MODELING THE MICROBIAL GROWTH OF TWO Escherichia coli STRAINS IN A MULTI-SUBSTRATE ENVIRONMENT

Transcription

1 Brazlan Journal of Chemcal Engneerng ISSN Prnted n Brazl Vol. 3, No. 0, pp , Aprl - June, 04 dx.do.org/0.590/ s MODELING THE MICROBIAL GROWTH OF TWO Eschercha col STRAINS IN A MULTI-SUBSTRATE ENVIRONMENT M. E. Pocca, A. J. Beccara and R. G. Dondo * Laboratoro de Fermentacones, Facultad de Boquímca y Cencas Bológcas, Unversdad Naconal del Ltoral Cudad Unverstara, Parae El Pozo (S3000ZAA), Santa Fe, Argentna. Insttuto de Desarrollo Tecnológco para la Industra Químca, INTEC, CONICET, Phone: , Fax: , Unversdad Naconal del Ltoral, Güemes 3450, (3000) Santa Fe, Argentna. * E-mal: rdondo@santafe-concet.gov.ar (Submtted: March 6, 03 ; Revsed: August, 03 ; Accepted: September 0, 03) Abstract - The mcrobal growth n mult-substrate envronments may be vewed as an optmal resources allocaton problem. The optmzaton ams at mzng some bologcal obectve lke the bomass growth. The models developed usng ths hypothess are called cybernetc and they represent the complex cell structure as an optmzng functon that regulates the ntracellular enzymatc machnery. In ths work, a cybernetc model was developed to represent the growth of two E. col strans (JM 09 and BL DE3 ) on a medum contanng glucose and glycerol as carbon and energy sources. The model was able to accurately smulate the bomass growth, the substrates consumpton and the growth-rate profles. Keywords: Cybernetc model; Optmal control; Eschercha col; Mult-substrates; Glycerol. INTRODUCTION Modelng the knetcs of a fermentaton process n complex meda remans a challenge for bochemcal engneers and botechnologsts. Although emprcal models such as the Monod equaton descrbe the growth knetcs on a sngle substrate, they are usually not useful for a mult-substrate envronment. Dfferent growth phenomena such as sequental utlzaton of substrates, smultaneous substrates consumpton and co-metabolsm of substrates may be observed n these envronments (Bapa-Dksht et al., 003). There have been numerous attempts to model these phenomena wth an unfed representaton. A successful approach to these phenomena was the cybernetc modelng concept developed by Ramkrshna and co-workers (Kompala et al., 984; Kompala et al., 986; Varner and Ramkrshna, 999; Song and Ramkrshna, 00). In bochemcal systems, many reactons are catalyzed by enzymes, whch may be nhbted or actvated by regulatory effects on the genes that code for these enzyme actvtes. As a result, the changes at the regulatory level may mpact the dynamcs of the bochemcal pathways. Cybernetc models expand the tradtonal ordnary dfferental equatons (ODE) models by ncorporatng control laws that manpulate the enzyme synthess and/or actvty n order to drve the pathways toward a stated bologcal goal. The framework s based on the postulate that bologcal systems operate accordng to an optmal resources allocaton obectve. Ths obectve s developed as a result of evolutonary pressures mposed on the organsm. Varous crtera have been proposed as the optmalty crteron for the allocaton of crtcal resources: mum bomass productvty (Kompala et al., 984), mzaton of the fnal bomass concentraton (Dhurat et al., 985) and matchng-law allocaton (Kompala et al., 986). The cybernetc modelng concept hypotheszes that the *To whom correspondence should be addressed

