Functions for microbial growth.

The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The functions considered were three-phase linear, logistic, Gompertz, Richards, Weibull and Baranyi. One data-set was used, comprising 34 curves stemming from viable count enumeration data of Yersinia enterocolitica grown on agar plates under different conditions of pH, temperature and carbon dioxide (time-constant conditions for each culture). Curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were based on goodness-of-fit: lowest residual mean square (RMS), extra residual variance F-test and Akaike's information criterion (AIC). The goodness-of-fit attained with all models was acceptable, with the Baranyi and three-phase linear functions showing best overall performance, followed by the Richards and Weibull, whilst the performances of the Gompertz and logistic were least satisfactory. Estimates of the maximum specific growth rate (µmax) and the lag time (T) were obtained with the six models, and then a multiple comparison was performed based on pairwise correlation analysis. Although Baranyi and three-phase linear gave lower estimates of µmax than the other four models, pairwise Pearson (R), Spearman rank-order (ρ) and Lin concordance (Rc) correlation coefficients were always greater than 0.998, 0.998 and 0.900, respectively, with a high level of statistical significance (P<0.001). These results indicate that all six models gave comparable estimates of µmax, and that all the curves were ranked in almost the same order according to the estimates of this growth attribute. However, the estimates of T varied considerably among the models, and in this case the pairwise correlation coefficients were not so high (R=0.700-0.999; ρ=0.486-0995 and Rc=0.222-0.983). In general, the Baranyi and three-phase linear gave the shortest, and the logistic model the longest, lag times. The position of the point of inflection and the different approaches used to estimate T by each model may explain the discrepancies observed among models. Our results indicate that general application of the Gompertz to describe microbial growth should be reconsidered critically, as other models showed a significantly superior ability to fit experimental data.