THE USE OF MODELS IN FERMENTATION CONTROL

THE USE OF MODELS IN FERMENTATION CONTROL T. Yoshida and H. Taguchi Depa men o Fe men a ion Technology Facul y o Enginee ing, Osaka Uni e si y Yamada-kami, Sui a-shi, Osaka 565, Japan Abs ac Recen ly he e has been a s ong in e es in he di ec digi al con ol o e men a ion p ocesses. Mo e e o on iden i ica ion o modeling o he p ocesses is indispensable o accomplish his highly sophis ica ed con ol. This pape was w i en o p esen a undamen al iew on model cons uc ion, p ocess iden i ica ion, pa ame e es ima ion, analysis o model and examples o uses o model. Models in he pape s su eyed in he las i e yea s a e p esen ed classi ying hem in o h ee ca ego ies: subcul u e, subcellula and submolecula models. Assessmen o model by means o sensi i i y analysis and se e al app oaches o modi y models o p ocess con ol a e discussed. 94 T. Yoshida, H. Taguchi 1. In oduc ion Con inuous de elopmen s in e men a ion echnology suppo ed by emendous ad ances in molecula biology and o he ela ed ields ha e esul ed in mo e p ecise unde - s anding o mic obial sys ems. Mo e quan i a i e exp essions o e men a ion p ocesses ha e accumula ed as a esul o p og ess in s udies o e men a ion kine ics by a ious kinds o app oaches, some o which ha e been de eloped in ela ed ields, mainly chemical enginee ing. The scale o e men a ion p oduc ion is g owing and mo e so- phis ica ed echnology o con ol enginee ing migh be applied o ou ield in conse- quence. A compu e aided con ol sys em is he mos c a ed aid o e men a ion sys ems, because he e men a ion sys em has a mul iplici y o a iables. P ocess con ol means no only he p ac ice o an op imal policy bu also sol ing he op imum p oblems. The iden i ica ion o he model is essen ial o p ocess con ol. Takama su [1] epo ed he esul s o an inquisi ion abou he way o aise he e iciency o a compu e coupled sys em. 27 o 83 answe s om he chemical enginee s wo king, o p ac ical use o a compu e , eques ed imp o emen o de elop- men o he ma hema ical model, and ano he 17 wan ed a mul iple a iable con ol sys em. The in es iga ion done wi h chemical enginee s may no ag ee wi h one wi h biochemical enginee s. The esul is, howe e , e y in e es ing and he au ho s expec ed a s onge eques o modeling o e men a ion p ocesses. Up o now many models ha e been p esen ed, analysed and applied. Fi s , a gene al concep o modeling o e men a ion p ocesses, is desc ibed and examples o models a e shown acco ding o a classi ica ion in o h ee ca ego ies. Some echnical me hods o p ocess iden i ica ion and pa ame e es ima ion a e lis ed. A e applica ions o sensi i i y analysis is in oduced, modi ica ion o simple models and e inemen o complex models a e discussed. Finally, a couple o examples o he uses o models in ac ual e men a ion con ol and in de eloping con ol algo i hms a e shown. 2. A Gene al Concep o Fe men a ion Models The pe o mance o e men a ion p ocesses is based on he biological ac i i y, which is an exhibi ion o in e ac i e enzyme ac ion. Main o ganisms employed o indus ial pu poses a e mic oo ganisms such as bac e ia, ungi and s ep omyces. Many echniques o modeling and simula ion de eloped in chemical enginee ing can be applied. Howe e mo e e o s a e s ill necessa y o he quan i a i e unde s anding o he esponse o mic oo ganisms o en i onmen al ac o s. Recen ly he e has been a e- mendous de elopmen o he ins umen a ion o de ec and con ol en i onmen al ac o s and his ad ance has allowed mo e de elopmen o modeling. A g ea a ie y o models ha e been de eloped, he amous Monod exp ession [2] o T. Yoshida, H. Taguchi 95 mic obial cell g ow h is he i s example. Acco ding o he li e a u es (e.g. Tsuchiya [3]) models we e classi ied in o wo ypes, namely uns uc u ed models in which a uni o mly dis ibu ed cell mass is conside ed along wi h he ela ionship be ween he biomass and p oduc ion and s uc u ed models in