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.

1. Introduction

Foods are frequently contaminated by bacteria or viruses from feces, such as Escherichia coli (E. coli), which can result in various diseases. To ensure food safety, quality standards have been established based on an in-depth understanding of microorganisms including their behavior, life cycles, and interactions throughout the product’s shelf life. Mathematical models are essential in this process [1].

Certain medicinal plants are utilized as remedies in traditional medicine. These plants can be found in popular markets, while others are commercially packaged as infusions. Many possess notable effects, such as acting as tranquilizers, alleviating insomnia, and demonstrating antimicrobial activity, which has been assessed through in vitro studies. Among these, Aracanto (Lessonia nigrescens), Clinopodium mexicanum, and Citrus aurantium stand out. The extracts and essential oils derived from these plants are frequently incorporated into packaging to ensure food safety [2,3,4].

Some secondary metabolites have demonstrated significant antimicrobial activity. Coumarin, a heterocyclic compound comprised of a benzene ring and a pyrone ring containing oxygen, along with its derivatives, is widely recognized for its diverse biological activities and clinical applications. These compounds are particularly effective due to their strong inhibitory properties and favorable selectivity ratios. Coumarins are regarded as a promising class of bioactive heterocyclic compounds that exhibit a broad spectrum of biological activities including antimicrobial, antiviral, antidiabetic, anticancer, antioxidant, antiparasitic, antihelminthic, antiproliferative, anticonvulsant, anti-inflammatory, and antihypertensive effects [5]. Coumarin mammea A/BA (1) and the triterpene friedelin (6) were extracted from the leaves of Calophyllum brasiliense. Coumarin mammea A/AA (2) was extracted using hexane at room temperature from the fruit peels (421 g) of Mammea americana L. (Clusiaceae) [6]. Tannins are natural phenolic compounds that can be categorized into two main types: hydrolyzable tannins, with a molecular weight ranging from 500 to 3000, and condensed tannins, which have a molecular weight between 1000 and 20,000. There are also other tannins that are combinations of these two basic chemical groups. Research has shown that these compounds have an inhibitory activity against Escherichia coli O157:H7 [7]. Tannins have been identified in several medicinal plants, with concentrations of 2.15% in Cinchona and 10.41% in Epilobium. Notably, the highest concentrations were observed in plants that are typically not classified as rich in tannins such as Epilobium, Rubus, Sanguisorba, and Filipendula. These plants also exhibit an inhibitory effect on E. coli [8]. Crude saponins were extracted from the ethanolic extract of green tea seeds. The results showed that the saponins from green tea seeds have strong antibacterial activity against E. coli [9]. In the seeds of black pepper (Piper nigrum L.), the presence of saponins has been identified, and extracts demonstrating antibacterial activity against Escherichia coli ATCC 25922 exhibited a minimum bactericidal concentration (MBC) of 50 mg/mL [10]. Flavonoids are a group of plant-derived organic compounds known for their heterocyclic structure. Many biological properties of flavonoids have been reported including antimicrobial, antioxidant, and vascular activities [8].

There are three main groups of tests for assessing the antibacterial activity of bioactive compounds: diffusion methods, dilution methods, and bio-autography. The evaluation of antimicrobial activity against various bacteria in vitro, as influenced by antimicrobials, is typically carried out using different variations of the dilution methods. The minimum inhibitory concentration (MIC) is measured in these methods and is considered a standard for comparing other methods that assess antimicrobial susceptibility [11]. However, these methods do not accurately reflect the microbial behavior in situ. Considering the antibacterial effectiveness that the secondary metabolites of Amm have demonstrated through in vitro methods, they may still be present throughout the shelf life [3]. Ensuring food safety at various temperatures is essential to accurately simulate the shelf-life conditions that are influenced by the environment. Additionally, understanding the inhibition rate and survival capacity of E. coli ATCC 25922 over time is critical. We propose a model to calculate the half-life, “t_1/2”, of CFU within Kraft paper bags, which are widely used for packaging a range of infusions including dried medicinal and aromatic plants like Amm.

Moreover, we suggest a mathematical model to adjust the BGIPEc in order to evaluate the change rate and the area under the curve. This framework aims to elucidate the microbial lethality synergistic effect (MLSE) arising from the combined effects of temperature stress, starvation stress, and the secondary metabolites found in Amm.

