Effect of sodium propionate on inhibition of Botrytis cinerea (in vitro) and a predictive model based on Monte Carlo simulation

Abstract

Botrytis cinerea is a ubiquitous fungal pathogen mainly found on citrus and stone fruits. The use of mathematical models to quantify and predict microbial growth curves has received much attention because of its usefulness in decision making for preventing risk to human and animal health. In this study, we used sodium propionate to inhibit mycelial growth of the pathogenic fungus B. cinerea in vitro and modeled the efficacy of sodium propionate using a mathematical model. The antifungal efficacy of different concentrations (0.1–2.2% w/v) of sodium propionate was evaluated by measuring mycelial growth. The higher the concentration of sodium propionate tested, the greater the inhibitory effect on B. cinerea. Three mathematical models were used as deterministic models: the modified logistic model, the modified Gompertz model, and the Baranyi and Roberts model. The modified logistic model showed the best performance with satisfactory statistical indices (root mean squared error: RMSE, and R²), indicating that it was a better fit than the other models tested in this study. Furthermore, a stochastic modified logistic model that assumes a multivariate normal distribution of two random kinetic parameters successfully described the growth behavior of B. cinerea mycelia at various concentrations of sodium propionate as a probability distribution. Although the performance of sodium propionate in inhibiting B. cinerea was not ideal, Monte Carlo simulation may be a useful tool for predicting the probability of events based on the variability of B. cinerea growth behavior.

Introduction

Botrytis cinerea is a ubiquitous fungal pathogen that is found mainly on fruits such as grape (Qin et al., 2010), jujube (Li et al., 2019a), and pear (Yu et al., 2007). B. cinerea occurs as visible gray mold rot on the surface of horticultural products and negatively impacts shelf life, resulting in significant economic losses. The use of synthetic fungicides, such as benzimidazoles and dicarboximides, to control fungal pathogens on agriculture products is now limited due to the risk of chemical residues remaining on the products, the emergence of resistant pathogenic strains, and public concern for environmental pollution and food safety (Ji et al., 2018; Sun et al., 2018).

Therefore, in order to tackle the above-described issues, alternative additives with properties equivalent to such fungicides are currently gaining attention. Numerous approaches, such as the use of biocontrol agents, natural antimicrobials, and Generally Regarded As Safe (GRAS) substances, have proven to be effective in controlling postharvest diseases (Ji et al., 2018).

Among GRAS substances, sodium propionate is widely used as a food preservative, animal feed additive, and in cosmetics with antimicrobial effects (Li et al., 2019b). There are numerous reports of the efficacy of sodium propionate against postharvest fungal pathogens, such as B. cinerea and Alternaria alternata (Fagundes et al., 2013), Monilinia fructicola (Karaca et al., 2014), and Sclerotium rolfsii (Türkkan and Erper, 2015). The addition of GRAS substances to produce coatings seems to be effective in quality preservation and shelf life extension of fresh and fresh-cut fruits, however, related scientific studies are limited or non-existent. In order to gain approval for the coating formulation, scientific findings on the efficacy and risk of sodium propionate are required. As for the coating treatment of fresh and fresh-cut fruits, coatings enriched with sodium propionate on fruit surfaces can be removed by washing with water or peeling, and the concentration of food additives in the coatings is extremely small in terms of the whole product. Before application of sodium propionate to coatings, it is necessary to carry out basic microbial experiments and quantify results with basic (deterministic) and advanced (stochastic) models to evaluate the antimicrobial effect at ambient temperature as a first step. In general, accelerated tests have been frequently used to rapidly determine the growth characteristics of microorganisms (Maringgal et al., 2020). Mohammadi et al. (2021) reported the preservation of strawberry fruit with edible coating at ambient temperature. As a result, the coated strawberries successfully retained their postharvest quality at ambient temperature (20 °C) for 7 d. Such coating treatments allow fruit and vegetables to be stored at room temperature, which may be useful in countries where the cold chain is not well developed.

Although the antimicrobial effect of sodium propionate on fungal pathogens has been confirmed, there are limited studies on the efficacy of sodium propionate in controlling postharvest fungal pathogens and the inactivation kinetics or predictive models of sodium propionate on postharvest fungal growth (De Oliveira et al., 2013). Prediction of fungal growth is of global interest for the improvement of food safety and quality (Basak and Guha, 2015; Ozcakmak and Gul, 2017). Further, accurate prediction by inactivation models benefits the food industry by facilitating the selection of optimum sterilization conditions to obtain the desired product, thus minimizing production costs and maximizing sensory and nutritional product quality (Chen et al., 2004).

