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Acta Crystallogr A. May 1, 2012; 68(Pt 3): 366–381.
Published online Mar 22, 2012. doi:  10.1107/S010876731200493X
PMCID: PMC3329770
Classifying and assembling two-dimensional X-ray laser diffraction patterns of a single particle to reconstruct the three-dimensional diffraction intensity function: resolution limit due to the quantum noise
Atsushi Tokuhisa,a Junichiro Taka,b Hidetoshi Kono,b and Nobuhiro Gobc*
aRiken Harima Institute, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo, 679-5148, Japan
bQuantum Beam Science Directorate, Japan Atomic Energy Agency, 8-1-7 Umemidai, Kidugawa-shi, Kyoto, 619-0215, Japan
cXFEL Division, Japan Synchrotron Radiation Research Institute, 1-1-1 Kouto, Sayo-cho, Sayo-gun, Hyogo, 679-5198, Japan
Correspondence e-mail: go.nobuhiro/at/spring8.or.jp
Received April 28, 2011; Accepted February 4, 2012.
A new two-step algorithm is developed for reconstructing the three-dimensional diffraction intensity of a globular biological macromolecule from many experimentally measured quantum-noise-limited two-dimensional X-ray laser diffraction patterns, each for an unknown orientation. The first step is classification of the two-dimensional patterns into groups according to the similarity of direction of the incident X-rays with respect to the molecule and an averaging within each group to reduce the noise. The second step is detection of common intersecting circles between the signal-enhanced two-dimensional patterns to identify their mutual location in the three-dimensional wavenumber space. The newly developed algorithm enables one to detect a signal for classification in noisy experimental photon-count data with as low as ~0.1 photons per effective pixel. The wavenumber of such a limiting pixel determines the attainable structural resolution. From this fact, the resolution limit due to the quantum noise attainable by this new method of analysis as well as two important experimental parameters, the number of two-dimensional patterns to be measured (the load for the detector) and the number of pairs of two-dimensional patterns to be analysed (the load for the computer), are derived as a function of the incident X-ray intensity and quantities characterizing the target molecule.
Keywords: biological macromolecules, classification of two-dimensional diffraction patterns, common intersecting circles, attainable structural resolution
New, intense X-ray free-electron laser (XFEL) light sources offer a new possibility in imaging single biological macromolecules. The main problems to be solved for realization of this possibility originate from the extreme weakness of the scattered light from a single molecule. One problem is severe damage of a target caused by a single shot of intense X-ray light used to compensate for the weakness. In this respect, a lower intensity of incident X-rays is preferred. Another problem due to the weakness is the quantum noise. Algorithms for structure determination must be developed to process the experimental data immersed in the quantum noise. From this perspective, a higher intensity of incident X-rays is preferred. This paper focuses attention on this latter problem. For this purpose we make a tentative assumption that the damage process can be neglected, and will clarify the mechanism of how the quantum noise sets a limit on the resolution of structure determination. In other words, we are interested in this paper only in the lower bound of the incident X-ray intensity.
The damage problem prevents the possibility of using the same molecule repeatedly as a target. Instead we assume that a target macromolecule assumes a well defined three-dimensional structure and a new molecule from an ensemble of the same molecules is placed repeatedly at the target position, but unfortunately in an unknown random orientation. A macromolecular complex with a definite molecular composition and three-dimensional structure can also be treated. The term ‘molecule’ is used to mean both a biological macromolecule and its complex. Extension of the results of this paper to cases of large-scale conformational fluctuations with a magnitude larger than the resolution of structure determination will be addressed in a future paper. Because of this limitation, we specifically exclude fibrous macromolecules and assume that molecules are globular with a more-or-less spherical shape. Note also that we develop analyses in this paper under idealizing (as compared with probable experimental realizations) assumptions about the state of the target molecule such as (i) ideally random orientations and (ii) the absence of hydrating water molecules. Adaptation to these realistic problems will also be treated in future papers.
A measurable two-dimensional diffraction intensity pattern depends on an unknown molecular orientation. This missing orientational information is to be recovered computationally during an analysis of a set of many two-dimensional intensity patterns. Also missing in a two-dimensional intensity pattern is the phase information necessary for derivation of a three-dimensional molecular structure. This missing phase information is also to be recovered computationally by the so-called oversampling method (Fienup, 1982 [triangle]; Elser, 2003 [triangle]).
The methods of single-particle imaging by XFEL can be classified into two paths, depending on which of the two types of missing information is recovered first. In the first, path A, method, a computational procedure is applied to a set of phase-missing two-dimensional intensity patterns to find their mutual locations in the three-dimensional wavenumber space. When a sufficient number of two-dimensional patterns are properly located, a three-dimensional diffraction intensity function can be constructed, to which the oversampling method is applied to recover the missing phase information. Together with this phase information, a three-dimensional real-space structure can be derived by an inverse Fourier transformation. In the second, path B, method, the oversampling method is applied to each of the measured two-dimensional intensity patterns. Together with the recovered phase information, a two-dimensional real-space structure is obtained by an inverse Fourier transformation, which is approximately a projection of a three-dimensional real-space structure along an axis of the incident X-ray beam. Such two-dimensional structures of a minivirus particle as revealed by a single-shot 6.9 Å hard-X-ray free-electron laser have been recently reported (Seibert et al., 2011 [triangle]). From many projected two-dimensional images thus obtained, a three-dimensional real-space structure can be constructed by applying the method of tomography. Such a three-dimensional human chromosome structure as revealed by coherent 2.5 Å X-rays from synchrotron radiation has been reported (Nishino et al., 2009 [triangle]).
Because of the weakness of scattered light from a single molecule, the quantum noise is a serious problem especially at high-angle pixels. The quantum noise appears to limit the resolution of a three-dimensional real-space structure to be obtained at the end, though by different mechanisms in paths A and B. In path A, the quantum noise is expected to set a resolution limit to locating a two-dimensional intensity pattern in the three-dimensional wavenumber space. Because photon-count data at many pixels can be considered integrally in a computational procedure to find a location, effective information seems extractable even from high-noise data at pixels with an expected mean photon count smaller than unity. Even though data at high-angle pixels are very noisy in each of the two-dimensional intensity patterns thus located, data at similar locations can be averaged to reduce the noise. The attainable structural resolution is determined by the wavenumber of a limiting pixel, from the data of which effective information can be extracted. In path B, where the oversampling method is applied directly to each two-dimensional intensity pattern, the quantum noise is expected to set a limit on the applicability of this method. When high-noise data from high-angle pixels with an expected photon count smaller than unity are included, the phase recovery procedure is expected to fail to converge and to cease to work. Therefore a pixel with an expected photon count of unity appears to be a limiting pixel, and its wavenumber appears to give the resolution. From such an analysis we expect that the path A analysis is better in extracting effective information from noisy data and therefore in deriving higher-resolution real-space structures. For this reason we are interested in this paper in developing a path A method. Of course, superior situations of the path B method are conceivable depending on problems of developing detecting devices and sample preparation, and also on the biological significance of the results obtained.