2 348 M. E. Pocca, A. J. Beccara and R. G. Dondo cellular obectves of the wld-type and the genetcally altered systems are dentcal (Varner and Ramkrshna, 998), supportng the use of the same obectve functon for predctng changes n dynamcs due to regulatory effects. Eschercha col (E. col) s perhaps the most studed bacterum. It can adapt tself to very dfferent growth condtons. Hgh densty cultures of several strans of E. col are currently used to produce recombnant protens due to ts hgh volumetrc productvty; see e.g. Shloach J. and Fass R. (005) and Huang et al. (0). Ths, n practce, has transformed E. col nto a factory of recombnant protens and many pharmaceutcal products are produced n ths way. E. col s capable of consumng ether glucose or glycerol as carbon and energy (C/E) sources. Also t may consume ether organc ntrogen (N) from sources such as peptones or norganc (N)-salts. It s, therefore, extremely versatle. Although glucose s the classcal (C/E)-source, glycerol s an attractve alternatve because t has become an nexpensve and abundant (C/E)-source due to ts generaton as an nevtable by-product of bodesel (Yazdan and Gonzalez, 007; Gonzalez et al., 008) and ts overproducton has become an ssue of concern (Albarell et al., 0). The replacement of glucose by glycerol as (C/E)-source also allows obtanng hgher bomass denstes, probably because of a hgher solublty and a slower catabolsm, whch avods acetate producton (Kortz et al., 995). Ths makes glycerol a cheap opton as the man substrate for numerous ndustral cultures. So, t s desrable to adust the E. col growth condtons to promote glycerol consumpton rather than glucose consumpton. Nevertheless, glucose may be useful as a growth ntator. Consequently, n ths work a cybernetc model has been used to descrbe the dynamcs of glucose and glycerol consumpton by two E. col strans (JM 09 and BL DE3-) n batch and batch + substrate-necton cultures. MATERIALS AND METHODS Mcroorgansms and Medum E. col BL (DE3) and JM 09 strans were grown on MEP medum adusted accordng to gudelnes proposed by Gordano et al. (00): acd hydrolysate of casen -AHC- (30 g/l); yeast extract (7.5 g/l); NaCl ( g/l); glycerol ( g/l) and glucose ( g/l). Sterlzaton was done at C for 5 mn. AHC and yeast extract were acqured from Brtana, Argentna. NaCl, glycerol and glucose were acqured from Ccarell Laboratoros, Argentna. Although the concentratons of AHC, yeast extract and NaCl are relatvely hgh, the ncrease n the yeld wth respect to produced bomass overcomes ths dsadvantage (Gordano et al., 00). Inoculum Preparatons and Culture Condtons A loopful of each stran (stored n MEP slant agar at 4 C) was noculated nto a 50 ml Erlenmeyer flask contanng 30 ml of sterlzed MEP medum and grown overnght on a rotary shaker at 00 RPM and 30 C. Alquots of these cultures were ppetted nto 50 ml Erlenmeyer flasks contanng 30 ml of the same medum to obtan an ntal optcal densty at 600 nm (OD600) of 0.. The cultures were ncubated at 00 rpm at 30 C. Several samples were wthdrawn from each culture. Samples Analyss OD600 was measured and used as an estmate of the cell mass ( OD = 0.0 g dry weght/l). Alquots were centrfuged (0 g, 5 mnutes) and supernatants were stored at -0 C for further analyss. Analytcal Glucose and glycerol concentratons were determned n thawed samples. The enzymatc colormetrc assays from Glcema Enzmátca AA kt and TG Color GPO/PAP AA kt were employed for glucose and glycerol quantfcaton, respectvely. Both were purchased from Wener Lab, Argentna. Determnaton accuracy s 90.7 mg/dl ±.0 mg/dl (C.V. =.39%) for glucose concentraton and.4 g/l ± g/l (C.V. =.8%) for the glycerol concentraton. Computatonal Tools ODE and dfferental-algebrac equatons (DAE) were solved by usng the MATLAB ODE 45 solver (Mathworks, Natck, MA, USA) and the optmal control problems were solved by the GPOPS 4. toolbox (Rao et al, 0) runnng on MATLAB 7.6. MODEL DEVELOPMENT The mcrobal growth s represented here n a way smlar to the cybernetc model developed by Kompala et al. (984): ( ) e X/ S X/ S X + Y S + Y X () Brazlan Journal of Chemcal Engneerng

