Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 91-100   | 1 | 2 |

Section Basic crystallographic computations

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Basic crystallographic computations

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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. Fourier synthesis of electron-density maps

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Bragg (1929)[link] was the first to use this type of calculation to assist structure determination. Progress in computing techniques since that time was reviewed in Section[link].

The usefulness of the maps thus obtained can be adversely affected by three main factors:

  • (i) limited resolution;

  • (ii) errors in the data;

  • (iii) computational errors.

Limited resolution causes `series-termination errors' first investigated by Bragg & West (1930)[link], who used an optical analogy with the numerical aperture of a microscope. James (1948b)[link] gave a quantitative description of this phenomenon as a convolution with the `spherical Dirichlet kernel' (Section[link]), which reflects the truncation of the Fourier spectrum by multiplication with the indicator function of the limiting resolution sphere. Bragg & West (1930)[link] 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 point-spread function. When the electron-density map is to be used for model refinement, van Reijen (1942)[link] suggested using Fourier coefficients calculated from the model when no observation is available, as a means of combating series-termination effects.

Errors in the data introduce errors in the electron-density maps, with the same mean-square value by virtue of Parseval's theorem. Special positions accrue larger errors (Cruickshank & Rollett, 1953[link]; Cruickshank, 1965a[link]). To minimize the mean-square electron-density error due to large phase uncertainties, Blow & Crick (1959)[link] introduced the `best Fourier' which uses centroid Fourier coefficients; the associated error level in the electron-density map was evaluated by Blow & Crick (1959)[link] and Dickerson et al. (1961a[link],b[link]).

Computational errors used to be a serious concern when Beevers–Lipson strips were used, and Cochran (1948a)[link] 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[link]). Fourier analysis of modified electron-density maps

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Various approaches to the phase problem are based on certain modifications of the electron-density map, followed by Fourier analysis of the modified map and extraction of phase information from the resulting Fourier coefficients. Squaring

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Sayre (1952a)[link] 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. Thus[F_{{\bf h}} = \theta_{{\bf h}} {\textstyle\sum\limits_{{\bf k}}}\, F_{{\bf k}} F_{{\bf h}-{\bf k}},]where [\theta_{{\bf h}} = f({\bf h})/f^{\rm sq} ({\bf h})] is the ratio between the form factor [f({\bf h})] common to all the atoms and the form factor [f^{\rm sq} ({\bf h})] 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[link], 1974[link]). Other nonlinear operations

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A category of phase improvement procedures known as `density modification' is based on the pointwise application of various quadratic or cubic `filters' to electron-density maps after removal of negative regions (Hoppe & Gassmann, 1968[link]; Hoppe et al., 1970[link]; Barrett & Zwick, 1971[link]; Gassmann & Zechmeister, 1972[link]; Collins, 1975[link]; Collins et al., 1976[link]; Gassmann, 1976[link]). These operations are claimed to be equivalent to reciprocal-space phase-refinement techniques such as those based on the tangent formula. Indeed the replacement of[\rho\llap{$-\!$} ({\bf x}) = {\textstyle\sum\limits_{{\bf h}}} \,F_{{\bf h}} \exp (-2 \pi i {\bf h} \cdot {\bf x})]by [P[\rho\llap{$-\!$} ({\bf x})]], where P is a polynomial[P(\rho\llap{$-\!$}) = a_{0} + a_{1} \rho\llap{$-\!$} + a_{2} \rho\llap{$-\!$}^{2} + a_{3} \rho\llap{$-\!$}^{3} + \ldots]yields[\eqalign{ P[\rho\llap{$-\!$}({\bf x})] =\ &a_{0} + {\textstyle\sum\limits_{{\bf h}}} \left[a_{1} F_{{\bf h}} + a_{2} {\textstyle\sum\limits_{{\bf k}}} \,F_{{\bf k}} F_{{\bf h} - {\bf k}}\right.\cr &\left.+\ a_{3} {\textstyle\sum\limits_{{\bf k}}} {\textstyle\sum\limits_{\bf l}} \,F_{{\bf k}} F_{\bf l} F_{{\bf h} - {\bf k} - {\bf l}} + \ldots \right] \exp (- 2 \pi i{\bf h} \cdot {\bf x})}]and hence gives rise to the convolution-like families of terms encountered in direct methods. This equivalence, however, has been shown to be rather superficial (Bricogne, 1982[link]) because the `uncertainty principle' embodied in Heisenberg's inequality (Section[link]) 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 [\rho\llap{$-\!$}] extends to resolution Δ and if the pointwise nonlinear filter involves a polynomial P of degree n, then P([\rho\llap{$-\!$}]) should be sampled at intervals of at most [\Delta/2n] to accommodate the full bandwidth of its spectrum. Solvent flattening

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Crystals of proteins and nucleic acids contain large amounts of mother liquor, often in excess of 50% of the unit-cell volume, occupying connected channels. The well ordered electron density [\rho\llap{$-\!$}_{\rm M}({\bf x})] corresponding to the macromolecule thus occupies only a periodic subregion [{\scr U}] of the crystal. Thus[\rho\llap{$-\!$}_{\rm M} = \chi_{\scr U} \times \rho\llap{$-\!$}_{\rm M},]implying the convolution identity between structure factors (Main & Woolfson, 1963[link]):[F_{\rm M}({\bf h}) = \sum\limits_{{\bf k}} \bar{{\scr F}} \left[{1 \over {\scr U}} \chi_{{\scr U}}\right] ({\bf h} - {\bf k}) F_{\rm M} ({\bf k})]which is a form of the Shannon interpolation formula (Sections[link],[link]; Bricogne, 1974[link]; Colman, 1974[link]).

It is often possible to obtain an approximate `molecular envelope' [{\scr U}] from a poor electron-density map [\rho\llap{$-\!$}], either interactively by computer graphics (Bricogne, 1976[link]) 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[link]). However, as it is a convolution of [\rho\llap{$-\!$}] with [\chi_{\rm S}], it can be speeded up considerably (Leslie, 1987[link]) by computing the moving average [\rho\llap{$-\!$}_{\rm mav}] as[\rho\llap{$-\!$}_{\rm mav}({\bf x}) = {\scr F}[\bar{{\scr F}}[\rho\llap{$-\!$}] \times \bar{{\scr F}}[\chi_{\rm S}]]({\bf x}).]

This remark is identical in substance to Booth's method of computation of `bounded projections' (Booth, 1945a[link]) described in Section[link], except that the summation is kept three-dimensional.

The iterative use of the estimated envelope [{\scr U}] for the purpose of phase improvement (Wang, 1985[link]) is a submethod of the previously developed method of molecular averaging, which is described below. Sampling rules for the Fourier analysis of envelope-truncated maps will be given there. Molecular averaging by noncrystallographic symmetries

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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)[link] recognized that intensity data collected from such structures are redundant (Sayre, 1952b[link]) 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)[link], and were generalized by Main & Rossmann (1966)[link]. Crowther (1967[link], 1969[link]) reformulated them as linear eigenvalue equations between structure factors, for which he proposed an iterative matrix solution method. Although useful in practice (Jack, 1973[link]), this reciprocal-space approach required computations of size [\propto N^{2}] for N reflections, so that N could not exceed a few thousands.

