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Bifurcation Analysis of Continuous Biochemical Reactor Models

Yongchun Zhang and Michael A. Henson*

Department of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803-7303

The validity of a biochemical reactor model often is evaluated by comparing transientresponses to experimental data. Dynamic simulation can be a rather ineff icient andineffective tool for ana lyzing bioreactor models tha t exhibit complex nonlinea r beha vior.Bifurcation analysis is a powerful tool for obtaining a more eff icient and completechara cterizat ion of the model behavior. To illustra te th e power of bifurca tion a na lysis,th e s tea dy -s ta t e a n d tr a n s i en t b eh a v i or of th r ee con ti n u ou s b ior ea cto r m odelsconsisting of a sma ll number of ordina ry differentia l equat ions a re investiga ted. Severalimporta nt feat ures, as well as potentia l l imita tions, tha t a re diff icult to ascertain viadyna mic simula tion ar e disclosed thr ough the bifurcation an alysis. The results motivat ethe use of dynamic simulation and bifurcation analysis as complementary tools foran alyzing the nonlinear behavior of bioreactor models.

1. Introduction

Biochemical reactors can be viewed as highly complexd y n a m i c s y s t e m s . T h e i n t r a c e l l u l a r a n d e x t r a c e l l u l a renvironments comprise ma ny chemical components, a ndeach cell has unique properties. A rigorous mechanisticmodel accounting for both these complexities is knowna s a s t r u ct u r e d a n d s e gr e g a t e d m od e l (9, 13, 33). Suchm od e ls con s i st of cou p le d s e t s of p a r t i a l d i ff er e n t ia lequat ions a nd ordinary differential equa tions. In additiont o b ei n g d i ff icu l t t o f or m u la t e , t h e s e m o d el s a r e n ota m e n a b le t o s y s t em a t i c a n a l ys is a s a r es u lt of t h e ircomplexity. Simplified models can be developed either byneglecting t he intra cellular chemical st ructure (unstruc-tured segregated models) (6 , 1 2 , 1 7 ) or by neglectingheterogeneity of the cell population (structured unseg-

regat ed models) (8, 20, 32). The simplest models a re bothunstructured and unsegregated (10). I f the int racellulare nv ir on m en t ca n b e ch a r a c t er iz ed b y a f ew cr it i ca lvaria bles (20, 32), then structured unsegregated models(l ike unstructured unsegregat ed models) consist of ar e a s on a b l y s m a l l n u m b er of n on l in e a r or d i n a r y d i f-f er e n t ia l e q u a t i on s . S u c h m od e ls a r e w e l l s u i t e d f o rr igorous a na lysis.

Three unsegregated models of different cell culturesare st udied in this paper . First t he structured model ofHybr idoma cells proposed by G ua rdia a nd co-workers (16)is considered. This mammalian cell culture is reportedto exhibit multiple steady states under certain operatingconditions (14, 38). The cybernetic modeling paradigmused to reproduce this beha vior involves t he ma ximiza-

tion of cel l growt h via competi t ion between a l ternat ivep a t h w a y s t h a t u t il iz e a p a ir of com pl em en t a r y a n dpartia lly substitutable substra tes. An unst ructured modelproposed by McLellan and co-workers (27) for contin uousfermentation of the microorganism Z y momonas mobi l i z is studied next. Continuous cultures of this microorgan-ism exhibit oscillatory behavior which adversely affectsetha nol productivity a nd r eactor opera bility. This beha v-ior is captured in the model by introducing a dynamic

specific growth ra te t ha t accounts for t he inhibitory effect

of the past etha nol rate of cha nge on th e current st at e ofthe rea ctor . The last model investigat ed is a structuredmodel proposed by J ones an d Kompa la (20) for the yea stSaccha rom yces cerevisiae. Continuous cultures of thism i cr oor g a n i s m h a v e b ee n s h ow n t o e xh i b it com p le xdynamic behavior that includes the sudden appearancean d disa ppeara nce of susta ined oscil lat ions (29, 34, 35).The cybernetic model reproduces this behavior throught h e c o m p e t i t i o n o f t h r e e m e t a b o l i c p a t h w a y s t h a t a r euti l ized to ma ximize the cell growth ra te.

Transient bioreactor models of ten are evaluated bycom p a r i n g e xp er i m en t a l d a t a a n d m od e l s i m u la t i o nresults. While dynam ic simulat ion is a very useful toolfor model val idation, several l imitations can be identi-

f ied: (i) i t is inef f icient an d potential ly inconclusive,especia lly wh en th e model possesses slow dyna mic modes;(i i) it is necessari ly incomplete since only a l imitednumber of simulat ion test s can be performed and impor-ta nt dyna mic beha viors ma y not be observed; a nd (ii i) i tdoes not easily reveal the model characteristics that leadto certain dynamic behaviors. Therefore, dynamic simu-lat ion should not be viewed a s th e only tool for evalua tingtra nsient bioreactor models.

Th e p ur pos e of t h i s p a pe r i s t o d em on s t r a t e t h a tb if u r ca t i o n a n a l y s is i s a p ow e r f ul t o ol f or e va l u a t i n gtra nsient models of cont inuous bioreactor. The objectiveof bi furcation theory is to characterize changes in theq u a l i t a t i v e d y n a m i c b e h a v i or of a n on l in e a r s y s t e m a skey para meters a re varied. The model equa tions are used

to locate steady -sta te solutions, periodic solutions, an dbifurcation points where the qua li tat ive dyna mic beha v-i or c h a n g es . B i fu r ca t i o n a n a l y s i s c a n b e m u c h m o r eeffective than simply integrating the model equationsov er t i m e a n d co mp a r i n g t h e t r a n s i e n t r e sp on s es t oexperimental data. Instead a complete picture of themodel behavior is obta ined in t he form of a bi furcationdiagram. This diagram can be used to determine i f themodel supports the steady-state and dynamic behaviorobserved experimentally. It also can guide the design ofexperiments aimed at validating unexpected model pre-dictions. Num erical bifurcat ion packages d eveloped by a

* P h: 225-388-3690. Fa x: 225-388-1476. E -ma il: [email protected].

647Biotechnol. Prog. 2001, 17, 647660

10.1021/bp010048w CCC: $20.00 2001 American Chemical Society and American Institute of Chemical EngineersPublished on Web 06/16/2001


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n u m b er of r e se a r ch e r s (1 1 , 2 1 , 2 3 ) make bifurcationan alysis of low-order nonlinear systems reasonably simple.

It is important to note that a number of other inves-tigat ors have a pplied bifurcation an alysis t o continuousbioreactor models (1, 26, 30, 37). Ma ny of these studiesf ocu s on t h e d y n a m i cs of t w o m i cr ob i a l p op u la t i o n scompeting for a common ra te-limiting subst ra te (2, 3, 24,31). In most studies the objective is to provide a veryd e t a i le d ch a r a c t er i za t i on o f t h e m od e l b e h a v i or w i t hminimal comparison to available experimental data. The

ob je ct i v e o f t h i s p a p er i s n ot a b l y d i ff er e n t . We a r econcerned primarily with the use of bifurcation analysisfor va lidat ion of tra nsient bioreactor models. We empha-s i ze t h e com p a r i s on of p h y s ic a l ly m e a n i n gf u l m od e lbehavior with experimental data rather than cataloginga ll possible model beha viors. A similar a pproach ha s beenp u rs u ed b y J on e s a n d K o m pa l a (20) t o d e t er m i n e u n -known para meters in t heir cybernetic yeast model . Thepresent contr ibution can be viewed as an extension ofour bifurcation studies on unstructured cell populationbalance models for continuous yeast bioreactors (37).

