4.3 Rate Expression and Metabolic Quotient

We have already discussed rate expressions for cell growth, Eq (4-1). Let us now examine rate expressions for other medium components in the growth reaction, Eq (2-1). Consider the growth reaction on the basis of one g of substrate consumed. It can be written as ,

The stoichiometric coefficients in growth reaction become yield coefficients on the basis of substrate. See Example 2-2. The general rate expression is then:

where ri is expressed in g of i L-1 h-1. Since rx is the most fundamental of the various rates, it is conventional to write the stoichiometric coefficient in terms of it. That is

Following the examples above, the rate expression for species i can be written as

Metabolic quotients are rate expressions on the basis of unit mass of biomass. That is

The metabolic quotient for oxygen is of special interest. This single property determines the upper limit of cell concentration that can be achieved in many bacterial fermentation systems. We will see further analysis in the next chapter. Typical values of metabolic coefficients are given below.


qglucose g glucose (g cell) -1 h-1

qO2 g glucose (g cell) -1 h-1

E. coli









Example 4-0

If specific growth rate of a bacteria is 0.35 h-1 and cell yield is 0.6, calculate glucose consumption rate.