## 2 Materials and Methods
### Mathematical Models
#### 2.2.2 Modeling Single Cultures  
Both one-step and two-step kinetic data analysis methods were applied for the growth prediction of single cultures of _S. aureus, E. coli_, and Fresco culture in milk. Using the two-step approach, each growth curve was analyzed using the HM (Equation (1)) to determine its parameters. Subsequently, suitable secondary models, Equations (2) and (3), were used to analyze the effect of temperature on the maximum growth rates and lag times calculated from the primary model. Once the primary and secondary models were used, all they were integrated in the prediction tertiary model, in our case, the two-step HM\({}_{2\cdot\text{S}}\). As the initial values \(x_{0}\) were slightly different in the specific temperature experiments, the parameter vector \(X_{0}\) used for forward prediction contained nine parameters (\(x_{0,10}\), \(x_{0,12}\), \(x_{0,15}\), \(x_{0,18}\), \(x_{0,21}\), \(x_{0,25}\), \(x_{0,30}\), \(x_{0,35}\), and \(x_{0,37}\)). The maximum population density \(x_{\rm max}\) for each single microorganism was found to be independent of temperature; therefore, the average of the values estimated at isothermal conditions was used. The growth parameters \(\mu_{\rm max}\) and \(t_{\lambda}\) were replaced with square root secondary models (Equations (2) and (3)), respectively. Subsequently, the tertiary HM\({}_{\rm 2-S}\) for growth predictions of single cultures of _S. aureus_, _E. coli_, and Fresco culture was found to have 13 parameters (Equation (4)). Applying the one-step integrated method reported by [PERSON] [12; 27], all isothermal growth curves of the given single bacterial populations were simultaneously analyzed using the HM (Equation (1)) and the secondary square root models (Equations (2) and (3)) in one step, to directly construct the tertiary HM\({}_{\rm 1-S}\) and minimize the global sum of square error (\(SSE\)). By analogy, the HM\({}_{\rm 1-S}\) contains the same 13 parameters (Equation (4)) as the tertiary HM\({}_{\rm 2-S}\); 9 different initial values \(x_{0}\) for each incubation temperature, the estimated average value of \(x_{\rm max}\), and parameters \(b_{T}\), \(T_{\rm min}\), and \(b_{\lambda}\) from both square root models (Equations (2) and (3)). \[p_{HM}=\{X_{0};x_{\rm max};b_{T};T_{\rm min};b_{\lambda}\} \tag{4}\]  
where \(p_{\rm HM}\) is the vector of the parameters for the growth prediction of _S. aureus_, _E. coli_, and Fresco culture. The goodness-of-fit of the tertiary HM\({}_{\rm 1-S}\) and HM\({}_{\rm 2-S}\) was evaluated with the global sum of square error (\(SSE\)), the adjusted determination coefficient (\(R_{adj}\)), and the root mean square error (\(RMSE\)) to test the suitability of the models to fit the whole set of observation points according to Equations (5)-(7), respectively:  
\[SSE=\sum\limits_{i=1}^{n}\left(x_{i}^{\rm exp}-x_{i}^{\rm cal}\right)^{2} \tag{5}\]  
\[R_{adj}^{2}=1-\frac{\frac{SSE}{\frac{SSE}{\delta ST}}}{\frac{\delta ST}{d_{T}}} \tag{6}\]  
\[RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(x_{i}^{\rm exp}-x_{i}^{\rm cal} \right)^{2}}{n-p}} \tag{7}\]  
where \(x_{i}^{\rm exp}\) and \(x_{i}^{\rm cal}\) correspond to the observed and predicted log CFU mL\({}^{-1}\) values, respectively, \(n\) is the total number of data points, \(p\) is the number of parameters, \(SST\) is the total sum of square errors, \(df_{\rm E}=n-p\) and \(df_{\rm T}=n-1\) are the degrees of freedom. Statistically significant differences between the one- and two-step approaches for growth prediction of each microbial population were evaluated by post hoc analyses using [PERSON]'s test [23] after obtaining a significant F ratio from the analysis of variance.