On the design of optimal dynamic experiments for parameter estimation of a Ratkowsky-type growth kinetics at suboptimal temperatures

It is generally known that accurate model building, i.e., proper model structure selection and reliable parameter estimation, constitutes an essential matter in the field of predictive microbiology, in particular, when integrating these predictive models in food safety systems. In this context, Versyck et al. (1999) have introduced the methodology of optimal experimental design techniques for parameter estimation within the field. Optimal experimental design focuses on the development of optimal input profiles such that the resulting rich (i.e., highly informative) experimental data enable unique model parameter estimation. As a case study, Versyck et al. (1999) [Versyck, K., Bernaerts, K., Geeraerd, A.H., Van Impe, J.F., 1999. Introducing optimal experimental design in predictive modeling: a motivating example. Int. J. Food Microbiol., 51(1), 39-51] have elaborated the estimation of Bigelow inactivation kinetics parameters (in a numerical way). Opposed to the classic (static) experimental approach in predictive modelling, an optimal dynamic experimental setup is presented. In this paper, the methodology of optimal experimental design or parameter estimation is applied to obtain uncorrelated estimates of the square root model parameters [Ratkowsky, D.A., Olley, J., McMeekin, T.A., Ball, A., 1982. Relationship between temperature and growth rate of bacterial cultures. J. Bacteriol. 149, 1-5] describing the effect of suboptimal growth temperatures on the maximum specific growth rate of microorganisms. These estimates are the direct result of fitting a primary growth model to cell density measurements as a function of time. Apart from the design of an optimal time-varying temperature profile based on a sensitivity study of the model output, an important contribution of this publication is a first experimental validation of this innovative dynamic experimental approach for uncorrelated parameter identification. An optimal step temperature profile, within the range of model validity and practical feasibility, is developed for Escherichia coli K12 and successfully applied in practice. The presented experimental validation result illustrates the large potential of the dynamic experimental approach in the context of uncorrelated parameter estimation. Based on the experimental validation result, additional remarks are formulated related to future research in the field of optimal experimental design.     

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.     

Estimating the parameters of the Baranyi model for bacterial growth

The identifiability properties of the Baranyi model for bacterial growth were investigated, both structurally and applied to real-life data. Using the Taylor-series approach, it was formally proven that the model is structurally identifiable, i.e. it is now ascertained that it is certainly possible to give unique values to all parameters of the model, provided the bacterial growth data are of sufficiently good quality. The model also has acceptable practical identifiability properties in the presence of realistic data, which means that the confidence intervals on the parameter values are reasonable. However, there was a relatively high correlation between the maximum specific growth rate μ_maxand the suitability indicator h₀. An optimal experimental design to improve parameter estimation uncertainty was worked out, using the sampling times of the microbial growth curve as experimental degree of freedom. Using a D-optimal design criterion, it could be shown that the optimal sampling times were concentrated in four time periods (initial, start and end of exponential growth, end of experiment), each providing maximum information on a particular parameter. Because the optimal experimental design requires a priori estimates of the parameters, the propagation of the parameter uncertainty into the experimental design was assessed with a Monte Carlo simulation. In this way, 95% confidence intervals could be established around the optimal sampling times to be used in the optimal experiment. Based on these intervals, a design was proposed and experimentally validated. The error on the parameter estimates was more than halved, their correlation diminished and the nonlinearity of the result improved.     

Accurate estimation of cardinal growth temperatures of Escherichia coli from optimal dynamic experiments

Prediction of the microbial growth rate as a response to changing temperatures is an important aspect in the control of food safety and food spoilage. Accurate model predictions of the microbial evolution ask for correct model structures and reliable parameter values with good statistical quality. Given the widely accepted validity of the Cardinal Temperature Model with Inflection (CTMI) [Rosso, L., Lobry, J. R., Bajard, S. and Flandrois, J. P., 1995. Convenient model to describe the combined effects of temperature and pH on microbial growth, Applied and Environmental Microbiology, 61: 610-616], this paper focuses on the accurate estimation of its four parameters (T(min), T(opt), T(max) and micro(opt)) by applying the technique of optimal experiment design for parameter estimation (OED/PE). This secondary model describes the influence of temperature on the microbial specific growth rate from the minimum to the maximum temperature for growth. Dynamic temperature profiles are optimized within two temperature regions ([15 degrees C, 43 degrees C] and [15 degrees C, 45 degrees C]), focusing on the minimization of the parameter estimation (co)variance (D-optimal design). The optimal temperature profiles are implemented in a computer controlled bioreactor, and the CTMI parameters are identified from the resulting experimental data. Approximately equal CTMI parameter values were derived irrespective of the temperature region, except for T(max). The latter could only be estimated accurately from the optimal experiments within [15 degrees C, 45 degrees C]. This observation underlines the importance of selecting the upper temperature constraint for OED/PE as close as possible to the true T(max). Cardinal temperature estimates resulting from designs within [15 degrees C, 45 degrees C] correspond with values found in literature, are characterized by a small uncertainty error and yield a good result during validation. As compared to estimates from non-optimized dynamic experiments, more reliable CTMI parameter values were obtained from the optimal experiments within [15 degrees C, 45 degrees C].     