2. Results

Table 1. Results of colorimetric reactions to find secondary metabolites of the Amm extract.

Metabolite and Reagent	Colorimetric Reaction	Result
Alkaloids Dragendorff Mayer Wagner	Turbidity or precipitate (red to orange, white to cream and brown)	− − −
Coumarins Bornträger	Yellow fluorescence (U.V.)	+++
Flavonoids Mg2+ and HCl	Red, orange, and violet color	++
Tannins Ferric chloride (FeCl3)	Hydrolyzable tannins (blue) Condensed tannins (green)	− +++
Confirmation
Solution of gelatin	Precipitate White	+++
Gelatine and saline solution	Precipitate White	+++
Saline solution	Precipitate White	−
Triterpenes/Sterols Liebermann–Buchard	Color blue, blue−green (sterols)	+
Salkowski	Red to purple (triterpene)	+
Saponins Water	Foam formation	++

(−) Not detected, (+) light positive reaction, (++) positive reaction, (+++) strong positive reaction.

displays the secondary metabolites identified through colorimetric reactions. Three different tests were conducted to confirm the presence of tannins in the Amm extracts. Coumarins and tannins exhibited the strongest positive reactions among the secondary metabolites.

The discrete points values were fitted using Equation (2), resulting in the bacterial growth inhibition profile of E. coli ATCC 25922 (BGIPEc), generated by MLSE acting on the substrates. Figure 1a,b shows the BGIPEc of the inoculation on Amm and food grade paper to 35.5 °C, whose $k$ values were 0.07283 and 0.05803 day⁻¹, respectively. In the same way, Figure 1c,d shows the BGIPEc of the inoculation on Amm and food grade paper to 8.12 °C, whose $k$ values were 0.05803 and 0.01306 day⁻¹, respectively. This means that the mortality of the CFUs is greater when $k$ is greater.

Assuming that BGIPEc is asymptotic for all experiments, then the best MLSE should achieve CFU = 0 in the shortest time; furthermore, the area under the curve should also be the lowest when there is more mortality and vice versa. To demonstrate this, the first derivative and the integral evaluated from $t_0$ to $t_1$ from Equation (2) were calculated, and the results are plotted in Figure 2b and Figure 3.

Figure 2a shows the BGIPEc for all of the experiments. The profiles tended to zero more quickly when the values of $k$ were higher and had less area under the curve. Therefore, it can be said that the value of $k$ is influenced by the MLSE.

Figure 2b illustrates the change rate of BGIPEc, which represents the value of the first derivative of Equation (2) (see Equation (12)). The CFUs were inhibited more rapidly when Amm was inoculated at higher temperatures, while inhibition occurred more slowly when food-grade paper was inoculated at lower temperatures. This indicates that MLSE is more effective in inhibiting CFUs when the secondary metabolites of Amm are present. This is further supported by Figure 2c,d, which shows the gradients of the BGIPEc change rate due to temperature effects (Δ(Amm-control)) and substrate effects (Δ(8.12–35.5 °C)), respectively.

Figure 3 shows the area under the curve between CFU and the survival time intervals calculated with Equation (13), revealing that the greater the survival time, the more area under the curve, even if the CFUs are few (see interval of time #4 for 8.12 °C).

3. Discussion

There are different types of mathematical models, especially for bacterial growth, whose exponential equations in general were used to fit the experimental data with the Gompertz and logistic equations for the in vitro inoculation experiments.

In the case of inactivation or bacterial inhibition, the models were made with the Weibull and Baranyi equations with MIC measurements at a bacterial density in vitro. Studies in situ have also been conducted on the survival of E. coli but without mathematical modeling [12,13,14,15,16,17].

The proposed model is innovative because it allows for the inhibition of CFUs of E. coli in situ, and it does not depend on the minimum inhibitory concentrations of phytochemical compounds.

