Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 9.3, pp. 234-235

Section 9.3.1. Introduction

D. Shapiroa*

aAdvanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, MS 2–400, Berkeley, CA 94720, USA
Correspondence e-mail:

9.3.1. Introduction

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The brightness of third-generation synchrotron X-ray sources removes the need to enhance a diffracted X-ray 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. Single-particle X-ray diffraction microscopy, also known as coherent X-ray diffraction microscopy (CXDM), treats an isolated non-crystalline sample as a crystallographer treats a crystal. The far-field 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 phase-measuring 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 constraint-based 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[link]). Algorithms being developed for electron microscopy in the early 1970s would establish the constraint-based paradigm of iterative phase retrieval from Fourier modulus measurements. In particular, the alternating projection algorithm of Gerchberg and Saxton, the error-reduction 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[link]). This algorithm was later modified by Fienup into the input–output algorithm to handle cases where only one intensity measurement is made (Fienup, 1978[link], 1982[link]; Fienup et al., 1982[link]; Miao et al., 1998[link]). In this case, the required real-space 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 error-reduction 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[link]; Bates, 1982[link]). The method of phase retrieval from diffraction patterns sampled between Bragg peaks became known as the oversampling phasing method (Sayre, 1991[link]; Sayre & Chapman, 1995[link]; Miao et al., 1998[link]). The oversampling method compensates for the information deficit by supplying a zero-density region of approximately known size in the object domain. The more finely the diffraction pattern is sampled, the larger this zero-density 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 X-rays (1–10 nm wavelength). Indeed, since the initial demonstration of CXDM (Fig.[link]), 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. Three-dimensional images of radiation-hard materials have been obtained at 15 nm resolution, while two-dimensional 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[link]; Shapiro et al., 2005[link]; Pfeifer et al., 2006[link]; Barty et al., 2008[link]; Schroer et al., 2008[link]; Nelson et al., 2010[link]). 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 two-dimensional case (Huang et al., 2009[link]; Lima et al., 2009[link]). Lens-based X-ray microscopes have achieved this landmark with moderate resolution (Wang et al., 2000[link]; Weiss et al., 2000[link]; Larabell & Le Gros, 2004[link]). Transmission X-ray microscopes (TXM) and scanning transmission X-ray 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 high-numerical-aperture 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[link]) at the highest spatial frequencies. Thus, at the cost of throughput and ease of use, a diffraction microscope provides increased X-ray efficiency and resolution.


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First soft X-ray demonstration of the CXDM method. (a) Diffraction pattern using 1.7 nm X-rays. (b) Reconstruction of (a) to 75 nm resolution. Reprinted by permission from Macmillan Publishers Ltd: Nature (Miao et al., 1999[link]), copyright (1999).

The following sections first report on the current standard single-particle phase retrieval techniques and then on recent experiments in CXDM, which establish the state of the art in whole-cell imaging by diffractive methods. Images of dry yeast at 11 nm resolution are presented, which represent the highest resolution X-ray 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 low-dose three-dimensional information is explored (Sayre, 2008[link]).


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