How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 594

Besides the substrate S, extracellular product P, and biomass X, there are two intermediate
species (that are part of the biomass): E and S $ E. The net rates of the system are given by

r E ¼k 1 S½ E þðk 1 þ k 1c Þ½S$ E þn E ðk 2 X k 2 ½ E Þ (11.78a)

r S$ E ¼ k 1 S½ E ðk 1 þ k 1c Þ½S$ E (11.78b)

r S ¼k 1 S½ E þk 1 ½S$ E (11.78c)

r P ¼ YF P=S k 1c ½S$ E (11.78d)

r X ¼ YF X=S k 1c ½S$ E (11.78e)

¼ ¼

r S $ E , the simplest reaction network results in the same rate expres-
sion as the Monod equation. However, without the pseuosteady-state (or balanced growth)
assumption, the rate Eqn (11.78) describes the temporal variations of the enzyme and thus the
lag phase in substrate consumption and biomass production.

By setting 0 r E and 0

11.14.4. Simplest Metabolic Pathway
The simplest metabolic pathway that can capture the lag phase and temporal transitional
effects is shown in Fig. 11.12 . Substrate uptake occurs as the cell converts intracellular-bound
substrate into an intermediate Y, and then to an extracellular product P and an intracellular
product P 2 , which is integrated into the cell biomass X. The enzyme required for each step is
also produced as part of P 2 . The stoichiometries along the pathway illustrated in Fig. 11.12
can be written as:

k 1
k 1 S$ E 1 /

k 1 c Y $ E 1

S þ E 1 % (11.79a)

k 2
k 2

E 2 þ Y $ E 1 % Y $ E 2 þ E 1 (11.79b)

k 2 c

Y $ E 2 / P$ E 2 (11.79c)

k P

P$ E 2 % P þ E 2 (11.79d)

k P

k 3
k 3

Y $ E 1 þ E 3 % Y $ E 3 þ E 1 (11.79e)

P

2

S Y

1

3

P 2

FIGURE 11.12 The simplest metabolic pathway for cell growth with one limiting substrate S.