The remainder of the paper is organized as fol lows.Basic concepts of bifurcation theory and an introductionto the bifurca tion packa ge AU TO ar e presented in S ection2. The dyna mic behavior a nd bifurcat ion a na lysis of theaforementioned bioreactor models are discussed in Sec-tions 3-5. A summary and conclusions are presented inSection 6.

2. Bifurcation Analysis

A n o n li n ea r d y n a m i c s y s t e m d i ff er s f r om a l in e a rd y n a m i c s y s t e m i n t h a t i t s q u a l i t a t i v e p r o p e r t i e s c a nchange under small perturbations of the model param-eters. These properties include t he num ber of equilibria,s t a b i l it y of t h e e q u il ib r ia , e xi s t en ce o f l im i t cy cl es ,multiple modes of behavior , and chaos (22). Below weprovide a very brief introduction to bifurcation analysis.The t extbook by Kuzn etsov (22) provides more completedescriptions of these concepts. A bifurca tion is in troduced

formally as follows.Definition 1. Two dynamical system F: R n fR t a n d

G: R n fR t are called topologically equivalent if thereexists a diffeomorphism h: R n f R n s u c h t h a t hF )Gh.

Two topological ly equivalent systems have the samequali tat ive dynamic behavior in the sense that they canbe mapped to each other .

Definition 2. The appearance of a topologically non-equivalent phase portrait under variation of a parameteris called a bi f ur c at i on.

Bifurcat ion a nalysis is the study of how the qua lita t iveproperties of a nonlinear dy na mic system change a s keyparameters are varied. Consider a continuous-time non-linear system depending on a para meter vector R:

w h e r e fis smooth with respect to both th e sta te vector xa n d t h e b i fu r ca t i on p a r a m e t e r v ect o r R. I f x0 is ahyperbolic equil ibrium point where a l l the real pa rts oft h e e ig en v a lu es of t h e J a c ob ia n m a t r i x D f (x0) a r enonzero, then a sma ll perturbat ion in t he model para m-e t er s w i ll n o t ch a n g e t h e q u a l i t a t i v e b e h a v ior of t h es y s t em , i .e ., a h y p er b ol ic e q u il ib r iu m i s s t r u ct u r a l l ystable. Bifurcations occur when some of the eigenvaluesapproach the ima ginary axis in the complex plane. Thesimplest bi furcations are associated with a single realeigenvalue becoming zero (1 ) 0) or a pair of complex

conjugate eigenvalues crossing the imaginary axis (1,2) (i0, 0 > 0).

Definition 3. The bifurcat ion where 1 ) 0 is called afold bifurcation.

Definition 4. The bifurcat ion w here 1,2 ) (i0, 0 >0 is called a H opf bifurcation.

These are the most common bifurcations present inn on l in e a r s y s t e m s. F o ld b if u r ca t i o n s u s u a l l y a r e t h ecause of multiple steady sta tes a nd hyst eresis behavior .

Hopf bifurcations a re responsible for th e appeara nce anddisappearance of periodic solutions.

While bifurcation theory is a powerful tool for modelchara cterizat ion, ana lytical treatm ent of physically basedmodels usua lly is intr acta ble because of the complexityof the nonlinear model equations. A number of softwarepackages including AUTO (11), LOCBI F (21), an d CON-TENT(23) have been developed for numerica l bifurcat ionanalysis of low-order nonlinear models. Numerical bi-furcation a na lysis involves an itera tive procedure knowna s continuation. First a stable steady-state or periodicsolution for a particular set of parameter values is locatedby dynamic simulation. Then one of the parameters isvaried to allow the continued calculation of solutions asa function of this bi furcation parameter . At each i tera-tion, a step in the bifurcation parameter is taken and apredictor-corrector method is utilized to locate the newsolution. The step size of each iteration is controlled bya convergence criteria. The procedure is repeated untila desired range of para meter values has been evalua ted.Therefore, continuation can provide a complete pictureof the nonlinear dyna mic beha vior .

The results of the continuation calculations typicallya r e p r e s e n t e d a s a b i f u r c a t i o n d i a g r a m w h e r e t h e b e -ha vior of a key model varia ble is shown a s a function ofthe bifurcation para meter. The stea dy-sta te an d periodicsolutions are mapped into this two-dimensional space.As compared to dyna mic simulation, a key adva nta ge ofcontinua tion is that unsta ble as well as st able solutions

can be located. Because i t provides a very concise andcomplete description of the syst em beha vior, a bifurcat iond i a g r a m i s i d ea l f or c om p a r i n g m od e l p r e d ic t ion s t oe xp er i m en t a l l y ob s er v ed b e ha v i or . F o r e xa m p l e, t h eran ge of model param eter values tha t support multiplesteady states or periodic solutions can be determined.This al lows a more meaningful analysis of a nonlinearmodel than is possible with dynamic simulation alone.In t he next t hree sections, th e applica bility of bifurcationan a lysis to continuous bioreactor models is demonstra tedvia t hree example systems.

The AUTO continua tion pa ckage d eveloped by Doedeland co-workers (11) i s p e r h a p s t h e m o s t w i d e l y u s e dnumerical bifurcation code. AUTO can perform bifurca-

tion ana lysis of nonlinear sy stems described by a lgebraicequations, ordinar y dif ferential equa tions, and para bolicpartial di f ferential equations. In addit ion to the simplefold and Hopf bifurcations described above, AUTO canlocat e tori and period doubling bifurcations a s a functionof t w o or m or e p a r a m et e r s. AU TO a l s o i n cl ud es agra phica l user interface (GU I) w hich simplifies specifica -tion of computational parameters required by the con-tinuation code. For these reasons, we use AUTO in thisp a pe r. I t i s i m por t a n t t o n o t e t h a t AU TO a n d ot h e rgenera l purpose bifurcat ion codes ca n be expected t o workonly for nonlinear models of moderate dimension. There-fore, they are best suited for analysis of unstructured,unsegregated a nd simple structured, unsegrega ted biore-actor models.

x ) f(x, R), xR n, RR l (1)

648 Biotechnol. Prog., 2001, Vol. 17, No. 4


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3. Hybridoma Cell Bioreactor Model

Background. Hybridoma cells uti l ize glucose andglutamine as complementary and pa rtial ly substi tuta blesubstrat es for growt h (4, 36). There exists severa l meta -bolic pathways, each of which is favored under certainculture condit ions, for cell growt h. The existen ce of th esemultiple pathways creates complex behavior when Hy-bridoma cells are grown in a continuous bioreactor . Inparticular , researchers have shown tha t di f ferent steady

s t a t e s c a n b e r e a c h e d w h e n c u l t u r e s w i t h t h e s a m eoperat ing conditions but different initial meta bolic stat esar e swit ched from ba tch or fed-ba tch mode to continu ousoperation (14, 38).

A structured, unsegregated model ha s been proposedb y G u a r d i a e t a l . (16) t o c a p t u r e t h e ob s er v ed s t e a d y -sta te mult iplicity. The cybernetic model account s for themultiple metabolic pathways created by the complemen-tary and partially substitutable substrate utilization. Thecy b er n e t i c m od e l p r ed i ct s t h e e xi s t en ce of m u l t ip lesteady-state solutions for some operating conditions.More specifica lly, it is shown tha t d ifferent initial condi-tions established by batch and fed-batch operation canl ea d t o d i ff er e n t s t e a d y -s t a t e s ol u t ion s . H o w e ve r , t h ea u t h or s n ot e t h a t t h i s p rop er t y i s v er y s en s it i ve t o

ch a n g es i n t h e m od el p a r a m et e rs a n d t h e r eg ion ofoperating conditions that support multiplici ty is quitesmall. It is not clear if this lack of robustness is attribut-able to the model structure or to the particular choice ofmodel para meters. Furthermore, a precise cha racteriza-tion of the opera ting ra nge which supports steady -sta temultiplicity has not been presented.

