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Journal of Bacteriology, October 2008, p. 6376-6383, Vol. 190, No. 19
0021-9193/08/$08.00+0 doi:10.1128/JB.00539-08
Copyright © 2008, American Society for Microbiology. All Rights Reserved.

Department of Geological Sciences, University of Saskatchewan, 114 Science Place, Saskatoon, Saskatchewan S7N 5E2, Canada,1 Department of Pediatrics, Stanford University School of Medicine, 300 Pasteur Drive, Stanford, California 94305,2 ExxonMobil Biomedical Sciences Inc., Annandale, New Jersey 08801,3 School of Life Sciences, University of Nevada at Las Vegas, Las Vegas, Nevada 891544
Received 19 April 2008/ Accepted 21 July 2008
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-sulfur (20). Known allotropes of sulfur include various ring forms, the most stable and common of which is cyclo-octasulfur (S8), and polymeric forms, known as catenasulfur, that consist of long helically wound chains (Sn). Various exotic forms of sulfur have been proposed to explain the properties of the sulfur globules, including micelles formed from long-chain polythionates [Sn(SO3)2]2– (34). Studies using polarizing microscopy and X-ray diffraction (16) have been used to argue that the globules contain sulfur in a liquid form. Conversely, laser Raman spectroscopy has given clear evidence that globule sulfur in Thioploca and Beggiatoa is present predominantly as S8 (21) and has even shown some evidence of microcrystallinity. Furthermore, globule sulfur is at least partly soluble in carbon disulfide and other organic solvents (reference 34 and references therein), a property exhibited by the ring forms of sulfur (such as S8) but not by polymeric sulfur. Sulfur K-edge X-ray absorption spectroscopy (XAS) (1) is a powerful in situ probe of sulfur biochemistry in intact cells and tissues. Under favorable circumstances, the technique can provide quantitative information on the chemical identities of the sulfur species that are present in a sample (11, 14, 23, 25, 33, 43). The data can be analyzed by using a curve-fitting approach that fits a linear combination of the spectra of standard compounds to that of the unknown (11, 23, 25). This type of analysis can provide quantitative estimates of the individual sulfur types in the sample but is obviously critically dependent upon the choice of reference spectra (23, 25).
Two independent groups have applied sulfur K-edge XAS in attempts to gain an understanding of the nature of the sulfur in bacterial sulfur globules but have come to totally different conclusions. Pickering, George, and coworkers (9, 23) concluded that the sulfur was principally in a form resembling cyclo-octasulfur but with predictable spectroscopic distortions caused by experimental artifacts. In contrast, Prange, Hormes, and coworkers (6, 7, 17, 26, 27, 28) concluded that the XAS data indicated mixtures of cyclo-octasulfur and significant amounts of polymeric sulfur and also concluded that there are substantial differences between the chemical forms of sulfur stored in the globules of different organisms. For example, they concluded that the globules of Beggiatoa alba and T. namibiensis contain predominantly cyclo-octasulfur (S8), while other organisms contain more polythionates (Acidithiobacillus ferrooxidans) (we note that this organism does not form globules and is not formally a sulfur bacterium) and polymeric sulfur (e.g., Allochromatium vinosum). In more recent work, they concluded that when A. vinosum is grown with elemental sulfur it shows a predilection for only polymeric forms (7). These studies are in contradiction to the earlier work of Pickering et al. (23), which concluded that the sulfur in all globule species examined resembled that expected for various-size spherical particles of S8.
Pivotal to the difference of opinion between the two groups is the fact that very different experimental spectra are reported for the same chemical form, in particular for the common yellow allotrope
-sulfur, which contains the thermodynamically most stable molecular entity, cyclo-octasulfur (S8). Pickering, George, and coworkers (9, 23) attribute this discrepancy to uncorrected experimental artifacts of the transmittance detection method used by Prange, Hormes, and coworkers (6, 7, 17, 27, 28), while Prange, Hormes, and coworkers contend that their data show no substantial distortions and suggest that different detection methods yield inherently different spectra (26). Central to their work is their observation that, when measured in transmittance, the spectrum of elemental cyclo-octasulfur (S8) lacks intense preedge absorption features, whereas polymeric sulfur has a significant peak in this region. We believe this observation is fundamentally flawed because of experimental artifacts. In this paper, we review the experimental artifacts expected to arise from the different measurement techniques, report experiments using all three detection methods for the same species, and discuss the utility and limitations of each detection technique for forming conclusions about biological samples. Finally, we return to the original question—the chemical nature of the sulfur in bacterial sulfur globules.
