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How to derive a Ratkowsky (square root model) for use in predictive microbiology

DRAFT Article.

This article is available On Line to enable editing. Its draft status is scheduled to be removed by September, 2015. In the meantime comments are invited from scientists and technologists familiar with the subject area to help improve this article. I wish to acknowledge the generous comments and advice received from workers in this area. As a result of the feedback I will include more information about the principles behind modelling bacterial growth and the limitations of models.

Michael Mullan. 17th April, 2015

The purpose of this article is to explain how to derive a simple Ratkowsky square root model (Ratkowsky et al., 1982) that describes the growth of a fault causing bacterium on cooked meat over the temperature range 10° - 35 °C. 

An example of the growth curve of the fault causing bacterium on tomatoes at 35°C is shown in figure 1.

The curve shows the 4-typical growth phases, lag, logarithmic, stationary and decline.


The specific growth rate (k) is calculated for the  logarithmic phase using equation 1.

Equation 1. k=log10(Nt) - log 10 (N0)
                          T x 0.301


Nt = is the number of bacteria at the end of the observation period.
N0 = is the number when the observation period started.
T = is the time that has elapsed over the growth period in hours.

k= 9.146-6.669
        2 x 0.301

k= 4.115 generations h-1 at 37 °C.

Next we need to construct a series of curves at 10°, 20°, and 30 °C to find the specific growth rate at each temperature. We have got the growth curve at 35 °C already (Figure 1 above). The specific growth rates at each temperature are then tabulated using equation 1 against time (Table 1). Note I have not shown the growth curves at the other temperatures. 

  Table 1. The effect of temperature on the specific growth rates of a bacterial isolate on tomatoes
 Temperature, °C  Specific growth rate, h-1
 10  0.023
 20  0.16
 30  0.384
 35 0.518

 We can now use the Ratkowsky square root model (Ratkowsky et al., 1982) to derive the relationship between the growth rate constant, temperature and the initial number of microorganisms.

The equation has been described previously, √r= b (T-To), where r is the growth rate constant.  In particular we need to calculate b, the slope of the square root of specific growth rate versus temperature plot, and, To, the value at which the square root of growth rate intercepts the x axis.

The curve produced using linear regression (black line) and the actual data (blue line) using the data in table 1 is shown in figure 2.


Using Excel, the linear regression equation that describes the trend line (black line in figure, for the growth of the fault causing bacterium on cooked meat over the temperature range 10° to 35 °C is:

√r= 0.0228 (T-0.0693) where √r is the square root of the growth rate constant and T is the temperature in °C.

Use of Combase Tools

While the calculations described previously are not difficult it is possible to automate the derivation of growth rate equations using free tools from Combase.  DMFit is an Excel add-in to fit log counts vs. time data and extract parameters such as growth rate. It can be downloaded from Combase.

Model validation

Validation is an important element in model development. The literature cited below provides useful insights into model development and may prove useful during validation. The USDA Pathogen Modelling Program (PMP) and ComBase should be consulted for pathogen-models.


To be added.

Literature cited

Baranyi, J., Pin, C. and  Ross, T. (1999). Validating and comparing predictive models. Int. J. Food Microbiol. 48:159-166.

Baranyi, J.,  Ross, T., Roberts, T.A. and  McMeekin, T. (1996). The effects of overparameterisation on the performance of empirical models used in predictive microbiology. Food Microbiol. 13:83-91.

 Pin, C.,  Sutherland, J. P. and Baranyi, J. (1999). Validating predictive models of food spoilage organisms. J. Appl. Microbiol. 87:491-499.


To be added.

How to cite this article

Mullan, W.M.A. (2015). [On-line]. Available from: https://www.dairyscience.info/index.php/technology/259-predictive-modelling.html . Accessed: 18 August, 2017.  



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