International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 91100

Fourier transform (FT) calculations play an indispensable role in crystallography, because the Fourier transformation is inherent in the diffraction phenomenon itself.
Besides this obligatory use, the FT has numerous other applications, motivated more often by its mathematical properties than by direct physical reasoning (although the latter can be supplied after the fact). Typically, many crystallographic computations turn out to be convolutions in disguise, which can be speeded up by orders of magnitude through a judicious use of the FT. Several recent advances in crystallographic computation have been based on this kind of observation.
Bragg (1929) was the first to use this type of calculation to assist structure determination. Progress in computing techniques since that time was reviewed in Section 1.3.4.3.1.
The usefulness of the maps thus obtained can be adversely affected by three main factors:
Limited resolution causes `seriestermination errors' first investigated by Bragg & West (1930), who used an optical analogy with the numerical aperture of a microscope. James (1948b) gave a quantitative description of this phenomenon as a convolution with the `spherical Dirichlet kernel' (Section 1.3.4.2.1.3), which reflects the truncation of the Fourier spectrum by multiplication with the indicator function of the limiting resolution sphere. Bragg & West (1930) suggested that the resulting ripples might be diminished by applying an artificial temperature factor to the data, which performs a further convolution with a Gaussian pointspread function. When the electrondensity map is to be used for model refinement, van Reijen (1942) suggested using Fourier coefficients calculated from the model when no observation is available, as a means of combating seriestermination effects.
Errors in the data introduce errors in the electrondensity maps, with the same meansquare value by virtue of Parseval's theorem. Special positions accrue larger errors (Cruickshank & Rollett, 1953; Cruickshank, 1965a). To minimize the meansquare electrondensity error due to large phase uncertainties, Blow & Crick (1959) introduced the `best Fourier' which uses centroid Fourier coefficients; the associated error level in the electrondensity map was evaluated by Blow & Crick (1959) and Dickerson et al. (1961a,b).
Computational errors used to be a serious concern when Beevers–Lipson strips were used, and Cochran (1948a) carried out a critical evaluation of the accuracy limitations imposed by strip methods. Nowadays, the FFT algorithm implemented on digital computers with a word size of at least 32 bits gives results accurate to six decimal places or better in most applications (see Gentleman & Sande, 1966).
Various approaches to the phase problem are based on certain modifications of the electrondensity map, followed by Fourier analysis of the modified map and extraction of phase information from the resulting Fourier coefficients.
Sayre (1952a) derived his `squaring method equation' for structures consisting of equal, resolved and spherically symmetric atoms by observing that squaring such an electron density is equivalent merely to sharpening each atom into its square. Thuswhere is the ratio between the form factor common to all the atoms and the form factor for the squared version of that atom.
Most of the central results of direct methods, such as the tangent formula, are an immediate consequence of Sayre's equation. Phase refinement for a macromolecule by enforcement of the squaring method equation was demonstrated by Sayre (1972, 1974).
A category of phase improvement procedures known as `density modification' is based on the pointwise application of various quadratic or cubic `filters' to electrondensity maps after removal of negative regions (Hoppe & Gassmann, 1968; Hoppe et al., 1970; Barrett & Zwick, 1971; Gassmann & Zechmeister, 1972; Collins, 1975; Collins et al., 1976; Gassmann, 1976). These operations are claimed to be equivalent to reciprocalspace phaserefinement techniques such as those based on the tangent formula. Indeed the replacement ofby , where P is a polynomialyieldsand hence gives rise to the convolutionlike families of terms encountered in direct methods. This equivalence, however, has been shown to be rather superficial (Bricogne, 1982) because the `uncertainty principle' embodied in Heisenberg's inequality (Section 1.3.2.4.4.3) imposes severe limitations on the effectiveness of any procedure which operates pointwise in both real and reciprocal space.
In applying such methods, sampling considerations must be given close attention. If the spectrum of extends to resolution Δ and if the pointwise nonlinear filter involves a polynomial P of degree n, then P() should be sampled at intervals of at most to accommodate the full bandwidth of its spectrum.
Crystals of proteins and nucleic acids contain large amounts of mother liquor, often in excess of 50% of the unitcell volume, occupying connected channels. The well ordered electron density corresponding to the macromolecule thus occupies only a periodic subregion of the crystal. Thusimplying the convolution identity between structure factors (Main & Woolfson, 1963):which is a form of the Shannon interpolation formula (Sections 1.3.2.7.1, 1.3.4.2.1.7; Bricogne, 1974; Colman, 1974).
