Biomedical Engineering Reference
In-Depth Information
concentrations decrease monotonously while the product concentrations increase monoto-
nously. However, the difference in the initial (fed) substrate concentrations and the sum of
the residual substrate and extracellular product concentrations increases initially, reaches
a maximum, and then decreases to its final value at long times. This is an interesting and
important observation for bioreactions, as the substrate uptake by cells and subsequent conver-
sion to intracellular and extracellular products pass through a number of intermediates. When
cell biomass is measured, substrate retained inside the cells are likely to be treated as biomass.
Therefore, one can expect a maximum biomass weight before all the substrate is consumed.
As we have learned from Chapters 8 and 9, the reaction system as described by Eqn (11.5)
may be simplified by one rate-limiting step (for each branch of parallel reactions) with rate
constants altered to fit the whole reaction network. It is not difficult to show that each
complete step of conversion from one species to another in Eqn (11.15) can be approximated
by the Michaelis e Menten equation, i.e.
r P1max S
r P1 ¼
K P1 þ S ð
1
þ c 0 P 2 Þ
(11.20a)
r P2max S
K P2 þ S ð
r P2 ¼
1
þ c 0 P 2 Þ
(11.20b)
and
r S ¼r P1 r P2 (11.20c)
Utilizing this simplification to approximate the rates for P 1 and P 2 , one can solve the problem
more easily based on Eqn (11.17) for S, P 1 , and P 2 . The solutions based on Eqn (11.20) with
r P1max ¼
0.20853 k 1 [E 1 ] 0 , K P1 ¼
0.014424 S 0 , r P2max ¼
0.10117 k 1 [E 1 ] 0 , K P2 ¼
0.07919 S 0 , and
c 0 ¼
9.81296/S 0 are plotted as dotted line in Fig. 11.4 .
One can observe from Fig. 11.4 that the approximate model, i.e. Eqn (11.20) gives reason-
able prediction for the reaction network in the product formation and the substrate consump-
tion. As one may infer from Chapter 10 that DNA and RNA constructions and manipulations
by cells also follow a kinetic equation similar to Eqn (11.20) , the approximation may actually
be applicable to cell growth and/or metabolism, in general. Therefore, it is reasonable to
approximate the growth rate of cells by an equation similar to Eqn (11.20) ,or
r X
X ¼
m max S
K S þ S
m G ¼
(11.21)
which is known as the Monod equation of growth. Equation (11.21) is an empirical rate func-
tion, and we have now seen its relevance in the kinetics of cell growth from an approximate
kinetic behavior point of view. Similarly, for extracellular product formation,
r P
X ¼
m P max S
K P þ S
m P ¼
(11.22)
Often, K P and K S are assumed to be identical, further simplifying the growth kinetics.
However, one can notice the difference between the simple approximation and the full
solution in Fig. 11.4 . There seemed to have a difference in substrate concentration due to
Search WWH ::




Custom Search