MATHEMATICAL MODEL FOR DIAUXIC GROWTH OF MICROORGANISMS IN MIXED SUBSTRATE MEDIUM
Tetsuji CHOHJI, Tatsuro SAWADAAND Yoshitoshi NAKAMURA Department of Chemical Engineering, Faculty of Technology, KanazawaUniversity, Kanazawa920 SlGERU KUNO Department of Biochemistry, School of Medicine, KanazawaUniversity, Kanazawa920
Key Words: Biochemical Engineering, Bacterial Growth, Diauxic Growth, Catabolite Repression, Enzyme Induction
The mathematical model for diauxic growth of microorganisms in the presence of two substrates, such as glucose and maltose, was proposed, being constructed in due consideration of catabolite repression and enzymeinduction. In the model, it was assumed that the first substrate is metabolized by a constitutively synthesized enzymesystem, while the second substrate is utilized by an inducibly synthesized enzyme. The synthesis of the inducible enzymeis dually controlled by catabolite repression, which is caused by the first substrate, and induction which is triggered by the second substrate as an inducer. The extent of catabolite repression was expressed as the inhibition of promoter activity of inducible gene. The promotor of inducible gene was assumed to be activated by a co factor, the synthesis of which is inhibited by the first substrate. The extent of induction was expressed as the activity of the operator, which is activated by the second substrate. The equations introduced from the model were applied to experiments carried out with a batch culture, with glucose and maltose as carbon sources. The calculated values were in satisfactory agreement with the experimental data, especially in the estimation of lag time between the first log phase state and the second one.
Several models1'4'6"10'15'1^ have been presented Introduction to date for the quantitative expression of diauxic Bacterial treatment of wastewater has been often growth. However, these models do not take ade- used for sanitary purposes. However, it is very diffi- quate.consideration of repression and induction, so cult to degrade nutrients completely, because waste- that they are somewhatdoubtful as to their applica- water usually contains many type of nutrients. bility for various types of diauxic growth. Although bacteria are known to degrade several The present investigation has aimed at the quanti- nutrients simultaneously, they occasionally utilize tative expression of diauxic growth by a mathematical preferentially only one type of nutrient among the model. The model has been constructed on the basis multiple components contained in media. In the latter of the notion of transcriptional control due to cata- case, other nutrients are degraded only after the bolite repression and induction. The proposed model preferentially utilized nutrient is almost exhausted. showed satisfactory agreement with experimental re- This phenomenon, termed "diauxie" by Monod,12) is sults in the diauxic growth of a kind of soil bacteria characterized by growth phases in which two or more with glucose and maltose as nutrients. logarithmic growth states are separated by one or more lag. The mechanism of diauxie growth has been 1. Mathematical Model extensively studied in Escherichia coli, especially for Intracellular enzymes can be classified into the the expression of the lac gene.2'5'13'17) The expression following classes: constitutive enzymes, which are of an inducible gene like the lac gene is dually synthesized constantly no matter what the environ- controlled by induction and catabolite repression mental conditions of the cell, so that enzyme content (glucose effect); that is, the transcription of inducible per cell mass is usually constant; and inducible en- gene requires an inducer, usually the substrate or its zymes, which are generally synthesized in trace analogue for the enzyme derived from the inducible amounts, but are synthesized rapidly in case of nec- gene, and is strongly repressed in the presence of essity, such as in the presence of a substrate for glucose or a good growth substrate. these enzymes. Whentwo substrates, St and S2, are simultaneously added to the medium, and S1 and Received November 29, 1983. Correspondence concerning this article should be addressed to T. Chohji. S2 are utilized with the action of constitutive and in-