It is often possible to obtain an approximate `molecular envelope' from a poor electrondensity map , either interactively by computer graphics (Bricogne, 1976) or automatically by calculating a moving average of the electron density within a small sphere S. The latter procedure can be implemented in real space (Wang, 1985). However, as it is a convolution of with , it can be speeded up considerably (Leslie, 1987) by computing the moving average as
This remark is identical in substance to Booth's method of computation of `bounded projections' (Booth, 1945a) described in Section 1.3.4.2.1.8, except that the summation is kept threedimensional.
The iterative use of the estimated envelope for the purpose of phase improvement (Wang, 1985) is a submethod of the previously developed method of molecular averaging, which is described below. Sampling rules for the Fourier analysis of envelopetruncated maps will be given there.
Macromolecules and macromolecular assemblies frequently crystallize with several identical subunits in the asymmetric metric unit, or in several crystal forms containing the same molecule in different arrangements. Rossmann & Blow (1963) recognized that intensity data collected from such structures are redundant (Sayre, 1952b) and that their redundancy could be a source of phase information.
The phase constraints implied by the consistency of geometrically redundant intensities were first derived by Rossmann & Blow (1963), and were generalized by Main & Rossmann (1966). Crowther (1967, 1969) reformulated them as linear eigenvalue equations between structure factors, for which he proposed an iterative matrix solution method. Although useful in practice (Jack, 1973), this reciprocalspace approach required computations of size for N reflections, so that N could not exceed a few thousands.
The theory was then reformulated in real space (Bricogne, 1974), showing that the most costly step in Crowther's procedure could be carried out much more economically by averaging the electron densities of all crystallographically independent subunits, then rebuilding the crystal(s) from this averaged subunit, flattening the density in the solvent region(s) by resetting it to its average value. This operation is a projection [by virtue of Section 1.3.4.2.2.2(d)]. The overall complexity was thus reduced from to N log N. The design and implementation of a generalpurpose program package for averaging, reconstructing and solventflattening electrondensity maps (Bricogne, 1976) led rapidly to the first highresolution determinations of virus structures (Bloomer et al., 1978; Harrison et al., 1978), with .
The considerable gain in speed is a consequence of the fact that the masking operations used to retrieve the various copies of the common subunit are carried out by simple pointwise multiplication by an indicator function in real space, whereas they involve a convolution with in reciprocal space.
The averaging by noncrystallographic symmetries of an electrondensity map calculated by FFT – hence sampled on a grid which is an integral subdivision of the period lattice – necessarily entails the interpolation of densities at nonintegral points of that grid. The effect of interpolation on the structure factors recalculated from an averaged map was examined by Bricogne (1976). This study showed that, if linear interpolation is used, the initial map should be calculated on a fine grid, of size Δ/5 or Δ/6 at resolution Δ (instead of the previously used value of Δ/3). The analysis about to be given applies to all interpolation schemes which consist in a convolution of the sampled density with a fixed interpolation kernel function K.
Let be a periodic function. Let K be the interpolation kernel in `normalized' form, i.e. such that and scaled so as to interpolate between sample values given on a unit grid ; in the case of linear interpolation, K is the `trilinear wedge'whereLet be sampled on a grid , and let denote the function interpolated from this sampled version of . Then:where , so that
The transform of thus consists of
Thus is not bandlimited even if is. Supposing, however, that is bandlimited and that grid satisfies the Shannon sampling criterion, we see that there will be no overlap between the different bands: may therefore be recovered from the main band by compensating its attenuation, which is approximately a temperaturefactor correction.
For numerical work, however, must be resampled onto another grid , which causes its transform to become periodized intoThis now causes the main band to become contaminated by the ghost bands of the translates of .
Aliasing errors may be minimized by increasing the sampling rate in grid well beyond the Shannon minimum, which rapidly reduces the r.m.s. content of the ghost bands.
The sampling rate in grid needs only exceed the Shannon minimum to the extent required to accommodate the increase in bandwidth due to convolution with , which is the reciprocalspace counterpart of envelope truncation (or solvent flattening) in real space.
