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Effect of storage temperature on the lag time of Geobacillus stearothermophilus individual spores
Accepted Manuscript Effect of storage temperature on the lag time of Geobacillus stearothermophilus individual spores Myrsini Kakagianni, Juan S. Aguirre, Alexandra Lianou, Konstantinos P. Koutsoumanis PII:
S0740-0020(16)31084-X
DOI:
10.1016/j.fm.2017.04.009
Reference:
YFMIC 2775
To appear in:
Food Microbiology
Please cite this article as: Myrsini Kakagianni, Juan S. Aguirre, Alexandra Lianou, Konstantinos P. Koutsoumanis, Effect of storage temperature on the lag time of Geobacillus stearothermophilus individual spores, Food Microbiology (2017), doi: 10.1016/j.fm.2017.04.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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The lag times (λ) of Geobacillus stearothermophilus single spores showed a significant variability The storage temperature affected both the position and the spread of the λ distributions The minimum mean value of λ (i.e. 10.87 h) was observed at 55°C Moving away from 55°C resulted in an increase for both the mean and standard deviation of λ A probabilistic growth model for individual spores was developed.
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Effect of storage temperature on the lag time of Geobacillus stearothermophilus
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individual spores
3 4 Myrsini Kakagianni1, Juan S. Aguirre2, Alexandra Lianou3 Konstantinos P.
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Koutsoumanis1*
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Laboratory of Food Microbiology and Hygiene, Department of Food Science and
Technology, School of Agriculture, Faculty of Agriculture, Forestry and Natural
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Environment, Aristotle University of Thessaloniki, Thessaloniki, Greece 54124
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El Líbano 5524, Macul, Santiago, Chile
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Science and Human Nutrition, School of Food, Biotechnology and Development,
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Agricultural University of Athens, Iera Odos 75, Athens 11855, Greece
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Laboratory of Microbiology and Biotechnology of Foods, Department of Food
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Laboratorio de Microbiología y Probioticos, INTA, Universidad de Chile, Avenida
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* Corresponding author.
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E-mail address: [email protected] (K.P. Koutsoumanis).
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The lag times (λ) of Geobacillus stearothermophilus single spores were studied at
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different storage temperatures ranging from 45 to 59°C using the Bioscreen C
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method. A significant variability of λ was observed among individual spores at all
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temperatures tested. The storage temperature affected both the position and the spread
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of the λ distributions. The minimum mean value of λ (i.e. 10.87 h) was observed at
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55°C, while moving away from this temperature resulted in an increase for both the
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mean and standard deviation of λ. A Cardinal Model with Inflection (CMI) was fitted
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to the reverse mean λ, and the estimated values for the cardinal parameters Tmin, Tmax,
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Topt and the optimum mean λ of G. stearothermophilus were found to be 38.1, 64.2,
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53.6 °C and 10.3 h, respectively. To interpret the observations, a probabilistic growth
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model for G. stearothermophilus individual spores, taking into account λ variability,
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was developed. The model describes the growth of a population, initially consisting of
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N0 spores, over time as the sum of cells in each of the N0 imminent subpopulations
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originating from a single spore. Growth simulations for different initial contamination
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levels showed that for low N0 the number of cells in the population at any time is
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highly variable. An increase in N0 to levels exceeding 100 spores results in a
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significant decrease of the above variability and a shorter λ of the population.
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Considering that the number of G. stearothermophilus surviving spores in the final
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product is usually very low, the data provided in this work can be used to evaluate the
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probability distribution of the time-to-spoilage and enable decision-making based on
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the “acceptable level of risk”.
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ACCEPTED MANUSCRIPT Keywords: Geobacillus stearothermophilus, individual spore lag time, lag phase
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distributions, variability, stochastic growth
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1. Introduction The presence of Geobacillus stearothermophilus spores in thermally processed foods constitutes one of the most important quality problems for the food industry.
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The high prevalence and concentration of spores in the raw materials, the adhesive
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characteristics of spores that enhance their persistence in industrial environments and,
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most importantly, the high heat resistance of the spores are among the major factors
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explaining the importance of this endospore-former (Yoo et al., 2006; Postollec et al.,
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2012; André et al., 2013). Thus, the estimation of the risk of G. stearothermophilus
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growth to spoilage levels constitutes a major target of quality managers.
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Recently, Kakagianni et al. (2016) developed a predictive model for the effect of
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storage temperature on the growth of G. stearothermophilus, which can be used to
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predict the time-to-spoilage as the time required by the contaminating spores to
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germinate and multiply from the initial to a spoilage level. The model was developed
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and successfully validated for evaporated milk based on data from experiments with
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high initial contamination levels. In practice, however, the number of spores surviving
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the heat treatment is usually very low. Depending on the use of the model, predictions
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based on high initial contamination levels can be on the “fail-safe” or the “fail-
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dangerous” side. For example, when the model is used to predict the time-to-spoilage,
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the above predictions represent a worst-case scenario. This is because the early
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germinated and outgrown spores within the population begin to multiply up to a
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microbial concentration sufficient to spoil the food leading to shorter estimations of
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the time-to-spoilage compared to low initial contamination levels (Baranyi, 1998,
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2002; Pin and Baranyi, 2006; Baranyi et al., 2009). In contrast, when the objective is
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to evaluate the duration of a quality control test in which products are stored at the
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optimum temperature for growth for a certain time and the percentage of spoiled
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items is evaluated, the above predictions can lead to “fail-dangerous” estimates. Thus,
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for an effective risk assessment of spoilage the growth model needs to be combined
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with quantitative data on the variability of individual spore behavior. Lag time is much more uncertain and difficult to predict compared to growth
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rate (Baranyi, 2002). However, the credibility of a mathematical model in predicting
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the conditions that lead to microbiological spoilage depends highly on its ability to
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describe the effect of the environment on lag time. This implies that a better
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understanding of the lag phase is of great importance for the effective application of
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predictive models in food quality management. For a complete understanding of lag
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time, the behaviour of single cells or spores has to be taken into account through the
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development of stochastic models which are able to deal with more “realistic” low
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contamination events (Koutsoumanis and Lianou, 2013).
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2012; Pandey et al., 2013; Zhou et al., 2013; van Melis et al., 2014; Aguirre et al.,
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2015; Pandey et al., 2015; Warda et al., 2015) or Clostridium (Stringer et al., 2005;
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Webb et al., 2007; Stringer et al., 2009; Wang et al., 2012; Smelt et al., 2013; Lenz et
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al., 2014) species, while data on G. stearothermophilus are very limited. Zhou, et al.
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(2013) studied the kinetics of germination of individual spores of G.
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stearothermophilus measured by Raman spectroscopy and differential interference
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contrast microscopy. Previous studies, however, demonstrated that germination and
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lag time are independent (Stringer et al., 2005). The variability of post-germination
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stages such as outgrowth and doubling is relatively large, and as a result, the
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germination time cannot be used to predict the overall lag time.
