The Poisson Distribution Is Applied to Improve the Estimation of Individual Cell and Micropopulation Lag Phases ()
1. Introduction
The measuring of any parameter characterizing the microbial growth is essential for any quantitative microbial risk assessment. Then, to know the microbial lag phase length of viable cells is critical, especially in RTE products, which nature and storing conditions may allow the growth of viable, pathogen or not, bacteria. In the case of populations of thousands or hundreds of viable cells, the lag phase, is quite reproducible if the pre-inoculation and growth conditions are constant. However, the lag phase of populations composed by few cells, or even by only individual cells, is inherently variable. Therefore, it is understandable that researchers [1-9] have paid attention to the distribution of single cells lag times and to the techniques that can measure them. Measuring the lag time of individual cells requires direct microscopic observation [4,8] or techniques to isolate single cells [10]. Cell isolation can be achieved by diluting [2], sorting by flow cytometry [11] or inactivating all organisms except one [9]. When growth is detected in some samples and not in others, it is commonly assumed that growth comes from one cell [12]. The number of samples must always be high for reliable mathematical treatment. It is recommended that approximately 100 samples show growth [13], and this figure must not be a high percentage of the samples. Guillier et al. [5] stated that if 35% of samples show growth, this should not significantly affect individual cell lag phase distributions because at least 80% of samples contain one cell, according to the Poisson distribution function.
“Growth/no growth sampling” has been widely applied to foods and opaque liquids; in the special case of translucent liquids, an apparatus called the Bioscreen C can be used to construct 200 growth curves simultaneously for the same temperature, on the basis of the turbidity resulting from microbial growth. If the specific growth rate (μ) under the experimental conditions is known, the lag phase is determined using the following equations. In the case of translucent liquids analyzed using Bioscreen, the equation is [14]:
(1)
where Td is the detection time, i.e. the time needed to reach an arbitrary absorbance (turbidity), Ln(Nd) is the natural logarithm of the number of cells generating such absorbance, Ln(N0) is the natural logarithm of the number of organisms in the inoculum, and μ is the specific growth rate. In the case of opaque samples [6], the equation is:
(2)
where Tcount is the time between inoculation and plating of the sample, Ln(xcount) is the natural logarithm of the cell number detected at Tcount, Ln(xinitial) is the natural logarithm of the initial number of bacteria and µ is the specific growth rate.
When a certain percentage of samples does not show growth, the assumption that growth in the other samples is due to one cell contravenes the predictions of the Poisson distribution. Several researchers have used the Poisson distribution to calculate the proportion of growthpositive samples initially containing more than one cell [5,11,15,16]. McKellar and Hawke [17] recognised that one of the limitations of the Bioscreen as a tool to study single cell behaviour is that it is difficult to ensure that the growth in any given positive well arose from a single cell. Earlier, some authors [2,15] performed a series of binary dilutions to have one cell per sample. Francois et al. [2] observed that single cells should be found in wells of Bioscreen microtitre plates where the mean cell number added to each well was less than one. These authors advocate pooling data from the last five binary dilution series to maximise the number of replicate wells; these series contained 0.7812, 0.3906, 0.1951, 0.0977, and 0.0977 cells per well, from a theoretical mean dilution range.
According to the Poisson probability function, if a determined number of occurrences (ρ is expected, then the probability that there are exactly k occurrences (k being a non-negative integer number, k = 0, 1, 2, ∙∙∙) is:
(3)
where e is the base of the natural logarithm; k is, in our case, the number of organisms in a sample, and the probability of k is given by the function; ρ is a positive real number, which expresses the average number of cells per sample; and k! is the factorial of k. To highlight the relevance of the data that equation (3) offers, the Table 1 has been built up. This table shows the percentages of samples predicted by equation (3) that would contain a determined number of viable cells as a function of the percentage of samples, in which growth was detected. The average number of cell per sample (m) is also shown in Table 1. This average is calculated by assuming that the number of cell per sample follows a Poisson distribution. Hence, the following equation was used:
(4)
where P is the probability of there is not any viable cell in a sample. Applying equation (3) to the data of Francois et al. [2], with an average number of cells per sample of 0.78, indicates that 65% of the positive samples contain one cell, 25.3% contain two, 6.6% three and 1.3% four. These figures suggest that the estimated lag phase determinations for individual cells will have a certain error. Indeed, Baranyi et al. [18] affirmed that the greater the Poisson parameter (ρ, average number of cells per sample), the less accurately equations (1) and (2) estimate the distribution of the single cell lag time. If samples are considered to contain only one cell, the value of Ln(N0) in equation (1) and Ln(Xinitial) in equation (2) is zero. However, if the predictions of the Poisson function are applied, we have to assume that some samples contain two, three or more cells, which is an undisputed fact in most real samples. In this case, Ln(N0) and Ln(Xinitial) are positive numbers that lengthen the lag phase of such samples.
