title: Monod equation
text: # Monod equation
  
  The Monod equation is a mathematical model for the growth of microorganisms. It is named for Jacques Monod (1910 – 1976, a French biochemist, Nobel Prize in Physiology or Medicine in 1965), who proposed using an equation of this form to relate microbial growth rates in an aqueous environment to the concentration of a limiting nutrient. The Monod equation has the same form as the Michaelis–Menten equation, but differs in that it is 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 empirical Monod equation is:
  
      $\mu =\mu _{\max }{[S] \over K_{s}+[S]}$
  
  where:
  
  * μ is the growth rate of a considered microorganism
  * $μ_{max}$ is the maximum growth rate of this microorganism
  * [S] is the concentration of the limiting substrate S 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 (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.
  
  ## Application notes
  
  The rate of substrate utilization is related to the specific growth rate as follows:
  
      $r_{s}=\mu X/Y$
  
  where:
  
  * X is the total biomass (since the specific growth rate, μ is normalized to the total biomass)
  * Y is the yield coefficient
  
  $r_s$ is negative by convention.
  
  In some applications, several terms of the form [S] / $(K_s$ + [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). 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.
  
  ## Graphical determination of constants
  
  As with the Michaelis–Menten equation graphical methods may be used to fit the coefficients of the Monod equation:
  
  * Eadie–Hofstee diagram
  * Hanes–Woolf plot
  * Lineweaver–Burk plot
  