Probabilistic Modeling of Saccharomyces cerevisiae Inhibition under the Effects of Water Activity, ph, and Potassium Sorbate Concentration

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1 9 Journal of Food Protection, Vol. 63, No., 2, Pages 995 Copyright, International Association for Food Protection Probabilistic Modeling of Saccharomyces cerevisiae Inhibition under the Effects of Water Activity, ph, and Potassium Sorbate Concentration A. LÓPEZ-MALO, * S. GUERRERO, 2 AND S. M. ALZAMORA 2 Departamento de Ingeniería Química y Alimentos, Universidad de las Américas-Puebla, Sta. Catarina Mártir, Puebla, 7282, México; and 2 Departamento de Industrias, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria 428, Buenos Aires, Argentina MS 98-3: Received 2 November 998/Accepted 27 August 999 ABSTRACT Probabilistic microbial modeling using logistic regression was used to predict the boundary between growth and no growth of Saccharomyces cerevisiae at selected incubation periods (5 and 35 h) in the presence of growth-controlling factors such as water activity ( ;.97,.95, and.93), ph (6., 5., 4., and 3.), and potassium sorbate (, 5,, 2, 5, and, ppm). The proposed model predicts the probability of growth under a set of conditions and calculates critical values of, ph, and potassium sorbate concentration needed to inhibit yeast growth for different probabilities. The reduction of ph increased the number of combinations of and potassium sorbate concentration with probabilities to inhibit yeast growth higher than.95. With a probability of growth of.5 and using the logistic models, the critical ph values were higher for 5 h of incubation than those required for 35 h. With lower values and increasing potassium sorbate concentration the critical ph values increased. Logistic regression is a useful tool to evaluate the effects of the combined factors on microbial growth. The actual desire for natural, healthier, and more convenient foods has often resulted in lower heat processes to improve quality, lower levels of preservatives, and changes in packaging methods and materials (6). Therefore, microbiological safety and stability margins are being reduced and improved control in manufacture and distribution are required to assure the recommended shelf-lives. Such improved control requires identification of key microorganisms and their response to factors that determine their survival or growth in food products. Microbial predictive models provide this information. The techniques for the development of mathematical models in the area of predictive microbiology have improved greatly, allowing better and more accurate descriptions of microbial responses to a particular environment (, 79). Predictive models can be useful tools for the food industry, allowing for the description of the interactions of a number of preservative factors when acting in combination (9, ). Probabilistic modeling focuses attention to the boundary between microbial growth and no growth, and probabilistic models must be useful to define and predict the combinations of conditions (factors or hurdles) that influence the probability of the target organism s ability to reproduce and grow (2, 3, 7). The objective of hurdle technology is to select and combine preservation factors or hurdles in such ay that microbial stability and safety can be assured, retaining sensory acceptance and nutritional characteristics. Thus, if we can pre- * Author for correspondence. Tel: (22) ; Fax: (22) 29 29; amalo@mail.udlap.mx. Member of Consejo Nacional de Investigaciones Científicas y Técnicas. dict with accuracy the interface between microbial growth and no growth for an identified target microorganism under several combinations of factors, the selection of such factors can be chosen scientifically and the selected hurdles can be kept at minimum levels. The aim of this paper is to present and discuss probabilistic modeling of Saccharomyces cerevisiae growth for a set combinations of water activity ( ), ph, and potassium sorbate concentration. MATERIALS AND METHODS Data. An ideal data set for probabilistic modeling would be one that contains conditions that give growth and no observable growth under several combinations of factors (3). The data reported by Cerrutti (2) and Cerrutti et al. (3) are appropriate for this purpose. These authors reported the effect of, ph, and potassium sorbate (KS) concentration on S. cerevisiae growth response in laboratory media at 27 C. From the results reported by Cerrutti (2) and Cerrutti et al. (3), we generated two data sets (Table ), one with the combinations that cause growth or no growth after 5 h of incubation and the other with the results after 35 h of incubation. Any increase of the initially inoculated yeast count was considered growth. Growth decline as well as no appreciable change in the initial inoculere considered no growth. We selected 5 and 35 h as representative incubation times for the yeast response under the evaluated conditions; the effect of lag time was noticed after 35 h of incubation. We assigned a value of if growth was observed for the particular combination of conditions at which the three factors were evaluated, or the value was assigned if growth was not detected. The model. Many distribution functions have been proposed for use in the analysis of a dichotomous outcome variable (7).