which he cell s uc u e is conside ed, he o al ni ogen con en is di e en ia ed (DNA, RNA, p o ein), and u he mo e enzyme ac i i ies in me abolic pa hways a e possibly aken in o accoun . Fe men a ion sys ems show complex nonlinea cha ac e is ics caused by se e al hun- d ed in e ac i e enzyma ic eac ions. I migh be impossible o desc ibe he sys em in ol ing all enzyme eac ions. The e o e one needs o ex ac a mas e ule om he many obse a ions on he esponses o he sys em o en i onmen al ac o s o o pick up a a e-limi ing one ou o he many eac ions in he sys em. Wi h his backg ound in mind we like o di ide he models o e men a ion p ocesses in o h ee ca ego ies in he same way as Koga e al. [4] sugges ed, namely: subcul u e, subcellula and submolecula models. | (1) Subcul u e model In subcul u e models he mic obiological sys em is deal wi h a an a e aged mac oscopic le el and many o hem a e accep ed as he p ocess model o p ac ical use in design and con ol o e men a ion p ocesses. I is assumed ha he a e o g ow h and me abolic ac i i y a e explici unc ions only o he s a e o he sys em, and no o ime as ollows, = CC) (1) whe e € is a a iable ec o . When he s a e in he model e e s o popula ion densi y, he g ow h a e is exp essed as a unc ion o popula ion densi y: us (x) (2) whe e , is he g ow h a e, namely he a e o p oduc ion o cell mass pe uni olume o cul u e and X is he popula ion densi y, o he cell mass pe uni olume o cul u e. Nyi i [5] has so ed ou many uns uc u ed models o he g ow h a e and he p oduc ion a e. The i s ype o he g ow h model shows exponen ial g ow h, = ux (3) x | whe e LU is a kine ic coe icien e med he speci ic g ow h a e, which is a unc ion : o en i onmen al ac o s as discussed la e . Khan and Ghose [6] p esen ed he same | kind o model ha Kono and Asai [7, 8] de eloped. Thei model is based on he ac ha he alue o coe icien U changes acco ding o he phase in a ba ch cul u e, namely, induc ion, ansien , exponen ial g ow h and declining g ow h. The nex is he logis ic law model: 96 T. Yoshida, H. Taguchi Bam HX(1 - BX) (4) This simple model seems o be use ul o he simula ion o a ba ch cul u e a cons an en i onmen al condi ions. Cons an inides ied o p edic he op imal empe a u e p o iles o ba ch e men a ion o penicillin using he logis ic model. Monod assumed he ollowing s oichiome ic ela ion be ween g ow h and consump ion o he subs a e which limi s g ow h: E --+Y (5) whe e s is he consump ion a e o he limi ing subs a e and Y is e med he yield cons an . Nex Monod ecognized ha he g ow h a e could ha e a maximum alue, and he pos ula ed ha he subs a e dependence o g ow h a e ollowed he Michaelis- Men en o m; su e 2 (6) whe e S is he concen a ion o he limi ing subs a e and un is he maximum speci ic g ow h a e ob ained when S is much g ea e han he cons an Ky: which is e med he hal - eloci y cons an . Eqs. (5) and (6) oge he a e a comple e s a emen o Monod's model, and hey may be applied o a ious cases, o op imiza ion, p ocess analysis and p ocess con ol. In he con inuous e men o , u_Ss dx m qe h gaa e ” s us as _ Lam (8) de an S R _ whe e Sn is he concen a ion o limi ing subs a e in he eed. Usabili y o Monod's model o e men a ion p ocesses has been in es iga ed by many au ho s. They ecognize he accep abili y o he model o he analysis o s eady s a e, hough wi h some modi ica ions, e.g. conside a ion o endogenous e m in equa ions. Ex ensi e appli- ca ion o he model o he p edic ion o ansien beha iou has been ied. Many au ho s [9-12] claimed, as desc ibed in a la e i em, ha modi ica ions should be made o he model o analysis o ansien s a e. Recen ly, mul i-subs a e limi ing sys ems we e subjec ed o s udy. Sinclai and Ryde [13] de eloped models o con inuous cul u e unde bo h ca bon and oxygen limi - ing condi ions. I was ound ha Monod kine ics we e applicable o he g ow h a e dependence on oxygen concen a ion bu Con oi's kine ics we e