Transient Model. Ramkrishna and co-workers (32)have proposed that microorganisms optimize utilizationof ava i lable substra tes to maximize their insta nta neousgrowth rate. While somewhat controversial, this hypoth-e s i s h a s b e e n d e r i v e d f r o m t h e a n a l y s i s o f e x t e n s i v eexperimental data. The cybernetic modeling approachhas been used to capture the partia l ly substi tuta ble an dcomplementary substrate uti l ization that leads to mul-

t i pl e m et a b ol ic p a t h w a y s i n H y b r id om a cu lt u r es . Ad e t a i l ed d e sc ri pt i on of t h e m e t a b o li c p a t h w a y s c a n b efound in the original reference (16). The cybernetic modeli n cl u d es t w o s u b st r a t e s , g l u cos e a n d g l ut a m i n e; t h r e ei n t er m e di a t e s ; a n d f iv e e n z y m es a s s o ci a t e d w i t h t h ev a r i ou s p a t h w a y s .

The cybernetic model equations are

w h e r e X is the cell mas s concentra tion; S1 a n d S2 a r e t h e

concentra tions of glucose a nd gluta mine, respectively; Mia n d ei ar e the concentra tions of the three intermediates

a nd five enzymes, respectively; Sif ar e the feed substra te

concentrations; D i s t h e d i lu t i on r a t e ; Yi a r e t h e y i e l d

coefficients; rei

/ i s t h e co ns t i t u t i ve s y n t h e s i s r a t e o f t h eenzyme ei; a n d bi is the degradation rate constant of the

enzyme ei. The cybernetic variables uic, ui

s, vic a n d vi

s a r ethe syn thesis a nd a ctivity coefficients of the enzymes forthe complementar y a nd substi tuta ble pat hwa ys, respec-tively. The rea ction rat es ri a nd rei a re as sumed t o follow

Monod-type kinetics. Their definitions can be found inthe origina l reference (16) and a re omitted h ere for sakeof brevity.

We experienced some difficulties producing multiplesteady-state solutions w ith t he model para meter valuesreported in ref 16. After modifying several parametervalues, we w ere able to generate the expected multiplesteady-state behavior. The parameter values used in oursimulation an d bifurcat ion st udies are l isted in Table 1

w h e r e t h e v a l u e s o f rima x a nd YMix ar e dif ferent f rom

those l isted in the original reference. The discrepancybetween the two sets of parameter values further moti-vates the need for a more detai led investigation of themodel beha vior using bifurcat ion an alysis.

Results and Discussion. Several dyna mic simulationtests w ere performed to study the st eady-sta te behaviorof the cybernetic model in different regions of the operat-i n g s p a ce . I n t h e f i r st t e s t , t h e r e a c t or i n i t i a l ly i s a t a

steady state corresponding to S1f

) 0.95 g/L , S2f

) 0.43g/L, a nd D ) 0.0295 h -1. The results obtained for stepchanges of various magnitudes in the di lution rate areshown in Figure 1 where the cell mass concentration (X)i s s e l ect e d a s a r e pr e se n t a t i v e o u t pu t v a r i a b l e f or t h eculture. The steady states obtained for D ) 0.0300 h -1

a n d D ) 0.0292 h -1 are close to the ini t ial steady state,while those for D ) 0.0291 h -1 a n d D ) 0.0290 h -1 a revery fa r removed. Also shown is t he response obta inedfor a step change to D ) 0.0300 h -1 a t t ) 2000 h fromt h e D ) 0.0290 h -1 steady state. The culture reaches a

dif ferent steady st at e tha n the one obta ined for the D)

0.0300 h -1 step change from the init ial steady state.Th e s e d y n a m i c s i m u la t i on r e s ul t s d e m on s t r a t e t h a t

the cybernetic model predicts the presence of multiplesta ble steady sta tes an d hysteresis behavior. On the otherha nd, this t ype of a na lysis provides a ra ther incompletep ict u r e of t h e m od e l b e h a v i or . F o r i n s t a n c e, d y n a m i csimulation is not a convenient tool for determining therange of dilution rates over which multiple steady statesexist. This is very va luable information for model valida-t i on . B e low w e s h ow t h a t b if ur ca t i on a n a l y s is i s apowerful tool for extracting this knowledge.

Figure 2 shows the one-para meter bifurcat ion dia gra mfor the cybernetic model where the di lution rate (D) ischosen as the bifurcation parameter and the cell mass

dX

d t) (rg - D)X

dS1

dt) -(r1v1

sv1

c+ r3v3

c)X + D(S1f

- S1)

dS2

dt) -r

2v

2

sX + D(S

2

f- S

2)

dM1

dt) Y1r1v1

sv1

c+ Y4r5v5

s- (Ym1x + M1)rg - r4v4

s

dM2

dt) Y2r2v2

s+ Y5r4v4

s- (Ym2x + M2)rg - r5v5

s

dM3

dt) Y3r3v3

c- (Ym3x + M3)rg

dei

d t) rei

/

+ reiui

cui

s- (rg + bi)ei; i ) 1, ..., 5 (2)

Table 1. Hybridoma Cell R eactor Model Parameters

pa r a m et er un it va lue

rima x g/gdw-h 0.05, 0.03, 0.01, 0.03, 0.01

Ki g /L 0 .0 01, 0 .0 01 , 0 .0 01 , 0. 01 , 0 .0 00 1

reima x g/gdw -h 0.001, 0.0005, 0.001, 0.0005, 0.001

Kei g /L 0. 001, 0. 0001, 0. 001, 1e-5, 1e-5

rei/ g/gdw -h 1e-6, 1e-6, 1e-6, 1e-6, 1e-6

bi gdw /gdw -h 0.05, 0.1, 0.1, 0.1, 0.05

Yi g /g 0.9, 0. 9, 0.8, 1. 0, 1.0

KgM i g /g d w 0 .0 005 , 0. 00 05 , 0 .0 005YMix g /g dw 0.7, 0. 99, 0. 1

rgma x gdw/gdw-h 0.0575

Biotechnol. Prog., 2001, Vol. 17, No. 4 649


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concentration (X) is the output variable. The feed sub-

strate concentrations are held constant at S1f

) 0.95 g/L

a nd S2f

) 0.43 g/L. The locus of st ead y-sta te opera tingpoints a s a function of D is determined using AUTO. Thesolid line (s) denotes sta ble steady-stat e solutions, whilet h e d a s h e d l in e (- - -) d e n ot e s u n s t a b l e s t e a d y -s t a t esolutions. The model exhibits fold bifurcations at D )0.02917 h -1 (point 1) and D ) 0.03068 h -1 (point 2), w hichd e li n ea t e t h e p a r a m e t er s p a ce w h e r e m u l t ip le s t e a d y -s t a t e s ol u t ion s e xi s t s . I n t h i s r e gi on t h e m od e l s h o w sh y s t e r es i s b e h a v i or d u e t o t h e S -s h a p e d s t e a d y -s t a t elocus. There only is a single stable steady-state solutionoutside this region.