The approach is based on the development of mathematical models that describe the effects of predominant control factors (such as temperature, pH, water activity, gaseous atmosphere, preservatives, etc.) on the lag phase and growth rate of pathogens and spoilage organisms (Andres et al., 2001).

There are various models (deterministic models) for predicting the behavior of microorganisms, which are classified as first-order, second-order, and third-order. Modified logistic, modified Gompertz, and Baranyi and Roberts equations are some of the widely used models that have been proposed to describe the nonlinear survival curves of microorganisms (Ozcakmak and Gul, 2017).

However, only a few studies have evaluated and modeled the effect of inoculum size on fungal growth. Gougouli et al. (2011) showed that while the inoculum level did not affect the radial growth rate, it had a significant effect on the duration of the lag time. To evaluate the effect of sodium propionate on the growth rate and lag time of fungi, the inoculum level should be kept constant. Due to the limitations of deterministic predictive models, which cannot account for the variability and uncertainty of bacterial phenomena (Abe et al., 2019), a Monte Carlo simulation can be introduced to overcome this problem using the law of large numbers (Aspridou and Koutsoumanis, 2015). In this study, a stochastic modeling approach was applied based on a model of B. cinerea mycelial growth, taking into account important variabilities and correlation between model parameters. The introduction of the stochastic model that assumes a multivariate normal distribution of growth variables makes it possible to predict a growth curve that takes into account the variation in the growth of B. cinerea and the correlation between model parameters, and characterizes a growth curve that varies with the strength of B. cinerea.

Therefore, the objectives of this study were to: (i) evaluate the effect of sodium propionate in controlling B. cinerea in vitro, (ii) predict mycelial growth using a deterministic model to evaluate the antifungal activity of sodium propionate, and (iii) develop a predictive model to estimate mycelial growth using the Monte Carlo method.

Materials and Methods

Fungal strain    The stock culture of B. cinerea (NRBC 100717) used in this study was obtained from the Biological Resource Center, National Institute of Technology and Evaluation (NBRC), Japan. The stock cultures were maintained on potato dextrose agar (PDA) slants at 25 ± 2 °C for further studies. The fungi were sub-cultured onto fresh PDA and incubated at 25 ± 2 °C for 3 d before the experiments.

Mycelial growth antifungal assays    The poisoned substrate method (dilution in solid media) was employed to investigate the effect of sodium propionate on radial mycelial growth of B. cinerea according to the method of Vilaplana et al. (2018) with slight modifications. PDA was autoclaved (KTS-3065, ALP, Inc., Tokyo) at 121 °C for 15 min, and then mixed with different amounts of sodium propionate to reach a concentration of 0.1–2.7% w/v at native pH (Table 1). During mixing, the medium was still warm (70–80 °C). Then, 10 mL (40–45 °C) of each concentration was poured into plastic petri dishes (90 mm diameter and 20 mm height). The same amount of PDA without sodium propionate was used as a control.

Table 1. pH dependency on the concentration of sodium propionate with PDA solution.

Concentration (%)	pH
0.0	5.55
0.1	6.73
0.2	6.83
0.3	6.95
0.4	7.01
0.5	7.09
1.1	7.12
1.6	7.15
2.2	7.18

The effect of sodium propionate on radial mycelial growth of B. cinerea was assessed according to a procedure modified from the method of De Oliveira et al. (2017). One agar plug (5 mm in diameter) containing mycelia of B. cinerea was taken from the edge of a 3-day old culture grown on PDA at 25 °C using a cork borer and transferred to the center of a petri dish containing PDA supplemented with sodium propionate. The petri dish was sealed with parafilm and incubated with B. cinerea in an incubator (CR-41, Hitachi, Japan) at 25 °C. The radial mycelial growth of the pathogen was measured using a caliper for 45 d or when the control sample was completely covered with mycelia. The radial mycelial growth diameter (mm) of the colonies was measured in two perpendicular directions to obtain the mean colony diameter. Five replicates of plates were used for each concentration. A fungicidal effect was reported when no mycelial growth was observed during incubation.

Modeling of fungal growth    Mathematical models (modified logistic model, modified Gompertz model, and Baranyi and Roberts model) were used to describe the growth behavior (lag, log, stationary, and decline phases) of B. cinerea on PDA supplemented with sodium propionate. The model was conducted using data from 0.1% to 2.2% w/v of sodium propionate treatments, as the highest concentration (2.7% w/v) of sodium propionate treatment did not result in fungal growth.