Green's theorem stated in terms of distributions (Section 1.3.2.3.9.1) is particularly well suited to the calculation of the Fourier transforms of indicator functions. Let f be the indicator function and let S be the boundary of U (assumed to be a smooth surface). The jump in the value of f across S along the outer normal vector is , the jump in the normal derivative of f across S is , and the Laplacian of f as a function is (almost everywhere) 0 so that . Green's theorem then reads:
The function satisfies the identity . Therefore, in Cartesian coordinates:i.e.where n is the outer normal to S. This formula was used by von Laue (1936) for a different purpose, namely to calculate the transforms of crystal shapes (see also Ewald, 1940). If the surface S is given by a triangulation, the surface integral becomes a sum over all faces, since n is constant on each face. If U is a solid sphere with radius R, an integration by parts gives immediately:
An atomic model of a crystal structure consists of a list of symmetryunique atoms described by their positions, their thermal agitation and their chemical identity (which can be used as a pointer to formfactor tables). Form factors are usually parameterized as sums of Gaussians, and thermal agitation by a Gaussian temperature factor or tensor. The formulae given in Section 1.3.4.2.2.6 for Gaussian atoms are therefore adequate for most purposes. Highresolution electrondensity studies use more involved parameterizations.
Early calculations were carried out by means of Bragg–Lipson charts (Bragg & Lipson, 1936) which gave a graphical representation of the symmetrized trigonometric sums Ξ of Section 1.3.4.2.2.9. The approximation of form factors by Gaussians goes back to the work of Vand et al. (1957) and Forsyth & Wells (1959). Agarwal (1978) gave simplified expansions suitable for mediumresolution modelling of macromolecular structures.
This method of calculating structure factors is expensive because each atom sends contributions of essentially equal magnitude to all structure factors in a resolution shell. The calculation is therefore of size for N atoms and reflections. Since N and are roughly proportional at a given resolution, this method is very costly for large structures.
Two distinct programming strategies are available (Rollett, 1965) according to whether the fast loop is on all atoms for each reflection, or on all reflections for each atom. The former method was favoured in the early times when computers were unreliable. The latter was shown by Burnett & Nordman (1974) to be more amenable to efficient programming, as no multiplication is required in calculating the arguments of the sine/cosine terms: these can be accumulated by integer addition, and used as subscripts in referencing a trigonometric function table.
Robertson (1936b) recognized the similarity between the calculation of structure factors by Fourier summation and the calculation of Fourier syntheses, the main difference being of course that atomic coordinates do not usually lie exactly on a grid obtained by integer subdivision of the crystal lattice. He proposed to address this difficulty by the use of his sorting board, which could extend the scale of subdivision and thus avoid phase errors. In this way the calculation of structure factors became amenable to Beevers–Lipson strip methods, with considerable gain of speed.
Later, Beevers & Lipson (1952) proposed that trigonometric functions attached to atomic positions falling between the grid points on which Beevers–Lipson strips were based should be obtained by linear interpolation from the values found on the strips for the closest grid points. This amounts (Section 1.3.4.4.3.4) to using atoms in the shape of a trilinear wedge, whose form factor was indicated in Section 1.3.4.4.3.4 and gives rise to aliasing effects (see below) not considered by Beevers & Lipson.
The correct formulation of this idea came with the work of Sayre (1951), who showed that structure factors could be calculated by Fourier analysis of a sampled electrondensity map previously generated on a subdivision of the crystal lattice Λ. When generating such a map, care must be taken to distribute onto the sample grid not only the electron densities of all the atoms in the asymmetric motif, but also those of their images under spacegroup symmetries and lattice translations. Considerable savings in computation occur, especially for large structures, because atoms are localized: each atom sends contributions to only a few grid points in real space, rather than to all reciprocallattice points. The generation of the sampled electrondensity map is still of complexity for N atoms and reflections, but the proportionality constant is smaller than that in Section 1.3.4.4.4 by orders of magnitude; the extra cost of Fourier analysis, proportional to , is negligible.
The idea of approximating a Fourier transform by a discrete transform on sampled values had already been used by Whittaker (1948), who tested it on the first three odd Hermite functions and did not consider the problem of aliasing errors. By contrast, Sayre gave a lucid analysis of the sampling problems associated to this technique. If the periodic sampled map is written in the form of a weighted lattice distribution (as in Section 1.3.2.7.3) asthen its discrete Fourier transform yieldsso that each correct value is corrupted by its aliases for .
To cure this aliasing problem, Sayre used `hypothetical atoms' with form factors equal to those of standard atoms within the resolution range of interest, but set to zero outside that range. This amounts to using atomic densities with builtin seriestermination errors, which has the detrimental effect of introducing slowly decaying ripples around the atom which require incrementing sample densities at many more grid points per atom.
