Square Root Model for Rates of Microbial Growth or Inactivation

Microbial growth or inactivation starts at a certain characteristic temperature, which is incorporated into temperature-dependent models for their rates. Ratkowski’s model, known as the "square root model" makes use of a minimum temperature. This model, in its original and modified forms, has been successful in describing the increase of a growth rate parameter in a large variety of microorganisms. This Demonstration provides graphical representations of these models’ properties near the onset of microbial growth or thermal inactivation, in which an inactivation rate parameter replaces the growth rate parameter.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Snapshot 1: growth rate parameter of a psychrophilic organism
Snapshot 2: growth rate parameter of a mesophilic organism
Snapshot 3: growth rate parameter of a thermophilic organism
Snapshot 4: inactivation rate parameter of a vegetative bacterial cell
Snapshot 5: inactivation rate parameter of a heat resistant bacterial endospore
Snapshot 6: inactivation rate parameter of a heat resistant bacterial endospore where
Ratkowski’s original model was written in the form , where is the growth rate parameter (usually the isothermal growth curve’s slope at its inflection point), is the temperature, and is the temperature below which growth ceases [1, 2]. The model can also be written in the general form: if , 0., else , where can but need not always be equal to 2 [2, 3].
In principle, the model can also be used for thermal inactivation, where is the inactivation rate parameter and is a temperature marking the onset of lethality [4].
This Demonstration provides a graphical representation of Ratkowski’s model in its original and expanded versions in the region around , that is, before growth is turned into inactivation or vice versa. Choose between pure growth and inactivation with the setter bar. You can use sliders to vary temperature in , and the magnitudes of , , and . The plot of the corresponding versus relationship has a blue arrow marking and a moving point on the curve. The numerical coordinates of the movable point, which are the values of and corresponding , are displayed above the plot.
When , the model yields versus relationships that are reminiscent of those produced by the log-logistic model.
The purpose of this Demonstration is only to illustrate the Ratkowski model and its modifications, not to match any particular organism or process. Therefore, not all parameter combinations allowed by the controls represent realistic growth or inactivation scenarios.
[1] D. A. Ratkowski, J. Olley, T. A. McMeekin, and A. Ball, "Relationship between Temperature and Growth Rate of Bacterial Cultures," Journal of Bacteriology, 149(1), 1982 pp. 1–5.
[2] T. Ross, "Bĕlehrádek-Type Models," Journal of Industrial Microbiology and Biotechnology, 12(3–5), 1993 pp. 180–189. doi:10.1007/BF01584188.
[3] L. Huang, C.-A. Hwang, and J. Phillips, "Evaluating the Effect of Temperature on Microbial Growth Rate—The Ratkowski and a Bĕlehrádek-Type Models," Journal of Food Science, 76(8), 2011 pp. M547–M557. doi:10.1111/j.1750-3841.2011.02345.x.
[4] M. M. Gil, T. R. S. Brandão, and C. L. M. Silva, "A Modified Gompertz Model to Predict Microbial Inactivation under Time-Varying Temperature Conditions," Journal of Food Engineering, 76(1), 2006 pp. 89–94. doi:10.1016/j.jfoodeng.2005.05.017.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+