We discuss the action mechanisms to inhibit the bacterial growth by every secondary metabolite: secondary metabolites exert different mechanisms of action on bacteria; some saponins bound with a sugar exert covalent bonds with lipid membranes forming steroid complexes, generating “pores” and altering the osmotic balance, producing bacterial lysis [18]. On the other hand, the mechanism of action of flavonoids takes place in the cell membrane with interactions of some peptidoglycans and lipopolysaccharides that inhibit the respiratory chain and ATP synthesis [19]. Meanwhile, some families of alkaloids contain bactericidal activity; it has been reported that their methoxyl functional groups act by inhibiting protein synthesis in furoquinoline-type alkaloids [20].

The antimicrobial mechanism of the flavonoids against E. coli ATCC 25922 was investigated through cell membranes and a liposome model. The release of bacterial protein and images from transmission electron microscopy demonstrated damage to the E. coli ATCC 25922 membrane [21]. Tannins directly inhibit bacteria by denaturation of bacterial proteins [8]. The high antibacterial activity of coumarin per se is due to both its lipophilic character and planar molecular structure, which contribute to penetration through the bacterial cell membrane or cell walls [22].

An analysis of Figure 3 reveals that the trend changed after the third time interval at 8.12 °C; the area under the curve at the fourth time period remained almost constant and increased significantly for food-grade paper and Amm, respectively. These values indicate that the inhibition of E. coli was smaller and survived for a longer time because the CFU trend was downward (see Figure 2a,b). Furthermore after this point, the area under the curve recovered its downward trend when the substrate was Amm (Figure 2a), however, when the substrate was food-grade paper, the trend took longer to decrease and did not reach zero (see Figure 3). The optimal temperature for E. coli growth is typically 35–40 °C. Growth is minimal below 10 °C, increases to about 25% of the maximum at room temperature, peaks at 35–40 °C, and declines sharply after 42 °C, dropping to nearly zero at 50 °C. However, in conditions of starvation stress and low water activity, like in the dry strata of Amm and food-grade paper, the temperature also becomes a stress factor. Research on the response of E. coli to abiotic stressors has shown that the bacteria often enter a viable but non-culturable state. In this state, they are unable to reproduce but still maintain the functionality of their electron transport chain. This is achieved through the structural integrity of their cytoplasmic membrane as well as the assembly of macromolecules and micromolecules such as proteins and ribosomes. Additionally, they retain the ability to reduce tetrazolium salts [23,24]. There was a direct relationship between temperature and the loss of culturability, whose interaction allows for non-viable cells to secrete proteins, amino acids, and carbohydrates into the surrounding medium, which surviving cells can utilize. This effect, stemming from a slowed metabolism, contributes to delaying cell degeneration when the temperature is below 10 °C [25]. Additionally, some food-grade papers contain corn starch on its surface coating [25,26].

The corn starch present in papers may contribute to this behavior, because these are a source of food and perhaps a higher water activity.

4. Material and Methods

4.1. Procedure to Obtain Plants of Amm

Plants were sown by means of seeds that underwent chemical priming pre-treatment for germination. The seeds were placed inside a laboratory in a sterilized box in Petri dishes on a double layer of paper towel moistened with distilled water and were subsequently watered on their surface by sprinkling. Once the seedlings had cotyledons (four leaves), they were sown in a tray with 200 cavities in a substrate of Canadian peat and vermiculite mixed in a 3/2 ratio, respectively, until they transformed into plants. During this process, the maximum and minimum temperature and relative humidity were recorded as 33.9 and 28.4 °C and 61.4 and 51.2%, respectively. When the plants reached a height of 8 to 12 cm, they were immersed in a solution of water and rooting agent. The substrate was a mixture in a ratio of 50/50 V/V of the humus of leaves and mountain clay loam soil of Taxco de Alarcón municipality, Guerrero México. The plants were placed under shade from the surrounding trees starting at 2 p.m. and from clouds typical of the mild environment. The temperature and relative humidity were measured every hour during the crop time with a programmed datalogger instrument.

Table 2. Irrigation doses during the experimental time.

Stage	Irrigation Water
1	2 L water
2	3 L water
3	3.5 L water

Stage 1 covers the months of August, September, October, November, and December and irrigation was applied only on Saturday. Stage 2 covers the month of January, and weekly irrigations were distributed between on Saturday. Stage 3 covers the month of February, and weekly irrigations were distributed between Tuesday, Thursday, and Saturday. These steps are in accordance with the manuscript published [27].

shows the monthly irrigation profiles during the crop season. To ensure the plants’ survival, the water quantity had to change over time to avoid the permanent wilting point.

