Light scattering, as it was first proposed and substantiated by Wyatt in 1968 (1), is a powerful physical method for bacteria identification and characterization (2–4). The intensity, polarization, and spatial distribution of light scattering by a particle depend on the overall particle morphology, including shape, internal distribution of the refractive index, and the particle orientation relative to the incident beam. Therefore, spatial distribution, i.e., angular dependence of light-scattering intensity provides valuable information on morphological properties of a particle (5). Under certain a priori assumptions, the morphology information can be used to identify microorganisms (distinguish them among several classes) and can also provide a real-time monitoring of bacterial growth (6). Moreover, light-scattering can be used in combination with other (e.g., fluorescent) methods to correlate morphological changes with cell cycle phases. However, the utility of any light-scattering method for this application largely depends on the precision of morphology assessment, e.g., of single-bacteria length measurement.
Angle-resolved light scattering was applied by many researchers in studies of bacterial cells, both on bacteria suspensions and single cells. The angular dependence of light scattered by suspension of spherical Staphylococcus albus was measured with “Differential I” light-scattering photometer, which agreed with calculations using the Mie theory (7). The measurements of the light scattering from randomly oriented heterogeneous cultures of E. coli in water were carried out with a special photometer in the angular range from 10° to 90° (8). The Rayleigh-Debye approximation applied to the homogeneous ellipsoid correctly predicted the observed positions of scattering minima. Bronk et al. (9) measured combinations of Mueller matrix elements for a randomly oriented suspension of E. coli in the scattering angles ranging from 10° to 150°, which were demonstrated to be in a good agreement with calculations performed using the coupled dipole approximation for cells modeled by a cylinder capped with hemispheres (10). In combination with measurements of the cell volume by Coulter method, they demonstrated rapid determination of both average diameter and average lengths of bacterial population (11).
Sequential analysis of individual cells generally provides better accuracy of characterization of population in comparison to measurement of whole cell suspension. The most powerful light-scattering technique for analysis of multiple individual bacteria is flow cytometry (12–14). An ordinary flow cytometer allows one to measure only two numbers from light scattering which are the intensities of forward scatter (FSC) and side scatter (SSC). However, the relationship between cell morphology and these two parameters is generally so complicated that little hope is left for a detailed characterization of bacterial population. This is exemplified by Müller and Nebe-von-Caron (15), who recently reviewed the problems with the flow cytometric analysis of bacterial cells. With regards to light scattering, they also stated that “the performance of most cytometers is insufficient to analyze the small differences in the signals or to obtain any signal at all due to lack of critical alignment and instrument noise.” These factors become critical, because the ordinary FSC and SSC light-scattering signals per se are instrument-dependent in contrast to angle-resolved light scattering of individual bacteria.
A logical development of standard flow-cytometric is to measure a multiangle scattering or even an entire light-scattering pattern (LSP) of individual particles in a wide angular range. The authors (16, 17) performed differentiation of four different bacterial species using multiangle (four-ring) FSC flow cytometer. The measurement of the LSP was realized with the scanning flow cytometer (SFC) (18, 19). There was shown that LSPs are very sensitive to a particle morphology, i.e., physical dimensions and consistence, as well as to an orientation of the particle within an incident laser beam. The high-sensitivity of LSPs to particle morphology improves cell identification from light scattering in flow cytometry substantially (20, 21). Moreover, an analysis of the LSP potentially allows one to determine morphological characteristics of biological particles (5). However, such characterization constitutes the inverse light-scattering (ILS) problem, which is a field of active research, see e.g., Refs.22 and23.
LSPs of individual E. coli cells were first experimentally measured by Shvalov et al. (24) in logarithmic and stationary phases of cell growth. They demonstrated qualitative differences of LSPs between these two phases, but have not addressed the characterization problem due to unbearable computational complexity. However, a number of methods have since been developed for the solution of the ILS problem for a single particle. In particular, global optimization methods were used for robust characterization of single- and multi-layered spheres, even with large experimental noise (22, 25). Unfortunately, optimization is not feasible for nonspherical particles due to large computational cost of the direct light-scattering problem. This can be alleviated by using a preliminary calculated database of LSPs and solving the inverse problem by the nearest-neighbor interpolation. This approach was demonstrated for spheres (26), spheroids (27), and biconcave disks (28).