The theory was then reformulated in real space (Bricogne, 1974[link]), 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[link](d)[link]]. The overall complexity was thus reduced from [N^{2}] to N log N. The design and implementation of a general-purpose program package for averaging, reconstructing and solvent-flattening electron-density maps (Bricogne, 1976[link]) led rapidly to the first high-resolution determinations of virus structures (Bloomer et al., 1978[link]; Harrison et al., 1978[link]), with [N \sim 200\,000].

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 [\chi_{U}] in real space, whereas they involve a convolution with [\bar{{\scr F}}[\chi_{U}]] in reciprocal space.

The averaging by noncrystallographic symmetries of an electron-density 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)[link]. 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 [\rho\llap{$-\!$}] be a [{\bb Z}^{3}]-periodic function. Let K be the interpolation kernel in `normalized' form, i.e. such that [\int_{{\bb R}^{3}} K ({\bf x}) \hbox{ d}^{3} {\bf x} = 1] and scaled so as to interpolate between sample values given on a unit grid [{\bb Z}^{3}]; in the case of linear interpolation, K is the `trilinear wedge'[K({\bf x}) = W(x) W(y) W(z),]where[\eqalign{ W(t) &= 1 - |t|\quad \hbox{ if } |t| \leq 1,\cr &= 0\phantom{= 1 - t} \hbox{ if } |t| \geq 1.}]Let [\rho\llap{$-\!$}] be sampled on a grid [{\scr G}_{1} = {\bf N}_{1}^{-1} {\bb Z}^{3}], and let [I_{{\bf N}_{1}}\rho\llap{$-\!$}] denote the function interpolated from this sampled version of [\rho\llap{$-\!$}]. Then:[I_{{\bf N}_{1}}\rho\llap{$-\!$} = \left[\rho\llap{$-\!$} \times {\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{3}}} \delta_{({\bf N}_{1}^{-1} {\bf m})}\right] * [({\bf N}_{1}^{-1})^{\#} K],]where [[(N_{1}^{-1})^{\#} K]({\bf x}) = K({\bf N}_{1}{\bf x})], so that[\eqalign{{\bar{\scr F}}[I_{{\bf N}_{1}} \rho\llap{$-\!$}] &= \left[\bar{{\scr F}}[\rho\llap{$-\!$}] * |\!\det {\bf N}_{1}| {\textstyle\sum\limits_{{\bf k}_{1} \in {\bb Z}^{3}}} \delta_{({\bf N}_{1}^{T}{\bf k}_{1})}\right]\cr &\quad \times \left[{1 \over |\!\det {\bf N}_{1}|} (N_{1}^{T})^{\#} \bar{{\scr F}}[K]\right]\cr &= \left[{\textstyle\sum\limits_{{\bf k}_{1} \in {\bb Z}^{3}}} \boldtau_{{\bf N}_{1}^{T}{\bf k}_{1}} \bar{{\scr F}}[\rho\llap{$-\!$}] \right] \times (N_{1}^{T})^{\#} \bar{{\scr F}}[K].}]

The transform of [I_{{\bf N}_{1}}\rho\llap{$-\!$}] thus consists of

  • (i) a `main band' corresponding to [{\bf k}_{1} = {\bf 0}], which consists of the true transform [\bar{{\scr F}}[\rho\llap{$-\!$}]({\boldxi})] attenuated by multiplication by the central region of [\bar{{\scr F}}[K][({\bf N}^{-1})^{T}{\boldxi}]]; in the case of linear interpolation, for example,[\bar{{\scr F}}[K] (\xi, \eta, \zeta) = \left({\sin \pi \xi \over \pi \xi}\right)^{2} \left({\sin \pi \eta \over \pi \eta}\right)^{2} \left({\sin \pi \zeta \over \pi \zeta}\right)^{2}\hbox{\semi}]

  • (ii) a series of `ghost bands' corresponding to [{\bf k}_{1} \neq {\bf 0}], which consist of translates of [\bar{{\scr F}} [\rho\llap{$-\!$}]] multiplied by the tail regions of [(N_{1}^{T})^{\#} \bar{{\scr F}} [K]].

Thus [I_{{\bf N}_{1}}\rho\llap{$-\!$}] is not band-limited even if [\rho\llap{$-\!$}] is. Supposing, however, that [\rho\llap{$-\!$}] is band-limited and that grid [{\scr G}_{1}] satisfies the Shannon sampling criterion, we see that there will be no overlap between the different bands: [\bar{{\scr F}}[\rho\llap{$-\!$}]] may therefore be recovered from the main band by compensating its attenuation, which is approximately a temperature-factor correction.

For numerical work, however, [I_{{\bf N}_{1}}\rho\llap{$-\!$}] must be resampled onto another grid [{\scr G}_{2}], which causes its transform to become periodized into[|\!\det {\bf N}_{2}| {\textstyle\sum\limits_{{\bf k}_{2}\in {\bb Z}^{3}}} \boldtau_{{\bf N}_{2}^{T}{\bf k}_{2}} \left\{\left[{\textstyle\sum\limits_{{\bf k}_{1}\in {\bb Z}^{3}}} \boldtau_{{\bf N}_{1}^{T}{\bf k}_{1}} \bar{{\scr F}}[\rho\llap{$-\!$}]\right] (N_{1}^{T})^{\#} \bar{{\scr F}}[K]\right\}.]This now causes the main band [{\bf k}_{1} = {\bf k}_{2} = {\bf 0}] to become contaminated by the ghost bands [({\bf k}_{1} \neq {\bf 0})] of the translates [({\bf k}_{2} \neq {\bf 0})] of [I_{{\bf N}_{1}}\rho\llap{$-\!$}].

Aliasing errors may be minimized by increasing the sampling rate in grid [{\scr G}_{1}] well beyond the Shannon minimum, which rapidly reduces the r.m.s. content of the ghost bands.

The sampling rate in grid [{\scr G}_{2}] needs only exceed the Shannon minimum to the extent required to accommodate the increase in bandwidth due to convolution with [\bar{{\scr F}}[\chi_{U}]], which is the reciprocal-space counterpart of envelope truncation (or solvent flattening) in real space. Molecular-envelope transforms via Green's theorem

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Green's theorem stated in terms of distributions (Section[link]) is particularly well suited to the calculation of the Fourier transforms [\bar{{\scr F}}[\chi_{U}]] of indicator functions. Let f be the indicator function [\chi_{U}] and let S be the boundary of U (assumed to be a smooth surface). The jump [\sigma_{0}] in the value of f across S along the outer normal vector is [\sigma_{0} = -1], the jump [\sigma_{\nu}] in the normal derivative of f across S is [\sigma_{\nu} = 0], and the Laplacian of f as a function is (almost everywhere) 0 so that [T_{\Delta f} = 0]. Green's theorem then reads:[\eqalign{ \Delta (T_{f}) &= T_{\Delta f} + \sigma_{\nu}\delta_{(S)} + \partial_{\nu} [\sigma_{0} \delta_{(S)}]\cr &= -\partial_{\nu} [\delta_{(S)}].}]