Methods hitherto proposed to find the locations of individual two-dimensional intensity patterns in the three-dimensional wavenumber space in path A methodology can be classified into two groups. In group 1 (Huldt et al., 2003 [triangle]; Bortel & Faigel, 2007 [triangle]; Shneerson et al., 2008 [triangle]; Bortel et al., 2009 [triangle]; Yang et al., 2010 [triangle]), a method of finding similarity between an arbitrary pair of two-dimensional intensity patterns is prepared. Then, a set of two-dimensional patterns are classified into groups of similar patterns according to this similarity, which are then averaged to reduce the quantum noise. Then, for an arbitrary pair of noise-reduced intensity patterns, their mutual location in the three-dimensional wavenumber space is identified by finding an intersecting circle between them. A three-dimensional diffraction intensity function can be constructed when a sufficient number of two-dimensional patterns are properly located in the three-dimensional wavenumber space. In the methods of group 2 (Fung et al., 2009 [triangle]; Loh & Elser, 2009 [triangle]; Elser, 2009 [triangle]; Loh et al., 2010 [triangle]), a tentative three-dimensional diffraction intensity function (or a function of similar mathematical setting) is assumed. Then, each two-dimensional intensity pattern is located so as to best fit in this three-dimensional intensity function. From a set of two-dimensional patterns thus located, the three-dimensional diffraction intensity function is updated. By repeating this cycle of best fitting and updating, an ultimate three-dimensional diffraction intensity function is obtained. Even though the methods of the second group appear promising, the demonstrated abilities of the individual methods proposed so far are limited.
We focus attention in this paper on developing an algorithm beyond existing methods of single-particle imaging belonging to the path A, group 1 methodology. New developments have been attained in two aspects. First, we will develop and improve methods of computational analyses and procedures to arrange a set of many experimentally measurable two-dimensional intensity patterns in the three-dimensional wavenumber space so that a three-dimensional intensity function can be constructed. The newly developed method enables us to attain higher-resolution structures. Second, we will derive explicit theoretical expressions for two main parameters which govern the number An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi1.jpg of two-dimensional patterns to be measured (the load for the measuring machine) and the number An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi2.jpg of pairs of two-dimensional patterns to be compared (the main computational load for analyses), as well as for the space resolution attainable from the analysis of the data, in terms of (a) the X-ray intensity used for the measurement and two types of quantities characterizing a target; (b) the Shannon molecular length (or, simply, molecular length) An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi3.jpg (the length of a side of the smallest cubic box that can contain a target globular molecule); and (c) the radial diffraction intensity density function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi4.jpg (the average of the the squared modulus of the structure-factor function on a sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi5.jpg in the wavenumber space). The achievement of the second aspect provides basic information for designing new experiments and experimental instruments.
The results in the two aspects are obtained in this paper by taking advantage of simulated diffraction intensity data for a protein, lysozyme, and a protein complex, HslUV complex, for which structural atomic coordinates are available from the Protein Data Bank (PDB) (Berman et al., 2000 [triangle]). The former (Weaver & Matthews, 1987 [triangle]) (PDB code 2lzm, number of residues 164, molecular length 60 Å) is chosen as a typical small globular protein. The latter (Sousa et al., 2002 [triangle]) (PDB code 1kyi, total number of residues 7416, molecular length 200 Å) is chosen from very large protein complexes in the PDB with more-or-less globular shape. But it is in a sense an atypical complex, because it has a big hollow space inside the structure. Simulations are carried out by assuming the wavelength of incident X-rays An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi6.jpg 1 Å and for various intensities. The Shannon molecular length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi3.jpg and radial diffraction intensity density function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi4.jpg are determined for these two targets from their respective PDB atomic coordinates, and are used to estimate numerical values of the two main experimental parameters, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi1.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi2.jpg, and also of the attainable resolution as functions of the incident X-ray intensity.
To make the results of this paper more useful for designing new experiments and experimental instruments, the theory must be extended so as to be applicable even for structure-unknown molecules. This objective is studied in a separate paper.
This paper is ordered as follows. In §2, we will discuss a method of finding similarity between an arbitrary pair of two-dimensional diffraction intensity patterns. This method is used to classify two-dimensional patterns into groups of similar patterns. In §3, we will treat the problem of finding relative orientations between groups of similar patterns, which is information to be used for assembling two-dimensional intensity patterns into a three-dimensional diffraction intensity density function. In §4, we will give a somewhat detailed summary. Readers may find it easier to comprehend §§2 and 3 by reading §4 in parallel. Appendices A , B and C describe mathematical derivations of relations used in §§2 and 3.
2.1. Two-dimensional diffraction pattern on an Ewald sphere  
Here we define notations of pertinent quantities. The experimentally observable diffraction intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi11.jpg, given in the unit of a number of photons arriving at a pixel of the detector of solid angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi12.jpg, is given, except for a phase factor, by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd1.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg is the incident X-ray intensity (given, in the following, in the unit of a number of photons per pulse of free-electron laser per mm2), An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi14.jpg is the classical electron radius, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi15.jpg is a coefficient given by these quantities, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi16.jpg is the structure factor, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi17.jpg is the momentum transfer and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi18.jpg is the diffraction intensity density. The magnitude of the momentum transfer is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd2.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi19.jpg is the X-ray wavelength and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg is the angle of diffraction. Note that, even though this angle is expressed as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi21.jpg in the usual literature, we nevertheless express it as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg because this quantity, having a meaning as part of a certain polar angle, has an important role in this paper. Even though the structure factor An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi16.jpg is a continuous function in the wavenumber space, its squared modulus is measured experimentally by a detector with an array of finite-sized pixels. When An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi24.jpg is discretely sampled at lattice points of a cubic lattice with a lattice constant of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi25.jpg, it corresponds to the use of the detector pixel size of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi26.jpg in solid angle, i.e.
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd3.jpg
In this expression, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi3.jpg is the length of a side of the smallest cubic box (Shannon box) that can contain a target globular molecule. We shall refer to this quantity as the Shannon molecular length or simply the molecular length. From the point of view of the oversampling method for phase retrieval (Fienup, 1982 [triangle]; Elser, 2003 [triangle]), the detector pixel size must be chosen so that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi28.jpg. The ratio An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi29.jpg is called the linear oversampling ratio. A pixel with An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi30.jpg will be referred to as a Shannon pixel.
The quantity of equation (1) is an expected number of photons arriving at a detector pixel. However, in real experiments, what is measured is an integral number of photons, given by the quantum-mechanical probability. In this paper we simulate this probability by replacing the function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi11.jpg of equation (1) with a stochastic function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi32.jpg which assumes only integral values according to the Poisson distribution. To distinguish these two functions, let us call the quantity of equation (1) the theoretical diffraction intensity, and the replaced stochastic function the experimental diffraction intensity.
