270
Chapter 7
XA- X G=0
;
v,=k,xc
(7.36)
In analogy with the models of Roels and co-workers (Nielsen
et a i,
1991a,b) also proposed a
simple two compartment model for simulation of the growth of lactic acid bacteria that is analogous
to the Williams model and a simplified version of this model is discussed in Example 7.6.
Example 7.5 Analysis of the model of Williams
With the formulation of the Williams (1967) model given by Eqs. (7.34)-(7.36), the stoichiometric matrices
become
(
y
1
11
o '
A =
0
;
r =
-1
Y u
v0 ;
k 1
- u
Using Eq. (7.11). the specific growth rate is calculated:
or
" I
l)
✓--------------------
N.
l
o
+
l vJ
M = Y n K - ~ ~ r - { \ - Y n ) k 2X AX G
.V + A r
(
2
)
(
3
)
The third reaction does not contribute to the specific growth rate since its sole function is to recycle
biomass. Using Eq. (7.14) we find the mass balance for the active compartment:
ClX
c
— - k2XAXG
+ k%X G
-
mX a
ut
S
+ A
s
(4)
and since
XQ
=
1
-
XA
, we obtain the following relation for steady-state chemostat operation
X A= - X ( D + k,)
(5)
y
22
k 2
Clearly the model predicts that
XA
increases with
D,
as it should according to Fig. 7.12, since the A
compartment contains the PSS. If the parameters in the correlation are estimated from the experimental
data,
ky
is found to be 0.5 h'1, which as pointed out by Esener
et al.
(1982), is far too high to be reasonable.
The rather arbitrarily postulated kinetics is probably the reason for the inaccurate result. From Eq. (5) it
follows that if degradation of the G compartment is not included (i.e.,
k-,
= 0), the model predicts
X* -
0
when
D
= 0 h 1. This does not correspond with the experimental data of Fig. 7.12; i.e., even resting cells (fl
= 0) must have ribosomes on stand-by if they are to develop into actively growing cells. Thus the