In this article, we studied light-scattering properties of individual E. coli cells. We developed a method to solve the ILS problem for individual rod-shaped bacteria, i.e., to characterize their morphology with the SFC, using a preliminarily calculated database of theoretical LSPs. The method itself is described in the “Materials and Methods” section, including standard sample preparation, measurements of LSPs with the SFC, database-based fit of experimental LSPs with theoretical ones, and uncertainties estimates. In the “Results” section, we illustrate the performance of this method on samples of two strains of E. coli and compare results to those obtained with the optical microscopy. Conclusion is given in the “Discussion” section.
Materials and Methods
Scanning Flow Cytometer
A detailed description of the SFC was given elsewhere (19). Here, we only briefly define the measured experimental signal. A 30 mW laser of 405 nm (Radius) was used for generation of LSP of individual particles. Another laser (660 nm, 30 mW) was used for generating trigger signal. The measured LSP is expressed as (19):
where S is the Mueller matrix (29), and θ and φ are polar and azimuth scattering angles, respectively. The operational angular range of the SFC was determined from analysis of polystyrene microspheres, as described in (22) to be from 10° to 40°.
Images of E. coli cells were obtained with optical microscope Carl Zeiss Axio Imager.A1 using 100× oil immersion objective with 1.3 numerical aperture. The microphotographs were processed using MATLAB-based software package MicrobeTracker (30), capable to automatically measure bacteria dimensions from microscope images (default parameter set alg4ecoli.set was used). Typical processed image is shown in Figure 1—one can see that the largest bacteria are ignored by MicrobeTracker with current settings. We have not investigated this problem due to low number of such bacteria (<4%). However, it does truncate the tail of the measured distribution of sample over length.
The XL2-Blue (Stratagene) and XL10-GOLD (ATCC 55962) strains of E. coli were used in this work. The bacterial cells were grown at 37°C in a thermostat with vigorous shaking. The standard growth media LB (31) (10 g NaCl, 10 g tryptone, 5 g yeast extract, 1 L distilled water, pH 7.5) was used as a broth. After preparation, the medium was autoclaved at 120°C. Both used strains are resistant to ampicillin, so it was added to the medium in concentration 0.05 g/L for growth suppression of extraneous cells. No additional treatment of the sample was carried out before the experiments.
Optical Model of E. coli Cell
To simulate the scattering from the individual E. coli cells, we use an optical model of a cylinder capped with hemispheres of the same radius, previously used for simulation in (10) and based on microscopic analysis of E. coli cells. This model is described by three morphological parameters (length l, diameter d, and refractive index n) and an auxiliary parameter (orientation angle ψ of cell in the flow of the SFC). Deviations from this model are discussed in “Light Scattering by Dividing E. coli Cells” section.
Light Scattering Simulation
To simulate light-scattering by a single E. coli cell, we used the discrete dipole approximation (DDA), a general method to simulate light scattering by particles of arbitrary shape and composition (32). In particular, we used open-source code ADDA v.1.0 (33), which can run on a cluster of computers, parallelizing a single DDA computation. LSPs of E. coli cells [see Eq. (1)] were simulated for polar angle θ from 10° to 40° using default discretization of 10–11 dipoles per wavelength and step of 5.5° for integration over azimuthal angle φ. The refractive index of the medium (0.9% saline) is 1.337. We estimated accuracy of DDA simulations for six typical bacteria, comparing with DDA results using much finer discretization (40 dipoles per wavelength). Relative accuracy is better than 4% for any θ in the parts of the LSP, where intensity itself is significant, but is much worse for parts with negligibly small intensity. Nevertheless, the overall norm of the simulation error, defined as square root of S [Eq. (2)], is less than 3% of the LSP norm itself. All simulations were run on the compute cluster of Supercomputing center of the Novosibirsk State University (34). Typical simulation time of single LSP for average bacteria cell (0.6 × 6.2 μm) is 80 s on a single-core of Intel X5355 processor (2.66 GHz).
