International
Tables for Crystallography Volume F Crystallography of biological macromolecules Edited by E. Arnold, D. M. Himmel and M. G. Rossmann © International Union of Crystallography 2012 
International Tables for Crystallography (2012). Vol. F, ch. 9.3, pp. 234239
doi: 10.1107/97809553602060000826 Chapter 9.3. Xray diffraction imaging of whole cells^{a}Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, MS 2–400, Berkeley, CA 94720, USA Coherent Xray diffraction microscopy (CXDM) is a method for the determination of noncrystalline structures through the inversion of Xray diffraction data. The method relies on a numerical phaseretrieval step which, utilizing oversampled diffraction data, requires that the diffracting object be finite and isolated. The high efficiency of this lensless technique means that it is well suited for imaging radiationsensitive biological matter. Furthermore, the penetrating power of Xrays combined with the natural contrast provided by `water window' photon energies make it feasible to image whole, unstained cells. The first two sections of this chapter provide a brief history of CXDM and an overview of the basic phaseretrieval methods. The following sections discuss the stateoftheart capabilities for wholecell imaging, the effects of radiation damage and the use of stereoscopic imaging for acquiring threedimensional information with a minimum of radiation dose. 
The brightness of thirdgeneration synchrotron Xray sources removes the need to enhance a diffracted Xray signal with crystallographic redundancy for certain classes of samples. This is particularly useful for samples for which there exists only one unique structure or for which arrangement into a crystalline form is exceedingly difficult. Singleparticle Xray diffraction microscopy, also known as coherent Xray diffraction microscopy (CXDM), treats an isolated noncrystalline sample as a crystallographer treats a crystal. The farfield diffraction intensity pattern of the sample is measured and the phase problem is solved computationally, allowing for structure recovery through a Fourier transform. Since Mother Nature restricts us to only measuring intensities, all methods of phase retrieval rely on decoding phase information that has been encoded in intensity measurements. Crystallographers, and the subsequent diffraction microscopists, have two basic methods for doing this. The first, the holographic method, mixes a known reference signal with the desired and unknown signal. The coherent superposition of these two signals results in measureable intensity variations that are directly related to the relative phase. This is, indirectly, a phasemeasuring method. The second method relies only on the signal from the unknown structure and additional information, supplied by the scientist, which is physically plausible. This is a constraintbased method. Both methods seek to fill the information deficit inherent in intensity measurements by adding something that may unlock the phase.
In 1980, David Sayre suggested that CXDM should be possible (Sayre, 1980). Algorithms being developed for electron microscopy in the early 1970s would establish the constraintbased paradigm of iterative phase retrieval from Fourier modulus measurements. In particular, the alternating projection algorithm of Gerchberg and Saxton, the errorreduction algorithm, was developed to reconstruct the phase information that is missing when intensity measurements are made in both real and reciprocal space (Gerchberg & Saxton, 1972). This algorithm was later modified by Fienup into the input–output algorithm to handle cases where only one intensity measurement is made (Fienup, 1978, 1982; Fienup et al., 1982; Miao et al., 1998). In this case, the required realspace constraint restricts the object to an area no larger than half the width of its autocorrelation. The combination of the input–output algorithm and the errorreduction algorithm was found to be a very robust method of image reconstruction using only Fourier domain intensities, but its success was not understood for several years. In 1982, Bates argued that the solutions to the phase problem are unique in two dimensions if the Fourier modulus is sampled on an interval at least twice as fine as the Bragg interval (Bates & Fright, 1983; Bates, 1982). The method of phase retrieval from diffraction patterns sampled between Bragg peaks became known as the oversampling phasing method (Sayre, 1991; Sayre & Chapman, 1995; Miao et al., 1998). The oversampling method compensates for the information deficit by supplying a zerodensity region of approximately known size in the object domain. The more finely the diffraction pattern is sampled, the larger this zerodensity region is, although this only adds useful information up to a certain point. This addition of information through Fourier space sampling results in an overdetermined inverse problem with a unique solution.
