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Appl Environ Microbiol. Sep 2010; 76(18): 6038–6046.
Published online Jul 23, 2010. doi:  10.1128/AEM.00613-10
PMCID: PMC2937510

Use of Logistic Regression for Prediction of the Fate of Staphylococcus aureus in Pasteurized Milk in the Presence of Two Lytic Phages [down-pointing small open triangle]

Abstract

The use of bacteriophages provides an attractive approach to the fight against food-borne pathogenic bacteria, since they can be found in different environments and are unable to infect humans, both characteristics of which support their use as biocontrol agents. Two lytic bacteriophages, vB_SauS-phiIPLA35 (phiIPLA35) and vB_SauS-phiIPLA88 (phiIPLA88), previously isolated from the dairy environment inhibited the growth of Staphylococcus aureus. To facilitate the successful application of both bacteriophages as biocontrol agents, probabilistic models for predicting S. aureus inactivation by the phages in pasteurized milk were developed. A linear logistic regression procedure was used to describe the survival/death interface of S. aureus after 8 h of storage as a function of the initial phage titer (2 to 8 log10 PFU/ml), initial bacterial contamination (2 to 6 log10 CFU/ml), and temperature (15 to 37°C). Two successive models were built, with the first including only data from the experimental design and a global one in which results derived from the validation experiments were also included. The temperature, interaction temperature-initial level of bacterial contamination, and initial level of bacterial contamination-phage titer contributed significantly to the first model prediction. However, only the phage titer and temperature were significantly involved in the global model prediction. The predictions of both models were fail-safe and highly consistent with the observed S. aureus responses. Nevertheless, the global model, deduced from a higher number of experiments (with a higher degree of freedom), was dependent on a lower number of variables and had an apparent better fit. Therefore, it can be considered a convenient evolution of the first model. Besides, the global model provides the minimum phage concentration (about 2 × 108 PFU/ml) required to inactivate S. aureus in milk at different temperatures, irrespective of the bacterial contamination level.

Staphylococcus aureus is one of the pathogenic bacteria considered a threat to food safety. Worldwide, it has a particular relevance to the food-processing industry because of the ability of some strains to produce heat-stable enterotoxins and other virulence factors responsible for staphylococcal food poisoning (14, 18). The ability of S. aureus to grow in different foodstuffs over a wide range of temperatures (10 to 45°C), pHs (4.5 to 9.3), and NaCl concentrations (up to 15%) explains its incidence in foods subjected to manipulation throughout the manufacturing process (1). Milk and dairy products have often been involved in several episodes of staphylococcal food poisoning (6). Food handlers and cattle are usually the main source of dairy product contamination, as humans and animals are the primary reservoirs for staphylococci (28).

S. aureus contamination can be avoided by heat treatment of food, but recontamination postpasteurization can occur if the hygienic conditions are inadequate. Alternative approaches to control S. aureus populations in dairy products, such as the use of bacteriocinogenic strains (24) and, more recently, bacteriophages (8), have been tested. Bacteriophages, as well as bacteriocins, are regarded as natural antibacterial agents since they are able to infect and lyse specific undesired target bacteria without disturbing the normal microbiota (19). Among the advantages of using phages as biocontrol strategies in foods, their ubiquity in the environment, their history of safe use, and their high host specificity can be quoted (10). Likewise, their potential use as biopreservatives in several food systems has recently been reviewed (13).

Factors affecting S. aureus growth and survival in foods (physicochemical characteristics of the products and conditions associated with food processing, storage, and distribution) have been studied, and mathematical models have been developed to estimate the microorganism's response (5, 30). However, the effect of lytic phages on S. aureus growth in milk has not been described by predictive models so far. There are several physical parameters which may influence bacterial inhibition by phages in milk. Among them, temperature is the first choice to be studied, as this parameter is always involved in milk processing and can easily be subjected to manipulation without markedly affecting the typical organoleptic characteristics of milk.

