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Chapter 7
substrates and the production of metabolic products and biomass is found by mass balances using
the yield coefficients of the black box model. A very general observation for cell growth on a
single limiting substrate is that at low substrate concentrations (s) the specific growth rate l1 is
proportional with
s,
but for increasing values of
s
the specific growth rate approaches an upper
limit. This verbal model can be described with many different mathematical models of which the
Monod (1942) model is the simplest, and most frequently used.
,
s
Monod m odel
:
p
=
---
(7.16)
s + K*
Fig. 7.4 shows the time course of a batch fermentation and a cross plot of I1 versus
s.
After a
lag
phase
of duration about 1 hour there is a long
exponential growth phase
where the biomass
concentration increases exponentially with time. At the end, where the substrate is used up,
growth stops quite abruptly. This is the typical behavior when the growth follows Monod
kinetics: The lag-phase is a period during which the biomass composition changes and the
specific growth rate increases. Here the assumption of the Monod model - or any other
unstructured model that assumes the growth to be an independent of the biomass composition -
is not satisfied. Since the batch cultivation starts with an initial substrate concentration
s
= .s0
much higher than
Ks
the specific growth rate P, which is the slope of the curve ln(x) versus time,
is almost constant for most of the fermentation period. When finally
s
decreases into the range of
AT, the specific growth rate declines and eventually growth stops when the substrate concentration
becomes close to zero.
In the Monod model
Ks
is the value of the limiting substrate concentration at which the specific
growth rate is half its maximum value. Roughly speaking, it divides the plot of
versus
s
into a
low-substrate-concentration range where the specific growth rate is strongly (almost linearly)
dependent on
s
, and a high-substrate-concentration range where P is more or less independent of
s.
This is illustrated in Fig. 7,4, where P is plotted versus the limiting substrate
s
for the batch
fermentation data The cross plot of P versus
s
has the typical shape of Eq. (7.16). Fig. 6.1 is of
course identical to Fig. 7.4B.
When glucose is the limiting substrate the value of
Ks
is normally in the micromolar range
(corresponding to the milligram per liter range), and it is therefore experimentally difficult to
determine. Some
Ks
values reported in the literature are compiled in Table 7.1. It should be stressed
that the
Ks
value in the Monod model does not represent the saturation constant for substrate uptake
but only an overall saturation constant for the whole growth process. However, since the substrate
uptake is often involved in the control of substrate metabolism, the value of
K
s is also often in the
range of the
Km
values of the substrate uptake system of the cells.