Modeling of Growth Kinetics
283
xs
X
C A P
+
c A M P
X
c A p W X c A M P
X
p
+ X CApm X cAMP
X pX CApinXcAMP
(7,53a)
(7.53b)
where
m
is a stoichiometric coefficient. Equilibrium between CAP and the promotor 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 individual components can be
used. Thus the association constants are
Y
_
\ * C A p m -^ cA M P
]
5 "
\ x aF \ x M
]"
^
_
\_XpX CAPm X cAMP
J
6 “ [
x X x ' c ^ m X ^ ]
and the total balances for CAP and promotor are
l ^ C A P
l = f r
C A P h
i
*
C A P r n ^
C A M P
]
+
p X
C A P c A M P
j
W ,
= [ ^ P\+ [ x pX aPm X CJMP\
We can now derive an expression for the fraction of promotors being activated:
=
[XPX
m XcAMP] _
K 5K b[X cAMP]m[X CAF]
2
R
1 +
^ K . l X ^ T l X ^ ]
(7.54)
(7.55)
(7.56)
(7.57)
(7.58)
The quantity
Q2
of Eq. (7.58) is used to model the repression effect of glucose, just as
Qx
in Eq.
(7.52) 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 Eq. (7.58), 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. Harder and Roels (1982) suggest the following empirical correlation
between
XcAw
and the extracellular glucose concentration 5g|C
:
X ^
= T C ~
<7-59'
K + S ,lc
With Eq. (7.59) the genetically structured model is linked up to the glucose concentration in the
medium, and the genetically structured model may be used to describe diauxic growth as discussed
in Example 7.7. Eq. (7.59) is a totally empirical description of all the different processes involved
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