Using Harker's [Harker (1953). Acta Cryst. 6, 731±736] idea of spherically averaged polyatomic groups or`globs' as the units of structure suitable for analyzing low-resolution diffraction data from protein crystals,`globbic' scattering factors have been calculated for main-chain peptide units and amino-acid side-chain groups to 3 A Ê resolution via Debye's [Debye (1915). Ann. Phys. (Leipzig), 46, 809±823] scattering formula. It is shown that the scattering factors are insensitive to intra-globbic conformational variation and can be approximated fairly well by a single-Gaussian formula, i.e. f g (s) = Z g exp(À1.7Z g s 2 ), where s = (sin)/! and Z g is the total electron count for the atoms of the glob. Phase errors due to the globbic approximation and their effect on electrondensity maps at 3.5 A Ê resolution have been assessed via calculations for the crambin structure; this analysis indicates that the globbic scattering factors will be useful in efforts to develop procedures for direct-methods phasing of diffraction data to $3.5 A Ê resolution from protein crystals.

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At 3 to 4 ,~, resolution, the electron density of a protein may be modeled by a continuous chain of 'globs' representing the amide region of the peptide backbone and the side-chain residues. Group scattering factors are derived from a trans planar CotC:ONCot backbone segment and most favored sidechain conformer for 18 different amino acids. Trial calculations indicate that the phase error and crystallographic residual comparing the atomic and 'globic' models rapidly decrease from high to low resolution. At 3 A resolution, the phase error is approximately 80 ° . These results indicate that the electron density of a protein composed of N amino acid residues may be adequately modeled by 2N globs at low resolution. IntroductionDavid Harker pointed out in 1953 that 'globs', i.e. the total electron density of clusters of atoms within a local region of the molecular envelope, may provide a more accurate representation of the average intensity for a protein at low resolution than the sum of the square of its atomic structure-factor magnitudes, and suggested using this concept in data reduction to obtain a better scale and temperature factor (Harker, 1953). This analysis is only appropriate for macromolecules that are well separated by solvent boundaries in large unit cells. Similar efforts to obtain better scaling of the normalized E values for the MULTAN program make use of much smaller chemically rigid molecular fragments of unknown position and orientation (Main, 1976); the contribution of these fragments to the average intensity utilizes an expression derived by Debye (1915). Podjarny and co-workers used three group scatterers (phosphate, ribose, nucleic acid bases) to assign peaks in lowresolution MIR maps of tRNA Met prior to using other methods to extend phases to data at higher resolution and fill in loworder terms that were not reliably determined by the MIR process (Podjarny & Yonath, 1977; Podjarny, Schevitz & Sigler, 1981;Podjarny & Faerman, 1982).The concept of atomicity for small structures is useful since, at atomic resolution, the atoms of the structure are recognizable in an electron-density map. For proteins, however, a more useful concept is 'globicity', which is based on the fact that 'globs', consisting of groups of atoms in the unit cell, are the only recognizable features in low-resolution electron-density maps. In the situation that one cannot confidently fit the known protein sequence to a low-resolution density map, globs may At low resolution, each glob may be treated as a spherically averaged cluster as its shape will be insufficiently resolved to determine the orientation of the underlying chemical group. The group scattering factors of these spherically averaged clusters may be analytically defined as a nine-coefficient exponential expression in sin 0/2 as shown previously (Cromer & Waber, 1965). Globs chosen from such a tabulation would logically correspond to the trans planar peptide segments in the backbone of polypeptide chain and the most favored conformations of the 20 amino ...

Probabilistic direct-methods phasing theory, originally based on a uniform atomic distribution hypothesis, is shown to be adaptable to a non-uniform bulk-solvent-compensated globbic approximation for protein crystals at low resolution. The effective number n g of non-H protein atoms per polyatomic glob increases with decreasing resolution; low-resolution phases depend on the positions of only N g = N a /n g globs rather than N a atoms. Test calculations were performed with measured structure-factor data and the re®ned structural parameters from a protein crystal with $10 000 non-H protein atoms per molecule and $60% solvent volume. Lowresolution data sets with d min ranging from 15 to 5 A Ê gave n g = ad min + b, with a = 1.0 A Ê À1 and b = À1.9 for the test case. Results of tangent-formula phase-estimation trials emphasize that completeness of the low-resolution data is critically important for probabilistic phasing.

It is shown that for crystals of large proteins at low diffraction resolution, with N approximately 10 000 independent non-H protein atoms and d(min) approximately 8 A, a simple bulk-solvent correction yields the Sayre equation in its classical form, F(h) = q summation operator(k)F(k)F(h - k). In the low-resolution protein case, the proportionality factor becomes q = 1/[( - rho(S))V], where V is the unit-cell volume, rho(S) is the assumed constant electron density in the solvent regions of the crystal and is the average electron density in the protein regions. The classical form of the tangent formula follows from the bulk-solvent-corrected Sayre equation and its validity at low resolution is verified in empirical calculations.