1 Departament d'Estadística, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007-Barcelona, Spain
2 Departament de Microbiologia, Universitat de Barcelona, Av. Diagonal, 645, 08028-Barcelona, Spain
Correspondence
Josep Vives-Rego
jvives{at}ub.edu
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ABSTRACT |
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INTRODUCTION |
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Flow cytometry has become an important tool in microbiology, as it combines direct and rapid assays to determine numbers, cell-size distribution and additional biochemical analysis of individual cells (Robinson, 1999; Shapiro, 2003
; Vives-Rego et al., 2000
). This makes it particularly attractive in the study of heterogeneous bacterial populations (Davey & Kell, 1996
; Vives-Rego et al., 2000
). Flow cytometry cell-size estimates are based on the intensity of forward light scatter (FS), which is used in preference to 90° scatter or side light scatter (SS) because of its high signal intensity and its insensitivity to subcellular structure conventionally described as granulosity. FS is normally assumed to be proportional to bacterial size (Christensen et al., 1995
; Julià et al., 2000
; Koch et al., 1996
; López-Amorós et al., 1994
), although the relationship between particle size and FS is not monotonic, as it is also affected by cell structure and chemical composition (Shapiro, 2003
).
Studies on the heterogeneity of bacterial axenic cultures are scarce, despite the fact that there is an obvious need to understand its morphological, biochemical and genetic bases. In addition to this, the understanding and monitoring of the internal heterogeneity of axenic cultures is crucial to an understanding of species evolution and diversity. Bacterial macromolecules are constantly synthesized and decomposed at variable rates, due to thermal fluctuations that affect cells. Even in carefully controlled experiments, the quantities and types of bacterial molecules vary from cell to cell, producing cellular-composition fluctuations that may be the origin of the detected spatial inhomogeneity in axenic cultures. Also, a general relationship has been reported between fluctuation and response in Escherichia coli clones that might prove useful in predicting evolution rate (Kaneko & Yomo, 1994; Ko et al., 1994
). It is assumed that such fluctuations are inevitable in living organisms.
To our knowledge, log-skew-Laplace has never been applied to the size distribution of bacteria, nor as a complement to flow-cytometric statistics. In this paper, we report fit to the log-skew-Laplace distribution for SS values in Gram-negative axenic cultures and we analyse various theoretical and applied considerations related to their biological properties.
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METHODS |
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Flow-cytometric analysis.
A Coulter Epics Elite flow cytometer equipped with an air-cooled 488 nm argon-ion laser at 15 mW power was used. Fluorescent beads (1 µm Fluoresbrite carboxylate microspheres; Polysciences Inc. and 4 µm latex fluorosphere beads; Molecular Probes) were used as an internal standard for scatter and fluorescence. The FS detector in the Elite flow cytometer is a photodiode that collects light between 1·5 and 19° from the laser axis and is able to detect particles as small as 0·5 µm in diameter. The SS detector is situated at a 90° angle from the laser axis. Due to the design of the closed flow chamber used, light for both SS and fluorescence is collected at an angle wider than 90°, using a combination of mirror and lens to improve efficiency. Data were analysed with Elitesoft version 4.1 (Coulter Corporation) and WinMDI version 2.5 (Trotter, 1999) software.
Cell-size determination.
Cell sizes were determined with an electronic particle-size analyser, Multisizer II (Coulter Corporation), with an aperture tube of 30 µm diameter and processing 100 µl cell suspension in 0·9 % NaCl, previously filtered through 0·2 µm pores. Three types of size measurement were obtained after the transformation of the electric pulses generated by the counter: diameter, volume and revolution surface. Data were analysed by AccuComp software version 1.15 (Coulter Corporation). Files generated by the particle-size analyser (Multisizer II) were exported in an ASCII (tab-delimited) format. Listmode files generated by the flow cytometer were opened with WinMDI software version 2.5 (Trotter, 1999). FS and SS were saved as a single parameter in an FCS ASCII format. The files thus generated were opened and formatted to a single column by using a tailored Microsoft Word macro.
Statistical theory and models.
Our goal was to find a satisfactory fit, should one exist, between the measurements yielded by the flow-cytometric scatters and the skew-Laplace distribution. The relationship between Multisizer II measures and the skew-Laplace distribution was also explored. Previous work has shown that size distributions given by the flow cytometer and other methods are not normal (Koch, 1987; Koch et al., 1996
; López-Amorós et al., 1994
; Vives-Rego et al., 1994
, 2000
; Wagensberg et al., 1988
). Other distributions have been tested in order to measure particle size in various settings (Bagnold & Barndorff-Nielsen, 1980
; Barndorff-Nielsen, 1977
; Fieller et al., 1992
) and the hyperbolic and skew-Laplace distributions stand out from amongst them.
Our data clearly feature asymmetrical tails, explaining why a fit to the normal distribution has not been found. The skew-Laplace distribution allows a different shape for each of its tails, suggesting its adequacy for the present problem. We prefer the skew-Laplace distribution over the hyperbolic one, due to its simpler formula and easy parameter estimation. Estimation of the four parameters of hyperbolic distributions has been shown to be unstable (Fieller et al., 1992), with nearly identical hyperbolic distributions being representable by different parametervalue combinations. An in-depth study on the skew-Laplace distribution was reported by Kotz et al. (1998)
. The skew-Laplace distribution has three parameters, as shown below:
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Maximum-likelihood estimation of µ.