478 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN ducible enzymes, respectively, bacteria are unable to of the co factor, z. Thus, the relationship between the synthesize the enzyme required for the utilization of promotor and co factor can be expressed as S2 until S1 is almost exhausted. Thus the profile of bacterial growth under these conditions is diauxic; promotor (inactive) + nyz --^ promotor (active) that is, the growth shows two logarithmic phases (3) intermitted with no growing state. This phenomenon Anequilibrium constant ( VOL 17 NO. 5 1984 479 (CJCJn- C Since the concentration of co factor, Cz, and cell mass dMv S2 M,(20) per unit culture volume, Cm, are expressed by the dx "y"y+(cjcjr»Kny zy+cS2 following equations (/(Y)=cell age density distri- bution function, and Nt=total cell number per unit And the synthetic rate of the inducible enzyme per culture volume), f unit culture volume, (dCJdt), is expressed as (CJCJ* c dC, S2 C=Nt MJ{x)dx (ll) Ktly c.(21) Jo it,+(CJCJ">-yiy $y+ CS2 f (12) 1.5 Equations of microbial growth rate Cm=Nt Jo MJ{x)dt Whenbacteria simultaneously use two substrates, S± and S2, and the reactions characteristic to the the synthetic rate of z per unit culture volume can be respective metabolic processes of these substrates act given from Eq. (10) as follows: as rate-limiting steps, the growth rate of bacteria in dCJdt=kzrizCJ{l +(Csi/ 480 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN 0.1 g/1 MgSO4à"7H2O.All media were adjusted to pH 7. Glucose and maltose, singly or in combinations, were added as carbon sources. 2.2 Growth conditions and measurements Cells from overnight cultures in glucose minimal medium were inoculated in fresh medium (10ml) and shaken at 37°C with a Monod-type shaker. Whenthe optical density at 660nm of the culture (UV-120-10, Shimadzu) reached 0.2, the culture was poured into 100ml of the fresh medium and shaken as above. Samples (2 to 3ml) withdrawn from the culture at intervals were used for measurementof optical den- sity and cell number. To determine the specific growth rate at a lower concentration of nutrient, the cell concentration in a batch culture was maintained at less than 107 cells/1 so that changes in nutrient Fig. 1. Lineweaver-Burk plots between specific growth concentrations were negligible during the experi- rates and concentrations of glucose or maltose as sole car- ments.14) At intervals, samples were withdrawn and bon source. viable cells were measured by the double agar layer method. Glucose concentration was measuredby the muta- rotase GODmethod (Glucose C-Test, Wako Pure Chemical). Maltose concentration was determined by subtracting the amount of glucose from the total amount of reducing sugar as measured by the method of Somogyi-Nelson. 3. Experimental Determination of Parameters When the concentration of Sx (glucose) and S2 (maltose) in the mediumis low, the terms of substrate inhibition of Eqs. (23) and (24) are negligible. Thus, /ix and pi2 can be expressed with following equations. /^/vQi/K + Qi) (27) Fig. 2. Inhibitory effects of higher concentration of sub- V2 = »naCS2/(aL2 + CS2) (28) strate on growth rate. Values of \iml and \im2 were obtained from the intercepts on ordinate in Fig. 1. According to these approximate equations, [iml9 fxm2, al5 and a2 were determined from Lineweaver-Burk was difficult to assess accurately the value of jS2, plots as shown in Fig. 1. Onthe other hand, whenthe concentration ofSx or because glucose is also utilized as nutrient by maltose- S2 is high, /ix and \i2 are obtained with the following grown cells. equations. The yields of cell growth on S1 and S2, Y1 and Y2, were determined from experiments of batch culture of ih =i^csl/(Csl +p1c^) (29) Sx or S2 as a sole carbon source by use of the /^ ^Qi/CQi +yiQf) (30) following equations, respectively. According to these approximate equations, fil9 y2, nn, ^i = (Cmoo - Cm0)/Csl0 (31) and n22, which are parameters for inhibition of sub- Y2 = (Cm^ - Cm0)/CS20 (32) strate uptake by substrate itself at higher concen- trations, were determined from the plots of aw//^ - 1 where Cm0is the initial concentration of cell mass and to CS1, and of Aim2/Ai2-1 to Q2? respectively, as Cwoois the final concentration of cell mass. CS1Oand CS20 are the initial concentrations of S1 and S2, shown in Fig. 2. #2> 7i5 ^i2> and «2iJ which are parameters for respectively. When #2 is usec* as sole carbon source in the inhibition of substrate utilization by other substrate, were determined from the growth profiles upon ad- mediumat low concentration, we can obtain Eqs. dition of maltose or glucose to the exponentially (33)-(35) from Eqs. (13), (21), (22), (23) and (24). growing culture with glucose or maltose, though it (\/Cz)(dCJdt) = kzrjzCJCz - K (33) VOL. 17 NO. 5 1984 481 Since the specific synthetic rates of cell mass, y, and z ( l / Cy)(dCy/dt) should be equal when bacteria are exponentially = kyt,y{ CS2/(Zy + CS2)}(CJCy) (34) grown, the following equation is introduced from Eqs. (39)-(41). (l/CJ(dCJdt) = nm2 CS2/(a2 + CS2) (35) ^2 = Mm2MyCS2/{(a2 + CS2)fi} If bacteria are grown exponentially, the specific syn- thetic rates of cell mass, y, and z would be equal. ikzt,j(fi + k'z){ i + (csl/ (CJCJy C 1 dCv S2 K% presence of maltose alone. The solid lines in these c, it "'å å figures are calculated from Eqs. (13), (21)-(26) by using the parameters presented in Table 1. It was Cs a1+C -+A C SI Cv m2 S2 'tyHCJCJ* £y+CS2 Cy found that the proposed model could simulate well Cm at a2+CS2 C, 482 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN Fig. 3. Effects of glucose on maltose-dependent growth rate, \i2. The \i2 values in medium containing 0.01 g/1 maltose and diverse concentrations of glucose were obtained by subtracting growth rate with glucose alone from growth rate in mixed medium. Solid line is calculated value from Eq. (40). Fig. 5. Growth profile of bacteria precultured in glucose medium. Solid lines are calculated results. Initial concen- tration: (a) Cslo=0.2g/l, CS20=0.7g/l. (b) Cslo=0.4g/l, CS20 =0.2g/l. Fig. 4. Contours of a for determination ofnz and tabolism was not affected by glucose. As shown in Fig. 6(b), when bacteria precultured in the presence of Table 1. Value of parameters maltose alone were transferred into the mediumcon- Mml=1.09b-1 ai=1.82x l(T4g/l /im2=1.04b-1 a2=5.56x l0~5g/l taining both glucose and maltose, the initial growth /in=2.5 A=3.2x lO-3(g/l)-1 5 rate was equal to the sumof jaml and \im2. However, «12=2.5 &<10-4(g/l)-1-5 the growth rate decreased from time to time, ap- proaching jjLml. Since the inducible synthesis of en- «21=2.5 y^U x lO-^g/l)-1-5 «22=2.5 y2=2.1 x l0-3(g/l)-1"5 zymefor maltose utilization is strongly repressed in /iy=2.0 ^<10"6g/l the presence of glucose, the above findings can be 0y=lO"3 0z=6.8x lO-6g/l »z=1.3 ^^lOO.Oh-1 explained by the decrease in enzymecontent per cell during the growth. The validity of this explanation ^ =0.66 72-0.74 was further confirmed in the following experiment. The microorganism precultured in the presence of the dynamic experimental data. maltose alone wastransferred into the mediumcon- Whencells precultured in the glucose mediumwere taining maltose plus a limiting amount of glucose inoculated into the mixed medium of glucose and (Fig. 6(a)). The initial growth rate was again equal to maltose, maltose was consumedonly after glucose (jiml +fim2) as expected. However, the rate decreased wascompletely consumed; this is a typical diauxic rather abruptly following the exhaustion of glucose, growth. On the other hand, when cells precultured in and the rate corresponding to \im2 was attained after maltose mediumwere inoculated into the mixed some time lag. The content of enzymefor maltose medium of glucose and maltose, both glucose and metabolismper cell at the time whenglucose was maltose were simultaneously consumed; that is, the exhausted is probably 30% of that of enzyme at time activity of the enzymeresponsible for maltose me- 0, because the bacteria were grown in the presence of VOL. 17 NO. 5 1984 483 Fig. 7. Effect of content of inducible enzyme on lag time between first and second logarithmic growth phases. Lag times were expressed as times at the intercept of logarithmic growth curve on the horizontal line at CJCm0= 1. Solid lines are calculated lines. in Fig. 8. It is very likely that maltose is metabolized through hydrolysis to glucose; that is, the metabolic pathways of glucose and maltose share partly the same deg- radative process. If this commonpathway included a rate-limiting step for glucose or maltose metabolism, the growth rate of the fully maltose-induced cells in Fig. 6. Growth profile of bacteria precultured in maltose the presence of glucose and maltose would no exceed medium.Solid lines are calculated results. Initial concen- the value of [iml or \xm2. Therefore, the fact that the tration: (a) Cslo=0.2g/l, CS20=0.7g/l. (b) C510=0.7g/1, growth rate of the induced cells with glucose plus CS20 =0.055 g/1. maltose was equal to (/iml +^w2) suggests strongly that the rate-limiting step of glucose or maltose glucose for 1.8 generation time. Therefore, the bac- metabolism lies at the respective unshared step, teria required a time lag to synthesize the enzymefor probably on the permeation of glucose or mal- attaining the growth rate, \im2\ that is, the duration of tose through the cell membrane. the lag time depends on the time required for reaching It is likely that the growth rate in the presence of the maximal steady