Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.3, pp. 255-257   | 1 | 2 |

Section 2.3.4. Anomalous dispersion

L. Tong,c* M. G. Rossmanna and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA,bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854–5638, USA, and cDepartment of Biological Sciences, Columbia University, New York 10027, USA
Correspondence e-mail:

2.3.4. Anomalous dispersion

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The physical basis for anomalous dispersion has been well reviewed by Ramaseshan & Abrahams (1975)[link], James (1965)[link], Cromer (1974)[link] and Bijvoet (1954)[link]. As the wavelength of radiation approaches the absorption edge of a particular element, then an atom will disperse X-rays in a manner that can be defined by the complex scattering factor [f_{0} + \Delta f' + i\Delta f'',]where [f_{0}] is the scattering factor of the atom without the anomalous absorption and rescattering effect, [\Delta f'] is the real correction term (usually negative), and [\Delta f''] is the imaginary component. The real term [f_{0} + \Delta f'] is often written as f′, so that the total scattering factor will be [f' + i\Delta f'']. Values of [\Delta f'] and [\Delta f''] are tabulated in IT IV (Cromer, 1974[link]), although their precise values are dependent on the environment of the anomalous scatterer. Unlike [f_{0}], [\Delta f'] and [\Delta f''] are almost independent of scattering angle as they are caused by absorption of energy in the innermost electron shells. Thus, the anomalous effect resembles scattering from a point atom.

The structure factor of index h can now be written as [{\bf F}_{{\bf h}} = {\textstyle\sum\limits_{j = 1}^{N}}\, f'_{j} \exp (2\pi i{\bf h}\cdot {\bf x}_{j}) + i {\textstyle\sum\limits_{j = 1}^{N}}\, \Delta f''_{j} \exp (2\pi i{\bf h}\cdot {\bf x}_{j}). \eqno(](Note that the only significant contributions to the second term are from those atoms that have a measurable anomalous effect at the chosen wavelength.)

Let us now write the first term as [A + iB] and the second as [a + ib]. Then, from (,[link] [{\bf F} = (A + iB) + i(a + ib) = (A - b) + i(B + a). \eqno(]Therefore, [|{\bf F}_{{\bf h}}|^{2} = (A - b)^{2} + (B + a)^{2}]and similarly [|{\bf F}_{{{\bar {\bf h}}}}|^{2} = (A + b)^{2} + (- B + a)^{2},]demonstrating that Friedel's law breaks down in the presence of anomalous dispersion. However, it is only for noncentrosymmetric reflections that [|{\bf F}_{{\bf h}}| \neq |{\bf F}_{{\bar{\bf h}}}|].

Now, [\rho ({\bf x}) = {1 \over V} {\sum\limits_{{\bf h}}^{\rm sphere}} {\bf F}_{{\bf h}} \exp (2\pi i{\bf h}\cdot {\bf x}).]Hence, by using ([link] and simplifying, [\eqalignno{\rho ({\bf x}) = &{2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} [(A \cos 2\pi {\bf h}\cdot {\bf x} - B \sin 2\pi {\bf h}\cdot {\bf x})\cr &+ i(a \cos 2\pi {\bf h}\cdot {\bf x} - b \sin 2\pi {\bf h}\cdot {\bf x})]. &(}]The first term in ([link] is the usual real Fourier expression for electron density, while the second term is an imaginary component due to the anomalous scattering of a few atoms in the cell. The [P_{\rm s}({\bf u})] function

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Expression ([link] gives the complex electron density expression in the presence of anomalous scatterers. A variety of Patterson-type functions can be derived from ([link] for the determination of a structure. For instance, the [P_{\rm s} ({\bf u})] function gives vectors between the anomalous atoms and the `normal' atoms.