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-sulfur was recrystallized at room temperature from xylene solutions of elemental sulfur. We note that while this is expected to yield predominantly
-sulfur containing predominantly S8 (13), small quantities of other rings, such as S7, might contaminate the sample (35). Polymeric sulfur was prepared by quenching boiling sulfur in ice water. Bacteria and growth conditions. A. vinosum was grown photoautotrophically with sulfide as an electron donor as previously described (23). The magnetotactic coccus strain MC-1 (30) was grown microaerobically under autotrophic conditions with 10 mM thiosulfate as the electron source. Autotrophy uses the reverse (reductive) tricarboxylic acid cycle, and further details can be found in Williams et al. (41). A complete description of the organism will be submitted (suggested name, Magnetococcus marinus) after a strain designation has been formally assigned by the American Type Culture Collection.
XAS. Spectra were recorded using beamline 6-2 of the Stanford Synchrotron Radiation Laboratory employing a Si(111) double-crystal monochromator and a downstream nickel-coated harmonic rejection mirror. Samples were enclosed in a helium-filled flight path to minimize absorption of the X-rays by air. X-ray fluorescence was measured using a Stern-Heald-Lytle fluorescent-ion chamber detector (EXAFS Company, Pioche, NV), which was filled with argon, and the electron yield was measured using a modified commercial detector system (EXAFS Company, Pioche NV), which was filled with helium. All X-ray windows were fabricated with 6.3-µm-thick polypropylene film (SPEX CertiPrep, Metuchen, NJ). Bacterial samples were run as dilute aqueous suspensions in acrylic cuvettes with polypropylene windows. Isolated globules were run frozen at a temperature of –20°C.
Data collection and processing. Data acquisition was done with the XAS Collect software (10). In order to avoid any X-ray-induced damage to samples, the sample was translated to a new (unexposed) position between individual sweeps. Energy calibration was performed by reference to the K-edge spectrum of a freshly prepared sodium thiosulfate standard that was recorded frequently during the experiments and assuming the lowest energy peak to be at 2,469.2 eV (32). Background removal and normalization of XAS data were performed using the EXAFSPAK suite of computer programs (http://ssrl.slac.stanford.edu/exafspak.html).
Calculation of fluorescence and transmittance effects.
In general, the intensity of transmitted light, IT, is given by
, where I0 is the incident light intensity,
is the density (typically given in g/cm3),
E is the absorption cross section (typically given in cm2/g) at the incident photon energy E, and t is the sample thickness (given in cm). The transmittance of any object can be generated by simply dividing the object into small-volume elements and performing a numerical integration. The fluorescence generated by each volume element (per unit incident intensity) is proportional to its absorbance and can be written as 

E, where
is a constant (proportional to the fluorescence efficiency). Finally, attenuation of outgoing fluorescence photons from a volume element by external parts of the object is proportional to
, where tout is the outgoing path and
F is the absorption cross section of the sphere material at the energy of the fluorescence. In our previous work (23), a computer program was developed to calculate the transmittance and fluorescence of spheres, although we note that it would be trivial to extend it to any three-dimensional shape.
For a collection of spheres, we need to account for the space between the spheres (the interstices). If the spheres are uniform, then the most efficient packing is close packing, and assuming normal illumination by the X-ray beam, the projected fractional area of the interstices is 1 –
/(2
3), or about 9.3%. In practice, microscopic examination of powder on tape samples shows a very much greater fractional area for the interstices—typically up to about 50%. The transmittance for a single sphere can show significant distortions due to the fact that the center can be effectively opaque with most of the signal passing through the periphery of the sphere, as previously described (23). The interstices serve to exacerbate these distortions.
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FIG. 1. Schematic diagram of different detection strategies. (A) Transmittance detection, where the incident and transmitted X rays (I0 and I1, respectively) are used to obtain the absorbance, A, directly: A = ln(I0/I1). (B) X-ray fluorescence detection, where the X-ray fluorescence, IF, is used to obtain the absorbance: A IF/I0. (C) Electron yield detection, where electrons from the sample are picked up by a grid and used to obtain the absorbance: A Ie/I0.
