330
Chapter 8
®(O) = ( l - 0 d)® (al + a 2)
(15)
® («r) = © ( 0 + a - O ® < a i + « ! )
(16)
With these boundary conditions, the steady-state solution of the population balance becomes
<
j>{a)
= <
L eD“ '*“‘> - l + 0„
0
< a <a,
D
__a,.
at < a< a1+a2
{
1
(17)
where
û i= 1 lnf
j
D
+ 0 J
(18)
From Eq. (18E6.3), it is observed that the length of the unbudded period decreases with increasing values of
and &. The reason is that when a large portion of the cells loses their plasmid at cell division (and
thereby die due to selection pressure from the environment), the culture has to speed up its specific growth
rate at the cellular level. This is observed as formation of larger daughter cells during the budding period,
and the length of the unbudded period for the daughter cells is therefore shortened. The decrease in a,
results in a narrower age distribution, with a larger fraction of the cells being mother cells;
In the model above, only the age distribution for the plasmid containing cells is considered. However, for
prediction of productivity of a recombinant product one needs information on the plasmid copy number
distribution, and Hjortso and Bailey therefore extended their model to describe the plasmid copy number
distribution (by an approach similar to that illustrated in Example 8.2). Two models for plasmid replication
during the cell cycle were examined: (1) Sufficient plasmids are synthesized during the cell cycle to ensure
that the copy number is always jVat cell division, and (2)
N
plasmids are synthesized during the cell cycle.
The plasmid copy number distribution was calculated, and the distribution is quite different for the two
models (one being bimodal and the Other looking qualitatively like an exponential decay).
Finally, Hjortso and Bailey (1984b) extended their model for the recombinant yeast to dynamic conditions
and examined a shift to a nonselective medium. In a chemostat, the fraction of the population that contains
plasmids approaches zero asymptotically as
growth proceeds. The decrease in the fraction of
plasmid-containing cells depends on the value of
au
which is a function of the specific growth rate. For any
d\
> 0, asymmetric division results in a faster decrease in the fraction of plasmid-containing cells than does
binary fission. This shows that the asymmetric division of budding yeast results in a different behavior than
that observed for recombinant bacteria._____
__________
Example 8.4
Population model for hyphal elements
In submerged cultures of filamentous fungi, there is a population of hyphal elements, which may be
characterized by many different properties (e.g., their length and the number of tips), but the relative content
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