) with eq . ( l l ) , m u lt ip ly ing by the m olecu la rweight of acetate, and rearrangement yields

v = 6 0 . 0 / 3 - ( ~- 0.007) (p - p,) ( 1 2 4

Qg E ~ X V 0.5pF + 0.007pXV

for p 2 p, = 1.36/3(er/VX)/(&- 0.007) (12b)which is equivalent to eq. ( lo b) if typical values of p , E ,and e,/VX can be assumed to correlate the data reason-ably well.The parameter, E , is simply related to the inverse of theproduct of the observed growth yield (Yrx)nd molecularweight of glucose. A typical value of Y,,, at rapid growthrates is 0.44 g/g; hence E = 0.0126. From Holms results,the maximum value of /3 can be estimated to be 0.50-0.54. Based on the typical value of Y,,,, /3 can be estimatedto be 0.5;hence, p = 0.5 is assumed. The term, e,/VX inthe critical growth rate corre spon ds to a characteristic mo-lar KDH flux per mass of cell. From Holms work, a typi-cal value for acetate-secret ing cel ls is 4 .3 mmol/h g .Using these estimates, v l nd p, are approximately 0.7 gacetate/g cell and 0.5 h-I, respectively. These estimatedvalues of v l nd p, agree surprisingly well with the valuesreported for E . coli by Bajpai3 and those which describeacetate secretion by E . coli-B/r2 considering the approxi-mations made.The alternative of the cr constraint being responsible foracetate overflow can be examined in a similar fashion. Byusing eqs. (9d) and ( I l ) , one obtains

v = l.8[ p - 16c,/XV] (13)

where the same values of E an d /3 were used as before.The quantity C,/XV corre spond s to the maximu m rate ofreduction equivalent turnover per mass of cell. Andersonand MeyenburgI3 found that the maximal respiration rateof E . cofi-B/r in batch culture was 20 mmol/h g cell forgrowth in minimal glucose-limited media. Thus, a reason-able estimate of c,/XV correspo nds to 0.04 mol NADH/hg cell (turnover of 1 mol NADH requires 1/2 mol 02)andeq. (13) becomes

(14)While there is uncertainty in the estimates made for theparameters, a comparison of eq. (14) and results obtainedfor the e , constraint case (v l= 0.7 and p , = 0.5 h-) doeslead to some interesting findings. First, more acetate ini-tially overflows for the e, constraint than the cT constraint.Thus, the e, constraint overrides the cr constraint in thatsatisfying the e , constraint will result in the cr constraintnot being violated. H owever, the predicted specific rates ofacetate overflow converge at p = 0.73 h-I. If the value ofc,/XV assumed is accurate, then the conve rgenc e suggeststhat at growth rates greater than 0.72 h-I, overflow of areduced product is required for the cr constraint to be satis-fied. Ethanol is one such product which is formed by re-ducing acetylCoA twice. In this case, fully utilizing eT andalso satisfying the cTconstraint results in the multioverflowsolution:

0.5 9 p 5 0.72; vA = 0 . 7 [ p - 0.51

v = l.8[p - 0,641

(15a)0.72 9 p 5 0.85; vA = 1.0[0.85 - p] (15b)0.72 5 p 5 p,,,,, u, = 1.3[p - 0 . 7 2 1 ( 1 5 ~ )

where vA and V, represent the specific production rates ofacetate and reduced overflow metabolite, respectively, andeq. (15) was obtained by accounting for another AcCoAbranch flow and performing an analysis s imilar to thatused to obtain eqs. (12) and (13).If. on the other han d, the value of C,/XV chosen is notrepresentative of the E . coli respiration rate, then the re-duced product overflow will differ in its extent. Based onthe work of Harrison and Lovele ss, i t is possible thatc,/XV was underestimated. The y found that the specificrespiration rate of K . aerogenes was approximately 43%higher for chemostat-cultivated cells near washout than ex-ponentially growing, batch-cultivated cells. Consequently,they postulated that the overall regulation of the potentialrespiration rate may differ for batch and chemostat-culti-vated cells. Thus, pc n eq. (14) could potentially be larger(p c= 0.92 h- if same regulatory effect occurs for E . colias for K . aerogenes ) . Therefore, the implications of theoverflow results are for aerobic growth of E . coli on glu-cose, acetate overflow is likely initiated by a constraint inthe Krebs cycle and if the maximal specific respiration rateexceeds 20 mmol 02/h g, the turnover of reducing equiva-lents wil l become cons train ing only near the maximalgrowth rate and acetate will be the dominant overflowproduct present in the medium.

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Although the corre spond ence betw een the specific pro-ducthity correlation and our results suggests that the regu-l a to r y a ssu m p t io n s m ad e m ay b e v a l id , t h e p r ed ic t edpattern for acetate overflo w and the assum ptions made mustalso conform with plausible mechanisms that regulate theacetate producing reactions. Alternately, if the mechanismsare known. then closure requires that they do not conflictwith the assumptions made. The reactions responsible foracetate synthesis are catalyzed by acetylCoA: orthophos-phate acet! ltransferase and acetate kinase. The reactionsare written below in their thermodynamically favorabledirection.

acetyl phosphate + CoA- cetylCoA + P,acetyl phosphate + A D P- cetate + ATP (16a)(16b)

When couple d. reversibility is high as indicated by the equi-librium cons tant being approx ima tely one.: It has beenreported that the amount of acetate kinase in chemostat-cultivated E . ~.oliglucose minimal media) does not dependsignificantly on growth rate.5 Thus . the available evide ncedoes not support the idea that the inductiodrepression con-trol of acetate kinase is respo nsible for turnin g on the cou-pling of the reactions. Instead, constitutive expression ofacetylCo.4: orthophosphate acetyltransferase or inductionat high growth rate is required for acetate formation. How-ever. once both enzymes are induced, the reversibility ofthe combined reactions suggests that the regulation is massaction in manner. That is, as triose flux increases and theKrebs cycle becomes a flux constraint, the elevation of theAcCoA pool and acetate export result in the combined re-actions producing a net flux to acetate. Additionally, the fluxwould increase as growth rate (or triose flux) increases.Therefore. i t appears that once the enzymes are induced,no conflict exists between the assumptions made and theregulation of the acetate producing reactions.