Examples of a simulated two-dimensional experimental and a theoretical diffraction intensity pattern are shown in Figs. 1 [triangle](a) and 1 [triangle](b), respectively, for the case of lysozyme. We see that, when theoretical diffraction intensities are much less than unity, experimental diffraction intensity values at most pixels vanish. From such noisy data, we have to guess the correct mean values. These are patterns for which we develop a method of analysis for structure determination. For this purpose we need some theoretical tools. A mathematical expression of a two-dimensional pattern for a molecule with a given orientation is explored in detail in Appendix A . A molecular orientation is described by a Eulerian angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg with its corresponding 3 × 3 orthogonal matrix An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg given by equation (27). The Eulerian angle is defined so that, out of a set of three angles, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi35.jpg are polar angles of the direction of the incident X-ray beam with respect to the molecule, and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg is an angle of rotation of the detector plane around the axis of the incident beam. A two-dimensional pattern is given by the quantity of equation (1) on the surface of an Ewald sphere. By introducing a polar coordinate An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi37.jpg on the surface of an Ewald sphere, the two-dimensional pattern is given from equations (40), (39), (35) and (42) by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd4.jpg
This equation means that the Ewald spheres corresponding to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi39.jpg are essentially the same spheres giving the same surface.
Figure 1
Figure 1
Examples of two-dimensional diffraction intensity patterns simulated for lysozyme by assuming the wavelength of the incident X-rays An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi19.jpg = 1 Å and the intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi290.jpg 5 × 1021 photons pulse−1 mm−2 (more ...)
2.2. High correlation line in a correlation pattern  
As outlined in §1 we adopt in this paper the basic strategy in which, after measurement of a large number of two-dimensional patterns for a molecule in unknown random orientations, we classify them into groups of similar patterns. In the following we consider a pair of Eulerian angles An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi40.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi41.jpg with corresponding matrices An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi42.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi43.jpg, and two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi44.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi45.jpg. For this pair we will be interested in an angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg between the two beam directions An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi47.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi48.jpg, which satisfies
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd5.jpg
This angle plays the role of a measure of similarity between a pair of two-dimensional patterns. When it is very small, we classify them into one group of similar patterns even for very different An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi49.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi50.jpg.
Our problem is to judge, for a given pair of experimental two-dimensional patterns such as those shown in Fig. 1 [triangle](a), whether or not they are realizations of a similar theoretical two-dimensional pattern. The starting point is the calculation of a correlation function as was originally proposed by Huldt et al. (2003 [triangle]). In this treatment we are interested in pixels on a circle with a fixed value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg. Let the number of pixels on a circle be An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi52.jpg. Bortel & Faigel (2007 [triangle]) proposed to pre-normalize the data on a circle to have uniform second moments for the calculation of a correlation function. More recently, they (Bortel et al., 2009 [triangle]) proposed further to pre-normalize so as to have vanishing mean and uniform second moment and also to compare axially rotated patterns. In our method we normalize the data on a circle to have a uniform mean and are interested in the correlation of their deviation from the mean after they are mutually rotated by an angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg. This means that we are concerned with the following correlation function:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd6.jpg
This quantity can also be expressed as follows:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd7.jpg
The correlation function of equation (6) is calculated as a two-dimensional function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg, which will be referred to as a correlation pattern. The merits of using the normalized quantity of equation (6) are to normalize for variations of intensities over different circles and for experimental variations of the pulse intensities of XFELs.
An example of a correlation pattern is shown in Fig. 2 [triangle](a). In this figure the pattern is shown not as a function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg but as a function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg, where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg is given by equation (2). In a general case where the angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg between the two beam directions is not very small, this correlation pattern appears to be a random function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg. Let us write such An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi64.jpg as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi65.jpg, where BG stands for background. When An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg is very small, there appears a line of high correlation for a certain value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg. In the particular case of Fig. 2 [triangle](a), the two beam directions are identical, and therefore both An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg vanish.
Figure 2
Figure 2
Correlation pattern for lysozyme by assuming the intensity of the incident X-rays An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi290.jpg 5 × 1021 photons pulse−1 mm−2. (a) Correlation pattern of equation (6) for a pair of two-dimensional experimental intensity (more ...)
The correlation pattern of equation (6) is heavily affected by the quantum noise. When the quantum noise is suppressed, this quantity is given approximately by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd8.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi70.jpg is a mean averaged over the quantum noise. To derive this expression we introduced an approximation of taking the means of three factors independently, an approximation which is good when standard deviations of the three factors are significantly smaller than their respective means. Examples of this quantity are shown in Figs. 2 [triangle](b) and 2 [triangle](c). Fig. 2 [triangle](c) shows a pair of diffraction patterns for which the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg is not small. This is an example of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi72.jpg. Even though Figs. 2 [triangle](b) and 2 [triangle](c) contain no effect of quantum noise, the pattern other than the high correlation line appears to be rather random. Such a behaviour should be a consequence of an appearance of the diffraction intensity density function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi18.jpg that can be captured as a stochastic function.
To characterize the diffraction intensity density function from such a point of view, we studied first how values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi74.jpg are distributed on a sphere of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi5.jpg around its mean An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi76.jpg. For this purpose, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi77.jpg points are sampled randomly with a uniform probability on each sphere, and the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi18.jpg is calculated at each sampled point. We confirmed that, except for small values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg, the distribution is given to a very good accuracy by the exponential distribution as was originally discovered by Wilson (1949 [triangle]). Moreover, as shown in Fig. 3 [triangle], it is found empirically that, except for small values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg, values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi18.jpg on the sphere of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi5.jpg are correlated in such a way as to satisfy the following simple relation,
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd9.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi83.jpg is an angle between the two An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi17.jpg vectors, the average is taken over all pairs of vectors with a given value An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi83.jpg, and the correlation length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi86.jpg has been found to be independent of the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg and is given approximately by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd10.jpg
Because, as shown by Wilson (1949 [triangle]), the exponential distribution is a consequence of the irregular three-dimensional structures of biopolymers at the atomic level, we shall refer to the empirically observed distribution as a structure irregularity distribution.
Figure 3
Figure 3
Normalized correlation function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi297.jpg of equation (9) for the space correlation of the values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi18.jpg on a sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi299.jpg of radius An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg for the HslUV complex. The angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi83.jpg is shown in the abscissa as a product with An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg. For An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg equal to or larger than 0.2 Å−1 (more ...)
To derive the mean behaviour of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi65.jpg, we average An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi72.jpg over the structure irregularity distribution. As shown in Appendix C , we see that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi90.jpg vanishes. This is reasonable because when there is no correlation between two points on two circles appearing in equation (6) an average can be taken on each of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi91.jpg to yield a vanishing result.