Modified logistic model    The modified logistic model is a sigmoidal growth model that is suitable for fitting growth curves with three phases: lag, exponential, and stationary phases (Tornuk et al., 2013). The modified logistic equation below was used to describe the nonlinear inactivation of microorganisms by Koide and Yasokawa (2008):   

where D(t) is the average colony diameter (mm) at time t (d), A is the maximum growth achieved during the stationary phase (mm), µ is the maximum specific growth rate (mm/d), and λ is the lag time (d).

a)    Modified Gompertz model The modified Gompertz model was developed primarily to describe the asymmetrical sigmoidal shape of microbial growth curves and was later used to fit inactivation kinetics (Chen and Zhu, 2011). This model is given as follows:   

b) Baranyi and Roberts model    To avoid omissions in the calculation of lag phase and the possibility of a final asymptote (Dagnas and Membré, 2013), we used the Baranyi and Roberts model as described in the following equation:   

  

where D₀ is the initial colony diameter (mm) (t = 0), µ (mm/d) is the radial proliferation rate, and λ (d) is the apparent lag time of the mold growth. D_max is the maximum colony diameter (often equal to the diameter of the culture dish).

Model validation    Model parameters of all the models were estimated by regression analysis using Microsoft Excel software (Excel 2016, Microsoft, Redmond, WA, USA). The RMSE was used to evaluate each model. The parameters provide information to identify differences between experimental data and the model estimates (Ozcakmak and Gul, 2017). The values of R² and RMSE were calculated as follows:   

  

where the value of the experimental data is given by D_exp, the value estimated by the model is given by D_cal, is the mean of D_exp, and n is the number of experimental observations.

Statistical analysis    One-way analysis of variance (ANOVA) was used to evaluate the effect of different concentrations of sodium propionate during the incubation period. When a significant (p < 0.05) effect was found, the mean values were further analyzed according to the Tukey-Kramer multiple range test.

Stochastic model based on the Monte Carlo method    The deterministic model can be transformed into a stochastic model using the Monte Carlo method (Gougouli and Koutsoumanis, 2017). The transformed model can predict the colony diameter of B. cinerea at each concentration of sodium propionate in the form of probability distributions. The modified logistic model was chosen as the target deterministic model to be transformed into a stochastic prediction model because it is a general model for predicting mold growth, as well as showing the best-fit model with statistical indices (low RMSE) in the previous study section.

The colony diameter D (t) (mm) at time t (d) and different concentrations was estimated stochastically based on a modified logistic model (Eq. 1) with two random numbers. Samples of random parameters were generated by a covariance decomposition algorithm using a random number generator in Microsoft Excel ( = 0, σ²=1), assuming a multivariate normal distribution for 1/µ (=µ′) and λ. A total of 10 000 Monte Carlo simulations were performed to estimate the B. cinerea colony diameter distribution during growth. A procedure for generating random numbers based on the covariance matrix obtained from the analysis of experimental data is provided by Tanaka et al. (2008).

Results and Discussion

Antifungal activity of sodium propionate on mycelial growth of B. cinerea (in vitro)    The inhibitory effect of sodium propionate was evaluated according to Fagundes et al. (2013). The mycelial diameter at each concentration of sodium propionate was measured after 5 d of incubation at 25 °C, and the inhibition of fungal growth was evaluated by comparing with the control sample. During storage at 25 °C, the inhibitory effect of sodium propionate on mycelial growth of B. cinerea increased with increasing concentration (Fig. 1). The effect of sodium propionate increased with increasing concentration as shown in Figs. 1, 2, and Table 2 (p < 0.05); 2.7% w/v sodium propionate showed a bactericidal effect of 100% inhibition of B. cinerea germination (data not shown). Similar results were also obtained by Fagundes et al. (2013), where the inhibition of B. cinerea growth by sodium propionate at concentrations of 0.2%, 1.1%, and 2.2% w/v were 57.39%, 87.07%, and 94.18%, respectively. However, the effect of sodium propionate on inhibition of molds varied depending on the target molds; complete inhibition of M. fructicola was observed at concentrations above 1.0% (Karaca et al., 2014). Furthermore, the antifungal effect of sodium propionate was assessed by various researchers. Droby et al. (2003) reported that the mycelial growth of B. cinerea was completely inhibited after treatment with 5% w/v calcium propionate, while (Mills et al. 2004). reported that mycelial growth was completely inhibited at a concentration of 0.2 M (3.7% w/v). The difference in pH of these solutions may have affected the results. Sodium propionate is categorized as a weak acid. The mechanism of action of weak acid preservatives has not yet been clarified (Palou et al., 2016).

