research papers
The oversampling phasing method
^{a}Department of Physics and Astronomy, State University of New York at Stony Brook, New York 11794, USA
^{*}Correspondence email: miao@ssrl.slac.stanford.edu
Sampling the diffraction pattern of a finite specimen more finely than the Nyquist frequency (the inverse of the size of the diffracting specimen) corresponds to surrounding the electron density of the specimen with a nodensity region. When the nodensity region is bigger than the electrondensity region, sufficient information is recorded so that the phase information can be retrieved from the oversampled diffraction pattern, at least in principle. By employing an iterative algorithm, the phase information from the oversampled diffraction pattern of a micrometresized test specimen has been successfully retrieved. This method is believed to be able to open a door for highresolution threedimensional i.e. whole cells and submicrometre molecular clusters and micrometresized protein crystals. With the possible appearance in the future of Xray freeelectron lasers, it may become possible to image single molecules by recording diffraction patterns before radiation damage manifests itself.
of complex and noncrystalline biological specimens,Keywords: oversampling.
1. Introduction
When an object is illuminated by a plane wave, the diffracted wave in the far field, within the Born approximation, is the Fourier transform of the object. While the magnitude of the Fourier transform can be measured by a detector, the phase is lost. To recover the structure of the object, however, one has to know the phase information. This constitutes the well known i.e. sampled more finely than the Nyquist frequency. That oversampling a diffraction pattern could be used to retrieve the phase information in Xray diffraction was suggested by Sayre (1991) on the basis of a similar method in optics (Bates, 1982). Historically, the oversampling method bore some relationship with solvent flattening (see, for example, Wang, 1985) and the use of for phase determination (see, for example, Crowther, 1969). Recently, we proposed a theory to explain the oversampling method. We showed that sampling a diffraction pattern more finely than the Nyquist frequency corresponds to surrounding the electron density of the specimen with a nodensity region. The finer the sampling, the bigger the nodensity region. When the nodensity region is bigger than the electrondensity region, sufficient information is recorded so that at least in principle the phase can be retrieved from an oversampled diffraction pattern (Miao et al., 1998). When the specimen is a crystal, the diffracted pattern is confined to discrete Bragg peaks owing to the constructive interference from many unit cells. These Bragg peaks are at the Nyquist frequency corresponding to a This may explain why from Bragg peaks alone and without any ab initio information, the phase is not unique. If both the Bragg peaks and the intensity between the Bragg peaks, i.e. an oversampled diffraction pattern, can be measured from a finite crystal, the phase can be retrieved. For large finite crystals the intensity between Bragg peaks may be too weak to be measured. When the crystal is small, such as a micrometresized protein crystal, the intensity between Bragg peaks is no longer negligible. This extension of the oversampling method from noncrystalline specimens to small crystals is suggested and explored elsewhere (Miao & Sayre, 2000). In this method, crystal defects, if present, are accurately imaged.
The is somewhat different for noncrystals and crystals. When the specimen is noncrystalline, the diffraction pattern is weak and continuous. This pattern can therefore be oversampled,2. Methods and results
Oversampling a diffraction pattern alone, however, cannot uniquely determine the phase since one cannot distinguish the correct phase and its conjugate from the diffraction pattern only. This twofold ambiguity can be eliminated by using positivity constraints. When the energy of the incident Xrays is high and away from the absorption edges, the electron density of the specimen is mostly real and positive. When the energy of incident Xrays is low, the electron density is complex, but the real and imaginary parts are both positive. These positivity constraints on the electron density can thus be used for phase determination. To obtain the phase information from an oversampled diffraction pattern, we have developed an iterative algorithm (Miao et al., 1998; Miao & Sayre, 2000) by modifying that of Fienup (1982). Each iteration consists of the following four steps.