Sayre considered that the real niche of CXDM would be imaging objects of a few microns at a resolution of a few nanometres using soft Xrays (1–10 nm wavelength). Indeed, since the initial demonstration of CXDM (Fig. 9.3.1.1), ten years ago at the time of writing, active research projects have developed at all major synchrotron facilities and the technique has been applied to a diverse set of scientific problems. Threedimensional images of radiationhard materials have been obtained at 15 nm resolution, while twodimensional images of a whole cell at 30 nm and of a gold nanoparticle at 5 nm resolutions have been reported, among many others (Miao et al., 2002; Shapiro et al., 2005; Pfeifer et al., 2006; Barty et al., 2008; Schroer et al., 2008; Nelson et al., 2010). CXDM researchers have not yet achieved the `holy grail' of cellular imaging, the imaging of a whole frozen hydrated cell in three dimensions, but at least two teams are pursuing this goal with recent success in the twodimensional case (Huang et al., 2009; Lima et al., 2009). Lensbased Xray microscopes have achieved this landmark with moderate resolution (Wang et al., 2000; Weiss et al., 2000; Larabell & Le Gros, 2004). Transmission Xray microscopes (TXM) and scanning transmission Xray microscopes (STXM) utilizing diffractive zone plate lenses have high throughput and the advantage of direct imaging, but are limited in resolution by the technological challenge of making efficient highnumericalaperture lenses. TXM can now routinely image in three dimensions at 50 nm resolution, and 12 nm resolution has been demonstrated in two dimensions, although the total efficiency of the optic used was of the order of 1% (Chao et al., 2009) at the highest spatial frequencies. Thus, at the cost of throughput and ease of use, a diffraction microscope provides increased Xray efficiency and resolution.
The following sections first report on the current standard singleparticle phase retrieval techniques and then on recent experiments in CXDM, which establish the state of the art in wholecell imaging by diffractive methods. Images of dry yeast at 11 nm resolution are presented, which represent the highest resolution Xray images of whole cells currently on record. The effects of radiation damage are discussed and Sayre's idea of using stereoscopic viewing as a means of obtaining quick and lowdose threedimensional information is explored (Sayre, 2008).
The problem of phase retrieval is solved through successive application of constraints on the recovered object in the data and object spaces. Using the language of convex optimization, the mathematical operators which act on the data are projectors. The projector in the reciprocal (data) space forces the Fourier components to have the correct magnitude, while in object space finite support is enforced. To calculate the Fourier magnitude projector, one first needs to propagate the object density, ρ, to the data space by a Fourier transform, then replace the estimated magnitudes with the measured ones, I^{1/2}, and finally propagate back to real space. Using these transforms one simplifies the calculation of the projection, which becomes an elementwise operation on each recovered Fourier component. The forward and inverse transforms must be incorporated into the operator defined in real space P_{m}, where the measured Fourier magnitudes are enforced in Fourier space by . Using the Fourier basis, one simply replaces the estimated magnitudes with the measured ones I^{1/2}, []. Similarly, in the object space the finite support constraint is applied on a per pixel basis through multiplication by the support mask. The corresponding projector isTable 9.3.2.1 lists the combination of projections used by the most popular algorithms.

The violation of the support constraint is used as an error metric to monitor the convergence towards the solution. The solution should have zero density outside the support mask, so the error can be defined as the total density outside the support area, Alternatively, the error metric can be defined in the data space as the difference between the measured and calculated magnitudes,
In reality, the measured intensities are subject to noise which prohibits exact compliance with the constraints, so the error metrics cannot drop to zero. Although in most cases the algorithm can locate the global minimum, random noise will force fluctuations around the minimum. Once the algorithm reaches this steadystate regime, any particular iterate chosen as the solution would have a misleading degree of detail. On the other hand, the average of many fluctuating iterates would have reduced intensity in those Fourier components which are not reliably phased. The ratio of the average Fourier magnitude to the measured magnitude provides a measure of the stability, or reproducibility, of the retrieved phase information. This ratio as a function of spatial frequency is the phase retrieval transfer function (PRTF), where is the Fourier transform of the final averaged image and I_{m} is the measured intensity pattern. The PRTF is analogous to the differential phase residual of electron microscopy and, following Chapman et al. (2006), the resolution of a reconstruction is chosen as the spatial frequency at which the PRTF falls below a value of 0.5.
The Xray dose required to image a given volume of protein is nearly independent of energy above the oxygen K edge. At the same time, the photon flux required to image the same volume increases with E^{2} because of the energy dependence of the scattering cross section (Howells et al., 2009). For this reason, it is advantageous to use the lowest energy commensurate with the desired resolution of 5–10 nm. Commercially available chargecoupled device (CCD) detectors can easily provide a scattering angle of 0.1 radians [a 1 inch detector placed 5 inches from the sample (1 inch = 2.54 cm)], which results in a halfperiod resolution of 8 nm when using 750 eV Xrays. Furthermore, a cell with a diameter of 3 µm would have an oversampling ratio (number of intensity samples per speckle) of at least ten in this geometry if the detector has 20 µm pixels.