In this regard, probabilistic models have been widely used to describe the survival/death interface as a function of environmental hurdles (22, 23), and the positions of these limits are of interest in establishing conditions for product stabilization. A procedure of forward or backward stepwise regression with some criteria to include or reject dummy or quantitative explanatory variables, their quadratic terms, or their interactions is included in most of the standard statistical software packages (11, 12).

The aim of the present work was to determine the effectiveness of a cocktail of two lytic phages, selected according to its wide host range (9), at reducing S. aureus contamination in pasteurized milk, using different environmental conditions, such as different temperatures, initial bacterial viable counts, and initial phage titers. For this purpose, the survival/death interface was deduced using a logistic regression model.

MATERIALS AND METHODS

Bacterial strains, bacteriophages, media, and culture conditions.

The strain S. aureus Sa9, isolated from a mastitic milk sample (8), has been used in this study for contaminating pasteurized milk. The strain was grown in 2× YT broth medium (25) at 37°C for 18 h with slight agitation. Baird-Parker agar supplemented with egg yolk tellurite (Scharlau Chemie, S.A. Barcelona, Spain) was used for staphylococcal counting.

A mixture of two bacteriophages, vB_SauS-phiIPLA35 (phiIPLA35) and vB_SauS-phiIPLA88 (phiIPLA88), lytic derivatives of temperate phages previously isolated from milk samples (8), were used as biocontrol agents. Both phages are able to infect S. aureus Sa9, usually used as the bacterial host in phage propagation assays. Lysed cultures of S. aureus Sa9 were centrifuged (10,000 × g, 15 min, 4°C). Concentrated phage suspensions were obtained by ultracentrifugation (100,000 × g, 90 min, 4°C) of the culture supernatants, followed by CsCl gradient centrifugation (8). The phage titer was determined by mixing 100 μl of a 1/10 overnight culture of S. aureus Sa9 and 100 μl of the appropriate dilution of phage suspension. The mixture was added to 5 ml of molten 2× YT (0.7% agar) and poured onto 2× YT agar plates. The plates were incubated at 37°C for 18 h, and the numbers of plaques as a result of lysis were counted.

Bacterium-phage challenge assays.

Commercial pasteurized whole milk was inoculated with an overnight culture of S. aureus Sa9 and a mixture (1:1) of the lytic phages (phiIPLA35 and phiIPLA88), and the mixture was subsequently incubated at different temperatures. For control purposes, pasteurized milk was inoculated only with the staphylococcal strain. The combined effect of the initial bacterial contamination, the initial phage titer, and the temperature of incubation on S. aureus survival was evaluated throughout a period of 8 h. For each treatment, bacterial survival was monitored with time by the plate count method. For this purpose, samples were taken periodically at 0, 2, 4, 6, and 8 h. Decimal dilutions of the milk samples were made in quarter-strength Ringer's solution (Merck, Damstadt, Germany). Appropriate dilutions were plated in duplicate on Baird-Parker agar supplemented with egg yolk tellurite, and the plates were incubated at 37°C for 24 h. In parallel, the phage titer in each sample was determined according to the procedure indicated above.

Experimental design.

Design-Expert software (version 6.0.6; Stat-Ease, Inc., Minneapolis, MN) was used to determine how the independent quantitative continuous variables (temperature, initial level of S. aureus contamination, and initial phage titer) affect S. aureus survival (the response, or the dependent variable) in pasteurized milk. The bacterial and phage concentrations were expressed as the numbers of log10 CFU/ml and log10 PFU/ml, respectively. The multiplicity of infection (MOI; ratio of phage concentration/bacterial concentration), expressed as log10, was established over a range (−0.03 to 5.32) that allows the availability of phage titers high enough to provide the highest MOI even in high volumes of highly contaminated milk (≥6 log10 CFU/ml) to be guaranteed. In addition, temperature limits of between 15°C and 37°C were established. Refrigeration temperatures (≈4°C) were not included because the lytic phage infection process occurs only at temperatures permissive for bacterial activity.

