Monod equation

<h2 id="equation">Equation</h2>

The empirical Monod equation is<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

μ
        =
        
          μ
          
            max
          
        
        
          
            
              [
              S
              ]
            
            
              
                K
                
                  s
                
              
              +
              [
              S
              ]
            
          
        
      
    
    {\displaystyle \mu =\mu _{\max }{\frac {[S]}{K_{s}+[S]}}}

where:

μ is the growth rate of a considered microorganism,
μmax is the maximum growth rate of this microorganism,
[S] is the concentration of the <a href="/facts/Limiting_factor/IYufWHvx">limiting</a> <a href="/facts/Enzyme_substrate_(biology)/6MnaV19h">substrate</a> S for growth,
Ks is the "half-velocity constant"—the value of [S] when μ/μmax = 0.5.
μmax and Ks are empirical (experimental) coefficients to the Monod equation. They will differ between microorganism species and will also depend on the ambient environmental conditions, e.g., on the temperature, on the pH of the solution, and on the composition of the culture medium.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

<h2 id="application-notes">Application notes</h2>
The rate of substrate utilization is related to the specific growth rate as<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

r
          
            s
          
        
        =
        μ
        X
        
          /
        
        Y
        ,
      
    
    {\displaystyle r_{s}=\mu X/Y,}

where

X is the total biomass (since the specific growth rate μ is normalized to the total biomass),
Y is the yield coefficient.
rs is negative by convention.
In some applications, several 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. <a href="/facts/Organic_matter/t17tVCvk">organic matter</a> and <a href="/facts/Oxygen/acyn6o8r">oxygen</a> are both necessary to <a href="/facts/Heterotrophic/wtzlgBtz">heterotrophic</a> bacteria). When the yield coefficient, being the ratio of mass of microorganisms to mass of substrate utilized, becomes very large, this signifies that there is deficiency of substrate available for utilization.

<h2 id="graphical-determination-of-constants">Graphical determination of constants</h2>
As with the <a href="/facts/Michaelis%25E2%2580%2593Menten_kinetics/lYFykW3p">Michaelis–Menten equation</a> graphical methods may be used to fit the coefficients of the Monod equation:<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<ul><li><a href="/facts/Eadie%25E2%2580%2593Hofstee_diagram/4LEEL2Mn">Eadie–Hofstee diagram</a></li>
<li><a href="/facts/Hanes%25E2%2580%2593Woolf_plot/4mpPhulp">Hanes–Woolf plot</a></li>
<li><a href="/facts/Lineweaver%25E2%2580%2593Burk_plot/JGqpK2lC">Lineweaver–Burk plot</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Activated_sludge_model/hQW5x37a">Activated sludge model</a> (uses the Monod equation to model bacterial growth and substrate utilization)</li>
<li><a href="/facts/Bacterial_growth/L0RPTRAm">Bacterial growth</a></li>
<li><a href="/facts/Hill_equation_(biochemistry)/97o6hV38">Hill equation (biochemistry)</a></li>
<li><a href="/facts/Archibald_Hill/ad0uLQWg">Hill contribution to Langmuir equation</a></li>
<li><a href="/facts/Langmuir_adsorption_model/U7eDnqIv">Langmuir adsorption model</a> (equation with the same mathematical form)</li>
<li><a href="/facts/Michaelis%25E2%2580%2593Menten_kinetics/lYFykW3p">Michaelis–Menten kinetics</a> (equation with the same mathematical form)</li>
<li><a href="/facts/Gompertz_function/9nYLlKvx">Gompertz function</a></li>
<li><a href="/facts/Victor_Henri/vJk1HYrX">Victor Henri</a>, who first wrote the general equation form in 1901</li>
<li><a href="/facts/Von_Bertalanffy_function/I54juBT2">Von Bertalanffy function</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Monod, Jacques (1949). "The growth of bacterial cultures". Annual Review of Microbiology. 3: 371–394. doi:10.1146/annurev.mi.03.100149.002103. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Monod, J. (1942). Recherches sur la croissance des cultures bactériennes (in French). Paris: Hermann. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Dochain, D. (1986). On-line parameter estimation, adaptative state estimation and adaptative control of fermentation processes (Thesis). Louvain-la-Neuve, Belgium: Université catholique de Louvain. <a href="/wiki/Universit%C3%A9_catholique_de_Louvain" target="_blank">/wiki/Universit%C3%A9_catholique_de_Louvain</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"ESM 219: Lecture 5: Growth and Kinetics" (PDF). Archived from the original (PDF) on December 29, 2009. <a href="https://web.archive.org/web/20091229043435/http://www.bren.ucsb.edu/academics/courses/219/Lectures/Lecture_4_ESM219_06.ppt.pdf" target="_blank">https://web.archive.org/web/20091229043435/http://www.bren.ucsb.edu/academics/courses/219/Lectures/Lecture_4_ESM219_06.ppt.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Graeme, Walker M. (2000). Yeast Physiology and Biotechnology. John Wiley & Sons. pp. 59–60. ISBN 978-0-471-96446-9. <a href="978-0-471-96446-9" target="_blank">978-0-471-96446-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Metcalf, Eddy (2003). Wastewater Engineering: Treatment & Reuse (4th ed.). New York: McGraw–Hill. ISBN 0-07-041878-0. <a href="0-07-041878-0" target="_blank">0-07-041878-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">"ESM 219: Lecture 5: Growth and Kinetics" (PDF). Archived from the original (PDF) on December 29, 2009. <a href="https://web.archive.org/web/20091229043435/http://www.bren.ucsb.edu/academics/courses/219/Lectures/Lecture_4_ESM219_06.ppt.pdf" target="_blank">https://web.archive.org/web/20091229043435/http://www.bren.ucsb.edu/academics/courses/219/Lectures/Lecture_4_ESM219_06.ppt.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Monod equation open-in-new

Monod equation