The aim of this study is to compare the individual cell and/or micropopulation lag phase distributions obtained by assuming that all samples with growth contain one
Table 1. Percentage of samples with a determined number of cells, as predicted by Poisson function (equation (3)).
cell, with the distributions obtained by assuming a different number of cells per sample according to the Poisson distribution function.
2. Material and Methods
2.1. Simulation
A simulation was generated considering a different average number of cells per sample (0.2 - 2.0). To create the simulation, 100 values of lag phases were randomly generated by assigning them values from 40 to 180 arbitrary time units, following a gamma distribution with the following parameters: shape = 5.5, scale = 16.5, mean = 91.7 and standard deviation = 29.4. A specific growth rate (μ) of 0.0693 h–1 was also considered. The resulting distribution data are those of Scenario I (see next section).
2.2. Scenarios
Four scenarios were used to calculate the lag phase distributions: Scenario I assumes that all samples contain one cell. Scenarios II-IV use the Poisson distribution function to assign a number of cells to each sample. In Scenario II, the sample with the shortest lag phase contains the highest number of cells, according to the average number of cells per sample and the Poisson table [19], the sample with the second shortest lag contains the second highest number of cells, and so on. In Scenario III, the number of cells is randomly distributed among samples, regardless of the lag phase length. Scenario IV is calculated like Scenario II, except that all samples with more than one cell are not considered. From the data of Scenario I, lag phases were recalculated according to the assumptions of Scenarios II, III and IV and the corresponding distributions were obtained.
2.3. Statistical Analysis
Pairwise comparisons of the variances of lag phase distributions were carried out using a permutation test to analyse homogeneity of the two variances; this bilateral test assumes that the variances ratio is one. Permutation tests are non-parametric significance tests based on permutation resampling without replacement, with observed lag times drawn at random from the original data and reassigned to the two groups being compared. The distribution of possible variance ratios is calculated for all samples assuming the null hypothesis of homogeneity, and the observed ratio is positioned along this distribution. Values falling outside the main distribution rarely occur by chance and therefore give evidence of heterogeneity of variances [20]. Since our study involves multiple comparisons of several groups, a p-value correction must be applied in order to minimise the probability of rejecting a true hypothesis. The Holm-Bonferroni p-value correction [21] was applied. This correction is less conservative than those of Bonferroni and Sidak [22], which are also applied in the permutation test program described in appendix A.
A permutation test routine including a multiple comparison test was programmed using R language [23], which is described in Annex 1.
2.4. Application of Scenarios to Experimental Data
To check how well the simulations mimic the reality, the four scenarios were applied to the lag phases of Enterococcus faecalis, Pseudomonas fluorescens, Salmonella enterica serovar Enteritidis and Listeria innocua subjected to different irradiation treatments in tryptic soy broth (TSB) and cooked ham and subsequently incubated at different temperatures (experimental data from Aguirre et al. [9]). Lag phases were estimated according to equations (1) and (2) after determining the percentage of samples without growth and considering the Poisson function predictions and the scenarios above described.
3. Results
3.1. Simulation
Figure 1(a) shows the increase in the mean lag and Figures 1(b) and (c) show, respectively, the decrease in the standard deviation and the coefficient of variation as a function of the average number of cells per sample. The dashed line in Figure 1 shows the mean and standard deviations of the data in Scenario I. As expected, the greater the number of cells per sample is, the larger the difference between the mean and standard deviation of Scenario I data and those of the others. The distributions of Scenario III were not considered further because the averages were identical to those of Scenario II and their standard deviations were very close to those of Scenario I (data not shown).
Comparison of the distributions obtained in Scenarios I and II, I and IV and II and IV (Figure 2) shows that the higher the more probable number per sample is, the smaller the p value for comparisons of the mean and standard deviation, according to the permutation test. Significant differences (α < 0.05) were found among the three comparisons for the mean and between Scenarios I and II and I and IV for the standard deviation.
3.2. Application of Scenarios to Experimental Data
Table 2 summarises the experimental data, including the expected inactivation according to the irradiation applied, the average number of surviving cells per sample and the