Introducing optimal experimental design in predictive modeling: a motivating example

Predictive microbiology emerges more and more as a rational quantitative framework for predicting and understanding microbial evolution in food products. During the mathematical modeling of microbial growth and/or inactivation, great, but not always efficient, effort is spent on the determination of the model parameters from experimental data. In order to optimize experimental conditions with respect to parameter estimation, experimental design has been extensively studied since the 1980s in the field of bioreactor engineering. The so-called methodology of optimal experimental design established in this research area enabled the reliable estimation of model parameters from data collected in well-designed fed-batch reactor experiments. In this paper, we introduce the optimal experimental design methodology for parameter estimation in the field of predictive microbiology. This study points out that optimal design of dynamic input signals is necessary to maximize the information content contained within the resulting experimental data. It is shown that from few dynamic experiments, more pertinent information can be extracted than from the classical static experiments. By introducing optimal experimental design into the field of predictive microbiology, a new promising frame for maximization of the information content of experimental data with respect to parameter estimation is provided. As a case study, the design of an optimal temperature profile for estimation of the parameters D(ref) and z of an Arrhenius-type model for the maximum inactivation rate kmax as a function of the temperature, T, was considered. Microbial inactivation by heating is described using the model of Geeraerd et al. (1999). The need for dynamic temperature profiles in experiments aimed at the simultaneous estimation of the model parameters from measurements of the microbial population density is clearly illustrated by analytical elaboration of the mathematical expressions involved on the one hand, and by numerical simulations on the other.     

Practical application of dynamic temperature profiles to estimate the parameters of the square root model

Optimal experimental design for parameter estimation (OED/PE) is a promising method to improve parameter estimation accuracy and minimise experimental effort in the field of predictive microbiology. In this paper, the OED/PE methodology was applied on two practical examples: the growth of Bacillus cereus and Enterobacter cloacae in liquid whole egg product. Both strains were recovered from samples of a commercial product. The goal of the modelling exercise was to quantify the influence of temperature on bacterial growth. The Baranyi-model for bacterial growth combined with the Ratkowsky square root model to describe temperature dependence was used. Using this model, a temperature step profile was calculated based on the optimal D-criterion. The model was then fitted against the experimental bacterial growth curve measured under the dynamic temperature conditions. This process was repeated until the parameters could be estimated with sufficient accuracy, apparent by the model prediction errors. For B. cereus, prior information could be extracted from the literature, allowing calculating a dynamic temperature profile directly. Two-step profiles were sufficient to obtain a good estimation for the model parameters. No prior information could be found for E. cloacae. Therefore, a limited series of static experiments had to be conducted to obtain usable prior model parameters estimates. Only one dynamic experiment was then needed to achieve a good estimation.     

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.     

Optimal experiment design for cardinal values estimation: guidelines for data collection

Optimal experiment design for parameter estimation (OED/PE) is an interesting technique for modelling practices when aiming for maximum parameter estimation accuracy. Nowadays, experimental designs for secondary modelling within the field of predictive microbiology are mostly arbitrary or based on factorial design. The latter type of design is common practice in response surface modelling approaches. A number of levels of the factor(s) under study are selected and all possible treatment combinations are performed. It is however not always clear which levels and treatment combinations are most relevant. An answer to this question can be obtained from optimal experiment design for—in this particular case—parameter estimation. This technique is based on the extremisation of a scalar function of the Fisher information matrix. The type of scalar function determines the final focus of the optimised design.

In this paper, optimal experiment designs are computed for the cardinal temperature model with inflection point (CTMI) and the cardinal pH model (CPM). A model output sensitivity analysis (depicting the sensitivity of the model output to a small change in the model parameters) yields a first indication of relevant temperature or pH treatments. Performed designs are: D-optimal design aiming for a maximum global parameter estimation accuracy (by minimising the determinant of the Fisher information matrix), and E-optimal design improving the confidence in the most uncertain model parameter (by maximising the smallest eigenvalue of the Fisher information matrix). Although lowering the information content of a set of experiments, boundary values on the design region need to be imposed during optimisation to exclude unworkable experiments and partly account for incorrect nominal parameter values.

