A study of the Gamma hypothesis: predictive modelling of the growth and inhibition of Enterobacter sakazakii

Although the temperature growth profile of the opportunistic pathogen Enterobacter sakazakii is known, few other environmental factors affecting growth have been analysed. Using a model based on the Gamma hypothesis--that antimicrobial factors in mixtures exert independent effects--a range of weak acids (lactic, acetic, propionic, citric, sorbic and benzoic), pH, salt and temperature and some of their combinations were examined. The weak acids examined inhibited principally with the acid-form of the weak acid, however, benzoic, sorbic and propionic acids also displayed an inhibitory contribution from their respective anionic forms. In all cases pH could be considered an independent inhibiting factor. The minimum pH and maximum salt concentration for growth were calculated to be 3.89 and 9.1% respectively. In combination, there was no suggestion of any interactive effect between them. Studies performed on combinations of Na acetate/pH between 25 and 41 degrees C showed that temperature did not affect the relative inhibitory effects of the weak acid/pH mixtures. The results of this study support the Gamma hypothesis suggesting that there are no synergistic interactions between inhibitory factors and that growth can be predicted from a library of known effects. More importantly to the food industry, the results can be used to design good quality shelf-life challenge tests by reducing the number of studies required.     

Effect of inoculum size on the combined temperature, pH and aw limits for growth of Listeria monocytogenes

The growth/no growth responses of Listeria monocytogenes inoculated at four levels (0.90, 2.58, 4.20 and 6.81 log cfu/well; 300 microl medium/well) into tryptic soy broth (TSB) were monitored at different combinations of temperature (4 to 30 degrees C), pH (3.76 to 6.44) and aw (0.888 to 0.997) for 60 days. The study was conducted in 96-well microtiter plates and growth was monitored visually and by recording the turbidity of the medium with an automated microplate reader. The growth limits of the pathogen and hence the position of the growth boundary were found to be affected by the size of the inoculum. For example, at 25 degrees C and aw 0.997 the minimum pH values that allowed growth were 4.45 and 3.94 for inoculum levels of 0.90 and 6.81 log cfu/well, respectively. Similarly, at 25 degrees C and pH 6.44 the minimum aw values where growth was observed ranged from 0.900 to 0.928 depending on the inoculum level. The results showed that at temperatures from 10 to 30 degrees C the difference in the growth limits between the tested inoculum levels was higher in environments where a single factor (pH or aw) was inhibitory than in environments where pH and aw together were inhibitory. The data were used to develop a growth/no growth interface model for each inoculum level tested, using logistic polynomial regression. The concordance indices of the models ranged from 99.8% to 99.9% and showed a good fit to the observed data in all models. This study indicates the importance of inoculum size for microbial growth initiation and provides quantitative data that show how the combinations of hurdles which prevent growth vary with inoculum size.     

Evaluation of the effects of sodium chloride,potassium sorbate,nisin and lysozyme on the probability of growth of Clostridium sporogenes

The aim of this study was to evaluate the combined effect of salt (sodium chloride, 0–8% w/v), sorbate (potassium sorbate, 0–4.5% w/v), nisin (0–500 ppm) and lysozyme (0–500 ppm) on the survival of Clostridium sporogenes as a non‐toxigenic surrogate of Clostridium botulinum in terms of the probability of growth by using a central composite rotatable design. The results indicated that salt and sorbate were the most effective factors in preventing the growth of Cl. sporogenes in high‐moisture (>95%) and low‐acid conditions. The probability of growth of Cl. sporogenes in broth was reduced by combinations of salt and sorbate. Nisin and lysozyme had insignificant effects on the probability of growth of Cl. sporogenes (P > 0.05). Lysozyme individually and in combination with nisin had no inhibitory effect on Cl. sporogenes. Overall, the addition of sorbate and lysozyme may allow the salt concentration to be reduced while preventing growth.     

