Predictive model for the growth of spoilage bacteria on modified atmosphere packaged Atlantic salmon produced in Australia.

Most existing models for the spoilage of modified atmosphere packed Atlantic salmon are based on the growth of the spoilage organism Photobacterium phosphoreum. However, there is evidence that this organism is not the specific spoilage organism on salmon produced and packaged in Australia. We developed a predictive model for the growth of bacteria in Australian-produced Atlantic salmon stored under modified atmosphere conditions (30-98% carbon dioxide in nitrogen) at refrigeration temperatures (0-10 °C). As expected, both higher levels of carbon dioxide and lower temperatures decreased the observed growth rates of the total population. A Belehrádek-type model for growth rate fitted the data best with an acceptably low root mean square error. At low temperatures (∼0 °C) the growth rates in this study were similar to those predicted by other models but at higher temperatures (∼10 °C) the growth rates were significantly lower in the current study.

Comparative evaluation of a new lactation curve model for pasture-based Holstein-Friesian dairy cows.

Fourteen lactation models were fitted to average and individual cow lactation data from pasture-based dairy systems in the Australian states of Victoria and Tasmania. The models included a new "log-quadratic" model, and a major objective was to evaluate and compare the performance of this model with the other models. Nine empirical and 5 mechanistic models were first fitted to average test-day milk yield of Holstein-Friesian dairy cows using the nonlinear procedure in SAS. Two additional semiparametric models were fitted using a linear model in ASReml. To investigate the influence of days to first test-day and the number of test-days, 5 of the best-fitting models were then fitted to individual cow lactation data. Model goodness of fit was evaluated using criteria such as the residual mean square, the distribution of residuals, the correlation between actual and predicted values, and the Wald-Wolfowitz runs test. Goodness of fit was similar in all but one of the models in terms of fitting average lactation but they differed in their ability to predict individual lactations. In particular, the widely used incomplete gamma model most displayed this failing. The new log-quadratic model was robust in fitting average and individual lactations, and was less affected by sampled data and more parsimonious in having only 3 parameters, each of which lends itself to biological interpretation.

Milk composition and growth in wild and captive Tasmanian bettongs, Bettongia gaimardi (Marsupialia).

Changes in milk composition (total solids, carbohydrate, protein, lipid and calculated gross energy content) during lactation in three groups of Tasmanian bettongs ( Bettongia gaimardi): free-living animals (wild group), captive animals offered a diet of dry dog food and apples ad libitum (ad lib group), and captive animals fed restricted amounts of the same diet (maintenance group) were related to growth rates (measured as body mass and head length) of their young. There were no significant differences in the concentration of milk solids among the three groups, but the wild group had higher lipid concentrations and the maintenance group had higher carbohydrate but lower protein concentrations. For all three groups, milk total solids increased through lactation from ca. 25% to ca. 45% and carbohydrate concentrations decreased from ca. 18% to about 3%. Protein concentrations increased from ca. 5% to ca. 10% in the wild and ad lib groups, but only from 4% to 8% in the maintenance group. Lipid concentrations increased in the wild and ad lib groups from ca. 4% to ca. 18%, but in the maintenance group only to ca. 7%. Calculated gross energy content of milk increased through lactation in the wild and ad lib groups (from ca. 500 kJ.100 ml(-1) to ca. 1,000 kJ.100 ml(-1)), but there was no significant increase in the maintenance group. The volume of milk produced increased to a peak just prior to permanent pouch vacation by the young, when the gross energy output in milk was 120-150 kJ.3 h(-1) in the wild and ad lib groups. On a daily basis this is equivalent to the milk energy output of larger wallabies, and helps to explain the relatively high growth rates of young Tasmanian bettongs. There were significant differences in growth rates among the groups, with the heaviest young always in the ad lib group. Thus differences in milk composition resulting from different planes of nutrition can lead to differences in growth rates of marsupial young.

Modelling the effects of temperature, water activity, pH and lactic acid concentration on the growth rate of Escherichia coli.

