Tables for
Volume F
Crystallography of biological macromolecules
Edited by E. Arnold, D. M. Himmel and M. G. Rossmann

International Tables for Crystallography (2012). Vol. F, ch. 13.4, pp. 357-358   | 1 | 2 |

Section 13.4.7. Finding the averaged density

M. G. Rossmanna* and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907–1392, USA, and  bBiomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854–5638, USA
Correspondence e-mail:

13.4.7. Finding the averaged density

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Electron density can be averaged (1) among the N NCS-related molecules in the p-cell (the real crystal unit cell), thus creating a new and improved map of the p-cell; (2) among the N NCS-related molecules in the p-cell and placing the results into a standard orientation in the h-cell; or (3) among the N NCS-related molecules in different unit cells and placing the results back into the original different unit cells or into a standard h-cell. Before averaging commences, the [M \times N] matrices [[\hbox{E}_{m,\, n}]] and translation vectors [{\bf e}_{m,\, n}] must be evaluated [see ([link]) and ([link])]. Here, N is the noncrystallographic redundancy and M is the number of molecules that impinge on the crystallographic asymmetric unit of the p-cell. Associated with each grid point in the p-cell asymmetric unit will be (1) the value of m designating which molecular centre is to be associated with that grid point (a special value of m is for solvent) and (2) the p-cell electron density at that point.

The grid points within the asymmetric unit are then examined one at a time. If the grid point is within the mask, it is averaged among the N noncrystallographically related equivalent positions belonging to molecule m. If the grid point is solvent, the density can be set to the average solvent density.

The N noncrystallographically equivalent non-integral grid points can be computed from ([link]). Some of these will lie outside the crystallographic asymmetric unit. These will, therefore, have to be operated on by unit-cell translations and crystallographic symmetry operations to bring them back into the asymmetric unit before the corresponding interpolated density can be calculated.

Averaging into the h-cell can be done by a procedure similar to averaging in the p-cell, except that the rotation and translation matrices are given by ([link]). Furthermore, no mask is required as all the averaging into the h-cell (from p-cell electron density) can be done with respect to the reference molecule centred at [{\bf s}_{p,\, m = 1}] in the p-cell. Each grid point is taken in turn in the h-cell. The electron density at any grid point that is further away from [{\bf s}_{\rm h}] than from [R_{\rm out}] is set to zero. Other grid-point positions are expanded into the N equivalent positions in the p-cell surrounding [{\bf s}_{p,\, m = 1}]. The interpolated density is then found, averaged over the N equivalent positions, and stored at the original h-cell grid point in successive sections, in the same way as in the p-cell averaging. As in averaging within the p-cell, a record is kept of [\langle\sigma(\rho)\rangle] as a function of [\langle\rho({\bf x})\rangle] (Table[link]). In general, the local NCS is valid only within the molecule. Hence, the h-cell density will show the molecular envelope and can be used to recompute an improved p-cell density mask. The rate of build up of signal within the molecule should be roughly proportional to N, while the rate outside the molecule should be proportional to about [N^{1/2}].

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Mean root-mean-square scatter between noncrystallographically related points

Example taken from ϕX174 structure determination. [\langle\rho_{8}\rangle] is proportional to the mean density (e Å−3) based on eight-point interpolation; n is number of grid points with [\langle\rho_{8}\rangle] in a given range; [\langle\sigma (\rho_{8})\rangle] is the root-mean-square deviation from [\rho_{8}] among noncrystallographic asymmetric points averaged over all points in the mask.

[\langle\rho_{8}\rangle]Density derived from an electron microscopy image at 25 Å resolutionDensity derived from a 3.3 Å crystal structure
n[\langle\sigma (\rho_{8})\rangle]n[\langle\sigma (\rho_{8})\rangle]
−375 to −325 1 44.7 0 0.0
−325 to −275 16 44.4 0 0.0
−275 to −225 22 39.5 41 31.4
−225 to −175 81 34.9 3493 25.5
−175 to −125 299 34.7 65049 20.5
−125 to −75 1119 33.1 290025 17.7
−75 to −25 16617 34.7 661386 15.0
−25 to 25 33818 46.9 1,016274 12.8
25 to 75 6008 31.9 344620 16.3
75 to 125 4512 32.0 215036 18.9
125 to 175 3050 32.1 146690 22.1
175 to 225 1562 32.6 58155 26.3
225 to 275 542 33.4 6032 32.2
275 to 325 213 35.6 227 40.6
325 to 375 33 34.7 9 46.8

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