The experimental approach consisted of three consecutive designs, whose characteristics and theoretical variables ranges are the following. The first design consisted of a 23 central composite with variable ranges for the parameter values: temperature, 17°C to 37°C; bacterial contamination, 2.81 to 6 log10 CFU/ml; phage titer, no phage to phage at 8 log10 PFU/ml. The second design was D optimal with variable ranges for the parameter values: temperature, 15°C to 37°C; bacterial contamination, 4 to 6 log10 CFU/ml; phage titer, 2 to 8 log10 PFU/ml. The third design was D optimal, centered in the intermediate values of variables with variable ranges for the parameter values: temperature, 15°C to 26°C; bacterial contamination, 4 to 6 log10 CFU/ml; phage titer, 4 to 6.5 log10 PFU/ml (Table (Table1).1). Validation experiments were performed at selected temperatures within the experimental region (15°C, 18°C, 21°C, 24°C, 27°C, 30°C, 33°C, and 37°C); the level of bacterial contamination ranged from 4 to 7 log10 CFU/ml, and the phage titer ranged from 6 to 8 log10 PFU/ml (Table (Table2).2). Within each experiment, the runs, including duplicates, were randomly performed. This circumstance led to slight differences between duplicates. Due to the difficulties of obtaining the exact concentrations of bacteria and phages demanded by the designs, Tables Tables11 and and22 show the observed real concentrations of these variables that were used for the further statistical analysis. The responses (the dependent variable) were tabulated as 1 (survival) and 0 (death) when viable cells were either detected or not detected, respectively, at 8 h.

TABLE 1.
Experimental designs used for development of the survival/death interface of Staphylococcus aureus (model 1) as a function of temperature, initial bacterial contamination, and initial phage titera
TABLE 2.
Conditions and responses for the validation experimentsa

Logistic model development and validation.

A logistic regression model relates the probability (p) of occurrence of an event (Y) conditional on a vector (x) of independent variables (11). The key quantity (called the “conditional mean”) is the mean value of the dependent variable (Y), given the value of the independent variable (x), when the logistic distribution is used. This quantity is expressed in the equation p(x) = E(Y/x) and read as the expected value of Y (for example, growth probability), given the value x (for example, environmental factors). Accordingly, the specific model built for the probability of S. aureus survival is as follows:

equation M1
(1)

where β0 and βi are the intercept and the coefficients of the polynomial function, respectively, and the xi terms are the respective variables.

The logit transformation of P(x) is defined as

equation M2
(2)

where epsilon is a term for error. The model may also include quadratic terms, interactions, etc., for the corresponding variables.

The predicted survival probability with each variable combination may be estimated by

equation M3
(3)

From equation 3, the survival/death interface for a selected probability (p) can be deduced by replacing logit(p) by the corresponding polynomial equation and plotting the resulting equation as a function of two or three variables while maintaining the rest of the variables at the predetermined levels.

Statistical analysis.

The logistic regression model was fitted to the survival/death S. aureus experimental data using SYSTAT (version 10.2) software (Systat Software Inc., 2002) and introducing the following model into the logit regression module:

equation M4
(4)

where T, B, and P stand for temperature, initial viable counts of bacteria, and initial phage titer, respectively. The output provided the intercept and the significant coefficients for the model (equation 4). The automatic variable stepwise selection (the maximum number of allowable runs was set at 100) with the backward option was used to choose the coefficients. The coefficients retained were those significant at a P value of <0.05. Two successive models were built; the first included only data from the experimental design, and a global model also included those results derived from the validation experiments.

The log likelihood ratio statistic (12) was used to assess the importance of each of the independent variables to the response (survival/death of S. aureus). This statistic indicates if the coefficients of the model are significantly different from zero. It takes into account the number of independent variables (degrees of freedom) used in the model and shows a chi-squared distribution. McFadden's rho-squared statistic and Naglekerke's R2 statistic (a modification of the Cox and Snell R-squared value ranging from 0 to 1) were used to measure the strength of the association between the data and the logistic model. The former is a transformation of the likelihood statistic and is intended to mimic the coefficient of determination, R2. As both statistics are between 0 and 1, values closer to 1 usually indicate a better fit of the model, although interpretation of those values must be made in combination with evaluation of the other parameters (27). Finally, the Hosmer-Lemeshow test, also called the chi-square test for overall fit, was also performed (26).