An extended square root-type model describing Escherichia coli growth rate was developed as a function of temperature (7.63-47.43 degrees C), water activity (0.951-0.999, adjusted with NaCl), pH (4.02-8.28) and lactic acid concentration (0-500 mM). The new model, based on 236 growth rate data, combines and extends previously published square root-type models and incorporates terms for upper and lower limiting temperatures, upper and lower limiting pH, minimum inhibitory concentrations of dissociated and undissociated lactic acid and lower limiting water activity. A term to describe upper limiting water activity was developed but could not be fitted to the E. coli data set because of the difficulty of generating data in the super-optimal water activity range (i.e. >0.998). All data used to generate the model are presented. The model provides an excellent description of the experimental data.

Some examples of, and some problems with, the use of nonlinear logistic regression in predictive food microbiology.

A new technique, nonlinear logistic regression, is described for modelling binomially distributed data, i.e., presence/absence data where growth is either observed or not observed, for applications in predictive food microbiology. Some examples of the successful use of this technique are presented, where the controlling factors are temperature, water activity, pH and the concentration of lactic acid, a weakly dissociating organic acid. Generally speaking, good-fitting models were obtained, as evidenced using various performance measures and goodness-of-fit statistics. As may be expected with a new statistical technique, some problems were encountered with the implementation of the modelling approach and these are discussed.

Predictive microbiology: towards the interface and beyond.

This review considers the concept and history of predictive microbiology and explores aspects of the modelling process including kinetic and probability modelling approaches. The "journey" traces the route from reproducible responses observed under close to optimal conditions for growth, through recognition and description of the increased variability in responses as conditions become progressively less favourable for growth, to defining combinations of factors at which growth ceases (the growth/no growth interface). Death kinetics patterns are presented which form a basis on which to begin the development of nonthermal death models. This will require incorporation of phenotypic, adaptive responses and may be influenced by factors such as the sequence in which environmental constraints are applied. A recurrent theme is that probability (stochastic) approaches are required to complement or replace kinetic models as the growth/no growth interface is approached and microorganisms adopt a survival rather than growth mode. Attention is also drawn to the interfaces of predictive microbiology with microbial physiology, information technology and food safety initiatives such as HACCP and risk assessment.

Modelling the combined temperature and salt (NaCl) limits for growth of a pathogenic Escherichia coli strain using nonlinear logistic regression.

A broth-based method is used to determine if exponential phase Escherichia coli R31, an STEC, is able to grow within 50 days under various combinations of sub-optimal temperatures and salt concentrations. From these data, the growth limits for combinations of temperature (7.7-37.0 degrees C) and water activity (0.943-0.987; NaCl as humectant) are defined and modelled using a nonlinear logistic regression model. That form of model is able to predict the combinations of salt concentration/water activity and temperature that will prevent the growth of E. coli R31 with selected levels of confidence. The model fitted the data with an approximate concordance rate of 97.3%. The minimum water activity that permitted growth occurred in the range 25-30 degrees C, the temperature range which optimises cell yield. At temperatures below this range the minimum water activity which allowed growth increased with decreasing temperature.

Growth limits of Listeria monocytogenes as a function of temperature, pH, NaCl, and lactic acid.

Models describing the limits of growth of pathogens under multiple constraints will aid management of the safety of foods which are sporadically contaminated with pathogens and for which subsequent growth of the pathogen would significantly increase the risk of food-borne illness. We modeled the effects of temperature, water activity, pH, and lactic acid levels on the growth of two strains of Listeria monocytogenes in tryptone soya yeast extract broth. The results could be divided unambiguously into "growth is possible" or "growth is not possible" classes. We observed minor differences in growth characteristics of the two L. monocytogenes strains. The data follow a binomial probability distribution and may be modeled using logistic regression. The model used is derived from a growth rate model in a manner similar to that described in a previously published work (K. A. Presser, T. Ross, and D. A. Ratkowsky, Appl. Environ. Microbiol. 64:1773-1779, 1998). We used "nonlinear logistic regression" to estimate the model parameters and developed a relatively simple model that describes our experimental data well. The fitted equations also described well the growth limits of all strains of L. monocytogenes reported in the literature, except at temperatures beyond the limits of the experimental data used to develop the model (3 to 35 degrees C). The models developed will improve the rigor of microbial food safety risk assessment and provide quantitative data in a concise form for the development of safer food products and processes.