supe io o he co - esponding dependence on ca bon subs a e concen a ion. Fujio e al. [14] also p esen ed a model o ed-ba ch cul u e unde ca bon and oxygen limi ing condi ions. Spi ze [15] analysed he sys em unde con ol o subs a e and pH by means o he model which includes he equa ion o change in concen a ion o hyd ogen ions. T. Yoshida, H. Taguchi 97 Some p oduc s cause a educ ion in g ow h a e. Se e al ela ionships ha e been de eloped o cha ac e ize his educ ion, one is an exponen ial educ ion in g ow h a e [16]: = UL exp (-kP,) (9) Eganbe die e al. [17] conside ha a ec angla hype bola i s he da a mo e co ec ly bed me (10) Hinshelwood [18] has shown linea inhibi ion o he o m I w= u, G.0-4 P,) (11) Zines [19] in es iga ed he s eady s a e and he dynamic p ope ies o he e men a i e bac e ium, K. ae ogenes, using he Hinshelwood model. F equency esponse analysis, using sinusoidal a ia ions in he dilu ion a e showed ha e hanol inc eased he ime cons an o he me abolic pa ame e . (2) Subcellula model Mic obial p oduc s ha e been b oadly classi ied in o wo g oups, "g ow h associa ed" and "nong ow h associa ed". This classi ica ion was made acco ding o whe he he p oduc is o med while he o ganism is in he exponen ial o in he s a iona y phase. An ibio ics (e.g., penicillin, no obiocin, s ep omycin), oxin, and an igens, which a e seconda y me aboli es, ha e been included in he g oup o nong ow h associa ed p oduc s. Models o seconda y me aboli e p oduc ion mus desc ibe he a ia ions o p ope ies, in o he wo ds, chemical composi ion and me abolic ac i i ies o indi- idual cells. Sawada and Kojima [20] u ilized he s uc u ed model de eloped by Tsuchiya e al. [3] o he in es iga ion o he e ec o mic omixing on a con inuous ope a ion. In he model, cells a e di ided in o wo kinds, iable cells and inac i e cells. The la e we e iable cells be o e he p oduc ion o an inhibi o y subs ance by he cells. This model can p edic he g ow h cu e including lag, exponen ial and declining g ow h and s a iona y phase and he e ec o age o innoculum on he g ow h o cul u e. Ano he example o a subcellula model is based on he age app oach associa ing he p oduc biosyn hesis a e wi h he age o he cul u e. Shu [21] exp essed he a e o p oduc ion as a unc ion o he age o he cell (6). eg z A, exp (-k, 9) (12) Blanch and Roge s [22] de eloped a model o g amicidin S acco ding o he concep ha each cell in a cul u e ollows a p ede e mined gene ic o de in ca ying ou i s 98 T. Yoshida, H. Taguchi me abolic unc ions. The o ma ion o p oduc s by an o ganism can he e o e be con- side ed as a unc ion o cell age. The cell age is a bi a ily di ided in o wo g oups, iz., "imma u e" and "ma u e", whe e ma u e cells o m he p oduc . B own and Vass [23] also p oposed a me hod o de e mine he ma u ing ime and applied i o se e al an ibio ic p oduc ions. Ma sumu a e al. [24] analysed cell beha iou in he ansien s a e o a con inuous cul u e wi h a sophis ica ed "appa en age" model conside ing he esidence ime dis ibu ion in a e men e . (3) Submolecula model P og ess in s udies on cell physiology and me abolism allows he cons uc ion o mo e p ecise models which desc ibe changes o chemicals in cells. Pe inge e al. [26] p esen ed a ma hema ical model using he concep o an equilib ium be ween he ATP equi emen s and he ATP p oduced by ca abolic p ocesses. An analy ical exp ession o he ime cou se o g ow h could be de i ed which p o ides a quan i a i e exp ession o he Pas eu e ec , p edic s he alue o Y ins an aneous alues o P/O a ios, ATP? he quan i y o subs a e and oxygen consumed and o biomass p oduced. K eme [27] de i ed a kine ic model exhibi ed by ace yl Co-A me abolism and ATP egene a ion o in es iga e possible con ol mechanisms in e acycline p oduc ion. Monod [25] has de eloped he mos a o ed hypo hesis on induc ion and ep ession o enzyme p oduc ion in cells. His ope on hypo hesis is a splendid example