Opposed to the frequently applied equidistant or arbitrary treatment placement, optimal design results show that typically four informative temperature or pH levels are selected and replicate experiments are to be performed at these points. Informative experiments are typically placed at points with an extreme model output sensitivity.     

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.     

Growth of Clostridium perfringens from spore inocula in cooked cured beef: development of a predictive model

The objective of this study was to develop a model to predict the growth of C. perfringens from spores at temperatures applicable to the cooling of cooked cured meat products. C. perfringens growth from spores was not observed at a temperature of 12 °C for up to 3 weeks. The two parameters: germination, outgrowth, and lag (GOL) time and exponential growth rate, EGR, were determined using a function derived from mechanistic and stochastic considerations and the observed relative growths at specified times. A general model to predict the amount of relative growth for arbitrary temperature was determined by fitting the exponential growth rates to a square root Ratkowsky function, and assuming a constant ratio of GOL and generation times. The predicted relative growth is sensitive to the value of this ratio. A closed form equation was developed that can be used to estimate the relative growth for a general cooling scenario and determine a standard error of the estimate. The equation depends upon microbiological assumptions of the effect of history of the GOL times for gradual changes in temperature. Applying multivariate statistical procedures, a confidence interval was computed on the prediction of the amount of growth for a given temperature. The model predicts, for example, a relative growth of 3.17 with an upper 95% confidence limit of 8.50 when cooling the product from 51 to 11 °C in 8 h, assuming a log linear decline in temperature with time.     

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.     

Predictive model for growth of Clostridium perfringens during cooling of cooked cured chicken

Estimates of the growth kinetics of Clostridium perfringens from spores at temperatures applicable to the cooling of cooked cured chicken products are presented. A model for predicting relative growth of C. perfringens from spores during cooling of cured chicken is derived using a nonlinear mixed effects analysis of the data. This statistical procedure has not been used in the predictive microbiology literature that has been written for microbiologists. However, recently software systems have been including this statistical procedure. The primary growth curves, based on the stages of cell development, identify two parameters: (1) germination, outgrowth, and lag (GOL) time, or lag phase time; and (2) exponential growth rate, egr. The mixed effects model does not consider GOL and egr as constants, but as random variables that would, in all likelihood, differ for different cooling events with the same temperature. As such, it is estimated that the egr, for a given temperature, has a CV of approximately 19%. The model obtained by the mixed effects model is compared to the one obtained by the more traditional two-stage approach. The estimated parameters from the derived models are virtually the same. The model predicts, for example, a geometric mean relative growth of about 9·4 with an upper 95% confidence limit of 21·3 when cooling the product from 51°C to 12°C in 8 h, assuming log linear decline in temperature with time. C. perfringens growth from spores was not observed at a temperature of 12°C for up to 3 weeks.     

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.     

Quantifying microbial lag phenomena due to a sudden rise in temperature: a systematic macroscopic study

The microbial lag phase is a complex and yet not completely understood phenomenon. Many studies on the microbial lag phase have been published but few report a systematic study; moreover, previous lag studies have involved the effect of multiple confounded factors. Here, the effect of sudden temperature rises on an exponentially growing Escherichia coli culture is systematically investigated. Experiments are performed in a computer-controlled bioreactor where E. coli K12 MG1655 is grown under aerobic conditions in Brain Heart Infusion (BHI) broth. This experimental set-up is used to characterise the effect of (i) the amplitude of the temperature shift, (ii) the pre-shift temperature level and (iii) the post-shift temperature level on the occurrence and length of a lag phase. Besides temperature, no other environmental changes take place at the moment of the temperature shift. To quantify the length of the induced lag phase, the experimental data are described with a common growth model.

Depending on the three factors tested, a lag phase of more or less 1 h is induced or not. This lag/no lag behaviour can largely be explained by the existence of a normal physiological temperature range but also the amplitude of the temperature rise plays a role. It can be concluded that for the microorganism under study the lower boundary of the normal range lies approximately between 22.78 and 23.86 °C. It is shown that this boundary is no cut-off point, but rather a transition zone. Even more, repeated experiments at this boundary have yielded different results (lag or no lag). This observation points out that the mechanism triggering this lag phase is not absolute but may be subject to biological variability.     