3 Modelng the Mcrobal Growth of Two Eschercha col Strans n a Mult-Substrate Envronment 349 In the assmlaton of the th substrate by the bomass X, t s assumed that there s one lumped keyenzyme whch plays the bottleneck role and hence must be syntheszed before the substrate s utlzed for growth. The synthess of the th key enzyme n the presence of the th substrate can be represented as follows: S X X + e () ds S = μ evx dt Y k S X/ S + de S = α u βe dt k ' + S (7) (8) The cybernetc varables u and v are computed from the followng algebrac equatons: The symbols X, S and e were used to represent the pseudo-reacton stochometry, but subsequently X wll represent the bomass concentraton, S wll represent the concentraton of the th substrate and e wll represent the normalzed ntracellular level of the enzyme necessary to consume the th substrate. The actual specfc growth μ wll arse from the sequental/smultaneous growth on all substrates. Accordng to the modfed Monod s expresson (Kompala et al., 984), μ wll be: u = v = μ μ μ S e k + S k S + S e S e k + S μ e k + S S (9) (0) S e (3) k + S μ= μ where s the substrate ndcator, μ s the mum specfc growth rate on the substrate and k s the Monod substrate saturaton constant. The ODE for the substrate consumpton and the specfc enzyme formaton rate n a batch envronment are gven by: ds S = μ X dt Y k S de dt e X/ S + (4) = au be (5) Some varatons n the shape of the above equaton were proposed (Kompala et al., 984 and 986; Dhurat et al., 985; Dosh et al., 997), but the fundamental structure of the ODE system remans unaltered. To model the E. col-stran growth on a medum wth glucose and glycerol, we selected the matchng-law allocaton model (Kompala et al., 986) and the whole model-system may be wrtten as: S evx (6) + dx = μ dt k S The parameters α and β stand respectvely for the specfc actvaton rate and degradaton rate of the key enzyme e. The rght-hand-sde of Eq. (8) neglects a term of ntracellular dluton of the enzymelevel due to the bomass growth that s presented n the model by Kompala et al. (986). Nevertheless, ths dluton effect can be lumped nto the β parameter and the DAE wll not lose predctve capablty at all. The mum specfc growth rates μ and the saturaton constants k are usually estmated by fttng data from the exponental phases correspondng to the respectve substrates. Also t s usually assumed that k = k. RESULTS AND DISCUSSION Expermental results for E. col strans growng n a medum wth glucose and glycerol are summarzed n Table A. of the appendx. In both cultures we observed three growth stages: a fast growth stage on glucose followed by a slower growth stage on glycerol and a very slow growth phase utlzng the peptdes from AHC as (C/E)-source. Hence, the whole culture seems to be trauxc. The model-parameters obtaned by mnmzng the least square errors between the expermental data and the model predctons are reported n Table. To avod an over-parametrzaton of the model, the values of parameters α and β were prefxed. So, growth parameters n the late culture phase, wthout glucose and glycerol, Brazlan Journal of Chemcal Engneerng Vol. 3, No. 0, pp , Aprl - June, 04