By employing this algorithm, we have retrieved the phase information from an experimental oversampled diffraction pattern (Miao et al., 1999). The diffraction pattern was recorded from a set of six letters with overall size about 5 µm. The letters, deposited on a silicon nitride membrane, were made up of gold dots each about 1000 Å in diameter and 800 Å in thickness. Fig. 1(a) shows a scanning transmission Xray microscope (Kirz et al., 1995) image of the specimen. The specimen was illuminated by a small and clean synchrotron Xray beam (λ = 17 Å). The detector, a backthinned and liquidnitrogencooled CCD with 512 × 512 pixels and a 24 × 24 µm pixel size, was placed about 25 cm downstream of the specimen. To eliminate any unwanted scattering from the air, both the specimen and the detector were mounted in a vacuum with a pressure of 10^{−5}–10^{−6} Torr (1 Torr ≃ 133.33 Pa). Fig. 1(b) shows an experimental diffraction pattern from the specimen in Fig. 1(a). In this figure a 19pixel radius circular area at the center, a part of the pattern lost owing to a beam stop to block the direct beam, was filled by a patch derived from the magnitude of the Fourier transform calculated from Fig. 1(a). Although the patch occupied less than 0.5% of the whole diffraction pattern area, it was critical to the convergence of our algorithm.^{1} We then input this diffraction pattern into our algorithm by employing a random initial phase set and a 7.5 × 7.5 µm square finite support. Figs. 2(a), 2(b), 2(c), 2(d) and 2(e) show the reconstructed density after 0, 100, 200, 300 and 400 iterations, respectively. One may notice that the density rotated 180° between Figs. 2(b) and 2(c), which was a consequence of the ambiguity of the correct phase and its conjugate. Fig. 2(f) shows the convergence of the algorithm where the error function was defined as the ratio of the total electron density outside the finite support to that of inside. The algorithm quickly pushed the electron density inside the finite support after 50 iterations, but took more iterations to find the correct phase set. After 400 iterations, a well reconstructed density (Fig. 3) was obtained. Fig. 3, evaluated more finely in the Fourier sum than Fig. 2(e) for display purposes, is consistent with the resolution limit, ∼600 Å, set by the angular extent of the CCD detector. The computing time for 400 iterations was about 15 min on a 450 MHz Pentium II workstation. We performed a few more reconstructions for the same diffraction pattern with different initial phase sets and found that the number of iterations required for convergence was somewhat different each time.
3. Conclusions
We believe that this method can open a door for highresolution threedimensional et al., 1998; Schneider & Niemann, 1998) show that biological specimens at liquidnitrogen temperature can tolerate a radiation dose of up to 10^{10} Gy dosage without observable morphological damage. This method may also be extended to determine the structure of micrometresized protein crystals if both the Bragg peaks and the intensity between the Bragg peaks can be measured. In the long run, it may become possible to image single molecules by combining a fourthgeneration Xray source, such as a freeelectron laser, with the oversampling phasing method. The challenge of this very intriguing goal is whether an atomic or nearly atomic resolution diffraction pattern can be recorded from single molecules before the structure is destroyed by radiation damage. The oversampling phasing method thus may shift the difficulty of growing crystals to overcoming the radiationdamage problem (Johnson & Blundell, 1999).
of biological specimens such as whole cells and submicrometre molecule clusters. The only limitation to the resolution is radiation damage, which can be mitigated by using cryogenic techniques. Experimental studies (MaserFootnotes
‡Mailing address: Stanford Synchrotron Radiation Laboratory, PO Box 4349 MS69, Stanford, California 943090210, USA.
^{1}We have found that the quality of the reconstruction is very sensitive to the accuracy of the data in the central patch. However, we have not performed any quantitative study of their relationship. In the long run, we plan to circumvent the problem by (i) recording the experimental diffraction pattern with the central patch as small as possible and (ii) enlarging our algorithm to allow it to iteratively fill in a small amount of data at the center.
Acknowledgements
The idea of applying Xray diffraction to threedimensional ). The decision to try oversampling as a phasing technique was arrived at in a conversation in the late 1980s with G. Bricogne. We are grateful to P. Charalambous, Kings College, for fabrication of the test specimens. We thank C. Jacobsen for his help and advice with the numerical reconstruction, C. Jacobsen and M. Howells for use of the apparatus in which the exposures were made and S. Wirick for her assistance in the data acquisition. One of us (JM) thanks S. Doniach and K. Hodgson, Stanford Synchrotron Radiation Laboratory, for stimulating discussions. The experiment was performed at beamline X1A of the National Synchrotron Light Source, which is supported by the Department of Energy. This work is supported in part by Grant No. DEFG0289ER60858 from the Department of Energy.
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