How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 555

pathways in the middle of a growth cycle (see Example 10-1). After one carbon source is
exhausted, the cells adapt their metabolic activities to utilize the second carbon source.
The first carbon source is more readily utilizable than the second, and the presence of
more readily available carbon source represses the synthesis of the enzymes required for
the metabolism of the second substrate.
The exponential growth phase is also known as the maximum growth phase and logarith-
mic growth phase. In this phase, the cells have adjusted to their new environment. After
this adaptation period, cells can multiply rapidly with maximum rate, and cell mass
and cell number density increase exponentially with time. This is a period of balanced
growth, in which all components of a cell grow at the same rate (pseudosteady state).
That is, the average composition of a single cell remains approximately constant during
this phase of growth. During balanced growth, the net specific growth rate determined
from either cell number or cell mass would be the same. The specific growth rate is
constant, from which a phenomenological model is proposed for the exponential growth
phase:

r X ¼ m net X (11.2)

with the net specific growth rate being constant during this growth phase. This simple rela-
tion of Eqn (11.2) is called the Malthus growth model. In the batch process, the rate of change
of biomass concentration is the same as the rate of generation of biomass (mass balance). Inte-
gration of the mass balance equation with Eqn (11.2) as the rate of generation of biomass
yields

ln X

X 0 ¼ m net t

(11.3a)

or

X ¼ X 0 e m net t (11.3b)

where X and X 0 are cell concentrations at time t and initial time t 0, respectively.

The time required to double the microbial mass can be computed using Eqn (11.3a) as

¼

ln2
m net

t d ¼ (11.4)

The doubling time is also the time required for a new generation of cells to appear during
exponential growth period.
The deceleration growth phase follows the exponential phase. In this phase, growth deceler-
ates due to either the depletion of one or more essential nutrients or the accumulation of toxic
by-products of growth. For a typical bacterial culture, these changes occur over a very short
period of time. The rapidly changing environment results in unbalanced growth. During
unbalanced growth, cell composition and size will change. In the exponential phase, the
cellular metabolic control system is set to achieve maximum rates of reproduction. In the
deceleration phase, the stresses induced by nutrient depletion or waste accumulation cause
a restructuring of the cell to increase the prospects of cellular survival in a hostile environ-
ment. These observable changes are the result of the molecular mechanisms of repression
and induction that we discussed in Chapter 10. Because of the rapidity of these changes,