Modeling of Growth Kinetics
269
Proteins and genetic material participate in the doubling of genetic material and external substrate is
consumed [Eq, (7.32)]. Enzymatic proteins are formed in the reaction in Eq. (7.33) with active
participation of genetic material, and again external substrate is consumed. The two stoichiometric
coefficients a u and a 2( are model parameters to be fitted from experiments. As for many of the
original structured models the original model was (as pointed out by Fredrickson (1976)) not
formulated in terms of intracellular concentrations
X G
and
X D
(grams per gram dry weight), but if
one reinterprets the variables in the model as being intracellular variables, one arrives at the kinetic
expressions in Eqs. (7.32) and (7.33). In this model the specific growth rate is the sum of v, and v2,
and it is clearly seen that besides a dependence of the substrate concentration the specific growth
rate is a function of the biomass composition. Depending of the model parameters it is possible to
describe an increase in the specific growth rate when
X D
increases.
In another simple structured model, Williams (1967) also divides the biomass into two
compartments. The interpretation of the compartments is, however, different from that used by
Ramkrishna
et al.
(1967). Williams includes small metabolites and ribosomes in one compartment,
which we designate as the A (or active) compartment. All macromolecules such as protein and
DNA are pooled in another compartment, the G compartment. Synthesis of the two compartments
is described by
r l}X A - s = 0
;
v ,= * ,—- —
s + K s
Tl2^C~^A
= 0
\
V2
X q
(7.34)
(7.35)
Thus the A compartment is formed directly from
s,
whereas the G compartment is formed from
building blocks present in A. With the definition of the compartments, A is certainly an active part
of the cell, and the rate of formation of G must depend on the size of A. The role of the G
compartment is less clear. Most of the G compartment is probably inactive, but a constant fraction
of G may contain enzymes necessary for growth of G. This must be the reason why Williams
postulated that the rate of growth of
X G
is proportional to both
X A
and
XG,
as in Eq. (7.35). In Eq.
(7.34), a Monod-type dependence of the substrate is used for the rate of formation of A, whereas
Williams used a first-order dependence of
s.
Furthermore, the original model was not formulated in
terms of intrinsic variables, and the interpretation of the Williams model given above includes the
corrections pointed out by Fredrickson (1976) and by Roels and Kossen (1978).
The Williams model has been used by Roels and co-workers to describe different fermentation
processes (Roels and Kossen, 1978; Esener
et al.,
1981b,c; Esener
et al.,
1982; Harder and Roels,
1982; Jobses
et al.,
1985), and their application of the model is discussed in Example 7.5.
Conversion of the G compartment to the A compartment, as described in Eq. (7.36), was included
in the modifications of Roels and co-workers. The argument for the applied stoichiometry is that G
is degraded to small molecules that are included in the A compartment without any loss. It is
reasonable to include some kind of turnover of cellular material, but it is less justified to let the
normally very stable structural material degrade to the active material present in A.
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