Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Model for combined effect of temperature and salt concentration/water activity on the growth rate of Staphylococcus xylosus

The combined effect of temperature and NaCl concentration/water activity on the growth rate of a strain of halotolerant Staphylococcus is described by the square-root models which had been used previously to model temperature dependence only. The model square root r = b(T-T min) is shown to be a special case of the B?lehrádek temperature function which is given by r = a(T-alpha)d. The constant alpha is the socalled 'biological zero' and equivalent to T min in the square-root models. This and the exponent d = 2 were unaffected by changing NaCl concentration/water activity. The B?lehrádek-type equations are preferable to the Arrhenius equation in that their parameters do not change with temperature. The constancy of T min allows derivation of a simple expression relating growth rate of strain CM21/3 to temperature and salt concentration/water activity within the range of linear response to temperature predicted by the square-root model.     

Combined effect of temperature and salt concentration/water activity on the growth rate of Halobacterium spp.

Growth responses of the halophilic bacteria, Halobacterium sp. strain HB9 and Halobacterium salinarium strain CM42/12, to temperature and water activity/sodium chloride concentration were described by the square root model and T _min (the theoretical minimum temperature for growth) was fixed. Little change in growth rate was observed in response to added NaCl at water activities below which cell lysis was avoided. Hence, growth of halobacteria on products such as salted, dried fish at water activities below 0.85 may be based on the square root temperature response without the need to incorporate a water activity term.     

Evaluation of data transformations used with the square root and schoolfield models for predicting bacterial growth rate.

A comparison was made between mathematical variations of the square root and Schoolfield models for predicting growth rate as a function of temperature. The statistical consequences of square root and natural logarithm transformations of growth rate use in several variations of the Schoolfield and square root models were examined. Growth rate variances of Yersinia enterocolitica in brain heart infusion broth increased as a function of temperature. The ability of the two data transformations to correct for the heterogeneity of variance was evaluated. A natural logarithm transformation of growth rate was more effective than a square root transformation at correcting for the heterogeneity of variance. The square root model was more accurate than the Schoolfield model when both models used natural logarithm transformation.     

Evaluation of data transformations used with the square root and schoolfield models for predicting bacterial growth rate.

A comparison was made between mathematical variations of the square root and Schoolfield models for predicting growth rate as a function of temperature. The statistical consequences of square root and natural logarithm transformations of growth rate use in several variations of the Schoolfield and square root models were examined. Growth rate variances of Yersinia enterocolitica in brain heart infusion broth increased as a function of temperature. The ability of the two data transformations to correct for the heterogeneity of variance was evaluated. A natural logarithm transformation of growth rate was more effective than a square root transformation at correcting for the heterogeneity of variance. The square root model was more accurate than the Schoolfield model when both models used natural logarithm transformation.     

Modelling the growth response of Staphylococcus xylosus to changes in temperature and glycerol concentration/water activity

The growth response of Staphylococcus xylosus strain CM21/3 to changes in temperature and water activity (glycerol concentration) was similar to that observed when water activity was adjusted by added NaCl. At each water activity level the effect of temperature on bacterial growth rate was described well by the square root model. T _MIN (the notional minimum temperature for growth) was found to be constant and was similar to the value obtained for the same organism grown in media containing NaCl. Growth rate was proportional to glycerol concentration/water activity allowing the combined effect of this factor and temperature to be modelled by substitution of the constant b in the basic square root model by a term for water activity. The observed minimum water activity for growth at the optimum temperature was close to that predicted by the model.     

Modelling the growth response of Staphylococcus xylosus to changes in temperature and glycerol concentration/water activity

The growth response of Staphylococcus xylosus strain CM21/3 to changes in temperature and water activity (glycerol concentration) was similar to that observed when water activity was adjusted by added NaCl. At each water activity level the effect of temperature on bacterial growth rate was described well by the square root model. TMIN (the notional minimum temperature for growth) was found to be constant and was similar to the value obtained for the same organism grown in media containing NaCl. Growth rate was proportional to glycerol concentration/water activity allowing the combined effect of this factor and temperature to be modelled by substitution of the constant b in the basic square root model by a term for water activity. The observed minimum water activity for growth at the optimum temperature was close to that predicted by the model.     

