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Mathematical Model for Diauxic Growth of Microorganisms in Mixed Substrate Medium

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Mathematical Model for Diauxic Growth of Microorganisms in Mixed Substrate Medium 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 (<py) of activation reaction of is caused by the repression and derepression of the promotor is obtained as follows: mRNAsynthesis from the inducible gene for the sec- ond substrate. cj>y = (MJMmy>(Mpr - Mpra)/Mpra (4) 1.1 Transcription of inducible genes where Mpr, Mpra, Mz, and Mmare the total number of Initiation frequency of transcription from inducible promotors per cell, the numberof active promotors gene is controlled dually by activity of "promotor," at per cell, the amountofcofactor per cell, and cell mass which RNA polymerase binds and then initiates per cell, respectively. Since biochemical details of the transcription, and "oeprator," which interacts with a reaction mechanisms of Eq. (3) are obscure, we repressor protein to control the expression of the assume that the activation of promotor is affected by gene. Efficiency of binding of RNApolymerase is the fty-order ofz. The activity of the promotor can be probably variable from promotor to promotor, and expressed by a ratio of amountof active promotor to somepromotors require a further activator protein total amountof promotor, and wehave such as CAMPreceptor protein in E. coli for attach- Epr = MpJMpr (5) ment of RNApolymerase. On the other hand, the transcription of the genes for inducible enzymes is Substituting Eq. (4) into Eq. (5), known to be regulated by a repressor protein. Epr = (MJMJ»y/{<l)y + (MJMJ»y} (6) Therefore, a frequency (E) of initiation of mRNA synthesis can be expressed as follows: If the value of MJMmis constant for whole cell ages, MJMmshould be equal to the ratio of the con- E= EprEop (1) centration of z per unit culture volume (Cz) to the where Epr and Eop are activities at promotor and concentration of cell mass per unit culture volume operator sites, respectively. It is clear that the rate of (Cm). Therefore, Epr can be expressed as follows: mRNAsynthesis from the specified operon is de- Epr = (CzICmTyl{4>y + (CJCmyy} (7) pendent on the amountof the operon within the cell, which is approximately proportional to DNAcontent The intracellular concentration of z maybe reg- per cell, together with activities of the promotor and ulated by the activity of the enzymeof z synthesis, as operator of the operon. Furthermore, mRNAin observed in the catabolite repression in E. coll. If the bacteria is unstable and is degraded with a half-life of enzyme (w) for z synthesis is reversibly inhibited by a few minutes. Then the differential rate of mRNA *Sl5 we have synthesis per cell is expressed by the following w (active) + nzS1 * w (inactive) (8) equation: Therefore, the content of active w per cell, Mwa,is dMRJdx = kaEprEopMD - kbMRm (2) expressed as where MRmis a content of specific mRNAper cell, %is Mwa = MJ{l + (CsJ<l>zy'} (9) cell age, and ka and kb are constants. 1.2 Activation of promotor where Mwis the total amount of enzyme (w) per cell, In E. coli, the promotor of inducible gene, such as <pz is an equilibrium constant, and nz is a reaction lac gene, is activated by the complex of CAMP order. Since the details of the reaction of Eq. (8) are receptor protein and CAMP,and the synthetic rate of unknown, we assume the inhibition of wz-order re- CAMPis repressed in the presence of glucose. Al- action of the Sx concentration, CS1. Assuming that though it is obscure whether the identical mech- the enzyme, w, is constitutively synthesized, the con- anism participates in the enzymeinduction in other tent of the enzymeper cell should be proportional to bacteria, it is reasonable to assume as a generalized cell mass per cell.3) The synthetic rate of z is pro- mechanism that the promotor of the inducible gene is portional to the content of active w per cell and the activated by the complex of a special activator and a reaction activity (rjz), which is a function of energy co factor with low molecular weight. If the activator supply from materials. The synthesized z may be lost protein is constitutively synthesized, the concen- from cells by diffusion and/or degradation at a rate tration of the activator protein is constant in growing proportional to the cellular z content. Therefore, the cells. Therefore, the activation of the promotor will be synthetic rate ofz per cell can be expressed as follows: entirely dependent on the intracellular concentration dMz/d?=kzrjzMm/{l +(CslM)zy*} -kzMz (10) 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"<i>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<I>,+(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.
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