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   

Dynamic predictive model for growth of Salmonella enteritidis in egg yolk

ABSTRACT:  Salmonella Enteritidis (SE) contamination of poultry eggs is a major human health concern worldwide. The risk of SE from shell eggs can be significantly reduced through rapid cooling of eggs after they are laid and their storage under safe temperature conditions. Predictive models for the growth of SE in egg yolk under varying ambient temperature conditions (dynamic) were developed. The growth of SE in egg yolk under several isothermal conditions (10, 15, 20, 25, 30, 35, 37, 39, 41, and 43 °C) was determined. The Baranyi model, a primary model, was fitted with growth data for each temperature and corresponding maximum specific growth rates were estimated. Root mean squared error (RMSE) values were less than 0.44 log₁₀ CFU/g and pseudo- R ² values were greater than 0.98 for the primary model fitting. For developing the secondary model, the estimated maximum specific growth rates were then modeled as a function of temperature using the modified Ratkowsky's equation. The RMSE and pseudo- R ² were 0.05/h and 0.99, respectively. A dynamic model was developed by integrating the primary and secondary models and solving it numerically using the 4th-order Runge–Kutta method to predict the growth of SE in egg yolk under varying temperature conditions. The integrated dynamic model was then validated with 4 temperature profiles (varying) such as linear heating, exponential heating, exponential cooling, and sinusoidal temperatures. The predicted values agreed well with the observed growth data with RMSE values less than 0.29 log₁₀ CFU/g. The developed dynamic model can predict the growth SE in egg yolk under varying temperature profiles.     

A new logistic model for bacterial growth

A new logistic model for bacterial growth was developed in this study. The model is based on a logistic model, which is often applied for biological and ecological population kinetics. The new model is described by a differential equation and contains an additional term for suppression of the growth rate during the lag phase, compared with the original logistic equation. The new model successfully described sigmoidal growth curves of Escherichia coli and Salmonella under various initial conditions. Data for E. coli were obtained from our experiments and data for Salmonella from the literature. When the new model was compared with a modified Gompertz model, which is widely used by many predictive microbiology researchers, it proved to be superior to the Gompertz model. Further, Salmonella growth at varying temperature could be well simulated by the new model. These results indicate that the new model will be a useful tool to predict bacterial growth under various temperature profiles.     

The mathematical properties of the quasi-chemical model for microorganism growth-death kinetics in foods

Knowledge of the mathematical properties of the quasi-chemical model [Taub, Feeherry, Ross, Kustin, Doona, 2003. A quasi-chemical kinetics model for the growth and death of Staphylococcus aureus in intermediate moisture bread. J. Food Sci. 68 (8), 2530-2537], which is used to characterize and predict microbial growth-death kinetics in foods, is important for its applications in predictive microbiology. The model consists of a system of four ordinary differential equations (ODEs), which govern the temporal dependence of the bacterial life cycle (the lag, exponential growth, stationary, and death phases, respectively). The ODE system derives from a hypothetical four-step reaction scheme that postulates the activity of a critical intermediate as an antagonist to growth (perhaps through a quorum sensing biomechanism). The general behavior of the solutions to the ODEs is illustrated by several examples. In instances when explicit mathematical solutions to these ODEs are not obtainable, mathematical approximations are used to find solutions that are helpful in evaluating growth in the early stages and again near the end of the process. Useful solutions for the ODE system are also obtained in the case where the rate of antagonist formation is small. The examples and the approximate solutions provide guidance in the parameter estimation that must be done when fitting the model to data. The general behavior of the solutions is illustrated by examples, and the MATLAB programs with worked examples are included in the appendices for use by predictive microbiologists for data collected independently.     

Development and validation of a dynamic growth model for Listeria monocytogenes in fluid whole milk

Listeria monocytogenes, a psychrotrophic microorganism, has been the cause of several food-borne illness outbreaks, including those traced back to pasteurized fluid milk and milk products. This microorganism is especially important because it can grow at storage temperatures recommended for milk (< or =7 degrees C). Growth of L. monocytogenes in fluid milk depends to a large extent on the varying temperatures it is exposed to in the postpasteurization phase, i.e., during in-plant storage, transportation, and storage at retail stores. Growth data for L. monocytogenes in sterilized whole milk were collected at 4, 6, 8, 10, 15, 20, 25, 30, and 35 degrees C. Specific growth rate and maximum population density were calculated at each temperature using these data. The data for growth rates versus temperature were fitted to the Zwietering square root model. This equation was used to develop a dynamic growth model (i.e., the Baranyi dynamic growth model or BDGM) for L. monocytogenes based on a system of equations which had an intrinsic parameter for simulating the lag phase. Results from validation of the BDGM for a rapidly fluctuating temperature profile showed that although the exponential growth phase of the culture under dynamic temperature conditions was modeled accurately, the lag phase duration was overestimated. For an alpha0 (initial physiological state parameter) value of 0.137, which corresponded to the mean temperature of 15 degrees C, the population densities were underpredicted, although the experimental data fell within the narrow band calculated for extreme values of alpha0. The maximum relative error between the experimental data and the curve based on an average alpha0 value was 10.42%, and the root mean square error was 0.28 log CFU/ml.     

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.