Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 2.3, pp. 258-260   | 1 | 2 |

Section 2.3.7. Translation functions

M. G. Rossmanna* and E. Arnoldb

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

2.3.7. Translation functions

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The problem of determining the position of a noncrystallographic symmetry element in space, or the position of a molecule of known orientation in a unit cell, has been reviewed by Rossmann (1972)[link], Colman et al. (1976)[link], Karle (1976)[link], Argos & Rossmann (1980)[link], Harada et al. (1981)[link] and Beurskens (1981)[link]. All methods depend on the prior knowledge of the object's orientation implied by the rotation matrix [C]. The various translation functions, T, derived below, can only be computed given this information.

The general translation function can be defined as [T({\bf S}_{x}, {\bf S}_{x'}) = {\textstyle\int\limits_{U}} \rho_{1} ({\bf x}) \cdot \rho_{2} ({\bf x}')\;\hbox{d}{\bf x},] where T is a six-variable function given by each of the three components that define [{\bf S}_{x}] and [{\bf S}_{x'}]. Here [{\bf S}_{x}] and [{\bf S}_{x'}] are equivalent reference positions of the objects, whose densities are [\rho_{1}({\bf x})] and [\rho_{2}({\bf x}')]. The translation function searches for the optimal overlap of the two objects after they have been similarly oriented. Following the same procedure used for the rotation-function derivation, Fourier summations are substituted for [\rho_{1}({\bf x})] and [\rho_{2}({\bf x}')]. It can then be shown that [\eqalign{ T({\bf S}_{x}, {\bf S}_{x'}) &= \int\limits_{U} \left\{{1 \over V_{\bf h}} {\sum\limits_{\bf h}} |{\bf F}_{\bf h}| \exp [i(\alpha_{\bf h} - 2\pi {\bf h} \cdot {\bf x})]\right\}\cr &\quad \times \left\{{1 \over V_{\bf p}} {\sum\limits_{\bf p}} |{\bf F}_{\bf p}| \exp [i(\alpha_{\bf p} - 2\pi {\bf p} \cdot {\bf x}')]\right\}\;\hbox{d}{\bf x}.}]

Using the substitution [{\bf x}' = [{\bi C}]{\bf x} + {\bf d}] and simplifying leads to [\eqalign{ T({\bf S}_{x}, {\bf S}_{x}') &= {1 \over V_{\bf h} V_{\bf p}} {\sum\limits_{\bf h}} {\sum\limits_{\bf p}} |{\bf F}_{\bf h}| |{\bf F}_{\bf p}|\cr &\quad \times \exp [i(\alpha_{\bf h} + \alpha_{\bf p} - 2\pi {\bf p} \cdot {\bf d})]\cr &\quad \times {\int\limits_{U}} \exp \{-2\pi i({\bf h} + [{\bi C}]^T{\bf p}) \cdot {\bf x}\}\;\hbox{d}{\bf x}.}] The integral is the diffraction function [G_{\bf hp}] ([link]. If the integration is taken over the volume U, centred at [{\bf S}_{x}] and [{\bf S}_{x'}], it follows that [\eqalignno{ T({\bf S}_{x}, {\bf S}_{x'}) &= {2 \over V_{\bf h} V_{\bf p}} {\sum\limits_{\bf h}} {\sum\limits_{\bf p}} |{\bf F}_{\bf h}| |{\bf F}_{\bf p}| G_{\bf hp}\cr &\quad \times \cos [\alpha_{\bf h} + \alpha_{\bf p} - 2\pi ({\bf h}\cdot {\bf S}_{x} + {\bf p}\cdot {\bf S}_{x'})]. &(}] Position of a noncrystallographic element relating two unknown structures

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The function ([link] is quite general. For instance, the rotation function corresponds to a comparison of Patterson functions [P_{1}] and [P_{2}] at their origins. That is, the coefficients are [F^{2}], phases are zero and [{\bf S}_{x} = {\bf S}_{x'} = 0]. However, the determination of the translation between two objects requires the comparison of cross-vectors away from the origin.