4 350 M. E. Pocca, A. J. Beccara and R. G. Dondo were forced to be dentcal n both least-squares fttngs. Fgures and compare the expermental data wth the model predctons. Table : Model parameters for the least squares fttng of expermental data from both batch runs. C/Esource Glucose Glycerol AHC Glucose Glycerol AHC μ k Y X/S α * E. col JM 09 stran *Prefxed to avod overparametrzaton Bomass concentraton (g/l; DO600) 0 0. E. col BL (DE3) stran Tme (h) β * (C/E)-sources concentraton (g/l) Glucose Glycerol Fgure : Model predctons (dotted lnes) vs. expermental data (square symbols) for the E. col JM 09 stran. Bomass concentraton (g/l; DO600) Tme (h) (C/E)-sources concentraton (g/l) Glucose Glycerol Fgure : Model predctons (dotted lnes) vs. expermental data (square symbols) for the E. col BL (DE3) stran. Snce expermental data for the growth on peptdes from AHC s scarce, the parameters for ths stage present a sgnfcant degree of uncertanty. Nevertheless, ths should not affect the desgn of a medum wth glucose and glycerol as the man (C/E)- sources because the growth on peptdes from AHC s much slower than the growth on the former substrates. As our am s to promote the utlzaton of glycerol whle usng glucose ust as a growth-ntator, to desgn a batch + necton culture, we solved an optmal control problem. In that problem, the glycerol concentraton may be changed by glycerol nectons up to the fnal culture-tme of t f = 0 h. The growth stage on peptdes from AHC was neglected n the optmzaton and the problem can be stated as follows: Maxmze ( f ) ( f ) Subect to: dv dt X t V t () = F () evx F (3) + dx S X = μ dt k S V F = μ evx + F F X/ S + ds S S S dt Y k S V V de S = α u βe dt k ' + S dq dt F (4) (5) = FS (6) Here F s the manpulated varable and the states are the culture volume (V), the bomass concentraton (X), the glucose concentraton (S ), the glycerol concentraton (S ), the normalzed ntracellular level of enzymes (e and e ) and the quantty Q of nected glycerol. Snce the necton s glucose-free S F = 0 g/l and Q = 0 g. The concentraton of glycerol n the feed s S F = 5 g/l. The obectve (Eq. ) s the mzaton, at the fnal run-tme t f = 0 h, of the bomass quantty subect to the constrants (Eqs. ()-(6)) of a fed-batch boreactor. The cybernetc varables u and v are computed from Eqs. (9) and (0). In addton, the constrant (7) mposes an upper bound Q to the quantty of nected glycerol: Q Q (7) Brazlan Journal of Chemcal Engneerng

5 Modelng the Mcrobal Growth of Two Eschercha col Strans n a Mult-Substrate Envronment 35 The optmal control problem s defned by the obectve functon (Eq. ), the dfferental equatons (Eqs. ()-(6)), the algebrac Equatons (9)-(0) and the state constrant (Eq. 7). It was coded on GPOPS 4. (Rao et al., 0) and solved n a.0 GHz 6 GRAM PC. We solved several nstances of ths optmal control problem by varyng the ntal states and the knetc parameter values. In all cases, the glycerol addton conssted of an necton ust before the start of the glycerol consumpton. The explanaton of ths feed-shape s farly smple. Snce μ for glucose s larger than for other substrates and the glucosesaturaton constant k s also very low, the glucose s preferred n the sequental substrate utlzaton. Because μ > μ and k < k, the optmal profle wll always yeld a sequental substrates utlzaton. So, an early necton of the slow substrate wll not ncrease the actual growth rate at all and t mght even brake the actvaton of the enzyme necessary for the degradaton of the fast substrate (.e., f μ s slghtly larger than μ ). Hence, the necton wll start ust after the glucose depleton and wll occur as early as possble. The goal s to mnmze the effect of the Monod saturaton-term for glycerol, amng at: μ S ev μ k + S (8) An optmal substrate-addton profle s llustrated n Fgure 3. The knetc parameters and ntal states of such a profle are reported n Table. The knetc and yeld parameters are those that ft the batch culture of the E. col BL (DE3) stran. After ths numercal experment, we performed a batch + necton culture to grow the E. col BL (DE3) stran under the envronmental condtons computed from the