The function [e_{{\bf H}}({\bf X}) = \exp (2\pi i{\bf H} \cdot {\bf X})] satisfies the identity [\Delta e_{{\bf H}} = - 4\pi^{2} \|{\bf H}\|^{2} e_{{\bf H}}]. Therefore, in Cartesian coordinates:[\eqalign{\bar{F} [\chi_{U}] ({\bf H}) &= \langle T_{\chi_{U}}, e_{{\bf H}}\rangle\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \langle T_{\chi_{U}}, \Delta e_{{\bf H}}\rangle\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \langle \Delta (T_{\chi_{U}}), e_{{\bf H}}\rangle \quad \quad [\hbox{Section }]\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \langle - \partial_{\nu} [\delta_{(S)}], e_{{\bf H}}\rangle\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \int\limits_{S} \partial_{\nu} e_{{\bf H}} \,{\rm d}^{2} S \quad \quad [\hbox{Section }]\cr &= - {1 \over 4\pi^{2} \|{\bf H}\|^{2}} \int\limits_{S} 2\pi i{\bf H} \cdot {\bf n} \exp (2\pi i{\bf H} \cdot {\bf X}) \hbox{ d}^{2} S,}]i.e.[\bar{{\scr F}}[\chi_{U}] ({\bf H}) = {1 \over 2\pi i\|{\bf H}\|^{2}} \int\limits_{S} {\bf H} \cdot {\bf n} \exp (2\pi i{\bf H} \cdot {\bf X}) \hbox{ d}^{2} S,]where n is the outer normal to S. This formula was used by von Laue (1936)[link] for a different purpose, namely to calculate the transforms of crystal shapes (see also Ewald, 1940[link]). 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:[\eqalign{{1 \over \hbox{vol} (U)} \bar{{\scr F}}[\chi_{U}] ({\bf H}) &= {3 \over X^{3}} [\sin X - X \cos X]\cr &\phantom{[X - X \cos X]} \hbox{with } X = 2\pi \|{\bf H}\| R.}] Structure factors from model atomic parameters

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An atomic model of a crystal structure consists of a list of symmetry-unique atoms described by their positions, their thermal agitation and their chemical identity (which can be used as a pointer to form-factor 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[link] for Gaussian atoms are therefore adequate for most purposes. High-resolution electron-density studies use more involved parameterizations.

Early calculations were carried out by means of Bragg–Lipson charts (Bragg & Lipson, 1936[link]) which gave a graphical representation of the symmetrized trigonometric sums Ξ of Section[link] The approximation of form factors by Gaussians goes back to the work of Vand et al. (1957)[link] and Forsyth & Wells (1959)[link]. Agarwal (1978)[link] gave simplified expansions suitable for medium-resolution 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 [\propto N{\scr N}] for N atoms and [{\scr N}] reflections. Since N and [{\scr N}] are roughly proportional at a given resolution, this method is very costly for large structures.

Two distinct programming strategies are available (Rollett, 1965[link]) 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)[link] 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. Structure factors via model electron-density maps

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Robertson (1936b)[link] 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)[link] 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[link]) to using atoms in the shape of a trilinear wedge, whose form factor was indicated in Section[link] 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)[link], who showed that structure factors could be calculated by Fourier analysis of a sampled electron-density map previously generated on a subdivision [\hbox{\bf N}^{-1} \Lambda] 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 space-group 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 reciprocal-lattice points. The generation of the sampled electron-density map is still of complexity [\propto N {\scr N}] for N atoms and [{\scr N}] reflections, but the proportionality constant is smaller than that in Section[link] by orders of magnitude; the extra cost of Fourier analysis, proportional to [{\scr N} \log {\scr N}], is negligible.

The idea of approximating a Fourier transform by a discrete transform on sampled values had already been used by Whittaker (1948)[link], 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[link]) as[\rho\llap{$-\!$}^{\rm s} = {\textstyle\sum\limits_{{\bf m} \in {\bb Z}^{3}}} \rho\llap{$-\!$}({\bf N}^{-1} {\bf m}) \delta_{({\bf N}^{-1}{\bf m})},]then its discrete Fourier transform yields[F^{\rm s} ({\bf h}) = {\textstyle\sum\limits_{{\boldeta} \in {\bb Z}^{3}}} F({\bf h} + {\bf N}^{T} {\boldeta})]so that each correct value [F({\bf h})] is corrupted by its aliases [F({\bf h} + {\bf N}^{T} {\boldeta})] for [{\boldeta} \neq {\bf 0}].

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 built-in series-termination 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[link]) 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)[link] 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 round-off 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 [B_{\rm eq}]. Let [\Delta = 1/d_{\max}^{*}] be the resolution to which structure factors are wanted. The Shannon sampling interval is [\Delta/2 = 1/2d_{\max}^{*}]. Let σ be the oversampling rate, so that the actual sampling interval in the map is [\Delta/2\sigma = 1/2\sigma d_{\max}^{*}]: then consecutive copies of the transform are separated by a distance [2\sigma d_{\max}^{*}] in reciprocal space. Let the artificial temperature factor [B_{\rm extra}] be added, and let[B = B_{\rm eq} + B_{\rm extra}.]The worst aliasing occurs at the outer resolution limit [d_{\max}^{*}], where the `signal' due to an atom is proportional to[\exp [( - B/4) (d_{\max}^{*})^{2}],]while the `noise' due to the closest alias is proportional to[\exp \{( - B/4)[(2\sigma - 1)d_{\max}^{*}]^{2}\}.]Thus the signal-to-noise ratio, or quality factor, Q is[\exp [B\sigma (\sigma - 1) (d_{\max}^{*})^{2}].]

If a certain value of Q is desired (e.g. [Q = 100] for 1% accuracy), then the equation[B = {\log Q \over \sigma (\sigma - 1) (d_{\max}^{*})^{2}}]defines B in terms of [\sigma, d_{\max}^{*}] and Q.

The overall cost of the structure-factor calculation from N atoms is then

  • (i) [C_{1} \times B^{2/3} \times N] for density generation,

  • (ii) [C_{2} \times (2\sigma d_{\max}^{*})^{3} \times \log [(2\sigma d_{\max}^{*})^{3}]] for Fourier analysis,

where [C_{1}] and [C_{2}] are constant depending on the speed of the computer used. This overall cost may be minimized with respect to σ for given [d_{\max}^{*}] and Q, determining the optimal B (and hence [B_{\rm extra}]) in passing by the above relation.

Sayre (1951)[link] did observe that applying an artificial temperature factor in real space would not create series-termination ripples: the resulting atoms would have a smaller effective radius than his hypothetical atoms, so that step (i)[link] would be faster. This optimality of Gaussian smearing is ultimately a consequence of Hardy's theorem (Section[link]). Derivatives for variational phasing techniques

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Some methods of phase determination rely on maximizing a certain global criterion [S[\rho\llap{$-\!$}]] involving the electron density, of the form [{\textstyle\int_{{\bb R}^{3}/{\bb Z}^{3}}} K[\rho\llap{$-\!$}({\bf x})] \hbox{ d}^{3}{\bf x}], under constraint of agreement with the observed structure-factor amplitudes, typically measured by a [\chi^{2}] residual C. Several recently proposed methods use for [S[\rho\llap{$-\!$}]] various measures of entropy defined by taking [K(\rho\llap{$-\!$}) = - \rho\llap{$-\!$} \log (\rho\llap{$-\!$}/\mu)] or [K(\rho\llap{$-\!$}) = \log \rho\llap{$-\!$}] (Bricogne, 1982[link]; Britten & Collins, 1982[link]; Narayan & Nityananda, 1982[link]; Bryan et al., 1983[link]; Wilkins et al., 1983[link]; Bricogne, 1984[link]; Navaza, 1985[link]; Livesey & Skilling, 1985[link]). Sayre's use of the squaring method to improve protein phases (Sayre, 1974[link]) also belongs to this category, and is amenable to the same computational strategies (Sayre, 1980[link]).

