International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 2.3, pp. 260-262   | 1 | 2 |

Section 2.3.8. Molecular replacement

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:  mgr@indiana.bio.purdue.edu

2.3.8. Molecular replacement

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2.3.8.1. Using a known molecular fragment

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The most straightforward application of the molecular-replacement method occurs when the orientation and position of a known molecular fragment in an unknown cell have been previously determined. The simple procedure is to apply the rotation and translation operations to the known fragment. This will place it into one `standard' asymmetric unit of the unknown cell. Then the crystal operators (assuming no further noncrystallographic operators are present in the unknown cell) are applied to generate the complete unit cell of the unknown structure. Structure factors can then be calculated from the rotated and translated known molecule into the unknown cell. The resultant model can be refined in numerous ways.

More generally, consider a molecule placed in any crystal cell (h), within which coordinate positions shall be designated by x. Let the corresponding structure factors be [{\bf F}_{{\bf h}}]. It is then possible to compute the structure factors [{\bf F}_{{\bf p}}] for another cell (p) into which the same molecule has been placed N times related by the crystallographic symmetry operators [[{\bi C}_{1}], {\bf d}_{1}\hbox{; } [{\bi C}_{2}], {\bf d}_{2}\hbox{; } \ldots\hbox{; } [{\bi C}_{N}], {\bf d}_{N}]. Let the electron density at a point [{\bf y}_{1}] in the first crystallographic asymmetric unit be spatially related to the point [{\bf y}_{n}] in the nth asymmetric unit of the p crystal such that [\rho ({\bf y}_{n}) = \rho ({\bf y}_{1}), \eqno(2.3.8.1)] where [{\bf y}_{n} = [{\bi C}_{n}]{\bf y}_{1} + {\bf d}_{n}. \eqno(2.3.8.2)] From the definition of a structure factor, [{\bf F}_{{\bf p}} = {\textstyle\sum\limits_{n = 1}^{N}}\; {\textstyle\int\limits_{U}} \rho ({\bf y}_{n}) \exp (2 \pi i{\bf p} \cdot {\bf y}_{n})\;\hbox{d}{\bf y}_{n}, \eqno(2.3.8.3)] where the integral is taken over the volume U of one molecule. But since each molecule is identical as expressed in equation (2.3.8.1)[link] and since (2.3.8.2)[link] can be substituted in equation (2.3.8.3)[link], we have [{\bf F}_{{\bf p}} = {\textstyle\sum\limits_{n = 1}^{N}}\; {\textstyle\int\limits_{U}} \rho ({\bf y}_{1}) \exp [2 \pi i{\bf p} \cdot ([{\bi C}_{n}]{\bf y}_{1} + {\bf d}_{n})]\;\hbox{d}{\bf y}_{1}. \eqno(2.3.8.4)] Now let the molecule in the h crystal be related to the molecule in the first asymmetric unit of the p crystal by the noncrystallographic symmetry operation [{\bf x} = [{\bi C}]{\bf y} + {\bf d}, \eqno(2.3.8.5)] which implies [\rho ({\bf x}) = \rho ({\bf y}_{1}) = \rho ({\bf y}_{2}) = .\ldots \eqno(2.3.8.6)] Furthermore, in the h cell [\rho ({\bf x}) = {1 \over V_{{\bf h}}} {\sum\limits_{{\bf h}}} {\bf F}_{{\bf h}} \exp (-2 \pi i{\bf h} \cdot {\bf x}), \eqno(2.3.8.7)] and thus, by combining with (2.3.8.5)[link], (2.3.8.6)[link] and (2.3.8.7)[link],[\rho ({\bf y}_{1}) = {1 \over V_{{\bf h}}} {\sum\limits_{{\bf h}}} {\bf F}_{{\bf h}} \exp [-2 \pi i({\bf h}[{\bi C}] \cdot {\bf y}_{1} + {\bf h} \cdot {\bf d})]. \eqno(2.3.8.8)] Now using (2.3.8.4)[link] and (2.3.8.8)[link] it can be shown that [{\bf F}_{{\bf p}} = {U \over V_{{\bf h}}} {\sum\limits_{{\bf h}}} {\bf F}_{{\bf h}} {\sum\limits_{n = 1}^{N}} G_{{\bf hp}_{n}} \exp [2 \pi i({\bf p} \cdot {\bf S}_{n} - {\bf h} \cdot {\bf S})], \eqno(2.3.8.9)] where [UG_{{\bf hp}_{n}} = {\textstyle\int\limits_{U}} \exp [2 \pi i(p[{\bi C}_{n}] - {\bf h}[{\bi C}]) \cdot {\bf u}]\;\hbox{d}{\bf u}. \eqno(2.3.8.10)] S is a chosen molecular origin in the h crystal and [{\bf S}_{n}] is the corresponding molecular position in the nth asymmetric unit of the p crystal.