From ([link] it is easy to show that [\eqalign{{\bf F}_{{\bf h}} {\bf F}_{{\bf h}}^{*} = &\ |{\bf F}_{{\bf h}}|^{2}\cr = &\ {\textstyle\sum\limits_{i, \, j}} (f'_{i} f'_{j} + f''_{i} f''_{j}) \cos 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j})\cr &+ {\textstyle\sum\limits_{i, \, j}} (f'_{i} f''_{j} - f''_{i} f'_{j}) \sin 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j}).}]Therefore, [|{\bf F}_{{\bf h}}|^{2} + |{\bf F}_{{\bar{\bf h}}}|^{2} = 2 {\textstyle\sum\limits_{i, \, j}} (f'_{i} f'_{j} + f''_{i} f''_{j}) \cos 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j})]and [|{\bf F}_{{\bf h}}|^{2} - |{\bf F}_{{\bar{\bf h}}}|^{2} = 2 {\textstyle\sum\limits_{i, \, j}} (f'_{i} f''_{j} - f''_{i} f'_{j}) \sin 2\pi {\bf h}\cdot ({\bf x}_{i} - {\bf x}_{j}).]

Let us now consider the significance of a Patterson in the presence of anomalous dispersion. The normal Patterson definition is given by [\eqalign{ P({\bf u}) &= {\textstyle\int\limits_{V}} \rho^{*} ({\bf x}) \rho ({\bf x} + {\bf u})\,\hbox{d}{\bf x}\cr &= {1 \over V^{2}} {\sum\limits_{{\bf h}}^{\rm sphere}} |{\bf F}_{{\bf h}}|^{2} \exp (-2 \pi i {\bf h} \cdot {\bf u})\cr &\equiv P_{\rm c} ({\bf u}) - iP_{\rm s} ({\bf u}),}]where [P_{\rm c} ({\bf u}) = {2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} (|{\bf F}_{{\bf h}}|^{2} + |{\bf F}_{{\bar{\bf h}}}|^{2}) \cos 2 \pi {\bf h} \cdot {\bf u}]and [P_{\rm s} ({\bf u}) = {2 \over V} {\sum\limits_{{\bf h}}^{\rm hemisphere}} (|{\bf F}_{{\bf h}}|^{2} - |{\bf F}_{{\bar{\bf h}}}|^{2}) \sin 2 \pi {\bf h} \cdot {\bf u}.]

The [P_{\rm c}({\bf u})] component is essentially the normal Patterson, in which the peak heights have been very slightly modified by the anomalous-scattering effect. That is, the peaks of [P_{\rm c}({\bf u})] are proportional in height to [(f'_{i} f'_{j} + f''_{i} f''_{j})].

The [P_{\rm s} ({\bf u})] component is more interesting. It represents vectors between the normal atoms in the unit cell and the anomalous scatterers, proportional in height to [(f'_{i} f''_{j} - f''_{i} f'_{j})] (Okaya et al., 1955[link]). This function is antisymmetric with respect to the change of the direction of the diffraction vector. An illustration of the function is given in Fig.[link]. In a unit cell containing N atoms, n of which are anomalous scatterers, the [P_{\rm s}({\bf u})] function contains only [n(N - n)] positive peaks and an equal number of negative peaks related to the former by anticentrosymmetry. The analysis of a [P_{\rm s}({\bf u})] synthesis presents problems somewhat similar to those posed by a normal Patterson. The procedure has been used only rarely [cf. Moncrief & Lipscomb (1966)[link] and Pepinsky et al. (1957)[link]], probably because alternative procedures are available for small compounds and the solution of [P_{\rm s}({\bf u})] is too complex for large biological molecules.


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(a) A model structure with an anomalous scatterer at A. (b) The corresponding [P_{\rm s}({\bf u})] function showing positive peaks (full lines) and negative peaks (dashed lines). [Reprinted with permission from Woolfson (1970, p. 293)[link].] The position of anomalous scatterers

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Anomalous scatterers can be used as an aid to phasing, when their positions are known. But the anomalous-dispersion differences (Bijvoet differences) can also be used to determine or confirm the heavy atoms which scatter anomalously (Rossmann, 1961a[link]). Furthermore, the use of anomalous-dispersion information obviates the lack of isomorphism but, on the other hand, the differences are normally far smaller than those produced by a heavy-atom isomorphous replacement.