Further examples of a correlation pattern of equation (6) are shown in Fig. 4 [triangle]. Because the mean of the correlation pattern vanishes except for a high correlation line, a mathematical expression of the high correlation line should be obtained as a mean of the correlation pattern over the two distributions, the Poisson distribution and the structure irregularity distribution. As shown in Appendix C it is given to a good approximation by the following expression:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd11.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg is a function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg through equation (2) and the direction of the high correlation line, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi94.jpg, is given to the zeroth order of the small quantity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd12.jpg
It should be noted that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg appears in equation (11) as a quantity normalized by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi86.jpg, or, because of equation (10), as a product An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi98.jpg.
Figure 4
Figure 4
Correlation patterns for the HslUV complex and by assuming the intensity of incident X-rays An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi305.jpg photons pulse−1 mm−2. (a), (b) and (c) are for the angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg between the two beam directions being 0, 1 and 3°, respectively. (more ...)
As mentioned earlier, Bortel et al. (2009 [triangle]) proposed to use a quantity pre-normalized to have a vanishing mean and uniform second moment. However, when we apply our analysis to their proposed correlation pattern, an expression similar to equation (11) is obtained but with an additional factor which is approximately An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi99.jpg and becomes smaller for larger An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg. Because of this additional factor, the quantity they employed for detecting similarity between two-dimensional patterns is less sensitive for high An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg values. This explains why our method is more sensitive for higher An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg values.
2.3. Identifying the high correlation line against the noisy background and attainable resolution  
In Fig. 4 [triangle] we see that, even though the mean value of the correlation pattern should vanish in the background, its actual values become very noisy for larger values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg. This is due to both the quantum noise and the structure irregularity distribution. Identification of the high correlation line would be affected by the noise. (Note that we are here treating the structure irregularity distribution as a part of the noise.) The level of noise in the quantity of equation (6) can be expressed by its standard deviation. As derived in Appendix C , it is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd13.jpg
where the function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi104.jpg is defined by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd14.jpg
and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi52.jpg is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd15.jpg
which means that we are assuming Shannon pixels. Fig. 5 [triangle] shows the graph of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi106.jpg. Fig. 6 [triangle] is a plot of the standard deviation given by equation (13), which is a globally increasing function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg in the high-An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg region. When averaged over the two distributions, the Poisson distribution and the structure irregularity distribution, the peak value of the high correlation line for a given value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg (or, equivalently An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg) is An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi111.jpg according to equation (11), which is a decreasing function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg. It is expected that the actual high correlation line is observable roughly up to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi113.jpg, where the mean peak value becomes equal to the standard deviation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg, i.e.
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd16.jpg
In fact, in Figs. 2 [triangle](a) and 4 [triangle](a) for cases of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi115.jpg and therefore where the mean peak value should stay unity, the actual high correlation lines are observed for lysozyme up to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi116.jpg 0.7 Å−1 and for the HslUV complex up to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi116.jpg 0.55 Å−1, where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi118.jpg according to Fig. 6 [triangle]. In Fig. 4 [triangle](b) for the case of the HslUV complex with An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg =  1°, the actual high correlation line is observable up to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi116.jpg 0.3 Å−1, where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi121.jpg according to Fig. 6 [triangle] and equation (16) is roughly satisfied.
Figure 5
Figure 5
Graph of the function of equation (14). The abscissa An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi161.jpg and ordinate An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi162.jpg mean physically an expected number An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg of photons arriving at a Shannon pixel at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi151.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi314.jpg, respectively.
Figure 6
Figure 6
Plot of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi4.jpg, the average of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi18.jpg of equation (1) on the sphere of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi299.jpg (black line), and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg of equation (13) (blue line) for lysozyme (a) and the HslUV complex (b). The intensities assumed are An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi319.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi320.jpg photons pulse−1 mm−2, respectively. (more ...)
From equation (16) and Fig. 6 [triangle], we see that we can derive the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg from the measured length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi123.jpg of the high correlation line. The longer the length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi123.jpg, the smaller the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg. From the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg, we judge the similarity of a pair of two-dimensional patterns. When judged similar, they are classified into the same group. After the classification, two-dimensional patterns classified into the same group are averaged in order to improve the signal-to-noise ratio. Because this averaging is done for patterns with slightly different directions of the incident beam, the resolution of the resulting three-dimensional structure will be affected. In order to attain the highest possible resolution, we should adopt a strategy in which we classify a pair of two-dimensional patterns into the same group when their high correlation line reaches the highest possible k region. Let us define An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg (subscript An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi128.jpg for noise) as the lower bound of such a region. This quantity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg, the limiting An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg value for correlation recognition, plays a central role in the method of single-particle imaging developed in this paper. In the case of Fig. 4 [triangle](a) we judge that the high correlation line extends up to such a region, where the line can no longer be distinguished from the background. In the case of Fig. 4 [triangle](b) the line appears to have faded away before reaching such a region. The limiting value An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg, to be determined purely operationally in real applications, appears more-or-less well defined. However, we need to interpret the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg in a more theoretical setting. Because it defines the lower bound of the noisy region, it should be characterized by its value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg. From Figs. 4 [triangle](a) and 4 [triangle](b) we see that it should be between 0.6 and 1.0. As a modest estimate, we assume that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg corresponds to the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg at which An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi136.jpg. Then, from equation (16) we see that the corresponding value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg is estimated to be within
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd17.jpg
Let us now assume that a classification group of similar two-dimensional patterns is constructed by a group of two-dimensional patterns with An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg within this angle from a certain reference two-dimensional pattern. Note that the average distance (root-mean-square distance) between a pair of two-dimensional patterns in this classification group is also given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi139.jpg. During the procedure of averaging, two-dimensional patterns rotated by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg around the origin are averaged. The magnitude of displacement in An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg space by this rotation is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi142.jpg, with its maximum value being An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi143.jpg. When this magnitude is smaller than the correlation length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi144.jpg of the diffraction intensity density, the averaging procedure works to attenuate the effect of the noise. When the product becomes larger than An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi144.jpg, the averaging procedure works to destroy the information in two-dimensional patterns. This means that the structural information is contained in An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg only up to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi147.jpg satisfying An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi148.jpg. Then, from equation (17), we see
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd18.jpg
A limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg, which is an expected number of photons arriving at a limiting pixel, a Shannon pixel at the limiting An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg value, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi151.jpg, is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi152.jpg. In equation (13), An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi153.jpg at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi151.jpg is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi155.jpg An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi156.jpg. This expression can be approximated as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi157.jpg, because An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi158.jpg is in most cases at least a few times larger than An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi19.jpg. In Fig. 5 [triangle] for a graph of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi106.jpg, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi161.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi162.jpg can also be interpreted as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi164.jpg, respectively. Note that the normalized resolution, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi165.jpg, is the number of independent structural descriptive elements along the molecular length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi3.jpg. For a method of single-molecule imaging to be useful, this number should be at least 20, hopefully An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi167.jpg. Note that this number is determined mainly by the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg. Fig. 5 [triangle] shows this dependence. We see that, to attain An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi169.jpg 20–100, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg should be in the range of 0.25–0.08. We have to measure and analyse such low-photon-number data. Also this number highlights a high sensitivity of the proposed method of analysis to extracting information from noisy data.