2.3.8.2. Using noncrystallographic symmetry for phase improvement

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The use of noncrystallographic symmetry for phase determination was proposed by Rossmann & Blow (1962[link], 1963[link]) and subsequently explored by Crowther (1967[link], 1969[link]) and Main & Rossmann (1966)[link]. These methods were developed in reciprocal space and were primarily concerned with ab initio phase determination. Real-space averaging of electron density between noncrystallographically related molecules was used in the structure determination of deoxyhaemoglobin (Muirhead et al., 1967[link]) and of α-chymotrypsin (Matthews et al., 1967[link]). The improvement derived from the averaging between the two noncrystallographic units was, however, not clear in either case. The first obviously successful application was in the structure determination of lobster glyceraldehyde-3-phosphate dehydrogenase (Buehner et al., 1974[link]; Argos et al., 1975[link]), where the tetrameric molecule of symmetry 222 occupied one crystallographic asymmetric unit. The improvement in the essentially SIR electron-density map was considerable and the results changed from uninterpretable to interpretable. The uniqueness and validity of the solution lay in the obvious chemical correctness of the polypeptide fold and its agreement with known amino-acid-sequence data. In contrast to the earlier reciprocal-space methods, noncrystallographic symmetry was used as a method to improve poor phases rather than to determine phases ab initio.

Many other applications followed rapidly, aided greatly by the versatile techniques developed by Bricogne (1976)[link]. Of particular interest is the application to the structure determination of hexokinase (Fletterick & Steitz, 1976[link]), where the averaging occurred both between different crystal forms and within the same crystal.

The most widely used procedure for real-space averaging is the `double sorting' technique developed by Bricogne (1976)[link] and also by Johnson (1978)[link]. An alternative method is to maintain the complete map stored in the computer (Nordman, 1980b[link]). This avoids the sorting operation, but is only possible given a very large computer or a low-resolution map containing relatively few grid points.

Bricogne's double sorting technique involves generating real-space non-integral points [(D_{i})] which are related to integral grid points [(I_{i})] in the cell asymmetric unit by the noncrystallographic symmetry operators. The elements of the set [D_{i}] are then brought back to their equivalent points in the cell asymmetric unit [(D'_{i})] and sorted by their proximity to two adjacent real-space sections. The set [I'_{i}], calculated on a finer grid than [I_{i}] and stored in the computer memory two sections at a time, is then used for linear interpolation to determine the density values at [D'_{i}] which are successively stored and summed in the related array [I_{i}]. A count is kept of the number of densities received at each [I_{i}], resulting in a final averaged aggregate, when all real-space sections have been utilized. The density to be assigned outside the molecular envelope (defined with respect to the set [I_{i}]) is determined by averaging the density of all unused points in [I_{i}]. The grid interval for the set [I'_{i}] should be about one-sixth of the resolution to avoid serious errors from interpolation (Bricogne, 1976[link]). The grid point separation in the set [I_{i}] need only be sufficient for representation of electron density, or about one-third of the resolution.

Molecular replacement in real space consists of the following steps (Table 2.3.8.1[link]): (a) calculation of electron density based on a starting phase set and observed amplitudes; (b) averaging of this density among the noncrystallographic asymmetric units or molecular copies in several crystal forms, a process which defines a molecular envelope as the averaging is only valid within the range of the noncrystallographic symmetry; (c) reconstructing the unit cell based on averaged density in every noncrystallographic asymmetric unit; (d) calculating structure factors from the reconstructed cell; (e) combining the new phases with others to obtain a weighted best-phase set; and (f) returning to step (a) at the previous or an extended resolution. Decisions made in steps (b) and (e) determine the rate of convergence (see Table 2.3.8.1[link]) to a solution (Arnold et al., 1987[link]).

Table 2.3.8.1| top | pdf |
Molecular replacement: phase refinement as an iterative process

(A)[{\bf F}_{\rm obs}, \alpha'_{n}, m'_{n} \rightarrow \rho_{n}]
(B) [\rho_{n} \rightarrow \rho_{n}](modified)
  (i) Use of noncrystallographic symmetry operators
  (ii) Definition of envelope limiting volume within which noncrystallographic symmetry is valid
  (iii) Adjustment of solvent density
  (iv) Use of crystallographic operators to reconstruct modified density into a complete cell
(C) [\rho_{n} \hbox{(modified)} \rightarrow {\bf F}_{{\rm calc}, \,  n + 1}\hbox{; } \alpha_{{\rm calc}, \,  n + 1}]
(D) [({\bf F}_{{\rm calc}, \,  n + 1}, \alpha_{{\rm calc}, \,  n + 1}) + ({\bf F}_{\rm obs}, \alpha_{0}) \rightarrow {\bf F}_{\rm obs}, \alpha'_{n + 1}, m'_{n + 1}]
  (i) Assessment of reliability of new phasing set [\alpha_{n + 1}] in relation to original phasing set [\alpha_{0}(w)]
  (ii) Use of figures of merit [m_{0}, m_{n + 1}] and reliability w to determine modified phasing set [\alpha'_{n + 1}, m'_{n + 1}]
  (iii) Consideration of [\alpha_{n + 1}] and [m_{n + 1}] where there was no prior knowledge of
   (a) [{\bf F}_{\rm obs}] (e.g. very low order reflections or uncollected data)
   (b) [\alpha_{0}] (e.g. no isomorphous information or phase extension)
(E) Return to step (A) with [\alpha'_{n + 1}, m'_{n + 1}] and a possibly augmented set of [{\bf F}_{\rm obs}].
Wang (1985)[link]; Bhat & Blow (1982)[link]; Collins (1975)[link]; Schevitz et al. (1981)[link]; Hoppe & Gassmann (1968)[link].
Rossmann & Blow (1961)[link]; Hendrickson & Lattman (1970)[link].