Sayre considered another cure in the form of an artificial temperature factor B (Bragg & West, 1930) applied to all atoms. This spreads each atom on more grid points in real space but speeds up the decay of its transform in reciprocal space, thus allowing the use of a coarser sampling grid in real space. He discounted it as spoiling the agreement with observed data, but Ten Eyck (1977) pointed out that this agreement could be restored by applying the negative of the artificial temperature factor to the results. This idea cannot be carried to extremes: if B is chosen too large, the atoms will be so spread out in real space as each to occupy a sizeable fraction of the unit cell and the advantage of atom localization will be lost; furthermore, the form factors will fall off so rapidly that roundoff error amplification will occur when the results are sharpened back. Clearly, there exists an optimal combination of B and sampling rate yielding the most economical computation for a given accuracy at a given resolution, and a formula will now be given to calculate it.
Let us make the simplifying assumption that all atoms are roughly equal and that their common form factor can be represented by an equivalent temperature factor . Let be the resolution to which structure factors are wanted. The Shannon sampling interval is . Let σ be the oversampling rate, so that the actual sampling interval in the map is : then consecutive copies of the transform are separated by a distance in reciprocal space. Let the artificial temperature factor be added, and letThe worst aliasing occurs at the outer resolution limit , where the `signal' due to an atom is proportional towhile the `noise' due to the closest alias is proportional toThus the signaltonoise ratio, or quality factor, Q is
If a certain value of Q is desired (e.g. for 1% accuracy), then the equationdefines B in terms of and Q.
The overall cost of the structurefactor calculation from N atoms is then
where and are constant depending on the speed of the computer used. This overall cost may be minimized with respect to σ for given and Q, determining the optimal B (and hence ) in passing by the above relation.
Sayre (1951) did observe that applying an artificial temperature factor in real space would not create seriestermination ripples: the resulting atoms would have a smaller effective radius than his hypothetical atoms, so that step (i) would be faster. This optimality of Gaussian smearing is ultimately a consequence of Hardy's theorem (Section 1.3.2.4.4.3).
Some methods of phase determination rely on maximizing a certain global criterion involving the electron density, of the form , under constraint of agreement with the observed structurefactor amplitudes, typically measured by a residual C. Several recently proposed methods use for various measures of entropy defined by taking or (Bricogne, 1982; Britten & Collins, 1982; Narayan & Nityananda, 1982; Bryan et al., 1983; Wilkins et al., 1983; Bricogne, 1984; Navaza, 1985; Livesey & Skilling, 1985). Sayre's use of the squaring method to improve protein phases (Sayre, 1974) also belongs to this category, and is amenable to the same computational strategies (Sayre, 1980).
These methods differ from the densitymodification procedures of Section 1.3.4.4.3.2 in that they seek an optimal solution by moving electron densities (or structure factors) jointly rather than pointwise, i.e. by moving along suitably chosen search directions [or ].
For computational purposes, these search directions may be handled either as column vectors of sample values on a grid in real space, or as column vectors of Fourier coefficients in reciprocal space. These column vectors are the coordinates of the same vector in an abstract vector space of dimension over , but referred to two different bases which are related by the DFT and its inverse (Section 1.3.2.7.3).
The problem of finding the optimum of S for a given value of C amounts to achieving collinearity between the gradients and of S and of C in , the scalar ratio between them being a Lagrange multiplier. In order to move towards such a solution from a trial position, the dependence of and on position in must be represented. This involves the Hessian matrices H(S) and H(C), whose size precludes their use in the whole of . Restricting the search to a smaller search subspace of dimension n spanned by we may build local quadratic models of S and C (Bryan & Skilling, 1980; Burch et al., 1983) with respect to n coordinates X in that subspace:The coefficients of these linear models are given by scalar products:which, by virtue of Parseval's theorem, may be evaluated either in real space or in reciprocal space (Bricogne, 1984). In doing so, special positions and reflections must be taken into account, as in Section 1.3.4.2.2.8. Scalar products involving S are best evaluated by realspace grid summation, because H(S) is diagonal in this representation; those involving C are best calculated by reciprocalspace summation, because H(C) is at worst blockdiagonal in this representation. Using these Hessian matrices in the wrong space would lead to prohibitively expensive convolutions instead of scalar (or at worst matrix) multiplications.
