Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 826

identical in growth rate and in the probability of plasmid loss. This assumption is the
same as assuming that all cells have exactly the same copy number. As we showed in
Example 16-4, the actual distribution of copy numbers can make a significant difference
on plasmid loss. Also, plasmid-encoded protein production is not a linear function of
copy number, so assuming that all cells have the same copy number may lead to incorrect
estimates of the growth rate of plasmid-bearing cells. The assumption of a single type of
plasmid-bearing cell is a weak assumption, but other assumptions would result in more
differential equations to solve (and thus OdexLims or other numerical integrator will defi-
nitely come in handy).
Let us further restrict our initial considerations to a single-stage chemostat as illustrated in
Fig. 16.16 . Mass balances of the three species (plasmid-containing cells, plasmid-free cells,
and substrate) lead to

d ð X þ V Þ
d t

QX þ þm G þ X þ V r V ¼ (16.39)

=þ

d ð X V Þ
d t

QX þm G X Vþ r =þ V ¼ (16.40)

QðS 0 SÞ m G þ

X þ
YF þ= S

V m G

X
YF = S

d ð SV Þ
d t

V ¼ (16.41)

where

r =þ ¼ P =þ m G þ X þ 16.42

is the rate of generation of plasmid-free cells from plasmid-containing cells. P / þ is the prob-
ability of forming a plasmid-free cell. YF þ /S and YF /S are yield factors for the plasmid-con-
taining cells and plasmid-fee cells, respectively.
P / þ can be estimated by Eqn (16.38) if the copy number is known or can be predicted with
a more sophisticated structured segregated model. Avalue for P / þ could be estimated from
an experimentally determined copy-number distribution as in Example 16-4c, which would
be more realistic than assuming a monocopy number. As we will soon see, P / þ can be deter-
mined experimentally without a knowledge of copy number.

Q, S = S 0

Feed

X + = X - = 0

V

Q

Effluent

S , X + , X -

FIGURE 16.16 A schematic of chemostat cell culture.