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temperature conditions on the lag time of individual G. stearothermophilus spores,
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and to demonstrate the role of individual spore heterogeneity in growth predictions
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using a stochastic modeling approach.
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2. Materials and methods
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2.1 Bacterial strain and inoculum preparation
The type strain G. stearothermophilus ATCC 7953 was used for all experiments
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in the present study. The stock culture of the strain was stored frozen (-70°C) onto
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MicrobankTM porous beads (Pro-Lab Diagnostics, Ontario, Canada). The working
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culture was stored refrigerated (5°C) on nutrient agar (NA; Lab M Limited,
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Lancashire, United Kingdom) slants and was renewed bimonthly. Prior to each
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experiment, the microorganism was activated by transferring a loopful from the NA
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slants into 10 ml nutrient broth (NB; Lab M Limited) and incubating at 55°C for 24 h.
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The 24-h G. stearothermophilus cultures were subjected to a heat shock treatment in
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order to eliminate vegetative cells and enhance the homogeneous activation and
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germination of dormant endospores of the organism (Antolinos et al., 2012; Yuan et
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al., 2012). Then, the heat-shocked cultures were centrifuged (6000 rpm for 20 min) in
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a refrigerated centrifuge (4°C) (model PK120R, ThermoElectron Corporation,
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Waltham, MA). The harvested cells were washed with 20 ml of quarter-strength
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Ringer’s solution (Lab M Limited) through centrifugation under the same conditions.
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The pellet was finally resuspended in 10 ml of tryptone soy broth (TSB; Lab M
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Limited) in order to obtain an initial concentration of ca. 106 spores/ml.
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2.2 Growth kinetic experiments
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2.2.1 Maximum specific growth rate
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The maximum specific growth rate (µmax) of G. stearothermophilus was evaluated in TSB at 45, 47.5, 50, 55 and 59°C. The µmax values corresponding to each
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storage temperature were estimated by means of absorbance detection times of
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serially decimally diluted cultures using the automated turbidimetric system
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Bioscreen C (Oy Growth Curves Ab Ltd., Raisio, Finland) as described previously
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(Lianou and Koutsoumanis, 2011; Kakagianni et al., 2016). In this study, the range of
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initial concentrations obtained in the microtiter plates was approximately 106-102
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cfu/well. Optical density (OD) measurements in the Bioscreen C were measured at
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15-min intervals using the wideband filter (420-580 nm) for a total time period such
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that a considerable OD change was observed, if possible, for all five decimally diluted
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cultures. The microtiter plates were agitated for 15 s at medium amplitude prior to the
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OD measurements. The absorbance time to detection (Tdet) was defined as the time
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required for the OD to reach 0.2 units. Specifically, proper dilutions of the content of
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each well of a microtiter plate (100 wells per plate) were surface plated onto tryptone
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soy agar (TSA; Lab M Limited) plates during the exponential growth phase of the
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cultures at Tdet (ODdet = 0.2 A420-580). Colonies were enumerated after incubation of
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the plates at 55 °C for 24 h allowing the calculation of the bacterial concentration at
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the chosen Tdet (Ndet). The detection times (h) of five serial decimal dilutions of the
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bacterial culture were plotted against the corresponding natural logarithm of their
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initial concentrations, and µmax values were determined by linear regression as the
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negative reciprocal of the slope of the regression line (Dalgaard and Koutsoumanis,
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2001). For each temperature, five samples (i.e. quintuple wells of five serially diluted
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cultures) were analysed (n = 5).
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2.2.2 Lag time of individual spores Assuming an exponential bacterial growth at a constant µmax, after the lag period
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and until the Tdet, the single-spore lag time (λ) values of the tested microorganism
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were estimated based on the generated turbidity growth curves and according to the
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Bioscreen C methodology (Baranyi and Pin, 1999; Baranyi et al., 2009; Aguirre et al.,
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2011). The culture of the tested strain was decimally and binary diluted in TSB to
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obtain a target concentration of ca. 1 spore/well. Two microtiter plates (100 wells per
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plate) were inoculated with 350 µl of the aforementioned diluted organism’s culture in
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each well, incubated in the Bioscreen C at different temperatures (45, 47.5, 50, 55 and
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59°C), and the OD was recorded at 420-580 nm at 15-min intervals. Microtiter plates
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were shaken at medium amplitude for 15 s before measuring OD. The average initial
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number of spores per well (m) was estimated by assuming that the probability of
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having one spore per well is described by the Poisson distribution (Baranyi et al.,
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2009), as a result of the dilution process, using the following equation:
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where P0 is the probability of having no growth in a well (estimated as the percentage
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of wells without detected growth). To ensure that most of the positive wells contained
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on average one spore, when microtiter plates showed a growth/no growth ratio less
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than 0.2 or higher than 0.7, were discarded and the experiment was repeated.
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The individual spore λ values were calculated using the following equation proposed by Baranyi and Pin (1999):
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(2)
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where Tdet is the time (h) required for the OD at 420-580 nm to reach 0.2 units, µmax is
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the maximum specific growth rate (1/h) determined under the experimental conditions
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as described in Section 2.2.1, LnNdet is the natural logarithm of the cell population (ln
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cfu/well) at Tdet, LnN0 is the natural logarithm of the average initial number of spores
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initiating growth (ln cfu/well) in the considered well estimated according to Eq. (1).
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For each tested condition, the experiments were repeated as many times needed to
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obtain about 200 values of λ.
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2.2.3 Statistical analysis and growth simulation
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Various theoretical distributions were fitted to the cumulative distributions of individual spore λ values using @RISK 6.0 Professional Edition (Palisade
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Corporation, Newfield, New York, USA). The goodness of fit was evaluated using
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RMSE values. Monte Carlo simulation was performed to predict growth for different
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N0 (1, 10 and 100 spores) taking into account the variability in individual spore λ
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using the best fitted distribution. The simulation was performed using 10,000
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iterations.
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3. Results The differences in the Tdet from OD curves of 199 individual spores growth at
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45°C are illustrated in Fig. 1. The observed Tdet, µmax and Ndet values were used to
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evaluate the variability of single G. stearothermophilus spores λ based on equation 2.
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The estimated µmax values (mean ± standard deviation) were 0.452 ± 0.022, 0.563 ±
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0.051, 0.883 ± 0.047, 1.666 ± 0.047 and 1.955 ± 0.091 h-1 for 45, 47.5, 50, 55 and 59
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°C, respectively. The bacterial concentration parameter Ndet was found to be
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unaffected by temperature (P>0.05) with a mean (± standard deviation) value of 6.99
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(± 0.36) log cfu/well.