4.2. Obtaining Aqueous Extracts and Phytochemical Tests of Amm

From the aerial parts, 46.2 g was separated for 48 h in a 1 L amber flask using distilled water as a solvent by covering the plant’s material. The extract was then filtered using Whatman No. 4 filter, gauze, and cotton wool to remove any remaining plant material. The solvent was removed using a B Rotavapor© R-100 (Buchi Labortechnick AG, Flawil, Switzerland) at a constant temperature of 40 °C under a reduced pressure of 0.8 bar, followed by drying in a borosilicate fume cupboard under a high vacuum to give 9.88 g of a completely dry green powder. The presence of secondary metabolites in the aqueous extract from the aerial parts of Amm was identified by chemical reaction using specific reagents as follows: Dragendorff, Mayer, and Wagnerfor alkaloids; Bornträger for coumarins; Mg²⁺ particles and HCl for flavonoids. The FeCl₃, gelatin, and saline solution assays were used to determine the presence of tannins. Additionally, the Liebermann–Burchard and Salkowski reactions were used to identify the presence of triterpenes and sterols, and finally, a foam formation assay was performed to determine the presence of saponins [28]. Note: Alkaloids: Wagner: Iodine: brand: Hycel, Potassium iodide (KI): brand: Meyer. Mayer: Mercury (II) iodide = Hgl2: Meyer A.C.S, KI: brand: Meyer. Dragendorff: Bismuth (III) nitrate Bi(NO₃)₃: brand: Fermont PA Cert, Tartaric acid (C₄H₆O₆): brand: Meyer A.C.S, Potassium iodide (KI): brand: Meyer. Tannins: Ferric chloride (FeCl₃): brand: Fermont PA Cert, Gelatin: brand: BD Bioxon. Flavonoids: Mg²⁺: brand: Meyer, Hydrochloric acid (HCl): brand: Meyer A.C.S. Coumarins: Bornträger = Potassium hydroxide (NaOH): brand: Meyer A.C.S. Triterpenes/sterols: Lieberman-Burchard = Acetic anhydride (C₄H₆O₃): brand: J.T. Baker, Sulfuric acid (H₂SO₄) = Salkowsky: brand: Meyer A.C.S. Saponins: Distilled water: brand: Meyer A.C.S.

Phytochemical Analysis of the Aqueous Extract

Identification of alkaloids. The technique to identify the presence of alkaloids was as follows: 50 mg of the aqueous extract was placed in test tubes and 6 mL of HCl (10%) was added. The mixture was heated for 10 min at 100 °C. The tube was then cooled and filtered using Whatman filter paper (No. 4). The filtrate was divided into three test tubes, and 5 to 10 drops (0.5 mL) of the reagents Dragendorff, Mayer, and Wagner were added. The presence of a slight turbidity or a red, orange, white, or brown precipitate is considered evidence of the presence of alkaloids [29]. Determination of volatile coumarins. Three milliliters of the aqueous extract was added to a test tube, the base of the test tube was coated with Whatman filter paper (No. 4), and 2 to 3 drops of 1 N NaOH solution (Borntraeger reagent) were added. The test tube was placed in a water bath at 100 °C for 10 min. The filter paper was observed under an ultraviolet light source (254–265 nm); the presence of yellow fluorescence in UV indicates the presence of coumarins [30].

Flavonoid test. 20 mg of the extract was added to a 25 mL glass bottle to which 5 mL of hydrochloric acid (36%) was added; magnesium chips of approximately 3 × 3 mm were deposited then incubated for 24 h at room temperature (25–28 °C). A color change (red) indicates the presence of flavonoids [31].

