Modeling of Growth Kinetics
285
With these four reactions, the mass balances for
and
XE
are
^
mRNA
dt
^ 2 ^
m R N A
P
^
m R N A
(5)
k 2X mRNA
k 4X
£
p X E
(6)
dt
The formation of residual biomass from either glucose or lactose is described with Monod-type kinetics, but
for the metabolism of lactose a dependence of
XE
is included. Thus
sgic + ?gicX
- 0 ;
rgjc
-
k glc
(7)
A glc
~t" A
g /c
Slac + Tlac^ ~
0 ’
Th c= ^iac
^
^E
(8)
Slac+K l*c
Since the reactions in Eqs. (l)-(4) do not contribute to a net formation of new biomass, the specific growth
rate is found to be
P
=
y glcrglc +Ylacriac
(9)
For a batch fermentation where the glucose and the lactose concentrations are both initially high,
is
high and, according to Eq. (7.52),
Q\
is therefore high. However, since the glucose concentration is also
high, JfcAMp is low, according to Eq. (7.59) and
Q2
is therefore low. Thus the rate of synthesis of mRNA is
small, and with the rapid turnover of mRNA
high) the intracellular concentration of mRNA tends to be
very low. This again results in a low level of
XE,
and consequently the last term in Eq. (9) is negligible, i.e.,
only the metabolism of glucose contributes to the formation of cell mass. When later in the batch
fermentation the glucose concentration decreases,
XcAMF
and thus
Q2
increases, and as a result the rate of
formation of mRNA becomes sufficiently high to ensure an increasing level of mRNA. The result is
de
novo
synthesis of enzymes, and this will lead to a larger and larger contribution of lactose metabolism to the
total formation of residual biomass. Finally, when the glucose concentration is zero, the cells grow only on
lactose. Since some time is needed for synthesis of the enzymes needed for metabolism of lactose, the
specific growth rate may be low in a period where both % and
XE
are low. Thus the model may predict a lag
phase between growth on glucose and on lactose.
Equations (7.52) and (7.58) are true mechanistic elements of the model. Unfortunately the number of
adjustable parameters is quite large, and the model is still empirical due to Eq. (7.59). It is, however, an
excellent example of how known mechanisms can be included in structured models, and it may be possible
to find values for some of the binding coefficients in the literature (see Note 7.6)._______________________
As mentioned earlier the lac-promoter is often applied as a tool for expression of heterologous
genes in connection with industrial production of recombinant proteins. Most industrial enzymes
exhibit a complex regulation with induction and carbon catabolite repression, and also here
genetically structured models can be used to gain insight into the expression of the gene encoding
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