Figure 2 and Table 1 present the cumulative probability distributions and the
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statistics of the individual spore λ at the different tested temperatures. As shown, λ
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varied significantly among individual spores at all temperatures tested. The minimum
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mean value of λ was 10.87 h and it was observed at 55°C. Moving away from the later
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temperature resulted in an increase for both the mean and the standard deviation of λ.
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The Gamma distribution was further fitted to the λ data, following the theory of
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Baranyi and Pin (2001). The estimated parameters and the RMSE values of the fitting
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are presented in Table 2. A comparison between the observed and fitted quantiles
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(Fig. 3) clearly revealed that the Gamma distribution is suitable for describing the
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individual spore λ for all the tested temperatures.
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In order to quantify the temperature dependence of individual spore λ, the
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following Cardinal Model with Inflection (CMI) (Rosso et al., 1993) was fitted to the
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reverse mean λ:
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=
∙
(3) 10
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=
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&
&& &&'( )
*+ &,-. [0&*+ &'( 10&&*+ 1 &*+ & &*+ 2&'( 3&]
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,
≤ #
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≥ #67
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where Tmin and Tmax are the theoretical minimum and maximum temperature for
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growth (λ becomes infinite), respectively, Topt is the theoretical optimum temperature
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for growth (λ reaches a minimum value), and λopt is the value of lag at T=Topt. In order
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to stabilise the variance, a square root transformation was used. Both the R2 (Table 3)
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and the graphical evaluation of the fitting (Fig. 4) indicated a satisfactory performance
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of the CMI in describing the effect of temperature on the mean λ of G.
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stearothermophilus individual spores. The estimated values for the cardinal
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parameters Tmin, Tmax and Topt were 38.2, 64.2 and 53.6 °C, respectively, while the
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mean λopt of G. stearothermophilus was estimated 10.3 h (Table 3).
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To interpret the observations, a stochastic model for G. stearothermophilus
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spores’ growth was developed by introducing the distribution of individual spores λ
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values into a simple exponential growth with lag model (Koutsoumanis and Lianou,
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2013) as follows:
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N 1 N t = ∑1 0 µ max (t − λi ) e
for
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t > λi
(5)
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where Nt is the total number of cells or spores in a population at time t, N0 is the initial
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number of spores at t=0, µmax is the maximum specific growth rate and λi is the lag
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time of individual spores following the gamma distribution. The model describes the
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growth of a bacterial population, initially consisting of N0 spores, over time as the sum 11
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of cells in each of the N0 imminent subpopulations originating from a single spore.
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The above approach allows for taking into account the heterogeneity in the growth
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dynamics of single spores by introducing λ variability in the model using Monte Carlo
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simulation. Figure 5 presents the output of the model (Εq. (5)) for spore levels N0 equal to 1
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(Fig. 5a), 10 (Fig. 5b) and 100 spores (Fig. 5c) at 45 °C based on a Monte Carlo
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simulation with 10,000 iterations and with a Uniform distribution for t [t ~ Uniform
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(0, 80)]. As shown, the output of the model for N0 = 1 spore is a stochastic growth
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curve in which the number of cells in the population at any time is a distribution. An
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increase in N0 to 100 spores resulted in a significant decrease of the above variability
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as well as in a shorter lag phase of the population. The model can be used to evaluate
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the probability distribution of the time-to-spoilage, with the latter being defined as the
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time required by the initial contaminating spores to germinate and multiply to a
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spoilage level (Kakagianni et al., 2016).
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4. Discussion
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The Bioscreen C method, applied in the present work to study the λ of single G.
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stearothermophilus spores, has been extensively used for monitoring single cell and
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spore kinetics (George et al., 2008; Métris et al., 2008; Dupont and Augustin, 2009;
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Miled et al., 2011; Aguirre et al., 2015; Blana et al., 2015). One of the main
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assupmtions of the method is that growth is initiated from a single spore in a well.
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The number of spores in each well can be estimated based on the ratio between
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positive wells (wells showing growth) and negative wells (wells showing no growth)
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according to the Poisson distribution. In all experiments carried out in this study, the
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number of spores in a well between 0.22 and 1.2. According to the Poisson
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distribution, the above values correspond to a percentage of wells with one spore
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ranging from 51.4% to 98.1%. Previous studies have shown that the above range
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allows for the estimation of the distributions of single cell parameters (Métris et al.,
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2003; Stringer et al., 2005). Indeed, Stringer et al. (2005) studied the lag times of
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Clostridium botulinum individual spores utilizing both the Bioscreen C and a
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microscopic method, and reported that the two methodologies produced distributions
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with very similar estimates of both lag mean value and variability.
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The findings of this study highlight a considerable heterogeneity in the λ of G.
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stearothermophilus single spores. In the case of spores, the lag time is mainly
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determined by the germination and outgrowth processes (Smelt et al., 2002; Stringer
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et al., 2005). Thus, the observed λ variability can be considered as a reflection of the
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differences in the rates of the above processes among spores. The stochastic
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fluctuations in the number of the nutrient germinant receptors (nGRs) per spore due to
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epigenetic variations among individual spores, appears to be one of the main reasons
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of the significant heterogeneity in spore germination capacity at the population level
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(Elowitz et al., 2002; Stringer et al., 2005; Chen et al., 2006; Webb et al., 2007;
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Ghosh and Setlow, 2009; Peng et al., 2009; Kong et al., 2010; Zhang et al., 2010).
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Moreover, the stochastic inactivation of nGRs during heat treatment can contribute to
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the differences in the germination rates among individual spores (Smelt et al., 2005).
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Finally the differences in the levels of specific lytic enzymes such as CwlJ and SleB,
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which initiate the spore’s peptidoglycan cortex degradation during germination can be
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an additional variability source (Setlow, 2003; Moir, 2006; Ghosh and Setlow, 2009;
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Peng et al., 2009). The storage temperature affected both the position and the spread of the λ
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distributions of G. stearothermophilus spores. In agreement to previous single cell
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studies (Métris et al., 2003; Guiller et al., 2005, 2006), the mean and the standard
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deviation of the λ values estimated in this study followed a similar trend in relation to
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temperature, with longer and more scattered λ as conditions move away from the
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optimum temperature. Moreover, at all tested temperatures, the λ distributions of G.
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stearothermophilus spores showed a positive skewness (Table 1) indicating an
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asymmetry with a tail extending towards longer times. This confirmed available
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microscopic and/or turbidimetric findings on other bacilli (Leuschner and Lillford,
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1999; Smelt et al., 2008; Pandey et al., 2013) and clostridia (Billon et al., 1997;
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Stringer et al., 2009, 2011), but also on non-sporeforming bacteria such as Listeria
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spp. and Escherichia coli O157:H7 (Métris et al., 2003; McKellar and Lu, 2005;
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Guillier and Augustin, 2006; Li et al., 2006; McKellar and Hawke, 2006). The above
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skewed shape was sucessfully described with a Gamma distribution which has been
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previously used for individual cell λ values (Métris et al., 2003). Information about
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the shape of λ distributions are of great importance in predicting microbial growth at
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low contamination levels due to the stochastic nature of microbial lag phase (Baranyi,
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1998, 2002).
