Maximum Likelihood based comparison of the specific growth rates for P. aeruginosa and four mutator strains (original) (raw)
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Modeling of the Bacterial Growth Curve
Applied and Environmental Microbiology, 1990
Several sigmoidal functions (logistic, Gompertz, Richards, Schnute, and Stannard) were compared to describe a bacterial growth curve. They were compared statistically by using the model of Schnute, which is a comprehensive model, encompassing all other models. The t test and the F test were used. With the t test, confidence intervals for parameters can be calculated and can be used to distinguish between models. In the F test, the lack of fit of the models is compared with the measuring error. Moreover, the models were compared with respect to their ease of use. All sigmoidal functions were modified so that they contained biologically relevant parameters. The models of Richards, Schnute, and Stannard appeared to be basically the same equation. In the cases tested, the modified Gompertz equation was statistically sufficient to describe the growth data of Lactobacillus plantarum and was easy to use.
Applied and Environmental Microbiology, 2010
Quantitative microbiological models predicting proliferation of microorganisms relevant for food safety and/or food stability are useful tools to limit the need for generation of biological data through challenge testing and shelf-life testing. The use of these models requires quick and reliable methods for the generation of growth data and estimation of growth parameters. Growth parameter estimation can be achieved using methods based on plate counting and methods based on measuring the optical density. This research compares the plate count method with two optical density methods, namely, the 2-fold dilution (2FD) method and the relative rate to detection (RRD) method. For model organism Bacillus cereus F4810/72, the plate count method and both optical density methods gave comparable estimates for key growth parameters. Values for the maximum specific growth rate ( max ) derived by the 2FD method and by the RRD method were of the same order of magnitude, but some marked differences between the two approaches were apparent. Whereas the 2FD method allowed the derivation of values for lag time () from the data, this was not possible with the RRD method. However, the RRD method gave many more data points per experiment and also gave more data points close to the growth boundary. This research shows that all three proposed methods can be used for parameter estimation but that the choice of method depends on the objectives of the research.
Statistical evaluation of mathematical models for microbial growth
2004
The aim of this study was to evaluate the suitability of several mathematical functions for describing microbial growth curves. The nonlinear functions used were: three-phase linear, logistic, Gompertz, Von Bertalanffy, Richards, Morgan, Weibull, France and Baranyi. Two data sets were used, one comprising 21 growth curves of different bacterial and fungal species in which growth was expressed as optical density units, and one comprising 34 curves of colony forming units counted on plates of Yersinia enterocolitica grown under different conditions of pH, temperature and CO 2 (time-constant conditions for each culture). For both sets, curves were selected to provide a wide variety of shapes with different growth rates and lag times. Statistical criteria used to evaluate model performance were analysis of residuals (residual distribution, bias factor and serial correlation) and goodness-of-fit (residual mean square, accuracy factor, extra residual variance F-test, and Akaike's information criterion). The models showing the best overall performance were the Baranyi, three-phase linear, Richards and Weibull models. The goodness-of-fit attained with other models can be considered acceptable, but not as good as that reached with the best four models. Overall, the Baranyi model showed the best behaviour for the growth curves studied according to a variety of criteria. The Richards model was the best-fitting optical density data, whereas the three-phase linear showed some limitations when fitting these curves, despite its consistent performance when fitting plate counts. Our results indicate that the common use of the Gompertz model to describe microbial growth should be reconsidered critically, as the Baranyi, three-phase linear, Richards and Weibull models showed a significantly superior ability to fit experimental data than the extensively used Gompertz.
A random effect multiplicative heteroscedastic model for bacterial growth
BMC Bioinformatics, 2010
Background: Predictive microbiology develops mathematical models that can predict the growth rate of a microorganism population under a set of environmental conditions. Many primary growth models have been proposed. However, when primary models are applied to bacterial growth curves, the biological variability is reduced to a single curve defined by some kinetic parameters (lag time and growth rate), and sometimes the models give poor fits in some regions of the curve. The development of a prediction band (from a set of bacterial growth curves) using non-parametric and bootstrap methods permits to overcome that problem and include the biological variability of the microorganism into the modelling process. Results: Absorbance data from Listeria monocytogenes cultured at 22, 26, 38, and 42°C were selected under different environmental conditions of pH (4.5, 5.5, 6.5, and 7.4) and percentage of NaCl (2.5, 3.5, 4.5, and 5.5). Transformation of absorbance data to viable count data was carried out. A random effect multiplicative heteroscedastic model was considered to explain the dynamics of bacterial growth. The concept of a prediction band for microbial growth is proposed. The bootstrap method was used to obtain resamples from this model. An iterative procedure is proposed to overcome the computer intensive task of calculating simultaneous prediction intervals, along time, for bacterial growth. The bands were narrower below the inflection point (0-8 h at 22°C, and 0-5.5 h at 42°C), and wider to the right of it (from 9 h onwards at 22°C, and from 7 h onwards at 42°C). A wider band was observed at 42°C than at 22°C when the curves reach their upper asymptote. Similar bands have been obtained for 26 and 38°C. Conclusions: The combination of nonparametric models and bootstrap techniques results in a good procedure to obtain reliable prediction bands in this context. Moreover, the new iterative algorithm proposed in this paper allows one to achieve exactly the prefixed coverage probability for the prediction band. The microbial growth bands reflect the influence of the different environmental conditions on the microorganism behaviour, helping in the interpretation of the biological meaning of the growth curves obtained experimentally.
