Biochemical Engineering Fundamentals, Part 1 - Bailey J., Ollis D. 1989

Kinetics of substrate utilization, metabolic product formation, and biomass accumulation in cell cultures
Kinetics of balanced growth
Other kinetics equations for cell growth

Other mathematical expressions defining the specific Cell Growth Rate have also been proposed; in some cases, these expressions fit the experimental data better. For instance, Teissier, Moser, and Contois suggested the following equations:

Teissier equation:

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Moser equation:

Contois equation:

Compared to the Monod equation, the algebraic solution of the first two cell growth equations is rather difficult. The Contois equation incorporates an apparent Michaelis constant proportional to the biomass concentration x; therefore, as the population density increases, the maximum growth rate must decrease, eventually making μ proportional to x^-1.

Certain medium components, such as a substrate or a metabolic product, can reduce the specific cell growth rate. Andrews proposed an equation that accounts for substrate inhibition of cell growth [11]:

An example of a process inhibited by metabolic products is Alcoholic Fermentation; Aiba, Shoda, and Nagatani proposed the following type of growth function to describe anaerobic glucose fermentation in the presence of Yeast [9, 10]:

There are situations where cell growth is limited by two (or more) substrates simultaneously. Although very few data are available to characterize such processes, it can be assumed that each of the limiting nutrients obeys the Monod equation:

In the absence of more convincing data to validate this equation, it can be regarded as a criterion of sorts, making it possible to determine whether cell growth is indeed governed by multiple nutrients.