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The temperature dependence of the mean G. stearothermophilus λ was
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considerably different from that reported for the growth rate of the same strain by
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Kakagianni et al. (2016). In the latter study, the authors reported an optimum
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temperature (mean value ± standard deviation) for growth Topt = 61.82 ± 0.20°C, 14
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estimated in the present study (Table 3). This difference was confirmed by the µmax
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values estimated in this study which showed a continuous increase as temperature
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increased from 47.5 to 59.0oC, in contrast to what was the case for the mean lag time
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which showed a minimum value at 55.0oC. These results indicate that the
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physiological state theorem (Baranyi, 1998) applied for vegetative cells, considering
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the lag as inversely proportional to the maximum specific growth rate, is not
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necessarily valid in the case of spores. This can be attributed to the different processes
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during the lag time of spores and vegetative cells. As described previously, in the case
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of spores the lag time is mainly determined by the germination and outgrowth
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process, the rate of which can be considerable different than the growth rate of
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vegetative cells. Indeed, Stringer et al. (2005) reported that the time intervals for
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germination, emergence, outgrowth and doubling of C. botulinum spores were all
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independent of each other. Consequently, although in the case of vegetative cells with
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known physiological state, growth at different temperatures can be predicted based on
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a model for the growth rate (Koutsoumanis et al., 2010), predicting growth of spores
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may require models for both lag time and growth rate since the effect of temperature
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on these two growth kinetic parameters can be substantially different.
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G. stearothermophilus is an important spoiler in various heat treated foods.
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Considering that the number of surviving spores in the final product is usually very
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low, knowledge of their lag time variability is important for predicting growth and
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spoilage in a probabilistic way (Fig. 5). Probabilistic modeling approaches that take
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into account the variability of the factors affecting microbial behavior can provide
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more realistic estimation of food quality. Most importantly, probabilistic models 15
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enable decision-making based on the “acceptable level of risk”, and can provide
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structured information allowing decision-makers to compare various actions and
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identify those that can lead to effective and economic reduction of quality risks.
334 Acknowledgements
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This study was carried out with the financial support of “Understanding the
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impact of manufacturing processes in the ecology of microorganisms that spoil-
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contaminate milk products (ESL, evaporated milk) and fresh fruit juices –
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Development of molecular methodologies and mathematical models for the prediction
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of their shelf-life” within the framework of the action “Cooperation” (NSRF 2007-
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2013), that was co-financed by the European Social Fund (ESF) and National
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Resources.
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Aguirre, J.S., de Fernando, G.G., Hierro, E., Hospital, X.F., Ordóñez, J.A., Fernández,
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530 van Melis, C., Den Besten, H., Groot, M.N., Abee, T., 2014. Quantification of the
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NaCl concentrations affect the duration and distribution of lag times from individual
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variability in time between addition of nutrient germinants and rapid dipicolinic acid
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release during germination of spores of Bacillus species. Journal of Bacteriology 192,
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3608-3619. 23
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spores of Geobacillus stearothermophilus as measured by Raman spectroscopy and
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differential interference contrast microscopy. PloS ONE 8, e74987.
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Table 1
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Statistics of the individual spore lag time (λ) of Geobacillus stearothermophilus
λα (h)
Storage temperature Mean
St. Deviation
Median
45
224
37.9
18.6
33.3
47.5
199
21.4
17.7
15.3
50
227
11.7
11.8
55
199
10.9
9.20
59
193
17.9
15.0
b
Number of spores tested.
Kurtosis
16.4
72.6
1.39
2.62
1.26
56.9
1.03
0.102
6.46
1.50
37.8
1.66
2.06
7.16
0.705
29.9
1.42
1.85
13.6
0.760
47.7
1.00
0.305
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Skewness
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Individual spore lag time.
95th Percentile
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5th Percentile
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Nb
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568
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ACCEPTED MANUSCRIPT Table 2 Parameter estimation of the Gamma distribution fitted to the individual spore lag times of Geobacillus stearothermophilus. The probability density function of the Gamma distribution in the shape-rate parametrization is:
45
4.69
47.5
1.30
50
1.28
59
RMSEa
7.84
0.0186
0.0316
7.81
0.0576
1.40
4.72
0.0603
1.10
17.1
0.0200
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Root Mean Square error
β
16.5
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Temperature (oC)
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Table 3
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Estimated values and statistics for the parameters of the Cardinal Model with Inflection (Equations 3 and 4) describing the effect of temperature on the reciprocal of the mean lag time (λ) of individual spores of Geobacillus stearothermophilus. JK (h)
Standard Error
R2a
10.3
0.5
0.977
Tmax (oC)
64.2
Tmin (oC)
38.2
5.1
Topt (oC)
53.6
0.8
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R2: coefficient of determination.
2.3
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Parameter
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Fig. 2. Cumulative distributions of Geobacillus stearothermophilus single spore lag times (λ) at different growth temperature conditions. The number of data were 224,
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199, 227, 199 and 193 for 45, 47.5, 50, 55 and 59 °C, respectively.
Fig. 3. Comparison between observed and fitted quantiles of the Gamma distribution
50°C (c), 55°C (d) and 59°C (e).
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for Geobacillus stearothermophilus single spore lag times (λ) at 45°C (a), 47.5°C (b),
Fig. 4. Effect of temperature on the reciprocal of the mean lag times of Geobacillus
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stearothermophilus single spores fitted to the Cardinal Model with Inflection (solid line) (Equations 3 and 4). Points (○) represent observed values.
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Fig. 5. Simulations of Geobacillus stearothermophilus growth at 45 °C for various
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initial contamination levels: 1 spore (a), 10 spores (b) and 100 spores (c). Growth is predicted by the stochastic model (Equation (5)) using Monte Carlo simulation with 10,000 iterations and a Uniform distribution for time t [t ~ Uniform (0, 80)].
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0,35
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0,30 0,25 0,20 0,15 0,10 10
20
Time (h)
1
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2
30
40
50
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0
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OD (420-580nm)
0,40
60
70
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
59°C
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55°C 50°C
47.5°C 45°C
0
20
60 Lag time (h)
80
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Fig. 2
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40
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Cumulative Probability
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120
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Fig. 3
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0.40 0.35
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0.25 0.20 0.15 0.10
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1/Lag^0.5 (1/h)
0.30
0.05
40
45
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55
Temperature (°C) 1
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60
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Fig. 5
S0740-0020(16)31084-X
DOI:
10.1016/j.fm.2017.04.009
Reference:
YFMIC 2775
To appear in:
Food Microbiology
Please cite this article as: Myrsini Kakagianni, Juan S. Aguirre, Alexandra Lianou, Konstantinos P. Koutsoumanis, Effect of storage temperature on the lag time of Geobacillus stearothermophilus individual spores, Food Microbiology (2017), doi: 10.1016/j.fm.2017.04.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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The lag times (λ) of Geobacillus stearothermophilus single spores showed a significant variability The storage temperature affected both the position and the spread of the λ distributions The minimum mean value of λ (i.e. 10.87 h) was observed at 55°C Moving away from 55°C resulted in an increase for both the mean and standard deviation of λ A probabilistic growth model for individual spores was developed.
