Enzyme Kinetics and Metabolic Control Analysis
199
Table 6.1 Enzym atic rate data
r
(g substrate converted L'1
h ') at four levels o f s and
p.
Product concentration (m g L ')
5 ( g L ')
3
9
27
81
0.1
0.073
0.058
0.036
0.017
0.4
0.128
0.102
0.064
0.030
1.6
0.158
0.126
0.079
0.037
6.4
0.168
0.134
0.084
0.039
The same advice is also valuable when parameters in expressions for cell kinetics are determined
from (hopefully) a large set of data representing an adequate variation of all the variables of the
model.
Example 6.1 A nalysis o f enzym atic reaction data.
In an enzym atic reaction substrate
S
is converted irreversibly to product
P.
It is suspected that the
product inhibits the reaction, and consequently the rate
r
o f the reaction is determ ined for four values o f s
at each o f four levels o f the product concentration
p.
T he 16 rate m easurem ents are collected in Table
6.1.
s
is in g L’1
and
p
in m g L '1.
Fig. 6.3 is a L inew aever-B urk plot o f the data. F or increasing level o f product concentration
p
the slope
of the lines increases (from A to D), and all four lines intersect the abscissa axis at m ore or less the sam e
point. C learly a n o n -com petitive product inhibition is indicated. T he value o f
Kcq
in eq (6.14) is
determined from the average o f the four intersection abscissas: ATeq = 0.142 gL .1. N ext the slopes o f the
four regression lines are p lotted against
p:
Slope
K .
k
{
1
+
V
\
p
K = J
(
1
)
Fig. 6.4 is a plot o f the slope vs. the product concentration. From the slope and the intersection w ith the
ordinate axis, and using the previously determ ined value for
Keq
the follow ing values are determ ined for
=k2e0
in eq. (6.14) and for
Krql:
rm„
= 0.212 g substrate L ‘h 1
and Aj,q] = 18.45 m g L 1
(2)
The data in T able 6.1 w ere generated w ith
Keq =
0.136, A'tq, = 20.92 and r m„ = 0.196. The data are
rem arkably accurate w ith a standard deviation o f 1
m g L 1
for
s
and 1
p gL 1
for
p.
Still the regressed
param eters are about 10% from their true values. T his is an indication o f the difficulties that are
encountered in experim ental studies o f enzym e kinetics: the param eters are strongly correlated and the
param eter estim ation can easily be sw am ped by even sm all experim ental errors. In the present case m ore
data should have been obtained for sm all s' values to obtain a better value for
Kcq,
w hile the values o f
p
were judiciously chosen to obtain a good spread o f points on Fig. 6.4. T he piece-w ise determ ination o f
param eters illustrated in Fig. 6.3 and 6.4 is a nice, intuitive approach, but a full non-linear regression o f
the 16 rate data is probably better, once the structure o f the m odel has been revealed by the graphical
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