Abstracts. Aline Metris

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Journal of Microbiological Methods, Volume 55, Issue 3 , December 2003, Pages 821-827

Distribution of turbidity detection times produced by single cell-generated bacterial populations

Aline Métris, Susan M. George, Michael W. Peck and József Baranyi

Institute of Food Research, Norwich Research Park, Colney, Norwich NR4 7UA, UK

Available online 19 September 2003.

ABSTRACT

The distributions of the times to turbidity for wells inoculated with single cells of Listeria innocua were determined in different environmental conditions (pH 4.5 to 7 and with 0.5% to 8% of NaCl at 30 °C). It was established by statistical analysis that the main source of the variability of the detection times, T, is the variability of individual lag times. A linear relation dev(T)~T was observed between the detection times and their standard deviation. At slow growth, other sources of variability became increasingly significant.

Author Keywords: Lag time; Single cell; Turbidity; Variability

1. INTRODUCTION

Turbidity measurements are used to estimate the growth parameters of bacteria as an alternative to traditional plate counts [McMeekin et al., 1993]. Their use has increased as new trends in predictive microbiology have started to focus on the quantification of the variability of bacterial responses to food environments. This is especially important in relation to microbiological risk assessment studies. Automated turbidity measurement systems allow the production of large amounts of replicate observations, which is vital for variance analysis calculations.

Turbidity is the ratio of intensities of the incident light intensity and the light scattered by the culture. At higher cell densities (between 10^6.0 and 10^7.5 CFU/ml), it follows Beer's law, i.e. turbidity is proportional to the cell concentration [McMeekin et al., 1993 and Begot et al., 1996]. The range of proportionality depends on the size and shape of the bacteria, which can in turn be affected by environmental conditions. For example, Listeria monocytogenes has a very narrow range of linearity in the presence of lactic acid [Le Marc, 2002]. The correlation between cell concentration and turbidity, also called calibration, depends on the bacterial species, and sometimes even on the specific strain used [Begot et al., 1996]. As the range where there is an acceptable relationship between turbidity and cell concentration is limited, the estimation of growth parameters is often inaccurate.

Some authors linked the parameters of optical density (OD) curves directly to the "real" (viable count) specific growth rate and the lag times(e.g. [Dalgaard et al., 1994, Begot et al., 1996, Augustin et al., 1999 and Dalgaard and Koutsoumanis, 2001]). They fitted growth curves to OD values and those curves were used, via calibration, to describe the increase in cell concentration. The results were reasonably good under most, but not all, conditions [Dalgaard and Koutsoumanis, 2001]. As an alternative, other authors proposed the use of the observed detection time, i.e. the time for a cell population to reach a detectable level of turbidity, instead of the whole turbidity growth curve [Cuppers and Smelt, 1993, McClure et al., 1993, Augustin et al., 1999, Baranyi and Pin, 1999 and McKellar and Knight, 2000]. When plotting the detection time against initial cell concentration, one should obtain a straight line with a slope inversely proportional to the specific growth rate. This technique was used in this study of the within-population variance of lag times of individual cells.

Traditionally, lag is defined on the "log(cell concentration) vs. time" curve. The end of the lag period is the time where the tangent drawn to the exponential phase of the growth curve crosses the level of inoculum. A more mechanistic definition was suggested by [Baranyi and Roberts, 1995]. An initial physiological state parameter (₀) was introduced and the lag was calculated as a function of ₀ and the subsequent specific growth rate.

The connection between the lag times of individual cells and the lag of the population is not straightforward. This was analysed by [Baranyi, 1998 and Baranyi, 2002] and [Baranyi and Pin, 2001] and a mathematical formula was derived to link the distribution of the individual lag times and the lag of the whole population. The formula implied that the average of the individual lag times can be much longer than the lag observed at population level, although the average of the individual physiological state parameters is the same as the physiological state parameter of the population. Furthermore, the transition curve from the lag to the exponential phase can be obtained by a transformation of the distribution of the individual lag times but, at the current accuracy of available data, it is impossible to deduce the distribution of the individual lag times from a population growth curve. Consequently, the lag time of individual cells cannot be studied using traditional viable counts. Automated turbidity measurements could provide a solution since they are suited to producing a large quantity of replicate detection time measurements. If each observed culture starts from a single cell, then the distribution of the detection times should be close to the distribution of the lag times of the initial individual cells assuming that the specific growth rate is the same and constant for each population engendered by a single cell.

In this paper, the individual lag times of Listeria innocua were determined from turbidity detection times produced by single-cell-generated subpopulations. Cells were subjected to different pH and NaCl concentrations and the effect of these environmental factors on the obtained distributions were analysed by statistical means.

2. MATERIAL AND METHODS

2.1. Experiments

2.1.1. Strain

L. innocua was used as a test organism, because it is safe to handle but has similar properties to the food pathogen L. monocytogenes, which is a major concern in the food industry. L. innocua, strain NCTC 11288, serotype 6a, was maintained on tryptone soya agar slopes (TSA, Oxoid) stored at 3 °C.