The bifurcation diagram clearly shows the existenceof multiple steady-state solutions and provides an ex-planation for the hysteresis behavior observed in Figure1 . E a c h s t a b l e s t e a d y s t a t e h a s a d o m a i n o f a t t r a c t i o nfrom w hich a l l ini t ial conditions converge to t ha t steadysta te. Ini t ial conditions corresponding to low dilutionr a t e s c o n v e r g e t o t h e u p p e r s t e a d y s t a t e w h i l e t h o s ecorresponding t o high dilution ra tes converge to the lowers t e a d y s t a t e . T h e b i f u r c a t i o n d i a g r a m a l s o s h o w s t h a tthe parameter region that supports multiple steady-statesolutions is quite small : D [0.02917 h-1, 0.03068 h-1].Th i s i s m u ch m or e v a l u a b l e i n f or m a t i o n f or m od e lv a l id a t i on t h a n s im pl y k n ow i n g t h e m od el e xh ib it s

Figure 1. Dynamic simulation of the Hybridoma reactor model.

Figure 2. Bifurcation diagram of the Hybridoma reactor model.



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multiple steady states. In particular , experimental de-termination of the di lution rates that support multiplesteady st at es can be used to a djust t he model para metersto obtain agreement. I f this is not possible, the models t r uct u r e m a y b e con cl ud ed t o b e i na d e q u a t e. S u chconclusions a re very dif ficult to obta in from dyn am icsimulation a lone.

4. Z ym o m o n a s m o b i l i s Reactor Model

Background.Z y momonas mobi l i s ha s been proposedas a more promising microorganism than conventionalbakers yea st for industr ial production of ethanol (5, 7).A m a j or d r a w b a c k o f t h i s m i cr oor g a n is m i s t h a t i te xh i bi t s s u s t a i n e d o s ci ll a t i on s ov er a w i d e r a n g e ofoperating conditions when grown in continuous culture.This leads to decreased ethanol productivi ty and lessefficient use of ava ilable substra te (7, 15). Various modelsha ve been proposed to describe the oscilla tory d yna micsof continuous Z y momonas mobi l i s cultures (10, 15, 18,25). Daugulis et al. (10) present an unstructured, unseg-regated model based on the concept of a dynamic specific

growth rate. The predictive capability of the model hasbeen evaluated experimental ly by McLellan et al . (27).A variety of simulation tests were performed, and modelpredictions were found to be in reasonable agreementw i t h e xp er im en t a l d a t a . H ow e ve r, a m or e t h or ou g ha n a l y s is of t h e m od el d y n a m ics w i t h r es pe ct t o t h ema thema tical cause of the oscillat ions a nd th e range overwhich periodic solutions exist has not been presented.B e l ow w e s h ow t h a t b if u r ca t i o n a n a l y s i s i s w e ll s u i t e dto answer these important questions.

Transient Model. D a u g u l i s e t a l . (10) a r g u e t h a tmany models for continuous Zymomonas mobil i sculturesrequire measurements of physiological q uant ities that ar edifficult a nd t ime-consuming t o obta in. As a n a lternat ive,they propose the concept of a dynamic specific growth

r a t e w h i ch e xp li ci t ly a c cou n t s f or t h e e ff ect of p a s tculture conditions on subsequent cell behavior . On theb a s i s o f t h i s c on ce pt , a s i m pl e t r a n s i en t m od e l t h a trequires only measurements of extracellular variablessuch as etha nol, substra te, and cell mass concentra tionsis proposed. The uns tructur ed, unsegrega ted model con-s is t s of m a t e r ia l b a l a n ce s on b iom a s s , e t h a n ol a n d

s u b st r a t e c om b in e d w i t h t w o a d d i t i on a l e q u a t i on s t h a tdescribe the inhibition of cell growth caused by the pastrat e of change of t he etha nol concentra tion. The modele q u a t i on s a r e

where X, S a nd P are th e biomass, substra te, and ethanolconcentra tions, r espectively; W a n d Z are the first-ordera nd second-order weight ed avera ges, respectively, of theethanol concentration change rate; D is the dilution ra te;Sf is the feed substrate concentration; (S, P, Z) is t h edyna mic specific growt h ra te; QP is the specific productionr a t e ; YP/S is the yield coeff icient between glucose ande t ha n o l; a n d is a pa r a m et er t h a t d et er m in es t h em a g n i t u d e o f t h e t i m e l a g f o r t h e d e l a y e d i n h i b i t i o neffect. More precise definitions of W, Z, , a n d QP can be

Figure 3. D y n a m i c s im u l at i on of t h e Zy mo mo n a s mo b il i s reactor model at high dilution rates.

dX

d t) [(S, P, Z) - D]X

dS

d t) -( 1Yp/s)QpX + D(Sf - S)

dP

d t) QpX - D P

dZ

d t) (W - Z)

dW

d t) (QpX - D P - W) (3)



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found in the original reference (27). The model param-eters used in the fol lowing simulation and bifurcationa n a l y s is a r e t a k en f r om t h e o r ig i n a l r e fe r en ce a n d a r elisted in Table 2.

Results and Discussion. D yna mic simulations wereperformed t o study the oscillat ory beha vior of the model

in different regions of the opera ting spa ce. Figure 3 showstw o simulat ion tests a t high dilution rat es. The cell ma ssconcentra tion (X) is chosen a s the output va riable for thecu lt u re. B ot h s im ul a t ion s s t a r t a t t h e s t ea d y s t a t ecorresponding to D ) 0.1120 h -1 a nd Sf ) 200 g/L . Thefeed substra te concentra tion is ma inta ined at t his valuean d the dilution rat e is changed to a slight ly lower va lue.For a step change to D ) 0.1116 h -1, the culture oscillatesw i t h v er y s m a l l a m p li t u d e a n d t h e os ci ll a t i on s s l ow l ydecay to a st eady -sta te solution tha t is close to the initialpoint. For D ) 0.1115 h -1, the culture exhibits sustainedos ci ll a t i on s o f a r a t h e r s m a l l a m p li t u d e. N o t e t h a t t h echange in dyna mic beha vior for the t wo dilution ra tes isn o t d r a m a t i c . W h i l e t h e s e t e s t s i m p l y t h a t t h e m o d e lexhibits a bifurca tion in th is region, it is a time-consum-

ing ta sk to f ind by dyna mic simulation the di lution rateat which t his bi furcat ion occurs.Figure 4 shows the results of three simulat ion t ests a t

lower dilution rat es and the sa me constan t feed substra teconcentra tion. The initia l condition for the first tw o testscorresponds t o the st eady sta te for D ) 0.033 h -1. For astep change t o D ) 0.0332 h -1, the culture oscillates w ithvery small amplitude unti l a new stea dy sta te is reacheda fter a very long period of time. For D ) 0.0333 h -1, t h ecu l t u r e i n i t ia l l y o sc il la t e s w i t h s m a l l a m p l it u d e b u teventually exhibits a transition to large amplitude oscil-lat ions. In the th ird test, the simulat ion is restart ed froman oscil latory ini t ial condition from the last test (D )0.0333 h -1) an d th e di lution rat e is stepped back to D )0.0332 h -1. Sustained oscillations are maintained despite

t h e f a c t t h a t t h e s a m e d i lu t i on r a t e p r od u ce d a s t e a d ys t a t e i n t h e f i rs t t e st .

