How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 566

presence of substrate inside the cells (substrate e enzyme complexes) and the intermediates.
The intermediates are not able to be captured by the simple approximation. Therefore, the
simple approximation is applicable for pseudosteady-state cases. Inclusion of the initial or
induction time results in noticeable differences (or errors). The difference between the dashed
line and the solid line in Fig. 11.4 exhibits a time lag when matching products formation and
substrate consumption. The time lag is due to the inability of the simple model to capture the
uptake of substrate in the cell and a “delay” in excrete products. This phenomenon is more
pronounced in real cultures as the metabolic pathway is much more complex than that exem-
plified by Fig. 11.3 .
Equation (11.21) indicates that when the limiting substrate concentration is high, i.e. S

[ K S ,

the specific growth rate m z m max ¼ constant. That is, the growth is at the maximum growth

rate. Thus, the exponential growth regime (of the batch cell growth, Fig. 11.1 ) is also termed
maximum growth regime. The change of growth rate is due to the change of substrate concen-
tration. There is no control on the growth. Therefore, the batch exponential growth is also
uncontrolled exponential growth.

11.5. CELL DEATH RATE

In section 11.4, cell growth rate is suggested based on metabolic pathway analysis. Cells
(or whole microorganisms) die due to environmental conditions (for example heat) or phys-
iological developments. The death rate of cells is proportional to the cell population, and thus
the specific death rate is not a function of cell population of concentration, which is confirmed
by the experimental observation, Eqn (7) . That is,

(11.23)
In single-celled microorganisms, cell death is the point at which reinitiation of division is no
longer possible. Dead cells cease nutrient uptake and cell functions stop.
The net specific cell growth rate is the growth rate subtracted cell death rate, that is

m d ¼ k d

m net ¼ m G k d (11.24)

During sterilization by heat, cells are destroyed due to excessive heat. The death rate increases
with increasing temperature. In normal growth conditions, the death rate is usually very low.
Beforewe exiting the discussions on the cell growth, let us go back to revisit the biomass yield

in 11.3. Solving the mass balance equations of biomass and substrate based on Eqns (11.24) ,

(11.21) , (11.10) ,and (11.11) , we obtain the apparent yield in batch cultivations as

x

YF X=S 1

X X 0 k d k d

m max

K S
S 0 S

ln S 0
S

Y X=S ¼

S 0 S ¼ m max

(11.25)

That is, the apparent growth yield Y X/S is always less than the growth yield factor YF X/S . The
higher the death rate in comparison to the growth rate, the lower the apparent growth yield.
We note that

K S
S 0 S

ln S 0

K S
ðS 0 ; SÞ logmean S ¼

(11.26)