Modeling of Growth Kinetics
271
degradation reaction of G compartment to active A compartment is taken to be the mechanism that ensures
that some A compartment is present at M
= 0. Despite the inconsistencies of the Williams model, it gives a
far better description of a fed-batch fermentation with
Klebsiella pneumoniae
than a traditional Monod
model including maintenance (Esener
et al.t
1981b,c; Esener
et al.,
1982), and the number of parameters is
not higher than that which can be estimated from steady-state growth measurements.
Jobses
et al.
(1985) also applied the modified Williams model for analysis of fermentations with
Zymomonas mobilis.
The experimental data for the substrate uptake at different specific growth rates in a
steady-state chemostat indicate a nonlinear increase in the volumetric glucose uptake rate
qs
(grams of
glucose per liter per hour) with
D.
This cannot be described by a traditional unstructured model, but the
two-compartment model fits the experimental data well.
It is normally claimed that the Williams model is able to describe a lag phase initially in a batch
fermentation (Bailey and Ollis, 1986; Roels and Kossen, 1978; Williams, 1967). Inserting
XQ
=
1 -
XK
in Eq.
(3) we find:
=
(
6
)
and since
y2 -
1 we find for high substrate concentrations (i.e.,
s » Kt),
that
XA
must be > 0.5 if the specific
growth rate is to increase with
XA.
In model simulations presented by Williams,
XA
increases from 0 to 0.75
at the beginning of a batch fermentation, and the specific growth rate therefore decreases until
XA
becomes
larger than 0.5. This is, however, not easily seen from the presented model simulations since the biomass
concentration is shown in a linear plot rather than a semi-logarithmic plot. Since
XA >
0.5 is not biologically
reasonable (Esener
et al.
(1982) uses a maximum of 0.3 for
XA)
the conclusion is that in reality the model
does not predict that the specific growth rate increases with
XA
but rather that P decreases when the culture
is inoculated with cells having a low
XA,
i.e., resting cells. This illustrates a general problem with
compartment models. Through the introduction of structure into the biomass it may be possible to describe
the specific growth rate as a function of the cellular composition and hereby describe some phenomena
quite well, but the compartments (or variables) used in the model may not necessarily relate to any
biological variables._________________________________________________________________________ *
Example 7.6 Two compartment model for lactic acid bacteria
Nielsen
et al.
(1991a,b) presented a two compartment model for the lactic acid bacterium
Lactococcus
cremoris.
The model is a progeny of the model of Williams with a similar definition of the two
compartments:
Active (A) compartment contains the PSS and small building blocks
Structural and genetic (G) compartment contains the rest of the cell material
The model considers both glucose and a complex nitrogen source (peptone and yeast extract), but in the
following presentation we discuss the model with only one limiting substrate (glucose). The model
considers two reactions for which the stoichiometry is:
(
1
)
YuX'a ~s = 0
y n X G~ X A
= 0
(2)
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