Consider a structure of many light atoms giving rise to the structure factor [{\bf F}_{{\bf h}} (N)]. In addition, it contains a few heavy atoms which have a significant anomalous-scattering effect. The non-anomalous component will be [{\bf F}_{{\bf h}} (H)] and the anomalous component is [{\bf F}''_{{\bf h}} (H) = i(\Delta f''/ f') {\bf F}_{{\bf h}} (H)] (Fig.[link]). The total structure factor will be [{\bf F}_{{\bf h}}]. The Friedel opposite is constructed appropriately (Fig.[link]). Now reflect the Friedel opposite construction across the real axis of the Argand diagram (Fig.[link]). Let the difference in phase between [{\bf F}_{{\bf h}}] and [{\bf F}_{{\bar{\bf h}}}] be ϕ. Thus [4 |{\bf F}''_{\bf h} (H)|^{2} = |{\bf F}_{\bf h}|^{2} + |{\bf F}_{{\bar{\bf h}}}|^{2} - 2|{\bf F}_{\bf h}| |{\bf F}_{{\bar{\bf h}}}| \cos \varphi,]but since ϕ is very small [|{\bf F}''_{\bf h} (H)|^{2} \simeq {\textstyle{1 \over 4}} (|{\bf F}_{\bf h}| - |{\bf F}_{{\bar{\bf h}}}|)^{2}.]Hence, a Patterson with coefficients [(|{\bf F}_{\bf h}| - |{\bf F}_{{\bar{\bf h}}}|)^{2}] will be equivalent to a Patterson with coefficients [|{\bf F}''_{\bf h} (H)|^{2}] which is proportional to [|{\bf F}_{\bf h} (H)|^{2}]. Such a Patterson (Rossmann, 1961a[link]) will have vectors between all anomalous scatterers with heights proportional to the number of anomalous electrons [\Delta f'']. This `anomalous dispersion' Patterson function has been used to find anomalous scatterers such as iron (Smith et al., 1983[link]; Strahs & Kraut, 1968[link]) and sulfur atoms (Hendrickson & Teeter, 1981[link]). The anomalous signal from Se atoms in selenomethionine-substituted proteins has been found to be extremely powerful and is now routinely used for protein structure determinations (Hendrickson, 1991[link]). Anomalous signals from halide ions or xenon atoms have also been used to solve protein structures (Dauter et al., 2000[link]; Nagem et al., 2003[link]; Schiltz et al., 2003[link]). The anomalous signal from sulfur atoms, though very small (Hendrickson & Teeter, 1981[link]), has recently been applied successfully to solve several protein structures (Debreczeni et al., 2003[link]; Ramagopal et al., 2003[link]; Yang et al., 2003[link]).


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Anomalous-dispersion effect for a molecule whose light atoms contribute [{\bf F}_{\bf h}(N)] and heavy atom [{\bf F}_{\bf h}(H)] with a small anomalous component of [{\bf F_h''}(H)], [90^{\circ}] ahead of the non-anomalous [{\bf F}_{\bf h}(H)] component. In (a) is seen the construction for [{\bf F}_{\bf h}] and [{\bf F}_{\bar{\bf h}}]. In (b) [{\bf F}_{\bar {\bf h}}] has been reflected across the real axis.

