Growth kinetics of Listeria monocytogenes in broth and beef frankfurters--determination of lag phase duration and exponential growth rate under isothermal conditions

ABSTRACT:  The objective of this study was to develop a new kinetic model to describe the isothermal growth of microorganisms. The new model was tested with Listeria monocytogenes in tryptic soy broth and frankfurters, and compared with 2 commonly used models—Baranyi and modified Gompertz models. Bias factor (BF), accuracy factor (AF), and root mean square errors (RMSE) were used to evaluate the 3 models. Either in broth or in frankfurter samples, there were no significant differences in BF (approximately 1.0) and AF (1.02 to 1.04) among the 3 models. In broth, the mean RMSE of the new model was very close to that of the Baranyi model, but significantly lower than that of the modified Gompertz model. However, in frankfurters, there were no significant differences in the mean RMSE values among the 3 models. These results suggest that these models are equally capable of describing isothermal bacterial growth curves. Almost identical to the Baranyi model in the exponential and stationary phases, the new model has a more identifiable lag phase and also suggests that the bacteria population would increase exponentially until the population approaches to within 1 to 2 logs from the stationary phase. In general, there is no significant difference in the means of the lag phase duration and specific growth rate between the new and Baranyi models, but both are significantly lower than those determined from the modified Gompertz models. The model developed in this study is directly derived from the isothermal growth characteristics and is more accurate in describing the kinetics of bacterial growth in foods.     

Comparison of the Baranyi model with the modified Gompertz equation for modelling thermal inactivation of Listeria monocytogenes Scott A

The Baranyi model was used to fit the four commonly observed survival curves: linear curves, those with a lag phase, those with a tailing phase and sigmoidal curves. It was validated by using published experimental data for thermal inactivation of Listeria monocytogenes Scott A heated in infant formula and compared with the modified Gompertz equation. For the prediction performance, the Baranyi model was better and more robust than the modified Gompertz equation.     

Description of growth of Clostridium perfringens in cooked beef with multiple linear models

The traditional linear model used in food microbiology employs three linear segments to describe the process of food spoilage and categorize a growth curve into three phases — lag, exponential, and stationary. The linear model is accurate only within certain portions of each phase of a growth process, and can underestimate or overestimate the transitional phases. While sigmoid functions (such as the Gompertz and logistic equations) can be used to fit the experimental growth data more accurately, they fail to indicate the physiological state of bacterial growth. The objective of this paper was to develop a new methodology to describe and categorize accurately the bacterial growth as a process using Clostridium perfringens as a test organism. This methodology utilized five linear segments represented by five linear models to categorize a bacterial growth process into lag, first transitional, exponential, second transitional, and stationary phases. Growth curves described in this paper using multiple linear models were more accurate than the traditional three-segment linear models, and were statistically equivalent to the Gompertz models. With the growth rates of transitional phases set to 1/3 of the exponential phase, the durations of the lag, first transitional, exponential, and second transitional phases in a growth curve described by the new method were correlated linearly. Since this linear relationship was independent of temperature, a complete five-segment growth curve could be generated from the maximum growth rate and a known duration of the first four growth phases. Moreover, the lag phase duration defined by the new method was a linear function of the traditional lag phase duration calculated from the Gompertz equation. With this relationship, the two traditional parameters (lag phase and maximum growth rate) used in a three-segment linear model can be used to generate a more accurate five-segment linear growth curve without involving complicated mathematical calculations.     

A new logistic model for Escherichia coli growth at constant and dynamic temperatures

A new logistic model for bacterial growth was developed in this study. The model, which is based on the logistic model, contains an additional term for expression of the very low rate of growth during a lag phase, in its differential equation. The model successfully described sigmoidal growth curves of Escherichia coli at various initial cell concentrations and constant temperatures. The model predicted well the bacterial growth curves, similar to the Baranyi model and better than the modified Gompertz model, especially in terms of the rate constant and the lag period of the growth curves. Using the experimental data obtained at the constant temperatures, the new logistic model was studied for growth prediction at a dynamic temperature. The model accurately described E. coli growth curves at various patterns of dynamic temperature. It also well described other bacterial growth curves reported by other investigators. These results showed that this model could be a useful tool for bacterial growth prediction from the temperature history of a tested food.     

