Revisiting secondary model features for describing the shoulder and lag parameters of microbial inactivation and growth models.

The Baranyi and Geeraerd models are two of the most reliable models for the description of, respectively, microbial growth and inactivation. They are defined as a system of differential equations, whose algebraic solution can describe the microbial response during isothermal conditions, especially when combined with suitable secondary models. However, there are still large uncertainties regarding the best functions to use as secondary models for the lag phase duration (λ) and the shoulder length (S). In this article, we revisit these models, focusing on the implications related to the assumption of an ideal substance whose dynamics define bacterial adaptation. We demonstrate that their link with the isothermal lag and shoulder leads to unique secondary models for the effect of temperature changes on λ and S. Namely, a log-linear relationship for S and a reverse cuadratic relationship for λ (considering a Ratkowsky model for μ). Furthermore, we observe a coupling between both secondary models (k and S for Geeraerd; λ and μ for Baranyi), reducing the number of unknown model parameters from four to three. Using data from the scientific literature, we illustrate the applicability of these results, being able to improve the robustness of parameter estimates. The identification of these links are of great relevance for the field of predictive microbiology, as they resolve the uncertainty regarding the functional form of secondary models. Our results also provide a way to assess the validity of those dynamic hypotheses using data gathered under isothermal conditions, something that was hardly possible using data gathered under dynamic conditions. Although this study is limited to the effect of temperature, the general approach and methodology are also applicable to other type of secondary models, so this article can be a blueprint for future studies.