Sustainability and Stability - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 827

Eqns (16.39) through (16.42) assume only simple competition between plasmid-containing
and plasmid-free cells. No selective agents are present, and the production of complementing
factors from the plasmid is neglected. The simplest assumption for cellular kinetics is that
Monod equation holds for both plasmid-containing and plasmid-free cells,
m G þ ¼ m þ max S (16.43a)

K S þ þS

m G ¼ m max S

(16.43b)

K S þS

Since the cultural volume is constant, Eqns (16.39) through (16.42) can be reduced to

d X þ
d t

ðDm G þ þP

=þ m G þ ÞX þ ¼

(16.44)

d X
d t

ðDm G ÞX þP

=þ m G þ

X þ ¼ (16.45)

DðS 0 SÞ m G þ X þ

YF þ= S m G X

d S
d t YF = S ¼

(16.46)

If we did choose to model the plasmid-containing cells by a multiple individual species,
we would have (similar to Eqns 16.44 and 16.45 ) more differential equations to deal with
than just these above three. The problem is further simplified if we assume that substrate
reached steady state, i.e., the concentration of substrate does not change with time. This
not only takes away on differential Eqn (16.46) , but leads to constant specific growth rates
as well.
Integrating Eqn (16.44) , we obtain

X þ ¼ X þ0 exp½ðDm G þ þP

=þ m G þ Þt

(16.47)

where X þ 0 is the concentration of the plasmid-containing cells in the chemostat at time t ¼ 0.
Substituting Eqn (16.47) into Eqn (16.45) and multiplying exp[(D e m G )t]dt on both sides,
we obtain

ðDm G ÞX exp½ðDm G Þt d t þP =þ m G þ X þ0 exp½ðm G m G þ þP =þ m G þ Þt d t
¼ exp½ðDm G Þt d X

(16.48)

Integrating Eqn (16.48) , we obtain

exp ½ðm G þ P =þ m G þ D Þ t exp ½ðm G D Þ t
m G m G þ þP =þ m G þ

X ¼ X 0 exp½ðm G DÞtP =þ m G þ X þ0

(16.49)