In the above relation between the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg and the normalized resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi165.jpg, the incident X-ray intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg is treated as an implicit variable parameter. To identify a particular value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg to attain a resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi175.jpg, we remember the relation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi176.jpg [equation (1)]. Then, by defining a function inverse to the function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi104.jpg of equation (14) as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi178.jpg, equation (13) can be transformed to
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd19.jpg
Fig. 7 [triangle] shows the resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi147.jpg as a function of intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg for lysozyme and the HslUV complex obtained by using this equation. When there is more than one value of resolution for a given value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg, the best value can be obtained. (The high correlation line may become visible again in a high-An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg but low-noise region after once becoming invisible in a low-An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg but high-noise region.)
Figure 7
Figure 7
Attainable resolution as a function of the incident X-ray intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg (photons pulse−1 mm−2) for lysozyme (dotted and solid right-hand lines) and the HslUV complex (dotted and solid left-hand lines). When there is more than (more ...)
Since the solid angle of the range of one classification group is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi184.jpg, and the total solid angle of the direction of the incident beam is An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi185.jpg because of the centrosymmetric property of the three-dimensional diffraction intensity function, the number of classification groups An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi186.jpg is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd20.jpg
After two-dimensional patterns are classified by the method described in the previous section, patterns classified into the same group are averaged to reduce the noise. When signal-enhanced patterns are obtained, they are to be placed in the three-dimensional wavenumber space by finding their relative orientations. The two-dimensional patterns exist on Ewald spheres. Because all these Ewald spheres have the same radii and their surfaces contain the origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi187.jpg of the wavenumber space, any pair of Ewald spheres either contact at the origin or have a circular intersection, which also contains the origin. Shneerson et al. (2008 [triangle]) studied the problem of placing two-dimensional patterns in the three-dimensional space by paying attention only to an approximately straight portion of the intersecting circles near the origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi187.jpg. Yang et al. (2010 [triangle]) refined the method of finding the tangential direction of the intersecting circle at the origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi187.jpg by explicitly paying attention to the curvature of the intersection. However, for the placement problem they used the method of Singer & Shkolnisky (2011 [triangle]) for cryo-electron microscopy in which only information on tangential directions is used. However, it is obvious that a relative orientation between a pair of Ewald spheres can be determined once their common circle is identified. In this section we first develop a method of identifying an intersecting circle for a given pair of signal-enhanced two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi190.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi191.jpg, and derive a mathematical expression for the relative orientation. Second, we ask what is the necessary number of patterns to be averaged for possible identification of common circles? The actual construction of a single three-dimensional diffraction intensity function from the data of relative orientations will be treated in a different paper.
We assume that a pair of signal-enhanced two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi190.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi191.jpg exist on Ewald spheres of as yet unknown orientations, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg. Because of the centrosymmetric property of the three-dimensional diffraction intensity function, any Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg has its centrosymmetric image characterized by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi197.jpg. Therefore, Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg should have an intersecting circle with each of the Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi200.jpg. This means that for any pair of two-dimensional patterns, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi190.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi191.jpg, there exist two common circles. A method of finding them is developed in Appendix B and here we describe only the result. Each of the two common circles exists as a circle of vanishing values in each of two groups of plots, (a) An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi203.jpg and (b) An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi204.jpg, where the parameters An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi205.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi206.jpg generate each of the two groups, respectively. When the plot (a) vanishes on a circle with its centre at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi207.jpg for a certain parameter value An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi205.jpg, the polar coordinates of the centres of intersecting circles are An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi209.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi210.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi211.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi212.jpg. When the plot (b) vanishes on a circle with its centre at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi213.jpg for a certain parameter value An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi206.jpg, the polar coordinates of the centres of intersecting circles are An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi215.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi210.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi217.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi212.jpg. Then, the Euler angle of the relative orientation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi219.jpg is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd21.jpg
Thus, the same set of Eulerian angles An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg is now determined from each of the intersecting circles between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg, and between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi200.jpg. Even though the result is redundant, the actual procedures of finding vanishing circles on plots (a) and (b) are much influenced by experimental noise. In this situation, finding the same quantity simultaneously by the two methods is a desirable numerical procedure.
Fig. 8 [triangle] shows an example of how circles of vanishing values become visible as the number of averaging patterns is increased. In the case of this example for the HslUV complex, in which the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg is An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi226.jpg, we see by inspection that averaging over about 61 patterns is necessary for the identification. This process has been done for lysozyme and the HslUV complex both for a series of values of the incident X-ray intensity. It has been found that a product of the number An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi227.jpg of necessary patterns and the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg (therefore, the intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg of the incident X-ray) is constant in either ‘molecule’. The analysis described in Appendix C indicates that the product An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi230.jpg must be 8 or larger for common circles to be identified. This is exactly the number observed in the case of Fig. 8 [triangle]. Therefore, the necessary number of patterns is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd22.jpg
Table 1 [triangle] summarizes the results obtained as applied to the two ‘molecules’.
Figure 8
Figure 8
Plots for detecting an intersecting circle between a pair of two-dimensional patterns for the case of the HslUV complex and for the incident X-ray intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi305.jpg photons pulse−1 mm−2. By careful examination of the plot, (more ...)
Table 1
Table 1
Resolution and necessary number of patterns and classification calculations expected for two sample ‘molecules’
Two aspects of a method of single-particle imaging belonging to the path A, group 1 methodology have been developed. First, a new, improved method has been developed for computational analyses and procedures to arrange a set of many experimentally measurable two-dimensional diffraction intensity patterns in the three-dimensional wavenumber space. Second, explicit theoretical expressions have been derived for important experimental parameters in terms of the incident X-ray intensity and two types of quantities characterizing a target.
The number An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi1.jpg of two-dimensional patterns to be measured is given by the product An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi232.jpg of the number of classification groups An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi186.jpg and the average number of two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi227.jpg to be averaged in each group for noise reduction. The number An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi2.jpg of pairs of two-dimensional patterns to be analysed is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi236.jpg, because the detection of similarity of patterns is to be carried out for each of the pairs, one from patterns representing each group and the other from all measured patterns. We derived theoretical expressions for the two parameters, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi186.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi227.jpg.