These methods differ from the density-modification procedures of Section[link] 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 [v_{i}({\bf x})] [or [V_{i}({\bf h})]].

For computational purposes, these search directions may be handled either as column vectors of sample values [\{v_{i}({\bf N}^{-1}{\bf m})\}_{{\bf m} \in {\bb Z}^{3}/{\bf N}{\bb Z}^{3}}] on a grid in real space, or as column vectors of Fourier coefficients [\{V_{i}({\bf h})\}_{{\bf h} \in {\bb Z}^{3}/{\bf N}^{T}{\bb Z}^{3}}] in reciprocal space. These column vectors are the coordinates of the same vector [{\bf V}_{i}] in an abstract vector space [{\scr V} \cong L({\bb Z}^{3}/{\bf N}{\bb Z}^{3})] of dimension [{\scr N} = |\hbox{det } {\bf N}|] over [{\bb R}], but referred to two different bases which are related by the DFT and its inverse (Section[link]).

The problem of finding the optimum of S for a given value of C amounts to achieving collinearity between the gradients [\nabla S] and [\nabla C] of S and of C in [{\scr V}], the scalar ratio between them being a Lagrange multiplier. In order to move towards such a solution from a trial position, the dependence of [\nabla S] and [\nabla C] on position in [{\scr V}] must be represented. This involves the [{\scr N} \times {\scr N}] Hessian matrices H(S) and H(C), whose size precludes their use in the whole of [{\scr V}]. Restricting the search to a smaller search subspace of dimension n spanned by [\{{\bf V}_{i}\}_{i = 1, \ldots, n}] we may build local quadratic models of S and C (Bryan & Skilling, 1980[link]; Burch et al., 1983[link]) with respect to n coordinates X in that subspace:[\eqalign{S({\bf X}) &= S({\bf X}_{0}) + {\bf S}_{0}^{T} ({\bf X} - {\bf X}_{0})\cr &\quad + {\textstyle{1 \over 2}}({\bf X} - {\bf X}_{0})^{T} {\bf H}_{0}(S) ({\bf X} - {\bf X}_{0})\cr C({\bf X}) &= C ({\bf X}_{0}) + {\bf C}_{0}^{T}({\bf X} - {\bf X}_{0})\cr &\quad + {\textstyle{1 \over 2}}({\bf X} - {\bf X}_{0})^{T} {\bf H}_{0}(C) ({\bf X} - {\bf X}_{0}).}]The coefficients of these linear models are given by scalar products:[\eqalign{[{\bf S}_{0}]_{i} &= ({\bf V}_{i}, \nabla S)\cr [{\bf C}_{0}]_{i} &= ({\bf V}_{i}, \nabla C)\cr [{\bf H}_{0}(S)]_{ij} &= [{\bf V}_{i}, {\bf H}(S){\bf V}_{j}]\cr [{\bf H}_{0}(C)]_{ij} &= [{\bf V}_{i}, {\bf H}(C){\bf V}_{j}]}]which, by virtue of Parseval's theorem, may be evaluated either in real space or in reciprocal space (Bricogne, 1984[link]). In doing so, special positions and reflections must be taken into account, as in Section[link] Scalar products involving S are best evaluated by real-space grid summation, because H(S) is diagonal in this representation; those involving C are best calculated by reciprocal-space summation, because H(C) is at worst [2 \times 2] block-diagonal in this representation. Using these Hessian matrices in the wrong space would lead to prohibitively expensive convolutions instead of scalar (or at worst [2 \times 2] matrix) multiplications. Derivatives for model refinement

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Since the origins of X-ray 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)[link], and was put into practice in the determination of the structures of topaz (Alston & West, 1929[link]) and diopside (Warren & Bragg, 1929[link]). Later, more systematic methods of arriving at a trial model were provided by the Patterson synthesis (Patterson, 1934[link], 1935a[link],b[link]; Harker, 1936[link]) and by isomorphous replacement (Robertson, 1935[link], 1936c[link]). The role of Fourier syntheses, however, remained essentially unchanged [see Robertson (1937)[link] 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)[link].

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. The method of least squares

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Hughes (1941)[link] was the first to use the already well established multivariate least-squares method (Whittaker & Robinson, 1944[link]) 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)[link], Sparks et al. (1956)[link], Busing & Levy (1961)[link] and others.

The Fourier relations between [\rho\llap{$-\!$}] and F (Section[link]) are used to derive the `observational equations' connecting the structure parameters [\{u_{p}\}_{p = 1, \ldots, n}] to the observations [\{|F_{{\bf h}}|^{\rm obs}, (\sigma_{{\bf h}}^{2})^{\rm obs}\}_{{\bf h} \in {\scr H}}] comprising the amplitudes and their experimental variances for a set [{\scr H}] of unique reflections.

The normal equations giving the corrections δu to the parameters are then[({\bf A}^{T}{\bf WA})\delta {\bf u} = - {\bf A}^{T}{\bf W}\Delta,]where[\eqalign{A_{{\bf h}p} &= {\partial | F_{{\bf h}}^{\,\rm calc}| \over \partial u_{p}}\cr \Delta_{{\bf h}} &= |F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}\cr {\bf W} &= \hbox{diag } (W_{{\bf h}}) \quad \hbox{with} \quad W_{{\bf h}} = {1 \over (\sigma_{{\bf h}}^{2})^{\rm obs}}.}]To calculate the elements of A, write:[F = |F| \exp (i\varphi) = \alpha + i\beta\hbox{\semi}]hence[\eqalign{ {\partial |F| \over \partial u} &= {\partial \alpha \over \partial u} \cos \varphi + {\partial \beta \over \partial u} \sin \varphi\cr &= {\scr Re} \left[{\partial F \over \partial u} \overline{\exp (i\varphi)}\right] = {\scr Re} \left[{\overline{\partial F} \over \partial u} \exp (i\varphi)\right].}]

In the simple case of atoms with real-valued form factors and isotropic thermal agitation in space group P1,[F_{{\bf h}}^{\,\rm calc} = {\textstyle\sum\limits_{j \in J}}\, g_{j} ({\bf h}) \exp (2\pi i{\bf h}\cdot {\bf x}_{j}),]where[g_{j} ({\bf h}) = Z_{j}\,f_{j} ({\bf h}) \exp [-{\textstyle{1 \over 4}} B_{j} (d_{{\bf h}}^{*})^{2}],][Z_{j}] being a fractional occupancy.