The coefficients of the logit model are the odds ratio (in ln units) that an event would happen (survival, p) or would not happen (death, 1 − p) and represent the changes in the log of the odds due to one unit change in the quantitative variable or due to the change from one level with respect to the reference level for qualitative variables.

The predicted survival probabilities and the survival/death interfaces for specific p levels were produced from logit(p) = ln[p/(1 − p)] with the STATISTICA (version 7.0) software package (2001; StatSoft Inc., Tulsa, OK). The limit was estimated for P equal to 0.01 and was expressed as a function of the quantitative variables in graphs of two or three dimensions.

RESULTS

Effect of bacteriophage cocktail on survival of S. aureus.

The evolution of the S. aureus population in pasteurized milk was determined in the presence and in the absence of the bacteriophage cocktail at different temperatures over a period of 8 h, as no relevant changes in either the level of bacterial contamination or the phage titer up to 24 h were previously observed (data not shown). Under certain combinations of the independent variables (24 out of 60), no surviving cells were detected after 8 h, when a phage cocktail was added to pasteurized milk (Table (Table1).1). However, different kinetics of staphylococcal growth were observed over time (Fig. (Fig.1).1). At high phage concentrations (≥6 log10 PFU/ml), the phage cocktail was very effective at inhibiting S. aureus when the initial level of bacterial contamination was high (≥6 log10 CFU/ml). In fact, no staphylococcal viable counts were detected after 2 h, regardless of the temperature of incubation (Fig. 1a and e). At a lower level of contamination (≤4 log10 CFU/ml), a slower inhibitory effect against the phages occurred (Fig. (Fig.1b),1b), particularly at a growth-restricted temperature (15°C) (Fig. (Fig.1f).1f). Temperature clearly affected the inhibitory effect against phages at a high level of bacterial contamination when low phage concentrations (2 to 3 log10 PFU/ml) were used since bacterial inhibition occurred only at the optimal growth temperature (Fig. (Fig.1c)1c) and no bacterial inhibition was observed at 15°C (Fig. (Fig.1g).1g). At a lower bacterial contamination level, a notable delay in the inhibitory effect of a low phage concentration was observed. Surviving cells were still detected at 8 h at 37°C (Fig. (Fig.1d),1d), and no effect on bacterial survival occurred at 15°C (Fig. (Fig.1h).1h). The limited effect of the phages at the lower temperature is not surprising, since the lytic phage infection process is very much dependent on the bacterial host metabolic machinery. Thus, the closer to the optimum that the bacterial growth temperature is, the higher that the phage inhibition activity is. On the other hand, in the absence of phages, the multiplication of the pathogen occurred over the 8-h period at 37°C, whereas bacterial proliferation was hardly detected at 15°C.

FIG. 1.
Inhibitory effect of phiIPLA35 and phiIPLA88 phage cocktail on the growth of Staphylococcus aureus Sa9 in pasteurized milk at growth temperatures that were optimal (37°C) and restricted (15°C). The assay was performed in whole pasteurized ...

Evaluation of goodness of fit of the logistic model.

Results from the 60 experiments proposed by the successive designs were used to build a first logistic regression model to predict the effects of temperature and the initial bacterial and bacteriophage levels on S. aureus growth at 8 h.

