Biomedical Engineering Reference
In-Depth Information
k
3
c
Y
$
E
3
/
P
2
$
E
3
(11.79f)
k
X
P
2
$
E
3
/
n
2X
X þ
E
3
(11.79g)
k
4
k
4
n
E
j
E
j
X
%
(11.79h)
This is a set of reaction network based on the simplest metabolic pathway approximation. It is
slightly more complex than the simplest reaction network. The brunching of extracellular
and intracellular products formation can capture the difference in rates (product and
biomass) as observed in many microbial systems.
The rate expressions for all the species in the simplest metabolic pathway are given by
r
S
¼ k
1
S½
E
1
þk
1
½
SE
1
(11.80a)
r
SE
1
¼ k
1
S½
E
1
ðk
1
þ k
1c
Þ½
SE
1
(11.80b)
r
E
1
¼ k
1
½
SE
1
k
1
S½
E
1
þk
2
½
YE
1
½
E
2
k
2
½
YE
2
½
E
1
þk
3
½
YE
1
½
E
3
k
3
½
YE
3
½
E
1
þn
E
1
ðk
4
X k
4
½
E
1
Þ
(11.80c)
r
YE
1
¼ k
1c
½
SE
1
k
2
½
E
2
½
YE
1
þk
2
½
E
1
½
YE
2
k
3
½
E
3
½
YE
1
þk
3
½
E
1
½
YE
3
(11.80d)
r
YE
2
¼ k
2
½
E
2
½
YE
1
k
2
½
YE
2
½
E
1
k
2c
½
YE
2
(11.80e)
r
PE
2
¼ k
2c
½
YE
2
þk
P
½
E
2
½Pk
P
½
PE
2
(11.80f)
r
E
2
¼k
2
½
E
2
½
YE
1
þk
2
½
YE
2
½
E
1
þk
P
½
PE
2
k
P
½P½
E
2
þn
E
2
ðk
4
X k
4
½
E
2
Þ
(11.80g)
r
YE
3
¼ k
3
½
YE
1
½
E
3
k
3
½
YE
3
½
E
1
k
3c
½
YE
3
(11.80h)
r
P
2
E
3
¼ k
3c
½
YE
3
k
X
½P
2
E
3
(11.80i)
r
E
3
¼k
3
½
E
3
½
YE
1
þk
3
½
YE
3
½
E
1
þk
X
½P
2
E
3
þn
E
3
ðk
4
X k
4
½
E
3
Þ
(11.80j)
r
P
¼ k
P
½PE
2
k
P
½P½E
2
(11.80k)
r
X
¼ k
X
n
X
½P
2
E
3
(11.80l)
The metabolic pathway usually results in a set in excess of 12 equations.
11.14.5. Cybernetic Models
Another modeling approach has been developed primarily to predict growth under condi-
tions when several substrates are available. Simple systems have been outlined in
11.14.2.2.
These substrates may be complementary (e.g. carbon or nitrogen) or substitutable. However,
there have been no models given for the fractional rate contribution
a
j
's in
Eqns (11.74)
through (11.76)
. Multiple substrate medium is common. For example, glucose and lactose
would be substitutable, as these compounds both supply carbon and energy. We discussed
x
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