Concerning the first aspect, we have improved a hitherto proposed method for judging whether or not an arbitrary pair of two-dimensional patterns are similar enough to belong to the same classification group. Also, we developed methods of finding common intersecting circles between an arbitrary pair of noise-reduced two-dimensional patterns, and thereby relatively locating them in the three-dimensional wavenumber space. After locating many two-dimensional diffraction patterns properly in the wavenumber space, we have to construct a single three-dimensional diffraction intensity function. This problem, as well as the problem of application of the phase retrieval procedure to such a three-dimensional function, will be treated in a different paper.
The judgment of similarity is based on a two-dimensional correlation pattern for each pair of two-dimensional diffraction intensity patterns. For the calculation of correlation patterns, a new normalization of measurable two-dimensional intensity patterns is employed, thereby enabling one to enhance the sensitivity of judgment to high-angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg values, and eventually to improve attainable space resolution.
A two-dimensional intensity pattern depends on the direction of the incident X-ray beam with respect to the molecule-fixed coordinate system and an angle of rotation of the detector plane placed perpendicularly to the beam axis. When an angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg between the directions of the incident beam for a pair of two-dimensional intensity patterns is small, a high correlation line is observed in the two-dimensional correlation pattern as a straight line extending radially from the centre. The angle of the line in the correlation pattern gives a relative angle of rotation of the detector plane. The intensity of a high correlation line is unity near An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi241.jpg and becomes weaker at higher angles. The intensity reduces faster for larger values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg.
The background of correlation patterns other than the high correlation line is characterized as a pattern of random appearance reflecting the irregular three-dimensional structures of biopolymers at the atomic level superimposed with the quantum noise. Owing to the deliberately adopted, new normalization of measurable two-dimensional intensity patterns, the mean value of the distribution in the background of two-dimensional correlation patterns turns out to vanish. The standard deviation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg of the distribution around its vanishing mean becomes globally, but not monotonically, larger as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg becomes larger. When the standard deviation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg becomes larger than the intensity of a high correlation line, the latter becomes no longer recognizable. The recognizable length of a high correlation line becomes longer as the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg becomes smaller. The latter can be determined from the former.
When the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg is smaller than a certain value, say, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi139.jpg (therefore, when the recognizable length of the corresponding high correlation line becomes longer than a certain value, say, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg), we classify the pair into the same group. To attain the best resolution, we should employ the largest possible value for An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg. Operationally we determine the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg, the limiting k value for correlation recognition, as the lower bound of the noise-dominant An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg region in two-dimensional correlation patterns. Such a value An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg can be characterized theoretically as the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg at which the standard deviation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg of the background distribution is An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi256.jpg. An analytic expression, equation (13), for the standard deviation is derived which is approximately a function of the wavenumber normalized by the Shannon length of the target molecule, i.e. An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi98.jpg, and an expected photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi258.jpg by a pixel at the position of the wavenumber An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg. The quantity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg plays a central role in the method of analysis developed in this paper. It is shown that the structural resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi147.jpg attainable by this method is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg.
In the method of identification of a high correlation line, upon which judgment of similarity of a pair of two-dimensional intensity patterns is based, effective information is extracted from the very noisy data in the range of wavenumbers up to the limiting value An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi127.jpg where the value of the standard deviation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg is An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi256.jpg. From the analytic expression for An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi114.jpg, we can derive the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg, an expected photon count at a limiting pixel, approximately as a function of the normalized resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi268.jpg (Fig. 5 [triangle]). For a method of structure determination to be useful, the value of the normalized resolution should be in the range of 20–100. The corresponding value of the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg turns out to be in the range of 0.25–0.08. The proposed method of analysis is sufficiently sensitive to enable one to extract information from such low-photon-count noisy data. This high sensitivity has been attained by employing a new correlation function. When the molecular length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi3.jpg and the radial diffraction intensity density function An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi4.jpg are known, the above relation between the normalized resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi165.jpg and the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi176.jpg can be transformed to a relation giving the intensity An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi13.jpg of the incident beam to be used to attain a resolution An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi147.jpg.
The angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi139.jpg to define a range of classification groups is given by an inverse of the normalized limiting wavenumber, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi277.jpg. As a result, the number of classification groups An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi186.jpg is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi279.jpg.
A method of identifying common circles between an arbitrary pair of noise-reduced two-dimensional patterns is developed. For an arbitrary Ewald sphere, there exists a conjugate Ewald sphere which is centrosymmetric with respect to the origin of the wavenumber space. In the proposed method, when a common circle between two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi280.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi281.jpg is searched, another common circle is searched at the same time between two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi280.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi283.jpg, where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi283.jpg is a two-dimensional pattern on an Ewald sphere conjugate to the one on which An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi281.jpg exists. The average number of two-dimensional patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi227.jpg to be averaged in each group for identification of common circles has been shown to be given in terms of the limiting photon count An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi149.jpg by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi288.jpg.
The obtained theoretical expressions are used to evaluate values of important parameters for the two sample ‘molecules’ by assuming, respectively, two typical intensities of the incident beam. The results are shown in Table 1 [triangle]. We should note the very low limiting photon counts, highlighting the strength of the method developed here. We should also note that the predicted attainable resolutions are remarkably high. This is partly due to the strength of the method of analysis developed here, but also due to the assumed high intensities of the incident beam. The assumed values of intensity in Fig. 7 [triangle] and Table 1 [triangle] are in the range of around 1021 photons pulse−1 mm−2, which is far larger than the peak value of 1.6 × 1016 photons pulse−1 mm−2 reported in the recent experiment (Seibert et al., 2011 [triangle]) carried out at the Linac Coherent Light Source (LCLS). Because the X-ray beam diameter reported in the experiment at LCLS is about 10 µm and a new technology (Mimura et al., 2010 [triangle]) is now available to focus it down to 10 nm, the values assumed in this paper appear realistic. Since we developed the analysis in this paper under a tentative assumption that damage processes can be neglected, the indicated intensity is the lower bound to attain a targeted resolution. We are now carrying out a study of the damage processes to assess the upper bound of employable intensity. The number of two-dimensional patterns to be measured in Table 1 [triangle] is not small, but appears tractable for real experiments. At the same time we should note that the number of classification calculations is not small.
Acknowledgments
This study has been supported by the ‘X-ray Free Electron Laser Utilization Research Project’ of the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT). We are heavily indebted to Dr Yasumasa Joti for his help during the revision of the manuscript.