Statistical evaluation of mathematical models for microbial growth

The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The nonlinear functions used were: three-phase linear, logistic, Gompertz, Von Bertalanffy, Richards, Morgan, Weibull, France and Baranyi. Two data sets were used, one comprising 21 growth curves of different bacterial and fungal species in which growth was expressed as optical density units, and one comprising 34 curves of colony forming units counted on plates of Yersinia enterocolitica grown under different conditions of pH, temperature and CO(2) (time-constant conditions for each culture). For both sets, curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were analysis of residuals (residual distribution, bias factor and serial correlation) and goodness-of-fit (residual mean square, accuracy factor, extra residual variance F-test, and Akaike's information criterion). The models showing the best overall performance were the Baranyi, three-phase linear, Richards and Weibull models. The goodness-of-fit attained with other models can be considered acceptable, but not as good as that reached with the best four models. Overall, the Baranyi model showed the best behaviour for the growth curves studied according to a variety of criteria. The Richards model was the best-fitting optical density data, whereas the three-phase linear showed some limitations when fitting these curves, despite its consistent performance when fitting plate counts. Our results indicate that the common use of the Gompertz model to describe microbial growth should be reconsidered critically, as the Baranyi, three-phase linear, Richards and Weibull models showed a significantly superior ability to fit experimental data than the extensively used Gompertz.     

Modeling survival of high hydrostatic pressure treated stationary- and exponential-phase Listeria innocua cells

High Hydrostatic Pressure (HHP) inactivation (325–400 MPa; 0–20 min; maximum temperature 30 °C) of cells of Listeria innocua CECT 910 was studied in two different growth phases (exponential and stationary), and the corresponding survival curves were obtained for each case. The curves were fitted to two nonlinear models, the modified Gompertz equation and the Baranyi model. The kinetic constants calculated for both models, µ_max and k_max, indicated that cells in exponential growth phase were more sensitive to pressure than those in stationary phase. Both mathematical models were suitable for describing L. innocua HHP survival curves, rendering kinetic constants that increased with increasing pressure. When considering the experimental models validation, both Gompertz and Baranyi predicted in a similar way, however Baranyi had slightly lower A_f (Accuracy factor) and B_f (Bias factor) values, which indicated better prediction values. In summary, both mathematical models were perfectly valid for describing L. innocua inactivation kinetics under HHP treatment.Industrial relevanceThe mathematical models for inactivation and growth of microorganisms are the foundation of predictive microbiology and are used in risk assessments procedures as part of the food safety management system. Besides, these models together with those applied to inactivation of enzymes and destruction of quality factors are essential to optimize processes and thus to lay the foundations for industrial processing. It is therefore necessary to identify generally applicable kinetic models that will produce primary and secondary kinetic parameters and are statistically reliable as a key tool to predict the behaviour of microorganisms, enzymes and quality factors after processing.     

Growth curve prediction from optical density data

A fundamental aspect of predictive microbiology is the shape of the microbial growth curve and many models are used to fit microbial count data, the modified Gompertz and Baranyi equation being two of the most widely used. Rapid, automated methods such as turbidimetry have been widely used to obtain growth parameters, but do not directly give the microbial growth curve. Optical density (OD) data can be used to obtain the specific growth rate and if used in conjunction with the known initial inocula, the maximum population data and knowledge of the microbial number at a predefined OD at a known time then all the information required for the reconstruction of a standard growth curve can be obtained. Using multiple initial inocula the times to detection (TTD) at a given standard OD were obtained from which the specific growth rate was calculated. The modified logistic, modified Gompertz, 3-phase linear, Baranyi and the classical logistic model (with or without lag) were fitted to the TTD data. In all cases the modified logistic and modified Gompertz failed to reproduce the observed linear plots of the log initial inocula against TTD using the known parameters (initial inoculum, MPD and growth rate). The 3 phase linear model (3PLM), Baranyi and classical logistic models fitted the observed data and were able to reproduce elements of the OD incubation-time curves. Using a calibration curve relating OD and microbial numbers, the Baranyi equation was able to reproduce OD data obtained for Listeria monocytogenes at 37 and 30°C as well as data on the effect of pH (range 7.05 to 3.46) at 30°C. The Baranyi model was found to be the most capable primary model of those examined (in the absence of lag it defaults to the classic logistic model). The results suggested that the modified logistic and the modified Gompertz models should not be used as Primary models for TTD data as they cannot reproduce the observed data.     

Model comparison for Escherichia coli growth in pouched food

We recently studied the growth characteristics of Escherichia coli cells in pouched mashed potatoes (Fujikawa et al., J. Food Hyg. Soc. Japan, 47, 95-98 (2006)). Using those experimental data, in the present study, we compared a logistic model newly developed by us with the modified Gompertz and the Baranyi models, which are used as growth models worldwide. Bacterial growth curves at constant temperatures in the range of 12 to 34 degrees C were successfully described with the new logistic model, as well as with the other models. The Baranyi gave the least error in cell number and our model gave the least error in the rate constant and the lag period. For dynamic temperature, our model successfully predicted the bacterial growth, whereas the Baranyi model considerably overestimated it. Also, there was a discrepancy between the growth curves described with the differential equations of the Baranyi model and those obtained with DMfit, a software program for Baranyi model fitting. These results indicate that the new logistic model can be used to predict bacterial growth in pouched food.     