Since the origins of Xray crystal structure analysis, the calculation of crystallographic Fourier series has been closely associated with the process of refinement. Fourier coefficients with phases were obtained for all or part of the measured reflections in the basis of some trial model for all or part of the structure, and Fourier syntheses were then used to complete and improve this initial model. This approach is clearly described in the classic paper by Bragg & West (1929), and was put into practice in the determination of the structures of topaz (Alston & West, 1929) and diopside (Warren & Bragg, 1929). Later, more systematic methods of arriving at a trial model were provided by the Patterson synthesis (Patterson, 1934, 1935a,b; Harker, 1936) and by isomorphous replacement (Robertson, 1935, 1936c). The role of Fourier syntheses, however, remained essentially unchanged [see Robertson (1937) for a review] until more systematic methods of structure refinement were introduced in the 1940s. A particularly good account of the processes of structure completion and refinement may be found in Chapters 15 and 16 of Stout & Jensen (1968).
It is beyond the scope of this section to review the vast topic of refinement methods: rather, it will give an account of those aspects of their development which have sought improved power by exploiting properties of the Fourier transformation. It is of more than historical interest that some recent advances in the crystallographic refinement of macromolecular structures had been anticipated by Cochran and Cruickshank in the early 1950s.
Hughes (1941) was the first to use the already well established multivariate leastsquares method (Whittaker & Robinson, 1944) to refine initial estimates of the parameters describing a model structure. The method gained general acceptance through the programming efforts of Friedlander et al. (1955), Sparks et al. (1956), Busing & Levy (1961) and others.
The Fourier relations between and F (Section 1.3.4.2.2.6) are used to derive the `observational equations' connecting the structure parameters to the observations comprising the amplitudes and their experimental variances for a set of unique reflections.
The normal equations giving the corrections δu to the parameters are thenwhereTo calculate the elements of A, write:hence
In the simple case of atoms with realvalued form factors and isotropic thermal agitation in space group P1,where being a fractional occupancy.
Positional derivatives with respect to are given byso that the corresponding subvector of the righthand side of the normal equations reads:
The setting up and solution of the normal equations lends itself well to computer programming and has the advantage of providing a thorough analysis of the accuracy of its results (Cruickshank, 1965b, 1970; Rollett, 1970). It is, however, an expensive task, of complexity , which is unaffordable for macromolecules.
It was the use of Fourier syntheses in the completion of trial structures which provided the incentive to find methods for computing 2D and 3D syntheses efficiently, and led to the Beevers–Lipson strips. The limited accuracy of the latter caused the estimated positions of atoms (identified as peaks in the maps) to be somewhat in error. Methods were therefore sought to improve the accuracy with which the coordinates of the electrondensity maxima could be determined. The naive method of peakshape analysis from densities recalculated on a grid using highaccuracy trigonometric tables entailed 27 summations per atom.
Booth (1946a) suggested examining the rapidly varying derivatives of the electron density rather than its slowly varying values. Ifthen the gradient vector of at can be calculated by means of three Fourier summations from the vector of Fourier coefficientsSimilarly, the Hessian matrix of at can be calculated by six Fourier summations from the unique elements of the symmetric matrix of Fourier coefficients:
The scalar maps giving the components of the gradient and Hessian matrix of will be called differential syntheses of 1st order and 2nd order respectively. If is approximately but not exactly a maximum of , then the Newton–Raphson estimate of the true maximum is given by:This calculation requires only nine accurate Fourier summations (instead of 27), and this number is further reduced to four if the peak is assumed to be spherically symmetrical.
The resulting positions are affected by seriestermination errors in the differential syntheses. Booth (1945c, 1946c) proposed a `backshift correction' to eliminate them, and extended this treatment to the acentric case (Booth, 1946b). He cautioned against the use of an artificial temperature factor to fight seriestermination errors (Brill et al., 1939), as this could be shown to introduce coordinate errors by causing overlap between atoms (Booth, 1946c, 1947a,b).
Cruickshank was able to derive estimates for the standard uncertainties of the atomic coordinates obtained in this way (Cox & Cruickshank, 1948; Cruickshank, 1949a,b) and to show that they agreed with those provided by the leastsquares method.
The calculation of differential Fourier syntheses was incorporated into the crystallographic programs of Ahmed & Cruickshank (1953b) and of Sparks et al. (1956).