Positional derivatives with respect to [{\bf x}_{j}] are given by[\eqalign{ {\partial F_{{\bf h}}^{\,\rm calc} \over \partial {\bf x}_{j}} &= (2\pi i{\bf h}) g_{j} ({\bf h}) \exp (2\pi i{\bf h}\cdot {\bf x}_{j})\cr {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial {\bf x}_{j}} &= {\scr Re} [(- 2\pi i{\bf h}) g_{j} ({\bf h}) \exp (-2 \pi i{\bf h}\cdot {\bf x}_{j}) \exp (i\varphi_{{\bf h}}^{\rm calc})]}]so that the corresponding [3 \times 1] subvector of the right-hand side of the normal equations reads:[\eqalign{&- \sum\limits_{{\bf h}\in {\scr H}} W_{{\bf h}} {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial {\bf x}_{j}} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs})\cr &\quad= - {\scr Re} \left [\sum\limits_{{\bf h}\in {\scr H}} g_{j} ({\bf h}) (-2\pi i{\bf h}) W_{{\bf h}} (|F_{{\bf h}}^{\rm calc}| - |F_{{\bf h}}|^{\rm obs})\right.\cr &\qquad \times \left.\exp (i\varphi_{{\bf h}}^{\rm calc}) \exp (-2\pi i{\bf h}\cdot {\bf x}_{j})\right ].}]

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[link], 1970[link]; Rollett, 1970[link]). It is, however, an expensive task, of complexity [\propto n \times |{\scr H}|^{2}], which is unaffordable for macromolecules. Booth's differential Fourier syntheses

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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 electron-density maxima could be determined. The naive method of peak-shape analysis from densities recalculated on a [3 \times 3 \times 3] grid using high-accuracy trigonometric tables entailed 27 summations per atom.

Booth (1946a)[link] suggested examining the rapidly varying derivatives of the electron density rather than its slowly varying values. If[\rho\llap{$-\!$} ({\bf x}) = {\textstyle\sum\limits_{{\bf h}}} F_{{\bf h}} \exp (-2\pi i{\bf h}\cdot {\bf x})]then the gradient vector [\nabla_{{\bf x}}\rho\llap{$-\!$}] of [\rho\llap{$-\!$}] at [{\bf x}^{0}][(\nabla_{{\bf x}}\rho\llap{$-\!$}) ({\bf x}^{0}) = {\textstyle\sum\limits_{{\bf h}}} \,F_{{\bf h}} (-2\pi i{\bf h}) \exp (-2\pi i{\bf h}\cdot {\bf x}^{0})]can be calculated by means of three Fourier summations from the [3 \times 1] vector of Fourier coefficients[(-2\pi i{\bf h}) F_{{\bf h}}.]Similarly, the Hessian matrix of [\rho\llap{$-\!$}] at [{\bf x}^{0}][[(\nabla_{{\bf x}} \nabla_{{\bf x}}^{T}){\rho\llap{$-\!$}}] ({\bf x}^{0}) = {\textstyle\sum\limits_{{\bf h}}} \,F_{{\bf h}} (-4\pi^{2} {\bf hh}^{T}) \exp (-2\pi i{\bf h}\cdot {\bf x}^{0})]can be calculated by six Fourier summations from the unique elements of the symmetric matrix of Fourier coefficients:[-4\pi^{2} \pmatrix{h^{2} &hk &hl\cr hk &k^{2} &kl\cr hl &kl &l^{2}\cr} F_{{\bf h}}.]

The scalar maps giving the components of the gradient and Hessian matrix of [\rho\llap{$-\!$}] will be called differential syntheses of 1st order and 2nd order respectively. If [{\bf x}^{0}] is approximately but not exactly a maximum of [\rho\llap{$-\!$}], then the Newton–Raphson estimate of the true maximum [{\bf x}^{*}] is given by:[{\bf x}^{*} = {\bf x}^{0} - [[(\nabla_{{\bf x}} \nabla_{{\bf x}}^{T}){\rho\llap{$-\!$}}] ({\bf x}^{0})]^{-1} [\nabla_{\bf x}\rho\llap{$-\!$} ({\bf x}^{0})].]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 series-termination errors in the differential syntheses. Booth (1945c[link], 1946c)[link] proposed a `back-shift correction' to eliminate them, and extended this treatment to the acentric case (Booth, 1946b[link]). He cautioned against the use of an artificial temperature factor to fight series-termination errors (Brill et al., 1939[link]), as this could be shown to introduce coordinate errors by causing overlap between atoms (Booth, 1946c[link], 1947a[link],b[link]).

Cruickshank was able to derive estimates for the standard uncertainties of the atomic coordinates obtained in this way (Cox & Cruickshank, 1948[link]; Cruickshank, 1949a[link],b[link]) and to show that they agreed with those provided by the least-squares method.

The calculation of differential Fourier syntheses was incorporated into the crystallographic programs of Ahmed & Cruickshank (1953b)[link] and of Sparks et al. (1956)[link]. Booth's method of steepest descents

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Having defined the now universally adopted R factors (Booth, 1945b[link]) 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 [\{{\bf x}_{j}\}_{j\in J}] by descending along the gradient of R in parameter space (Booth, 1947c[link],d[link]). This `steepest descents' procedure was compared with Patterson methods by Cochran (1948d)[link].

When calculating the necessary derivatives, Booth (1948a[link], 1949[link]) used the formulae given above in connection with least squares. This method was implemented by Qurashi (1949)[link] and by Vand (1948[link], 1951[link]) with parameter-rescaling modifications which made it very close to the least-squares method (Cruickshank, 1950[link]; Qurashi & Vand, 1953[link]; Qurashi, 1953[link]). Cochran's Fourier method

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Cochran (1948b[link],c[link], 1951a[link]) undertook to exploit an algebraic similarity between the right-hand side of the normal equations in the least-squares 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[link]). In doing so he initiated a remarkable sequence of formal and computational developments which are still actively pursued today.

Let [\rho\llap{$-\!$}_{\rm C} ({\bf x})] be the electron-density map corresponding to the current atomic model, with structure factors [|F_{{\bf h}}^{\,\rm calc}| \exp (i\varphi_{{\bf h}}^{\rm calc})]; and let [\rho\llap{$-\!$}_{\rm O} ({\bf x})] be the map calculated from observed moduli and calculated phases, i.e. with coefficients [\{|F_{{\bf h}}|^{\rm obs} \exp (i\varphi_{{\bf h}}^{\rm calc})\}_{{\bf h}\in {\scr H}}]. If there are enough data for [\rho\llap{$-\!$}_{\rm C}] to have a resolved peak at each model atomic position [{\bf x}_{j}], then[(\nabla_{{\bf x}} \rho\llap{$-\!$}_{\rm C})({\bf x}_{j}) = {\bf 0} \quad \hbox{for each } j \in J\hbox{\semi}]while if the calculated phases [\varphi_{\bf h}^{\rm calc}] are good enough, [\rho\llap{$-\!$}_{\rm O}] will also have peaks at each [{\bf x}_{j}]:[(\nabla_{{\bf x}} \rho\llap{$-\!$}_{\rm O})({\bf x}_{j}) = {\bf 0} \quad \hbox{for each } j \in J.]It follows that[\eqalign{[\nabla_{{\bf x}} (\rho\llap{$-\!$}_{\rm C} - \rho\llap{$-\!$}_{\rm O})] ({\bf x}_{j}) &= {\textstyle\sum\limits_{{\bf h}}} (-2 \pi i{\bf h}) [(|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) \exp (i\varphi_{\bf h}^{\rm calc})]\cr &\quad \times \exp (-2 \pi i{\bf h} \cdot {\bf x}_{j})\cr &= {\bf 0} \hbox{ for each } j \in J,}]where the summation is over all reflections in [{\scr H}] or related to [{\scr H}] by space-group and Friedel symmetry (overlooking multiplicity factors!). This relation is less sensitive to series-termination errors than either of the previous two, since the spectrum of [\rho\llap{$-\!$}_{\rm O}] could have been extrapolated beyond the data in [{\scr H}] by using that of [\rho\llap{$-\!$}_{\rm C}] [as in van Reijen (1942)[link]] without changing its right-hand side.