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Effect of storage temperature on the lag time of Geobacillus stearothermophilus
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individual spores
3 4 Myrsini Kakagianni1, Juan S. Aguirre2, Alexandra Lianou3 Konstantinos P.
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Koutsoumanis1*
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Laboratory of Food Microbiology and Hygiene, Department of Food Science and
Technology, School of Agriculture, Faculty of Agriculture, Forestry and Natural
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Environment, Aristotle University of Thessaloniki, Thessaloniki, Greece 54124
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El Líbano 5524, Macul, Santiago, Chile
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Science and Human Nutrition, School of Food, Biotechnology and Development,
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Agricultural University of Athens, Iera Odos 75, Athens 11855, Greece
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Laboratory of Microbiology and Biotechnology of Foods, Department of Food
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Laboratorio de Microbiología y Probioticos, INTA, Universidad de Chile, Avenida
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* Corresponding author.
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E-mail address: [email protected] (K.P. Koutsoumanis).
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ACCEPTED MANUSCRIPT ABSTRACT
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The lag times (λ) of Geobacillus stearothermophilus single spores were studied at
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different storage temperatures ranging from 45 to 59°C using the Bioscreen C
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method. A significant variability of λ was observed among individual spores at all
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temperatures tested. The storage temperature affected both the position and the spread
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of the λ distributions. The minimum mean value of λ (i.e. 10.87 h) was observed at
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55°C, while moving away from this temperature resulted in an increase for both the
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mean and standard deviation of λ. A Cardinal Model with Inflection (CMI) was fitted
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to the reverse mean λ, and the estimated values for the cardinal parameters Tmin, Tmax,
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Topt and the optimum mean λ of G. stearothermophilus were found to be 38.1, 64.2,
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53.6 °C and 10.3 h, respectively. To interpret the observations, a probabilistic growth
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model for G. stearothermophilus individual spores, taking into account λ variability,
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was developed. The model describes the growth of a population, initially consisting of
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N0 spores, over time as the sum of cells in each of the N0 imminent subpopulations
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originating from a single spore. Growth simulations for different initial contamination
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levels showed that for low N0 the number of cells in the population at any time is
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highly variable. An increase in N0 to levels exceeding 100 spores results in a
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significant decrease of the above variability and a shorter λ of the population.
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Considering that the number of G. stearothermophilus surviving spores in the final
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product is usually very low, the data provided in this work can be used to evaluate the
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probability distribution of the time-to-spoilage and enable decision-making based on
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the “acceptable level of risk”.
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ACCEPTED MANUSCRIPT Keywords: Geobacillus stearothermophilus, individual spore lag time, lag phase
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distributions, variability, stochastic growth
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1. Introduction The presence of Geobacillus stearothermophilus spores in thermally processed foods constitutes one of the most important quality problems for the food industry.
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The high prevalence and concentration of spores in the raw materials, the adhesive
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characteristics of spores that enhance their persistence in industrial environments and,
53
most importantly, the high heat resistance of the spores are among the major factors
54
explaining the importance of this endospore-former (Yoo et al., 2006; Postollec et al.,
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2012; André et al., 2013). Thus, the estimation of the risk of G. stearothermophilus
56
growth to spoilage levels constitutes a major target of quality managers.
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Recently, Kakagianni et al. (2016) developed a predictive model for the effect of
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storage temperature on the growth of G. stearothermophilus, which can be used to
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predict the time-to-spoilage as the time required by the contaminating spores to
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germinate and multiply from the initial to a spoilage level. The model was developed
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and successfully validated for evaporated milk based on data from experiments with
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high initial contamination levels. In practice, however, the number of spores surviving
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the heat treatment is usually very low. Depending on the use of the model, predictions
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based on high initial contamination levels can be on the “fail-safe” or the “fail-
65
dangerous” side. For example, when the model is used to predict the time-to-spoilage,
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the above predictions represent a worst-case scenario. This is because the early
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germinated and outgrown spores within the population begin to multiply up to a
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microbial concentration sufficient to spoil the food leading to shorter estimations of
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the time-to-spoilage compared to low initial contamination levels (Baranyi, 1998,
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2002; Pin and Baranyi, 2006; Baranyi et al., 2009). In contrast, when the objective is
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to evaluate the duration of a quality control test in which products are stored at the
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optimum temperature for growth for a certain time and the percentage of spoiled
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items is evaluated, the above predictions can lead to “fail-dangerous” estimates. Thus,
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for an effective risk assessment of spoilage the growth model needs to be combined
75
with quantitative data on the variability of individual spore behavior. Lag time is much more uncertain and difficult to predict compared to growth
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rate (Baranyi, 2002). However, the credibility of a mathematical model in predicting
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the conditions that lead to microbiological spoilage depends highly on its ability to
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describe the effect of the environment on lag time. This implies that a better
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understanding of the lag phase is of great importance for the effective application of
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predictive models in food quality management. For a complete understanding of lag
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time, the behaviour of single cells or spores has to be taken into account through the
83
development of stochastic models which are able to deal with more “realistic” low
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contamination events (Koutsoumanis and Lianou, 2013).
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Available studies on single spore lag time focus on either Bacillus (Aguirre et al.,
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2012; Pandey et al., 2013; Zhou et al., 2013; van Melis et al., 2014; Aguirre et al.,
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2015; Pandey et al., 2015; Warda et al., 2015) or Clostridium (Stringer et al., 2005;
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Webb et al., 2007; Stringer et al., 2009; Wang et al., 2012; Smelt et al., 2013; Lenz et
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al., 2014) species, while data on G. stearothermophilus are very limited. Zhou, et al.
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(2013) studied the kinetics of germination of individual spores of G.
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stearothermophilus measured by Raman spectroscopy and differential interference
92
contrast microscopy. Previous studies, however, demonstrated that germination and
93
lag time are independent (Stringer et al., 2005). The variability of post-germination
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stages such as outgrowth and doubling is relatively large, and as a result, the
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germination time cannot be used to predict the overall lag time.
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temperature conditions on the lag time of individual G. stearothermophilus spores,
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and to demonstrate the role of individual spore heterogeneity in growth predictions
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using a stochastic modeling approach.