Having defined the now universally adopted R factors (Booth, 1945b) as criteria of agreement between observed and calculated amplitudes or intensities, Booth proposed that R should be minimized with respect to the set of atomic coordinates by descending along the gradient of R in parameter space (Booth, 1947c,d). This `steepest descents' procedure was compared with Patterson methods by Cochran (1948d).
When calculating the necessary derivatives, Booth (1948a, 1949) used the formulae given above in connection with least squares. This method was implemented by Qurashi (1949) and by Vand (1948, 1951) with parameterrescaling modifications which made it very close to the leastsquares method (Cruickshank, 1950; Qurashi & Vand, 1953; Qurashi, 1953).
Cochran (1948b,c, 1951a) undertook to exploit an algebraic similarity between the righthand side of the normal equations in the leastsquares method on the one hand, and the expression for the coefficients used in Booth's differential syntheses on the other hand (see also Booth, 1948a). In doing so he initiated a remarkable sequence of formal and computational developments which are still actively pursued today.
Let be the electrondensity map corresponding to the current atomic model, with structure factors ; and let be the map calculated from observed moduli and calculated phases, i.e. with coefficients . If there are enough data for to have a resolved peak at each model atomic position , thenwhile if the calculated phases are good enough, will also have peaks at each :It follows thatwhere the summation is over all reflections in or related to by spacegroup and Friedel symmetry (overlooking multiplicity factors!). This relation is less sensitive to seriestermination errors than either of the previous two, since the spectrum of could have been extrapolated beyond the data in by using that of [as in van Reijen (1942)] without changing its righthand side.
Cochran then used the identityin the formto rewrite the previous relation as(the operation [] on the first line being neutral because of Friedel symmetry). This is equivalent to the vanishing of the subvector of the righthand side of the normal equations associated to a leastsquares refinement in which the weights would beCochran concluded that, for equalatom structures with for all j, the positions obtained by Booth's method applied to the difference map are such that they minimize the residualwith respect to the atomic positions. If it is desired to minimize the residual of the ordinary leastsquares method, then the differential synthesis method should be applied to the weighted difference mapHe went on to show (Cochran, 1951b) that the refinement of temperature factors could also be carried out by inspecting appropriate derivatives of the weighted difference map.
This Fourier method was used by Freer et al. (1976) in conjunction with a stereochemical regularization procedure to refine protein structures.
Cruickshank consolidated and extended Cochran's derivations in a series of classic papers (Cruickshank, 1949b, 1950, 1952, 1956). He was able to show that all the coefficients involved in the righthand side and normal matrix of the leastsquares method could be calculated by means of suitable differential Fourier syntheses even when the atoms overlap. This remarkable achievement lay essentially dormant until its independent rediscovery by Agarwal in 1978 (Section 1.3.4.4.7.6).
To ensure rigorous equivalence between the summations over (in the expressions of leastsquares righthand side and normal matrix elements) and genuine Fourier summations, multiplicitycorrected weights were introduced by:where Gh denotes the orbit of h and its isotropy subgroup (Section 1.3.4.2.2.5). Similarly, derivatives with respect to parameters of symmetryunique atoms were expressed, via the chain rule, as sums over the orbits of these atoms.
Let be the label of a parameter belonging to atoms with label j. Then Cruickshank showed that the pth element of the righthand side of the normal equations can be obtained as , where is a differential synthesis of the formwith a polynomial in (h, k, l) depending on the type of parameter p. The correspondence between parameter type and the associated polynomial extends Booth's original range of differential syntheses, and is recapitulated in the following table.Unlike Cochran's original heuristic argument, this result does not depend on the atoms being resolved.
Cruickshank (1952) also considered the elements of the normal matrix, of the formassociated with positional parameters. The block for parameters and may be writtenwhich, using the identitybecomes(Friedel's symmetry makes redundant on the last line). Cruickshank argued that the first term would give a good approximation to the diagonal blocks of the normal matrix and to those offdiagonal blocks for which and are close. On this basis he was able to justify the `nshift rule' of Shoemaker et al. (1950). Cruickshank gave this derivation in a general space group, but using a very terse notation which somewhat obscures it. Using the symmetrized trigonometric structurefactor kernel of Section 1.3.4.2.2.9 and its multiplication formula, the above expression is seen to involve the values of a Fourier synthesis at points of the form .
Cruickshank (1956) showed that this analysis could also be applied to the refinement of temperature factors.
