Example: The Production of cx-Galadosidase by Monascus (A Structured Model) The production of the enzyme a-galactosidase is subject to catabolite repression; when

the mold Monascus is grown on glucose as a carbon and energy source, no a-galactosidase enzyme is made. However, if both glucose and galactose are present, glucose is consumed first. When the glucose concentration is reduced to a low leve!, cx-galactosidase is induced and it is produced about 80 minutes later in batch culture. Similarly, when glucose is added to a system producing a-galactosidase, enzyme production is repressed over a 40 minute period. A structured model for the production of the enzyme a-galactosidase by Monascus has been formulated by Imanaka et al. 36 and is described here as it illustrates the coupling of substrate transport and regulation of gene expression.

We shall denote extracellular glucose and galactose concentrations as SA and Se; the interna! galactose concentration is Se; By assuming that galactose inhibits the consumption of glucose in a competitive manner, the specific growth rate of the mold is assumed to depend on both substrates in the fol!owing manner:

mASA = +---K +S +K_.., S Kss+Ss

SA A K, 8

(3.96)

Transport of galactose into the cell is assumed to be governed by an active transpon mechanism. The maximum concentration of galactose-transporter binding sites is Ge. Externa! galactose binds to these sites by an adsorption isothenn that follows a Langmuir dependence on Se. The rate of galactose transport is described by the following expression.

( GsSs )

rate of galactose transport =U Km+ 58

5 8; (3.97)

Here U is a transport rate coefficient in hr" 1, Ge is in g galactose per mg cell mass and the intracellularconcentration of galactose is in g/mg cells. Intracellular galactose is consumed ata first order rate given by k1Se; Thus a mass balance on intracellular galactose gives

d(S8 ;X) ( G8S8; lv dt =U Km+Ss Ss;r-k,Ss;X (3.98)

The effect of glucose repression is modelled by considering that when the externa! glucose concentration exceeds a critica! value SAc the rate of galactose transport into the cell immediately ceases, i.e., when SA;:: SAc then U= O. As a consequence, the specific growth rate m8 decreases to zero. Experimental data, shown in Figure 3.26 indicates tha_t in the presence of 5 gmlliter glucose, galactose uptake is negligible. Galactose transport can also be seen to be constitutive, as there is no time delay in uptake.

o _.,l:l- - 0--

0 0 ............... 10_._~2~~ -;o"-'-'-=.50 T1m~ ( min )

Figure 3.26. Galactose uptake by Monascus. Uptake of galactose at 35"C, with a galactose concentration of 5 gmlliter containing 1 ci 14C-galactose. Open circles represent cells grown on giucose, washed and incubated on 5 gmlliter galactose medium. Closed circles indicare incubation in the presence of 5 gmlliter glucose. No galactose uptake is evident under glucose repression. (From lmanaka et. al.)

A simple model for enzyme induction by galactose, based on the Monod-Jacob operon model, can be developed. In the absence of galactose, a repressor (R) acts to inhibit the synthesis of mRNA for a-galactosidase. When galactose is present, the repressor and galactose combine to form [RS8 ] which can no longer bind to the DNA and enzyme synthesis can occur. The amount of the (unidentified) intracellular repressor is found by considering it to be produced at a constant rate k2, degraded by a first arder process (k3R) and reacting reversibly with intracellular galactose to form a complex [RS8;) by mass action kinetics. The mass balance on repressor R yields

dRX dr = (k2 -"fsR -k4R S8 ; + k5[RS8])X (3.99) The action of the repressor on the transcription of DNA and production of mRNA is as follows. The synthesis of mRNA depends on the concentration of free repressor; when galactose is present and repressor is bound to it, the concentration of free re pres sor is reduced and it is no longer as effective in blocking mRNA synthesis. The rate of mRNA synthesis is assumed to be proportional to the reduction in free repressor from sorne maximum value R.,. The mRNA is assumed to decay with a first arder rate constant k7 The synthesis of mRNA (M) is thus described by

dMX --={k6(Rc-R)-kM}X .(3 .100)

This scheme for regulation of enzyme production is illustrated in Figure 3.27. The rate of a-galactosidase formed by the mold depends on the concentration of mRNA:

dEX =k MX dt 8

(3.101 )

prevents binding of repressor to operan

DNA

,,--- ------ DNA transcribed in /' ---,, absence ot repressor

/ ' I '

I ', ...

