- Averaging Matrix.

The averaging, or 'RT' operators are a rotation and translation which describe the mapping between one area of density and another. For DM they can be given as matrices in CCP4 or O format, or as Euler/Polar angles and translations. They are always given in orthogonal angstrom coordinates. - Averaging Mask.

This covers the region of the unit cell that obeys the non-crystallographic symmetry. Note that it does not cover a unit cell or an asymmetric unit, it does not obey the cell repeat, and it does not obey the crystallographic symmetry. It is fundamentally different from a solvent mask. If you do not understand this you had better go back to the textbooks, or hope that the 'automask' facility works right first time.

You must get the matrices the right way round. For example, if you calculate a matrix which maps an A chain onto a B chain, then the averaging mask must cover the volume occupied by the A chain.

Alternatively you can make your averaging matrices if you can build some corresponding fragments of structure in symmetry related molecules. Heavy atom coordinates can also be used in the determination of averaging matrices, especially if the rotation function has been solved. In this case you will need to run a program such as LSQKAB.

lsqkab REFRCD chmi.pdb \ WORKCD chmi.pdb \ LSQOP junk.pdb \ << 'END-lsqkab' OUTPUT XYZ FIT WRESIDU MAIN 2 TO 126 WCHAIN B MATCH RRESIDU 2 TO 126 RCHAIN A END 'END-lsqkab'The output of this run included the following information:

SUM DISPLACEMENTS**2 = 78.070 SQRT(SUM DISPLACEMENTS**2)= 0.395 AVERAGE DISPLACEMENT = 0.236 MAXIMUM DISPLACEMENT = 5.001 ROTATION MATRIX: -0.43671 0.05443 0.89796 -0.62780 0.69647 -0.34754 -0.64432 -0.71551 -0.26998 TRANSLATION VECTOR IN AS 43.63491 38.05914 62.72586 ......... CROWTHER ALPHA BETA GAMMA 158.84190-105.66331 132.00317 SPHERICAL POLARS OMEGA PHI CHI 113.28130 103.41944 120.33858 DIRECTION COSINES OF ROTATION AXIS -0.21318 0.89350 -0.39524The small average displacement is a good indication that a correct match has been found. The symmetry operators may then be input to DM, either as matrices using the ROTA MATRIX and TRANS cards, or as angles and translations. Alternatively, the matrix may be transposed and input in O/RAVE format.

Thus in this case, the 'AVER' card in the DM command file was as follows (note that a second run of LSQKAB was used to determine the third symmetry, between chains A and C):

AVER REF ROTA POLAR 0.0 0.0 0.0 TRANS 0.0 0.0 0.0 AVER REF ROTA MATRIX -0.43671 0.05443 0.89796 -0.62780 0.69647 -0.34754 - -0.64432 -0.71551 -0.26998 TRANS 43.635 38.059 62.726 AVER REF ROTA MATRIX -0.42948 -0.62559 -0.65130 0.06496 0.69793 -0.71322 - 0.90074 -0.34862 -0.25911 TRANS 82.989 15.401 -8.928or

AVER REF ROTA POLAR 0.0 0.0 0.0 TRANS 0.0 0.0 0.0 AVER REF ROTA EULER 158.84190 -105.66331 132.00317 TRANS 43.635 38.059 62.726 AVER REF ROTA EULER 47.59828 -105.01736 21.15850 TRANS 82.989 15.401 -8.928or

AVER REF ROTA POLAR 0.0 0.0 0.0 TRANS 0.0 0.0 0.0 AVER REF ROTA POLAR 113.28130 103.41944 120.33858 TRANS 43.635 38.059 62.726 AVER REF ROTA POLAR 66.58067 -76.78019 119.69176 TRANS 82.989 15.401 -8.928or

AVER REF OMAT 1.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 AVER REF OMAT -0.437 -0.628 -0.644 0.054 0.696 -0.716 0.898 -0.348 -0.270 43.635 38.059 62.726 AVER REF OMAT -0.429 0.065 0.901 -0.626 0.698 -0.349 -0.651 -0.713 -0.259 82.989 15.401 -8.928

You need to specify how many monomers will map onto matching density under the non-crystallographic symmetry transformations. In the simplest case - purely improper ncs - the answer is 1. In the case of purely proper ncs (e.g. 3-fold rotation, or 2-2-2 ncs), then it is the number of monomers in the crystallographic asymmetric unit (3 and 4 in these cases).

More complex cases occur when ncs operators are related to crystallographic operators, usually to build up larger multimers. For example, in Insulin there is a 2-fold ncs axis perpendicular to the crystallographic 3-fold and intersecting it, with the result that the entire hexamer obeys the noncrystallographic symmetry, so <nmer>=6. If fact this case is further complicated because neighbouring cells up and down the c-axis are related onto each other, so the CLIM card must also be used.

Given the complexity of the task, it is often better to make your own mask. The local-correlation map can also be generated by MAPROT and used in conjunction with graphics programs, MAPROT, and NCSMASK, to generate an averaging mask.

Bones are generally generated and then pruned in 'O'. The output bones file can be turned into a pseudo-pdb file using the utility BONDES2PDB.

In either case, the .pdb file can be converted directly into an averaging mask using the NCSMASK program.