Fig. 1.

The mycelium growth of B.cinerea on PDA medium mixed with sodium propionate at various concentrations (0.0–2.2%) during storage at 25 ºC for 19 days.

Fig. 2.

Changes in the mycelium diameter of B.cinerea on PDA medium mixed with sodium propionate at various concentrations (0–2.2%) during storage at 25 ºC for 19 days.

Table 2. The mycelium diameter (mm) of B.cinerea growing on PDA mixed with sodium propionate at various concentrations (0.0–2.2%) during storage at 25 °C.

Storage time (day)	Concentration (%)
	0.0	0.1	0.2	0.3	0.4	0.5	1.1	1.6	2.2
0	4.50Ac	4.50Ae	4.50Af	4.50Ah	4.50Ah	4.50Ag	4.50Ag	4.50Af	4.50Ab
1	10.28Ac	5.86Be	4.74Bf	4.50Bh	4.50Bh	4.50Bg	4.50Bg	4.50Bf	4.50Bb
3	58.60Ab	22.55BCde	24.00Be	13.28BCDgh	12.08BCDg	10.39CDf	7.37Dfg	5.95Def	4.50Db
5	86.50Aa	29.81Bd	23.78Be	16.13Cgf	14.16Cfg	12.05CDf	9.40CDef	6.50Ddef	4.88Dab
7	86.50Aa	52.05Bc	37.42Cde	21.39Defg	20.18Def	17.30DEe	11.47DEFde	7.76EFcde	5.88Fab
9	86.50Aa	67.35Bbc	47.65Ccd	24.93Ddef	22.46DEde	19.91DEFde	12.89DEFcd	8.64EFbcd	6.31Fab
11	86.50Aa	82.74Aab	58.82Bbc	28.43Ccde	25.41CDcde	22.04CDEcde	13.98CDEcd	8.80DEbcd	6.37Eab
13	86.50Aa	86.50Aa	74.83Bab	33.26Ccd	28.34CDcd	24.73Dbcd	14.83Ebc	9.40EFbc	6.65Fab
15	86.50Aa	86.50Aa	85.08Aa	39.00Bbc	31.74Cbc	26.97Cabc	15.69Dbc	10.01DEabc	6.96Eab
17	86.50Aa	86.50Aa	86.50Aa	48.48Bb	35.97Cab	29.73Cab	17.81Dab	10.65DEab	7.26Eab
19	86.50Aa	86.50Aa	86.50Aa	59.86Ba	42.68Ca	30.43Da	21.00Ea	12.19Fa	7.74Fa

Means having same letters are not significantly difference (p< 0.05). Capital letters are for comparing concentrations in each time (column) and small letters are for comparing times in each concentration (row).

A variety of mechanisms of action have been proposed, including disruption of cell membranes, inhibition of essential metabolic functions, stress on pH homeostasis, and accumulation of toxic anions in cells. However, the main mechanism of action of weakly acidic preservatives is thought to be the passage of undissociated compounds through the cell membrane. (Mills et al., 2004). Thus, the use of sodium propionate to control molds should be based on the appropriate concentration for use and the target mold.

Predictive modelling

a) Deterministic modelling    The relationship between diameter and time follows a sigmoidal pattern of microbial growth (lag, log, stationary, and decline phases) and can be described using an empirical growth function (López et al., 2004). Fig. 3 shows the growth curves of B. cinerea at different concentrations of sodium propionate. The dotted curve was obtained by fitting a nonlinear regression model to the experimental data on colony diameter of B. cinerea (indicated by symbols). As the concentration of sodium propionate increased, there was a decrease in the maximum growth rate (µ_max) of the fungus and an increase in the apparent lag time (λ). The estimated model parameters are shown in Table 3. The results are in good agreement with previous studies conducted by Basak and Guha (2015), and Basak and Guha (2017), where they found similar trends in µ_max and λ in the growth curve of Penicillium expansum on medium containing betel leaf essential oil at a concentration of 0–0.6 µg·mL⁻¹. µ_max and λ of colony diameters of B. cinerea on PDA mixed with sodium propionate were estimated using the modified logistic model, modified Gompertz model, and Baranyi and Roberts model. Satisfactory statistical index (RMSE) was the lowest for the modified logistic model for predicting the mycelial growth of B. cinerea. In nonlinear regression analysis, R² is used only as a reference, and the residuals are more important for evaluating the fit of the model b) to the experimental data. The measured values were compared with the calculated values as true values; however, no bias was found and, therefore, the RMSE was used as the evaluation criterion. This means that the parameters of the modified logistic model were more accurate than the other models.

