Modeling of Growth Kinetics
253
- called the lag phase, is one example of this. Whereas a change in substrate
concentration in the reactor takes place within seconds, both in pulse addition of substrate
and by changing the feed rate, the change in biomass composition resulting from up - or
down regulation of genes has a time constant of one to several hours. Consequently
application of simple unstructured kinetic models results in underestimation of time
constants for dynamic changes by several orders of magnitude.
7.3.2 Multiple Reaction Models
In the black box model all the yield coefficients are taken to be constant. This implies that all the
cellular reactions are lumped into a single overall growth reaction where substrate is converted to
biomass. A requirement for this assumption is that there is a constant distribution o f fluxes
through all the different cellular pathways at different growth conditions. In the present section
we shall start to put some biochemical structure into kinetic models, moving from “unstructured,
non-segregated” towards “structured non-segregated” on Fig. 7.2. The truly structured models
will be the topic of Sections 7.4 to 7.6, and the models treated below are basically unstructured in
the sense that the influence of the biomass is still expressed solely through the biomass
concentration
x
. However, biochemical information will be included in a semi-quantitative way,
and it will be acknowledged that the overall kinetics is the result of several overall reactions
operating in the cell.
The first and obvious example is the substrate consumption for maintenance of Section 5.2.1, a
process that runs independently of the growth process. From eq. (5.8) we find:
~ rs ~ Y 1™ fj. + ms
(7.24)
Here
Y ffie
is referred to as the
true yield coefficient
and ms as the
maintenance coefficient.
The
extra substrate consumption is accompanied by synthesis of extra metabolic products and
rp = r~ ffiym p
(7.25)
Equations (7.24) and (7.25) can be used together with any black-box model for
ffis,p)
described
in Section 7.3.1. One simply expands the model with a constant term for substrate consumption
and product formation. This may in principle give rise to a conflict since the black box rate
expressions are zero for
s
= 0 and in some cases it might be necessary to specify ms as a function
ofs.
With the introduction o f the linear correlations the yield coefficients can obviously not be
constants. Thus for the biomass yield on the substrate:
(7.26)
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