Modelling the growth limits (growth/no growth interface) of Escherichia coli as a function of temperature, pH, lactic acid concentration, and water activity.

The form of a previously developed Belehrádek type of growth rate model was used to develop a probability model for defining the growth/no growth interface as a function of temperature (10 to 37 degrees C), pH (pH 2.8 to 6.9), lactic acid concentration (0 to 500 mM), and water activity (0.955 to 0.999; NaCl was used as the humectant). Escherichia coli was unable to grow in broth in which the undissociated lactic acid concentration exceeded 11 mM or, with two exceptions, at a pH of 3.9 or less with no lactic acid present. Under experimental conditions at which the pH and the undissociated acid concentrations were the major growth-limiting factors, the growth/no growth interface was essentially independent of temperature at temperatures ranging from 15 to 37 degrees C. The interface between conditions that allowed growth and conditions at which growth did not occur was abrupt. The inhibitory effect of combinations of water activity and pH varied with temperature. Predictions of the model for the growth/no growth interface were consistent with 95% of the experimental data set.

Quantitative microbiology: a basis for food safety.

Because microorganisms are easily dispersed, display physiologic diversity, and tolerate extreme conditions, they are ubiquitous and may contaminate and grow in many food products. The behavior of microbial populations in foods (growth, survival, or death) is determined by the properties of the food (e.g., water activity and pH) and the storage conditions (e.g., temperature, relative humidity, and atmosphere). The effect of these properties can be predicted by mathematical models derived from quantitative studies on microbial populations. Temperature abuse is a major factor contributing to foodborne disease; monitoring temperature history during food processing, distribution, and storage is a simple, effective means to reduce the incidence of food poisoning. Interpretation of temperature profiles by computer programs based on predictive models allows informed decisions on the shelf life and safety of foods. In- or on-package temperature indicators require further development to accurately predict microbial behavior. We suggest a basis for a "universal" temperature indicator. This article emphasizes the need to combine kinetic and probability approaches to modeling and suggests a method to define the bacterial growth/no growth interface. Advances in controlling foodborne pathogens depend on understanding the pathogens' physiologic responses to growth constraints, including constraints conferring increased survival capacity.

Modelling the growth rate of Escherichia coli as a function of pH and lactic acid concentration.

The growth rate responses of Escherichia coli M23 (a nonpathogenic strain) to suboptimal pH and lactic acid concentration were determined. Growth rates were measured turbidimetrically at 20 degrees C in the range of pH 2.71 to 8.45. The total concentration of lactic acid was fixed at specific values, and the pH was varied by the addition of a strong acid (hydrochloric) or base (sodium hydroxide) to enable the determination of undissociated and dissociated lactic acid concentrations under each condition. In the absence of lactic acid, E. coli grew at pH 4.0 but not at pH 3.7 and was unable to grow in the presence of > or = 8.32 mM undissociated lactic acid. Growth rate was linearly related to hydrogen ion concentration in the absence of lactic acid. In the range 0 to 100 mM lactic acid, growth rate was also linearly related to undissociated lactic acid concentration. A mathematical model to describe these observations was developed based on a Belehrádek-like model for the effects of water activity and temperature. This model was expanded to describe the effects of pH and lactic acid by the inclusion of novel terms for the inhibition due to the presence of hydrogen ions, undissociated lactic acid, and dissociated lactic acid species. Preliminary data obtained for 200 and 500 mM total lactic acid concentrations show that the response to very high lactic acid concentrations was less well described by the model. However, for 0 to 100 mM lactic acid, the model described well the qualitative and quantitative features of the response.

Effects of postmilking teat treatment on the colonization of Staphylococcus aureus on chapped teat skin.