Modelling the effect of environmental factors on the growth of Aspergillus parasiticus and mycotoxin production in paddy during storage

The simulated experiment of A. parasiticus isolated from the paddy was carried out during the paddy storage for 20 days. The growth and mycotoxin data were collected for constructing kinetic and probability models of moulds. The Baranyi and Gompertz model was employed as the primary model and estimated the lag phase and maximum specific growth rate. Secondary models, such as polynomial, Davey and Gibson model were used and completely evaluated under different conditions. The polynomial equation was highly rated compared with Gibson and Davey model and gave realistic temperatures and a_w for mould growth. Logistic model showed promising results on the prediction of growth boundary and AFB₁ production. Employed models showed promising predicted results, indicating that it is an effective tool for describing and predicting the growth of moulds under different temperatures and a_w. The results can be applied to develop the optimal strategy to prevent fungal spoilage and aflatoxin production during paddy storage.     

Develop Mechanistic Models of Transition Periods between Lag/Exponential and Exponential/Stationary Phase

A continuing goal in predictive microbiology is models directly based on physiological behavior. Buchanan et al.¹ hypothesized that (1) the curvilinear lag/exponential transition represents the variability of cells in the adjustment (t_a) and metabolic (t_m) periods, and (2) the exponential/stationary transition is determined by limiting nutrient diffusion rates. Nutritional shift trials were conducted to estimate E.coli K-12 growth. Lactase production time suggest that lactase gene translation occurs after completion of lag phase. Agitation rates and inoculum sizes both influenced the shape of the exponential/stationary phase transition. Monte Carlo simulations allowed the generation of sigmoidal growth curves while considering physiological events.     

GROWTH KINETICS OF CLOSTRIDIUM PERFRINGENS IN COOKED BEEF1

The objective of this work was to investigate the growth kinetics of a three‐strain cocktail of Clostridium perfringens in cooked beef. The study was conducted by growing the heat‐activated spores in ground beef under isothermal conditions between 17–50C. A four‐parameter Gompertz equation was used as a primary model to fit the growth curves along with a modified Ratkowsky model to analyze the temperature dependence of the bacterial growth. Results indicated that the Gompertz model could accurately describe the growth of C. perfringens in cooked beef. The estimated theoretical minimum, optimum, and maximum growth temperatures of this organism in cooked beef were 9.8, 47.1, and 50.8C, respectively. A linear relationship between the durations of the lag and exponential phases of growth curves was observed in this study. Such a linear relationship can be used to generate a linear isothermal growth curve complete with the lag, exponential, and stationary phases without complex mathematical computation. The kinetic models and growth parameters obtained from this study potentially can be applied to the food industry to design appropriate cooling schedules and estimate the growth of C. perfringens in thermally processed beef products under temperature abuse conditions.     

Comparison of Mathematical Models of Lactic Acid Bacteria Growth in Vacuum‐Packaged Raw Beef Stored at Different Temperatures

The lactic acid bacteria grown in vacuum‐packaged raw beef under 7, 10, 15, and 20 °C has been studied in this paper. Four primary models, the modified Gompertz, logistic, Baranyi, and Huang model were used for data fitting. Statistical criteria such as the bias factor and accuracy factor, mean square error, Akaike's information criterion, and the residual distribution were used for comparing the models. The result showed that all of the 4 models can fit the data well and they were not significantly different in the performance. They were equally capable of describing bacterial growth, but the growth rate and lag time estimated from the modified Gompertz model were a little higher than other models. The estimate for the lag time was not accurate as the growth rate.     