Identification of tannins. The presence of tannins was determined by dissolving 50 milligrams of the aqueous extract in 10 mL of distilled water and then filtering on Whatman paper (No. 4). In three test tubes, 3 mL of each sample was placed, and 1 to 3 drops of ferric chloride solution (10%) were added. The presence of a blue color indicates the presence of hydrolyzable tannins, and a green color indicates the presence of condensed tannins. To confirm the presence of these compounds, the remaining extract was divided into three equal parts in 10 mL glass vials; 1 mL of gelatin solution, gelatin solution + saline, or saline solution was placed in each vial. The presence of a white precipitate in the vials with gelatin solution and gelatin + salt solution indicated the presence of tannins. If this color change was observed in the vials with saline solution, it was considered negative [30].

Determination of triterpenes and steroids. To identify the presence of triterpenes and steroids, 30 × 100 mm test tubes were used. For this test, 50 mg of the extract was deposited. Ten milliliters of chloroform was added, and the mixture was filtered using Whatman paper (No. 4). The mixture was divided into two test tubes. In one of these tubes, 5 to 10 drops (0.5 mL) of Liebermann–Burchard reagent was added; in another test tube, 5 to 10 drops (0.5 mL) of Salkowski reagent was added to the sample, the latter reaction had to be used in a water bath on ice to avoid a violent reaction due to the sulfuric acid. If there was a blue/green color change, the reaction was considered positive for the presence of steroids, and a red to purple coloring indicated the presence of triterpenes [32].

Identification of saponins. The presence of saponins was determined as follows: 5 mL of the extract was added to a test tube and then placed in a boiling water bath for 2 to 5 min. The sample was then allowed to cool and finally shaken vigorously until foaming. Permanent foaming within 10–15 min was considered positive for the presence of saponins [29].

4.3. Microbiological Tests

4.3.1. Inoculum Preparation

A roasted E. coli ATCC 25922 (Thermo Scientific™ BactiDisks™, Waltham, MA, USA) was taken from glycerol and inoculated in 100 mL of Luria Bertani broth and incubated at 35 °C for 24 h. Then, the inoculum was poured into 300 mL of sterile isotonic NaCl solution (0.85% m/V), mixed by shaking, and then added to a new spray bottle.

E. coli strain ATCC 25922 is a recommended reference strain for antibiotic susceptibility testing. Broth passage has been found to often result in a shift in MIC levels. Therefore, it is best to maintain it on agar and prepare stocks for storage. For the experiments, the same glycerol was always used, thus reducing the probability of mutations in the E. coli genome. The optimal growth temperature for E. coli is considered to be between 35 and 40 °C. The agar plate method recommends taking measurements between 24 and 48 h, because at 24 h, all CFUs that have formed are visible; therefore, is guaranteed to inoculate. The isotonic NaCl solution works to maintain osmotic pressure into and outside the cells and avoids stress and/or cellular death.

4.3.2. Inoculation of Amm and Food-Grade Paper

Under aseptic conditions, three burners were lit on a table in the laboratory, and the foliage and aerial parts of Amm and 2 × 2 cm² food-grade white Kraft paper previously sterilized at 121 °C for 15 min were spread out. Both substrates were sprayed with the E. coli ATCC 25922 inoculum using a spray bottle for 34 and 28 min respectively. The substrates were constantly stirred to ensure a homogeneous distribution of the inoculum and were dried for 48 h at room temperature, taking care not to have air currents that could contaminate them (see Figure 4). Kraft paper bags with a plastic window were prepared under aseptic conditions: (a) 30 bags with 10 g of the inoculated substrate of Amm, and (b) 30 bags with 10 g of the inoculated substrate of food grade white Kraft paper. Then, 15 bags with each substrate were stored in a fridge with an average temperature of 8.12 °C and 15 bags with each substrate in a plastic box with Styrofoam blocks on its walls at an average temperature of 35.50 °C.

4.3.3. Measurement of E. coli ATCC 25922 CFU on Substrates

During the experimental period, a total of eight CFU measurements at different times of 0, 9, 18, 28, 48, 76, 120, and 167 days were taken based on the rule “Secretaria de Salud 1995. Norma Oficial Mexicana NOM-092-SSA1-1994. Método para la cuenta de bacterias aerobias en placa. Diario Oficial de la Federación. 12 de diciembre de 1995. México”. The technique also is called the agar plate method. Two replicates were performed for each treatment and time, and the CFU values were averaged according to the standard. This strategy minimizes the error in each measurement. The variability in each scenario is handled by the estimation method, which minimizes the sum of squared errors to obtain a fitted model that passes through the center of the CFU for each treatment.