In this particular case, a Princeton Instruments CCD (PIMTE:1300) is placed 136 mm downstream of a freezedried yeast cell using the CXDM instrument on Beamline 9.0.1 of the Advanced Light Source (ALS). The yeast cell is illuminated by a coherent beam of 750 eV Xrays defined by a 5 µm pinhole located 25 mm upstream. The incident intensity of 4 × 10^{6} photons s^{−1} µm^{−2} is high enough to cause rapid structural changes to the cell (discussed in the next section), so the sample requires preirradiation for about 30 minutes prior to collection of the final data set intended for reconstruction. The final data set, shown in Fig. 9.3.3.1, is a 1024 × 1024 pixel subset of the full CCD and extends to a resolution of 11 nm at the edge and 7.8 nm in the corner. Speckles extend to the corner of the data set after a total exposure of 406 s and to the edge after 226 s.
The full reconstruction of this type of data set can only proceed once a highfidelity support is determined. Algorithmic support determination using Shrinkwrap, a variation on the hybrid input–output algorithm, is straightforward for objects with sharp boundaries, as is often the case in the material sciences (Marchesini et al., 2003; Chapman et al., 2006). However, biological samples which often have soft boundaries require manual intervention in the early stages of reconstruction. The softedge problem is exacerbated by the loss of low spatial frequency information behind the beamstop. A combination of Shrinkwrap support adjustments and intuitively reasonable manual adjustments are made until the algorithm is stable to automatic adjustment. This point is found when further Shrinkwrap adjustments no longer alter the shape of the support but just its tightness. Highfidelity reconstructions of complexvalued objects are not possible with a loose support (Fienup, 1987). Once the support is found, the final image is produced as the average over many reconstructions, all using the same support, which are started from different random phase sets. This averaging procedure, discussed earlier, reduces features that are primarily due to noise and provides a measure of the reproducibility of the recovered phases and therefore an estimate of the resolution. The final reconstruction shown in Fig. 9.3.3.1 had PRTF values above 0.5 for spatial frequencies extending to the edge of the data set or 11 nm resolution. The inset image clearly shows features of the order of 15 nm in size. In general, it is not possible to identify cell organelles without labelling specific proteins. However, a correlative study of yeast which combines highresolution CXDM with optical fluorescence would be a powerful tool for the cellbiology community. This technique has already been demonstrated at lower resolutions using a transmission Xray microscope (Le Gros et al., 2009).
Highresolution imaging of single particles with Xrays requires a large radiation dose because of the very strong dependence of the scattering cross section on spatial frequency and because, in the singleparticle case, there is none of the coherent amplification one obtains when many identical copies of the particle are arranged into a crystal. This large dose means that either the sample must be protected from morphological changes induced by radiation exposure, most effectively through cryogenic techniques, or that the obtained images will represent an altered form of the original sample. Cryoprotection of hydrated cells has been successfully used in electron microscopy and lensbased Xray microscopy for some time, but the sample preparation requirements of diffractive imaging are more severe, so the development of cryotechniques has been slower. The requirements of a finite sample support and that the sample be maintained on a zeroscattering background are extraordinarily difficult to achieve when that background consists of a micronthick layer of ice. This is an active area of research by several Xray diffraction microscopy groups.
For the case of dry cells, various preparation techniques (freeze drying or chemical fixation and dehydration) can preserve the largescale internal structure, such as the size and shape of large organelles, but the ultrastructure will inevitably show artifacts of the drying process. Furthermore, exposure to ionizing radiation results in the well known shrinkage problem. Studies with a transmission Xray microscope indicate that Xray induced shrinkage primarily produces a higherdensity but smaller version of the original cell (Jearanaikoon & AbrahamPeskir, 2005). The effect of cell shrinkage on the Xray diffraction pattern is shown in Fig. 9.3.3.2. A freezedried cell is repeatedly exposed to 750 eV Xrays from Beamline 9.0.1 of the ALS. Each exposure is 30 s and delivers an Xray dose of approximately 5 × 10^{8} Gy. The sample is stable for the first two exposures but then experiences a rapid collapse, followed by a slow but continuous shrinkage. The collapse is apparent from the elongated speckles, which indicate a sample for which the diameter is changing during the exposure while its relative structure is maintained. Overall, the sample loses about 25% of its volume prior to the final exposure used for reconstruction (Shapiro et al., 2005; Thibault et al., 2006). Over the course of this exposure series the total scattered signal does not change, indicating that the total mass of the sample remains intact. The rapid change in the diffraction pattern during the early exposures means that successful imaging experiments require preirradiation of the sample. The slow but continuous shrinkage of dry samples with further dose means that the resolution of the threedimensional images will necessarily be reduced. Indeed, Nishino et al. (2009) observed reduced resolution in their threedimensional reconstruction of a dry chromosome because of morphological changes which occurred during data collection. Hence, full threedimensional imaging of cells by diffractive methods requires cryogenic protection against radiation damage. Predictions based on a calculation of the cross section for coherent scattering by a smooth dielectric indicate that 10 nm resolution imaging of frozen hydrated organic matter should be possible using soft Xrays at currently available synchrotron sources (Howells et al., 2009). This limit is arrived at through a comparison of the radiation dose required for imaging and the dose at which radiation damage has been empirically observed at different length scales. However, it seems plausible that the presence of many identical particles within a cell could be exploited to provide superresolution information.