Dynamic model for predicting growth of Salmonella spp. in ground sterile pork

A predictive model for Salmonella spp. growth in ground pork was developed and validated using kinetic growth data. Salmonella spp. kinetic growth data in ground pork were collected at several isothermal conditions (between 10 and 45 °C) and Baranyi model was fitted to describe the growth at each temperature, separately. The maximum growth rates (μ_max) estimated from the Baranyi model were modeled as a function of temperature using a modified Ratkowsky equation. To estimate bacterial growth under dynamic temperature conditions, the differential form of the Baranyi model, in combination with the modified Ratkowsky equation for rate constants, was solved numerically using fourth order Runge-Kutta method. The dynamic model was validated using five different dynamic temperature profiles (linear cooling, exponential cooling, linear heating, exponential heating, and sinusoidal). Performance measures, root mean squared error, accuracy factor, and bias factor were used to evaluate the model performance, and were observed to be satisfactory. The dynamic model can estimate the growth of Salmonella spp. in pork within a 0.5 log accuracy under both linear and exponential cooling profiles, although the model may overestimate or underestimate at some data points, which were generally < 1 log. Under sinusoidal temperature profiles, the estimates from the dynamic model were also within 0.5 log of the observed values. However, underestimation could occur if the bacteria were exposed to temperatures below the minimum growth temperature of Salmonella spp., since low temperature conditions could alter the cell physiology. To obtain an accurate estimate of Salmonella spp. growth using the models reported in this work, it is suggested that the models be used at temperatures above 7 °C, the minimum growth temperature for Salmonella spp. in pork.     

低温条件下冷却猪肉中假单胞菌生长模型的比较分析

为了确定拟合冷却猪肉中假单胞菌低温下生长的最适模型,分别对低温(0、5、10℃)条件下托盘和真空包装冷却猪肉中假单胞菌的生长特点进行分析,应用修正的Gompertz、Baranyi及Huang模型对其进行拟合,通过残差和拟合度(RSS、AIC、RSE)等统计指标比较3种模型的拟合能力,分析不同模型拟合假单胞菌生长的差别。结果表明：低温托盘和真空包装条件下假单胞菌在延滞期出现了明显的菌数下降现象,随后呈现“S”形生长;0℃条件下Baranyi模型拟合出最小的RSS、AIC、RSE值,分别是5.2933、－54.0428、0.1708;而修正的Gompertz模型和Huang模型分别在5℃和10℃条件下拟合出最小的RSS、AIC、RSE值,分别是17.7372、－18.9098、0.5068和13.0410、－22.4848、0.4207。拟合冷却猪肉中假单胞菌生长的最适模型0℃是Baranyi模型,5℃是修正的Gompertz模型,10℃是Huang模型。因此,在冷却猪肉腐败菌预测时,不同温度条件下应该选择最适合的模型而不是单一的模型来预测假单胞菌的生长。     

Mathematical modeling on the growth of Staphylococcus aureus in sandwich

The growth of Staphylococcus aureus in sandwich fillings at different incubation temperatures was tested. These growth data were fitted into the Gompertz model, Logistic model, and Baranyi model in order to compare the goodness-of-fit of the 3 primary models using several factors such as coefficient of determination (R²), the standard deviation (S_y.x), and the Akaike’s information criterion (AIC). The Gompertz model showed the best statistical fit. Hence, growth parameters such as specific growth rate (SGR) and lag time (LT) obtained from the Gompertz model were used to construct the secondary models. Further, developed models were evaluated by bias factor (B_f) and accuracy factor (A_f). For the SGR, the Bf value was 0.993 and A_f value was 1.156 which indicated conservative predictions. While for LT, a clear deviation was observed between predictions and observations (B_f=0.635 and A_f=1.592). The results, however, were also considered acceptable after comparing with previous publications.     

Estimation of the survival curve of Listeria monocytogenes during non-isothermal heat treatments

Published survival curves of Listeria monocytogenes under several constant temperatures in the range of 50–65°C could be described by the model Log₁₀[S(t)]=−b(T)t^n(T), where S(t) is the momentary survival ratio, and b(T) and n(T) coefficients whose temperature dependence was expressed by empirical models. When the temperature history T(t) is also expressed algebraically, b(T) and n(T) are transformed into time dependent terms, b[T(t)] and n[T(t)] respectively. If there is no growth and damage repair during the heating process, and the momentary inactivation rate only depends on the momentary temperature and survival ratio, then the solution of the differential equation dLog₁₀[S(t)]/dt=−b[T(t)]*n[T(t)]*{−Log₁₀[S(t)]/b[T(t)]}^∧{(n[T(t)]−1)/n[T(t)]} provides the survival curve under the specified non-isothermal conditions. The validity of this model is demonstrated by the agreement of its predictions, calculated numerically using Mathematica®, to reported survival data of Listeria during heating at a constant and varying rates. Unlike in the traditional calculation methods of microbial survival, the one employed here does not require that microbial mortality be a process following a first or any other order kinetics model.     