Cochran then used the identity[{\partial F_{{\bf h}}^{\,\rm calc} \over \partial {\bf x}_{j}} = (2 \pi i{\bf h}) g_{j} ({\bf h}) \exp (2 \pi i{\bf h} \cdot {\bf x}_{j})]in the form[(-2 \pi i{\bf h}) \exp (-2 \pi i{\bf h} \cdot {\bf x}_{j}) = {1 \over g_{j} ({\bf h})} {\overline{\partial F_{{\bf h}}^{\,\rm calc}} \over \partial {\bf x}_{j}}]to rewrite the previous relation as[\eqalign{&[\nabla_{{\bf x}} (\rho\llap{$-\!$}_{\rm C} - \rho\llap{$-\!$}_{\rm O})] ({\bf x}_{j})\cr &\quad= \sum\limits_{{\bf h}} {1 \over g_{j} ({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) {\scr R}{e} \left[{\overline{\partial F_{{\bf h}}^{\,\rm calc}} \over \partial {\bf x}_{j}} \exp (i \varphi_{{\bf h}}^{\rm calc})\right]\cr &\quad= \sum\limits_{{\bf h}} {1 \over g_{j} ({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial {\bf x}_{j}}\cr &\quad= {\bf 0} \quad \hbox{for each } j \in J}](the operation [{\scr Re}][] on the first line being neutral because of Friedel symmetry). This is equivalent to the vanishing of the [3 \times 1] subvector of the right-hand side of the normal equations associated to a least-squares refinement in which the weights would be[W_{{\bf h}} = {1 \over g_{j} ({\bf h})}.]Cochran concluded that, for equal-atom structures with [g_{j} ({\bf h}) = g ({\bf h})] for all j, the positions [{\bf x}_{j}] obtained by Booth's method applied to the difference map [\rho\llap{$-\!$}_{\rm O} - \rho\llap{$-\!$}_{\rm C}] are such that they minimize the residual[{\textstyle{1 \over 2} }\sum\limits_{{\bf h}} {1 \over g({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs})^{2}]with respect to the atomic positions. If it is desired to minimize the residual of the ordinary least-squares method, then the differential synthesis method should be applied to the weighted difference map[\sum\limits_{{\bf h}} {W_{{\bf h}} \over g({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) \exp (i \varphi_{{\bf h}}^{\rm calc}).]He went on to show (Cochran, 1951b[link]) 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)[link] in conjunction with a stereochemical regularization procedure to refine protein structures. Cruickshank's modified Fourier method

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Cruickshank consolidated and extended Cochran's derivations in a series of classic papers (Cruickshank, 1949b[link], 1950[link], 1952[link], 1956[link]). He was able to show that all the coefficients involved in the right-hand side and normal matrix of the least-squares 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[link]).

To ensure rigorous equivalence between the summations over [{\bf h} \in {\scr H}] (in the expressions of least-squares right-hand side and normal matrix elements) and genuine Fourier summations, multiplicity-corrected weights were introduced by:[\eqalign{ w_{{\bf h}} &= {1 \over |G_{{\bf h}}|} W_{{\bf h}}\quad \hbox{ if } {\bf h} \in G{\bf h}\quad \hbox{ with } {\bf h} \in {\scr H},\cr w_{\bf h} &= 0\phantom{{241 \over |G_{{\bf h}}{12 \over 1}|}}\quad \hbox{otherwise},}]where Gh denotes the orbit of h and [G_{{\bf h}}] its isotropy subgroup (Section[link]). Similarly, derivatives with respect to parameters of symmetry-unique atoms were expressed, via the chain rule, as sums over the orbits of these atoms.

Let [p = 1, \ldots, n] be the label of a parameter [u_{p}] belonging to atoms with label j. Then Cruickshank showed that the pth element of the right-hand side of the normal equations can be obtained as [D_{p, \, j} ({\bf x}_{j})], where [D_{p, \, j}] is a differential synthesis of the form[\eqalign{ D_{p, \, j} ({\bf x}) &= {\textstyle\sum\limits_{{\bf h}}} P_{p} ({\bf h}) g_{j} ({\bf h}) w_{{\bf h}} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs})\cr &\quad \times \exp (i\varphi_{{\bf h}}^{\rm calc}) \exp (-2 \pi i{\bf h} \cdot {\bf x})}]with [P_{p}({\bf h})] 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.[\eqalign{ &\quad\hbox{Parameter type}\qquad P (h, k, l)\cr &\overline{\hskip13pc}\cr &\quad x \hbox{ coordinate} \quad \qquad - 2 \pi ih\cr &\quad y \hbox{ coordinate} \quad \qquad - 2 \pi ik\cr &\quad z \hbox{ coordinate} \quad \qquad - 2 \pi il\cr &\quad B \hbox{ isotropic} \quad \qquad \,- {\textstyle{1 \over 4}} (d_{{\bf h}}^{*})^{2}\cr &\quad B^{11} \hbox{ anisotropic} \qquad - h^{2}\cr &\quad B^{12} \hbox{ anisotropic} \qquad - hk\cr &\quad B^{13} \hbox{ anisotropic} \qquad - hl\cr &\quad B^{22} \hbox{ anisotropic} \qquad - k^{2}\cr &\quad B^{23} \hbox{ anisotropic} \qquad - kl\cr &\quad B^{33} \hbox{ anisotropic} \qquad - l^{2}.}]Unlike Cochran's original heuristic argument, this result does not depend on the atoms being resolved.

Cruickshank (1952)[link] also considered the elements of the normal matrix, of the form[\sum\limits_{{\bf h}} w_{{\bf h}} {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial u_{p}} {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial u_{q}}]associated with positional parameters. The [3 \times 3] block for parameters [{\bf x}_{j}] and [{\bf x}_{k}] may be written[\eqalign{&{\textstyle\sum\limits_{{\bf h}}} w_{{\bf h}} ({\bf hh}^{T}) {\scr Re} [(-2 \pi i) g_{j} ({\bf h}) \exp (-2 \pi i{\bf h} \cdot {\bf x}_{j}) \exp (i \varphi_{{\bf h}}^{\rm calc})]\cr &\quad \times {\scr Re} [(-2 \pi i) g_{k} ({\bf h}) \exp (-2 \pi i{\bf h} \cdot {\bf x}_{k}) \exp (i \varphi_{{\bf h}}^{\rm calc})]}]which, using the identity[{\scr Re}(z_{1}) {\scr Re}(z_{2}) = {\textstyle{1 \over 2}}[{\scr Re} (z_{1} z_{2}) + {\scr Re} (z_{1} \overline{z_{2}})],]becomes[\eqalign{&2\pi^{2} {\textstyle\sum\limits_{{\bf h}}} w_{{\bf h}} ({\bf hh}^{T}) g_{j}({\bf h})g_{k}({\bf h})\cr &\quad \times \{\exp [-2\pi i{\bf h} \cdot ({\bf x}_{j} - {\bf x}_{k})]\cr &\quad - \exp (2i\varphi_{{\bf h}}^{\rm calc}) \exp [ - 2\pi i{\bf h} \cdot ({\bf x}_{j} + {\bf x}_{k})]\}}](Friedel's symmetry makes [{\scr Re}] 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 off-diagonal blocks for which [{\bf x}_{j}] and [{\bf x}_{k}] are close. On this basis he was able to justify the `n-shift rule' of Shoemaker et al. (1950)[link]. Cruickshank gave this derivation in a general space group, but using a very terse notation which somewhat obscures it. Using the symmetrized trigonometric structure-factor kernel [\Xi^{-}] of Section[link] and its multiplication formula, the above expression is seen to involve the values of a Fourier synthesis at points of the form [{\bf x}_{j} \pm S_{g} ({\bf x}_{k})].