These two results made it possible to obtain all coefficients involved in the normal equations by looking up the values of certain differential Fourier syntheses at or at . At the time this did not confer any superiority over the standard form of the leastsquares procedure, because the accurate computation of Fourier syntheses was an expensive operation. The modified Fourier method was used by Truter (1954) and by Ahmed & Cruickshank (1953a), and was incorporated into the program system described by Cruickshank et al. (1961). A more recent comparison with the leastsquares method was made by Dietrich (1972).
There persisted, however, some confusion about the nature of the relationship between Fourier and leastsquares methods, caused by the extra factors which make it necessary to compute a differential synthesis for each type of atom. This led Cruickshank to conclude that `in spite of their remarkable similarities the leastsquares and modifiedFourier methods are fundamentally distinct'.
Agarwal (1978) rederived and completed Cruickshank's results at a time when the availability of the FFT algorithm made the Fourier method of calculating the coefficients of the normal equations much more economical than the standard method, especially for macromolecules.
As obtained by Cruickshank, the modified Fourier method required a full 3D Fourier synthesis
Agarwal disposed of the latter dependence by pointing out that the multiplication involved is equivalent to a realspace convolution between the differential synthesis and , the standard electron density for atom type j (Section 1.3.4.2.1.2) smeared by the isotropic thermal agitation of that atom. Since is localized, this convolution involves only a small number of grid points. The requirement of a distinct differential synthesis for each parameter type, however, continued to hold, and created some difficulties at the FFT level because the symmetries of differential syntheses are more complex than ordinary spacegroup symmetries. Jack & Levitt (1978) sought to avoid the calculation of difference syntheses by using instead finite differences calculated from ordinary Fourier or difference Fourier maps.
In spite of its complication, this return to the Fourier implementation of the leastsquares method led to spectacular increases in speed (Isaacs & Agarwal, 1978; Agarwal, 1980; Baker & Dodson, 1980) and quickly gained general acceptance (Dodson, 1981; Isaacs, 1982a,b, 1984).
Lifchitz [see Agarwal et al. (1981), Agarwal (1981)] proposed that the idea of treating certain multipliers in Cruickshank's modified differential Fourier syntheses by means of a convolution in real space should be applied not only to , but also to the polynomials which determine the type of differential synthesis being calculated. This leads to convoluting with the same ordinary weighted difference Fourier synthesis, rather than with the differential synthesis of type p. In this way, a single Fourier synthesis, with ordinary (scalar) symmetry properties, needs be computed; the parameter type and atom type both intervene through the function with which it is convoluted. This approach has been used as the basis of an efficient generalpurpose leastsquares refinement program for macromolecular structures (Tronrud et al., 1987).
This rearrangement amounts to using the fact (Section 1.3.2.3.9.7) that convolution commutes with differentiation. Letbe the inversevariance weighted difference map, and let us assume that parameter belongs to atom j. Then the Agarwal form for the pth component of the righthand side of the normal equations iswhile the Lifchitz form is
A very simple derivation of the previous results will now be given, which suggests the possibility of many generalizations.
The weighted difference map has coefficients which are the gradients of the global residual with respect to each :By the chain rule, a variation of each by will result in a variation of R by withThe operation is superfluous because of Friedel symmetry, so that may be simply written in terms of the Hermitian scalar product in :If is the transform of , we have also by Parseval's theoremWe may therefore writewhich states that is the functional derivative of R with respect to .
The righthand side of the normal equations has for its pth element, and this may be writtenIf belongs to atom j, thenhenceBy the identity of Section 1.3.2.4.3.5, this is identical to Lifchitz's expression . The present derivation in terms of scalar products [see Brünger (1989) for another presentation of it] is conceptually simpler, since it invokes only the chain rule [other uses of which have been reviewed by Lunin (1985)] and Parseval's theorem; economy of computation is obviously related to the good localization of compared to . Convolutions, whose meaning is less clear, are no longer involved; they were a legacy of having first gone over to reciprocal space via differential syntheses in the 1940s.