rule derved from the above optmal control problem. That rule conssts of nectng glycerol ust after glucose exhauston. Table : Model parameters and ntal states for the optmal control problem solved to desgn the glycerol necton profle. Tme horzon t f = 0 h Intal states X(0) S (0) S (0) e (0) e (0) V(0) Q(0) Yeld and knetc parameters C-source μ k Y X/S α β Glucose Glycerol Feed rate (l/h) Bomass concentraton (g/l) Tme (h) Tme (h) (b) (a) (C/E)-sources concentraton (g/l) Glucose Glycerol Fgure 3: An optmal glycerol necton profle (a) and the evoluton of the bomass and substrates concentratons (b). The expermental results for such a run are summarzed n Table A of the Appendx. The least squares ft of the model to the data yelded the parameters reported n Table 3 and Fgure 4 compares the model predctons wth the expermental data. The expermental results ndcate a growth rate on glucose smlar to the growth rate observed n the batch culture, but also ndcate a smultaneous consumpton of glucose and glycerol after t = 3 h and up to the tme of the glycerol-necton. The smultaneous substrates-consumpton was not caused Table 3: Model parameters for the least squares fttng of the expermental results presented n Table A. C-source Glucose Glycerol Inecton tme Inected volume F S μ k Y X/S * α E. col BL (DE3) stran *Prefxed to avod overparametrzaton β * 6 h 0. ml/ml culture 5 g/l Brazlan Journal of Chemcal Engneerng Vol. 3, No. 0, pp , Aprl - June, 04

6 35 M. E. Pocca, A. J. Beccara and R. G. Dondo Bomass concentraton (g/l; DO600) Tme (h) 3 0 (C/E)-sources concentraton (g/l) Glucose Glycerol Fgure 4: Model predctons (dotted lnes) vs. expermental (square symbols) data for the batch + necton culture wth the E. col BL (DE3) stran. by the necton because t occurred before. Before ths unexpected behavor and after the glycerol necton, the agreement s qute good. Therefore, the remanng expermental data seems to valdate the model also n a batch + necton mode. CONCLUSIONS A cybernetc model based on the mzaton of the nstantaneous specfc growth rate hypothess s able to represent a varety of expermental growth data nvolvng dauxc and trauxc growth. The model utlzes a lumped representaton of the cell populatons, and the mcrobal growth on varous (C/E)-sources can be treated as a multvarable constraned optmzaton problem, as proposed by Kompala et al (986). Ths model was chosen to predct the growth of two E. col strans on a medum wth glucose and glycerol. It can accurately represent the growth dynamcs of both strans n batch runs. To desgn a batch + necton culture, an optmal control problem was subsequently solved. As a consequence, an necton rule was derved and an addtonal experment was performed. In that case, the agreement between the model predctons and the expermental data was not as good as for batch cultures, although ths cannot be attrbuted to the glycerol necton. So, the emprcal rule remans vald. ACKNOWLEDGEMENTS Ths work was supported by grants from Unversdad Naconal del Ltoral (CAI+D and CAI+D ). We also want to thank the revewer for helpful observatons. NOMENCLATURE a Specfc actvaton rate for the th enzyme b Specfc nactvaton rate for the th enzyme e Normalzed ntracellular level of the th enzyme k ; k Monod substrate saturaton g/l constant F Feed rate L/h Q Mass of the th substrate nected to the culture S Concentraton of the th g/l substrate F S Concentraton of the th g/l substrate n the feed u Cybernetc varable for controllng the synthess of the th enzyme v Cybernetc varable for regulatng the actvty of the th enzyme V Culture volume L X Bomass concentraton g/l Y X/S Yeld of bomass on the th substrate g/g Greek Symbols α Specfc actvaton