Cruickshank (1956)[link] 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 [{\bf x}_{j}] or at [{\bf x}_{j} \pm S_{g} ({\bf x}_{k})]. At the time this did not confer any superiority over the standard form of the least-squares procedure, because the accurate computation of Fourier syntheses was an expensive operation. The modified Fourier method was used by Truter (1954)[link] and by Ahmed & Cruickshank (1953a)[link], and was incorporated into the program system described by Cruickshank et al. (1961)[link]. A more recent comparison with the least-squares method was made by Dietrich (1972)[link].

There persisted, however, some confusion about the nature of the relationship between Fourier and least-squares methods, caused by the extra factors [g_{j} ({\bf h})] 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 least-squares and modified-Fourier methods are fundamentally distinct'. Agarwal's FFT implementation of the Fourier method

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Agarwal (1978)[link] 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

  • – for each type of parameter, since this determines [via the polynomial [P_{p} ({\bf h})]] the type of differential synthesis to be computed;

  • – for each type of atom [j \in J], since the coefficients of the differential synthesis must be multiplied by [g_{j}({\bf h})].

Agarwal disposed of the latter dependence by pointing out that the multiplication involved is equivalent to a real-space convolution between the differential synthesis and [\sigma\llap{$-$}_{j} ({\bf x})], the standard electron density [\rho\llap{$-\!$}_{j}] for atom type j (Section[link]) smeared by the isotropic thermal agitation of that atom. Since [\sigma\llap{$-$}_{j}] 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 space-group symmetries. Jack & Levitt (1978)[link] 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 least-squares method led to spectacular increases in speed (Isaacs & Agarwal, 1978[link]; Agarwal, 1980[link]; Baker & Dodson, 1980[link]) and quickly gained general acceptance (Dodson, 1981[link]; Isaacs, 1982a[link],b[link], 1984[link]). Lifchitz's reformulation

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Lifchitz [see Agarwal et al. (1981)[link], Agarwal (1981)[link]] 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 [g_{j} ({\bf h})], but also to the polynomials [P_{p} ({\bf h})] which determine the type of differential synthesis being calculated. This leads to convoluting [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] with the same ordinary weighted difference Fourier synthesis, rather than [\sigma\llap{$-$}_{j}] 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 [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] with which it is convoluted. This approach has been used as the basis of an efficient general-purpose least-squares refinement program for macromolecular structures (Tronrud et al., 1987[link]).

This rearrangement amounts to using the fact (Section[link]) that convolution commutes with differentiation. Let[D({\bf x}) = {\textstyle\sum\limits_{{\bf h}}} w_{{\bf h}}(|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) \exp (i\varphi_{{\bf h}}^{\rm calc}) \exp (-2\pi i {\bf h} \cdot {\bf x})]be the inverse-variance weighted difference map, and let us assume that parameter [u_{p}] belongs to atom j. Then the Agarwal form for the pth component of the right-hand side of the normal equations is[\left({\partial D \over \partial u_{p}} * \sigma\llap{$-$}_{j}\right)(x_{j}),]while the Lifchitz form is[\left(D * {\partial \sigma\llap{$-$}_{j} \over \partial u_{p}}\right)({\bf x}_{j}).] A simplified derivation

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A very simple derivation of the previous results will now be given, which suggests the possibility of many generalizations.

The weighted difference map [D({\bf x})] has coefficients [D_{{\bf h}}] which are the gradients of the global residual with respect to each [F_{{\bf h}}^{\,\rm calc}]:[D_{{\bf h}} = {\partial R \over \partial A_{{\bf h}}^{\rm calc}} + i {\partial R \over \partial B_{{\bf h}}^{\rm calc}}.]By the chain rule, a variation of each [F_{{\bf h}}^{\,\rm calc}] by [\delta F_{{\bf h}}^{\,\rm calc}] will result in a variation of R by [\delta R] with[\delta R = \sum\limits_{{\bf h}} \left[{\partial R \over \partial A_{{\bf h}}^{\rm calc}} \delta A_{{\bf h}}^{\rm calc} + {\partial R \over \partial B_{{\bf h}}^{\rm calc}} \delta B_{{\bf h}}^{\rm calc}\right] = {\scr Re} \sum\limits_{{\bf h}} [\overline{D_{{\bf h}}} \delta F_{{\bf h}}^{\,\rm calc}].]The [{\scr Re}] operation is superfluous because of Friedel symmetry, so that [\delta R] may be simply written in terms of the Hermitian scalar product in [\ell^{2}({\bb Z}^{3})]:[\delta R = ({\bf D}, \delta {\bf F}^{\rm calc}).]If [\rho\llap{$-\!$}^{\rm calc}] is the transform of [\delta {\bf F}^{\rm calc}], we have also by Parseval's theorem[\delta R = (D, \delta \rho\llap{$-\!$}^{\rm calc}).]We may therefore write[D ({\bf x}) = {\partial R \over \partial \rho\llap{$-\!$}^{\,\rm calc} ({\bf x})},]which states that [D({\bf x})] is the functional derivative of R with respect to [\rho\llap{$-\!$}^{\rm calc}].

The right-hand side of the normal equations has [\partial R/\partial u_{p}] for its pth element, and this may be written[{\partial R \over \partial u_{p}} = \int_{{\bb R}^{3}/{\bb Z}^{3}} {\partial R \over \partial \rho\llap{$-\!$}^{\,\rm calc}({\bf x})} {\partial \rho\llap{$-\!$}^{\rm calc}({\bf x}) \over \partial u_{p}} \hbox{d}^{2}{\bf x} = \left(D, {\partial \rho\llap{$-\!$}^{\rm calc} \over \partial u_{p}}\right).]If [u_{p}] belongs to atom j, then[{\partial \rho\llap{$-\!$}^{\rm calc} \over \partial u_{p}} = {\partial (\tau_{{\bf x}_{j}} \sigma_{j}) \over \partial u_{p}} = \tau_{{\bf x}_{j}} \left({\partial \sigma\llap{$-$}_{j} \over \partial u_{p}}\right)\hbox{\semi}]hence[{\partial R \over \partial u_{p}} = \left(D, \tau_{{\bf x}_{j}} \left({\partial \sigma\llap{$-$}_{j} \over \partial u_{p}}\right)\right).]By the identity of Section[link], this is identical to Lifchitz's expression [(D * \partial \sigma\llap{$-$}_{j}/\partial u_{p})({\bf x}_{j})]. The present derivation in terms of scalar products [see Brünger (1989)[link] 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)[link]] and Parseval's theorem; economy of computation is obviously related to the good localization of [\partial \rho\llap{$-\!$}^{\rm calc}/\partial u_{p}] compared to [\partial {F}^{\,\rm calc}/\partial u_{p}]. 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[link]) 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[link]; Brünger et al., 1987[link]; Brünger, 1988[link]; Brünger et al., 1989[link]; Kuriyan et al., 1989[link]) or stereochemical (Konnert, 1976[link]; Sussman et al., 1977[link]; Konnert & Hendrickson, 1980[link]; Hendrickson & Konnert, 1980[link]; Tronrud et al., 1987[link]) 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 second-order approximation, albeit only in a subspace, hence the ability of detecting negative curvature and the resulting bifurcation behaviour (Bricogne, 1984[link]). Current methods are unable to do this because they use only first-order models, and this is known to degrade severely the overall efficiency of the refinement process. Discussion of macromolecular refinement techniques