2 92 LÓPEZ-MALO ET AL. J. Food Prot., Vol. 63, No. TABLE. Effect of water activity ( ), ph, and potassium sorbate (KS) concentration on growth of S. cerevisiae after 5 and 35 h of incubation at 27 C a 5 h 35 h KS concentration (ppm) KS concentration (ppm) ph 5 2 5, 5 2 5, a Data from Cerrutti (2) and Cerrutti et al. (3)., Growth observed;, no growth observed;, factor combination not evaluated. There are two primary reasons for choosing the logistic distribution. These are (i) from a mathematical point of view, it is an extremely flexible and easily used function, and (ii) it lends itself to a biologically meaningful interpretation. Statistically, a logistic regression model relates the probability of occurrence of an event, Y, conditional on a vector, x, of explanatory variables. The quantity (x) E(Y x) represents the conditional mean of Y given x when the logistic distribution is used. The specific model of the logistic regression is as follows: [ i ] i i exp x i (x) () exp x [ ] [ ] The logit transformation of (x) is defined as follows i (x) logit(p) g(x) ln x i (2) (x) The importance of this transformation is that g(x) has many of the desirable properties of a linear regression model. The logit, g(x), is linear in its parameters, may be continuous, and may vary depending on the range of x (7). For our particular case,, ph, and KS concentration are the independent variables and the outcome or dependent variable is the response of S. cerevisiae. We chose as a growth response and as a no-growth response under the evaluated conditions. In order to fit the logistic model the following equation was selected: g(x) 2 ph 3 KS 4 ph 5 KS 6 ph KS (3) where the coefficients ( i ) are the parameters to be estimated by fitting the model to experimental data. The logistic regression was performed employing the logistic subroutine in SPSS 6. for Macintosh (SPSS Inc., Chicago, Ill.). The significance of the coefficients was evaluated and eliminated from the model if the probability of being zero was greater than.. After fitting the logistic regression model, predictions of the interface (growth/no growth) were made at a probability level of.5, by substituting the value of logit (p) in the model and finding the value of one independent variable maintaining the other two as fixed. Also the probability of growth was calculated using the logistic model under the evaluated conditions. TABLE 2. Coefficients for reduced logistic models Source Intercept ph KS ph KS Estimate 5 h RESULTS The results of Cerrutti (2) and Cerrutti et al. (3), summarized in Table, demonstrate that yeast inhibition (no observable growth) can be obtained with several combinations of, ph, and KS concentration. It is also apparent that for several cases, the conditions in which no growth was observed after 5 h of incubation allow growth for longer incubation time (35 h), reflecting the effect of the combined preservation factors on the lag time. Some synergistic combinations can be detected in Table that permit inhibition of yeast growth at relatively high values. This is the principle of the hurdle technology approach, the search for those minimal combinations of factors that inhibit microbial growth and that is the purpose of the present article. Fitting equation 3 to the data in Table by logistic regression and eliminating nonsignificant (P.) terms resulted in the reduced models shown in Table 2. For both cases, 5 and 35 h of incubation, main effects were significant as well as the phks concentration interaction. Variables included in the model are significant in both biological and statistical senses. The models goodness of fit was tested by the log-likelihood ratio and chi-square test being in both cases significant, indicating that models are Estimate 35 h Coefficient Significance Significance