淡腌黄鱼微生物生长动力学参数的初步研究

对淡腌黄鱼产品在不同温度条件下进行贮藏性实验,研究贮藏期间产品的微生物生长情况及产品的品质变化,根据实验数据用运Gompertz动力学方程,得出产品中微生物生长动力学参数。从产品感官、TVBN评价和微生物生长动态分析中,得出产品在不同贮藏条件下的货架期。通过温度对微生物生长速率的影响,分析栅栏效应。食品在联合栅栏作用下,存在着微生物生长／非生长界面,探讨通过从微生物生长动力学模型的参数来限定微生物生长/非生长界面的可能性,即量化栅栏技术。为开发一种最低限度影响产品质量和保障产品安全的新一代食品保藏技术提供一些有用的实验参数。     

Growth/no growth interfaces of Bacillus cereus, Staphylococcus aureus and Salmonella enteritidis in model systems based on water activity, pH, temperature and ethanol concentration

This study focuses its attention on the boundary between the growth and no growth of three strains ofSalmonella enteritidis , Bacillus cereus and Staphylococcus aureus in the presence of growth controlling factors such as temperature, pH, water activity (A_w) and ethanol concentration. Preliminarly, the minimal values of pH, A_wand temperature, and the maximum ethanol concentrations allowing the growth of the considered micro-organisms were determined. The calculation of these values enabled the use of logistic model to evaluate the growth/no growth boundary for the bacteria in relation to the considered independent variables. The location of the growth/no-growth boundaries for S. enteritidis and Staph. aureus were strongly affected, at the same ethanol concentration, by temperature, pH and A_w. Among the considered species, Staph. aureus was endowed with the highest sensitivity to low pH values whileB. cereus's growth/no growth interface, was quite unaffected by the combination of the stresses, when the physico–chemical conditions were above the minimum for growth. The effects of temperature, A_wand ethanol on the limitation of growth of the considered species were not merely additive. It was possible to identify the combinations of such factors preventing the growth of Salmonella enteritidis, Staph. aureus and B. cereus.     

The effects of temperature, pH, sodium chloride and sodium nitrite on the growth of Listeria monocytogenes.

An automated turbidimetric system using multiwelled plates was used to examine the effects of different combinations of NaCl (0.5-8.0% w/v), NaNO2 (0-400 micrograms/ml) pH (4.6-7.4) and temperature (5-30 degrees C) on the growth of Listeria monocytogenes in tryptone soya broth. The data presented clearly illustrate the combinations that permit visible growth of the organism. The ability of L. monocytogenes to grow at low pH levels was strongly influenced by incubation temperature as well as NaNO2 concentration. At 20 degrees C and below, no visible growth was detected, even with 50 micrograms/ml NaNO2 at pH 5.3 (or below) within 21 days. At pH 6.0 and above, NaNO2 had little effect in delaying visible growth except at higher concentrations and also at lower incubation temperatures.     

Growth/no growth model of Listeria monocytogenes as a function of temperature, pH, citric acid and ascorbic acid

A linear logistic regression model was built to describe the growth/no growth boundaries of Listeria monocytogenes as a function of temperature (4–30 °C), pH (4.5–6), citric acid (0–0.4%) and ascorbic acid (0–0.4%). A fractional factorial design was followed among the factors considered and an inoculum size of 10⁵ CFU/ml was used. Evaluation of growth was performed by optical density measurements in Bioscreen C (Labsystems, Finland), during 21 days. Data of optical density were transformed to log CFU/ml by using a calibration curve. Among the 232 combinations of factors tested, growth was observed in 133 and no growth in 99. The degree of agreement between predictions and observations was 97.8% concordant and 2.2% discordant. An internal validation with additional data within the interpolation region of the model was performed. The predictions were concordant in 94% of the cases, and all the wrong cases failed to the safe side of the boundary region. The probability of growth was strongly influenced at low temperatures (<15 °C). The effect of pH was more notorious below 5.3, since the minimum temperature that inhibited growth was higher. Citric acid was more effective than ascorbic acid when the analysis was based on the undisssociated acid concentration (u.a.c.). However, when dealing with the undissociated fraction (u.a.f.), ascorbic acid presented more inhibitory effect. Organic acids also accentuate the temperature and pH inhibition of bacterial growth limits, and increased the minimum pH at which growth was detected. These results have a practical implication for stakeholders and risk managers in order to identify the treatments that can be applied to food and ensure that no growth of this pathogenic microorganism will occur.     