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2. Materials and methods
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2.1 Bacterial strain and inoculum preparation
The type strain G. stearothermophilus ATCC 7953 was used for all experiments
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in the present study. The stock culture of the strain was stored frozen (-70°C) onto
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MicrobankTM porous beads (Pro-Lab Diagnostics, Ontario, Canada). The working
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culture was stored refrigerated (5°C) on nutrient agar (NA; Lab M Limited,
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Lancashire, United Kingdom) slants and was renewed bimonthly. Prior to each
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experiment, the microorganism was activated by transferring a loopful from the NA
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slants into 10 ml nutrient broth (NB; Lab M Limited) and incubating at 55°C for 24 h.
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The 24-h G. stearothermophilus cultures were subjected to a heat shock treatment in
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order to eliminate vegetative cells and enhance the homogeneous activation and
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germination of dormant endospores of the organism (Antolinos et al., 2012; Yuan et
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al., 2012). Then, the heat-shocked cultures were centrifuged (6000 rpm for 20 min) in
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a refrigerated centrifuge (4°C) (model PK120R, ThermoElectron Corporation,
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Waltham, MA). The harvested cells were washed with 20 ml of quarter-strength
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Ringer’s solution (Lab M Limited) through centrifugation under the same conditions.
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The pellet was finally resuspended in 10 ml of tryptone soy broth (TSB; Lab M
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Limited) in order to obtain an initial concentration of ca. 106 spores/ml.
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2.2 Growth kinetic experiments
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2.2.1 Maximum specific growth rate
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The maximum specific growth rate (µmax) of G. stearothermophilus was evaluated in TSB at 45, 47.5, 50, 55 and 59°C. The µmax values corresponding to each
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storage temperature were estimated by means of absorbance detection times of
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serially decimally diluted cultures using the automated turbidimetric system
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Bioscreen C (Oy Growth Curves Ab Ltd., Raisio, Finland) as described previously
127
(Lianou and Koutsoumanis, 2011; Kakagianni et al., 2016). In this study, the range of
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initial concentrations obtained in the microtiter plates was approximately 106-102
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cfu/well. Optical density (OD) measurements in the Bioscreen C were measured at
130
15-min intervals using the wideband filter (420-580 nm) for a total time period such
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that a considerable OD change was observed, if possible, for all five decimally diluted
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cultures. The microtiter plates were agitated for 15 s at medium amplitude prior to the
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OD measurements. The absorbance time to detection (Tdet) was defined as the time
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required for the OD to reach 0.2 units. Specifically, proper dilutions of the content of
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each well of a microtiter plate (100 wells per plate) were surface plated onto tryptone
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soy agar (TSA; Lab M Limited) plates during the exponential growth phase of the
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cultures at Tdet (ODdet = 0.2 A420-580). Colonies were enumerated after incubation of
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the plates at 55 °C for 24 h allowing the calculation of the bacterial concentration at
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the chosen Tdet (Ndet). The detection times (h) of five serial decimal dilutions of the
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bacterial culture were plotted against the corresponding natural logarithm of their
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initial concentrations, and µmax values were determined by linear regression as the
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negative reciprocal of the slope of the regression line (Dalgaard and Koutsoumanis,
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2001). For each temperature, five samples (i.e. quintuple wells of five serially diluted
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cultures) were analysed (n = 5).
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2.2.2 Lag time of individual spores Assuming an exponential bacterial growth at a constant µmax, after the lag period
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and until the Tdet, the single-spore lag time (λ) values of the tested microorganism
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were estimated based on the generated turbidity growth curves and according to the
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Bioscreen C methodology (Baranyi and Pin, 1999; Baranyi et al., 2009; Aguirre et al.,
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2011). The culture of the tested strain was decimally and binary diluted in TSB to
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obtain a target concentration of ca. 1 spore/well. Two microtiter plates (100 wells per
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plate) were inoculated with 350 µl of the aforementioned diluted organism’s culture in
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each well, incubated in the Bioscreen C at different temperatures (45, 47.5, 50, 55 and
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59°C), and the OD was recorded at 420-580 nm at 15-min intervals. Microtiter plates
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were shaken at medium amplitude for 15 s before measuring OD. The average initial
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number of spores per well (m) was estimated by assuming that the probability of
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having one spore per well is described by the Poisson distribution (Baranyi et al.,
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2009), as a result of the dilution process, using the following equation:
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161 162 163
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where P0 is the probability of having no growth in a well (estimated as the percentage
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of wells without detected growth). To ensure that most of the positive wells contained
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on average one spore, when microtiter plates showed a growth/no growth ratio less
167
than 0.2 or higher than 0.7, were discarded and the experiment was repeated.
169 170 171
The individual spore λ values were calculated using the following equation proposed by Baranyi and Pin (1999):
= −
(2)
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where Tdet is the time (h) required for the OD at 420-580 nm to reach 0.2 units, µmax is
174
the maximum specific growth rate (1/h) determined under the experimental conditions
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as described in Section 2.2.1, LnNdet is the natural logarithm of the cell population (ln
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cfu/well) at Tdet, LnN0 is the natural logarithm of the average initial number of spores
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initiating growth (ln cfu/well) in the considered well estimated according to Eq. (1).
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For each tested condition, the experiments were repeated as many times needed to
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obtain about 200 values of λ.
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2.2.3 Statistical analysis and growth simulation
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Various theoretical distributions were fitted to the cumulative distributions of individual spore λ values using @RISK 6.0 Professional Edition (Palisade
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Corporation, Newfield, New York, USA). The goodness of fit was evaluated using
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RMSE values. Monte Carlo simulation was performed to predict growth for different
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N0 (1, 10 and 100 spores) taking into account the variability in individual spore λ
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using the best fitted distribution. The simulation was performed using 10,000
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iterations.
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3. Results The differences in the Tdet from OD curves of 199 individual spores growth at
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45°C are illustrated in Fig. 1. The observed Tdet, µmax and Ndet values were used to
193
evaluate the variability of single G. stearothermophilus spores λ based on equation 2.
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The estimated µmax values (mean ± standard deviation) were 0.452 ± 0.022, 0.563 ±
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0.051, 0.883 ± 0.047, 1.666 ± 0.047 and 1.955 ± 0.091 h-1 for 45, 47.5, 50, 55 and 59
196
°C, respectively. The bacterial concentration parameter Ndet was found to be
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unaffected by temperature (P>0.05) with a mean (± standard deviation) value of 6.99
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(± 0.36) log cfu/well.
Figure 2 and Table 1 present the cumulative probability distributions and the
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statistics of the individual spore λ at the different tested temperatures. As shown, λ
201
varied significantly among individual spores at all temperatures tested. The minimum
202
mean value of λ was 10.87 h and it was observed at 55°C. Moving away from the later
203
temperature resulted in an increase for both the mean and the standard deviation of λ.