3 J. Food Prot., Vol. 63, No. PROBABILISTIC MODELING OF S. CEREVISIAE INHIBITION 93 TABLE 3. Cross-classification of observed and predicted values using the logistic model for S. cerevisiae growth Predicted response Observed response 5 h No growth Growth () () % correct No growth () 35 h Growth () % correct No growth () Growth () Overall FIGURE. Probabilities (P) of growth of S. cerevisiae after 35 h of incubation in laboratory media formulated with selected water activities, phs, and potassium sorbate concentrations. useful to predict the outcome variable (growth or no growth). Table 3 presents the cross-classification results; the overall correct predictions were 92%. A low number of missclassified observations (Table 3) was obtained. The estimated coefficients for the independent variables and their interactions represent the rate of change of a function of the dependent variable per unit of change in the independent variable. In our case, it is apparent (Table 2) that the main differences between models are in the coefficient ( ), reflecting the higher influence of on the yeast response at short incubation periods (5 h). Figure presents the predicted probability of growth for S. cerevisiae after 35 h of incubation on the variable range tested. At ph 6., KS addition and reduction have no effect on yeast inhibition with probabilities of growth.5. The ph reductions to 5., 4., or 3. gradually increase the number of combinations of and KS concentration with probabilities of inhibiting yeast growth higher than.95. With the models obtained (Table 2), predictions for critical ph,, or KS concentration necessary to inhibit or permit growth with a selected probability can be calculated. This probability can be a relaxed value like.5 in which there is a 5:5 chance that growth occurs, or it can be tight with a probability of growth of. or.5. Choosing a probability of growth of.5 and using the logistic model, critical ph values can be calculated. Table 4 shows, for 5 and 35 h of incubation, the minimum or critical ph values needed to inhibit yeast growth (no observable growth) in combination with selected values and KS concentrations. As decreased and KS concentration increased, the critical ph value increased. For.965 and KS concentrations 2 ppm, lower ph values were needed to achieve no growth (P.5) after 35 h than those required for 5 h of incubation. DISCUSSION Predictive microbiology is usually divided in kinetic and probabilistic approaches (5). Probabilistic microbial models are less common than models based on kinetic data. A great number of probabilistic modeling reports are for pathogenic bacteria, especially for Clostridium botulinum growth and toxin production. Probabilistic models are used to estimate the probability of growth or toxin formation using polynomial expressions incorporating the effects of environmental factors (6). Model construction is based on the logistic, log-logistic equations or on the Weibull distribution. In many cases, the observed responses are time-totoxicity or time-to-reach a population level (8, 4, 6, 2). Probability values are calculated as functions of selected preservation factors (incubation temperature, ph, sodium

4 94 LÓPEZ-MALO ET AL. J. Food Prot., Vol. 63, No. TABLE 4. Predicted critical ph values with a probability of.5 on the boundary of the growth/no growth interface for S. cerevisiae Potassium sorbate concentration (ppm) , 5 h of incubation h of incubation chloride concentration, water activity, among others) using quadratic expressions (polynomials) resulting from response surface designs and analysis. Another approach to construct probabilistic microbial models, as used in the present report, is the application of logistic regression. Models based on logistic regression have been reported for Shigella flexneri (3), Escherichia coli (2), Listeria monocytogenes (), and Zygosaccharomyces bailii (4). These reports demonstrate logistic regression flexibility to construct the model, taking into account square root-type kinetic models (2, 3) or polynomial-type models (, 4) and the model presented in our work. However, in every case growth/no-growth observations were taken at fixed storage times that limited probabilistic models to microbial response within that period of time, in our case, limited to the results observed after 5 or 35 h of incubation. Mathematical modeling and prediction of fungal growth have not received a similar degree of interest as bacterial growth modeling (5), especially with a probabilistic approach. Cole et al. (4), using logistic regression, reported Z. bailii behavior under the effects of selected preservation factors and observed that yeast probability of growth depended on the individual effects of ph, Brix, sorbic acid, benzoic acid, and sulfite concentrations as well as upon complex interactions among preservation factors. Logistic regression