Light Scattering by Dividing E. coli Cells
The division of E. coli is known to be accompanied by formation of constriction in the middle of cell longer axis (35). To estimate the effect of cell division on LSPs and hence on the results of characterization, we compared LSPs of five different bacteria with and without constrictions. Here, we present only a single typical result in Figure 2. The difference between the two calculated LSPs is less than 10%, which leaves little hope to distinguish dividing from nondividing cells based on the noisy experimental LSPs. However, it also shows that our characterization algorithm (“Inverse Light-Scattering Problem” section) should be perfectly applicable to dividing cells. Thus, in the rest of the manuscript, we assume that dividing cells are characterized as a single cell with doubled length.
Inverse Light-Scattering Problem
To solve the ILS problem, we use a method previously developed in (22, 23) and briefly describe it below. The problem is transformed into the global minimization of the weighted sum of squares:
where β is a vector of four model parameters, Ith and Iexp are theoretical and experimental LSP, respectively, N = 64 number of LSP points (in the range of θ from 10° to 40°), and w(θ) is weighting function to reduce an effect of the noise on the fitting results (22):
Global minimization is performed by the nearest-neighbor interpolation using a precomputed database of 80,000 theoretical LSPs (36). Model parameters corresponding to the theoretical LSPs were chosen randomly from the ranges l ∈ [1.2, 8.0] μm, d ∈ [0.5, 1.2] μm, n ∈ [1.39, 1.41], ψ ∈ [0°, 30°], which amply cover the range of E. coli cells of the studied strains. For the refractive index, we advisedly selected a relatively narrow range of [1.39, 1.41] using existing information on refractive index for E. coli cells (37–40) to overcome the problem of parameter compensation (36). In other words, we restrict the additional information on refractive index obtained from the experimental LSP (because original bounds are already tight), but significantly increase the accuracy of measurement of all other bacteria characteristics, especially its length. The range of ψ is based on the hydrodynamic orientation of elongated cells in Poiseuille flow (18), which makes larger values of ψ physically improbable. Total one-time computational effort to calculate the database of LSPs is about 3,000 core-hours on the supercomputer.
Comparing an experimental LSP with all theoretical LSPs from the database, we do not only find the best-fit theoretical LSP with parameters β0 that minimizes S(β), but also obtain an approximate description of the whole surface of S(β). The latter is used to calculate probability density function P(β) over parameter space for a given experimental LSP through the Bayesian approach. P(β) is further used to calculate mathematical expectation μ = <β> (generally different from β0), standard deviations of parameter estimates, and 95% highest-posterior density confidence region (see (36) for details). The width Δ of the projection of the latter on a certain parameter is a robust measure of characterization accuracy. Finally, we note that typical processing time for a single measured bacterium is about 0.2 s on a standard desktop computer.
We measured LSPs of individual E. coli cells, total 2,918 and 7,367 for XL2-Blue and XL10-GOLD, respectively. The global optimization algorithm described in “Inverse Light-Scattering Problem” section was applied to each cell. Typical results of this procedure for three random bacteria cells from each of two strains are shown in Figures 3a–3c and Figures 3d–3f, respectively. We note that typical results for refractive index are noninformative because for 30% of E. coli cells the 95% confidence range (± two standard deviations) covers the whole range used for database construction (Fig. 3b). This agrees with the design goal of characterization method (see “Inverse Light-Scattering Problem” section).