Consider, for instance, the determination of the precise translation vector parallel to a rotation axis between two identical molecules of unknown structure. For simplicity, let the noncrystallographic axis be a dyad (Fig.[link]. Fig.[link] shows the corresponding Patterson of the hypothetical point-atom structure. Opposite sets of cross-Patterson vectors in Fig.[link] are related by a twofold rotation and a translation equal to twice the precise vector in the original structure. A suitable translation function would then compare a Patterson at S with the rotated Patterson at [-{\bf S}]. Hence, substituting [{\bf S}_{x} = {\bf S}] and [{\bf S}_{x'} = - {\bf S}] in ([link], [T({\bf S}) = {2 \over V^{2}} {\sum\limits_{\bf h}} {\sum\limits_{\bf p}} |{\bf F}_{\bf h}|^{2} |{\bf F}_{\bf p}|^{2} G_{\bf hp} \cos [2\pi ({\bf h} - {\bf p})\cdot {\bf S}]. \eqno(]


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Crosses represent atoms in a two-dimensional model structure. The triangles are the points chosen as approximate centres of molecules A and B. [\Delta^{AB}] has components t and s parallel and perpendicular, respectively, to the screw rotation axis. [Reprinted from Rossmann et al. (1964)[link].]


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Vectors arising from the structure in Fig.[link]. The self-vectors of molecules A and B are represented by + and ·; the cross-vectors from molecules A to B and B to A by × and ○. Triangles mark the position of [+\Delta^{AB}] and [-\Delta^{AB}]. [Reprinted from Rossmann et al. (1964)[link].]

The opposite cross-vectors can be superimposed only if an evenfold rotation between the unknown molecules exists. The translation function ([link] is thus applicable only in this special situation. There is no published translation method to determine the interrelation of two unknown structures in a crystallographic asymmetric unit or in two different crystal forms. However, another special situation exists if a molecular evenfold axis is parallel to a crystallographic evenfold axis. In this case, the position of the noncrystallographic symmetry element can be easily determined from the large peak in the corresponding Harker section of the Patterson.

In general, it is difficult or impossible to determine the positions of noncrystallographic axes (or their intersection at a molecular centre). However, the position of heavy atoms in isomorphous derivatives, which usually obey the noncrystallographic symmetry, can often determine this information. Position of a known molecular structure in an unknown unit cell

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The most common type of translation function occurs when looking for the position of a known molecular structure in an unknown crystal. For instance, if the structure of an enzyme has previously been determined by the isomorphous replacement method, then the structure of the same enzyme from another species can often be solved by molecular replacement [e.g. Grau et al. (1981)[link]]. However, there are some severe pitfalls when, for instance, there are gross conformational changes [e.g. Moras et al. (1980)[link]]. This type of translation function could also be useful in the interpolation of E maps produced by direct methods. Here there may often be confusion as a consequence of a number of molecular images related by translations (Karle, 1976[link]; Beurskens, 1981[link]; Egert & Sheldrick, 1985[link]).

Tollin's (1966)[link] Q function and Crowther & Blow's (1967)[link] translation function are essentially identical (Tollin, 1969[link]) and depend on a prior knowledge of the search molecule as well as its orientation in the unknown cell. The derivation given here, however, is somewhat more general and follows the derivation of Argos & Rossmann (1980)[link], and should be compared with the method of Harada et al. (1981).[link]

If the known molecular structure is correctly oriented into a cell (p) of an unknown structure and placed at S with respect to a defined origin, then a suitable translation function is [T({\bf S}) = {\textstyle\sum\limits_{{\bf p}}} |{\bf F}_{{\bf p}, \,  {\rm obs}}|^{2}|{\bf F}_{{\bf p}} ({\bf S})|^{2}. \eqno(] This definition is preferable to one based on an R-factor calculation as it is more amenable to computation and is independent of a relative scale factor.