Fig. 3.

The mycelium diameter of B.cinerea growing on PDA mixed with sodium propionate at various concentrations (0–2.2%) during storage at 25 ºC for 45 days fitting to mathematic models (modified logistic, modified Gompertz and Bayanri and Robert model). The dotted line corresponds to the calculated values from the models distribution function. The symbols correspond value from the experiment.

Table 3. Estimated values and statistics for the parameters of deterministic model describing the effect of sodium propionate at various concentration on growth rate of B.cinerea

Concentration (%)	Modified logistic model				Modified Gompertz model				Baranyi and Robert model
	µ	λ	RMSE	R2	µ	λ	RMSE	R2	µ	λ	RMSE	R2
0.0	31.894	1.268	0.653	0.999	38.388	1.100	1.294	0.997	19.339	0.570	4.582	0.979
0.1	10.140	1.743	2.465	0.991	10.213	1.371	3.295	0.985	7.304	0.752	3.848	0.979
0.2	7.445	2.090	2.741	0.990	7.570	1.641	3.725	0.982	5.705	1.082	3.519	0.982
0.3	4.478	4.685	4.143	0.982	4.508	3.837	5.792	0.964	3.403	2.912	4.736	0.971
0.4	3.270	4.759	4.527	0.977	3.300	3.673	6.263	0.957	2.594	2.736	3.999	0.981
0.5	2.689	12.683	6.419	0.954	2.949	5.595	8.373	0.922	2.444	5.235	6.410	0.949
1.1	2.401	12.683	5.493	0.950	2.429	10.321	7.531	0.909	2.400	13.007	5.674	0.956
1.6	1.322	15.656	0.916	0.992	0.976	7.218	1.674	0.975	0.937	10.487	2.023	0.964
2.2	0.585	32.830	0.455	0.970	0.333	8.966	0.471	0.966	0.222	5.474	0.491	0.967

As mentioned above, not all mathematical models can be used to predict fungal growth. Therefore, the goodness of fit of the model needs to be considered before using the model to characterize the growth of the fungus.

Stochastic model    The establishment of quantitative microbial risk assessment (QMRA) as a basis for food safety management has increased the need for stochastic bacterial growth models (Abe et al., 2018, Koutsoumanis and Lianou, 2013). The limitations of deterministic models, which cannot describe the heterogeneity of individual cells, were overcome using the concept of probability, i.e., Monte Carlo simulations (Abe et al., 2018, Aspridou and Koutsoumanis, 2015, Hiura et al., 2020). The probability of colony diameter of B. cinerea at time t after treatment with different concentrations of sodium propionate by Monte Carlo simulation (10 000 runs) is shown in Fig. 4. The results show that there is a large variation in the colony diameter due to the variability in mycelial growth under different concentrations. It is notable that the largest variations in data were found for PDA supplemented with a low concentration of sodium propionate. This may be due to the low sodium propionate concentration and adaptation of the mold to suboptimal conditions. Since the total RMSE and correlation coefficient R between experimental and predicted values for variation in mycelial diameter were 1.6 mm and 0.71, respectively, the Monte Carlo simulation was able to predict not only the mean diameter of the mycelium but also its variability. The probabilistic concept of the Monte Carlo simulation technique can be applied to estimate the colony diameter distribution of Aspergillus niger in yogurt with reasonable accuracy and reliability according to QMRA (Gougouli and Koutsoumanis, 2017). Monte Carlo simulation has been successfully used to estimate variability and uncertainty phenomena in the survivor behavior of Bacillus simplex during thermal processing (Abe et al., 2019). With Monte Carlo simulation we can predict the variability of microbial growth, i.e., the intensity of the individual activity of a living organism. Also, the deterministic model can only predict the day of disposal, while the stochastic model can predict the food loss rate on a given day. Thus, Monte Carlo simulation is a useful technique for assessing food loss rate with more reliable determination.

Fig. 4.

