7.8. Electron density averaging Electron density averaging requires clear boundaries between protein and solvent volumes, it should be almost inevitably preceded by a solvent flattening in the case of multiple isomorphous replacement phase evaluation. There are two types of non-crystallographic or local symmetry: proper (also called spherical) and improper symmetry. Molecules of an asymetric unit are related by proper symmetry when they can be superimposed upon each other by a single rotation about a local symmetry axis, while in the case of improper local symmetry, superposition of the molecules require other operations (rotation usually combined with translation). Therefore, procedures for proper and improper symmetry averaging differ. For proper symmetry averaging, equivalent areas do not have to be separated, while for improper averaging it is necessary to distinguish between them. Besides improving the phases (and the electron density maps) of the starting resolution range, it is possible to evaluate phases of higher resolution reflections by gradually increasing the resolution range. The procedure is called phase extension. The larger the number of molecules in an asymmetric unit, the better the results which can be obtained with phase extension, with however the condition that the initial phases are sufficiently correct for the procedure to converge properly ( Bricogne, 1974, Podjarny, 1990). To perform real space electron density averaging some initial set of phases, equivalent areas and geometric transformations (rotation and translation parameters) are required. In a molecular replacement procedure, the equivalent areas are defined from the initial model placement. Transformations between them can be easily constructed by superimposing the molecular models (as demonstrated in the cases of cathepsin B and riboflavin synthase). In the case of a single or multiple isomorphous replacement procedure, it is possible to construct the transformation parameters from the heavy atom positions when they fulfill the local symmetry conditions. When they do not, it is necessary to construct the rotational parameters by an autocorrelation of a Patterson map and then to find the center of rotation and corresponding translational components by autocorrelating electron density (as demonstrated in the case of carbamoylsarcosine hydrolase). To recognize the boundaries of equivalent areas, it is recommended first to average the initial electron density map, whereby the uncorrelated areas are supposed to smear out, and in this averaged density to then find the borders (envelope) of the equivalent areas interactively at a graphical display by going through all the layers of an asymmetric unit (2-dimensional construction using a program such as X-CONTOUR (Buchberger, 1990)) or by preparing a 3-dimensional map representation of a whole asymmetric unit and determining the equivalent parts from its 3-dimensional skeletonized image (this can be done with MAIN). The theoretical basis for electron density averaging or real space averaging was established in late 1960s and early 1970s (Rossman, 1972). Bricogne has written a review (Bricogne, 1974) of the method, its application and limitations. There it was shown that the averaging in direct space is equivalent to the procedure in reciprocal space and that averaging in direct space has advantages over reciprocal space procedures. The reason seems to lie in the greater computational inaccuracies in reciprocal space calculations. In 1976 he published a description of his program (Bricogne, 1976). Besides Bricogne's program, other attempts have also been made, but none of them (Johnson in Rayment et al., 1978; Nordman, 1980) is used so widely and in so many variations as Bricogne's. Recently, Lawrence (Lawrence, 1991) reviewed the method and its applications to {\it de novo} determined structures. 7.8.1. Averaging of electron density: 2 molecules See "MAIN_DOC:nmol/nmol.txt". If you are intersted in the syntax run the "menu_dens_mod.sh" and corresponding "create_...pl" scripts and inspect the outcoming macros. 7.8.2. Averaging density between 2 different crystal forms The configuration scripts are yet not capable to create macros for averaging of electron density between several crystal forms. A runable case for averaging and refinement of a molecule in two crystal forms can be found in "MAIN:cases/2cryst/". The files therein should be edited. This procedure differs from the above described single crystal form averaging in the following: Since MAIN is not capable to deal with several refelction sets and symmetry groups simoultaneously these have to be read each time when they are referenced. To make file manipulation easier (and error free) "MAIN:cases/2cryst/get_data_hex.com" and "MAIN:cases/2cryst/get_data_ortho.com" files were created. They are only seting values of string variable to apprpriate file names. A standard "MAIN:cases/2cryst/rms_fit.cmds" calculates the rotation and translational parameters of the procedure for all equivalent molecules. The standard "MAIN:cases/2cryst/make_masks.cmds" is a bit modified, because there is no overlap between molecules from different crystal forms. Of course there is also the "MAIN:cases/2cryst/read.com" file. The "MAIN:cases/2cryst/dm_prep.cmds" prepares averaging and "MAIN:cases/2cryst/dm_prep.cmds" continues and calls "MAIN:cases/2cryst/aver_loop.com", which finaly does it including a map calculation for each map separately. The new unit cell generation and a new 2Fobs-Fcalc map calculation for each crystal form independently: > read file FILE_CELL cell > read file FILE_SYMM symm > read file FILE_REFL reflect init resol RESOL_HEX 100. limit 0 0 0 friedel > set vari MAP_CELL = 1 > make map MAP_CELL zero > make map MAP_CELL set -1 1 -9999. > make map MAP_CELL from 3 cell > read file FILE_CELL cell > read file FILE_SYMM symm > read file FILE_REFL reflect init resol RESOL_ORT 100. limit 0 0 0 friedel > set vari MAP_CELL = 2 > make map MAP_CELL zero > make map MAP_CELL set -1 1 -9999. > make map MAP_CELL from 4 cell > rewind file 7.8.1. Averaging: yeast proteasome schell script Lars Ditzel gaved us his script "yeast_prot/aver-ypr.com" for the yeast proteasome averaging. It runs in background with no graphics involved.