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The Gamma distribution was further fitted to the λ data, following the theory of
205
Baranyi and Pin (2001). The estimated parameters and the RMSE values of the fitting
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are presented in Table 2. A comparison between the observed and fitted quantiles
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(Fig. 3) clearly revealed that the Gamma distribution is suitable for describing the
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individual spore λ for all the tested temperatures.
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In order to quantify the temperature dependence of individual spore λ, the
209 210
following Cardinal Model with Inflection (CMI) (Rosso et al., 1993) was fitted to the
211
reverse mean λ:
212 213
=
∙
(3) 10
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216
=
0
&
&& &&'( )
*+ &,-. [0&*+ &'( 10&&*+ 1 &*+ & &*+ 2&'( 3&]
0
217
,
≤ #
%$, #$% < < #67 (4) ,
≥ #67
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where Tmin and Tmax are the theoretical minimum and maximum temperature for
219
growth (λ becomes infinite), respectively, Topt is the theoretical optimum temperature
220
for growth (λ reaches a minimum value), and λopt is the value of lag at T=Topt. In order
221
to stabilise the variance, a square root transformation was used. Both the R2 (Table 3)
222
and the graphical evaluation of the fitting (Fig. 4) indicated a satisfactory performance
223
of the CMI in describing the effect of temperature on the mean λ of G.
224
stearothermophilus individual spores. The estimated values for the cardinal
225
parameters Tmin, Tmax and Topt were 38.2, 64.2 and 53.6 °C, respectively, while the
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mean λopt of G. stearothermophilus was estimated 10.3 h (Table 3).
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To interpret the observations, a stochastic model for G. stearothermophilus
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spores’ growth was developed by introducing the distribution of individual spores λ
229
values into a simple exponential growth with lag model (Koutsoumanis and Lianou,
230
2013) as follows:
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N 1 N t = ∑1 0 µ max (t − λi ) e
for
t ≤ λi
for
t > λi
(5)
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where Nt is the total number of cells or spores in a population at time t, N0 is the initial
233
number of spores at t=0, µmax is the maximum specific growth rate and λi is the lag
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time of individual spores following the gamma distribution. The model describes the
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growth of a bacterial population, initially consisting of N0 spores, over time as the sum 11
ACCEPTED MANUSCRIPT 236
of cells in each of the N0 imminent subpopulations originating from a single spore.
237
The above approach allows for taking into account the heterogeneity in the growth
238
dynamics of single spores by introducing λ variability in the model using Monte Carlo
239
simulation. Figure 5 presents the output of the model (Εq. (5)) for spore levels N0 equal to 1
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(Fig. 5a), 10 (Fig. 5b) and 100 spores (Fig. 5c) at 45 °C based on a Monte Carlo
242
simulation with 10,000 iterations and with a Uniform distribution for t [t ~ Uniform
243
(0, 80)]. As shown, the output of the model for N0 = 1 spore is a stochastic growth
244
curve in which the number of cells in the population at any time is a distribution. An
245
increase in N0 to 100 spores resulted in a significant decrease of the above variability
246
as well as in a shorter lag phase of the population. The model can be used to evaluate
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the probability distribution of the time-to-spoilage, with the latter being defined as the
248
time required by the initial contaminating spores to germinate and multiply to a
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spoilage level (Kakagianni et al., 2016).
250 251
4. Discussion
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The Bioscreen C method, applied in the present work to study the λ of single G.
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stearothermophilus spores, has been extensively used for monitoring single cell and
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spore kinetics (George et al., 2008; Métris et al., 2008; Dupont and Augustin, 2009;
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Miled et al., 2011; Aguirre et al., 2015; Blana et al., 2015). One of the main
256
assupmtions of the method is that growth is initiated from a single spore in a well.
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The number of spores in each well can be estimated based on the ratio between
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positive wells (wells showing growth) and negative wells (wells showing no growth)
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according to the Poisson distribution. In all experiments carried out in this study, the
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261
number of spores in a well between 0.22 and 1.2. According to the Poisson
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distribution, the above values correspond to a percentage of wells with one spore
263
ranging from 51.4% to 98.1%. Previous studies have shown that the above range
264
allows for the estimation of the distributions of single cell parameters (Métris et al.,
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2003; Stringer et al., 2005). Indeed, Stringer et al. (2005) studied the lag times of
266
Clostridium botulinum individual spores utilizing both the Bioscreen C and a
267
microscopic method, and reported that the two methodologies produced distributions
268
with very similar estimates of both lag mean value and variability.
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The findings of this study highlight a considerable heterogeneity in the λ of G.
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stearothermophilus single spores. In the case of spores, the lag time is mainly
271
determined by the germination and outgrowth processes (Smelt et al., 2002; Stringer
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et al., 2005). Thus, the observed λ variability can be considered as a reflection of the
273
differences in the rates of the above processes among spores. The stochastic
274
fluctuations in the number of the nutrient germinant receptors (nGRs) per spore due to
275
epigenetic variations among individual spores, appears to be one of the main reasons
276
of the significant heterogeneity in spore germination capacity at the population level
277
(Elowitz et al., 2002; Stringer et al., 2005; Chen et al., 2006; Webb et al., 2007;
278
Ghosh and Setlow, 2009; Peng et al., 2009; Kong et al., 2010; Zhang et al., 2010).
279
Moreover, the stochastic inactivation of nGRs during heat treatment can contribute to
280
the differences in the germination rates among individual spores (Smelt et al., 2005).
281
Finally the differences in the levels of specific lytic enzymes such as CwlJ and SleB,
282
which initiate the spore’s peptidoglycan cortex degradation during germination can be
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an additional variability source (Setlow, 2003; Moir, 2006; Ghosh and Setlow, 2009;
284
Peng et al., 2009). The storage temperature affected both the position and the spread of the λ
286
distributions of G. stearothermophilus spores. In agreement to previous single cell
287
studies (Métris et al., 2003; Guiller et al., 2005, 2006), the mean and the standard
288
deviation of the λ values estimated in this study followed a similar trend in relation to
289
temperature, with longer and more scattered λ as conditions move away from the
290
optimum temperature. Moreover, at all tested temperatures, the λ distributions of G.
291
stearothermophilus spores showed a positive skewness (Table 1) indicating an
292
asymmetry with a tail extending towards longer times. This confirmed available
293
microscopic and/or turbidimetric findings on other bacilli (Leuschner and Lillford,
294
1999; Smelt et al., 2008; Pandey et al., 2013) and clostridia (Billon et al., 1997;
295
Stringer et al., 2009, 2011), but also on non-sporeforming bacteria such as Listeria
296
spp. and Escherichia coli O157:H7 (Métris et al., 2003; McKellar and Lu, 2005;
297
Guillier and Augustin, 2006; Li et al., 2006; McKellar and Hawke, 2006). The above
298
skewed shape was sucessfully described with a Gamma distribution which has been
299
previously used for individual cell λ values (Métris et al., 2003). Information about
300
the shape of λ distributions are of great importance in predicting microbial growth at
301
low contamination levels due to the stochastic nature of microbial lag phase (Baranyi,
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1998, 2002).
