Diffraction imaging of single particles and biomolecules
Introduction
Emerging radiation sources offer exciting new possibilities in biomolecular imaging. X-ray free-electron lasers will provide femtosecond X-ray pulses with a peak brilliance more than 10 orders of magnitude higher than that currently available from synchrotrons. Such light sources may permit non-crystalline biological samples to be imaged with X-rays, and could thus remove a current bottle-neck in structure determination (Neutze et al., 2000). Unfortunately, the intense radiation pulse emitted by the laser will destroy any biological sample in a single shot, precluding the collection of multiple diffraction patterns from a single particle or molecule. We therefore assume that the sample is reproducible, and that single-shot diffraction images can be collected from individual sample particles exposed to the beam one-by-one in unknown orientations. The mathematical treatment of this problem is not unique to planned experiments with X-ray free-electron lasers, but can be extended to diffraction studies with electrons, neutrons, and other types of scattering probes. As a consequence, this paper has a broder scope than simply anticipating experiments with X-ray lasers.
The diffraction pattern of an object is proportional to the squared modulus of the molecular transform (the three-dimensional Fourier transform of the electron density). The coordinates of the diffraction space, usually called reciprocal space, are those of the scattering vector (or momentum transfer vector) between the incident and scattered X-rays. In order to reconstruct the electron density, reciprocal space must be sampled with sufficient density and the diffracted intensities must be known with an acceptable accuracy. In view of this, there are two reasons why a large number of diffraction patterns need to be collected. As a single diffraction image samples only a spherical slice through the origin of reciprocal space (see Fig. 1), so the sample must be imaged in multiple orientations for the space to be adequately covered. In addition, the signal-to-noise ratio of raw diffraction images will probably be insufficient for a high-resolution reconstruction, and it will be necessary to obtain a redundant data set so that the signal can be enhanced by averaging.
When the orientation of the samples is unknown, the images must be classified according to the view of the sample that they present before signal averaging is possible. Methods to sort and average images have been developed for single-particle electron microscopy (Mueller et al., 2000; Saxton and Frank, 1977; van Heel, 1987, van Heel et al., 1996, van Heel et al., 1997), and have produced substantially increased resolution even for irregular objects like the ribosome (Mueller et al., 2000). With particles displaying high symmetry, the resolution can be extended further by exploiting the symmetry of the structure (Bottcher et al., 1997; Stowell et al., 1998).
There are important differences between the task of classifying tomographic images of electron microscopy (micrographs) and diffraction patterns of single molecules. Some of these stem from differences between planar (tomography) and spherical sectioning (diffraction), while others reflect differences in the way the images are formed, which also affects their statistical properties. Perhaps the most prominent difference is that the diffraction pattern has a known center, whereas in the micrograph, the molecular image has to be located and centered. Equally significant are the differences in background: in the micrograph the molecular image and the background are separate (although the background contributes to the noise in the image), but in a diffraction pattern there is no obvious way to distinguish the background from the diffraction pattern. Also important is that the micrograph has to be corrected for imperfections of the microscope (the contrast transfer function), whereas diffraction images are perfect in that sense and need no correction. We note that diffraction patterns can also be obtained in electron microscopes, with similar advantages and disadvantages as discussed here.
Averaging techniques are based on the assumption that the data set is redundant. The images can thus be sorted into classes that correspond to a distinct view (orientation) of the sample. Images within each class are then averaged; if the classification is correct, the signal adds constructively but the noise does not. We note that, according to the sampling theorem (Jerri, 1977), a finite set of views of the sample is sufficient for full reconstruction; thus, it is sufficient if the data set is redundant with respect to a number of views satisfying this condition. Errors in classification as well as heterogeneity in the samples degrade the signal to noise ratio and the intrinsic resolution of the class averages. It is important, therefore, that the number of classes be adapted to the signal-to-noise ratio of the raw images. Methods to accurately classify diffraction images with extremely low signal-to-noise ratio, as well as methods to identify wrongly classified images, need to be developed.
Once a complete set of averaged images is obtained, they can be used as reference images to check and correct the original classification of each noisy diffraction image in an iterative process. Images that are significantly different from any of the class averages can be removed at this stage. Also, the procedure must at some point include a search for images that present the same view of the sample, but which are rotated with respect to one another around the axis of the beam.
After classification and averaging, the mutual three-dimensional orientation of the class averaged images must be determined in order to assemble a three-dimensional data set. This may be possible through the method of common lines (see e.g. Frank, 1996, van Heel et al., 2000), a technique widely used in electron microscopy, where the micrographs represent planar sections through the center of the molecular transform. Diffraction images are different and represent spherical sections. Each pair of images will intersect in an arc that also passes through the origin of the molecular transform (Fig. 1). If the signal (after averaging) is strong enough for the line of intersection to be found in two averaged images, it will then be possible to establish the relative orientation of these images. We note that due to the curvature of the sections, the common arc will provide a three-dimensional fix rather than a hinge-axis. Moreover, the centric symmetry of the modulus of the molecular transform ensures that we obtain 2×2 independent repeats of the common lines in the two images. This feature provides redundancy for determining sample orientation, and is unique to diffraction images.