The procedure to inoculate was as follows:

(a) Preparation of standard bead agar: In a 1 L Erlenmeyer flask, 350 mL of distilled water was mixed with 8.2 g of standard bead agar, heating with oscillating movements over the lit burner until the boiling point. Once dissolved, it was sterilized int an autoclave for 15 min at 120 °C.

(b) Preparation of 0.1% m/V peptone water: 1 g of casein peptone and 8.5 g of NaCl were mixed in 1 L of distilled water and stirred until a homogeneous mixture was obtained. Using a 5 mL pipette, 9 mL of the solution was emptied into each test tube with a screw cap and covered with aluminum foil and sterilized in an autoclave for 15 min at 120 °C. The samples were homogenized in order to redistribute the bacteria present.

(c) An initial suspension was prepared by grinding 10 g of Amm into 90 mL of peptone water at 0.1% m/V (1 × 10⁻¹ dilution) and another initial suspension of 10 g of food-grade paper in 360 mL of diluent (2.7 × 10⁻²) in a blender previously sterilized in an autoclave for 15 min at 120 °C.

(d) From the initial suspension, decimal dilutions were made from 10⁻² to 10⁻⁹. The dilutions were made with 0.1% m/V peptone water, taking 1 mL of the initial suspension with 9 mL of diluent. After the second suspension, we took 1 mL of the 10⁻² dilution, and so on.

(e) 1 mL of each dilution was added to sterile Petri dishes.

(f) 12 mL of previously melted and rested standard bead agar at a maximum of 30 °C was added to each plate. This is the most important to avoid variability.

(g) Gentle rotary movements were applied to the plates for 10 s to the left and 10 s to the right to evenly distribute the sample with the agar and allowed to solidify. Then, the plates were inverted and incubated at 37 °C in aerobiosis for 48 h.

(h) Only the CFUs in the plates that had between 25 to 250 CFUs were counted, and taking into account the dilution, calculations were made to determine the CFUs in the sample.

Table 3. CFU/g of E. coli by every treatment.

T/°C	Sample	t−1	t0
		(CFU/g)	(CFU/g)
35.5 °C	Control	0	51.35 × 10−7
	Amm	50 × 10−10	15.20 × 10−6
8.12 °C	Control	0	26.54 × 10−5
	Amm	83 × 10−10	89.50 × 10−4

displays the CFU/g of E. coli before and after being inoculated. The CFU values were higher in Amm compared to Kraft paper.

4.4. Mathematical Models

The proposed ungrowth (or exponential decay equation) equation to fit the experimental results is:

$$y\left(x\right)=a·e^{-kx}$$

(1)

Since the results through time can be evaluated like this:

$${CFU}_t\left(t\right)={CFU}_0·e^{-k·t}$$

(2)

We changed of variable: y(x) $y\left(x\right)$ by ${CFU}_t\left(t\right)$, $a$ by ${CFU}_0$, $x$ by $t$, which can be seen in Equation (2), where ${CFU}_t\left(t\right)$ represents the number of E. coli populations as a function of time, with ${CFU}_0$ indicating the colony forming units for time equal to zero. Here, $t$ refers to the measurement time in days, and $k$ is the constant rate of decrease in E. coli populations.

Table 4. Description of equation elements of Equation (2).

Element	Variable	Description	Units
${CFU}_t$	Dependent	CFU; t ≠ 0	Units
${CFU}_0$	Ordered to the origin Maximum amplitude $t_0$ CFU in a time interval since $t_0$ from $t_1$	CFU; t = 0	Units
$t$	Independent	Time	Day
$k$	Curvature of the function	Shows which experimental condition approaches zero most quickly	Day−1
$e$	Base of natural logarithm	How microbial growth is inhibited	Dimensionless

shows the elements and variables of Equation (2). When the CFU was measured at the initial experimental time, then $t$ was equal to zero; when the CFU was measured during the experimental time, then $t$ was different than zero. When $e$ was elevated to $-kt$, then we can obtain the quantity of CFUs of E. coli decreasing through time.