2.1.2. Growth curves

L. innocua was subcultured from a stock slope to tryptone soya broth (TSB) and incubated at 30 °C for 24 h. The culture was diluted 1:1000 in peptone salt dilution fluid (PSDF) and 50 l inoculated into 50 ml test medium. This was TSB supplemented with 1% glucose and 0.3% yeast extract (TSYGB, Oxoid). The pH was adjusted with 1 M HCl to 7.0, 5.5, 5.0 and 4.5. NaCl was added to pH 7.0 broth to give final concentrations of 4, 6 and 8% NaCl (w/v).

Broths were sampled immediately after inoculation and at intervals for up to 72 h whilst incubated in a water bath at 30 °C. Samples were diluted in PSDF and plated onto triplicate TSA plates for viable counts.

2.1.3. Growth experiments

The aim of the experiment was to have many separate volumes of media inoculated with, on average, one to two cells.

L. innocua was subcultured from a stock slope to TSB and grown to stationary phase at 30 °C for 24 h. The culture was diluted in PSDF to give 20 cells ml⁻¹. The diluted culture was inoculated into the wells (50 l well⁻¹) of two one hundred well plates. Each well was then topped up with 350 l of test medium (TSYGB pH 7.0 with or without added NaCl or pH 5.5, 5.0, 4.5 without added NaCl). Filled plates were placed in a Bioscreen C automatic reader (Labsystems) and incubated at 30 °C. Turbidity was monitored at a wavelength of 600 nm at regular intervals for up to 7 days.

To estimate the number of cells inoculated into each well, dilutions of the inoculum culture were plated onto TSA plates incubated at 30 °C for 2 days.

Experiments were repeated two or three times for each environmental condition.

2.1.4. Detection times

Detection time was the time taken for the well to reach a turbidity of 0.11. This corresponded to a concentration of 10⁷ cells ml⁻¹ and was checked for each environmental condition by viable counts.

2.1.5. Accuracy of the readings

The accuracy of the Bioscreen readings was checked by carrying out an experiment under the same conditions as the growth experiments, but using 10⁷ cells ml⁻¹ instead of 20. The lag phase for large populations in optimum conditions should be constant, so the variability between wells is that of the Bioscreen readings.

2.2. Theory

Suppose a well is inoculated with N₀ cells and that, after the lag period, the bacterial population grows at a constant specific growth rate , reaching N_det, the detection level, at the time T_det, at which point the culture is still in the exponential phase. Then the detection time T_det can be calculated by the following formula

(1)

where (N₀) denotes the lag time of the culture in the well with N₀ cells. The distribution of the individual lag times when N₀=1 should be reflected by the distribution of the detection times (see Fig. 1). However, the number of cells per well obtained by serial dilution, N₀, is not necessarily exactly 1, and the variability of N₀ contributes to the variability of the observed T_det values. When analysing the sources of the variability observed in the detection times, we can safely assume that:

–the specific growth rate is homogeneous in the population and constant after the first division of an initial cell;
–N₀, the initial number of cells in a well, follows the Poisson distribution.

Fig. 1. For a subpopulation generated by a single cell (N₀=1), the distribution of the detection times is the same as the distribution of the lag times of the individual cells, assuming the same growth rate.

We checked experimentally (see above) that:

–The detection level corresponded to the same cell density whatever the conditions studied.
–The inaccuracy of the OD measurement (noise) of the Bioscreen was not significant.
–The detection time values, in identical growth conditions, were replicable.

2.3. Simulation

Eq. (1) was used to analyse the dependence of the distribution of the T_det detection times on the four (more or less random) variables, N₀, N_det, and (distribution of which depends on N₀). The simulation studies were carried out in Excel and analysed with @Risk (Risk analysis add-in to Microsoft Excel™, Version 4.05, Professional Edition, Palisade, USA, 2000) with the following input:

–The initial number of cells in a well followed a Poisson distribution with an average of 1 cell per well;
–The lag times of single cells, (1), were assumed to be Gamma distributed, following the theory of [Baranyi and Pin, 2001]. A Gamma function is defined by two parameters and . Under optimum growth conditions (30 °C, pH 7, 0.5% NaCl), was taken equal to 5 and =0.5 h, leading to an average of 2.5 h for the expected lag time (note that 2.5 h is the average of the lag times of the single cells, which is different from the population lag time) and a standard deviation of 1.12 h. When there was more than one cell per well, the lag time was estimated by the formula derived by [Baranyi and Pin, 2001]. With the same notation as in Eq. (1),

(2)

–The specific growth rate was assumed to follow the normal distribution, with 1 h⁻¹ mean and 0.03 h⁻¹ standard deviation. This corresponds to about 3% relative deviation, which was established by generating independently replicated viable count growth curves. Fig. 2 shows four of these curves, from cultures grown under optimal conditions. Similar curves were also generated under less favourable environmental conditions. They showed that the maximum specific growth rate is a well reproducible parameter, with 1–5% error margins in growth supporting environments.

–The distribution of the number of cells at the detection level was assumed to be uniform, between 5×10⁶ and 1.5×10⁷ cell/ml.