The structure factor [{\bf F}_{{\bf p}} ({\bf S})] can be calculated by modifying expression ([link] (see below). That is, [{\bf F}_{{\bf p}} ({\bf S}) = {U \over V_{{\bf h}}} {\sum\limits_{n = 1}^{N}} \exp (2 \pi i{\bf p} \cdot {\bf S}_{n}) \left[{\sum\limits_{{\bf h}}} {\bf F}_{{\bf h}} G_{{\bf hp}_{n}} \exp (-2 \pi i{\bf h} \cdot {\bf S})\right],] where [V_{{\bf h}}] is the volume of cell (h) and [{\bf S}_{n}] is the position, in the nth crystallographic asymmetric unit, of cell (p) corresponding to S in known cell (h). Let [A_{p, \,  n} \exp (i\gamma_{n}) = {\textstyle\sum\limits_{{\bf h}}} {\bf F}_{{\bf h}} G_{{\bf hp}_{n}} \exp (-2 \pi i{\bf h} \cdot {\bf S}),] which are the coefficients of the molecular transform for the known molecule placed into the nth asymmetric unit of the p cell. Thus [{\bf F}_{{\bf p}} ({\bf S}) = {U \over V_{{\bf h}}} {\sum\limits_{n = 1}^{N}} A_{{\bf p}, \,  n} \exp [i(\gamma + 2 \pi {\bf p} \cdot {\bf S}_{n})]] or [{\bf F}_{{\bf p}} ({\bf S}) = {U \over V_{{\bf h}}} {\sum\limits_{n = 1}^{N}} A_{{\bf p}, \,  n} \exp [i(\gamma_{n} + 2 \pi {\bf p}_{n} \cdot {\bf S})],] where [{\bf p}_{n} = [{\bi C}_{n}^{T}]{\bf p}] and [{\bf S} = {\bf S}_{1}]. Hence [\eqalign{ |{\bf F}_{{\bf p}} ({\bf S})|^{2} &= \left({U \over V_{{\bf h}}}\right)^{2} {\sum\limits_{n}} {\sum\limits_{m}} \big(A_{{\bf p}, \,  n}A_{{\bf p}, \,  m}\cr &\quad \times \exp \{i[2 \pi ({\bf p}_{n} - {\bf p}_{m}) \cdot {\bf S} + (\gamma_{n} - \gamma_{m})]\}\big),}] and then from ([link] [\eqalignno{T({\bf S}) &= \left({U \over V_{{\bf h}}}\right)^{2} {\sum\limits_{{\bf p}}} {\sum\limits_{n}} {\sum\limits_{m}} \Big(|{\bf F}_{{\bf p}, \, {\rm obs}}|^{2} A_{{\bf p}, \,  n}A_{{\bf p}, \,  m} &\cr &\quad \times \exp \{i[2 \pi ({\bf p}_{n} - {\bf p}_{m}) \cdot {\bf S} + (\gamma_{n} - \gamma_{m})]\}\Big), &(}] which is a Fourier summation with known coefficients [\{|{\bf F}_{{\bf p}, \,  {\rm obs}}|^{2} A_{{\bf p}, \,   n}A_{{\bf p}, \,  m} \times \exp [i(\gamma_{n} - \gamma_{m})]\}] such that T(S) will be a maximum at the correct molecular position.

Terms with [n = m] in expression ([link] can be omitted as they are independent of S and only contribute a constant to the value of T(S). For terms with [n \neq m], the indices take on special values. For instance, if the p cell is monoclinic with its unique axis parallel to b such that [{\bf p}_{1} = (p, q, r)] and [{\bf p}_{2} = (\bar{p}, q, \bar{r})], then [{\bf p}_{1} - {\bf p}_{2}] would be (2p, 0, 2r). Hence, T(S) would be a two-dimensional function consistent with the physical requirement that the translation component, parallel to the twofold monoclinic axis, is arbitrary.

Crowther & Blow (1967)[link] show that if [{\bf F}_{M}] are the structure factors of a known molecule correctly oriented within the cell of the unknown structure at an arbitrary molecular origin, then (altering the notation very slightly from above) [T({\bf S}) = {\textstyle\sum\limits_{{\bf p}}} |{\bf F}_{\rm obs} ({\bf p})|^{2} {\bf F}_{M} ({\bf p}) {\bf F}_{M}^{*} ({\bf p} [{\bi C}]) \exp (-2 \pi i{\bf p} \cdot {\bf S}),] where [C] is a crystallographic symmetry operator relative to which the molecular origin is to be determined. This is of the same form as ([link] but concerns the special case where the h cell, into which the known molecule was placed, has the same dimensions as the p cell.

