Full‐matrix least squares is taken as the basis for an examination of protein structure precision. A two‐atom protein model is used to compare the precisions of unre­strained and restrained refinements. In this model, restrained refinement determines a bond length which is the weighted mean of the unrestrained diffraction‐only length and the geometric dictionary length. Data of 0.94 Å resolution for the 237‐residue protein concanavalin A are used in unrestrained and restrained full‐matrix inversions to provide standard uncertainties σ(r) for positions and σ(l) for bond lengths. σ(r) is as small as 0.01 Å for atoms with low Debye B values but increases strongly with B. The results emphasize the distinction between unrestrained and restrained refinements and between σ(r) and σ(l). Other full‐matrix inversions are reported. Such inversions require massive calculations. Several approximate methods are examined and compared critically. These include a Fourier map formula [Cruickshank (1949). Acta Cryst.2, 65–82], Luzzati plots [Luzzati (1952). Acta Cryst.5, 802–810] and a new diffraction‐component precision index (DPI). The DPI estimate of σ(r, Bavg) is given by a simple formula. It uses R or Rfree and is based on a very rough approximation to the least‐squares method. Many examples show its usefulness as a precision comparator for high‐ and low‐resolution structures. The effect of restraints as resolution varies is examined. More regular use of full‐matrix inversion is urged to establish positional precision and hence the precision of non‐dictionary distances in both high‐ and low‐resolution structures. Failing this, parameter blocks for representative residues and their neighbours should be inverted to gain a general idea of σ(r) as a function of B. The whole discussion is subject to some caveats about the effects of disordered regions in the crystal.

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In the XO, . -tetrahedral ions (X = Si, P, S, or C1) two strong x-bonding molecular orbitals are formed with the 3 d z ~-y ~ and 3 d , ~ orbitals of X and the appropriate 2fix and 2pd-orbitals of oxygen. This explains the contraction of the S-0 bond, say, from the presumed 1-69 A for a single bond to 1.49 A in SO, , -.In ethyl sulphate the ester-oxygen still has a share in one x-orbital, which explains its intermediate S-O value of 1.60 A. The ability of doubly linked oxygen and triply linked nitrogen to share in the x-bond systems generated by tetrahedrally co-ordinated X is widespread. An interpretation is given of the structures of [S20,]2--, [NH(S0,),]2-, [CH(S0,),]2-, SO,NH,, [SO,NH,]-, SO,(NH,),, and (SO,), in terms of the n-bond systems. Structures with tetrahedral phosphorus and silicon are similarly discussed, with special attention to those with XOX angles of 140-180". In ions containing hydroxyl groups, the bonds are intermediate in length between those for the ion without the proton and for the ion with R in place of H, apparently owing to hydrogen-bond formation. The use of dzr-yS and d,r orbitals for x-bonding is of potential importance whenever X is tetrahedrally co-ordinated and must be considered in, say, RSO,-, R,S02, and RS02-. X-F bonds and the x-bond systems in [PNF,], and [PN(CH,),],, etc., are also discussed.THE single system of x-orbitals in planar aromatic molecules is well known. This paper presents evidence for a double system of x-orbitals in sulphates, phosphates, silicates, etc., the orbitals extending throughout the molecules. The discussion is restricted to molecules in which silicon , phosphorus, sulphur, or chlorine atoms are tetrahedrally co-ordinated. The bonding in the X04nions is considered first, and the description is extended successively to molecules in which the oxygens are linked also to other atoms or to other tetrahedra, to molecules in which nitrogen and other atoms are ligands of X, and to molecules in which lone pairs of X occupy some of the tetrahedral positions. Thus, ions such as [C,H,-SO,]-, P20,6-, SO*NH,+, and C10,-are discussed, but not SO, or SO,.The theme of the paper is thus summarised in its title. The x-orbital system is a double one in the sense that two d-orbitals of X, dza-ya and dza, are used simultaneously when X is tetrahedrally co-ordinated. The experimental evidences appealed to in this paper are the observed bond lengths in a large number of molecules, many of which have been recently determined or refined in this Department by X-ray crystallographic methods. The main features of the paper were reported briefly at the Congress of the International Union of Crystallography at Cambridge in August, 1960.Estimated standard deviations (e.s.d.'s) for the observed bond lengths quoted in this paper are shown in the style '' &0.01 k" Some of these e.s.d.'s are larger than those quoted by the original authors. Such increases have been made to cover more recently discovered types of systematic error, etc. The rotational-oxillation effect,, in particular, ...

Transmission Laue diffraction photographs can be recorded with short exposure times from stationary macromolecular and small-molecule crystals. With the use of a broad wavelength band a very large number of reflections is stimulated in a single 'snapshot' of large regions of reciprocal space. Processing software has been developed which allows quantitation of the Laue data without resort to monochromatic data. The procedures have been developed and the software strategies optimized by using test data recorded on the SRS wiggler from a protein, pea lectin, and small-molecule crystals. These latter include an organic molecule, trimethyl-lH-2,1,3-benzophosphadiazine-4(3H)-thione 2,2-disulfide, referred to as BPD, and a rhodium complex, [Rh6(CO)~4(dppm)], where dppm is Ph2PCH2PPh2, referred to as RHCOP. Monochromatic data were available for comparison.