These simulations veri fy tha t the model predicts theappearance and disappearance of sustained oscil lat ionas observed experimental ly. However, the results alsoraise questions about (i) the large dif ference in ampli-tudes observed wh en susta ined oscil lat ions a re ini t iatedat low an d high dilution rat es; and (i i) the tw o differentsta ble solutions present at the di lution ra te D ) 0.0332h -1. A n ot h e r i s s u e t h a t w a r r a n t s f u r t h er i n ve s t ig a t i o n

i s t h e u s e o f m u lt i pl e s e t s of p a r a m et e r v a l ue s f ors t a t i o n a r y a n d o s c i l l a t o r y s i m u l a t i o n s i n t h e o r i g i n a lreference (27). Below we show that bi furcation analysisprovides a convenient framework to address these ques-tions.

Figure 5 shows a one-para meter bifurcat ion dia gramo f t h e Z y momonas mobi l i s m od e l w h e r e t h e f ee d s u b -stra te concentra tion (Sf) is held consta nt a t 200 g/L. Thedilution ra te (D) is chosen a s t he bifurcat ion pa ra meter ,a n d t h e c e l l m a s s c o n c e n t r a t i o n (X) i s s e l e c t e d a s t h eou t p u t v a r i a b l e. A t l ow d i lu t i on r a t e s t h e r e o n ly i s as i n gl e s t a b l e s t e a d y -s t a t e s ol u t ion (s). At p oi n t 2 , abifurcation occurs where the steady-state solution be-com e s u n s t a b l e (- - -) a n d a p er i od i c s ol u t ion o f t h ea m p l it u d e i n d ica t e d i s c r e a t e d . A t p oi n t 1 , a n o t h er

Figure 4. D y n a m i c s im u l at i on of t h e Zy mo mo n a s mo b il i s reactor model at low dilution rates.

Table 2. Z y m o m o n a s m o b i l i s Reactor Model Parameters

pa r a m et er va lue pa r a m et er va lue

ma x 0.41 h -1 Qp, max 2.613 h -1

Pob 50. 0 g /L Ks 0. 5 g /LPm a 217. 0 g /L Kmp 0. 5 g /LPm b 108. 0 g /L Ki 200. 0 g /LPm e 127. 0 g /L YP/S 0.4 95 g /gSi 80. 0 g /L 0.0366 h -1

0.8241 21.05R 8.77 a 0.3142b 1.415



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bifurcation occurs where the periodic solution disappearsand the st eady-sta te solution becomes sta ble. At higherd il ut i on r a t e s , t h e r e i s a s in g le s t a b le s t ea d y -s t a t esolution.

The bifurcation that occurs at point 1 (D ) 0.11151h -1) is known as a supercrit ical Hopf bifurcation wh ereloss of st abil i ty of the st eady-sta te solution is a ccompa-nied by th e a ppeara nce of very sma ll amplitude oscil la-tions. For the range of di lution rates between points 1an d 2, a st able periodic solution coexists w ith a n unst a blesteady-state solution. The bifurcation that occurs at point2 (D ) 0.03322 h -1) i s k n ow n a s a s u bcr it i ca l H o pfbifurcation where large amplitude oscil lat ions appearwhen the steady-state solution becomes unstable. Points3 a n d 4 (D ) 0.03095 h -1) represent a fold bifurcat ion ofl imit cycles. They denote the di lution ra te a t which th eperiodic solution changes stabil i ty. For di lution rates

b e t w e e n p o i n t s 2 a n d 3 , s t a b l e a n d u n s t a b l e p e r i o d i csolutions coexist. Below point 3, the steady-state solutionis th e only sta ble solution of the model .

The one-para meter bifurcat ion diagra m shows tha t th eappearance and disappearance of periodic solutions is duet o t h e e xi s t en ce of t w o H o pf b i fu r ca t i on p oi n t s . Th ea n a l y s i s a l s o a n s w e r s t h e q u e s t i o n s r a i s e d a b o u t t h edifferent amplitudes of sustained oscillations at low andh i g h d i l u t i o n r a t e s . S i n c e t h e H o p f b i f u r c a t i o n a t t h elower dilution rate is subcritical, stable periodic solutionsar e cha ra cterized by large magnit ude oscillations and t hestable periodic solution coexists with a stable steady-states o l u t i o n o v e r a v e r y s m a l l r a n g e o f d i l u t i o n r a t e s . B ycontra st, the H opf bifurcation at the higher dilution ra tesis supercritical a nd t he a ssociat ed periodic solutions h ave

very small amplitude.Furth er chara cterizat ion of the model behavior can beobtained by computing th e locus of each H opf bifurcationp oi nt i n t h e D a nd Sf pl a n e. Th is is k now n a s at w o - p a r a m e t e r b i f u r c a t i o n d i a g r a m , a n d i t a l l o w s t h era nge of operat ing conditions u nder w hich periodic solu-tions exist to be determined. The tw o-para meter bifurca-t i on d i a g r a m i s s h o w n i n F i g u r e 6 . Th e u p p er b r a n c hb et w e en p oi nt s 1 a n d 4 r ep re se nt s t h e l ocu s of t h esupercri t ical bi furcation point , while the lower branchb e t w e e n p o i n t s 5 a n d 6 i s t h e l o c u s o f t h e s u b c r i t i c a lbifurcation point . These t wo branches def ine a closedregion in th e D-Sf plane in which stable periodic solu-tions exist a nd sta ble stea dy-sta te solutions ca nnot exist.Outside this region, sustained oscillations occur over a

very sma ll range of operat ing conditions an d coexist w ithsta ble steady-stat e solutions. The diagr am a llows concisedetermination of the operating conditions tha t supportsustained oscil lat ions and is well suited for val idatingthe model aga inst da ta . I t is very dif f icult t o obta in thistype of informa tion using only dyna mic simulat ion.

The two-parameter bifurcation analysis also providesinsights into the structural l imitations of the model . Inthe origina l reference (27), three sets of model parametersw e r e e st i m a t e d t o f i t d a t a w h e n e i t h er t h e s t e a d y -s t a t esolution or the periodic solution was stable. The param-e t er v a l u es u s e d i n t h i s s t u d y w e r e o bt a i n ed i n r e f 2 7f r om os ci ll a t o r y d a t a . Wh i le e a c h s e t of p a r a m e t e r sprovides a reasonable f i t to the corresponding da ta , thelack of validation tests raises concerns about the simplem od e l s t r u c t u r e. I n on e s e t of e xp er i m en t s (27) , t h eculture reaches a steady state for D ) 0.133 h -1 a nd Sf)

150 g/L. By contra st , the tw o-para meter bifurcationdiagra m in Figure 6 indicat es tha t a periodic solution isexpected for these values. This indicates that the modelca n n o t b e r e con ci le d a g a i n s t e xp er i m en t a l d a t a a n d amore sophistica ted model structure is needed. In fa ct, thea u t h o r s (27) note t ha t osci l lat ory behavior is associat edwith a change in cell morphology. This suggests that asegregat ed model may be more appropriate.