state level of enzyme content per two substrates is sometimes lower than the sum of cell. The relationship between lag time and amount of respective growth rates in the presence of each sub- the enzymepreexisting at the initiation of induction is strate alone. In such case, a minor modification ofEq. shown in Fig. 7. The log phase cells grown in the (22) will be required. However, from the successful presence of maltose (thus, fully induced state as to application of the model in the present investigation, maltose utilization) were transferred into a medium webelieve that the model is also applicable to other containing glucose and shaken at 37°C. At 0.5, 1, 2, types of diauxic and practical wastewater treatment. and 3 generations (38 min per generation), the cells were collected by centrifugation, suspended in the Conclusion maltose containing medium, and shaken at 37°C. A mathematical model for diauxic growth of mi- Since the synthesis of the enzyme for maltose utili- croorganismsin the presence of twosubstrates, such zation is repressed in the presence of glucose, the as glucose and maltose, wasproposed. The model enzymecontent per cell would reduce to half with was constructed in due consideration of catabolite every generation. As shown in the figure, the length of repression and enzyme induction. Upon analysis the lag time increased with the increase in incubation of the experimental results with the model the fol- time with glucose. The relationship between the pre- lowing findings were obtained. sumptivelevel of the enzymefor maltose metabolism, (1) The values calculated from the model showed Cy0/Cm0, and the length of the lag time, tL, is indicated satisfactory agreement with the experimental data, 484 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN cr = criterion function [h 2] t = cell age [h] (Subscript) m = cell mass op = operator opa = active operator pr = promotor pra = active promotor Rm = mRNA S = carbon source (substrate) w =enzymefor z synthesis wa = active enzyme for z synthesis y = inducible enzyme Fig. 8. Relationship between inducible enzyme content per z = co factor cell mass at t=0 and lag time. 0 = attime0 1 = glucose 2 = maltose especially in the estimation of lag time between the first log phase and the second one. Literature Cited (2) The rate-limiting step of maltose utilization 1) Baipai, R. K. and T. K. Ghose: Biotechnol. Bioeng., 20, 927 appeared to lie on the reaction of the inducibly (1978). synthesized enzyme. Therefore, the duration of the 2) Chambers, D. A. and G. Zubay: Proc. Natl. Acad. Sci. diauxic lag time depends on the time required for U.S.A., 63, 118 (1969). reaching the maximal steady-state level of enzyme 3) Chohji, T., T. Sawada and S. Kuno: Appl. Environ. content per cell. Microbiol, 31, 864 (1976). 4) Dedem, G. V. and M. Moo-Young: Biotechnol. Bioeng., 15, (3) The model appears to be widely applicable to 419 (1973). other types of diauxic growth, provided that rather 5) Emmer, M., B. de Crombrugghe, I. Pastan and R. L. Perlman: minor modifications are introduced. Proc. Natl. Acad. Sci. U.S.A., 66, 480 (1970). 6) Gondo, S., K. Venkatasubramanian, W. R. Vieth and A. Constantinides: Biotechnol. Bioeng., 20, 1797 (1978). Nomenclature 7) Imanaka, T. and S. Aiba: Biotechnol. Bioeng., 14, 754 (1977). C = concentration per unit culture volume [g/1] 8) Imanaka, T., T. Kaieda, K. Sato and H. Taguchi: /.'Ferment. E = activity [-] Technol, 50, 633 (1972). k = rate constant [h"1] 9) Imanaka, T., T. Kaieda and H. Taguchi: /. Ferment. Technol., M =content per cell [g] 51, 423 (1973). n = order of reaction [-] 10) Kono, T. and T. Asai: /. Ferment. Technol., 49, 128 (1971). t = time [h] ll) Kubitsuchek, H. E.: Biophys. J., 14, 119 (1974). tL = lag time [h] 12) Monod, J.: "Recherches sur la croissance des cultures Y = yield of cell growth [-] bacteriennes," Herman et Cie, Paris (1942). 13) Riggs, A. D., R. F. Newbyand S. Bourgeois: /. Mol. Biol, 51, a = saturation constant [g/1] 303 (1970). /?, y = parameters for inhibition of substrate uptake 14) Sawada, T., T. Chohji and S. Shimada: Kagaku Kogaku at higher concentration of S1 and S2, Ronbunshu, 4, 372 (1978). respectively Kg/1) " 1 '5] 15) Toda, K. and I. Yabe: Biotechnol. Bioeng., 21, 487 (1979). y\ = efficiency of reaction [-] 16) Yagil, G. and E. Yagil: Biophys. J., ll, ll (1971). fi = specific growth rate [h^1] 17) Yamakawa, A. and S. Kuno: /. Biochem., 93, 281 (1983). fj,m = maximumspecific growth rate [h"1] 18) Yoon, H., G. Klinzing and H. W. Blanch: Biotechnol. Bioeng., £ = equilibrium constant [g/1] 14, 1193 (1977). VOL.17 NO.5 1984 485