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FIG. 2. Effects of sample charging upon the X-ray absorption spectra of elemental -sulfur using electron yield detection. The solid and broken lines indicate spectra obtained with and without grinding with graphite, respectively.
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Transmittance detection suffers from a problem that is variously known as "pinhole," "leakage," or "transmittance thickness" effects. It is often convenient to divide X-ray measurements into hard- and soft-X-ray regimes, which we define according to the photon energy—hard X rays correspond to energies above 6 keV, and soft X rays are below this energy. For most hard-X-ray measurements, these distortions are trivial to avoid, although some clearly distorted hard-X-ray spectra are still published. However, with soft X rays (such as those at the sulfur K edge), the very high X-ray absorption coefficients mean that these effects can be severe, especially with X-ray absorption spectra of particulate samples. The problem arises from spatial variations in absorption due to inhomogeneity in thickness over the part of the sample that is irradiated by the X-ray beam. Distortions occur both with thin areas of the sample and with actual holes. As with electron yield and fluorescence, these effects distort spectra by attenuating intense features (12, 23). Particular care must thus be taken to prepare samples of uniform thickness. This requirement is especially stringent with elemental sulfur for two reasons: unlike sulfur compounds, the sulfur is not diluted with other atoms, and because of the especially large X-ray absorption cross section, which peaks at about 8.04 x 103 cm2 g–1 at 2,469.8 eV for
S8 (estimated from electron yield measurements and tabulated X-ray cross sections [19], as described by Weng et al. [40]). Because of this high absorption coefficient, a variation in sample thickness of only fractions of a micrometer can result in significant distortion of the spectra (9). We have previously discussed these distortions in a quantitative manner using an idealized spherical particle shape (23) and pointed out that our calculations indicated that it might be essentially impossible to produce a sample of elemental sulfur that is finely divided enough to produce distortion-free transmission spectra. Nevertheless, Prange, Hormes, and coworkers disagree, reassured that they are able to fit the experimental data and driven by the maxim that if their fitting procedure appears to work, then the standard spectra must be undistorted (26).
Figure 3 shows calculated spectra (normalized to the edge jump) for uniform spheres of
-sulfur. It can be seen that radii of greater than 1 µm give significantly distorted spectra. Figure 4 A shows a comparison of the experimental transmittance spectra of elemental sulfur obtained with manual grinding using an agate pestle and mortar and with very extensive grinding using a mechanical grinding mill ("Wig-L-Bug") set to its highest setting for 2 h. The mechanical mill was located in a cold room (4°C) to minimize sample heating (which might partly transform
-sulfur to catenasulfur), and following the grinding, the sample was observed to be cool to the touch. In both cases grinding was performed with boron nitride. As expected, the transmittance spectrum of the more extensively ground sample (Fig. 4A, b) is less distorted, although still clearly distorted relative to the (undistorted) electron yield spectrum (Fig. 4A, c). The fluorescence spectra, measured simultaneously with the transmittance, are shown in Fig. 4B and show the expected distortions, corresponding to particle diameters of 7.5 and 2.5 µm, for Fig. 4B, a and b, respectively (assuming spherical particles) (23). Although examination of the entire area irradiated in the experiment was not practical, microscopic examination of representative areas of both samples showed particle sizes that agreed very closely with these computed values. The greater distortion for the transmittance spectra relative to fluorescence on the same sample is explained by the space between the particles, which produces additional distortions above those from particle size effects. Using the methods we have previously described (23), the spectra of Fig. 4A, a and b, are consistent with the particle sizes estimated from fluorescence (7.5 and 2.5 µm) with a percentage coverage of 65 to 60%. Again, microscopic examination of the samples indicated that these values for coverage were reasonable.
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FIG. 3. Computed transmittance spectra for -sulfur spheres of different radii. The spectra were calculated assuming 65% coverage—the best possible coverage for uniform spheres is for the close-packed arrangement (90.7% coverage).
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FIG. 4. Effects of particle size upon transmittance and fluorescence X-ray absorption spectra. Traces a and b show the spectra obtained from -sulfur after grinding (a) and extensive grinding (b) measured with transmittance (A) and fluorescence (B). In both panels A and B, trace c shows the undistorted electron yield spectrum for comparison.