5. S a c c h a r om y c es c e r ev i s i a e Reactor Model

Background.Saccharomyces cerevisiae(bakers yeast)is an importan t microorganism in a n umber of industriesi n cl u d in g b r ew i n g , b a k i n g , f o od m a n u f a c t u r in g , a n dgenetic engineering. Many investiga tors have shown tha tcont inuous cultures of Saccharomyces cerevisiaeexhibit

susta ined oscil lat ions in glucose l imited environmentsunder aerobic growth conditions (28, 29, 34, 35). Theunderlying cellular mechanisms that cause oscil latoryyeast dyna mics are controversial a nd ha ve been a subjectof three decades of intensive research. A large numberof transient models have been proposed to explain theexistence of sust ained oscillations (20, 34, 39). J ones a ndKompala (20) have proposed a cybernetic model t ha t isa b l e t o p r ed i ct t h e a p p ea r a n ce a n d d i sa p p ea r a n c e ofsustained oscillations in continuous yeast bioreactors. Avariety of simulation tests were performed to evaluatethe dynamic behavior of the model . Such tests providean incomplete chara cterizat ion of the model propertiestha t is biased towa rd t he specif ic behaviors being inves-t i ga t e d . B e low w e s h ow t h a t a d et a i le d b i fu r ca t i on

Figure 5. One-parameter bifurcation diagram of the Zy mo mo n a s mo b il i s reactor model.



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a n a l y s i s c a n r e v e a l c o m p l e x d y n a m i c b e h a v i o r t h a t i sunlikely to be discovered using dyn a mic simulat ion alone.

Transient Model. O n t h e b a s is of t h e cy b er n et i cmodeling framework (32), J ones and Kompala (20) havedeveloped a structur ed, unsegregat ed model for continu-ous yeast bioreactors. Susta ined oscil lat ions are vieweda s t h e r e su l t o f c om p et i t i on b et w e e n t h r e e m e t a b ol icpathw a ys: glucose fermenta tion, ethan ol oxidation, andglucose oxidat ion. Deta iled modeling of the int ra cellularregulatory processes is replaced by cybernetic variablesui a n d vi representing the optimal st ra tegies for enzymesynthesis and activity, respectively. Denoting the instan-t a n e o u s g r o w t h r a t e a l o n g t h e i-t h p a t h w a y s a s ri, t h eoptima l stra tegies for ui a n d vi a re

I f t h e g r ow t h r a t e a l on g t h e i-t h p a t h w a y i s l a r g e , t h e nthe associa ted synth esis (ui) an d a ctivity (vi) will be large.The growth rate ri along each pathway is modeled withm od if ie d M on od r a t e e q ua t i on s i n w h i ch t h e r a t e i sproportional t o the intra cellular concentra tion of a keyenzyme ei controlling the i- t h p a t h w a y :

w h e r e G, E, a n d O represent the concentra tions ofglucose, ethanol and dissolved oxygen, respectively; i a rem a x i m u m g r ow t h r a t e con s t a n t s ; Ki a nd KO i a r e s a t u r a -tion constants for the substrate and dissolved oxygen,respectively.

Th e cy b er n e t ic m od e l e q u a t i on s p r es e n t ed i n t h eoriginal reference (20) c on t a i n s e ve r a l t y p og r a p h i ca l

errors. The corrected ma ss ba lances are writ t en a s:

w h e r e X a n d C a r e t h e ce ll m a s s c on ce n t r a t i on a n d

intracellular storage carbohydrate mass fraction, respec-tively; D i s t h e d i l u t i o n r a t e ; G0 is the glucose feedconcentration; kLa is the dissolved oxygen mass transfercoefficient; O* is the saturation concentration of oxygen;Yi ar e yield coefficients; i a n d i are stoichiometriccoeff icients for substrate and intracellular storage car-bohydrate synthesis and consumption, respectively; R a nd a r e e n z y m e s y n t h e s i s a n d d e c a y r a t e c o n s t a n t s , r e -spectively; and R* i s a c o n s t a n t p a r a m e t e r a s s o c i a t e dw i t h con s t i t u t i ve e n zy m e s y n t h e si s . A m or e d e t a i le ddescription of the model can be found in ref 20. It is wort hn ot i n g t h a t t h e c yb e rn e t ic m o d el d oe s n o t g en e r a t esusta ined oscil lat ions with t he para meter values exactlya s g i v e n i n r e f 2 0 . T h e p a r a m e t e r v a l u e s u s e d i n t h e

Figure 6. Two-parameter bifurcation diagram of the Z y mo mo n a s mo b il is reactor model.

dX

dt) (

i

(rivi) - D)X

dC

d t) 3r3v3 - (1r1v1 + 2r2v2)C -

i

(rivi)C

dG

d t) D(G0 - G) - (

r1v1

Y1+

r3v3

Y3)X - 4(CdXd t + XdCd t)

dE

d t) -D E + (1

r1v1

Y1-

r2v2

Y2)X

dO

d t) kLa(O* - O) -

(2

r2v2

Y2+ 3

r3v3

Y3

)X

de1

d t) Ru1

G

K1 + G- (

i

(rivi) + )e1 + R*

de2

d t) Ru2

E

K2 + E- (

i

(rivi) + )e2 + R*

de3

d t) Ru3

G

K3 + G- (

i

(rivi) + )e3 + R* (4)

ui )ri

jrj

vi )ri

m a xjrj

r1 ) 1e1G

K1 + G

r2 ) 2e2E

K2 + E

O

KO2+ O

r3 ) 3e3G

K3 + G

O

KO3+ O



9/14

subsequent simulation and bifurcation studies are givenin Ta ble 3. The par am eters a re obta ined from ref 20 withthe exception tha t R* ) 0.03 g/g-h .

Results and Discussion. As mentioned previously,the cybernetic model does not produce susta ined oscil-lat ions with t he para meter values given exactly a s in ref

20. Because th e value of the pa ra meter R* is not specifiedin th e origina l reference, a primitive search w as con-ducted to determine combina tions of the para meters tha tsupport periodic solutions. The following combinationsof D a nd G0 were found by dyna mic simulat ion t o producesusta ined oscil lat ions when R* ) 0.03 g/g-h:

Clearly this is a very inefficient method for determiningt h e r a n g e of p a r a m et e r v a l u es . B e low w e s h ow t h a tbifurcation a na lysis is a much more powerful tool fordetermining such information. I t is worth n oting that inthe original reference (20) bi furcation analysis is usedto determine the range of four unknown para meters (3,K3, KO2, KO3) that support periodic solutions.

First t he tra nsient beha vior of the model is studied viadyna mic simulat ion. Figure 7 shows tw o simulat ion testsat high dilution rates. The cell mass concentration (X) i sch os en a s t h e ou t pu t v a r ia b l e f or t h e cu lt u r e. B o t hsimulations are ini t iated with the steady-state solutioncorresponding to D ) 0.165 h -1 a nd G0 ) 10 g/L. The

feed glucose concentration is maintained at this valuew h i l e t h e d i l u t i o n r a t e i s c h a n g e d s l i g h t l y . W h e n t h ed i l u t i o n r a t e i s c h a n g e d t o D ) 0.1648 h -1 the modelm ov es t o a n e w s t e a d y -s t a t e s ol u t ion , w h i l e f or D )0.1646 h -1 the model exhibits sustained oscillations of a

sma ll amplitude. Figure 8 shows some simulation resultsa t l ow e r d il ut ion r a t es a n d t h e s a m e f eed g lu cos ec o n c e n t r a t i o n . T h e s i m u l a t i o n s a r e i n i t i a t e d w i t h t h estea dy-sta te solution for D ) 0.13 h -1. When the dilutionr a t e i s c h a n g e d t o D ) 0.1366 h -1, the model ini t ial lyos ci ll a t es a n d t h e n r ea c h es a n ew s t ea d y s t a t e a t a p -proximately 200 h. When D ) 0.1367 h -1, t h e m o d e lexhibits sust a ined oscillat ions wit h a considerably largera m p l i t u d e t h a n t h a t s h o w n i n F i g u r e 7 . A t t ) 200 h,the di lution ra te of the lat ter simulation is reduced to D) 0.1366 h -1. While the amplitude is decreased, sus-ta ined oscil lat ions a re maint ained even t hough t he samedilution ra te previously produced a stea dy-sta te solution.