In summary. by hypothesizing that the objective of max-imizing ATP synthesis must be met subject to two differentconstraints that are associated with the respiratory system,we have shou n how the redirection of metabolic fluxes toacetate production can occur. A correlation for the specificrate of acetate production was also derived from the re-sults. Fo r the case of a l imiting enzyme activity in theKrebs cycle. the form and constants were similar to thatreported by Bajpai. When the constraint of limiting elec-tron transfer through the cytochrome chain was examined,it was found that this constraint likely becomes importantonly as the maximal growth rate is approached. In general,a flow network analysis which inco rporates the phenom eno-logical effect of mechanistic details could prove useful foran initial screening of hypotheses. One possible extensionof this work would be to examine the acetate and lactateoverflows oi aerobically grow ing B . subtilis.NOMENCLATURE

AcCoA acet) I coenzyme ACIT citratecs citrate synthase

K I TFn ; MKDHMA LMD HOA APE PPYRSD HsuccSuccCoAVxY

Q,

YO LYSLabdec

fCT

eTf K D HPE

ccc c c

cc-

Iyo

v,

v

isocitratetotal triose fluxfumaratea-ketoglutarate dehydrogenasemalatemalate dehydrogenaseoxaloacetatephosphoenol pyruvatepynwatetotal molar glucose consumption ratesuccinate dehydrogenasesuccinatesuccinyl coenzyme Avolume of cell suspensioncell m ass concentrationtotal production rate of ATP (sum of substrate and oxida-tive level phosphorylation)total ATP synthesis at oxidative leveltotal ATP synthesis at substrate levelnet a-ketoglutarate flux to biosynthesisnet oxaloacetate flux to biosynthesisnet flux of phosphoenol pyruvate to oxaloacstatenet acetyl CoA flux to acetatenet acetyl CoA f l u x to biosynthesisnet pyruvate flux to biosynthesistotal capacity of the cytochrome chain to turnover reducingequivalents (e.g., moles per unit time)maximal activity of an enzymatic bottleneck in respiratorysystemflux through a-ketoglutarate dehydrogenasefraction of triose carbon dissimilatedproportionality constant between total molar rate of glucoseconsumption and rate of total cell mass increasespecific rate of cell mass increasecritical value of specific growth rate at which a metabolicoverflow occursmaximal specific growth ratespecific rate of acetate formationconstant in the specific rate of acetate formation correlationspecific rate of acetate overflow when oxidized and re-duced overflows are both possiblespecific rate of reduced metabolite overflow when oxidizedand reduced overflows are both possible

References1. R. J . Brittan, Science. 119, 578 (1954).2. J. Freschko and T. Ritch, C h e m . E n g . Commun., 45. 229 (1986).3 . R. Bajpai, An n . NY Acad. Sci., 506, 446 (1987).4. K. Y. San and G . Stephanopoulos. Biotechnol . Bioeng. , 21, 11785. J. Staniskis and R. Simutis. Biorechnol Bioen g. . 27. 362 (1986).6. D. Ramkrishna. in Foundarions of Biochemical Enqineering. AC S7. P. Dhurjati. Ph. D. disserta tion, Purdue University. W est Lafaye tte.8 . K. W alsh and D. E. Koshland, J . Bio l . C h e m . , 260. 8340 (1985).9. W. H. Holms, Current Topics C e l l . R e g . , 28 , 69 (1986).

10. B. Sonnleitner and 0 . Kappeli, Biotechnol . Bioeng . . 28, 927 (1986).11 . C. Burstein. L. Tiankova, and A. Kepes, Eur. J . Biochem. . 94, 38712. A. H. Stouthammer and C . W. Bettenhaussen, F E M S Microbiol .13 . K. B. Anderson and K . Vo n Meyenburg. J . Bac re r io l . . 144, 11414. W. J . Ingledew and R. K. Poole, Microbiol.Re v . . 48. 222 (1984).15 . D . E. F. Harrison and J . E. Loveless, J . Genl. Mic robio l . . 68, 35

(1984).

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(1971).

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16. N. Hollywood and H. W. Doelle, Microbios. 17, 23 (1976).17 . C. T.Gray, J . W. T. Wimpenny, and M. R. Mo ssman, Biochem. Bio-18 . C . R . Amarasingham and B. D. Davis , J . B i d . Chem.. 240, 366419 . L. J. Reed and B. B. Mukherjee, Methods Enzymol., 13, 55 (1969).20 . K . I . Steginsky. K . J . Truys , an d P. A . Frey, J . Bio l . C h e m . . to

ph!s. Acta. 117, 33 (1966).(1965).

appear.

21 . N . Sakarnoto, A. Kotre, and M . A. Savageau, J . Bacteriol . . 124,22. H. E. Reiling, H. Laurila, and A. Fiechter, J . Biotechnol.. 2 , 19123 . M. Attaii, personal communication, Department of Chemical Engi-24 . D. K. Fox and S . Roseman, J . B i d . C h e m . , 261, 13487 (1986).25. M. . Koplove and C. L. Cooney, J . Bacteriol . , W, 992 (1978).

775 (1975).(1985).neering, University of Pittsburgh, Pittsburgh, PA.

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majewski opt acetate

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