How Cells Grow - Bioprocess Engineering: Kinetics, Biosystems, Sustainability and Reactor Design - page 587

For many systems, segregation is not a critical component of culture response, so nonseg-
regated models will be satisfactory under many circumstances. An important exception is the
prediction of the growth responses of plasmid-containing cultures.
In this text, we focus more on the metabolic and genetic approaches and their
simplification/approximations. Therefore, our discussions in this text will be confined to
very simple phenomenological models only.

11.14.1. Unstructured Growth Models
The simplest growth kinetic equations are unstructured models. These models view the
cell as a single species in solution and attempt to describe the kinetics of cell growth based
on cell and nutrient concentration profiles. The simplest model is that of Malthus (

x

11.2):

r X ¼ m net X (11.2)
Upon examining Eqn (11.2) , one may conclude that this is the definition for specific
growth rate and thus valid in general. However, this is not the case in the context of
Malthus model, as m net is assumed to be constant, not dependent on substrate concentra-
tion. This is based on the empirical observation of batch growth during the maximum
growth phase ( 11.2).

It is clear that the application of Malthus model is limited as it provides no limit to the cell
growth. The only limit would be the exhaust of substrate, which is not built into the model.
Even for the simplest batch growth case, the deceleration phase would not be fully described
with the Malthus model. The relation to the substrate can be found via the growth yield.
To provide means to limit growth, Verhulst model inserted an apparent biomass inhibition
term to the Malthus equation:

x

(11.5)
which is also known as the logistic equation. The Verhulst model can describe the batch
growth quite accurately for a wide range of growth periods.
Apart from the ability of the logistic equation to correlate the deceleration and stationary
phases on top of the Malthus model, it has the appeal that the maximum cell concentration
can be considered. Although the carrying capacity X

r X ¼ kXð 1 X=X N Þ

was a parameter related to the
maximum nutrient level available in the medium, it can also be the maximum (packing)
density of the cells as well (when the nutrients are unlimited).

N

11.14.2. Simple Growth Rate Model: Monod Equation
The simple growth rate model can mechanistically describes microbial growth in close
approximation is the one that approximates the whole metabolic pathway and cell genera-
tion with a single rate-limiting step. In this case, the Monod equation prevails. This is also
the simplest model that incorporates the effects of both the cell concentration and substrate
concentration:

r X
X ¼

m max S
K S þ S

m G ¼ (11.21)