ALLSPACE
By Vinzenz Unger and Anchi Cheng
For a given structure factor list of an untilted crystal, this program is capable of determining its 2-sided plane group from the 17 groups that are possible for chiral molecules. The phases of the symmetry-related reflections have defined relationships. For example, the Friedel mates (h, k, 0) and (-h,-k, 0) should have identical phases (in fact, 0 or 180 °) if there is a 2-fold axis parallel to the z-axis and passing the phase origin of the untilted image. The accumulated phase residual derived from this symmetry is therefore the sum of the absolute phase difference for each pair of (h, k, 0) and (-h, -k, 0). In some cases, the phase relationship predicted by the symmetry is not identity. For instance, for a 2- fold screw axis along the y-axis, the phases of the reflections (h, 2n+1,0) and (-h,2n+1,0) are 180˚ apart. The phase residual for such symmetry will have to be modified accordingly so that the residual corresponds to the phase difference ± 180°. A complete list of the phase relationships considered for the individual space group can be found in the program header. The averaged internal phase residual for data assuming a particular space group is therefore the sum of all symmetry related phase comparisons divided by the total number of comparison made in the data set. To determine which space group best describes the symmetry that is present in the data set, the averaged phase residual for each space group is compared with a target residual based on the histogram of the signal to noise ratio of the reflections involved in calculating the residual. With proper use of this program, the 2-sided plane group with the highest symmetry that has a phase residual as good as or better than its target phase residual can be safely called the 2-sided plane group of the 2-D crystal. However, we have a few caution notes that will be discussed later.
For pairing reflections and the calculation of the phase residual, ALLSPACE operates on the assumptions that: (1) z*=l=0 for all reflections, i.e., the specimen is untilted and viewed in the direction of the z axis; (2) the phase origin is chosen at the conventional crystallographic origin for the particular space group; (3) the symmetry-related phases are not distorted by effects such as strong astigmatism and, at high resolution, beam tilt. If any of these assumptions is not true in the data, the phase residual can be significantly higher than the target or the predicted 2-sided plane group does not correspond to the real symmetry of the crystal. Hence only data from images of truly untilted crystals should be used to determine the symmetry of the crystal using this program, especially if the specimen is thick (see main text for examples). Strong astigmatism, if exist in the data, should be corrected.
Usually, the input data will not be centered at the proper origin for a particular 2-sided plane group. Therefore, the program also includes options for searching (SEARCH = T) and refining (REFINE = T) the proper phase origin, and for beam tilt refinement (TILT = T). The search and refinement are based on the assumption that the calculated phase residual will be the minimal when the parameters are optimal.
A few caution notes: (1) While the distribution of the quality of the input reflection is taken into account for the calculation of target residual, the actual phase residual calculation is not weighted by the quality of the individual data points. Consequently, to obtain reliable results only the stronger data should be included. This can be achieved by properly choosing IQMAX. (3) A slightly tilted image may suggest a lower symmetry than the true symmetry of the specimen. Possibility of higher symmetry should be examined carefully at a later stage of 3-D data. (4) For a high symmetry two-sided plane group to be choosen as the symmetry of the crystal, its sub-symmetry should also have acceptable phase residual. For example, p4 contains p2 symmetry. Therefore, both should have high rank in the choice of symmetry if p4 is to be the correct symmetry. If not, it may indicate that the high symmetry is an artifact, which may associate with insufficient number of data in the calculation. (5) ALLSPACE does not compare amplitudes of the reflection, nor does it check for systematic absences. Users should check for the consistency in the symmetry suggested by the phases and amplitude to avoid error. (6) Strictly speaking, ALLSPACE determines the projection symmetry rather than the 2-sided plane group symmetry. In the unlikely but possible case, one may find that a lower symmetry better describes the true 2-sided plane group symmetry when the data is expanded to 3-D (Amos et al., 1982). It is always a good idea at a later stage of 3-D data merging to check carefully if the symmetry constraints to phases as well as to amplitudes are met at all z*. (7) Although ALLSPACE usually predicts the correct 2-sided plane group for cryo images, the 2-sided plane group so determined for a negatively stained specimen classifies the stain exclusion pattern, which may not correspond to the 2-sided plane group for the real specimen.
Other input parameters that were not yet mentioned:
- SYMM: This parameter controls the space groups that are to be considered. The most useful options are:
- - ALL - all space groups
- - HEXA - all hexagonal space groups plus p2 and c2
- - SQUA - all non-hexagonal space groups
- - RECT - all rectangular, but not square nor hexagonal space groups
- - OBLI - p2
- STEP, ISIZE: These two parameters [˚] control the phase origin search which is performed in ISIZE steps using an increment for the individual phase shift specified by STEP.
- A, B, GAMMA, RIN, ROUT, CS, KV: These are the cell dimensions and the resolution range in Å, the spherical aberration in mm, and the microscope operating voltage in kV.
- ROT180=1 enables an effective 180˚ turn of the unit cell along z-axis to match index convention used by other programs for p3. More details can be found in comments on ORIGTILTD.