Appendix A . Relations between molecular orientation and the Ewald sphere
We define two right-handed coordinate systems, one fixed to the molecule and the other fixed to the experimental detector. They are defined, respectively, in terms of a set of mutually orthogonal unit vectors An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi342.jpg fixed to the molecule and in terms of another set of mutually orthogonal unit vectors An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi343.jpg fixed to the detector. The position of any point in the molecule can be expressed either in terms of coordinates An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi344.jpg (here the superscript t means transpose) in the molecule-fixed coordinate system (MFCS) as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi345.jpg or in terms of coordinates An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi346.jpg in the detector-fixed coordinate system (DFCS) as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi347.jpg. Thus
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd23.jpg
The relative orientation between the MFCS and DFCS can be described in terms of a 3 × 3 orthogonal matrix An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd24.jpg
This is a shorthand notation for
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd25.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi349.jpg is an element of the matrix An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg. This is an orthogonal matrix, i.e. An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi351.jpg. By introducing equation (24) into equation (23), we have the following relation between coordinates in the two coordinate systems:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd26.jpg
It is convenient to express the orthogonal matrix in terms of a Eulerian angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd27.jpg
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd28.jpg
where
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd29.jpg
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd30.jpg
The range of variation of the Eulerian angle is taken as follows:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd31.jpg
Molecular structure is described by its electron density:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd32.jpg
When the molecule is in the orientation described by an orthogonal matrix An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg with respect to the detector, the electron density in the DFCS is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd33.jpg
The structure factor is calculated in the MFCS as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd34.jpg
The structure factor in the DFCS is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd35.jpg
This equation has the same structure as equation (33), indicating that the electron density and the structure factor behave in the same way for rotation. It follows from this that the wavenumber vectors An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi17.jpg in the MFCS and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi355.jpg in the DFCS are related to each other similarly as in equation (26):
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd36.jpg
Experimentally, the diffracted X-rays from a single molecule in a certain specific orientation are measured by a two-dimensional detector as a continuous diffraction pattern. This diffraction pattern is given by the squared modulus of the structure factor on the following Ewald sphere in the wavenumber space in the DFCS,
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd37.jpg
Here An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi19.jpg is the wavelength of X-rays used in the experiment. The incident X-rays are assumed to proceed to the negative direction of the third axis An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi357.jpg of the DFCS. The first and second axes are taken on the surface of the detector. Equation (37) describes a sphere in the An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi355.jpg space. The surface of the sphere contains the origin of the wavenumber space and its centre is located at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi359.jpg.
We now introduce a polar coordinate An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi37.jpg on the surface of this Ewald sphere by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd38.jpg
The origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi361.jpg of the polar coordinate corresponds to the origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi362.jpg of the wavenumber space. This polar coordinate system is adopted so that when the detector is viewed from the direction of the incident X-rays, the detector’s positive horizontal and vertical axes coincide with the directions of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi363.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi364.jpg, respectively. The observable diffraction pattern is given by equation (1) as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd39.jpg
Because the orthogonal matrix An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg can be specified by a Eulerian angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg, we sometimes write
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd40.jpg
We also identify various Ewald spheres by their corresponding orthogonal matrix An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg.
Let us now locate the Ewald sphere in the MFCS. It is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd41.jpg
The last equality means
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd42.jpg
This equation means that the Ewald spheres corresponding to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi39.jpg are essentially the same spheres giving the same surface. The former is obtained from the latter by rotating the latter anticlockwise around its third axis by angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg.
Here, the different ways in which a Eulerian angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi33.jpg appears in the two different coordinate systems may be worth noting. In the DFCS, a set of three angles describes the spatial orientation of the molecule. In the MFCS, the angles An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi35.jpg are nothing but the polar coordinates of the incident X-ray beam axis An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi357.jpg and the angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg describes the angle of rotation of the detector around this beam axis. Because of these clear meanings of the Eulerian angles in the MFCS, we employ them to describe molecular orientations rather than often mathematically better behaved quaternions.
Equation (42) suggests an experimental procedure of classifying various observable diffraction patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi375.jpg into groups only with respect to values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi376.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi377.jpg. Those with the same values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi376.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi377.jpg but with different values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi36.jpg are to be classified into the same group.
Because the electron density An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi381.jpg is a real function, the structure factor satisfies the relation
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd43.jpg
Because of this relation, the same diffraction pattern is observed on a pair of different Ewald spheres. From equations (40), (39), (35) and (43) we have
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd44.jpg
The last equation indicates that two Ewald spheres, characterized by rotation matrices An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi197.jpg, respectively, exist always in a pair. They occupy positions centrosymmetric to each other with respect to the origin of the wavenumber space, and they are in touch with each other at the origin. When we introduce a right-handed coordinate system to the Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg with its origin at the centre of the sphere, the corresponding coordinate system of the centrosymmetric Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi197.jpg is now left handed. This fact is understood as the two Ewald spheres having different handedness or parity.
Appendix B . Finding the relative orientation between a pair of unknown orientations from their corresponding diffraction intensity patterns
Real, experimentally observable diffraction patterns are subject to severe quantum noise. To cope with such noise, the same patterns should be measured many times and their means should be calculated to reduce the noise. In the following, the quantity of equation (39) is used under the assumption that such averaging has already been done so that the effect of noise can be neglected.
Let the two unknown orientations be given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd45.jpg
The corresponding Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi42.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi387.jpg are given, respectively, by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd46.jpg
and
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd47.jpg
Because all these Ewald spheres have the same radii and their surfaces contain the origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi187.jpg of the wavenumber space, they either contact at the origin or have a circular intersection, which also contains the origin.
Let us now study the intersecting circle between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi390.jpg. Let An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi391.jpg on the first Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi393.jpg on the second Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg be the same point on the intersecting circle,
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd48.jpg
Therefore, by setting
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd49.jpg
we have
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd50.jpg
These equations mean that the polar coordinates An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi391.jpg on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg for points on the intersecting circle with Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg are also polar coordinates on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg for points on the intersecting circle with Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg. Similarly, the polar coordinates An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi393.jpg on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg for points on the intersecting circle with Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg are also polar coordinates on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg for points on the intersecting circle with Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi404.jpg.
We now calculate the intersecting circle between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg, and that between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi404.jpg. We assume that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg is given by equation (27). The centres of the three Ewald spheres, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi410.jpg, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi411.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi412.jpg, are given in the MFCS by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd51.jpg
Therefore
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd52.jpg
Let the centre of the intersecting circle between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi34.jpg on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg be An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi416.jpg. Similarly, let the centre of the intersecting circle between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi418.jpg on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi398.jpg be An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi420.jpg. They are given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd53.jpg
We now proceed to describe a procedure to find the intersecting circles for a given pair of experimental diffraction patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi210.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi212.jpg. When we move clockwise along the intersecting circle on the Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg, it appears as an anticlockwise motion along the intersecting circle on the Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg. Therefore, we calculate the following quantity for all assumed values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi205.jpg in the range of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi426.jpg:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd54.jpg
This quantity should vanish on a certain circle for a certain value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi205.jpg. Because the intersecting circle contains the polar origin An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi361.jpg, it can be uniquely specified by the polar coordinate of its centre. When the quantity of equation (54) vanishes on a circle with its centre at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi207.jpg, the polar coordinates of the centres of intersecting circles are An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi209.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi210.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi211.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi212.jpg. These polar coordinates of the centres are to be compared with equation (53). Therefore,
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd55.jpg
From these relations we have a half of equation (21) in the main text.
