Research with bioprocesses that have highly dilute media is difficult because of problems in measuring low concentrations and the analytical interferences of other unmeasured dilute nutrients. Survival of organisms in dilute media consumes most of the nutrients, leaving little for growth. Some microbiology at low nutrient concentrations.
Equations that work well for continuous cultivation with more concentrated media are:
Mass balance for cells: dx/dt = µ
x - Dx (1)
Mass balance for growth-limiting nutrient:
dS/dt = DSo - DS - µ
x /Y - Mx (2)
Monod equation:
Although actual data show patterns that agree fairly well with this theory, the Yield coefficient, Y, and the Monod coefficient, Ks, need not be constant and have been shown to be functions of cell physiology.
Worst of all is the concept that maintenance energy is constant. As concentration of limiting nutrient approaches zero, the term for endogenous metabolism in Equation 2 can consume more substrate than is available.
The need for better models inspired the use of a modified Monod model where the term for respiration forms a Monod-like equation:
Consider first the upper equation where Em is the maintenance energy, and the hat over it designates its maximum value. This equation works in the range of very low concentrations of S. At higher concentrations at which the usual Monod equation applies, it is a good assumption that Em is constant.
Further improvement comes from abandoning the Monod model for a model
in which there is no growth below some low nutrient concentration, Se. This is the second
equation in the above figure. Note the restriction that µ
= 0 when S is less than Se.
This gives a graph of µ
versus S that is almost the same as for the
Monod equation, but it does not pass through zero.
Speculation about a better graph