In both samples, we have discovered a subpopulation of bacteria with unacceptably large characterization errors. This is best illustrated by the confidence regions in coordinates of the length, diameter, and orientation angle, which consist of two or more separated domains—typical example is shown in Figures 4b and 4c. The corresponding bacteria are characterized by high values of confidence range width: Δψ, Δr, and Δl, as illustrated by the corresponding maps (Fig. 5), and large standard errors of length, diameter, and orientation angle. For the comparison, the typical normal confidence region is shown in Figure 4a. Discontinuous confidence regions appear due to combination of factors that deform measured LSP, including imperfect alignment of SFC and deviation of a realistic shape from the used model, and peculiar dependence of the LSP on model parameters.
Because maps of Δψ versus Δl, Δd versus Δl (Fig. 5) allow unambiguous discrimination of normal and large-error events, we choose this criterion to remove the bacteria with Δψ > 20°, Δl > 2 μm, and Δr > 0.15 μm from further consideration (at least one of this condition). In other words, only the cells satisfying both gates G1 and G2 remain (Fig. 5), which constitute 38% and 17% of the original samples for XL2-Blue and XL10-Gold, respectively (1,104 and 1,235 cells, respectively). Because experimental deformations do not depend on model parameters, such a filter should not introduce a bias into the distribution of the whole bacteria population over the model parameters. For the remaining bacteria, the median uncertainty (precision) of determined length and diameter is 135 nm and 15 nm, respectively.
The characterization results for each strain are presented as distributions over length and diameter (best-fit values) in Figure 6 in comparison with corresponding distributions obtained with the optical microscopy (see “Optical Microscope” section), showing a good agreement. Moreover, our results fall within the typical ranges reported for other E. coli strains for the length (9–11, 41) and diameter (10, 11). The only significant difference is a minor fraction of cells from XL-10 GOLD sample, for which the SFC-measured diameter is unusually small (close to 0.5 μm), which is caused by relatively large uncertainties of diameter determination (median is 25 nm) for these particular cells. Also, distributions over SFC-measured parameters are narrower (inside the main mode of the distributions) due to performed gating strategy, which leaves the more accurately characterized part of events, although still representative of the whole population. By contrast, the uncertainty of microscopy measurements is larger (about 250 nm), which results in broadening of distributions near its mean values. The absence of large bacteria in distributions obtained with optical microscopy is caused by features of the computer program used for processing of microscope images, as described in “Optical Microscope” section, whereas the SFC measurements confirm the presence of these cells in populations.
Distributions over refractive index and orientation angle obtained from the characterization results are presented in Figure 7. One can see that results for refractive index (Figs. 7a and 7b) are noninformative, spanning the whole range that was originally chosen for database construction. This is consistent with single-cell results discussed above.
This article describes a new method for characterization of E. coli morphology, which allows high-precision determination of length and diameter of single bacteria in a flow from angle-resolved LSPs measured with SFC. We modeled E. coli as a cylinder capped with hemispheres and used the DDA to calculate a database of their LSPs in a wide range of model parameters. To solve the ILS problem, we performed the nearest-neighbor interpolation on this database. This allowed us to calculate the probability density function over parameter ranges for a given experimental LSP and to estimate the mathematical expectations and standard deviations for each model parameter. Single-cell measurements allow one to reliably measure the whole distribution of the E. coli sample over morphological characteristics.
The method was applied to two strains of E. coli cells, showing 135 and 15 nm median precision in determination of length and diameter of single cells, respectively, which is very good for optical methods. We also compared population distributions over model parameters to optical microscope measurements and obtained good agreement for both diameter and length. Unfortunately, the method does not allow the determination of refractive index of individual cells, thus we could not narrow the confidence range based on the literature data.
It is important to note that the SFC-based method is not specific to E. coli and can be directly applied to any rod-shaped bacteria. The only additional effort may be needed for extension of the database to larger or smaller bacteria sizes. Therefore, this method is promising for precise control of bacteria morphology during cell cycle studies or for monitoring changes in cell growth rates due to external affects.
E.A. Uvarova and V.S. Fishman (Institute of Cytology and Genetics, Novosibirsk, Russia) have kindly provided the studied E. coli strains. We also thank Susann Muller and two anonymous reviewers for helpful comments on this manuscript.