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.
of two-dimensional patterns to be measured is given by the product
of the number of classification groups
and the average number of two-dimensional patterns
to be averaged in each group for noise reduction. The number
of pairs of two-dimensional patterns to be analysed is given by
, 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,
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
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
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
and becomes weaker at higher angles. The intensity reduces faster for larger values of
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
of the distribution around its vanishing mean becomes globally, but not monotonically, larger as
becomes larger. When the standard deviation
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
becomes smaller. The latter can be determined from the former.
When the value of
is smaller than a certain value, say,
(therefore, when the recognizable length of the corresponding high correlation line becomes longer than a certain value, say,
), we classify the pair into the same group. To attain the best resolution, we should employ the largest possible value for
. Operationally we determine the value of
, the limiting k value
for correlation recognition, as the lower bound of the noise-dominant
region in two-dimensional correlation patterns. Such a value
can be characterized theoretically as the value of
at which the standard deviation
of the background distribution is
. 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.
, and an expected photon count
by a pixel at the position of the wavenumber
. The quantity
plays a central role in the method of analysis developed in this paper. It is shown that the structural resolution
attainable by this method is given by
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
where the value of the standard deviation
. From the analytic expression for
, we can derive the limiting photon count
, an expected photon count at a limiting pixel
, approximately as a function of the normalized resolution
(Fig. 5). 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
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
and the radial diffraction intensity density function
are known, the above relation between the normalized resolution
and the limiting photon count
can be transformed to a relation giving the intensity
of the incident beam to be used to attain a resolution
to define a range of classification groups is given by an inverse of the normalized limiting wavenumber,
. As a result, the number of classification groups
is given by
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
is searched, another common circle is searched at the same time between two-dimensional patterns
is a two-dimensional pattern on an Ewald sphere conjugate to the one on which
exists. The average number of two-dimensional patterns
to be averaged in each group for identification of common circles has been shown to be given in terms of the limiting photon count
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. 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 and Table 1 are in the range of around 1021
, which is far larger than the peak value of 1.6 × 1016
reported in the recent experiment (Seibert et al.
) 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.
) 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 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.