276
Chapter 7
particular substrate, and this key enzyme must be synthesized before growth can occur on that
substrate. In reality this key enzyme may represent several enzymes, e.g., lactose permease and P-
galactosidase in case of lactose metabolism. The reaction scheme for the growth process on each
substrate can therefore be summarized by the following three reactions:
1
II
o
v =
rw
=
k
— - —
X E
w
s,+ K ,
£‘
(7.38)
X£i-X =
0
;
,
si
£'
E'‘ S i+ K f ‘
(7.39)
X -X ^
= °
;
v* u =kw X S
i
(7.40)
The index
i
indicates the substrate, and the model may consider
N
different substrates. The first set
of reactions (7.38) is the formation of all biomass components except the enzyme
E\
used for uptake
of the ith substrate, and the second set of reactions (7.39) is the formation of the particular enzyme.
The enzymes are synthesized from the general biomass compartment. Finally, the third set of
reactions represents degradation of enzymes. In the model the “potential” substrate uptake rate is
given by
r*,
i.e., the substrate uptake rate that occurs when there is no limitation in the enzyme
system involved in substrate uptake. The kinetics for the potential substrate uptake is described by
Monod-kinetics. Similarly the formation of the key enzymes is given by Monod-type expressions
with respect to the substrate. Degradation o f the enzyme is described by a first-order reaction.
The production of the ith enzyme cannot proceed without some critical cellular resources, which
must be suitably allocated for different enzyme synthesis reactions. This feature is included in the
kinetics through the cybernetic variable
uif
which may be regarded as the fractional allocation of
resources for the synthesis of the ith enzyme, and it can be interpreted as a controller of the enzyme
production. The kinetics for substrate assimilation (and hereby biomass growth) is determined by
another cybernetic variable wr This variable ensures that the growth takes place primarily on the
best-suited substrate, and it may be interpreted as a control mechanism at the enzyme level. It is
doubtful whether there are control mechanisms that work directly on the transport enzyme, but with
the complex interactions between different intracellular pathways it is reasonable to include this
control function in the model.
Several different models for the cybernetic variables have been described (Dhuijati
et ai,
1985;
Kompala
et a i,
1984; Kompala
et al
., 1986), but based on an examination of the various models
Kompala
et al.
(1986) conclude that the best model is obtained when Eqs. (7.41) and (7.42) are
used for the two cybernetic variables.
u,
=
(7.41)
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