The regression was performed with the stepwise method, which selects the most significant variables affecting S. aureus growth. The coefficient estimates and statistics with significant effects (P < 0.05) are shown in Table Table3.3. There were only three terms which contributed significantly to the model prediction (log likelihood statistic = 47.158). Similarly, the Hosmer-Lemeshow statistic (4.670) indicated a nonsignificant lack of fit. In addition, the values of the McFadden's rho-squared (0.584) and Naglekerke's R2 (0.736) statistics indicate a high degree of association and, when they are assessed in combination with the other statistics, an apparent good fit of the model to the observations. Apart from the intercept, the initial level of bacterial contamination and the interaction of this independent variable with temperature and phage titer were retained (model 1). Then, the equation that expresses the logit(p) was

equation M5
(5)

A mathematical interpretation like that given to β in the linear regressions is not straightforward in logit regression. In this, the coefficients represent the effect of each variable or their interactions on the ln of the odds of survival (p) [ratio of p/(1 − p)]. So, the increase in the initial level of bacterial contamination increases the ln of the odds of survival, while the interactions of initial bacterial contamination with temperature or phage titer decrease it. The graphical representation of the probability of bacterial growth as a function of the values (concentrations) of these variables provides a clear description of their real effects. As there are three variables, a three-dimensional graphical representation as a function of two of them can be obtained by fixing the third at selected values. A plot showing the predicted probability of survival for Staphylococcus aureus Sa9 as a function of the initial phage titer from model 1, when the initial level of bacterial contamination was fixed at 6.0 log10 CFU/ml, is shown in Fig. Fig.22.

FIG. 2.
Contour plot showing the predicted probability (model 1) of survival for Staphylococcus aureus as a function of the initial phage titer at a fixed level of bacterial contamination of 6 log10 CFU/ml and at temperatures of 15, 20, 25, 30, 35, and 37°C. ...
TABLE 3.
Parameters used in the first logistic regression model for survival and death of S. aureus in pasteurized milk after bacteriophage treatmenta

For a similar staphylococcal contamination level, the effect of temperature is clear. As the temperature goes higher, a lower level of phage titer is required for inactivation. On the contrary, a low temperature (e.g., 15°C) requires a higher phage titer.

The survival/death interface for a selected level of probability (e.g., 0.05) may also be deduced from equation 5 by replacing logit(p) by ln[p/(1 − p)], solving for one variable (for example, temperature), and giving selected levels to one of the other variables. After reordering, it can read as follows:

equation M6
(6)

Figure Figure33 shows the contour plot of the survival/death interfaces for S. aureus predicted by the model and the responses of S. aureus obtained from the validation experiments. It shows the effects of temperature and initial phage titer (log10 PFU/ml) at different initial concentrations of bacteria (3.0, 4.0, 5.0, and 6.0 log10 CFU/ml) and at a level of probability of 0.05. Differences in the responses among the diverse levels of initial bacterial counts were observed. The interfaces become closer as the initial staphylococcal contamination level increases. With respect to the comparison of this survival/death interface and the results obtained from the validation experiments, it must be emphasized that there was not any case of survival in the region of death, although cases of death were observed in the area of survival. Therefore, the interfaces deduced may be considered fail-safe.

FIG. 3.
Survival/death interface for Staphylococcus aureus predicted by model 1 at a 0.05 probability level. The initial levels of bacterial contamination were fixed at 6.0, 5.0, 4.0, and 3.0 log10 CFU/ml. Superimposed are the results obtained in the validation ...

The cases of death observed at low temperatures in the region of growth may be due to lysis from without, which can occur even at suboptimal temperatures for bacterial growth when a high phage titer is used (20).

Due to the high number of validation experiments and the agreement of their results with those of the model deduced earlier, it was considered to be of interest to build a new model taking into account all the experiments, because in this case it could be deduced from larger degrees of freedom. As result, a new logit regression considering all the data (107 experiments) was accomplished. The coefficients retained are shown in Table Table4.4. In this case, only two variables (temperature and initial phage titer), along with their interactions, contributed significantly to the model prediction (log likelihood statistic = 97.985).