Enlarge Image

Fig. 2. Replicate viable count growth curves. The specific growth rate is a very well reproducible parameter, with a relative standard error of less than 3%.

3. RESULTS AND DISCUSSION

3.1. Simulation

Ten thousand iterations with a Monte Carlo sampling method gave a standard deviation of 1.25 h for the T_det detection times compared with the 1.12 h deviation for the individual lag times. In analysis of variance (ANOVA) terms, 1.12²/1.25²=80% of the variance of the detection times was explained by the variance of the lag times. Note that the lag time variability was relatively small in this simulation. Had the cells been stressed, their lag times would have been more scattered and the contribution of the variability of the individual lag times to the overall variability would have become even more dominant.

The simulated T_det detection times kept the shape of the Gamma distribution. All the distributions available in @Risk (lognormal, etc.) were fitted to the T_det values. The Gamma distribution gave a good fit, with a Chi-square value up to 93%. The variance of T_det was somewhat higher than that of the individual lag times.

It was concluded that the Gamma distribution was suitable to fit the distribution of the T_det values, and that most of the variability of the detection times could be attributed to the variability of the lag times.

The simulations also showed that the number of positive wells, necessary to get a good estimation for the distribution of the detection times, is around 100.

3.2. Experimental results

The Gamma distribution proved to be suitable to fit the measured T_det detection times (Fig. 3). This is not in agreement with the findings of [Wu et al., 2000], who fitted a normal distribution to the distribution of the lag time of individual cells. However, they reported on 40 replicates at each dilution, which, in our experience, is not sufficient to establish which distribution is the best.

Fig. 3. Observed and fitted distribution of the detection times (in hour) of a culture grown at 30 °C, pH 7 and 0.5% NaCl. Columns: histogram of observed detection times. Continuous curve: frequencies generated by the Gamma distribution fitted to the data by the maximum likelihood method.

Out of the 200 wells of the Bioscreen microtitre plate, the average number of positive wells was always between 125 and 175. Based on the Poisson assumption, this means that the mean of N₀ was between 1 and 2 in all experiments.

We have established that the main source of variability in the detection times was due to the variability of the individual lag times. To test if the effect of growth conditions was consistent, two or three replicate experiments were carried out in each environmental condition. The total variance of the detection times in one set of environmental conditions was split into three independent variances

V_Total=V_R+V_E+V_H,

where (i) V_Ris due to the Bioscreen reading inaccuracy; (ii) V_E is due to the variability of the effect of environmental conditions; (iii) V_H is due to the heterogeneity of the cells' physiological state when they are inoculated. This is what is measured by the distribution of the detection times (its main source is the variability the individual lag times).

The Bioscreen reading inaccuracy, V_R, was evaluated with the high level inoculation experiment and was assumed to be the same in any environmental condition. The variability due to the environmental conditions, V_E, was the variance of the means of the detection times in experiments carried out in the same conditions. The means were weighed with the standard deviation of the detection times. Finally, the variability due to the population heterogeneity, V_H, was obtained by subtracting the former variances from the average total variance in a given set of environmental conditions. The result was expressed as an R² percentage, the proportion of the total variability due to the heterogeneity of the cells' physiological state at inoculation (Table 1). In most cases, the replication of the environmental conditions was satisfactory (variability due to the population heterogeneity higher than 95%). This was, however, not true for experiments done at pH 5.0 or 5.5; results could not be repeated consistently at these low pH values.

Table 1. Sources of variability of the detection times at different environmental conditions

The main source is generally the heterogeneity of the cells' initial physiological state, V_H, except at low pH, when the variability of the effect of the environmental factors, V_E, can be of equal weight.

As expected, the more inhibiting the growth condition was, the longer were the detection times and the more scattered their distribution. In Fig. 4, the standard deviation of the detection times is plotted against their average for each growth condition. It can be seen that, in the studied region of environmental factors, the relation between the standard deviation and the average of the detection times is more or less linear. It is notable, however, that the standard deviation remained almost at the same level with 4% NaCl as with 0.5% before increasing. This is in agreement with the results of [Patchett et al., 1992], who found that increases of potassium and betaine, among other osmoprotectants accumulating in the cell as the NaCl concentration increases, step up significantly from a concentration of NaCl of about 5%.

Fig. 4. Standard deviation of the distributions of the detection times vs. their mean value, for all the studied growth conditions.

4. CONCLUSION

It would be desirable to control the variability of the bacterial lag time of pathogens to produce safe for example minimally processed food. However, technically, it is difficult to get sufficient and sufficiently accurate data on the lag times of individual cells. Turbidity detection times may provide a means to deal with the problem. As we have shown, the main source of the variability of detection times is close to the variability of individual lag times, if the initial number of cells is low. As a first approximation, a linear relation dev(T)~T can be established between the detection times and their standard deviation. However, if the growth is slow, then other sources ("noise") can match the proportion of the original variability and the linear relation may not be easily observed/reproduced.

ACKNOWELEDGEMENTS

This work was prepared partly under the project CSG 434.1213A of the Institute of Food Research and partly funded by the EU project QoL 2000-01145 (BACANOVA).

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