Modeling of Growth Kinetics
251
jyi are the concentrations of growth-enhancing substrates, and
s}
are the concentration of substrates
that are essential for growth. The presence of growth-enhancing substrates results in an increased
specific growth rate, whereas the essential substrates must be present for growth to take place.
Application of Eq. (7.17) may be quite successful, but in the presence of many substrates there will
of course be a large number of parameters, and the fit of the model to the experimental data may be
coincidental. A special case ofEq. (7.18) is growth in the presence of two essential substrates, i.e.,
_
№ m ta .,i№ m tu .,2 S \ S 2
(S| +
Ks
\
X‘V2 +
(7.18)
Equation (7.18) may, for example, be used as a model for the growth of methanotrophic bacteria on
the two substrates 0
2
and CH
4
- an important process leading to single-cell protein (see Example
3.4). If the concentrations of both substrates are at levels where the specific growth rate for each
substrate reaches 90% of its maximum value, i.e.,
= 9
Ku
then the total rate of growth is limited to
81% of the maximum possible value. This is hardly reasonable, and Roels (1983) has therefore
proposed two alternatives to Eq. (7.18), both of which may be generalized for application to more
than two limiting substrates:
M
- = min
5
] +
Kl s2
+
K2
j
(7.19)
u
1
(K tl
KM2
_ ] + _
+
s-2
M
5,
(7.20)
Both models will give growth at 90% of the maximum value in the situation mentioned above.
Growth on two or more substrates that may substitute for each other, e.g., glucose and lactose,
cannot be described by any of the unstructured models described above. Consider, e.g., growth of
E. coli
on glucose and lactose. Glucose is metabolized first since it is the “best” carbon source, and
metabolism on lactose will only start after glucose is used up. The bacterium needs one of the
sugars to grow, but in the presence of glucose there is not even a growth-enhancing effect of
lactose. Application of Eq. (7.17) to this example of multiple substrates for glycolysis will clearly
not be feasible. To describe the so-called diauxic growth it is necessary to apply a structured model,
as illustrated in Section 7.4. Baltzis and Fredrickson (1988) offer a systematic procedure for
treatment of growth limitation by two different substrates that are either complementary (such as O,
and CH
4
in the SCP process) or substitutable. See also Egli (1991) for some interesting
experimental results with dual limitation by C and N.
In some cases, growth is inhibited either by high concentrations of the limiting substrate or by the
presence of a metabolic product. In order to account for these possibilities, the Monod kinetics is
often extended with additional terms. Thus, for inhibition by high concentrations of the limiting
substrate,
previous page 274 Bioreaction Engineering Principles, Second Edition  read online next page 276 Bioreaction Engineering Principles, Second Edition  read online Home Toggle text on/off