When is simple good enough: a comparison of the Gompertz,Baranyi, and three-phase linear models for fitting bacterial growth curves

The use of primary mathematical models with curve fitting software is dramatically changing quantitative food microbiology. The two most widely used primary growth models are the Baranyi and Gompertz models. A three-phase linear model was developed to determine how well growth curves could be described using a simpler model. The model divides bacterial growth curves into three phases: the lag and stationary phases where the specific growth rate is zero (gm=0), and the exponential phase where the logarithm of the bacterial population increases linearly with time (gm=constant). The model has four parameters: N_o(Log₈of initial population density), NMAX(Log₈of final population density), tLAG(time when lag phase ends), and tMAX(time when exponential phase ends). A comparison of the linear model was made against the Baranyi and Gompertz models, using established growth data forEscherichia coli0157:H7. The growth curves predicted by the three models showed good agreement. The linear model was more ‘robust' than the others, especially when experimental data were minimal. The physiological assumptions underlying the linear model are discussed, with particular emphasis on assuring that the model is consistent with bacterial behavior both as individual cells and as populations. It is proposed that the transitional behavior of bacteria at the end of the lag phase can be explained on the basis of biological variability.     

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.     

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.     

A new mechanistic growth model for simultaneous determination of lag phase duration and exponential growth rate and a new Bĕlehdrádek-type model for evaluating the effect of temperature on growth rate

A new mechanistic growth model was developed to describe microbial growth under isothermal conditions. The new mathematical model was derived from the basic observation of bacterial growth that may include lag, exponential, and stationary phases. With this model, the lag phase duration and exponential growth rate of a growth curve were simultaneously determined by nonlinear regression. The new model was validated using Listeria monocytogenes and Escherichia coli O157:H7 in broth or meat. Statistical results suggested that both bias factor (B_f) and accuracy factor (A_f) of the new model were very close to 1.0. A new B?lehdrádek-type rate model and the Ratkowsky square-root model were used to describe the temperature dependence of bacterial growth rate. It was observed that the maximum and minimum temperatures were more accurately estimated by a new B?lehdrádek-type rate model. Further, the inverse of square-roots of lag phases was found proportional to temperature, making it possible to estimate the lag phase duration from the growth temperature.     

Effect of sub-lethal heating and growth temperature on expression of the ribosomal RNA rrnB P(2) promoter during the lag phase of Pseudomonas fluorescens

Current models for the lag phase of food-borne pathogens are limited by our poor understanding of the physiological changes taking place as bacterial cells prepare for exponential growth. In a previous paper in this series, a strain of Pseudomonas fluorescens containing the Tn7-luxCDABE gene cassette regulated by the rRNA promoter rrnB P(2) was used to measure the influence of starvation on the lag phase duration (LPD(OD)) and growth rate (R(OD)). rrnB P(2) promoter activity increased exponentially during the lag phase, and was characterized by lag (LPD(Exp)) and rate (R(Exp)) parameters. In the present study, this work was expanded to include the influence of growth temperature (10 to 30 degrees C) and exposure to sub-lethal heating at 47 degrees C. With these additional datasets, the LPD(Exp) was often more pronounced than had been noted with starvation, so the original exponential association model (EXP) was compared to logistic and Gompertz (GOM) models. Based on root mean square error, the GOM model gave the better fit for some of the sub-lethal heating and growth temperature datasets; however, the EXP model was assessed as best overall. Increased growth temperature and decreased time of sub-lethal heating produced significant decreases in LPD(OD) and LPD(Exp) and increases in R(OD) and R(Exp). The results suggest that different stressors have differential effects on gene expression and subsequent growth.     

