**4.1 Growth Kinetics**

If a viable inoculum is introduced into a medium that contains a carbon source, suitable nitrogen source, other nutrients necessary for growth, and physiologic temperature and pH are maintained, it will grow. The rate of biomass synthesis is proportional to biomass present. That is

where r_{x} is the amount of cells synthesized in g L^{-1}
h^{-1}, X is cell concentration in g L^{-1}. The parameter
µ is called specific growth rate, analogous to the specific rate constant
in chemical reaction rate expressions. Recall the treatment of chemical
reactions, summarized below for ease of reference.

In the above C_{A} is concentration of A ( mol A L^{-1}),
-r_{A} is reaction rate ( mol A L^{-1}
h^{-1}) and k is rate constant ( h^{-1}). The negative sign
in front of -r_{A} is to comply with the definition of r_{A},
which is the rate of generation of A. In Eq(4-1), the negative sign is not
necessary as X increases with time.

Consider cell balance over a batch bioreactor:

Cells in - Cells out + Generation of cells = Accumulation of cells in Bioreactor

Substituting for r_{x} from Eq(4-1) and
noting that volume of reactor is constant gives,

The term, DX/X, represents fractional increase in cell amount and Dt
is the time over which the fractional increase was accomplished. That is,
µ can be interpreted as fraction of biomass formed per unit time.
For example if µ is 0.3 h^{-1}, every hour the biomass will
approximately increase by 30%. We use the term "approximately"
because we are using finite quantities to describe the rule which applied
at infinitesimal scale.

Treating µ as a constant for now, Eq (4-1) can be integrated to give

where X_{0} is the initial (inoculum) cell concentration. The
time, t, refers to the time since the inoculum emerged from lag phase. Eq(4-3)
can be rearranged setting the conditions for doubling of biomass. That is
= 2 and t = the doubling times, t_{d}.

The values of doubling time and specific growth rate have been reported by many researchers. Given below is a sample of typical values.

Organism |
Growth Rate, µ [h^{-1}] |
Doubling time, t_{d} [h] |

E. coli | 2.0 |
0.35 |

Yeast | 0.3 |
2.3 |

Hybridoma | 0.05 |
13.9 |

Insect Cells | 0.06 |
11.6 |