Monte Carlo simulation results (10 000 simulations) and experimental data for the growth of B.cincerea after being treated with sodium propionate at various concentrations for 45 days. The solid and dotted lines correspond to the average and SD values calculated from the model distribution function respectively. The symbols correspond value from the experiment.

Conclusions

The antifungal efficacy of different concentrations (0.1–2.2% w/v) of sodium propionate was evaluated by measuring mycelial growth. The higher the concentration of sodium propionate, the more effectively it inhibited the growth of B. cinerea mycelia. To quantify and predict the growth behavior of B. cinerea on PDA with sodium propionate, three general models were employed, and their agreement was evaluated. As a result, the modified logistic model fit well and could be described with satisfactory RMSE and R² values compared to other models. Monte Carlo simulation based on statistical kinetic data was proposed to estimate the mycelial growth stochastically. In particular, given the fact that the stochastic prediction is based on the covariance of kinetic parameters, the matrix of the maximum specific growth rate and lag time, it has the potential to contribute to more practical applications. The future challenge arising from this study will be to apply the treatment conditions and those derived from the modeling to actual food systems.

Acknowledgements    The work was supported by a JICA Innovative Asia scholarship [Project no. D1706943] and The Toyo Suisan Foundation.

Conflict of interest    There are no conflicts of interest to declare.

Credit authorship contribution statement    Passakorn Kingwascharapong: Experiment, Methodology, Writing – review & editing. Fumina Tanaka and Fumihiko Tanaka: Supervision, Conceptualization, Analyzing and Writing – review & editing, Arisa Koga: Analyzing and Writing – review & editing, Supatra Karnjanapratum: Writing – review.