Diffraction imaging in three dimensions proceeds as it does in standard Xray tomography. That is, twodimensional data are recorded from many angular orientations of the sample and then assembled into a threedimensional data set. In this case, however, the data are recorded in reciprocal space and the individual data sets only need to be registered with respect to the angular coordinate due to the Fourier shift theorem. In the absence of any additional information, the angular sampling of reciprocal space is determined by the Crowther resolution,where D is the object diameter and Δθ is the angular separation of the twodimensional data sets. This is the spatial frequency at which the unmeasured Fourier components, those in between the measured Ewald sphere segments, can be properly interpolated from the measured data. Diffraction microscopy, however, requires the addition of information in the form of realspace constraints. This additional information allows for the calculation of not only the missing reciprocalspace phases but also a limited number of missing magnitudes. Chapman et al. (2006) showed that Δθ could in fact be up to four times larger than required by the Crowther relation, with k_{C} matching the numerical aperture of the imaging system. Thus, threedimensional reconstructions could take place with nearly isotropic diffractionlimited resolution with only about 150 angular orientations of the sample.
Stereoscopic viewing can provide a significant degree of threedimensional perception of an extended object while only increasing the total radiation exposure by a factor of two. In principle, according to the dosefractionation theorem of Hegerl and Hoppe, full threedimensional visualization of a given resolution element should not require a dose any higher than twodimensional visualization of the same element with the same statistical accuracy (Hegerl & Hoppe, 1976). This theorem provides hope that highresolution imaging in threedimensions, perhaps even of dry specimens, is possible, but in practice this is very difficult to achieve and lowdose imaging techniques are only now being explored by the CXDM community. Stereoscopic viewing should be considered the preliminary lowdose technique of choice. One particular advantage is the rapid reconstruction (compared with full threedimensional reconstructions) which makes possible in situ sample inspection. Fig. 9.3.3.3 shows a stereo image of a chemically dried budding yeast cell. When viewed stereoscopically, with the viewer's focus in front of the image, the threedimensional arrangement of a group of vesicles in the mother cell can be visualized.
CXDM promises to be a highly efficient imaging methodology which can deliver highresolution and highcontrast images of large noncrystalline biological structures. Radiationinduced shrinkage of dry cells will probably prohibit threedimensional imaging of such cells at high resolution. However, significant threedimensionality can be achieved through stereoscopic viewing of a cell, which only doubles the necessary Xray dose. Even so, in the absence of lowdose diffraction techniques the sample must undergo considerable morphological change prior to imaging, although this change does not seem to alter the relative arrangement of organelles. The development of lowdose techniques will allow for the direct observation of these radiationinduced changes. In the long run, it is cryogenic protection that provides the most valuable structural information, since the cells are maintained in a near living state. Ultrahighresolution threedimensional imaging will still require the development of lowdose techniques, as cryoprotected samples have also been observed to suffer from mass loss with the accumulation of very high Xray doses (>10^{10} Gy). These imaging techniques are currently under development at the ALS in collaboration with Stony Brook University and elsewhere. Alternatively, Xray freeelectron lasers (FELs) promise the highest resolution imaging of living cells that is possible by any means; indeed, it has been proposed that subnanometre resolution is possible (Bergh et al., 2008). The ultrashort and ultrabright pulses of an Xray FEL will encode the structural information from a living cell before it is destroyed by the pulse.
Acknowledgements
The author would like to thank Johanna Nelson and Xiaojing Huang of Stony Brook University for preparing the samples and providing the diffraction data for reconstruction. This work was supported under the Seaborg Fellowship at Lawrence Berkeley National Laboratory and by grants from the National Institutes of Health, the National Science Foundation and the US Department of Energy to Stony Brook University. The Advanced Light Source at Lawrence Berkeley National Laboratory is supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences Division, of the US Department of Energy.
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