Repressor (R) mRNA (M)

Galactose (88 ) Galactosidase

Figure 3.27. Schematic of the gene leve[ regulation of a-galactosidase by galactose. The repressor R is unknown, but is probably related to glucose or its metabolites.

A complete set of mass balances can now be written, including a balance for repressor-galactose complex [RS8]. We introduce the individual yield coefficients Y A and Y 8 , these being the cell mass yields on glucose and galactose substrates, respectively. In writing the mass balances, the terms describing the intracellular concentrations have been expanded viz.

dS8;X dS8; dX dS8; dt =X dt + Ss; dt =X dt + Ss; (3.102) Thus the intrace!lular concentrations are effectively diluted by the expanding volume of the cell.

(3.103)

(3.104)

(3.105)

U=O (3.106)

(3.107)

dM dr = {k6(R,. -R)-k.,M}-M where dE -;=k8M-E

(3.108)

(3. 109)

(3.110)

The kinetic and microbial parameters in the above set of equations were determined by Imanaka et al. from batch and steady state continuous culture data. The intracellular parameters were estimated, whereas the yields and specific growth rates were obtained from the experimental data. The values of the constants are listed in Table 3.14.

Table 3.14. Values of constants in the model of Imanaka et al. Estimated Experimental values

k1 =40 hfI mA ,g = 0.215 hfl k1 =l g/mg cells-hr mB .g = 0.208 hfl

k3 = l hr"' mA .p = 0.190 h(I k4 = 0.1 mg cells/g-hr mB .p = 0.162 hr1

ks = l X 10.J hr" ' K,A.g = l.54 X JO.J gm/ml ~ = 1 hr" ' K,s., = 2.58 X 10.J grnlml k, =8 hfl KsA.p = l.54 X 10-J gm/ml ks =4.0 units/g M-hr K.s.p = 3.07 X 10.J gm/ml kN.p = 6.67 units/g M-hr K = 1.39 X WJ gm/ml SAc = 2.25 X 10'4 gm/ml u = 100 h(I YA., = 0.530 grnlgm Gs = 3.5 g /mg cells Ya., = 0.516 gm/gm Km = 1 X 108 g/mg cells YA.p = 0.377 grnlgm R: = 0.803 g/mg cells Ys.p = 0.361 gm/gm The subscripts g and p denote values during the growth phase and enzyme production phase repectively.

The predictions of this model compared to the experimental batch culture data are illustrated in Figure 3.28. The model can succesfully predict the catabolite repression of galactose uptake by glucose and the slow rise in the concentration of a-galactosidase once glucose is consumed and galactose is transported into the cell. The key feature of this model of enzyme production is the incorporation of a simple model of enzyme repression aceording to the scheme proposed by Jacob and Monod. It is able to effectively describe the observed production kinetics. We shall examine sorne more complex models of gene expression in Section 3.6.

The ability of this model to predict the dynamics of a -galactosidase repression by glucose is considered in Problem 16. The halflife of mRNA is a key constantin determining the strength of glucose repression.

o.~

~

8 10 o

o 2 4 6 8 10 12 t4 16 o Cullivolion lime (hr)

Figure 3.28. Time course of cell growth and cx-galactosidase production by Monsacus. The inoculum contained glucose grown cells. The medium composition was 10 gmll glucose, 3 gmll galactose, 5 gmll NH4N03 , KH2P04 5 gmll, MgS04 7Hp 1 gmll, yeast extract 0.1 gmll. Open squares, glucose; closed squares, galactose; open circles, cx-galactosidase; closed circles, cell mass. The initial conditions for the model were: X= 0.5 gmll; SA = JO gmll; S8 = 3 gmll; S8 =O glgm cells; R = 0.910 glgm cells; M =O glmg cells; E= O units/mg cells; [RS8;] =O glgm cells.