Then, transposing “t”

$$t={\displaystyle \frac{ln\left(\frac{{CFU}_t}{{CFU}_0}\right)}{-k}}$$

(3)

Equation (2) was evaluated in the border conditions obtained from the experimental results. The discrete points remained close to the curve (high values of determination coefficient R² > 0.95), therefore the model was a good fit.

Then, we have:

$$\underset{t\to \infty }{lim}{CFU}_t(t)=\underset{t\to \infty }{lim}{CFU}_t\left(t\right)={CFU}_0·e^{-k·t}=0$$

(4)

$$\underset{t\to 0}{lim}{CFU}_t={CFU}_0$$

(5)

$$\underset{k\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{CFU}_t=\underset{t\to 0}{lim}{CFU}_t={CFU}_0$$

(6)

To find the rate change over time, the first derivative was calculated, and then a variable change as follows:

$$u=-kt;y\left(t\right)={UFC}_t;a={UFC}_0$$

(7)

By substituting, deriving, and applying the chain rule,

$$y^{\prime }={\displaystyle \frac{da}{dt}}{e}^{-kt}+a{\displaystyle \frac{{de}^{-kt}}{dt}}$$

(8)

$${\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{t}}}={\displaystyle \frac{\mathrm{d}\mathrm{y}}{\mathrm{d}\mathrm{u}}}·{\displaystyle \frac{\mathrm{d}\mathrm{u}}{\mathrm{d}\mathrm{t}}}={\mathrm{e}}^{-\mathrm{k}\mathrm{t}}·{\displaystyle \frac{\mathrm{d}\left(-\mathrm{k}\mathrm{t}\right)}{\mathrm{d}\mathrm{t}}}$$

(9)

$$y^{\prime }=0·e^{-kt}+a\left(-k{·e}^{-kt}\right)$$

(10)

$$y^{\prime }=-k·a·e^{-kt}=-k·y$$

(11)

Then,

$$y^{\prime }+k·y=0$$

(12)

In the same way, to obtain the relationship between CFU and its survival time by every experimental time interval, we proposed an integrating operation. To calculate the area under the curve by every time interval or gradient, ${CFU}_0$ is the initial amplitude for $t_0$. Furthermore, $t_1$ is the final time according to the interval selected as follows:

$$Area={\int }_{t_0}^{t_1}{CFU}_0·e^{-k·t}dt={CFU}_0{\int }_{t_0}^{t_1}{e}^{-k·t}dt$$

Then by changing the variable and derivative:

$$u=-kt;du=-kdt;{\displaystyle \frac{du}{-k}}=dt$$

Therefore,

$${\displaystyle \frac{{CFU}_0}{-k}}{\int }_{t_0}^{t_1}{e}^{u}du$$

Integrating

$${\displaystyle \frac{{CFU}_0}{-k}}·e^{u1}-{\displaystyle \frac{{CFU}_0}{-k}}·e^{u0}$$

$$\mathrm{Where}u1=-k{t}_1;u0=-k{t}_0$$

Finally,

$$Area={\int }_{t_0}^{t_1}{CFU}_0·e^{-k·t}dt={\displaystyle \frac{{CFU}_0}{-k}}(e^{-k·t_1}-e^{-k·t_0})$$

(13)

If the area under the curve change is always following a decay tendency, then the inhibition process has an MLSE constant.

5. Conclusions

The colorimetric reactions of aqueous extracts showed a strong reaction of coumarins, tannins, saponins, and flavonoids that hypothetically have a positive and negative charge on E. coli (and vice versa); this interaction reduces their populations. With the simple exponential model, we described the decline of E. coli populations on paper and the aerial parts of Amm substrates and found that their BGIPEc occurred more rapidly at higher temperatures, with a higher $k$ value for Amm. Using the first derivative, we calculated the change rate CFU of E. coli across the substrates, which was the highest on Amm and at 35.5 °C. The commercial bags behaved like an adiabatic and isobaric system because they were placed at the same temperature and filled with the same amount of air. Gradients of the CFUs were calculated, allowing us to visualize the effect of substrates and temperatures separately. A mathematical model was established to calculate the area under the curve, which was set as the inhibitory microbial lethality synergistic.