The dominant microbes responsible for the oxidation of biodegradable carbonaceous pollutants in biotreaters are chemoheterotrophic bacteria. Such bacteria reproduce by binary fission. When the necessary requirements for bacterial growth are satisfied, the growth rate of a bacterial culture can be expressed as:

 dx/dt = µ x

where x = microbial cell mass
µ = specific growth rate coefficient

This equation is acceptable for describing binary fission; the rate of increase is proportional to the mass of organisms. However, this equation lets the mass increase without limit. This must be false because no living things can grow after their supply of food is exhausted. We can advance our understanding of microbial growth by making the specific growth rate a function of the concentration of some nutrient.

Growth-limiting Nutrient

Living cells require several types of nutrients. These may supply carbon-energy, building blocks for biochemicals, or trace factors such as vitamins or hormones. All cells require relatively large amounts of carbon, nitrogen, sulfur, phosphorus, oxygen, hydrogen, etc. Except in very rare cases in which the recipe for the medium has been carefully adjusted, one nutrient will be exhausted before the others; this is termed the "growth-limiting nutrient".

It is not simply mass that determines the growth-limiting nutrient, but rather the proportion. Something present at low concentrations can be growth-limiting if it is the first to be exhausted. Of several equations that relate specific growth rate coefficient to concentration of limiting nutrient, the Monod equation is most popular and usually fits actual data quite well:

The red curve is a plot of specific growth rate coefficient versus concentration of growth-limiting substrate when there is no inhibition (MonoD). The white line on the graph is one possibility for a substrate that inhibits growth). 

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