These simulation results demonstra te the cyberneticm od el i s ca p a b l e of p re di ct i n g t h e a p pe a r a n ce a n d

Figure 7. D y n a m i c s im u l at i on of t h e Sacchar omyces cer evi siaereactor model at high dilution rates.

D ) 0.16 h -1 a nd G0 [9.5 g /L, 10.5 g/L ]

D ) 0.14 h -1 a n d G0 [9.7 g /L , 12.3 g/L]

G0 ) 10 g/L a nd D [0.14 h-1, 0.16 h -1]

G0 ) 12.3 g/L a nd D [0.118 h-1, 0.14 h -1]

Table 3. S a c c h a r o my c e s c er e v i s i a e Reactor ModelParameters

pa r a m et er unit va lue

i, max h -1 0.44, 0.19, 0.36

Ki g/L 0.05, 0.01, 0.001

Yi g/g 0.16, 0.75, 0.60

i g/g 0.403, 2.0, 1.0, 0.95i g/g 10, 10, 0.8

R g/g-h 0.3

R* g/g-h 0.03

h -1 0.7

KO2 m g/L 0.01

KO3 m g/L 2.2

kLa h -1 225

O* m g/L 7.5



10/14

disappeara nce of susta ined oscil lat ion over a ran ge ofop er a t i n g p a r a m e t e r s . H ow e v er , t h e r e su l t s i n d ica t esignificant differences between experiment and simula-tion with respect to the oscil lat ion amplitude at low an dh i g h d i l u t i o n r a t e s a s w e l l a s t h e h y s t e r e s i s b e h a v i o robserved a t low d ilution rat es. We propose to investiga tethese issues via bi furcation analysis. I t is well-knowntha t continuous yeast bioreactors exhibit sust ained oscil-lations only for a specific range of dilution rates (29). Thissuggests that the di lution rat e (D) is a reas onable choicef or t h e b i fu r ca t i on p a r a m e t e r . I n t h i s ca s e , t h e f e edglucose concentra tion G0 is fixed a t 10 g/L.

The one-parameter bifurcation diagram with D a s t h eb if u r ca t i o n p a r a m e t er i s s h o w n i n F i g u re 9 w h e r e t h eglucose concentration (G) is chosen as the output vari-able. The solid l ines represent stable steady-state andperiodic solutions, and the dashed l ines represent un-sta ble solutions. The middle bra nch betw een points 1 -4represents steady-state solutions for di f ferent di lutionrates. The upper branch def ined by points 1, 5, and 2represents t he largest G value obtained for the periodics o l u t i o n a t t h e g i v e n d i l u t i o n r a t e . T h e l o w e r b r a n c hconsist ing of points 1, 6, and 2 defines the lower G limitof the periodic solutions. There is a single sta ble stea dy-

Figure 8. D y n a m i c s im u l at i on of t h e Sacchar omyces cer evi siaereactor model at low dilution rates.

Figure 9. One-parameter bifurcation diagram of the Sacchar omyces cer evi siaereactor model with D as t h e b i f u r cat i on p a r am e t e ra n d G0 ) 10 g/L .



11/14

state solution at low dilution rates. When D is increasedto 0.13664 h -1 (point 1), a subcritical Hopf bifurcationoccurs a nd the steady -sta te solution becomes un sta ble.The periodic solution generated initially is unstable andbends back toward low dilution rates until a fold bifurca-tion occurs at D ) 0.13654 h -1 (point 5). Past this pointthe periodic solution is stable.

For a wide ra nge of di lution rat es, the sta ble periodicsolution coexists w ith a n unst able stea dy-sta te solution.When D is increased to 0.16496 h -1 (point 2), a super-critical Hopf bifurcation occurs, which causes the periodics ol u t ion t o d i s a p p ea r a n d t h e s t e a d y -s t a t e s ol u t ion t oregain i ts st abil i ty. The steady -sta te solution ma inta insits sta bil ity unt i l a bra nching point (22) a t D ) 0.20901h -1 (point 3) is reached. Then the steady-state solutionturns back in the direction of decreasing di lution rateand become unstable. Another fold bifurcation occurs atD ) 0.18341 h -1 (point 4) wh ere the stea dy-sta te solutionbecomes stable once again. Although not shown, higherdilution ra tes wil l cause th e steady st at e to lose sta bili ty

and the wash-out steady state to become stable.The bifurcation structure shown in Figure 9 providesa concise explan at ion of the previous dyna mic simulationresults. The a ppeara nce and disappearan ce of susta inedoscil lat ions is at tr ibutable to two Hopf bifurcations, asalso wa s noted by J ones et a l. (19). How ever, considera blymore information can be extracted from t he bifurcat iond i a g r a m :

Th e f ir s t H o pf b i fu r ca t i on i s s u b cr i t i ca l , a n d t h eas sociat ed periodic solution und ergoes a fold bifurcat ion.Consequently, t here exists a very sma ll range of dilutionrates [0.13654 h -1, 0.13664 h -1] where t here coexists asta ble steady sta te, an unsta ble periodic solution, a nd asta ble periodic solution. For these para meter va lues, theran ge is too small for the multiple stable solutions t o be

observed experimenta l ly. Nevertheless, t he subcrit icalf o r m o f t h e b i f u r c a t i o n a g r e e s w i t h t h e e x p e r i m e n t a lobservation, and the simulation results (Figure 8) thatsustained oscillations at lower dilution rates have a largeam plitude once established.

The second Hopf bifurcation point is supercritical.Therefore, th e periodic solution th a t ema na tes from th isbifurcation ha s very small a mplitude. This explains thedyna mic simulat ion results in Figure 7. To our knowledget h e r e a r e n o e x p e r i m e n t a l s t u d i e s t h a t d e m o n s t r a t e adif ference between t he oscil lat ion amplitudes tha t ar isefrom bifurcations at low and high dilution rates. How-ever, this m odel prediction could be investiga ted experi-mental ly.

There is hysteresis for dilution rates in the range ofD [0.18341 h -1, 0.20901 h -1] due to the presence ofmultiple st able steady-state solutions. Hyst eresis couldh a v e a n a d v e r s e e ff ect on t h e r e a ct o r o pe r a b il it y . I ta p p e a r s t h a t s u c h b e h a v i o r h a s n o t b e e n r e p o r t e d b yother theoreticians or experimentalists; further experi-

menta l studies are required to test t his model prediction.Additional knowledge can be obta ined by investigatingthe bifurcat ion behavior with respect to tw o para meterss i m ul t a n e ou s ly . F i g u re 1 0 s h o w s t h e t w o -p a r a m e t e rb if ur ca t i on d ia g r a m w i t h d il ut i on r a t e (D) a n d f e e dglucose concentration (G0) as the bifurcation parameters.The figure depicts t he locus of the H opf bifurcat ion point swh ere HB 1 and HB 2 refer to points 1 a nd 2, respectively,in F igure 9. The r egion conta ined inside t hese tw o locirepresents t he operating space tha t supports susta inedoscil lat ions. Unlike the two-parameter bifurcation dia-gram for the Z y momonas mobi l i s model in Figure 6, thepara meter region is not closed an d th ere is a multiplicityphenomena as shown in the inset . The locus for HB 2 ismonotonic, wh ile the locus for HB 1 has a n S sh a pe. This

Figure 10. Two-parameter bifurcation diagram of the Sacchar omyces cer evi siae reactor model with D a n d G0 as t h e b i fu r cat i onp ar am e t e r s .