Growth Inhibition of Common Food Spoilage and Pathogenic Microorganisms in the Presence of Brown Seaweed Extracts

The possibility of using extracts from brown seaweed, Himanthalia elongata, as a natural antimicrobial agent for food preservation is presented. The effect of different concentrations of seaweed extract on the growth kinetics of four common food spoilage (Pseudomonas aeruginosa and Enterococcus faecalis) and food pathogenic microorganisms (Listeria monocytogenes and Salmonella abony) was examined. Seaweed extract at a concentration of 6% inhibited the growth of all four of the studied organisms. Lower concentrations of seaweed extract prolonged the lag phase and reduced both the exponential growth rate and final population densities of the culture. Suitability of three kinetic models, Baranyi–Roberts, modified Gompertz and logistic, for describing the growth/survival of organisms in the presence of different concentrations of the extract, was evaluated. Root mean square error (RMSE) and correlation coefficient (R ²) were used to evaluate the model performance. The R ² value was greater than 0.95 for most of the cases indicating that the models could provide a good fitting to the experimental data. The RMSE and residual sum of squares were very low for all the three models, and no significant difference was observed in the goodness of fit between the three models as indicated by the F test.     

真空包装狮子头货架期预测模型的建立

为建立真空包装狮子头货架期预测模型,分析不同温度贮藏期间狮子头中菌落总数的变化情况,分别用线性模型、修正的Gompertz模型、修正的Logistic模型和Baranyi模型对狮子头中菌落总数进行一级模型的拟合,在此基础上使用平方根模型建立二级模型。通过比较各模型的评价参数选择最优模型,并进一步建立货架期预测模型。结果表明在一级模型中,修正的Gompertz模型对真空包装狮子头中菌落总数生长曲线的拟合优度最高;基于修正的Gompertz模型建立的平方根模型可较好地描述温度对狮子头最大比生长速率和迟滞期的影响。在4、10、15、20、25℃条件下贮藏狮子头的货架期分别为80.79、45.22、10.96、4.96、4.01 d,货架期实测值与预测值的相对误差值均在10%以内,表明建立的模型可以较准确地对贮藏在4~25℃条件下的狮子头进行货架期预测。     

乙型副伤寒沙门氏菌在鲜切黄瓜中的生长行为及预测模型建立

为建立不同温度条件下鲜切黄瓜中乙型副伤寒沙门氏菌的生长预测模型,将新鲜黄瓜切丁,添加乙型副伤寒沙门氏菌,分别在10、15、20、25、30和35℃下的恒温条件下贮藏,以观察细菌的生长。使用USDA综合病原体建模程序(USDA-IPMP)拟合每个温度下每种细菌的生长曲线,以找出描述该细菌生长的最适初级生长模型,并拟合得到最大比生长速率。通过温度对初级模型中最大比生长速率的生长动力学拟合,分别建立Ratkowsky、Huang rate、Cardinal、Arrhenius-type二级生长模型,并进行数学评估和实测样品验证。结果表明,实验数据和生长曲线显示乙型副伤寒沙门氏菌的生长表现出三个阶段,包括延滞期,指数期和稳定期。乙型副伤寒沙门氏菌的延滞期时间随着孵育时间的增加而降低。相反,乙型副伤寒沙门氏菌的生长速率随着孵育温度而增加,由此表明风险随温度的升高而增加。使用Baranyi和Huang初级模型分析两种病原体的生长曲线,使用Ratkowsky、Huang平方根模型、Cardinal和Arrhenius模型描述温度对贮藏时间细菌生长的影响,同时应用实验数据和样品实测验证评估所建立的预测模型。从该研究中获得的结果和预测模型可用于预测鲜切黄瓜产品中乙型副伤寒沙门氏菌的生长。     

Modeling the inactivation of Salmonella typhimurium by dense phase carbon dioxide in carrot juice

The inactivation of Salmonella typhimurium inoculated into acidified carrot juice subjected to dense phase carbon dioxide (DPCD) was investigated. The pressures in the study were 10, 20 and 30 MPa, the temperatures were 32, 37 and 42 °C, and the treatment time was 5–90 min. The inactivation effect of DPCD was enhanced by increasing pressure and temperature. The sigmoid inactivation curves were characterized with the lag phase, exponential inactivation phase, and resistant phase. The inactivation curves were fitted to the modified Gompertz equation and the modified Logistic equation, the modified Gompertz equation was superior since its lowest residual sum of squares (RSS) was lower although there was no significant difference of goodness-of-fit between both models as indicated by F-test. The λ (the duration of the lag phase) and t_4-D (the time necessary to achieve 4-log cycles reduction) decreased with increasing pressure or temperature. The k_dm (the maximum specific value of the inactivation rate, min⁻¹) increased with increasing temperatures, and decreased with increasing pressures. The activation energy (Ea) and the activation volume (Va) necessary for inactivating S. typhimurium by DPCD were 19.06–29.39 kJ mol⁻¹ and 18.89–58.27 cm³ mol⁻¹.