Predictions under Isothermal and Dynamically Changing Conditions

Abstract

We would like to comment on a point in the paper of Fujikawa and Morozumi (2). The authors demonstrated that the model of Baranyi and Roberts (1) and their model (called model III) gave almost identical specific growth rates under isothermal conditions. Then they obtained predictions under fluctuating temperature conditions by numerically solving the respective differential equations. They found that the predictions based on the Baranyi model significantly overestimated the observed growth, while their model III worked at an acceptable accuracy in that case, too. We point out below that this is a mathematical contradiction. Both models assume that the instantaneous slope of the growth curve is determined by the temperature at that moment, with no delay term in the equation. The model describing the effect of the temperature on the specific growth rate (secondary model) is derived from those isothermal experiments for which the Baranyi model and the authors’ model III gave almost identical rates. Therefore, the two models cannot give significantly different results for dynamic situations. In Fig. 1A, the broken line represents the increase in the bacterial population as the authors produced it by means of the Baranyi model. It shows that between the sixth and the ninth hours of the experiment, the total increase is ca. 3 to 4 log10 units. This corresponds to an overall growth rate of ca. 1 to 1.3 log10 units in an hour, or a ca. 2.3- to 3-h 1 specific rate (measured on the natural-log scale). But such high specific rates were only obtained (by fitting either the authors’ or the Baranyi model to log count data) at optimum temperatures, whereas in the period in question the temperature varied between ca. 20 and 25°C! Figure 1B shows similar inconsistency. We reproduced the authors’ procedure by using their data. Our predicted curves are much slower than the authors’ analogous predictions that they claim they obtained by using the Baranyi model. In fact, our curves were very close to the authors’ predictions that they produced by means of their model III, similar to the isothermal situation.