130
Chapter 5
(on a C-mole basis):
lactic acid + 0.333 ATP - glucose = 0
(2)
Since the stoichiometric coefficient for lactic acid is 1 the forward rate of this reaction is equal to the
specific rate of lactic acid production
rp
(C-moles of lactic acid (C-mole biomass h)'1). In addition to these
two reactions ATP is consumed for maintenance, and this may be considered simply as a hydrolysis
reaction of ATP with forward reaction rate
mAT
p.
With the above reactions a balance for ATP directly gives:
0
=
~^xA
T P ^
~
‘rrp ~
m ATP
(3)
which can easily be re-written to Eq. (5.6) since rp equals 3 rATP. The balance equation (3) can also be
rewritten as a linear rate equation for the specific product formation rate, and in fact the first presentation of
a linear rate equation like Eq. (5.5) was empirically derived from analysis of the production of lactic acid by
Lactococcus delbrueckii
at different specific growth rates (Luedeking and Piret, 1959).
With the simple model given above the glucose uptake rate can also be derived:
-rs =afi + rp
(4)
which upon elimination of
rp
by using Eq. (3) gives:
rs
=
(a
+
Y^ tp)m + m ATP
= O
+
ms
(5)
Thus, the linear rate equation (5.5) is seen to be a consequence of the balancing of ATP in the cell, and this
explains the general applicability of this linear relationship between the specific substrate uptake rate and
the specific growth rate.
To illustrate the validity of the simple model used here we will consider experimental data by Benthin
et al.
(1994), who made a meticulous analysis of the growth and product formation of
L. cremoris.
Fig. 5.3A
shows results from a batch fermentation of
L. cremoris.
The medium contained 20 gram per liter of glucose
and 7 gram per liter of a complex N source (a mixture of 50% yeast extract and 50% caseine peptone). The
biomass concentration was monitored by flow injection analysis (Benthin
et al,
1991), and the specific total
acid production was measured by monitoring the amount of alkali added to keep the pH constant in the
bioreactor. Throughout the batch fermentation the glucose concentration is high, and consequently only
lactic acid is produced. It is observed that there are two distinct growth phases. First a phase with rapid
growth, about P=0.3 - 0.7 h'1. This is followed by a phase where the specific growth rate first decreases to
about 0.3 h'1
and then recovers to about 0.45 h 1. After 12 hours the specific growth rate decreases
monotonously to a very low value.
By plotting the specific production rate
rp
versus the specific growth rate 4 (Fig. 5.3B) it is observed that
two straight lines appear, one for H>0.3 h'1
and one for 4<0.3 h '. The lactic acid production is closely
correlated to the ATP production (1 mole of ATP per mole of lactic acid produced), and the plot in Fig.
5.3B can therefore be used to estimate T,ATP and mATP from, respectively, the slope and the intercept. The
results are:
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