rate of the th enzyme β Specfc nactvaton rate of the th enzyme e μ Actual specfc growth rate h - μ Maxmum growth rate on h - the th substrate REFERENCES Albarell, J., Santos, D. and Holanda, M., Energetc and economc evaluaton of waste glycerol cogeneraton n Brazl. Brazlan Journal of Chemcal Engneerng, 8(4), (0). Bapa-Dksht, J., Suresh, A. and Venkatesh, K., An optmal model for representng the knetcs of growth and product formaton by Lactobacllus rhamnosus on multple substrates. Journal of Boscence and Boengneerng, 96(5), (003). Brazlan Journal of Chemcal Engneerng

7 Modelng the Mcrobal Growth of Two Eschercha col Strans n a Mult-Substrate Envronment 353 Dhurat, P., Ramkrshna, D., Flcknger, M. C. and Tsao, G. T., A cybernetc vew of mcrobal growth: Modelng of mcrobes as optmal strategsts. Botechnology and Boengneerng, 7, -9 (985). Dosh, P., Rengaswamy, R. and Venkatesh, K., Modellng of mcrobal growth for sequental utlzaton n a multsubstrate envronment. Process Bochemstry, 3(8), (997). Gordano, P., Martínez, H., Iglesas, A., Beccara, A. and Gocoechea, H., Applcaton of response surface methodology and artfcal neural networks for optmzaton of recombnant Oryza satva non-symbotc haemoglobn- producton by Eschercha col n medum contanng byproduct glycerol. Boresource Technology, 0, (00). Gonzalez, R., Murarka, A., Dharmad, Y. and Yazdan, S., A new model for the anaerobc fermentaton of glycerol n enterc bactera: Trunk and auxlary pathways n Eschercha col. Metabolc Engneerng, 0, (008). Huang, C-Jr, Ln, H., Yang, X., Industral producton of recombnant therapeutcs n Eschercha col and ts recent advancements. Journal of Industral Mcrobology and Botechnology, 39, (0). Kompala, D., Ramkrshna, D., Jansen, N. and Tsao, G., Investgaton of bacteral growth on mxed substrates: Expermental evaluaton of cybernetc models. Botechnology and Boengneerng, 8, (986). Kompala, D., Ramkrshna, D. and Tsao, G., Cybernetc modelng of mcrobal growth on multple substrates. Botechnology and Boengneerng, 7, 7-8 (984). Kortz, D., Rnas, U., Hellmuth, K., Sanders, E. and Deckwer, W., Smple fed-batch technque for hgh cell densty cultvaton of Eschercha col. Journal of Botechnology, 39() (995). Rao, A. V., Benson, D., Darby, C., Francoln, C., Patterson, M., Sanders, I. and Huntngton, G. T., User s Manual for GPOPS Verson 4.0 (0). Shloach, J., Fass, R., Growng E. col to hgh cell densty. A hstorcal perspectve on method development. Botechnology Advances, 3(5), (005). Song, H. S. and Ramkrshna, D., Predcton of metabolc functon from lmted data: Lumped Hybrd Cybernetc Modelng (L-HCM). Botechnology and Boengneerng, 06(), 7-84 (00). Varner, J. and Ramkrshna, D., Metabolc engneerng from a cybernetc perspectve.. Theoretcal prelmnares. Botechnology Progress, 5(3), (999). Varner, J. and Ramkrshna, D., Applcaton of cybernetc models to metabolc engneerng: Investgaton of storage pathways. Botechnology and Boengneerng, 58(-3), 8-9 (998). Yazdan, S. and Gonzalez, R., Anaerobc fermentaton of glycerol: a path to economc vablty for the bofuels ndustry. Current Opnons n Botechnology, 8, 3-9 (007). Brazlan Journal of Chemcal Engneerng Vol. 3, No. 0, pp , Aprl - June, 04

8 354 M. E. Pocca, A. J. Beccara and R. G. Dondo APPENDIX Table A.: Expermental growth data for batch culture of E. col strans JM 09 and BL (DE3) on a mxed medum of glucose and glycerol. Tme (h) Bomass (g/l) Glucose (g/l) Glycerol (g/l) E. col JM 09 stran E. col BL (DE3) stran Table A.: Expermental growth data for batch + necton culture of E. col BL (DE3) stran on a mxed medum of glucose and glycerol. Tme (h) Bomass (g/l) Glucose (g/l) Glycerol (g/l) E. col BL (DE3) NA NA Brazlan Journal of Chemcal Engneerng