Cell Metabolism - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 496

to promote helix destabilization downstream. This in turn facilitates the binding of the RNA
polymerase and hereby stimulates gene expression. The carbon catabolite repression can be
described by the following equilibria:

k 5
k 5

þ m CAP $ð cAMP Þ m (10.19)

CAP cAMP %

k 6
k 6

E p þ CAP $ð cAMP Þ m % E p CAP $ð cAMP Þ m (10.20)

where m is a stoichiometric coefficient. Equilibrium between CAP and the promotor (E P )
is not considered, since this binding coefficient is taken to be very small. Again, we apply
an assumption of a pseudo-steady state and assume that the concentrations of the indi-
vidual components can be used. Thus, the association (or rather the dissociation)

constants are

m

K 5 ¼ k 5
k 5

½ CAP ½ cAMP

¼ (10.21)

½ CAP $ð cAMP Þ m

K 6 ¼ k 6
k 6

¼ ½

E P ½ CAP $ cAMP

(10.22)

½ E P $ CAP $ cAMP

and the total balances for CAP and promotor are

½ CAP T ¼½ CAP þ½ CAP $ð cAMP Þ m þ½ E P $ CAP $ð cAMP Þ m (10.23)

½ E P T ¼½ E P þ½ E P $ CAP $ð cAMP Þ m (10.24)

We can now derive an expression for the fraction of promoters being activated:

m

½ E P $ CAP $ð cAMP Þ m ½ CAP ½ cAMP

¼ (10.25)

m ½ E P T K 5 K 6 þ½ CAP ½ cAMP

The ratio in Eqn (10.25) can be used to model the repression effect of glucose, just as the ratio
in Eqn (10.18) is used to describe the induction of lactose on gene expression and hereby
synthesis of enzymes necessary for lactose metabolism. However, in order to apply Eqn
(10.25) , one needs to know the intracellular level of CAP (which in a simple model may be
assumed to be constant) and also the level of cAMP. There are many routes one may take
to model the cAMP level, such as the adenosine triphosphate (ATP) concentration, specific
growth rate, and the level of currently utilized substrate concentration. Harder and Roels
(1982) suggested the following empirical correlation between [cAMP] and the extracellular
glucose concentration [S G ],

K
K þ½

½ ¼ (10.26)

cAMP

S G

With Eqn (10.26) [cAMP] is linked up to the glucose concentration in the medium, and the
genetic model may be used to describe diauxic growth. Equation (10.26) is a totally empirical
description of all the different processes involved in determining the cAMP level in the cell at