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-sulfur (27, 28), and these are reproduced in Fig. 5, a and b. They have attributed the differences to differing spectroscopic resolutions (26), but this explanation cannot quantitatively account for the difference. Approximating resolution effects by convolution with a Gaussian, we note that a Gaussian of full width half maximum of around 10 eV is required to match the peak intensity of spectrum a in Fig. 5 to that of b, and this is far larger than any reasonable difference in spectroscopic resolutions. Furthermore, if such a large broadening were present, then the other spectra presented by Prange, Hormes, and coworkers (6, 7, 17, 26, 27, 28) would show very obvious differences in broadening. We contend that the differences in the spectra shown in Fig. 5, a and b, are simply due to different particle sizes. Figure 5, a' and b', show simulated distorted transmittance spectra of
-sulfur for particle sizes of 2.2 and 10.0 µm and show good correspondence with Fig. 5, a and b, respectively. We have demonstrated that simply by using different levels of grinding, quite different transmittance spectra can be generated. Prange, Hormes, and coworkers (6, 7, 17, 26, 27, 28) prepared their samples as finely ground powders on adhesive tape, and as we have previously pointed out (9, 23), samples prepared in this way are particularly susceptible to the artifacts that we have discussed here. Indeed, in order to obtain undistorted transmittance spectra of elemental sulfur, particle diameters of around 0.25 µm would be needed (23). Particle diameters smaller than about 10 µm are very difficult to obtain by mechanical grinding alone (29) (see above), and we conclude that the required diameter of 0.25 µm is essentially impossible to achieve. Thus, undistorted transmittance spectra of mechanically ground samples of elemental sulfur will also be essentially impossible to obtain, and the resulting data are likely to be nonquantitative and difficult to experimentally reproduce, which Fig. 5 shows directly.
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FIG. 5. Different X-ray absorption spectra for -sulfur reported by Prange et al. (27, 28). The spectra were taken from published figures, suitably digitized, and shifted by 1.7 eV to account for different calibration methods. Traces a and b show the experimental spectra (taken from references 28 and 27, respectively), while a' and b' show computed transmittance spectra obtained from our undistorted electron yield data with spherical particle diameters of 2.2 µm and 10.0 µm, respectively, as described in the text.
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We now turn to the suggestion of Prange et al. (26) that, in the absence of spectroscopic distortions, the different detection methods might give inherently different spectra. This, they suggest (26), might explain the difference between our electron yield and their transmittance spectra.
Do the different methods of detection yield identical results?
We sought to validate experimentally what has been widely assumed but rarely reported, that correctly conducted experiments yield transmittance, fluorescence, and electron yield spectra that are identical. Preliminary experiments using
S8 and cellulose nitrate in an acetone-xylene solvent gave films of nonuniform thickness (as determined by microscopic examination). Thus, rather than demonstrate this with elemental sulfur, we chose to use a more readily soluble molecular compound that could be cast into essentially uniform plastic films that should not exhibit transmittance pinhole effects. We selected dibenzyldisulfide because of its excellent solubility in acetone. Spectra obtained from dibenzyldisulfide-cellulose nitrate film samples (approximately 5 µm thick) using electron yield detection, transmittance, and fluorescence yield are shown in Fig. 6. Apart from differences attributable to the small amount of noise in the spectra, the three methods yielded identical spectra and unambiguously demonstrated that, when no experimental distortions are present, the three detection methods yield essentially identical results.
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FIG. 6. Experimental spectra of cellulose nitrate films of dibenzyldisulfide obtained with different detection methods—a, electron yield; b, X-ray fluorescence; c, transmission.