TABLE 4.
Parameters used in the global logistic regression model for survival and death of S. aureus in pasteurized milk after bacteriophage treatmenta

This global model also showed a good fit to the observed data, as indicated by the different statistics: Hosmer-Lemeshow = 1.548, McFaden's rho-squared = 0.684, and Naglekerke's R2 = 0.814. The equation of the expanded model was then

equation M7
(7)

The interpretation of the coefficients of this equation is similar to that for the first model. The greater effect on the log of odds changes is due to the initial phage titer (−5.195), followed by temperature (−0.966), with the interaction of both factors being fairly low (0.098). The probability that S. aureus would survive as a function of the two retained variables was calculated using the following equation:

equation M8
(8)

As expected, the graphical representation of the equation showed that the phage titer had a greater effect than temperature on the probability that S. aureus would survive (Fig. (Fig.4).4). A contour plot of this surface as a function of the initial phage titer for different temperatures is shown in Fig. Fig.5.5. At high temperatures, the slope of the probability is lower than that at low temperatures. This indicates that at high temperatures the effect of the initial phage titer is progressive and that lower levels are required. However, a sharper effect is observed at low temperatures. In this case, only a 1-log10-unit increase in the phage titer resulted in a change from survival to death. The similarities between the results presented in Fig. Fig.22 and Fig. Fig.55 must be emphasized, with the only difference being that Fig. Fig.22 shows a more uniform contour distribution of the temperature curves. Such a behavior provides a visual check of the agreement between the responses of both models, although the global model should be considered a natural evolution of the first one.

FIG. 4.
Response surface plot showing the predicted probability of survival for Staphylococcus aureus as a function of the temperature and initial phage titer (global model).
FIG. 5.
Contour plot showing the predicted probability of survival for Staphylococcus aureus as a function of the initial phage titer and temperatures of 15, 20, 25, 30, 35, and 37°C.

As earlier, the survival/death interface can be deduced by rearranging equation 8 and expressing temperature versus log10 PFU/ml. The equation of the interface reads as follows:

equation M9
(9)

Figure Figure66 shows the contour plot of the survival/death interfaces for S. aureus predicted by the model as function of temperature and log10 PFU/ml at probability levels of 0.05 and 0.10. The two interfaces remain approximately parallel during the entire experimental region. In order to visualize the model performance, the observed data were also superimposed. As occurred in the initial model, there was not any case of survival in the death region, although some cases of death were observed in the survival area. The number of the latter cases decreases as the probability of the interface increases. Therefore, the global model may be considered fail-safe. Besides, the global model provides the minimum phage concentration (about 2 × 108 PFU/ml) required to inactivate S. aureus in milk at different temperatures, irrespective of the bacterial contamination level. Nevertheless, the model developed is applicable only within the limits of the experimental region, and any extrapolation must be considered with caution.

FIG. 6.
Survival/death interface for Staphylococcus aureus predicted by the global model at probability levels of 0.10 and 0.05. Superimposed are the data from all experimental designs and the validation experiments. □, S. aureus survival; ○, ...

DISCUSSION

To ensure the microbiological safety of foods, processing criteria must be focused to achieve a required log reduction of the microbial contamination level (2). Regarding this, a number of physical and chemical decontamination treatments are usually applied. However, there is some concern over their undesirable effects, such as changes in sensory quality or the presence of residues in foods. Decontamination treatments based on natural antimicrobials, such as bacteriophages applied to inhibit or eradicate pathogen or spoilage bacteria in foods, could help to overcome the undesirable effects of classical treatments (13). The great potential of the use of the phage control strategy for foods has fuelled research into this subject and the creation of several companies devoted to phage applications. At present, two commercial products targeting Listeria monocytogenes in foods, Listex P100 and LMP 102, are already available. An additional proof of the increasing interest in the use of phages as food preservatives is the report recently delivered by the European Food Safety Authority (EFSA) BIOHAZ Panel on the use and mode of action of bacteriophages in food production (7).