Modeling the lag phase of Listeria monocytogenes

An estimate of the lag phase duration is an important component for predicting the growth of a bacterium and for creating process models and risk assessments. Most current research and data for predictive modeling programs initiated growth studies with cells grown to the stationary phase in a favorable pH, nutrient and temperature environment. In this work, Listeria monocytogenes Scott A cells were grown in brain heart infusion (BHI) broth at different temperatures from 4 to 37 degrees C to the exponential growth or stationary phases. Additional cells were suspended in a dilute broth, desiccated or frozen. These cells were then transferred to BHI broth at various temperatures from 4 to 37 degrees C and the lag phase durations were determined by enumerating cells at appropriate time intervals. Long lag phases were observed for cells initially grown at high temperatures and transferred to low temperatures. In general, exponential growth cells had the shortest lag phases, stationary phase and starved cells had longer, frozen cells had slightly longer and desiccated cells had the longest lag phases. These data were from immediate temperature transitions. When a computer-controlled water bath linearly changed the temperature from 37 to 5 degrees C over a 3.0- or 6.0-h period, the cells had short lags and grew continuously with declining growth rates. Transitions of 0.75 or 1.0 h had 20-h lag phases, essentially that of immediate transitions. When the transition was 1.5 h, an intermediate pattern of less than 1 log of growth followed by no additional growth for 20 h occurred.     

Effects of carbon dioxide on bacterial growth parameters in milk as measured by conductivity

Inhibition of bacterial growth by dissolved carbon dioxide (CO2) has been well established in many foods including dairy foods. However, the effects of dissolved CO2 on specific growth parameters such as length of lag phase, time to maximum growth rate, and numbers of organisms at the stationary phase have not been quantified for organisms of concern in milk. The effect of dissolved CO2 concentrations of 0.6 to 61.4 mM on specific bacterial growth parameters in raw or single organism inoculated sterile milk was determined at 15 degrees C by conductance. Commingled raw or sterile milks were amended to a final concentration of 0.5 mg/ml each of urea and arginine HCl. Sterile milks were inoculated singly with one of six different microorganisms to a final concentration of approximately 10(2) to 10(3) cfu/ml; raw milk was adjusted to a final indigenous bacterial population of approximately 10(3) cfu/ml. Conductivity of the milk was recorded every 60 s over 4 to 5 d in a circulating apparatus at 15 degrees C. Conductivity values were fit to Gompertz equations and growth parameters calculated. Conductance correlated with plate counts and was satisfactory for monitoring microbial growth. Data fit the Gompertz equation with high correlation (R2 = 0.96 to 1.00). In all cases, dissolved CO2 significantly inhibited growth of raw milk bacteria, influencing lag, exponential, and stationary growth phases as well as all tested monocultures.     

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.     

Correlation between the change in the kinetics of the ribosomal RNA rrnB P2 promoter and the transition from lag to exponential phase with Pseudomonas fluorescens

Developing accurate mathematical models to describe the pre-exponential lag phase in food-borne pathogens presents a considerable challenge to food microbiologists. While the growth rate is influenced by current environmental conditions, the lag phase is affected in addition by the history of the inoculum. A deeper understanding of physiological changes taking place during the lag phase would improve accuracy of models, and in earlier studies a strain of Pseudomonas fluorescens containing the Tn7-luxCDABE gene cassette regulated by the rRNA promoter rrnB P2 was used to measure the influence of starvation, growth temperature and sub-lethal heating on promoter expression and subsequent growth. The present study expands the models developed earlier to include a model which describes the change from exponential to linear increase in promoter expression with time when the exponential phase of growth commences. A two-phase linear model with Poisson weighting was used to estimate the lag (LPDLin) and the rate (RLin) for this linear increase in bioluminescence. The Spearman rank correlation coefficient (r=0.830) between the LPDLin and the growth lag phase (LPDOD) was extremely significant (P相似文献   

Effect of starvation on expression of the ribosomal RNA rrnB P2 promoter during the lag phase of Pseudomonas fluorescens

