Modeling of Growth Kinetics
243
negative, the specific conversion rate of the rth substrate in the y'th reaction is given by
When we want to calculate the overall production or consumption of a compound, we have to
sum the contributions from the different reactions. We can therefore write the net specific
consumption
rate for the z'th substrate as the sum of its rate of consumption in all / reactions:
(a
= - t a
on
J
j
=i
(7.3)
and similarly for the net specific rate of
formation
of the rth metabolic product:
j
r . —
V"
B
v.
p.i
L - t r ji j
j
=
i
(7.4)
Equations (7.3) and (7.4) specify important relationships between what can be directly measured,
namely, the specific uptake rates of substrates and the specific formation rate of products on one
hand and the rates of the reactions considered in the model on the other. Notice that in Chapter 5
we used the term
fluxes
for these intracellular reactions, but there we also considered well
defined chemical reactions, whereas in growth models the reactions are typically lumped
reactions of empirical nature.
Similar to eqs. (7.3) and (7.4) we find for the biomass constituents:
j
rX J
(7.5)
j'=i
These rates are not as easy to determine experimentally as the specific substrate consumption
rates and the specific product formation rates, but they are related to the specific growth rate of
the biomass. Thus, the rates in Eq. (7.5) are the net specific formation rates of the individual
biomass constituents, and the specific growth rate is therefore given as the sum of net formation
rates of all the biomass constituents:
Q
M
(7.6)
The summation equations (7.3) - (7.5) can be formulated in matrix notation as
( O
.
= -A rv
V
s / consum ption
(7.7)
rp
= B rv
(7.8)
r , = r v
(7.9)
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