鱼源腐败希瓦氏菌生长/非生长界面模型的建立和验证

选取鱼源腐败希瓦氏菌为研究对象,研究室温(25℃)条件下pH、aw及盐分(NaCl)对腐败希瓦氏菌生长概率的交互影响。采用二阶线性Logistic回归方程和PNN人工神经网络算法构建环境因子交互作用下腐败希瓦氏菌生长/非生长界面模型,对两种模型的拟合优度和预测力进行比较和验证。结果表明,二阶线性Logistic腐败希瓦氏菌生长/非生长模型的测试集和验证集一致率分别为93.80%和100.00%,能预测腐败希瓦氏菌生长/非生长概率;PNN人工神经网络模型的测试集和验证集一致率分别为100.00%和86.67%,最优时间为0.1s,能对腐败希瓦氏菌生长/非生长数据进行快速分类。随着盐分增长,腐败希瓦氏菌生长/非生长界限小幅度向高aw、高pH方向移动,高盐分对腐败希瓦氏菌有生长抑制作用。aw≤0.91时,菌株均不生长,aw=0.92,0.94和0.96时,腐败希瓦氏菌生长概率随pH增大而增大,上升陡峭。pH=4.5时,腐败希瓦氏菌基本不生长,随pH升高,高aw情况下的生长概率增至100%。通过构建鱼源腐败希瓦氏菌环境因子下生长/非生长界面模型,PNN人工神经网络模型可对其生长/非生长数据进行快速分类,二阶线性Logistic能为定量评估pH,aw和盐分范围水产品品质保障提供理论支持。     

An investigation of the Gamma hypothesis: a predictive modelling study of the effect of combined inhibitors (salt, pH and weak acids) on the growth of Aeromonas hydrophila

The Gamma hypothesis, that multiple inhibitory factors combine independently, is the underlying hypothesis for the quantification of the Hurdle concept used in food manufacture. The literature, however, is confused as to whether interactive effects exist and under which circumstances they occur, if at all. Using the method of time to detection (TTD), the inhibitory effect of pH, salt and specific weak acids (acetic, propionic, sorbic and benzoic) and combinations of these with respect to the growth of Aeromonas hydrophila (ATCC 7966) were analysed. A model based on the relative rate to detection described all combinations analysed as having independent effects on the TTD. No synergistic interactions were found between pH and salt, between pH and individual weak acids or between combinations of weak acids and pH for any of the systems under study. This study supports the validity of the Gamma concept -- that individual environmental effects act independently and should, in turn, facilitate attempts to model the growth of other microorganisms under a variety of conditions.     

Probabilistic modeling of Saccharomyces cerevisiae inhibition under the effects of water activity, pH, and potassium sorbate concentration

Probabilistic microbial modeling using logistic regression was used to predict the boundary between growth and no growth of Saccharomyces cerevisiae at selected incubation periods (50 and 350 h) in the presence of growth-controlling factors such as water activity (a(w); 0.97, 0.95, and 0.93), pH (6.0, 5.0, 4.0, and 3.0), and potassium sorbate (0, 50, 100, 200, 500, and 1,000 ppm). The proposed model predicts the probability of growth under a set of conditions and calculates critical values of a(w), pH, and potassium sorbate concentration needed to inhibit yeast growth for different probabilities. The reduction of pH increased the number of combinations of a(w) and potassium sorbate concentration with probabilities to inhibit yeast growth higher than 0.95. With a probability of growth of 0.05 and using the logistic models, the critical pH values were higher for 50 h of incubation than those required for 350 h. With lower a(w) 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 combined effects of environmental conditions related to meat fermentation on growth and lipase production by the starter cultureStaphylococcus xylosus

The ability of the meat starter cultureStaphylococcus xylosusto grow and produce lipase at the environmental conditions relevant to fermented meat production was examined. The combined effect of temperature (10–30°C), pH (5.1–6.0), salt concentration (1–9% w/v) and glucose (0 and 1% w/v) on the maximum specific growth rate ofS. xylosuswas studied in a microbiological medium. Growth was estimated by optical density measurements and growth curves were generated by fitting a modified Gompertz equation to the growth data using non-linear regression analysis. Maximum specific growth rates were derived and submitted to response surface analysis to generate a quadratic model to predict the maximum specific growth rate ofS. xylosusat any combination of the variables. Results from the combined effects of temperature, pH, salt concentration and age of culture on the lipase production ofS. xylosusaccording to a similar statistical analysis showed that lipase production was influenced by pH, salt concentration and age of the culture, whereas it was not significantly affected by the temperature. Generally, in cases with vigorous growth, the level of lipase production was high, and it was limited to a more narrow range of environmental combinations than growth.     

