Mathematical Modeling of Inhibitory Microbial Lethality Synergistic: Secondary Phytocompounds from Purple Toronjil, Temperature, and Harvest Stress Effects on Escherichia coli

Abstract

This research investigated the inhibition of *Escherichia coli* ATCC 25922 (E. coli) bacterial growth in situ, specifically on the stems and aerial parts of *Agastache mexicana* subsp. mexicana (Amm) or “purple toronjil” and on food-grade paper, both contained within Kraft paper bags with a plastic window. The qualitative phytochemical profile of an aqueous extract of Amm revealed the presence of various compounds including alkaloids, coumarins, tannins, flavonoids, saponins, triterpenes, and sterols. The results indicate that these secondary metabolites exhibit a synergistic bactericidal effect, especially when combined with temperature and starvation stress. This was quantified using a decay equation referred to as the bacterial growth inhibition profile of E. coli (BGIPEc). Calculations, which included first derivatives, gradients based on substrate effects and temperature as well as the area under the curve of BGIPEc, demonstrated that higher temperatures led to the greater inhibition of colony forming units (CFUs), further enhanced by the presence of secondary metabolites. Additionally, a shorter half-life corresponded to a faster change rate and a lower area under the curve, indicating a reduced survival rate over time. At lower temperatures, E. coli exhibited a survival effect, which was corroborated by the preceding calculations.