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The '''Monod equation''' is a [[mathematical model]] for the growth of microorganisms. It is named for [[Jacques Monod]] who proposed using an equation of this form to relate microbial growth rates in an aqueous environment to the concentration of a limiting nutrient.<ref>Jacques Monod. 1949. The Growth of Bacterial Cultures. ''Annual Review of Microbiology'', v. 3, p. 371.</ref><ref>Monod, J. 1942. Recherches sur la croissance des Cultures Bactériennes. Hermann. Paris</ref><ref>Dochain, D. 1986. On-line parameter estimation, adaptave state estimation and adaptave control of fermentation processes. Thése. Leuven University. Louvain la Neuve. Belgium</ref> The Monod equation has the same form as the [[Michaelis–Menten kinetics|Michaelis–Menten equation]], but differs in that it is [[Empirical relationship|empirical]] while the latter is based on theoretical considerations.

The Monod equation is commonly used in [[environmental engineering]]. For example, it is used in the [[activated sludge model]] for [[sewage treatment]].

==Equation==

The Monod equation is:<ref>http://www.bren.ucsb.edu/academics/courses/219/Lectures/Lecture_4_ESM219_06.ppt.pdf</ref>

: <math> \mu = \mu_\max {S \over K_s + S} </math>

[[Image:Monod 3.svg|thumb|200px|The specific growth rate μ as a function of substrate concentration S.]]

where:
* ''μ'' is the specific growth rate of the microorganisms
* ''μ''max is the maximum specific growth rate of the microorganisms
* ''S'' is the concentration of the [[Limiting factor|limiting]] [[Enzyme substrate (biology)|substrate]] for growth
* ''K''''s'' is the "half-velocity constant"—the value of ''S'' when ''μ''/''μ''max = 0.5

''μ''max and ''K''''s'' are empirical coefficients to the Monod equation. They will differ between species and based on the ambient environmental conditions{{citation needed|date=January 2012}}.

==Application notes==

The rate of substrate utilization is related to the specific growth rate as follows:<ref>Metcalf & Eddy, Inc. 2003. ''Wastewater Engineering: Treatment & Reuse'', 4th ed. McGraw–Hill: New York.</ref>

: ''r''''s''u = −''μX''/''Y''

where:
* ''X'' is the total biomass (since the specific growth rate, μ is normalized to the total biomass)
* Y is the yield coefficient

rsu is negative by convention.

In some applications, multiple terms of the form [S/(Ks + S)] are multiplied together where more than one nutrient or growth factor has the potential to be limiting (e.g. organic matter and oxygen are both necessary to [[Heterotrophic]] bacteria).

==Graphical determination of constants==

As with the [[Michaelis–Menten kinetics|Michaelis–Menten equation]] graphical methods may be used to fit the coefficients of the Monod equation:<ref>http://www.bren.ucsb.edu/academics/courses/219/Lectures/Lecture_4_ESM219_06.ppt.pdf</ref>

* [[Eadie–Hofstee diagram]]
* [[Lineweaver–Burk plot]]

==See also==
* [[Activated sludge model]] (uses the Monod equation to model bacterial growth and substrate utilization)
* [[Michaelis–Menten kinetics]]
* [[Jacques Monod]]

==References==
<references/>

[[Category:Environmental engineering]]
[[Category:Civil engineering]]
[[Category:Sewerage]]

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