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(S—S)
* and 1s
(S—S)
* transitions (5). The spectroscopic analogy between selenium and sulfur K-edge spectra of isostructural sulfur and selenium compounds has previously been reported (24). Sulfur K-edge spectra are invariably much more sensitive to chemical form than their selenium analogues because the longer core hole lifetimes yield much sharper spectra, and the fact that the spectra of the analogous selenium species show significant differences (4) means that we expect larger differences in the case of sulfur. In agreement with this, Armand et al. (2) used electron yield detection and reported quite distinct spectra for polymeric sulfur (washed with CS2 to remove
S8) and
S8. We note in passing that their
S8 spectrum (2) is in excellent agreement with ours (23) and in disagreement with the spectra reported by Prange et al. (27, 28). Our attempts to measure the XAS of polymeric sulfur using transmittance and electron yield resulted in spectra that were essentially indistinguishable from that of S8. We attribute this to the spontaneous conversion at the surface of polymeric sulfur to cyclo-octasulfur, which is known to occur (20). Microscopic examination of the surface revealed a powdery material, which is unlikely to be catenasulfur. The spectrum of polymeric sulfur reported by Prange et al. (28) strongly resembles a more subtly distorted spectrum of cyclo-octasulfur. Indeed, the spectrum can be accurately reproduced by using a mixture of computed distorted spectra corresponding to two different particle sizes (1.0 µm and 10.0 µm), as shown in Fig. 7. The close agreement between the computed distorted spectra and the experimental spectrum of Prange et al. (28) strongly suggests that no valid measurement of the sulfur K-edge XAS of polymeric sulfur has been reported to date by this group and that further work is needed. Even the spectra of Armand et al. (2), which show well-defined differences from
S8, can be rationalized as a mixture of the spectrum of
S8 and another, unknown form, presumably the spectroscopically elusive catenasulfur (not illustrated). For the present, a more tractable approach may be that used in our previous work (23), which was to employ solutions of stable organic polysulfides as models for the spectra, but further work is clearly required.
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FIG. 7. X-ray absorption spectrum of polymeric sulfur reported by Prange et al. (27, 28) compared with a distorted spectrum of -sulfur. The spectrum was taken from the published figures, suitably digitized, and shifted by 1.7 eV to account for different calibration methods. The solid line shows the published data of Prange et al. (27, 28), while the broken line shows the computed distorted spectrum of -sulfur, computed as described in the text.
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-sulfur. All of the XAS data that have been reported can be explained by the experimental distortions that we have discussed here. Thus, the data of Prange et al. show that the globules of B. alba and T. namibiensis show sulfur spectra with less pronounced peaks than those of A. ferrooxidans and A. vinosum (28), and this is entirely consistent with a greater spectral distortion resulting from the larger globule sizes of the former organisms. The same group reported that when A. vinosum was grown upon elemental sulfur, the spectra developed less intense peaks (7). To us, this indicates an overall increase in the mean particle size of elemental sulfur. This in turn is consistent with smaller particles being more readily consumed, leaving the larger ones. Other techniques clearly support the assignment of S8. In particular, Pasteris et al. (21) have reported a laser Raman spectroscopic study of the globules of Thioploca and Beggiatoa that clearly indicates the presence of S8. One concern with laser Raman spectroscopy is that local heating by the intense laser beam necessary for the technique might transform sulfur forms. Pasteris et al. (21) tested for such problems by deliberately heating areas of the sample using high laser powers and found that S8 predominated only when low powers were used on fresh samples of the microorganisms. Their conclusions are in complete agreement with ours—that the molecular form of sulfur stored in globules is cyclo-octasulfur, S8. The globules of the purple sulfur bacterium A. vinosum have become the de facto standard for research into sulfur globules, and we therefore reexamine their spectra here. Figure 8 shows the sulfur K-edge X-ray absorption spectra of isolated globules from A. vinosum, which are very similar to those reported earlier (23). Two different sample dilutions are shown in Fig. 8, illustrating an additional problem with sulfur fluorescence—that if samples are insufficiently dilute, then additional distortions may result. This may also be a potential problem when analyzing data from globules within intact cells, which are held in close proximity by their confinement within the cell, yielding an effectively high local concentration. This will effectively increase the distortions of the spectra and artifactually increase the estimated globule size. Figure 8 shows the estimated sizes of the globules from curve-fitting analysis, as previously described (23). Thus, for diluted globules, we estimate a particle size of 0.72 µm (which is in reasonably good agreement with our previous estimate of 0.65 µm (23), but for globules at 10 times the concentration, we estimate a particle size of 0.84 µm. Because of these difficulties, the globule diameters estimated from cells using our fitting method should be considered an upper bound, rather than a quantitative estimate.
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FIG. 8. Sulfur K-edge X-ray absorption spectra of A. vinosum globules taken at two different dilutions—as prepared () and diluted 10 times ( ). The lines show the fitting results, and the inset shows the fit error versus the radius (broken line, as prepared; solid line, diluted).