Cast in this form, the calculation of derivatives by FFT methods appears as a particular instance of the procedure described in connection with variational techniques (Section 1.3.4.4.6) to calculate the coefficients of local quadratic models in a search subspace; this is far from surprising since varying the electron density through a variation of the parameters of an atomic model is a particular case of the `free' variations considered by the variational approach. The latter procedure would accommodate in a very natural fashion the joint consideration of an energetic (Jack & Levitt, 1978; Brünger et al., 1987; Brünger, 1988; Brünger et al., 1989; Kuriyan et al., 1989) or stereochemical (Konnert, 1976; Sussman et al., 1977; Konnert & Hendrickson, 1980; Hendrickson & Konnert, 1980; Tronrud et al., 1987) restraint function (which would play the role of S) and of the crystallographic residual (which would be C). It would even have over the latter the superiority of affording a genuine secondorder approximation, albeit only in a subspace, hence the ability of detecting negative curvature and the resulting bifurcation behaviour (Bricogne, 1984). Current methods are unable to do this because they use only firstorder models, and this is known to degrade severely the overall efficiency of the refinement process.
The impossibility of carrying out a fullmatrix leastsquares refinement of a macromolecular crystal structure, caused by excessive computational cost and by the paucity of observations, led Diamond (1971) to propose a realspace refinement method in which stereochemical knowledge was used to keep the number of free parameters to a minimum. Refinement took place by a leastsquares fit between the `observed' electrondensity map and a model density consisting of Gaussian atoms. This procedure, coupled to iterative recalculation of the phases, led to the first highly refined protein structures obtained without using fullmatrix least squares (Huber et al., 1974; Bode & Schwager, 1975; Deisenhofer & Steigemann, 1975; Takano, 1977a,b).
Realspace refinement takes advantage of the localization of atoms (each parameter interacts only with the density near the atom to which it belongs) and gives the most immediate description of stereochemical constraints. A disadvantage is that fitting the `observed' electron density amounts to treating the phases of the structure factors as observed quantities, and to ignoring the experimental error estimates on their moduli. The method is also much more vulnerable to seriestermination errors and accidentally missing data than the leastsquares method. These objections led to the progressive disuse of Diamond's method, and to a switch towards reciprocalspace least squares following Agarwal's work.
The connection established above between the Cruickshank–Agarwal modified Fourier method and the simple use of the chain rule affords a partial refutation to both the premises of Diamond's method and to the objections made against it:
The calculation of the inner products from a sampled gradient map D requires even more caution than that of structure factors via electrondensity maps described in Section 1.3.4.4.5, because the functions have transforms which extend even further in reciprocal space than the themselves. Analytically, if the are Gaussians, the are finite sums of multivariate Hermite functions (Section 1.3.2.4.4.2) and hence the same is true of their transforms. The difference map D must therefore be finely sampled and the relation between error and sampling rate may be investigated as in Section 1.3.4.4.5. An examination of the sampling rates commonly used (e.g. one third of the resolution) shows that they are insufficient. Tronrud et al. (1987) propose to relax this requirement by applying an artificial temperature factor to (cf. Section 1.3.4.4.5) and the negative of that temperature factor to D, a procedure of questionable validity because the latter `sharpening' operation is ill defined [the function exp does not define a tempered distribution, so the associativity properties of convolution may be lost]. A more robust procedure would be to compute the scalar product by means of a more sophisticated numerical quadrature formula than a mere grid sum.
Certain correlation functions can be useful to detect the presence of multiple copies of the same molecule (known or unknown) in the asymmetric unit of a crystal of unknown structure.
Suppose that a crystal contains one or several copies of a molecule in its asymmetric unit. If is the electron density of that molecule in some reference position and orientation, thenwhere describes the placement of the jth copy of the molecule with respect to the reference copy. It is assumed that each such copy is in a general position, so that there is no isotropy subgroup.
The methods of Section 1.3.4.2.2.9 (with replaced by , and by ) lead to the following expression for the autocorrelation of :
If μ is unknown, consider the subfamily σ of terms with and :The scalar product in which R is a variable rotation will have a peak wheneversince two copies of the `selfPatterson' of the molecule will be brought into coincidence. If the interference from terms in the Patterson other than those present in σ is not too serious, the `selfrotation function' (Rossmann & Blow, 1962; Crowther, 1972) will show the same peaks, from which the rotations may be determined, either individually or jointly if for instance they form a group.
If μ is known, then its selfPatterson may be calculated, and the may be found by examining the `crossrotation function' which will have peaks at , . Once the are known, then the various copies of may be Fourieranalysed into structure factors:The cross terms with in then contain `motifs'with Fourier coefficientstranslated by . Therefore the `translation functions' (Crowther & Blow, 1967)will have peaks at corresponding to the detection of these motifs.
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