How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 592

growth. Before cells can divide, it needs to have space available for the new cells. Just like the
reactants in an elementary reaction, the free space is necessary as one cell will take one cell's
space. Therefore, the effect of cell inhibition can be added the same way as the logistic equa-
tion as it reflects the concentration of free space in terms of the (imaginary) cells (available):

1

m max S
K S þ S

X
X N

m G ¼ (11.70)

where X is the maximum cell concentration or the packing density of the cells. The value of

N

X depends on the type of cells: size, shape, and physiological conditions. When the cell

concentration reaches its maximum packing density, cell growth stops.

N

11.14.2.1.4. INHIBITION BY TOXIC COMPOUNDS
Toxic compounds can affect the metabolic behavior of cells via poisoning and/or
rendering the necessary enzymes less effective. While poisoning can be modeled with cell
death in extreme cases, the deactivation of enzymes is similar to inhibition in enzymatic reac-
tions. The following rate expressions are used for competitive, noncompetitive, and uncom-
petitive inhibition of growth in analogy to enzyme inhibition.
Competitive inhibition:

m max S

1

m G ¼ (11.71)

I
K I

K S þ þ S

Noncompetitive and uncompetitive inhibitions:

m max S
ðK S þ SÞ

1

m G ¼ (11.72)

I
K P

þ

In some cases, the presence of toxic compounds in the medium results in the inactivation of
cells or death.

11.14.2.2. Multiple Limiting Substrates
So far, we have been focused on cases where only one substrate is limiting. When more
than one substrate is limiting, the microbial growth rate changes with the concentrations
of these substrates that can change significantly in the medium.

11.14.2.2.1. COMPLEMENTARY SUBSTRATES
When two or more substrates that are complimentary, all these substrates are necessary for
the microbial growth. In this case, product rule applies:

m G ¼ m max Y

N SC

S j
K S j þ S j

(11.73)

j¼ 1

where N SC is the number of limiting substrates that are complementary. For example, energy
source, nitrogen source, sulfur source, and carbon source can be complementary, if only one
of each is supplied and are not in excess.