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The temperature dependence of the mean G. stearothermophilus λ was
304
considerably different from that reported for the growth rate of the same strain by
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Kakagianni et al. (2016). In the latter study, the authors reported an optimum
306
temperature (mean value ± standard deviation) for growth Topt = 61.82 ± 0.20°C, 14
ACCEPTED MANUSCRIPT which is about 9°C higher than the corresponding parameter for the mean lag time
308
estimated in the present study (Table 3). This difference was confirmed by the µmax
309
values estimated in this study which showed a continuous increase as temperature
310
increased from 47.5 to 59.0oC, in contrast to what was the case for the mean lag time
311
which showed a minimum value at 55.0oC. These results indicate that the
312
physiological state theorem (Baranyi, 1998) applied for vegetative cells, considering
313
the lag as inversely proportional to the maximum specific growth rate, is not
314
necessarily valid in the case of spores. This can be attributed to the different processes
315
during the lag time of spores and vegetative cells. As described previously, in the case
316
of spores the lag time is mainly determined by the germination and outgrowth
317
process, the rate of which can be considerable different than the growth rate of
318
vegetative cells. Indeed, Stringer et al. (2005) reported that the time intervals for
319
germination, emergence, outgrowth and doubling of C. botulinum spores were all
320
independent of each other. Consequently, although in the case of vegetative cells with
321
known physiological state, growth at different temperatures can be predicted based on
322
a model for the growth rate (Koutsoumanis et al., 2010), predicting growth of spores
323
may require models for both lag time and growth rate since the effect of temperature
324
on these two growth kinetic parameters can be substantially different.
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G. stearothermophilus is an important spoiler in various heat treated foods.
326
Considering that the number of surviving spores in the final product is usually very
327
low, knowledge of their lag time variability is important for predicting growth and
328
spoilage in a probabilistic way (Fig. 5). Probabilistic modeling approaches that take
329
into account the variability of the factors affecting microbial behavior can provide
330
more realistic estimation of food quality. Most importantly, probabilistic models 15
ACCEPTED MANUSCRIPT 331
enable decision-making based on the “acceptable level of risk”, and can provide
332
structured information allowing decision-makers to compare various actions and
333
identify those that can lead to effective and economic reduction of quality risks.
334 Acknowledgements
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This study was carried out with the financial support of “Understanding the
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impact of manufacturing processes in the ecology of microorganisms that spoil-
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contaminate milk products (ESL, evaporated milk) and fresh fruit juices –
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Development of molecular methodologies and mathematical models for the prediction
340
of their shelf-life” within the framework of the action “Cooperation” (NSRF 2007-
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2013), that was co-financed by the European Social Fund (ESF) and National
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Resources.
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ACCEPTED MANUSCRIPT 560 Zhou, T., Dong, Z., Setlow, P., Li, Y.-q., 2013. Kinetics of germination of individual
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differential interference contrast microscopy. PloS ONE 8, e74987.
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Table 1
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Statistics of the individual spore lag time (λ) of Geobacillus stearothermophilus
λα (h)
Storage temperature Mean
St. Deviation
Median
45
224
37.9
18.6
33.3
47.5
199
21.4
17.7
15.3
50
227
11.7
11.8
55
199
10.9
9.20
59
193
17.9
15.0
b
Number of spores tested.
Kurtosis
16.4
72.6
1.39
2.62
1.26
56.9
1.03
0.102
6.46
1.50
37.8
1.66
2.06
7.16
0.705
29.9
1.42
1.85
13.6
0.760
47.7
1.00
0.305
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Skewness
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Individual spore lag time.
95th Percentile
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a
5th Percentile
SC
Nb
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(°C)
566
568
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ACCEPTED MANUSCRIPT Table 2 Parameter estimation of the Gamma distribution fitted to the individual spore lag times of Geobacillus stearothermophilus. The probability density function of the Gamma distribution in the shape-rate parametrization is:
45
4.69
47.5
1.30
50
1.28
59
RMSEa
7.84
0.0186
0.0316
7.81
0.0576
1.40
4.72
0.0603
1.10
17.1
0.0200
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Root Mean Square error
β
16.5
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a
a
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Temperature (oC)
= > ; ? @ AB :DE ; > 0 GHI <, = > 0 C <
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: ;, <, = =
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Table 3
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Estimated values and statistics for the parameters of the Cardinal Model with Inflection (Equations 3 and 4) describing the effect of temperature on the reciprocal of the mean lag time (λ) of individual spores of Geobacillus stearothermophilus. JK (h)
Standard Error
R2a
10.3
0.5
0.977
Tmax (oC)
64.2
Tmin (oC)
38.2
5.1
Topt (oC)
53.6
0.8
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R2: coefficient of determination.
2.3
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Estimated Value
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Parameter
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ACCEPTED MANUSCRIPT Figure legends Fig. 1. OD growth curves during growth of 199 single spores of Geobacillus stearothermophilus in tryptone soy broth at 45 °C.
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Fig. 2. Cumulative distributions of Geobacillus stearothermophilus single spore lag times (λ) at different growth temperature conditions. The number of data were 224,
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199, 227, 199 and 193 for 45, 47.5, 50, 55 and 59 °C, respectively.
Fig. 3. Comparison between observed and fitted quantiles of the Gamma distribution
50°C (c), 55°C (d) and 59°C (e).
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for Geobacillus stearothermophilus single spore lag times (λ) at 45°C (a), 47.5°C (b),
Fig. 4. Effect of temperature on the reciprocal of the mean lag times of Geobacillus
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stearothermophilus single spores fitted to the Cardinal Model with Inflection (solid line) (Equations 3 and 4). Points (○) represent observed values.
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Fig. 5. Simulations of Geobacillus stearothermophilus growth at 45 °C for various
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initial contamination levels: 1 spore (a), 10 spores (b) and 100 spores (c). Growth is predicted by the stochastic model (Equation (5)) using Monte Carlo simulation with 10,000 iterations and a Uniform distribution for time t [t ~ Uniform (0, 80)].
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0,35
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0,30 0,25 0,20 0,15 0,10 10
20
Time (h)
1
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2
30
40
50
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0
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OD (420-580nm)
0,40
60
70
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
59°C
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55°C 50°C
47.5°C 45°C
0
20
60 Lag time (h)
80
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1
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Fig. 2
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40
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Cumulative Probability
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100
120
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Fig. 3
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0.40 0.35
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0.25 0.20 0.15 0.10
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1/Lag^0.5 (1/h)
0.30
0.05
40
45
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0.00 50
55
Temperature (°C) 1
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60
65
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1 2
Fig. 5