Coupled Logistic Models to Describe Culture pH, Growth, Product Formation, and Substrate Consumption in Batch Cultures of Lactic Acid Bacteria

The coupled logistic models developed in this study, across various media (concen-trated whey (CW), diluted whey (DW), MRS, and TGE broths) and nutrient (glutamic acid or glucose) supplementation levels, revealed clear patterns in model performance for six culture variables. Biomass growth, lactic acid (LA) production, total antibacte-rial activity (TAA) synthesis, and pH evolution were generally well described by both uncoupled and coupled logistic models, with R² values above 0.9700. Coupling pro-vided moderate improvements for LA and TAA, particularly in glucose-supplemented DW media, by linking metabolite formation to biomass growth and pH decline, result-ing in more mechanistically and physiologically coherent models. The most pro-nounced benefits of coupling were observed for the consumption of total sugars (TS) and nitrogen (TN). In uncoupled models, R² values for TS and TN were often low or highly variable, whereas coupled models substantially increased R² (up to ~0.9990). These findings demonstrate that TS and TN consumption are strongly growth-associated processes and cannot be adequately modeled as independent phe-nomena. Overall, while uncoupled models are sufficient to describe growth, acidifica-tion, and metabolite formation, the coupled modeling approach provides a robust, in-tegrated framework that captures the interactions among biomass growth, nutrient utilization, and metabolite production.