A New Secondary Model Developed for the Growth Rate of <Emphasis Type="Italic">Escherichia coli</Emphasis> O157:H7 in Broth

This study was attempted to develop a new exponential sum model to describe the effect of temperature on growth rate (GR) of Escherichia coli O157:H7 in broth. The growth rates of E. coli O157:H7 at different storage temperatures (4, 10, 15, 20, 25, 30, and 35°C) estimated by fitting with the modified Gompertz model were used to develop secondary models such as square root model, Ratkowsky model and exponential sum model. Measures of coefficient of determination (R ²), root mean square error (RMSE) and the sum of squares due to error (SSE) were employed to compare the performances of these three secondary models. Based on these criteria, the developed exponential sum model showed the better goodness-of-fit and performance.     

Comparison of Arrhenius-type and Bêlehrádek-type models for prediction of bacterial growth in foods

D.A. RATKOWSKY, T. ROSS, T.A. WCMEEKIN AND J. OLLEY. 1991. The development of Arrhenius-type ('Schoolfield') and Bêlehrádek-type (square root) models that describe microbial growth rates is briefly described. Both types of model have been advocated for use in predictive microbiology. On the basis of published data sets for the growth of bacteria, the consequences of mathematical transformation of data and the use of invalid stochastic assumptions upon model predictions are demonstrated. Mean square error is shown to be an inappropriate criterion by which to compare the performance of predictive models. The data show that bacterial growth responses such as generation time and lag time become more variable as their mean magnitude increases. The practical consequences of such variability for predictive microbiology are discussed.     

副溶血性弧菌温度-盐度双因素预测模型的建立

本文以副溶血性弧菌VP BJ1.1997为研究对象, 采用均匀设计试验方法, 建立并验证了温度范围为7°C~43°C, 盐度范围为0.5%~9.5%NaCl的生长动力学模型。结果表明, 所选一级模型的拟合效果优劣依次为Logistic方程>Gompertz方程>Linear方程, 以Logistic方程为一级模型计算生长参数; 二级模型采用平方根模型进行拟合, 得到模型相关系数r为0.9863, 最低生长温度T min为9.0506°C, 最高生长盐度为5.93%NaCl(对应最低生长水分活度Aw min     

Relationship between temperature and growth rate of bacterial cultures.

The Arrhenius Law, which was originally proposed to describe the temperature dependence of the specific reaction rate constant in chemical reactions, does not adequately describe the effect of temperature on bacterial growth. Microbiologists have attempted to apply a modified version of this law to bacterial growth by replacing the reaction rate constant by the growth rate constant, but the modified law relationship fits data poorly, as graphs of the logarithm of the growth rate constant against reciprocal absolute temperature result in curves rather than straight lines. Instead, a linear relationship between in square root of growth rate constant (r) and temperature (T), namely, square root = b (T - T0), where b is the regression coefficient and T0 is a hypothetical temperature which is an intrinsic property of the organism, is proposed and found to apply to the growth of a wide range of bacteria. The relationship is also applicable to nucleotide breakdown and to the growth of yeast and molds.     

应用近红外光谱预测水稻叶片氮含量

以水稻(Oryza sativa)新鲜叶片和干叶粉末两种状态的样品为研究对象, 基于近红外光谱(NIRS)技术, 应用偏最小二乘法(PLS)、主成分回归(PCR)和逐步多元回归(SMLR), 建立并评价了水稻叶片氮含量(NC)近红外光谱模型。结果表明, 基于PLS建立的模型表现最好, 鲜叶氮含量近红外光谱校正模型校正决定系数R_C²为0.940, 校正标准误差RMSEC为0.226; 干叶粉末氮含量的近红外光谱校正模型R_C²为0.977, RMSEC为0.136。模型的内部交叉验证分析表明, 预测鲜叶氮含量内部验证决定系数R_CV²为0.866, 内部验证标准误差RMSECV为0.243; 预测干叶粉末氮含量R_CV²为0.900, RMSECV为0.202。模型的外部验证分析表明, 预测水稻鲜叶氮含量的外部验证决定系数R_V²大于0.800, 外部验证标准误差RMSEP小于0.500, 预测干叶粉末氮含量的R_V²为0.944, RMSEP为0.142。说明, 近红外光谱分析技术与化学分析方法一致性较好, 且基于干叶粉末建立的近红外光谱预测模型的准确性和精确度较新鲜叶片高。     