Mathematical modelling of food-borne pathogen survival and growth is an important and expanding area of food microbiology. Effective models have been developed for growth rate as influenced by the environment; however, reliable models which describe the lag phase prior to exponential growth are more difficult to obtain. In order to improve our understanding of the physiological changes that take place in the microbial cell during this adaptation period, the effect of starvation on the expression of a gene for ribosomal RNA (rRNA) synthesis-an important step in preparing the cells for growth-was examined. A strain of Pseudomonas fluorescens containing the Tn7-luxCDABE gene cassette regulated by the rRNA promoter rrnB P(2) was used as a model system. Growth was measured as optical density at 600 nm (OD(600)), and fitting was achieved with a two-phase linear model to obtain the parameters growth rate (R(OD)) and lag phase duration (LPD(OD)). The increase in bioluminescence (measured as natural log [ln] relative light units per unit OD(600)) after inoculation of stationary phase cells into fresh tryptic soy broth (TSB) followed an exponential association model, with lag (LPD(Exp)) and rate (R(Exp)) parameters. Starvation of cells in either spent TSB or in MOPS buffer resulted in time-dependent linear increases in both lag parameters and, in the case of TSB, a decrease in the R(Exp) parameter. The results show that models can be developed for expression of genes during the lag phase, which will improve our ability to make accurate predictions of food-borne pathogen growth.     

A combined discrete-continuous model describing the lag phase of Listeria monocytogenes

Food microbiologists generally use continuous sigmoidal functions such as the empirical Gompertz equation to obtain the kinetic parameters specific growth rate (mu) and lag phase duration (lambda) from bacterial growth curves. This approach yields reliable information on mu; however, values for lambda are difficult to determine accurately due, in part, to our poor understanding of the physiological events taking place during adaptation of cells to new environments. Existing models also assume a homogeneous population of cells, thus there is a need to develop discrete event models which can account for the behavior of individual cells. Time to detection (t(d)) values were determined for Listeria monocytogenes using an automated turbidimetric instrument, and used to calculate mu. Mean individual cell lag times (tL) were calculated as the difference between the observed t(d) and the theoretical value estimated using mu. Variability in tL for individual cells in replicate wells was estimated using serial dilutions. A discrete stochastic model was applied to the individual cells, and combined with a deterministic population-level growth model. This discrete-continuous model incorporating tL and the variability in tL (expressed as standard deviation; S.D.(L)) predicted a reduced variability between wells with increased number of cells per well, in agreement with experimental findings. By combining the discrete adaptation step with a continuous growth function it was possible to generate a model which accurately described the transition from lag to exponential phase. This new model may serve as a useful tool for describing individual cell behavior, and thus increasing our knowledge of events occurring during the lag phase.     

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.     

冷鲜梅条肉中热杀索丝菌生长预测模型建立与检验

研究冷鲜梅条肉中热杀索丝菌在0℃、5℃、10℃、15℃、20℃不同温度下生长变化情况,利用Modified Gompertz模型建立热杀索丝菌一级生长预测模型(R2＞0.99);利用平方根模型描述温度与最大比生长速率和延滞期的关系,得到热杀索丝菌的生长预测二级模型,验证模型的数学参数准确因子Af、Bf在1左右.表明数学模型可用于预测0℃～20℃范围内热杀索丝菌的变化情况,为冷鲜肉的货架期预报提供了基础数据.     

2D motility tracking of Pseudomonas putida KT2440 in growth phases using video microscopy

Pseudomonas putida KT2440 is a gram negative motile soil bacterium important in bioremediation and biotechnology. Thus, it is important to understand its motility characteristics as individuals and in populations. Population characteristics were determined using a modified Gompertz model. Video microscopy and imaging software were utilized to analyze two dimensional (2D) bacteria movement tracks to quantify individual bacteria behavior. It was determined that inoculum density increased the lag time as seeding densities decreased, and that the maximum specific growth rate decreased as seeding densities increased. Average bacterial velocity remained relatively similar throughout the exponential growth phase (~20.9 μm/s), while maximum velocities peak early in the exponential growth phase at a velocity of 51.2 μm/s. P. putida KT2440 also favors smaller turn angles indicating that they often continue in the same direction after a change in flagella rotation throughout the exponential growth phase.     