Modelling the effect of temperature and water activity in the growth boundaries of Aspergillus ochraceus and Aspergillus parasiticus

The aim of this work was to model the growth of Aspergillus parasiticus and Aspergillus ochraceus, both mycotoxin producers, near to the growth/no growth boundaries and validate those models in sterile maize grain, peanuts and coffee beans. Malt extract agar was adjusted to six different water activities: 0.93, 0.91, 0.89, 0.87, 0.85 and 0.80. Plates were incubated at 10, 15, 20, 25, 30, 37 and 42 °C. For each of the 42 conditions, 10 Petri dishes were inoculated. Both kinetic and probability models were applied to colony growth data. The results of the present study indicate that the developed probability modelling approach could be satisfactorily employed to quantify the combined effect of temperature and water activity on the growth responses of A. ochraceus and A. parasiticus. However, validation of kinetic results led to poor goodness of prediction. In this study, the validation samples were placed near to the expected boundaries of the models in order to test them under the worst situation. Probability of growth prediction under extreme growth conditions was somewhat compromised, but it can be considered acceptable.     

A probability of growth model for Escherichia coli O157:H7 as a function of temperature, pH, acetic acid, and salt.

Data accumulated on the growth of Escherichia coli O157:H7 in tryptic soy broth (TSB) were used to develop a logistic regression model describing the growth-no growth interface as a function of temperature, pH, salt, sucrose, and acetic acid. A fractional factorial design with five factors was used at the following levels: temperature (10 to 30 degrees C), acetic acid (0 to 4%), salt (0.5 to 16.5%), sucrose (0 to 8%), and pH (3.5 to 6.0). A total of 1,820 treatment combinations were used to create the model, which correctly predicted 1,802 (99%) of the points, with 10 false positives and 8 false negatives. Concordance was 99.9%, discordance was 0.1%, and the maximum rescaled R2 value was 0.927. Acetic acid was the factor having the most influence on the growth-no growth interface; addition of as little as 0.5% resulted in an increase in the observed minimum pH for growth from 4.0 to 5.5. Increasing the salt concentration also had a significant effect on the interface; at all acetic acid concentrations, increasing salt increased the minimum temperature at which growth was observed. Using two literature data sets (26 conditions), the logistic model failed to predict growth in only one case. The results of this study suggest that the logistic regression model can be used to make conservative predictions of the growth-no growth interface of E. coli O157:H7.     

蜡样芽胞杆菌生长/非生长界面模型的建立和评价

蜡样芽胞杆菌是软烤虾仁产品的主要变质菌,它是一种条件致病菌,通过产生腹泻毒素和呕吐毒素导致食物中毒。该研究旨在建立一种概率模型来预测出蜡样芽胞杆菌的生长/非生长情况或者生长概率。用lo-gistic回归模型建立不同温度、水分活度和pH环境因子作用下蜡样芽胞杆菌的生长/非生长界面模型。实验结果表明蜡样芽胞杆菌在脑心浸液肉汤培养基中生长的最低温度为9.99℃,最低水分活度为0.931,最小pH值为4.5。在此基础上建立的蜡样芽胞杆菌生长/非生长界面模型的χ2=49.73,P<0.000 1。用logistic回归模型建立的生长/非生长模型拟合效果达到极显著水平。模型的预测值同时很好地量化了环境因子对蜡样芽胞杆菌的协同作用,为软烤虾仁产品中蜡样芽胞杆菌的生长/非生长界面模型的建立提供了参考。     

The boundary for growth of Zygosaccharomyces bailii in acidified products described by models for time to growth and probability of growth

Models to predict days to growth and probability of growth of Zygosaccharomyces bailii in high-acid foods were developed, and the equations are presented here. The models were constructed from measurements of growth of Z. bailii using automated turbidimetry over a 29-day period at various pH, NaCl, fructose, and acetic acid levels. Statistical analyses were carried out using Statistical Analysis Systems LIFEREG procedures, and the data were fitted to log-logistic models. Model 1 predicts days to growth based on two factors, combined molar concentration of salt plus sugar and undissociated acetic acid. This model allows a growth/no-growth boundary to be visualized. The boundary is comparable with that established by G. Tuynenburg Muys (Process Biochem. 6:25-28, 1971), which still forms the basis of industry assumptions about the stability of acidic foods. Model 2 predicts days to growth based on the four independent factors of salt, sugar, acetic acid, and pH levels and is, therefore, much more useful for product development. Validation data derived from challenge studies in retail products from the U.S. market are presented for Model 2, showing that the model gives reliable, fail-safe predictions and is suitable for use in predicting growth responses of Z. bailii in high-acid foods. Model 3 predicts probability of growth of Z. bailii in 29 days. This model is most useful for spoilage risk assessment. All three models showed good agreement between predictions and observed values for the underlying data.     