Sixteen Holstein cows were used to test the effect of postmilking teat treatment on colonization and intramammary infection by Staphylococcus aureus on chapped teats. Treatments were (1) chapping the teat and using 1% I2/10% glycerin postdip solution, (2) 1% I2/10% glycerin postdip solution on nonchapped teats, (3) chapping the teat and using 10% glycerin postdip solution, (4) chapping the teat and not using a postdip solution. All mammary glands were free of S aureus teat skin colonization and intramammary infection at the start of the study. Teats selected for chapping were dipped in 1N NaOH prior to 3 applications of S aureus broth culture; cultures were applied at 12-hour intervals on all teats. Treatments were applied after each milking for 30 days and were initiated after the second broth dip. Teat skin swab specimens and milk samples were collected before treatment application. Teat skin condition was scored daily. Nonchapped teats (treatment 2) did not support skin or orifice colonization by S aureus. Treatment-1 teats healed most rapidly and supported less colonization in skin and orifice than did treatment-3 and -4 teats. Teat skin scores and skin colonization were lower for treatment-3 than treatment-4 teats. A correlation between teat skin colonization and teat skin conditions was found. Two intramammary infections were found in treatment-4 quarters and 1 in a treatment-3 quarter.(ABSTRACT TRUNCATED AT 250 WORDS)

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 Belehrádek 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 Belehrádek-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.

A suitable parameterization of the Michaelis-Menten enzyme reaction.

It is shown here that a suitable form for estimation and inference using the Michaelis-Menten [(1913) Biochem Z. 49, 333-369] model for simple enzymic reactions is one in which the two parameters appear in the denominator of the equation. In this form, convergence to the least-squares estimates using the Gauss-Newton method [see Kennedy & Gentle (1980) Statistical Computing, Marcel Dekker, New York] is virtually ensured, or, as the model in this form is a member of the class of 'generalized linear models', it may be fitted by packages such as those of Rothamsted Experimental Station [(1977) GENSTAT (A General Statistical Program), Rothamsted Experimental Station, Harpenden] and the Numerical Algorithms Group [(1978) GLIM (Generalised Linear Interactive Modeling), Numerical Algorithms Group, Oxford]. Furthermore, the parameters-in-denominator principle is readily extended to more complicated catalytic models. With all parameters in the denominator, the least-squares estimators are close to being unbiased and normally distributed, whereas severe bias and non-normality may result from use of the standard formulations.

Choosing near-linear parameters in the four-parameter logistic model for radioligand and related assays.

Ten parameters extracted from six currently used parametrizations of the four-parameter logistic model, and one new proposal, were examined for their statistical behavior in nonlinear least-squares estimation in combination with ELISA and RIA data. Those which are adequately near-linear on the basis of the Lowry-Morton lambda statistic were identified and can be recommended for use in practice.

Longitudinal growth study of Tasmanian children, 1967-1983. The 13-, 14- and 15-year-old girls.

This study updates the heights and weights of sexually mature and immature girls aged 13, 14 and 15 years in the Tasmanian Longitudinal Growth Study. Problems arose when their heights and weights were plotted on the NHMRC standard distributions (1970-1972 NSW percentiles) in order to assess whether or not they were normal. The current NHMRC standards are not adequate for adolescent girls and should be replaced by separate charts for mature and immature girls, respectively.

Breastfeeding and oral contraceptives: Tasmanian survey.

A survey of 772 Tasmanian mothers was carried out to determine whether there was any connection between the mother's use of oral contraceptives and her ability to breastfeed her child. It was found that 80% of the 243 mothers combining breastfeeding with the use of oral contraceptives were able to breastfeed their children for at least 3 months. Of these, all but 15 mothers used pills containing only progestagen. It is concluded that the progestagen-only oral contraceptive is compatible with lactation and that a mother using this type of pill will probably be able to breastfeed her baby.

Model for bacterial culture growth rate throughout the entire biokinetic temperature range.

The "square-root" relationship proposed by Ratkowsky et al. (J. Bacteriol. 149:1-5, 1982) for modeling the growth rate of bacteria below the optimum growth temperature was extended to cover the full biokinetic temperature range. Two of the four parameters of this new nonlinear regression model represent minimum and maximum temperature bounds, respectively, for the predicted growth of the culture. The new model is easy to fit and has other desirable statistical properties. For example, the least-squares estimators of the parameters of the model were almost unbiased and normally distributed. The model applied without exception to all bacterial cultures for which we were able to obtain data. Results for 30 strains are reported.

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.