o ecen ad ances in molecula biology and e y help ul o in es iga e mic obial beha iou e en om a mac oscopic poin o iew. The hypo hesis is cons uc ed wi h he heo ies o gene ical egula ion o enzyme p oduc ion wi h chemical e ec o s, namely, ep esso and induce . Models based on he ope on hypo hesis ha e been de eloped o quan i a i e exp ession o indus ial e men a ion p ocesses. P e e en ial syn hesis o a enzyme, a-galac osidase, was desc ibed in submolecula model by Imanaka e al. [28]. Enzyme p oduc ion a e is p opo ional o he concen a ion o mRNA in cells (M): dE eae keM - LE (13) whe e E deno es enzyme con en in cells and he second e m on he igh hand side o he equa ion ep esen s dilu ion by g ow h. m-RNA is syn hesised wi h nega i e co - ela ion o he con en o ep esso (R) and is subjec o i s o de decomposi ion: dM de ke (R, - R) - k7M - uM (14) R. is he c i ical concen a ion o ep esso o s op he syn hesis o m-RNA. The change o ep esso con en , R, and a complex o ep esso and induce , Rön; > we e exp essed as ollows: dR _ —— ae denen ue eas kuRS,, + ksRS, - uR (15) T. Yoshida, H. Taguchi 99 BL a e O Da HE = kuRS, KsBS,. URS, (16) whe e Sai is he in acellula concen a ion o induce , galac ose. This model ga e a good p edic ion o he ba ch and con inuous cul u e o a-galac osidase p oduc ion [29] and was u ilized o he op imiza ion o he enzyme p oduc ion p ocess [30]. Van Dedem [31] p oposed he same kind o model desc ibing he deg ee o ep ession o induc ion o ex acellula enzyme syn hesis as a unc ion o nu ien concen a ion. 3. P ocess Iden i ica ion and Pa ame e Es ima ion Iden i ica ion o dominan biological o chemical a e p ocesses is essen ial o he op imal ope a ion and he bes pe o mances o he e men a ion p ocess, al hough in o ma ion abou mass and hea ans e and mixing cha ac e is ics is some imes necessa y. The esponses o he p ocesses o he change o ci cums ances could be in es iga ed in wo ways: expe imen s wi h he ac ual p ocesses o simula ions o he p ocesses by means o models. (1) Simula ion o e men a ion p ocesses Simula ion is de ined in he p esen a icle as he desc ip ion o eal sys ems o e men a ion p ocesses by use o ma hema ical equa ions; compu e me hods a e gene ally equi ed o hei solu ion. Blanch and Dunn [32] p esen ed a ull commen a y on modeling and simula ion in biochemical enginee ing. Se e al impo an s eps can be ecognized in any simula ion de elopmen as shown in he igu e. Fi s , he pu pose o Theo e ical Backg ound ee eee — | Chemical Kine ics N | Enzymology | Ma hema ical] „| Pa ame e Cell Physiology Model Es ima ion Popula ion Dynamics T anspo Phenomena A De e mina ion o Expe imen al Condi ions S eps o simula ion o e men a ion p ocesses simula ion mus be clea ly de ined and a good insigh in o eal e men a ion phenomena is qui e help ul o modeling. The second s ep is o w i e he equa ions. The e a e wo s a ing poin s o his s ep: one is heo e ical analysis and he o he is analy- sis o expe imen al esul s. Fundamen al knowledge abou e men a ion and ela ed 100 T. Yoshida, H. Taguchi ields is a need o bo h cases. The hi d is he pa ame e es ima ion by expe imen s wi h he aid o ma hema ical o nume ical echniques. The expe imen al condi ions depend on he me hod o pa ame e es ima ion. Finally, he alidi y o he model should be jus i ied wi h expe imen s, and econs uc ion o model should be done i necessa y. Chi and Howell [10] p esen ed a ypical example o a e p ocess iden i- ica ion compa ing wo compe ing models. They compa ed a modi ied Powell-Ie usalenskii bo le neck model o a Monod-Haldane model o he p edic ion o he ansien beha iou o a con inuous cul u e. They used Pseudomonas which has a cha ac e is ic o subs a e inhibi ion. Applying Monod-Haldane model, us le ONE an Ss 4; Following Powell [33], hey in oduced Q subs a e which is a quan i a i e exp ession o he physiological s a e o he mic oo ganisms and go e ns he g ow