Modeling the effect of storage atmosphere on growth-no growth interface of Listeria monocytogenes as a function of temperature, sodium lactate, sodium diacetate, and NaCl

The effect of aerobic and anaerobic conditions on growth initiation by a 10-strain composite of Listeria monocytogenes (10(4) CFU/ml) was evaluated in tryptic soy broth with 0.6% yeast extract (TSBYE) as a function of 220 combinations of pH (3.82 to 7.42), sodium lactate (SL) (0 to 10%, vol/vol), and sodium diacetate (SD) (0 to 0.5%, wt/vol) at 10 or 30 degrees C (a slightly abusive and the optimal growth temperature, both above the growth limiting range of 0 to 3 degrees C for L. monocytogenes) in 96-well microplates. In addition, four probability-of-growth models were developed to quantify the effect of 346 aerobic and 346 anaerobic combinations of temperature (4 to 30 degrees C), SL (0 to 6%, vol/vol), and SD (0 to 0.5%, wt/vol) in the presence of NaCl (0.5 or 2.5%, wt/vol) on the growth-no growth responses of the same L. monocytogenes strain composite, with a microplate reader. Growth responses were evaluated turbidimetrically (620 nm) every 5 days for a total of 40 days. Data were modeled with logistic regression to determine the growth-no growth interfaces. The minimum pH values at which growth of L. monocytogenes occurred were higher under anaerobic than under aerobic conditions, and this difference was more evident at 10 degrees C or at higher SL and SD concentrations. The MIC of SD decreased with increasing SL levels. Anaerobic storage reduced the levels of SL-SD, allowing the growth of L. monocytogenes compared with aerobic storage, especially at low temperatures. In the presence of 2.5% NaCl, the MICs for SD were lower than those obtained with 0.5% NaCl, especially at 4 and 10 degrees C, or in the presence of 5 to 6% SL. The developed models for anaerobic incubation showed good performance (80% successful predictions; i.e., in 40 of 50 comparisons) with independent data from studies on survival-growth of L. monocytogenes on meat products. The study provides quantitative data on the antimicrobial activity of SL (0 to 10%) and SD (0 to 0.5%), temperature (4 to 30 degrees C), and pH (3.82 to 7.42) and on the probability of growth of L. monocytogenes under anaerobic or aerobic conditions in the presence of 0.5 or 2.5% NaCl, and hence, addresses important needs for risk assessment activities.     

A simplified approach for modelling the bacterial growth/no growth boundary

A simplified growth/no growth (G/NG) model, conceptually derived from the Gamma model and making direct and explicit use of growth limits of bacteria through a normalization constant (η), was proposed. The η value, which quantifies the product of the cardinal optimal distances for growth probability, is a species-independent constant. This is of importance when experimental data is missing or insufficient. The simplified G/NG model was developed including the effect of temperature, pH and water activity, and was expanded incorporating the preservative effects. As a practical application, the model was investigated for its ability to describe published data. The successful validation of the simplified G/NG model is discussed in regard to its potential applicability as a first estimate method for the development of safe food products.     

Time-to-detection,percent-growth-positive and maximum growth rate models for Clostridium botulinum 56A at multiple temperatures

We previously developed models for the influence of inoculum size on the growth kinetics (time-to-detection and maximum growth rate) and percent-growth-positive samples of Clostridium botulinum 56A with factors of inoculum size (1, 100, and 10,000 spores/sample). pH (5.5. 6.0 and 6.5) and sodium chloride concentration (0.5%, 2% and 4%) at 30 degrees C. In this present study, data were collected at two more temperatures (15 and 22 degrees C), making the final design a complete 3 X 3 X 3 X 3 factorial with a total of 81 conditions. Growth was followed hourly as change in A620. The Gompertz equation was fit to the growth data, and the parameters derived were used to calculate the maximum growth rate and time-to-detection. Linear regression with polynomial terms was used to analyze the effect of environmental factors on time-to-detection and maximum growth rate. Logistic regression with polynomial terms was used to analyze the data for percent-growth-positive. Despite the fact that the variance is larger in this extended data set (which includes two temperatures that are further away from the optimum), the inoculum size effect is clearly demonstrated. When inoculum size increased, the percent-growth-positive samples increased and the time-to-detection decreased. When the inoculum was 1000 spores/sample or higher, little additional effect on time-to-detection was observed. Inoculum size might influence results through simple probability or quorum sensing. Our results show that the observed effect of inoculum size from the previous report at a single temperature is not restricted to a specific growth condition, but rather a general phenomenon. The maximum growth rate was independent of inoculum levels, confirming our previous results.