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implies that for a certain region of operating conditions,HB 1 bifurcat es into multiple bifurcat ion points a t a f ixedG0 or a fixed D. Therefore, a one-parameter bifurcationanalysis with the other parameter fixed at an appropriatev a l u e w i l l y i e ld t w o o r t h r e e b i f u rca t i on p oi n t s t h a toriginate from point 1 in Figure 9. As shown in Figure10, there is a single bifurcation point for G0 ) 10 g/L.

The one-parameter bifurcation diagram with D a s t h eb if u r ca t i on p a r a m e t e r a n d G0 ) 9.25 g/L is shown inFigure 11. Similar to Figure 9, the bra nch between points1-4 represents a steady -sta te solution contained w ithin

a periodic solution appearing from the subcritical Hopfb if u r ca t i on p oi n t (p oi n t 1 ) a n d d i s a p pe a r i n g a t t h esupercritica l H opf bifurcat ion (point 4). The periodicsolution eman at ing from point 1 is unst a ble until the foldbifurcation (point 6).

U nlike Figure 9, there now exists a n unst able periodicsolution w ithin t he lar ge-a mplitude sta ble periodic solu-tion. The unstable periodic solution emanates from as u pe r cr i t ic a l H o pf b i fu r ca t i on (p oi n t 2 ) a n d e n d s a ta nother supercritica l H opf bifurcat ion (point 3). With inthis periodic solution, the steady-state solution regainsits stabil i ty. Therefore, within the range D [0.15100h -1, 0.15883 h -1] there is a sta ble steady-sta te solution,a n u n s t a b l e p e r io di c s ol u t ion , a n d a s t a b l e p er i od i csolution. The cybernet ic model can predict t he coexistence

o f a s t a b l e s t e a d y - s t a t e s o l u t i o n a n d a s t a b l e p e r i o d i csolution over a meaningful ra nge of di lution rat es. I t isimportant to emphasize that the existence of multiplea t t r a c t or s w o u ld b e q u i t e d i f fi cu l t t o ob s er v e u s i n gd y n a m i c s i m u l a t i on a l o n e. Th i s a g a i n s u pp or t s ou rcontention that bi furcation analysis is a very powerfultool for model analysis and val idation.

Th e b eh a v i or s h ow n i n F i g u re 1 1 i s p a r t i cu l a r l yinteresting a s we ha ve observed experimenta lly multipleat tra ctors in continuous cultures of Saccharomyces cer-evisiae(37). Figure 12 shows the results of an experiment

d e si g n ed t o e xa m i n e t h i s p h en om e n on . Th e e vol ve dcarbon dioxide signal is used a s a representat ive outputv a r i a b l e f or t h e c ul t u r e. Th e e xp er i m en t s t a r t s w i t hoscil latory dynamics that are obtained by switching theculture from batch to continuous operation. At t ) 20 ht h e d i l u t i o n r a t e i s s l o w l y r a m p e d d o w n o v e r a 2 4 hperiod unti l the oscil lat ions disappear . The stat ionarys t a t e i s p r e s er v ed f or 2 d a y s w h i l e t h e d i l u t ion r a t e i sm a i n t a i n e d a t t h e l o w v a l u e . A t t ) 92 h the di lutionr a t e i s s l ow l y r a m p ed u p a t t h e s a m e r a t e a s w a s u s e dfor t he negat ive ramp. Oscillations a re not observed w henthe di lution ra te is mainta ined at t he high value despitet h e f a c t t h a t a s l ig h t l y l ow e r d i lu t i on r a t e p r od u ce dos ci ll a t i on s a t t h e b eg i n n in g of t h e e xp er i m en t . Anenlarged view of the dynamic behavior during the f irst

Figure 11. One-para meter bifurcat ion diagr am of the Sacchar omyces cer evi siaereactor model with D as the bifurcation parametera n d G0 ) 9.25 g/L.

Figure 12. Experimentally observed transient behavior in a continuous culture of Sacchar omyces cer evi siae for ramp changes int h e d i l u t i on r at e .



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and last parts of the experiment is shown in Figure 13.The upper plot shows the large amplitude oscil lat ionsobtained at the beginning of the experiment, while thelower plot shows the stationary response observed at theend of the experiment. The bifurcation a na lysis in Figure1 1 s u g ge s t s t h a t t h e cy b e rn e t i c m od e l c a n p r ed i ct t h e

existence of such multiple attractors.A one-parameter bifurcation diagram with D a s t h ebifurcation parameter and G0 fixed a t 8.7 g/L is pres ent edi n F i g u r e 1 4 . A s s h o w n i n F i g u r e 1 0 , t h e r e a r e t w obifurcation points that emanate from point 1 in Figure

9. In th is case, both of the Hopf bifurcat ions a re subcriti-cal . The ra nge of di lution rat es tha t support periodicsolution is quite small , a nd sust ained oscil lat ions mightbe difficult to observe experimentally. The bifurcationstructure for di lution rat es outside the region sh own isv er y s i m i la r t o t h a t i n F i g u r e 9 a n d h a s b ee n o m it t e d .

6. Summary and Conclusions

We have studied the dynamic behavior of three con-

tinuous bioreactor models tha t exhibit complex steady-s t a t e a n d t r a n s ie nt b eh a v ior . B i fu r ca t i on a n a l y s is i ss h ow n t o p r ov id e a m or e co mp le t e p ict u r e o f m o d elbehavior tha n is possible wit h dyna mic simulat ion a lone.The determination of bifurcation points when the quali-ta t ive model beha vior changes a nd t he cha ra cterizat ionof the range of parameter values that supports certainbehaviors is va luable informa tion for model va l idation.S e v er a l i m por t a n t c h a r a c t er i s t ics of t h e t h r e e m o d e lss t u d i ed t h a t a r e n ot ob s er v ed i n p r ev iou s s i m ul a t i onstudies are revealed through bifurcation analysis. Thesecharacteristics include lack of model robustness to smallpara meter va riat ions, a ppar ent inconsistencies betweenmodel structure and experimenta l dat a, a nd t he coexist-ence of multiple stable solutions under the same operat-ing conditions. These case studies suggests that bi fur-cation analysis is a very powerful tool for analyzing low-dimensional bioreactor models and should be used inconjunction w ith dyna mic simulat ion for model va l ida-tion.

Acknowledgment

Financial support from the LSU Depart ment of Chemi-cal Eng ineering is gra tefully acknowledged. The aut horswould l ike to a cknowledge Prof . Mart in H jortso for hisa s s is t a n ce i n a c q ui ri n g t h e Sacchar omyces cer evisia eexperimenta l dat a a nd P rof. Eusebius Doedel (ConcordiaUniversity) for providing a ssistance with the continua tion

packag e AUTO. The a uthors a lso would like to acknowl-e d ge P r of . D o r a i s w a m i R a m k r i s h n a (P u r d u e ) a n d D h i -na kar K ompala (Colorad o) for th eir a ssistance with theHybridoma and Sacchar omyces cer evi saereactor models,respectively.

Figure 13. Experimentally observed multiple attractors in acontinuous culture of Sacchar omyces cer evi siae.

Figure 14. One-para meter bifurcat ion diagr am of the Sacchar omyces cer evi siaereactor model with D as the bifurcation parametera n d G0 ) 8.75 g/L.



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Accepted for publication May 4, 2001.

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