References

•  Abe,  H.,  Koyama,  K.,  Kawamura,  S., and  Koseki,  S. (2018). Stochastic evaluation of Salmonella enterica lethality during thermal inactivation. Int. J. Food Microbiol., 285, 129-135. https://doi.org/10.1016/j.ijfoodmicro.2018.08.006.
•  Abe,  H.,  Koyama,  K.,  Kawamura,  S., and  Koseki,  S. (2019). Stochastic modeling of variability in survival behavior of Bacillus simplex spore population during isothermal inactivation at the single cell level using a Monte Carlo simulation. Food Microbiol., 82, 436-444. https://doi.org/10.1016/j.fm.2019.03.005.
•  Andres,  S.,  Giannuzzi,  L., and  Zaritzky,  N. (2001). Mathematical modeling of microbial growth in packaged refrigerated orange juice treated with chemical preservatives. J. Food Sci., 66, 724-728. https://doi.org/10.1111/j.1365-2621.2001.tb04628.x.
•  Aspridou,  Z. and  Koutsoumanis,  K.P. (2015). Individual cell heterogeneity as variability source in population dynamics of microbial inactivation. Food Microbiol., 45, 216-221. https://doi.org/10.1016/j.fm.2014.04.008.
•  Basak,  S. and  Guha,  P. (2015). Modelling the effect of essential oil of betel leaf (Piper betle L.) on germination, growth, and apparent lag time of Penicillium expansum on semi-synthetic media. Int. J. Food Microbiol., 215, 171-178. https://doi.org/10.1016/j.ijfoodmicro.2015.09.019574.
•  Basak,  S. and  Guha,  P. (2017). Betel leaf (Piper betle L.) essential oil microemulsion: Characterization and antifungal activity on growth, and apparent lag time of Aspergillus flavus in tomato paste. LWT, 75, 616-623. https://doi.org/10.1016/j.lwt.2016.10.021.
•  Chen,  H.,  Joerger,  R.D.,  Kingsley,  D.H., and  Hoover,  D.G. (2004). Pressure inactivation kinetics of phage λ cI 857. J. Food Prot., 67, 505-511. https://doi.org/10.4315/0362-028X-67.3.505.
•  Chen,  Z. and  Zhu,  C. (2011). Modelling inactivation by aqueous chlorine dioxide of Dothiorella gregaria Sacc. and Fusarium tricinctum (Corda) Sacc. spores inoculated on fresh chestnut kernel. Lett. Appl. Microbiol., 52, 676-684. https://doi.org/10.1111/j.1472-765X.2011.03061.x.
•  Dagnas,  S. and  Membré,  J.M. (2013). Predicting and preventing mold spoilage of food products. J. Food Prot., 76, 538-551. https://doi.org/10.4315/0362-028X.JFP-12-349.
•  De Oliveira,  K.Á.R.,  Berger,  L.R.R.,  De Araújo,  S.A.,  Câmara,  M.P.S., and  De Souza,  E.L. (2017). Synergistic mixtures of chitosan and Mentha piperita L. essential oil to inhibit Colletotrichum species and anthracnose development in mango cultivar Tommy Atkins. Food Microbiology, 66, 96-103. https://doi.org/10.1016/j.fm.2017.04.012.
•  De Oliveira,  T.L.C.,  De Araújo Soares,  R., and  Piccoli,  R.H. (2013). A Weibull model to describe antimicrobial kinetics of oregano and lemongrass essential oils against Salmonella Enteritidis in ground beef during refrigerated storage. Meat science, 93, 645-651. https://doi.org/10.1016/j.meatsci.2012.11.004.
•  Droby,  S.,  Wisniewski,  M.,  El Ghaouth,  A., and  Wilson,  C. (2003). Influence of food additives on the control of postharvest rots of apple and peach and efficacy of the yeast-based biocontrol product Aspire. Posthavest Biol. Tec., 27, 127-135. https://doi.org/10.1016/S0925-5214(02)00046-7.
•  Fagundes,  C.,  Pérez-Gago,  M.B.,  Monteiro,  A.R., and  Palou,  L. (2013). Antifungal activity of food additives in vitro and as ingredients of hydroxypropyl methylcellulose-lipid edible coatings against Botrytis cinerea and Alternaria alternata on cherry tomato fruit. Int. J. Food Microbiol., 166, 391-398. https://doi.org/10.1016/j.ijfoodmicro.2013.08.001.
•  Gougouli,  M.,  Kalantzi,  K.,  Beletsiotis,  E., and  Koutsoumanis,  K.P. (2011). Development and application of predictive models for fungal growth as tools to improve quality control in yogurt production. Food Microbiol., 28, 1453-1462. https://doi.org/10.1016/j.fm.2011.07.006.
•  Gougouli,  M. and  Koutsoumanis,  K.P. (2017). Risk assessment of fungal spoilage: A case study of Aspergillus niger on yogurt. Food Microbiol., 65, 264-273. https://doi.org/10.1016/j.fm.2017.03.009.
•  Hiura,  S.,  Abe,  H.,  Koyama,  K., and  Koseki,  S. (2020). Transforming kinetic model into a stochastic inactivation model: Statistical evaluation of stochastic inactivation of individual cells in a bacterial population. Food Microbiol., 103508. https://doi.org/10.1016/j.fm.2020.103508.
•  Ji,  D.,  Chen,  T.,  Ma,  D.,  Liu,  J.,  Xu,  Y., and  Tian,  S. (2018). Inhibitory effects of methyl thujate on mycelial growth of Botrytis cinerea and possible mechanisms. Posthavest Biol. Tec., 142, 46-54. https://doi.org/10.1016/j.postharvbio.2018.04.003.