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The impossibility of carrying out a full-matrix least-squares refinement of a macromolecular crystal structure, caused by excessive computational cost and by the paucity of observations, led Diamond (1971)[link] to propose a real-space refinement method in which stereochemical knowledge was used to keep the number of free parameters to a minimum. Refinement took place by a least-squares fit between the `observed' electron-density 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 full-matrix least squares (Huber et al., 1974[link]; Bode & Schwager, 1975[link]; Deisenhofer & Steigemann, 1975[link]; Takano, 1977a[link],b[link]).

Real-space 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 series-termination errors and accidentally missing data than the least-squares method. These objections led to the progressive disuse of Diamond's method, and to a switch towards reciprocal-space 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:

  • (i) it shows that refinement can be performed through localized computations in real space without having to treat the phases as observed quantities;

  • (ii) at the same time, it shows that measurement errors on the moduli can be fully utilized in real space, via the Fourier synthesis of the functional derivative [\partial R/\partial \rho\llap{$-\!$}^{\rm calc}({\bf x})] or by means of the coefficients of a quadratic model of R in a search subspace. Sampling considerations

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The calculation of the inner products [(D, \partial \rho\llap{$-\!$}^{\rm calc}/\partial u_{p})] from a sampled gradient map D requires even more caution than that of structure factors via electron-density maps described in Section[link], because the functions [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] have transforms which extend even further in reciprocal space than the [\sigma\llap{$-$}_{j}] themselves. Analytically, if the [\sigma\llap{$-$}_{j}] are Gaussians, the [\partial \sigma\llap{$-$}_{j}/\partial u_{p}] are finite sums of multivariate Hermite functions (Section[link]) 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[link] An examination of the sampling rates commonly used (e.g. one third of the resolution) shows that they are insufficient. Tronrud et al. (1987)[link] propose to relax this requirement by applying an artificial temperature factor to [\sigma\llap{$-$}_{j}] (cf. Section[link]) 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 [(\|{\bf x}\|^{2})] 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. Miscellaneous correlation functions

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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 [{\scr M}] in its asymmetric unit. If [\mu({\bf x})] is the electron density of that molecule in some reference position and orientation, then[\rho\llap{$-\!$}^{0} = {\textstyle\sum\limits_{j \in J}} \left[{\textstyle\sum\limits_{g \in G}} S_{g}^{\#} (T_{j}^{\#} \mu)\right],]where [T_{j}: {\bf x} \,\longmapsto\, {\bf C}_{j} {\bf x} + {\bf d}_{j}] 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[link] (with [\rho\llap{$-\!$}_{j}] replaced by [C_{j}^{\#} \mu], and [{\bf x}_{j}] by [{\bf d}_{j}]) lead to the following expression for the auto-correlation of [\rho\llap{$-\!$}^{0}]:[\eqalign{ \breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0} &= {\textstyle\sum\limits_{j_{1}}} {\textstyle\sum\limits_{j_{2}}} {\textstyle\sum\limits_{g_{1}}} {\textstyle\sum\limits_{g_{2}}} \boldtau_{{S_{g_{2}}} ({\bf d}_{j_{2}}) - s_{g_{1}} ({\bf d}_{j_{1}})}\cr &\quad \times [(R_{g_{1}}^{\#} C_{j_{1}}^{\#} \breve{\mu}) * (R_{g_{2}}^{\#} C_{j_{2}}^{\#} \mu)].}]

If μ is unknown, consider the subfamily σ of terms with [j_{1} = j_{2} = j] and [g_{1} = g_{2} = g]:[\sigma = {\textstyle\sum\limits_{j}} {\textstyle\sum\limits_{g}} \,R_{g}^{\#} C_{j}^{\#} (\breve{\mu} * \mu).]The scalar product [(\sigma, R^{\#} \sigma)] in which R is a variable rotation will have a peak whenever[R = (R_{g_{1}} C_{j_{1}})^{-1} (R_{g_{2}} C_{j_{2}})]since two copies of the `self-Patterson' [\breve{\mu} * \mu] of the molecule will be brought into coincidence. If the interference from terms in the Patterson [\pi = r * \breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0}] other than those present in σ is not too serious, the `self-rotation function' [(\pi, R^{\#} \pi)] (Rossmann & Blow, 1962[link]; Crowther, 1972[link]) will show the same peaks, from which the rotations [\{C_{j}\}_{j \in J}] may be determined, either individually or jointly if for instance they form a group.

If μ is known, then its self-Patterson [\breve{\mu} * \mu] may be calculated, and the [C_{j}] may be found by examining the `cross-rotation function' [[\pi, R^{\#} (\breve{\mu} * \mu)]] which will have peaks at [R = R_{g} C_{j}], [g \in G, j \in J]. Once the [C_{j}] are known, then the various copies [C_{j}^{\#} \mu] of [{\scr M}] may be Fourier-analysed into structure factors:[M_{j} ({\bf h}) = \bar{{\scr F}}[C_{j}^{\#} \mu] ({\bf h}).]The cross terms with [j_{1} \neq j_{2}, g_{1} \neq g_{2}] in [\breve{\rho\llap{$-\!$}}^{0} * \rho\llap{$-\!$}^{0}] then contain `motifs'[(R_{g_{1}}^{\#} C_{j_{1}}^{\#} \breve{\mu}) * (R_{g_{2}}^{\#} C_{j_{2}}^{\#} \mu),]with Fourier coefficients[\overline{M_{j_{1}} ({\bf R}_{g_{1}}^{T} {\bf h})} \times M_{j_{2}} ({\bf R}_{g_{2}}^{T} {\bf h}),]translated by [S_{g_{2}} ({\bf d}_{j_{2}}) - S_{g_{1}} ({\bf d}_{j_{1}})]. Therefore the `translation functions' (Crowther & Blow, 1967[link])[\eqalign{ {\scr T}_{j_{1} g_{1}, j_{2} g_{2}} ({\bf s}) &= {\textstyle\sum\limits_{{\bf h}}} |F_{{\bf h}}|^{2} \overline{M_{j_{1}} ({\bf R}_{g_{1}}^{T} {\bf h})}\cr &\quad \times M_{j_{2}} ({\bf R}_{g_{2}}^{T} {\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf s})}]will have peaks at [{\bf s} = S_{g_{2}} ({\bf d}_{j_{2}}) - S_{g_{1}} ({\bf d}_{j_{1}})] corresponding to the detection of these motifs.


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