We define the following functions:
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The maximum-likelihood estimation of µ is the value xi, which minimizes function h evaluated at the points x1, x2, , xn, given that this minimum is not reached at either extreme, i.e.
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Maximum-likelihood estimations of and
.
Once the maximum-likelihood estimation of µ has been determined, we obtain the estimations of the remaining two parameters by using the expressions:
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The Laplace distribution can be regarded as a mixture of normal distributions. A symmetrical Laplace (=
) random variable X is represented as follows (Kotz et al., 1998
):
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In order to assess the adequacy of fit, we used two techniques, graphical and numerical. Plots of data quantile versus skew-Laplace quantile (quantilequantile plot) were obtained. Graphically, the closer this plot is to a straight line, the better the fit. To quantify the suitability of the skew-Laplace distribution, we calculated the critical size, Ncrit, as proposed by Fieller et al. (1992). This statistic can be interpreted as the critical sample size required to just detect a lack of fit at the 5 % level. The critical size, Ncrit, is a statistic based on the
2 goodness-of-fit test, which, as we have shown (Vives-Rego et al., 2003
), is more appropriate in our context (flow-cytometer or Multisizer II data) than other tests of goodness of fit, such as Kolmogorov. A comprehensive description of goodness-of-fit tests was described by Conover (1971)
. Ncrit is defined as
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RESULTS |
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In general, the SS values fit better to the log-skew-Laplace distribution than to the arithmetic skew-Laplace distribution. Fieller et al. (1992) have already commented that it is more appropriate to consider models based on log-size because of the wide range of sizes, but also because of the multiplicative process of breakage underlying particle production in general. These arguments are applicable specifically to bacterial cultures as, in addition to the variability of sizes that always exists in a culture, the binary transversal division of bacteria involves a breakage process of mother cells, producing smaller daughter cells but preserving ribosome density.
Interpretations of the skew-Laplace distribution in bacterial cultures
Our interpretation of the correlation between skew-Laplace distribution and SS values in Gram-negative bacteria is as follows. SS values reflect cell granulosity, which in turn is an artefact of ribosome density. The more ribosomes, the higher a cell's metabolic activity and cellular performance is. Following this, a good skew-Laplace fit would suggest that an axenic culture is made of two ribosome-density subpopulations that are independent of each other. In other words, the population with high SS (or granulosity, ribosome density or metabolic activity) would be dynamically distinct from the subpopulation with low SS. This situation is present at various points during the incubation period, meaning that, irrespective of the cell-cycle phase, the two subpopulations with different SS coexist in the culture due to their differing biological potential. Consequently, in Gram-negative axenic cultures of any physiological state (young or old), the dominant population is more metabolically active than the non-dominant one, which is an indication that potential cellular changes or reactions towards survival or the mechanisms that generate diversity rely on the existence of two independent populations.
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DISCUSSION |
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The skew-Laplace distributions were originally proposed as a pragmatic alternative to the four-parameter hyperbolic family, which had proved notoriously unstable in numerical estimation. This latter family is a natural generalization of the normal distribution and there are theoretical arguments indicating that it might be an appropriate model for size distribution in a wide variety of cases. The skew-Laplace family is a subclass of this more general set. Reliable software is available for fitting hyperbolic distributions, but we have ruled out this family because it is too complex and because similarity of distribution does not necessarily imply that the parameters are similar.
A practical consequence of this newfound property of axenic cultures is that a flow-cytometric definition of high-granulosity subpopulations would be useful in the obtention and selection of mutants (experimentally or naturally). Once the high-granulosity subpopulation is sorted cytometrically, any genetic process affecting the sorted population will be more productive than in cultures containing both high- and low-granulosity populations. Spontaneous and naturally induced mutations would also be more effective in exclusively high-granulosity subpopulations than in conventional mixed cultures containing both high- and low-granulosity populations. Any experiment or natural process within axenic cultures that is intended to obtain high numbers of mutations will be more successful if applied to or occurring in the subpopulation with high granulosity, rather than in that with lower granulosity.
One biological interpretation of the fitting is that bacteria share a general mathematical distribution with small, repetitive biological and non-biological materials. This shared mathematical behaviour probably also reflects a general physical law that applies to all small particles, irrespective of whether they have a biological origin. Another biological interpretation refers to the two independent subpopulations detected by the Laplace distribution. The Laplace fitting confirms that bacterial axenic cultures are made up of two subpopulations, as reported previously by others (Koch, 1987; López-Amorós et al., 1994
; Vives-Rego et al., 2003
; Wagensberg et al., 1988
). In addition, the Laplace fitting shows that the distributions of the two subpopulations are mutually independent. This mathematical independence indicates that the two corresponding biological subpopulations may also be independent. If this is demonstrated, then the concept of axenic culture will need to be re-examined.
Finally, a key question is whether the fit of SS values to the skew-Laplace distribution that we have observed in three Gram-negative small bacilli is applicable to all micro-organisms or whether it is only the case for Gram-negative bacteria. To answer this question, more cross-species comparative work is necessary.
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ACKNOWLEDGEMENTS |
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REFERENCES |
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Received 2 July 2004;
revised 18 November 2004;
accepted 24 November 2004.
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