Phage-bacterium interactions have recently been modeled to facilitate phage therapy against Campylobacter jejuni (4). The influence of the bacterial and phage concentrations on the inactivation of Campylobacter and Salmonella in culture medium was also modeled (3). To our knowledge, however, predictive modeling procedures to determine bacterial behavior in foods when phages are used as biocontrol agents have not been developed so far. Consequently, in the present work, we have developed probabilistic models in order to facilitate the successful application of phages as biocontrol agents against S. aureus in milk. For this purpose, we have applied a survival/death interface model that describes the conditions that limit bacterial survival. This type of model is derived from a logistic regression procedure, which uses a binary response variable (S. aureus survival or death) and three independent variables (initial bacterial contamination, initial phage titer, and temperature of incubation). A fortuitous breakdown in the cold chain and the temperatures to which milk is subjected during dairy product processing have been taken into account when the temperature range was selected. The bacterial inoculum range represents low, medium, and high contamination levels.

To be effective as biocontrol agents, phages should be able to encounter the bacterial cell host and proceed efficiently with infection. In fact, larger numbers of phages are required to ensure efficient infection at low cell densities (15). It should be noticed that MOI (as phage concentration/bacterial concentration ratio) has not been used as an independent variable in this study because similar MOI values are obtained from different concentrations of phages and bacteria. However, the need to keep the concentration of phages within a reasonable range (i.e., a range based on the highest achievable phage titers) was taken into account.

Regarding this, two models that predict staphylococcal behavior after bacteriophage addition to milk as a probability of survival or death have been developed. They are not based on the kinetics of inactivation but provide a nice method for predicting S. aureus inactivation induced by phages. Previous studies with hurdles other than phages corroborate the suitability of the survival/death interface models to predict the inactivation of pathogens such as Escherichia coli O157:H7 (29), L. monocytogenes (17), and Enterobacter sakazakii (16) under different processing conditions.

The two predictive models have taken into account both the individual and the joint effects of the diverse independent variables on S. aureus inactivation, but differences between the models were observed. In the first model, the individual effect of initial bacterial contamination and the joint effects of this variable with temperature and initial phage titer were retained, while only the individual and the joint effects of temperature and phage titer were taken into account by the backward-selection option in the second model. Both models provide fail-safe predictions, and the results obtained with the two agree fairly well when the predicted values for the fate of S. aureus (survival/death) are compared with the observed values obtained in the challenge assays.

The dependence of model 1 on the three variables is in agreement with the fact that the kinetics of inactivation of S. aureus by phages show a certain influence of bacterial contamination on the effectiveness of phages (Fig. (Fig.1).1). This is probably related to the bacterial proliferation threshold and the phage inundation threshold required for bacterial suppression (15, 21). On the other hand, model 2 was developed by also taking into account the validation assays (deduced from a larger number of experiments), had a slightly better fit (statistical and apparent), depended on only two variables (temperature and initial phage titer, which makes its application easier), and enjoys the principle of parsimony. Overall, model 2 should be regarded as a natural evolution of model 1 and appropriate to estimate the survival/death of the S. aureus population in milk when lytic phages are used in a cocktail as biocontrol agents at different temperatures. This global model provides the minimum phage concentration (about 2 × 108 PFU/ml) for inactivating S. aureus in milk at different temperatures. This phage concentration is fairly similar to that found by Bigwood et al. (3) required to control the food-borne pathogenic bacteria Salmonella and Campylobacter. Finally, it should be noticed that these results could have practical biocontrol application, as they identify the concentration of phages that should be added to pasteurized milk to ensure that no survival of S. aureus will occur.

Acknowledgments

This work was supported by grants AGL2006-03659/ALI (from the Ministerio de Educación, Spain) and IB08-052 (from the Plan de Ciencia, Tecnología e Innovación, Principado de Asturias, Spain). J. M. Obeso was the recipient of a predoctoral fellowship from I3P Programme (CSIC). P. García was a fellow of the Ramón y Cajal Postdoctoral Programme (Ministerio de Educación, Spain), while F. N. Arroyo-López is a fellow of the Juan de la Cierva Programme (Ministerio de Educación, Spain).

Footnotes

[down-pointing small open triangle]Published ahead of print on 23 July 2010.

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