Modeling of Growth Kinetics
291
segregated model for cellular performance resulted. For some microbial systems differentiation of
the cells in the culture does, however, play an important role in the overall performance of the
culture, and both growth kinetics and productivity are effected by the presence of more than one
cell type in the culture. In Chapter 8 we are going to consider complete segregation, and instead of
considering a finite number of distinguishable cell types the culture is going to be characterized by
a continuous distribution of an important cell property, e.g. the cell age. Obviously the model for a
culture with only a few distinguishable cell types (e.g. cells that produce a desired protein and cells
that have lost this property) is much simpler than a model that has to take a continuous distribution
of a property into account. We shall refer to the crudely segregated models as
morphologically
structured models.
These models are particularly relevant for description of the growth of
filamentous fungi, where cellular differentiation takes place in connection with hyphal extension,
but they also find application for description o f other cellular systems, e.g., cultures with bacteria
containing unstable plasmids and to explain why yeast cultures sometimes exhibit oscillatory
behavior in several variables.
In the morphologically structured models the cells are divided into a finite number
Q
of cell
states Z (or morphological forms), and conversion between the different cell states is determined
by a sequence of empirical
metamorphosis reactions
. Ideally these metamorphosis reactions can
be described as a set of intracellular reactions, but the mechanisms behind most morphological
conversions are largely unknown. Thus, it is not known why filamentous fungi differentiate into
cells with a completely different phenotype than that of their origin. It is therefore not possible to
set up detailed mechanistic models describing these changes in morphology, and empirical
metamorphosis reactions have to be used. The stoichiometry of the metamorphosis reaction
where the
jth
form is converted to the r'th form is given by very simple relations:
Z,-Zj=
0
(7.65)
Zq will be used in the following to describe both the
qth
morphological form itself and the
fraction of cell mass that is Zq (g
qth.
morphological form (g D W )l). In the metamorphosis
reaction one morphological form is spontaneously converted to another form. This is of course
an extreme simplification since the conversion between morphological forms is the sum of many
small changes in the intracellular composition of the cell. Clearly there may be many different
metamorphosis reactions, and the stoichiometry for these reactions can be summarized in
analogy with the matrix equation (7.2) for intracellular reactions:
AZ =
0
(7.66)
where A is a stoichiometric matrix. It is assumed that the metamorphosis reactions do not involve
any change in the total mass, and the sum of all stoichiometric coefficients in each reaction is
therefore equal to zero.
To describe the rate of the metamorphosis reactions a forward reaction rate u{
is introduced for
the ith reaction, and the rates of all the metamorphosis reactions are collected in the rate vector
u.
Besides formation from other morphological forms, a given morphological form may also be
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