Modeling the Growth of Bacteria Streptococcus sobrinus Using Exponential Regression
Pesquisa Brasileira em Odontopediatria e Clínica Integrada
Objective: To build an exponential regression model based on parameter estimation. Material and Methods: We developed a simple mathematical model to simulate the growth of bacteria and the exponential growth is often used to model population growth as such cell growth while the exponential decay is portraying a declining or decreases in the size of the population. An exponential regression method was used to fit the data and estimate growth parameter values Streptococcus sobrinus using statistical software SPSS version 20. Results: Based on the results of the parameter estimates, which is constant are 83.039 and b1 is 0.005 while R-square is 0.952. According to the R-Square results obtained, the model is good and appropriate. Conclusion: The model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. This study helps researchers to understand the specific growth rate(s), which can be used to best grow the organism.
Accounting for inherent variability of growth in microbial risk assessment
International Journal of Food Microbiology, 2005
Risk assessments of pathogens need to account for the growth of small number of cells under varying conditions. In order to determine the possible risks that occur when there are small numbers of cells, stochastic models of growth are needed that would capture the distribution of the number of cells over replicate trials of the same scenario or environmental conditions. This paper provides a simple stochastic growth model, accounting only for inherent cell-growth variability, assuming constant growth kinetic parameters, for an initial, small, numbers of cells assumed to be transforming from a stationary to an exponential phase. Two, basic, microbial sets of assumptions are considered: serial, where it is assume that cells transform through a lag phase before entering the exponential phase of growth; and parallel, where it is assumed that lag and exponential phases develop in parallel. The model is based on, first determining the distribution of the time when growth commences, and then modelling the conditional distribution of the number of cells. For the latter distribution, it is found that a Weibull distribution provides a simple approximation to the conditional distribution of the relative growth, so that the model developed in this paper can be easily implemented in risk assessments using commercial software packages.
Food illness is a serious health threat and has significant economic consequences for people in both the developing and developed world. Salmonella genus is one of the most common pathogens and a major cause of foodborne illness in humans worldwide. Nowadays, the application of mathematical models and functions to describe the microorganism growth kinetics provides a new behavioral vision of the interaction between microorganisms and the environment. Lately the studies on the subject have been gathering interest in the elaboration and application of mathematical modeling and equations over the last years to be used in biotechnological and industrial process, therefore being a most useful tool, with the intent of reducing time and expenses associated with the conventional tests. The purpose of the present study was to compare the Baranyi and Roberts (1994) model with quadratic function generated from data experimentally obtained of the Salmonella typhimurium growth in vitro. It was o...
International Journal of Food Microbiology, 2011
Strain variability of the growth behavior of foodborne pathogens has been acknowledged as an important issue in food safety management. A stochastic model providing predictions of the maximum specific growth rate (μ max ) of Salmonella enterica as a function of pH and water activity (a w ) and integrating intra-species variability data was developed. For this purpose, growth kinetic data of 60 S. enterica isolates, generated during monitoring of growth in tryptone soy broth of different pH (4.0-7.0) and a w (0.964-0.992) values, were used. The effects of the environmental parameters on μ max were modeled for each tested S. enterica strain using cardinal type and gamma concept models for pH and a w , respectively. A multiplicative without interaction-type model, combining the models for pH and a w , was used to describe the combined effect of these two environmental parameters on μ max . The strain variability of the growth behavior of S. enterica was incorporated in the modeling procedure by using the cumulative probability distributions of the values of pH min , pH opt and a wmin as inputs to the growth model. The cumulative probability distribution of the observed μ max values corresponding to growth at pH 7.0-a w 0.992 was introduced in the place of the model's parameter μ opt . The introduction of the above distributions into the growth model resulted, using Monte Carlo simulation, in a stochastic model with its predictions being distributions of μ max values characterizing the strain variability. The developed model was further validated using independent growth kinetic data (μ max values) generated for the 60 strains of the pathogen at pH 5.0-a w 0.977, and exhibited a satisfactory performance. The mean, standard deviation, and the 5th and 95th percentiles of the predicted μ max distribution were 0.83, 0.08, and 0.69 and 0.96 h − 1 , respectively, while the corresponding values of the observed distribution were 0.73, 0.09, and 0.61 and 0.85 h − 1 . The stochastic modeling approach developed in this study can be useful in describing and integrating the strain variability of S. enterica growth kinetic behavior in quantitative microbiology and microbial risk assessment.
On parameter estimation of Monod's bacterial growth model from batch culture data
The Journal of General and Applied Microbiology, 1983
This is a report of a method to estimate the three parameters of Monod's bacterial growth model, , KS and Y, by linearization and integration of data from batch culture experiments. This method can be used with either substrate or biomass data. The technique is tested on three sets of Parameter Estimation of Monod's Bacterial Growth 93