冷鲜肉馅中热杀索丝菌预测模型的建立

通过建立数学模型,来快速准确预测和监控瘦肥比例为7:3的冷鲜猪肉馅中优势腐败菌生长情况,选取0、5、10、15、20℃五种不同的温度,建立和验证0～20℃条件下热杀索丝菌的生长预测模型。结果表明,利用Gompertz模型拟合0～20℃条件下热杀索丝菌的生长,判定系数R2均大于0.99,计算得到总的偏差因子和准确因子分别为1.16和0.88;利用平方根模型描述了温度与最大比生长速率和延滞期的关系,二者都呈现了良好的线性关系;由此建立的0～20℃条件下7:3冷鲜猪肉馅中热杀索丝菌的生长预测模型有比较好的预测效果。     

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。     

冷鲜猪排骨假单胞菌生长预测模型的建立与验证

为快速预测和监控冷鲜猪肉中微生物的生长,建立和验证冷鲜排骨中0℃～20℃温度条件下假单胞菌的生长预测模型.结果表明:Gompertz方程能很好地描述不同温度下假单胞菌的生长,得到的假单胞菌一级生长预测模型,且其偏差因子和准确因子都在1左右;利用平方根模型描述温度与最大比生长速率和延滞期的关系,且呈现良好的线性关系,R2分别为0.9934和0.9263,从而得到假单胞菌生长预测的二级模型.初步说明生长预测模型能有效地预测0℃～20℃冷鲜猪排骨中假单胞菌的生长.     

米粉生产过程中大米发酵液的菌相变化研究

为了探究米粉生产过程中大米发酵液的菌相变化规律,以不同发酵时间的大米发酵液为研究材料,测定pH值的变化;采用选择培养基进行细菌、乳酸菌、霉菌酵母、球菌及肠杆菌的菌落计数和形态观察,并运用修正的Gompertz模型对菌落总数和乳酸菌进行生长动力学模型的拟合。结果表明,大米发酵液主要含有乳酸菌、霉菌酵母、球菌,其中乳酸菌数量最高,达2.56×108 CFU/mL,采用修正的Gompertz模型能较好地拟合大米发酵液中菌落总数及乳酸菌的生长情况,获得最大比生长速率分别为0.657 d-1、1.418 d-1,延滞期（LPD）分别为0.023 d、0.224 d,稳定期最大菌落数对数值分别为8.40、8.41,且模型的判定系数R2均＞0.95。此结果为进一步研究米粉发酵剂、保证米粉发酵质量、维护食品安全提供依据。     

不同温度条件下三文鱼中荧光假单胞菌生长预测模型的建立

荧光假单胞菌SBW25(Pseudomonas fluorescence SBW25)作为三文鱼的特定腐败菌(specific spoilage organisms,SSOs),在三文鱼腐败过程中起着主导作用。实验选择以三文鱼的鱼肉和鱼汁为生长介质,采用修正的Gompertz方程和Belehradek方程拟合三文鱼特定腐败菌之一的荧光假单胞菌SBW25在不同温度条件下的生长动力学模型,同时探究鱼汁中蛋白酶活力对动力学参数的影响,并对模型的适用性进行评价。结果显示,修正的Gompertz方程所拟合出的各温度下货架期方程的决定系数(R~2)都达到0.999,适用于描述三文鱼鱼肉和鱼汁中微生物的生长曲线。随着温度升高,鱼肉和鱼汁中的荧光假单胞菌的最大比生长速率和延滞期出现上升和缩短。鱼汁中温度和最大比生长速率、延滞期的Belehradek平方根方程取得较高的决定系数,分别达到0.930 3和0.988 7,高于鱼肉中取得的。在鱼汁中蛋白酶活力和对应时期荧光假单胞菌的最大比生长速率出现相同变化趋势。基于Belehradek方程的鱼汁不同温度模型偏差度和准确度都更接近于1.00,说明鱼汁中SSOs的生长曲线能较好地反映出各温度条件下的三文鱼货架期。     

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。