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FIG. 9. Sulfur K-edge X-ray absorption spectra of the magnetotactic marine coccus strain MC-1 (points) and curve-fitting analysis (upper line). The fit components are shown in the lower set of curves: S8, with a spherical particle diameter of 0.32 µm (solid line); thiol (dotted line); disulfide (dashed line); and sulfate (dot-dashed line).
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FIG. 10. Dark-field scanning transmission electron micrograph of three clumped cells of the magnetotactic marine coccus strain MC-1. The arrows indicate sulfur-rich globules, which were confirmed by energy-dispersive X-ray analyses (not illustrated). The sizes of the sulfur globules in strain MC-1 are quite variable and likely reflect the stage of growth. In addition, the globules are sensitive to the electron beam and eventually evaporate. The larger globules (x) are phosphorus rich and likely represent polyphosphate deposits.
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-sulfur. Prange et al. stated that in our previous studies we concluded that the globules contain
-sulfur (26). This is not completely accurate, and we have never made such a claim; instead we proposed that the globules consist of a core of fragments with local structures resembling the S8 crowns of
-sulfur, with a modified globule surface conferring hydrophilic properties (23). The presence of S8 crowns is not surprising, as they are thermodynamically the most stable. Large-scale sulfur crystallites can be excluded from previous X-ray diffraction results (16). The hydrophilic surface might be due to the proteins that are known to be associated with the globules or to modification of surface sulfur by incorporation of polar groups, such as thionates (34). In either case, the surface sulfur content must constitute such a small fraction of the total that it is unobservable by XAS. We now discuss aspects of the mechanisms by which elemental sulfur is thought to be consumed by sulfur-oxidizing bacteria and their relevance to the structures of the forms stored by the bacteria. Bacteria harbor the Sox genes, and several proteins involved in sulfur oxidation have been described (8), while archaea possess a sulfur oxygenase system, upon which significant progress has been made in terms of understanding molecular mechanisms. Urich et al. (38) and, very recently, Li et al. (18) have determined the structures of a sulfur oxygenase reductase from the thermoacidophilic archaea Acidianus ambivalens and Acidianus tengchongensis. These enzymes are very similar, and both have an active-site pocket containing an essential cysteine and an iron coordinated by two histidines and a glutamate. The iron site is thought to be the site of oxygenase activity (18, 38), and in the A. ambivalens enzyme, the essential cysteine is present as persulfide (38), while in A. tengchongensis, no persulfide group was found (18). In both enzymes, the active-site pocket is elongated (18, 38), and Urich et al. (38) reported that the enzyme accommodates linear polysulfides, such as [S8]2–, but not cyclo-octasulfur, S8. We note in passing that the active-site pocket (18, 38) will not permit access by long-chain polymeric sulfur species. Thus, sulfur-sulfur bond scission and possibly reduction (to form polysulfides) is needed before the oxygenase system can act upon elemental sulfur, irrespective of the form. Polar organic solvents, such as methanol, promote equilibrium between the ring forms S6, S7, and S8 (37). Such equilibria necessarily involve sulfur-sulfur bond scission, and Steudel and coworkers have discussed possible mechanisms by which this might occur (36).
In this work, we have addressed the controversy surrounding the chemical nature of the sulfur in bacterial sulfur globules. To this end, we have experimentally demonstrated that the use of transmittance detection for sulfur K-edge XAS measurements is highly prone to spectroscopic distortions, and thus, that much of the published work on sulfur bacteria must be based on distorted data. We have also demonstrated that all three detection methods used for X-ray absorption experiments do indeed yield essentially identical spectra when measurements are carried out under conditions where no experimental distortions are expected. We have built on our previous work with A. vinosum and have shown that for strain MC-1 the XAS indicates the presence of a chemical form of sulfur resembling S8 in particles of a size commensurate with electron microscopy. Nearly a decade has passed since the first reports of in situ XAS measurements of sulfur globules, yet our understanding of the chemical nature of bacterial sulfur has remained clouded by experimental artifacts. If misleading conclusions are to be avoided, then spectroscopic measurements that are used to describe the biochemistry of sulfur in living things must be at least as rigorous as the limits of the biochemical techniques.
Published ahead of print on 1 August 2008. ![]()
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* resonances in electron transmission (ETS) and X-ray absorption (XAS) spectroscopies of dimethyl(poly)sulfides (CH3)2Sx (x = 1, 2, 3). J. Physics B 31:L497-L502.[CrossRef]
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