h as ollows: lL = Q YG (S) (18) and he change o Q was exp essed in he ollowing equa ion: aQ _ s Y s d Um K #8487 7K, a CR_+5+057/R, (19) whe e C is he a io o he lowe limi o Q, Q, o he uppe limi A The obse ed ansien beha iou was p edic ed by he new model, bu could no be p edic ed by a Monod-Haldane model. (2) Techniques o pa ame e es ima ion Pa ame e es ima ion is one o he mos impo an p oblems in he iden i ica ion o p ocesses. Values o pa ame e s mus be es ima ed by analysis o expe imen al esul s o he mos a ional explana ion o he beha iou o he p ocess. Obse a ions o Measu emen s ob ained by expe imen s a e a he andom a he han de e mis ic, since unde ac ual condi ions, p ocess noise, cycling, signal noise and o he phenomena in e e e wi h all measu emen s. Linea leas squa es es ima ion me hod has been employed o pa ame e es ima ion o a e p ocess in e men a ion. Many pa ame e es ima ion me hods ha e been de eloped o he nonlinea pa ame e sys em. The leas squa es pa ame e es ima ion does no equi e p io knowledge o he dis ibu ion o unobse ed e o s, and i yields unbiased es ima es, and esul s in he minimum a iance among all he linea unbiased es ima ion me hods, he leas squa es echnique is used ex ensi ely. Values o pa ame e s a e es ima ed o minimize n o=,2, [¥-G@lwiy-6] (20) whe e Y is he ma ix o obse a ion and G is he ma ix o p edic ion. I Wis a diagonal ma ix, becomes a "weigh ed leas squa es" c i e ion, i W is a uni ma ix is he "o dina y leas squa es" c i e ion. Box and D ape 's c i e ion [34] was applied by Johnson and Be ho ex [35] o es ima e wo pa ame e s in a Monod model o con inuous T. Yoshida, H. Taguchi 101 cul u e. They showed ha he con idence egion o es ima ed pa ame e s wi h bo h he da a o cell concen a ion and subs a e concen a ion is much smalle han wi h he da a o any o he wo alone. Sunds om e al. [36] s udied esponse o biological eac o s o he sinusoidal a ia ions o subs a e concen a ion and employed he ollowing c i e ion. Ss - oN 2 xX - X, 2 o=2 ica ee e l, (21) e ie whe e subsc ip s p and e deno e p edic ion and expe imen , espec i ely. Bode diag am has been u ilized o es ima e pa ame e s o analyse he esponse o sinusoidal a i- a ion o inpu signal in a e men a ion sys em [19, 37, 38]. 4. Reassessmen o Con en ional Models o P ocess Con ol A la ge numbe o models ha e been p esen ed as su eyed in p e ious i ems, bu i is a big p oblem o choose which model should be employed o "p ocess con ol". C i e ia o choice o model may be lis ed as ollows: 1. Objec o con ol: wha kind o con ol is desi ed; holding con ol o s a e a iable cons an , p og am con ol o op imal adap i e con ol? Mo e sophis ica ed models mus be p epa ed o mo e ine con ol. 2. A ailabili y o equipmen o acili ies o con ol: a ailable model may be limi ed wi h ex en o ins umen a ion, al hough e e y e o should be made o sea ch o a de ec o o con olle . 3. A ailabili y o an algo i hm o con ol: a con ol algo i hm is ob iously essen ial o ac ual compu e con ol, he e o e a complex model can no be employed. Howe e , i is no so di icul o de elop a con ol algo i hm i he e is adequa e echnical alen . Any models de eloped p e i- ously o o he pu poses, (e.g., kine ic s udy, undamen al analysis o mic obial beha iou o design o plan ) migh be analysed and modi ied wi h a iew o applica ion o p ocess con ol. (1) Applica ion o sensi i i y analysis o assess models The echnique o sensi i i y analysis has been applied o s udy he e ec o pa ame e pe u ba ion on s a e a iables and objec i e unc ions o op imal con ol. Yoshida and Wada [39] analysed he beha iou o a con inuous cul u e o implemen a ion o p ocess con ol by means o a sensi i i y analysis. In he con inuous e men a ion sys em whe e he pe o mance equa ion is exp essed as Eqs. (7) and (8), he ollowing equa ions can be ob ained o he small pe u ba ions, x1 and xz o he wo s a e a iables X and S: dxı _ 42 = Yabxz (22) L —