The molecular transform is related to the electron density simply and directly by a three-dimensional Fourier transform. Unfortunately, the formation of diffraction images is associated with a loss of information: the molecular transform is a complex, continuous function, whereas the diffraction data are real, discrete, and irregularly spaced in reciprocal space. This leads to a reconstruction problem where the data contain less information then the solution. Such a problem is ill-posed, and as a consequence a very broad set of solutions may fit the data within experimental error. To cure the ill-posedness, we need to include additional information about the sample that constrains the solutions to those that are physically acceptable, and thus allows us to discriminate between spurious solutions and those that are realistic. Classical crystallography has a similar problem.
It was surmised by Sayre (Sayre, 1980) that if the amplitudes of the molecular transform could be over-sampled, there would be enough information to replace the lost phases and reconstruct the electron density. The idea has its basis in sampling theory, which states that a band-limited function, such as the molecular transform of a finite-size molecule, can be fully represented by a set of discrete equidistant samples (Jerri, 1977). By sampling the amplitudes more finely than the sampling theorem requires, it may be possible to compensate for the missing phases.
In a recent publication (Szőke, 1999), Szőke has shown that the electron density can, indeed, be reconstructed from a simulated, oversampled continuous diffraction pattern, obtained from a crystal that is made of two similar molecules. He used an approach based on principles used in holography (encoded in the EDEN package, Szőke 1997) and some a priori information. Miao, Hodgson and Sayre used the iterative Gerchberg–Saxton–Fienup algorithm (Gerchberg and Saxton, 1972) to successfully reconstruct electron densities from both simulated (Miao et al., 2001) and real (Miao et al., 2002) diffraction images from non-crystalline samples. A third demonstration, by Oszlányi and Faigel (unpublished) used a maximum likelihood optimizer for this purpose. These approaches represent major developments in phasing, and could be applied to obtain three-dimensional structures from oversampled diffraction images like those of single particles and molecules.
In classical crystallography, the set of Bragg reflections constitute a uniform three-dimensional grid/lattice in reciprocal space. (Actually, both Szőke (1999) and Miao et al., 2001, Miao et al., 2002 used diffraction intensities measured on a three-dimensional regular grid.) Diffraction data sets derived from samples without translational symmetry, on the other hand, yield a highly non-uniform sampling of the molecular transform with a decreasing sampling density at higher resolutions. This is also true for tomograms. One could limit the analysis to those samples that lie on a regular grid, but this would be a very inefficient use of data and seems incompatible with the idea of a highly over-sampled diffraction pattern. Interpolating onto a regular grid does not improve the situation; it moves the problem of ill-posedness from real space to reciprocal space, but does not change its nature. Reconstruction algorithms will have to deal intelligently with the above problems.
Fortunately, there are extensive mathematical treatments of matrix inversion (Golub and Loan, 1996), image processing (Bertero and Boccaci, 1998), and of reconstruction in computed tomography (Natterer, 1986, Natterer and Wübbeling, 2001). One can state with some confidence that those inverse problems have similar difficulties, but are “easier.” Therefore, reconstruction algorithms for single particle diffraction will be a subset of those that work well for matrix inversion or tomography. We have recently extended EDEN, the holographic method for reconstructing the electron density in crystals, to deal with diffraction patterns from single particles (Hau-Riege et al., in preparation), and expect to be able to find the optimum electron density under conditions of incomplete, noisy measurements on an irregular set of points in reciprocal space.
Section snippets
Classification of diffraction images
The first step in the reconstruction process is to classify the diffraction images according to the view of the sample that they present. The images within each class can then be averaged to produce the set of high-quality views of the sample that is required for an atomic-resolution reconstruction of the structure. It is the precision and noise-tolerance of the classification procedure, rather than that of the reconstruction method, that sets the lower limit on the quality of the raw
Conclusions
We have presented a simple but realistic statistical model for the classification of diffraction images. Our quantitative conclusions are presented in Eqs. , , and Fig. 5, Fig. 6, which connect the number of incident X-ray photons, the particle size and the achievable resolution. We have shown that less then one photon per independent pixel can be enough for classification, even in the presence of a Poisson-type photon noise. As expected, the larger the particle and the larger the incident
Acknowledgements
We are grateful to Marin van Heel for discussions. This work was supported by the Swedish Research Councils STINT and VR.
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