Population Balance Equations
325
furthermore assumed that 2
M
plasmids are evenly distributed to the daughter cells, whereas the rest
are distributed at random.
Model 3. In this model it is assumed that all the plasmids are distributed randomly and that
K
plasmids are synthesized for each cell cycle.
Obviously, Model 2 does not result in instability of the recombinant culture, since each daughter cell always
receives
M
plasmids from the dividing cell. From the age distribution specified for each subpopulation of
cells with
^
plasmids at birth, it is possible to find the distribution of plasmid content
p
in cells with
pi
plasmids at birth:
/ b O
) = / b O
)
da
~dp
(
8
)
By assuming zeroth-order kinetics for the plasmid replication within each cell, the relationship between the
plasmid content and the cell age is given by
p = p b+ca
(
9
)
where
c
is the plasmid replication rate constant. Combining Eqs. (7), (8), and (9), we find
AC?) = / b(«)~ = c b- e x p ( - D ^ )
(10)
c
c
Finally, the distribution function for all plasmid-containing cells is found from summation of all the
subpopulations, i.e.,
/ O ) = £ / ( / > )
;
(ID
Figure 8.4 The calculated distribution of cells with varying plasmid copy number for Model 1 and Model 3.
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