Modeling of Growth Kinetics
293
illustrative since it shows that when one unit of recombinant cells divides, a fraction 6 of plasmid-free cells
is formed. However, since no new cell mass is formed by the metamorphosis reaction 5 Zp = S
and in
terms of stoichiometry (2) is simplified to
Zt, ~ Z p
= 0
(2)
However, the rate of the metamorphosis reaction (2) is different from that of reaction (1). Since the
metamorphosis reaction (1) specifies growth of the plasmid containing cells the forward rate of this reaction
is equal to (^Zp, whereas the rate of the metamorphosis reaction is equal to ÖMpZp.
In order to apply the general mass balance for the two morphological forms, we first specify the
stoichiometric matrix A;
A = (-1
1)
(3)
Now we set up the mass balance for the plasmid containing cells:
d*Zi
=
-fypZP
+
(ftp - v)Zp
= [(! - <*K -
p
(4)
and for the plasmid free cells:
^ = S p rZ„+(ph-p)Zh
(5)
For each morphological form, there is a contribution from the metamorphosis reaction, a contribution from
growth of the form, and finally a contribution accounting for dilution due to the expansion of the total
biomass. The term for the dilution is analogous to the dilution term in the intracellularly structured models,
whereas the term accounting for formation of intracellular components in intracellularly structured models
is replaced by two terms in the morphologically structured models; i.e., one for exchange between forms
and one for growth.
The specific growth rate for the total biomass is given by:
V=VPZP
+HhZh
(6)
Normally the specific growth rate for plasmid containing cells is lower than that of plasmid free cells, and
the specific growth rate for the total biomass will therefore also be smaller than
Consequently both terms
on the right hand side in the mass balance (5) are positive, and the fraction of plasmid free cells will
continuously increase due to two factors: there is a constant formation of plasmid free cells from plasmid
containing cells, and as )\ > dp the plasmid free cells will outgrow the plasmid containing cells.
With the formation of plasmid-free cells a culture with two different strains develops, namely the
recombinant strain and the parental (or host) strain. Thus the culture is
mixed,
with two different strains of
the same species. The metamorphosis reaction (2) may also be used to describe the spontaneous occurrence
of mutants during cellular growth, but here the forward reaction rate is not necessarily similar to the specific
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