It is then apparent that a Patterson with coefficients [\Delta F_{\rm ANO}^{2} = (|{\bf F}_{\bf h}| - |{\bf F}_{{\bar{\bf h}}}|)^{2}](Rossmann, 1961a[link]), as well as a Patterson with coefficients [\Delta F_{\rm ISO}^{2} = (|{\bf F}_{\rm NH}| - |{\bf F}_{\rm H}|)^{2}](Rossmann, 1960[link]; Blow, 1958[link]), represent Pattersons of the heavy atoms. The [\Delta F_{\rm ANO}^{2}] Patterson suffers from errors which may be larger than the size of the Bijvoet differences, while the [\Delta F_{\rm ISO}^{2}] Patterson may suffer from partial lack of isomorphism. Hence, Kartha & Parthasarathy (1965)[link] have suggested the use of the sum of these two Pattersons, which would then have coefficients [(\Delta F_{\rm ANO}^{2} + \Delta F_{\rm ISO}^{2})].

However, given both SIR and anomalous-dispersion data, it is possible to make an accurate estimate of the [|{\bf F}_{\rm H}|^{2}] magnitudes for use in a Patterson calculation [Blundell & Johnson (1976, p. 340)[link], Matthews (1966)[link], Singh & Ramaseshan (1966)[link]]. In essence, the Harker phase diagram can be constructed out of three circles: the native amplitude and each of the two isomorphous Bijvoet differences, giving three circles in all (Blow & Rossmann, 1961[link]) which should intersect at a single point thus resolving the ambiguity in the SIR data and the anomalous-dispersion data. Furthermore, the phase ambiguities are orthogonal; thus the two data sets are cooperative. It can be shown (Matthews, 1966[link]; North, 1965[link]) that [F^{2}_{\rm N} = F^{2}_{\rm NH} + F^{2}_{\rm N} \mp {2 \over k} (16k^{2} F^{2}_{\rm P} F^{2}_{\rm H} - \Delta I^{2})^{1/2},]where [\Delta I = {F^{+}_{\rm NH}}^{2} - {F^{-}_{\rm NH}}^{2}] and [k = \Delta f''/f']. The sign in the third-term expression is − when [|(\alpha_{\rm NH} - \alpha_{\rm H})| \,\lt\, \pi/2] or + otherwise. Since, in general, [|{\bf F}_{\rm H}|] is small compared to [|{\bf F}_{\rm N}|], it is reasonable to assume that the sign above is usually negative. Hence, the heavy-atom lower estimate (HLE) is usually written as [F^{2}_{\rm HLE} = F^{2}_{\rm NH} + F^{2}_{\rm H} - {2 \over k} (16 k^{2} F^{2}_{\rm P} F^{2}_{\rm H} - \Delta I^{2})^{1/2},]which is an expression frequently used to derive Patterson coefficients useful in the determination of heavy-atom positions when both SIR and anomalous-dispersion data are available. Computer programs for automated location of atomic positions from Patterson maps

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Several programs are currently used for automated systematic interpretation of (difference) Patterson maps to locate the positions of heavy atoms and/or anomalous scatterers from isomorphous replacement and anomalous-dispersion data (Weeks et al., 2003[link]). These include Solve (Terwilliger & Berendzen, 1999[link]), CNS (Brünger et al., 1998[link]), CCP4 (Collaborative Computational Project, Number 4, 1994[link]) and Patsol (Tong & Rossmann, 1993[link]). In these programs, sets of trial atomic positions (seeds) are produced based on one- and two-atom solutions to the Patterson map (see Section[link]) (Grosse-Kunstleve & Brunger, 1999[link]; Terwilliger et al., 1987[link]; Tong & Rossmann, 1993[link]). Information from a translation search with a single atom can also be used in this process (Grosse-Kunstleve & Brunger, 1999[link]). Scoring functions have been devised to identify the likely correct solutions, based on agreements with the Patterson map or the observed isomorphous or anomalous differences, as well as the quality of the resulting electron-density map (Terwilliger, 2003b[link]; Terwilliger & Berendzen, 1999[link]). The power of modern computers allows the rapid screening of a large collection of trial structures, and the correct solution is found automatically in many cases, even when there is a large number of atomic positions (Weeks et al., 2003[link]).


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