•  Karaca,  H.,  Pérez-Gago,  M.B.,  Taberner,  V., and  Palou,  L. (2014). Evaluating food additives as antifungal agents against Monilinia fructicola in vitro and in hydroxypropyl methylcellulose-lipid composite edible coatings for plums. Int. J. Food Microbiol., 179, 72-79. https://doi.org/10.1016/j.ijfoodmicro.2014.03.027.
•  Koide,  S. and  Yasokawa,  D. (2008). Growth prediction of mycelial mat and fruiting zone diameters of Aspergillus niger subjected to temperature changes. J. Japan Scoc. Food Sci. Technol., 55, 338-344. https://doi.org/10.3136/nskkk.55.338.
•  Koutsoumanis,  K.P. and  Lianou,  A. (2013). Stochasticity in colonial growth dynamics of individual bacterial cells. Appl. Environ. Microbiol., 79, 2294-2301. http://dx.doi.org/10.1128/AEM.03629-12.
•  Li,  H.,  He,  C.,  Li,  G.,  Zhang,  Z.,  Li,  B., and  Tian,  S. (2019a). The modes of action of epsilon-polylysine (ε-PL) against Botrytis cinerea in jujube fruit. Posthavest Biol. Tec., 147, 1-9. https://doi.org/10.1016/j.postharvbio.2018.08.009.
•  Li,  L.,  Tang,  X.,  Ouyang,  Q., and  Tao,  N. (2019b). Combination of sodium dehydroacetate and sodium silicate reduces sour rot of citrus fruit. Posthavest Biol. Tec., 151, 19-25. https://doi.org/10.1016/j.postharvbio.2019.01.006.
•  López,  S.,  Prieto,  M.,  Dijkstra,  J.,  Dhanoa,  M.S., and  France,  J. (2004). Statistical evaluation of mathematical models for microbial growth. Int. J. Food Microbiol., 96, 289-300. https://doi.org/10.1016/j.ijfoodmicro.2004.03.026.
•  Maringgal,  B.,  Hashim,  N.,  Mohamed Amin Tawakkal,  I.S., and  Muda Mohamed,  M.T., 2020. Recent advance in edible coating and its effect on fresh/fresh-cut fruits quality. Trends Food Sci. Technol. 96, 253-267. https://doi.org/10.1016/j.tifs.2019.12.024
•  Mills,  A.,  Platt,  H., and  Hurta,  R.A. (2004). Effect of salt compounds on mycelial growth, sporulation and spore germination of various potato pathogens. Posthavest Biol. Tec., 34, 341-350. https://doi.org/10.1016/j.postharvbio.2004.05.022.
•  Mohammadi,  L.,  Tanaka,  F., and  Tanaka,  F. (2021). Preservation of strawberry fruit with an Aloe vera gel and basil oil coating at ambient temperature. J. Food Proc. Pres., 45, e15836 (2021). https://doi.org/10.1111/jfpp.15836
•  Ozcakmak,  S. and  Gul,  O. (2017). Inhibition kinetics of Penicillium verrucosum using different essential oils and application of predictive inactivation models. Int. J. Food Prop., 20, S684-S692. https://doi.org/10.1080/10942912.2017.1308953.
•  Palou,  L.,  Ali,  A.,  Fallik,  E., and  Romanazzi,  G. (2016). GRAS, plant-and animal-derived compounds as alternatives to conventional fungicides for the control of postharvest diseases of fresh horticultural produce. Posthavest Biol. Tec., 122, 41-52. https://doi.org/10.1016/j.postharvbio.2016.04.017.
•  Qin,  G.,  Zong,  Y.,  Chen,  Q.,  Hua,  D., and  Tian,  S. (2010). Inhibitory effect of boron against Botrytis cinerea on table grapes and its possible mechanisms of action. Int. J. Food Microbiol., 138, 145-150. https://doi.org/10.1016/j.ijfoodmicro.2009.12.018.
•  Sun,  C.,  Fu,  D.,  Lu,  H.,  Zhang,  J.,  Zheng,  X., and  Yu,  T. (2018). Autoclaved yeast enhances the resistance against Penicillium expansum in postharvest pear fruit and its possible mechanisms of action. Biological control, 119, 51-58. https://doi.org/10.1016/j.biocontrol.2018.01.010.
•  Tanaka,  F.,  Maeda,  Y.,  Uchino,  T.,  Hamanaka,  D., and  Atungulu,  G.G. (2008). Monte Carlo simulation of the collective behavior of food particles in pneumatic drying operation. LWT - Food Sci. Tech., 41, 1567-1574. https://doi:10.1016/j.lwt.2007.10.020
•  Tornuk,  F.,  Ozturk,  I.,  Sagdic,  O.,  Yilmaz,  A., and  Erkmen,  O. (2013). A application of predictive inactivation models to evaluate survival of Staphylococcus aureusin fresh-cut apples treated with different plant hydrosols. Int. J. Food Prop., 17, 587-598. https://doi.org/10.1080/10942912.2011.650340.
•  Türkkan,  M. and  Erper,  İ. (2015). Inhibitory influence of organic and inorganic sodium salts and synthetic fungicides against bean root rot pathogens. Gesunde Pflanzen, 67, 83-94. https://doi.org/10.1007/s10343-015-0339-z.
•  Vilaplana,  R.,  Pazmino,  L., and  Valencia-Chamorro,  S. (2018). Control of anthracnose, caused by Colletotrichum musae, on postharvest organic banana by thyme oil. Posthavest Biol. Tec., 138, 56-63. https://doi.org/10.1016/j.postharvbio.2017.12.008.
•  Yu,  T.,  Chen,  J.,  Chen,  R.,  Huang,  B.,  Liu,  D., and  Zheng,  X. (2007). Biocontrol of blue and gray mold diseases of pear fruit by integration of antagonistic yeast with salicylic acid. Int. J. Food Microbiol., 116, 339-345. https://doi.org/10.1016/j.ijfoodmicro.2007.02.005.