A visco-hyperelastic-damage constitutive model for the analysis of the biomechanical response of the periodontal ligament

The periodontal ligament (PDL), as other soft biological tissues, shows a strongly non-linear and time-dependent mechanical response and can undergo large strains under physiological loads. Therefore, the characterization of the mechanical behavior of soft tissues entails the definition of constitutive models capable of accounting for geometric and material non-linearity. The microstructural arrangement determines specific anisotropic properties. A hyperelastic anisotropic formulation is adopted as the basis for the development of constitutive models for the PDL and properly arranged for investigating the viscous and damage phenomena as well to interpret significant aspects pertaining to ordinary and degenerative conditions. Visco-hyperelastic models are used to analyze the time-dependent mechanical response, while elasto-damage models account for the stiffness and strength decrease that can develop under significant loading or degenerative conditions. Experimental testing points out that damage response is affected by the strain rate associated with loading, showing a decrease in the damage limits as the strain rate increases. These phenomena can be investigated by means of a model capable of accounting for damage phenomena in relation to viscous effects. The visco-hyperelastic-damage model developed is defined on the basis of a Helmholtz free energy function depending on the strain-damage history. In particular, a specific damage criterion is formulated in order to evaluate the influence of the strain rate on damage. The model can be implemented in a general purpose finite element code. The accuracy of the formulation is evaluated by using results of experimental tests performed on animal model, accounting for different strain rates and for strain states capable of inducing damage phenomena. The comparison shows a good agreement between numerical results and experimental data.     

Effects of temperature on the life history parameters and population growth rates of Hyalopterus pruni (Hemiptera: Aphididae)

The mealy plum aphid, Hyalopterus pruni (Geoffroy) (Hemiptera: Aphididae) is a pest of prune trees in California. The impact of aphids as pests is well characterized by their population growth rate, a parameter integrating their age-specific development, survivorship, and fecundity. These population parameters were measured at five constants temperatures on potted prune trees. Development rates increased with temperature up to an optimum. The relationship between development rate and temperature was described by linear and nonlinear models. Developmental threshold temperature was greater for the nonlinear model than for the linear model. Thermal requirement for development and maximum lethal temperature determined by these models were similar to those for other aphids. The greatest proportional survivorship of nymphs occurred at 26 degrees C. Mean daily fecundity was lowest at 14 degrees C and highest at 22 degrees C. Adult longevity decreased with temperature. Population growth rates for H. pruni were estimated from measurements of fecundity and development time and were highest at 22 degrees C. This is the first study to document the temperature dependence of the life history parameters for H. pruni and the first to generate a degree-day model for the prediction of phenological events.     

Applicability of an Arrhenius model for the combined effect of temperature and CO(2) packaging on the spoilage microflora of fish

The temperature behavior of the natural microflora on the Mediterranean fish red mullet (Mullus barbatus) was examined as a case study. The growth of the spoilage bacteria Pseudomonas spp., Shewanella putrefaciens, Brochothrix thermosphacta, and lactic acid bacteria was modeled as a function of temperature and the concentration of carbon dioxide in modified atmosphere packaging. Combined models were developed and comparatively assessed based on polynomial, Belehradek, and Arrhenius equations. The activation energy parameter of the Arrhenius model, E(A), was independent of the packaging atmosphere and ranged from 75 to 85 kJ/mol for the different bacteria, whereas the preexponential constant decreased exponentially with the packaging CO(2) concentration. We evaluated the applicability of the models developed by using experimental bacterial growth rates obtained from 42 independent experiments performed with three Mediterranean fish species and growth rates predicted from the models under the same temperature and packaging conditions. The accuracy factor and bias factor were used as statistical tools for evaluation, and the developed Arrhenius model and the Belehradek model were judged satisfactory overall.     