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Angular oscillations of molecules cause displacements of the electron-density peaks. A formula is given relating the displacements to the amplitudes of small angular oscillations.The preceding paper (Cruickshank, 1956a) has described how the thermal anisotropic vibrations of atoms in molecular crystals may be interpreted in terms of the anisotropic translational and rotational oscillations of the molecules. In the course of detailed analyses of several crystals, particularly of benzene (Cox, Cruickshank & Smith, 1956), angular oscillations with r.m.s, amplitudes as large as 8 ° have been found. An immediate consequence of these angular oscillations is that the maxima of the atomic peaks in the electron density are closer to the centre of rotation than they would be otherwise. The purpose of the present note is to give a simple formula relating the errors in position to the amplitudes of angular oscillation.In Fig the maximum of the electron-density map will remain at P (supposing that finite-series and peak-overlapping effects are allowed for as usual). If the molecule makes rotational oscillations with 0 as centre, the atom will vibrate over the surface of the sphere through P whose centre is O. This causes a positional error, whose order of magnitude can easily be obtained from a highly simplified theory. Suppose that the motion of the atom is confined to the plane of the figure and that it vibrates along the arc of the circle, spending half its time at Q and half at Q'. Then the time average electron density will have its maximum at B, and the atom will appear too close to 0 byFor 0 = 8 ° and r = 1.39 A, the error is 0.014/~. Such an error is important in accurate structure determinations. A more complete calculation for small oscillations will now be given. We may suppose that the shape of the atomic peak, including finite-series effects and the spread due to translational but not rotational oscillations of the molecule, is represented by a Gaussian functionwhere x is the distance from the centre of the atom and q9 is a breadth parameter for the peak. The final formula for the error will show that this assumption of a spherical peak is a sufficiently good approximation. The use of a Gaussian function is certainly satisfactory, as Fourier map peaks have often been shown to be approximately Gaussian for at least ½ /~ from their centres.If the molecule makes harmonic angular oscillations the atom will move on the surface of the sphere with P as its middle position and with the maximum of its time average density lying along OP. The problem is thus to calculate the variation of the time average density along OP and to find its maximum.If the atom is stationary at Q the density at a point A along OP will be oc exp -(AQ~/2q~}.(2)We need be concerned only with the oscillations of the molecule about axes in the plane through O perpendicular to OP, since angular oscillations about OP do not move an atom at P. Let q and yJ be the angles of rotation about the principal axes in this plane which move the atom from P to ...

The protonation states of aspartic acids and glutamic acids as well as histidine are investigated in four X-ray cases: Ni,Ca concanavalin A at 0.94 A, a thrombin-hirugen binary complex at 1.26 A resolution and two thrombin-hirugen-inhibitor ternary complexes at 1.32 and 1.39 A resolution. The truncation of the Ni,Ca concanavalin A data at various test resolutions between 0.94 and 1.50 A provided a test comparator for the ;unknown' thrombin-hirugen carboxylate bond lengths. The protonation states of aspartic acids and glutamic acids can be determined (on the basis of convincing evidence) even to the modest resolution of 1.20 A as exemplified by our X-ray crystal structure refinements of Ni and Mn concanavalin A and also as indicated in the 1.26 A structure of thrombin, both of which are reported here. The protonation-state indication of an Asp or a Glu is valid provided that the following criteria are met (in order of importance). (i) The acidic residue must have a single occupancy. (ii) Anisotropic refinement at a minimum diffraction resolution of 1.20 A (X-ray data-to-parameter ratio of approximately 3.5:1) is required. (iii) Both of the bond lengths must agree with the expectation (i.e. dictionary values), thus allowing some relaxation of the bond-distance standard uncertainties required to approximately 0.025 A for a '3sigma' determination or approximately 0.04 A for a '2sigma' determination, although some variation of the expected bond-distance values must be allowed according to the microenvironment of the hydrogen of interest. (iv) Although the F(o) - F(c) map peaks are most likely to be unreliable at the resolution range around 1.20 A, if admitted as evidence the peak at the hydrogen position must be greater than or equal to 2.5 sigma and in the correct geometry. (v) The atomic B factors need to be less than 10 A(2) for bond-length differentiation; furthermore, the C=O bond can also be expected to be observed with continuous 2F(o) - F(c) electron density and the C-OH bond with discontinuous electron density provided that the atomic B factors are less than approximately 20 A(2) and the contour level is increased. The final decisive option is to carry out more than one experiment, e.g. multiple X-ray crystallography experiments and ideally neutron crystallography. The complementary technique of neutron protein crystallography has provided evidence of the protonation states of histidine and acidic residues in concanavalin A and also the correct orientations of asparagine and glutamine side chains. Again, the truncation of the neutron data at various test resolutions between 2.5 and 3.0 A, even 3.25 and 3.75 A resolution, examines the limits of the neutron probe. These various studies indicate a widening of the scope of both X-ray and neutron probes in certain circumstances to elucidate the protonation states in proteins.

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