Modeling of Growth Kinetics
245
or
dX.
~dt
„7 -
1
dm v
r 7
v
r v —
x ,
= r. v -
uX.
m dt
(7.14)
It should be noted that (7.14) is independent of the type of reactor operation that is used to produce
the biomass. Eq. (7.14) shows that the rate of formation by all the
J
reactions of a biomass
constituent must at least be
to preserve the same fraction of the constituent / in the biomass.
This is a consequence o f expansion of the biomass upon growth and this results in dilution of all
biomass components. As the concentrations of all biomass constituents sum to 1, Eq. (7.11) can
easily be derived from Eq. (7.14).
13
Unstructured Growth Kinetics
In unstructured models, all cellular components are pooled into a single biomass component
represented by the total biomass concentration
x.
Initially unstructured models were based on a
single reaction describing the overall conversion of substrate into biomass, and typically the
kinetics of this overall reaction was represented by the specific growth rate of the biomass l1. Many
different kinetic expressions have been proposed for the specific growth rate of the biomass, and we
will start our discussion of unstructured kinetic models with a description of these simple models,
and then move on to models considering more than one reaction and also look into the effect of
temperature and pH. Many of the unstructured models described in the following look similar to the
mechanistic based models for enzyme kinetics described in Chapter 6, but for description of overall
cell growth they can only be regarded as (often very useful) date fitters.
7.3.1 The Black Box Model
The simplest mathematical presentation of cell growth is the so-called
black box
model of
Section 3.3, where all the cellular reactions are lumped into a single overall reaction. This
implies that the yield of biomass on the substrate (as well as the yield of all other compounds
consumed and produced by the cells) is constant. Consequently the specific substrate uptake rate
is proportional with the specific growth rate of the biomass:
-rt =Yafi
(7.15)
Similar relations describe the specific uptake rate of other substrates,
e.g.
uptake of oxygen, and
the formation rate of metabolic products. Thus, in the black box model kinetic modeling reduces
to a description of the specific growth rate as function of the variables in the system. In the
simplest model it is assumed that there is only one limiting substrate, typically the carbon source
(which is often glucose), and no influence of other substrates on rs. Hence the specific growth
rate is a function of the concentration o f this substrate only. The consumption of all other