Let us now proceed to study the intersecting circle between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi200.jpg. By following similar procedures as above we have the following relations from equation (47):
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd56.jpg
By introducing similar quantities and following similar procedures, we have the following results:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd57.jpg
To find the intersecting circle between Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi200.jpg for a given pair of experimental diffraction patterns An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi210.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi212.jpg, we calculate the following quantity for all assumed values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi206.jpg in the range of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi441.jpg:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd58.jpg
When we move clockwise along the intersecting circle on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi194.jpg, it appears as an anticlockwise motion along the intersecting circle on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi200.jpg. Because Ewald spheres An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi200.jpg have opposite parities, this motion appears as a clockwise motion along its centrosymmetric circle on Ewald sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi195.jpg. This is the reason why the sign in front of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi447.jpg in An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi448.jpg in equation (58), unlike in equation (54), is now plus. When the quantity of equation (58) vanishes on a circle with its centre at An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi213.jpg, the polar coordinates of the centres of intersecting circles are An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi215.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi210.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi217.jpg on An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi212.jpg. These polar coordinates are to be compared with equation (57). Therefore
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd59.jpg
From these relations we have another half of equation (21) in the main text.
Appendix C . Derivation of equations (11), (12), (13) and (22)
The mean of the quantities of equations (6) and (7) with respect to the Poisson distribution is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd60.jpg
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd61.jpg
When the quantity of equation (60) is averaged over the structure irregularity distribution, we have from equation (9) and the relation An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi454.jpg due to the exponential distribution
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd62.jpg
where An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi455.jpg is an angle between the two An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi17.jpg vectors given explicitly as follows according to equation (41):
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd63.jpg
When the two An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi17.jpg vectors are always significantly different, equation (62) reduces to
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd64.jpg
The average of the quantity of equation (61) over the structure irregularity distribution is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd65.jpg
By introducing an approximation
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd66.jpg
we see from equations (64) and (65) that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi90.jpg vanishes as mentioned in the main text. Also, from equations (62), (65) and (66) we have
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd67.jpg
This is an expected expression when all background features having random appearance due to the quantum noise and the structure irregularity distribution are erased from such high correlation lines as observed in Figs. 2 [triangle](a) and 2 [triangle](b).
We now proceed to simplify equation (67). At first we calculate An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi455.jpg, an angle between the two vectors of equation (63). The magnitude of the two vectors is given by equation (2). To calculate the inner product between the two vectors, we introduce the relative Eulerian angle An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi460.jpg by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd68.jpg
Here An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg is the angle between beam directions given by equation (5). In terms of the relative Eulerian angle, we have
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd69.jpg
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd70.jpg
The quantity of equation (67) is to be obtained by substituting An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi455.jpg from this equation. Let us write the result of the integration with respect to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi447.jpg in equation (67) by writing three independent variables explicitly as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi464.jpg. Here, remember that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi58.jpg are related by equation (2). Even though it is difficult to derive an analytic form of this quantity, we can calculate it numerically. Fig. 9 [triangle](a) shows an example of this quantity obtained numerically. For practical applications we want to have an analytic form of this quantity even when it may be somewhat approximate. To derive an approximate but analytic form we at first notice that the integrand of equation (67) has a significant contribution only when An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi325.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg are small. By using this fact we Taylor-expand both sides of equation (69) in terms of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi455.jpg, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi325.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg, and retain only up to the second-order terms to obtain
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd71.jpg
The quantity of equation (67), obtained numerically by using this expression, was found to have almost no difference from An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi472.jpg, meaning that equation (71) is a very good approximation to equation (69). Even for this expression of equation (71), the derivation of an analytic form for the integral of equation (67) is difficult. In this situation we introduce a drastic approximation of replacing the quantity of equation (71) by its mean
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd72.jpg
and use the value of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi83.jpg thus obtained in equation (67). Writing the result thus obtained as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi474.jpg we have
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd73.jpg
For practical applications we can approximate this expression further as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd74.jpg
Fig. 9 [triangle](b) is a plot of this quantity. By comparing it with Fig. 9 [triangle](a), we see that this simple analytic form is a reasonably good approximation.
Figure 9
Figure 9
The quantity of equation (67) shown as a function of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi325.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg by assuming An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi327.jpg Å−1 [value corresponding to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi20.jpg = 10° and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi6.jpg 1 Å in equation (2)] and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi330.jpg 200 Å (the value for the HslUV complex). (a) Exact (more ...)
Equation (74) is equation (11) of the main text. Equation (12) can be obtained from equations (68) and (70) by assuming that An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi46.jpg and therefore both An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi476.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi477.jpg are small quantities of the same order.
For the purpose of deriving equation (13), we evaluate the standard deviations of the quantities appearing in equations (6) and (7). After some calculations they are obtained as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd75.jpg
During the derivation of this equation we assumed that the values of the diffraction intensity density at the neighbouring Shannon pixels on a circle are not correlated. This means that the distance in the wavenumber space between the neighbouring Shannon pixels must be larger than the correlation length An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi144.jpg of equation (10). Because the correlation fades quickly beyond An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi144.jpg, we assume that equation (75) still holds for this distance. Therefore we have equation (15). The standard deviation of the correlation pattern is then given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd76.jpg
from which we have equation (13).
We now proceed to derive equation (22). We study how the value of the product An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi230.jpg is determined for common circles to be identified. At first we note that a sum of Poisson distributions is a Poisson distribution. Therefore, a sum of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi227.jpg two-dimensional patterns is equal to one sample of a two-dimensional pattern whose expected photon number is given by An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi482.jpg, which we shall write as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi483.jpg in equation (77) below. We shall also write the mean of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi483.jpg on a sphere An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi5.jpg as An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi486.jpg. Now we consider the distribution of values of the difference An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi487.jpg (1) on and (2) off the common circle. (1) On the common circle, An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi488.jpg is given by a difference of two integers, both given by a Poisson distribution. Its mean vanishes of course. The mean of its absolute value may be estimated approximately as
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd77.jpg
(2) Off the common circle, the values of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi190.jpg and An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi490.jpg are both distributed according to the two distributions, the Poisson distribution and the exponential distribution. The composite distribution is given by
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd78.jpg
For this distribution, the mean of the absolute value is estimated approximately as follows:
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd79.jpg
Let us assume that, for clear recognition of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi491.jpg out of An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi492.jpg, the following condition must be satisfied up to An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi151.jpg: An external file that holds a picture, illustration, etc.
Object name is a-68-00366-efi494.jpg, which means
A mathematical equation, expression, or formula.
 Object name is a-68-00366-efd80.jpg
Equation (22) follows directly from this equation.
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