using polynomial-type models are useful to determine independent variable effects as well as their interactions and demonstrate the effects of preservation factors, when used in combination, on yeast growth. Predictive models can provide decision support in different areas relevant to the food industry. In many cases models are empirical, interpreting only the response of the organism without understanding the mechanism of the response. However, if the models are used properly, predictive probabilistic models are helpful tools for evaluating microbial responses and can help to identify potential problems for a product or process. Logistic regression is a useful tool to model the boundary between growth and no growth. Food development, formulation, and processing based on the hurdle concept can find in probabilistic microbial modeling a practical approach to evaluate the effects of the combined factors. ACKNOWLEDGMENTS The authors acknowledge financial support from Universidad de las Américas-Puebla (México), Universidad de Buenos Aires and CONICET (Argentina), and CYTED XI.3 Project. We also extend our gratitude to Dr. Luis G. Vazquez de Lara for his excellent advice and support. REFERENCES. Bolton, L. F., and J. F. Frank Defining the growth/no-growth interface for Listeria monocytogenes in Mexican-style cheese based on salt, ph, and moisture content. J. Food Prot. 62: Cerrutti, P Efectos combinados de, ph, aditivos y tratamiento térmico en el crecimiento y supervivencia de Saccharomyces cerevisiae. Ph.D. dissertation. Universidad de Buenos Aires, Argentina. 3. Cerrutti, P., S. M. Alzamora, and J. Chirife. 99. A multi-parameter approach to control the growth of Saccharomyces cerevisiae in laboratory media. J. Food Sci. 55: Cole, M. B., J. G. Franklin, and M. H. J. Keenan Probability of growth of the spoilage yeast Zygosaccharomyces bailii in a model fruit drink system. Food Microbiol. 4: Gibson, A. M., and A. D. Hocking Advances in the predictive modelling of fungal growth in food. Trends Food Sci. Technol. 8: Gould, G. W Overview, p. xvxix. In G. W. Gould (ed.), New methods of food preservation. Blackie Academic and Professional, New York. 7. Hosmer, D. W., and S. Lemeshow Applied logistic regression. John Wiley and Sons, New York.

5 J. Food Prot., Vol. 63, No. PROBABILISTIC MODELING OF S. CEREVISIAE INHIBITION Maas, M. R Development and use of probability models: the industry perspective. J. Ind. Microbiol. 2: McClure, P. J., C. de W. Blackburn, M. B. Cole, P. S. Curtis, J. E. Jones, J. D. Legan, I. D. Ogden, M. W. Peck, T. A. Roberts, J. P. Sutherland, and S. J. Walker Modelling the growth, survival and death of microorganisms in foods: the UK Food Micromodel approach. Int. J. Food Microbiol. 23: McClure, P. J., M. B. Cole, and K. W. Davies An example of the stages in the development of a predictive mathematical model for microbial growth: the effects of NaCl, ph and temperature on the growth of Aeromonas hydrophila. Int. J. Food Microbiol. 23: McMeekin, T. A., J. N. Olley, T. Ross, and D. A. Ratkowsky Predictive microbiology: theory and application. Research Studies Press, Ltd., England. 2. Presser, K. A., T. Ross, and D. A. Ratkowsky Modelling the growth limits (growth/no growth interface) of Escherichia coli as a function of temperature, ph, lactic acid concentration, and water activity. Appl. Environ. Microbiol. 64: Ratkowsky, D. A., and T. Ross Modelling the bacterial growth/no growth interface. Lett. Appl. Microbiol. 2: Razavilar, V., and C. Genigeorgis Prediction of Listeria spp. growth as affected by various levels of chemicals, ph, temperature and storage time in a model broth. Int. J. Food Microbiol. 4: Ross, T., and T. A. McMeekin Predictive microbiology. Int. J. Food Microbiol. 23: Schaffner, D. W., W. H. Ross, and T. J. Montville Analysis of the influence of environmental parameters on Clostridium botulinum time-to-toxicity by using three modeling approaches. Appl. Environ. Microbiol. 64: Skinner, G. E., and J. W. Larkin Mathematical modeling of microbial growth: a review. J. Food Saf. 4: Whiting, R. C Microbial modeling in foods. Crit. Rev. Food Sci. Nutr. 35: Whiting, R. C., and R. L. Buchanan Microbial modeling. Food Technol. 48: Whiting, R. C., and J. C. Oriente Time-to-turbidity model for non-proteolytic type B Clostridium botulinum. Int. J. Food Microbiol. 35:496.