R-factor calculations are sometimes used to determine the position of a known molecular fragment in an unknown cell, particularly if only one parameter is being searched. Such calculations are computationally less convenient than the Fourier methods described above, but can be more sensitive. All these methods can be improved by simultaneous consideration of packing requirements of the molecular fragments (Harada et al., 1981[link]; Hendrickson & Ward, 1976[link]; Rabinovich & Shakked, 1984[link]). Indeed, packing considerations can frequently limit the search volume very considerably. Position of a noncrystallographic symmetry element in a poorly defined electron-density map

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If an initial set of poor phases, for example from an SIR derivative, are available and the rotation function has given the orientation of a noncrystallographic rotation axis, it is possible to search the electron-density map systematically to determine the translation axis position. The translation function must, therefore, measure the quality of superposition of the poor electron-density map on itself. Hence [{\bf S}_{x} = {\bf S}_{x'} = {\bf S}] and the function ([link] now becomes [T({\bf S}) = {2 \over V_{{\bf h}}^{2}} {\sum\limits_{{\bf h}}} {\sum\limits_{{\bf p}}} |{\bf F}_{{\bf h}}| |{\bf F}_{{\bf p}}| G_{{\bf hp}} \cos [\alpha_{{\bf h}} + \alpha_{{\bf p}} - 2 \pi ({\bf h} + {\bf p}) \cdot {\bf S}].] This real-space translation function has been used successfully to determine the intermolecular dyad axis for α-chymotrypsin (Blow et al., 1964[link]) and to verify the position of immunoglobulin domains (Colman & Fehlhammer, 1976[link]).


Argos, P. & Rossmann, M. G. (1980). Molecular replacement methods. In Theory and practice of direct methods in crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 361–417. New York: Plenum.
Beurskens, P. T. (1981). A statistical interpretation of rotation and translation functions in reciprocal space. Acta Cryst. A37, 426–430.
Blow, D. M., Rossmann, M. G. & Jeffery, B. A. (1964). The arrangement of α-chymotrypsin molecules in the monoclinic crystal form. J. Mol. Biol. 8, 65–78.
Colman, P. M. & Fehlhammer, H. (1976). Appendix: the use of rotation and translation functions in the interpretation of low resolution electron density maps. J. Mol. Biol. 100, 278–282.
Colman, P. M., Fehlhammer, H. & Bartels, K. (1976). Patterson search methods in protein structure determination: β-trypsin and immunoglobulin fragments. In Crystallographic computing techniques, edited by F. R. Ahmed, K. Huml & B. Sedlacek, pp. 248–258. Copenhagen: Munksgaard.
Crowther, R. A. & Blow, D. M. (1967). A method of positioning a known molecule in an unknown crystal structure. Acta Cryst. 23, 544–548.
Egert, E. & Sheldrick, G. M. (1985). Search for a fragment of known geometry by integrated Patterson and direct methods. Acta Cryst. A41, 262–268.
Grau, U. M., Rossmann, M. G. & Trommer, W. E. (1981). The crystallization and structure determination of an active ternary complex of pig heart lactate dehydrogenase. Acta Cryst. B37, 2019–2026.
Harada, Y., Lifchitz, A., Berthou, J. & Jolles, P. (1981). A translation function combining packing and diffraction information: an application to lysozyme (high-temperature form). Acta Cryst. A37, 398–406.
Hendrickson, W. A. & Ward, K. B. (1976). A packing function for delimiting the allowable locations of crystallized macromolecules. Acta Cryst. A32, 778–780.
Karle, J. (1976). Partial structures and use of the tangent formula and translation functions. In Crystallographic computing techniques, edited by F. R. Ahmed, K. Huml & B. Sedlacek, pp. 155–164. Copenhagen: Munksgaard.
Moras, D., Comarmond, M. B., Fischer, J., Weiss, R., Thierry, J. C., Ebel, J. P. & Giegé, R. (1980). Crystal structure of yeast tRNAAsp. Nature (London), 288, 669–674.
Rabinovich, D. & Shakked, Z. (1984). A new approach to structure determination of large molecules by multi-dimensional search methods. Acta Cryst. A40, 195–200.
Rossmann, M. G. (1972). The molecular replacement method. New York: Gordon & Breach.
Tollin, P. (1966). On the determination of molecular location. Acta Cryst. 21, 613–614.
Tollin, P. (1969). A comparison of the Q-functions and the translation function of Crowther and Blow. Acta Cryst. A25, 376–377.

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