Dynamic modeling of Listeria monocytogenes growth in pasteurized milk

AIMS: The development and validation of a dynamic model for predicting Listeria monocytogenes growth in pasteurized milk stored at both static and dynamic temperature conditions. METHODS AND RESULTS: Growth of inoculated L. monocytogenes in a commercial pasteurized whole milk product was monitored at various isothermal conditions from 1.5 to 16 degrees C. The kinetic parameters of the pathogen were modelled as a function of temperature using a square root type model, which was further validated using data from 92 published growth curves from eight different milk products. Compared to four published models for L. monocytogenes growth, the model developed in this study performed better, with a per cent discrepancy and bias of 49.1 and -1.01%, respectively. The performance of the model in predicting growth at dynamic temperature conditions was evaluated at four different fluctuating temperature scenarios with periodic temperature changes from -2 to 16 degrees C. The prediction of growth at dynamic storage temperature was based on the square root model in conjunction with the differential equations of the Baranyi and Roberts model, which were numerically integrated with respect to time. The per cent relative errors between the observed and the predicted growth of L. monocytogenes were less than 10% for all temperature scenarios tested. CONCLUSIONS: Available models from experiments conducted in laboratory media may result in significant overestimation of L. monocytogenes growth in pasteurized milk because they do not take into account factors such as milk composition (e.g. natural antimicrobial compounds present in milk) and the interactions of the pathogen with the natural microflora. The product-targeted model developed in the present study showed a high performance in predicting growth of L. monocytogenes in pasteurized milk under both static and dynamic temperature conditions. SIGNIFICANCE AND IMPACT OF THE STUDY: Temperature fluctuations often occur during the transportation and storage of pasteurized milk. A high performance, dynamic model for the growth of L. monocytogenes can be a useful tool for effective management and optimization of product safety and can lead to more realistic estimations of pasteurized-milk related safety risks.     

Effect of incubation temperature on growth parameters of Pseudoalteromonas antarctica NF and its production of extracellular polymeric substancesdagger

Aim: To evaluate the effect of temperature on growth parameters and on extracellular polymeric substance (EPS) production for Pseudoalteromonas antarctica NF₃. Methods and Results: For this purpose, three growth parameters, lag time (λ), maximum growth rate (μ) and maximum population density (A), were calculated with the predictive Gompertz model. To evaluate the variations in μ with respect to temperature, the secondary Arrhenius and the square root models were used. Below the optimal growth temperature (17·5°C), the growth of P. antarctica was separated into two domains at the critical temperature of 12°C. Within the suboptimal domain (12–17·5°C), the temperature characteristic was the lowest (5·29 kcal mol^?1). Growth population densities were maintained over the entire physiological portion assayed (5–17·5°C). Higher crude EPS production was found at temperatures included in the cold domain (5–12°C). Conclusions: All calculated parameters revealed an optimal adaptation of this strain to cold temperatures. Significance and Impact of the Study: The knowledge of the influence of temperature on growth parameters of P. antarctica NF₃ and on EPS production could improve the production of this extracellular polymeric substance that is currently being used in the cosmetic and pharmaceutical industries.     

Evaluation of Data Transformations and Validation of a Model for the Effect of Temperature on Bacterial Growth

The temperature of chilled foods is an important variable for controlling microbial growth in a production and distribution chain. Therefore, it is essential to model growth as a function of temperature in order to predict the number of organisms as a function of temperature and time. This article deals with the correct variance-stabilizing transformation of the growth parameters A (asymptotic level), μ (specific growth rate), and λ (lag time). This is of importance for the regression analysis of the data. A previously gathered data set and model for the effect of temperature on the growth of Lactobacillus plantarum (M. H. Zwietering, J. T. de Koos, B. E. Hasenack, J. C. de Wit, and K. van 't Riet, Appl. Environ. Microbiol. 57:1094-1101, 1991) is extended with new data. With the total data set (original and new data), a variance-stabilizing transformation is selected in order to determine which transformation should precede fitting. No transformation for the asymptote data, a square root for the growth rate, and a logarithmic transformation for the lag time were found to be appropriate. After these transformations, no significant correlation was found between the variance and the magnitude of the variable. Model corrections were made and model parameters were estimated by using the original data. With the new data, the models were validated by comparing the lack of fit of the models with the measurement error, using an F test. The predictions of the models for μ and λ were adequate. The model for A showed a systematic deviation, and therefore a new model for A is proposed.