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Chapter 7
synthesized from substrate through intracellular reactions (and different metabolic products may
also be formed by different morphological forms). An intracellularly structured model may
describe these reactions, but in order to reduce the model complexity one will normally use a
simple unstructured model for description of the growth and product formation of each cell type,
e.g.
the Monod model describes the specific growth rate of the
qth
morphological form. When
the specific growth rate has been specified for each morphological form, the specific growth rate
of the total biomass is given as a weighted sum of the specific growth rates of the different
morphological forms:
The rate of formation of each morphological form is determined both by the metamorphosis
reactions and by the growth associated reactions for each form, and the mass balance for the
qth
morphological form can be derived in analogy with Eq. (7.14) (see Nielsen and Villadsen (1992)
for details):
The first term accounts for the net formation of the #th morphological
form by the
metamorphosis reactions (the vector Aq specifies the stoichiometric coefficients for the
qth
morphological form in all the metamorphosis reactions). The second term accounts for growth of
the
qth
morphological form and the last term accounts for dilution due to growth of the biomass
(this is a consequence of the normalization of the concentrations of the morphological forms).
The concept of morphologically structured models is illustrated in Example 7.8.
Example 7.8 A simple morphologically structured model describing plasmid instability
A potential obstacle to commercial application of recombinant bacteria and yeasts is plasmid instability.
Sometimes a daughter cell that does not contain plasmids is formed upon cell division, and since the
metabolic burden is higher for plasmid-containing cells, the plasmid-free cell will grow faster than the
plasmid-containing cells. Even a small plasmid instability will therefore ultimately result in the appearance
of a large fraction of nonprotein-producing cells. Plasmid stability can be improved by increasing the
plasmid copy number or by designing the host-plasmid system in a way that ensures that plasmid-free cells
cannot survive. Modeling of plasmid instability can be done using the concept of morphologically
structured models. Thus, we assume that when plasmid containing cells Zp are dividing a certain fraction 5
of the cells are converted to plasmid free cells Zj,, whereas the remaining fraction of the cells maintains the
plasmid. This can be described by the metamorphosis reaction:
e
(7.67)
/=1
(7.68)
(l -
รข )Z p + SZh - Z p = 0
(
1
)
The stoichiometric coefficient 5 (often called the
segregation parameter)
is equal to the probability of
formation of a plasmid-free cell upon growth of plasmid containing cells. The stoichiometry in Eq. (1) is
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