The power of the molecular-replacement procedure for either phase improvement or phase extension depends on the number of noncrystallographic asymmetric units, the size of the excluded volume expressed in terms of the ratio [(V - UN)/V] and the magnitude of the measurement error on the structure amplitudes. Crowther (1967[link], 1969[link]) and Bricogne (1974)[link] have investigated the dependence on the number of noncrystallographic asymmetric units and conclude that three or more copies are sufficient to ensure convergence of an iterative phase improvement procedure in the absence of errors on the structure amplitudes. As with the analogous case of isomorphous replacement in which three data sets ensure reasonable phase determination, additional copies will enhance the power of the method, although their usefulness is subject to the law of diminishing returns. Another example of this principle is the sign determination of the h0l reflections of horse haemoglobin (Perutz, 1954[link]) in which seven shrinkage stages constituted the sampling of the transform of a single copy.

Procedures for real-space averaging have been used extensively with great success. The interesting work of Wilson et al. (1981)[link] is noteworthy for the continuous adjustment of molecular envelope with increased map definition. Furthermore, the analysis of complete virus structures has only been possible as a consequence of this technique (Bloomer et al., 1978[link]; Harrison et al., 1978[link]; Abad-Zapatero et al., 1980[link]; Liljas et al., 1982[link]). Although the procedure has been used primarily for phase improvement, apparently successful attempts have been made at phase extension (Nordman, 1980b[link]; Gaykema et al., 1984[link]; Rossmann et al., 1985[link]). Ab initio phasing of glyceraldehyde-3-phosphate dehydrogenase (Argos et al., 1975[link]) was successfully attempted by initially filling the known envelope with uniform density to determine the phases of the innermost reflections and then gradually extending phases to 6.3 Å resolution. Johnson et al. (1976)[link] used the same procedure to determine the structure of southern bean mosaic virus to 22.5 Å resolution. Particularly impressive was the work on polyoma virus (Rayment et al., 1982[link]; Rayment, 1983[link]; Rayment et al., 1983[link]) where crude initial models led to an entirely unexpected breakdown of the Caspar & Klug (1962)[link] concept of quasi-symmetry. Ab initio phasing has also been used by combining the electron-diffraction projection data of two different crystal forms of bacterial rhodopsin (Rossmann & Henderson, 1982[link]).

2.3.8.3. Equivalence of real- and reciprocal-space molecular replacement

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Let us proceed in reciprocal space doing exactly the same as is done in real-space averaging. Thus [\rho_{\rm AV} ({\bf x}) = {1 \over N} {\sum\limits_{n = 1}^{N}} \rho ({\bf x}_{n}),] where [{\bf x}_{n} = [{\bi C}_{n}] {\bf x} + {\bf d}_{n}.] Therefore, [\rho_{\rm AV} ({\bf x}) = {1 \over N} {\sum\limits_{N}} {1 \over V} \left[{\sum\limits_{\bf h}} {\bf F}_{\bf h} \exp (2 \pi i{\bf h \cdot x}_{n})\right].] The next step is to perform the back-transform of the averaged electron density. Hence, [{\bf F}_{\bf p} = {\textstyle\int\limits_{U}} \rho_{\rm AV} ({\bf x}) \exp (-2 \pi i{\bf p \cdot x}) \;\hbox{d}{\bf x},] where U is the volume within the averaged part of the cell. Hence, substituting for [\rho_{\rm AV}], [{\bf F}_{\bf p} = \int\limits_{U} \left[{1 \over NV} {\sum\limits_{N}} {\sum\limits_{\bf h}} {\bf F}_{\bf h} \exp (2 \pi i{\bf h \cdot x}_{n})\right] \exp (-2 \pi i{\bf p \cdot x}) \;\hbox{d}{\bf x},] which is readily simplified to [{\bf F}_{\bf p} = {U \over NV} {\sum\limits_{\bf h}} {\bf F}_{\bf h} {\sum\limits_{N}} G_{{\bf hp}n} \exp (2 \pi i{\bf h} \cdot {\bf d}_{n}).] Setting [{\bf B}_{\bf hp} = {U \over NV} {\sum\limits_{N}} G_{{\bf hp}n} \exp (2 \pi i{\bf h} \cdot {\bf d}_{n}),] the molecular-replacement equations can be written as [{\bf F}_{\bf p} = {\sum\limits_{\bf h}} {\bf B}_{\bf hp} {\bf F}_{\bf h} \eqno(2.3.8.11)] (Main & Rossmann, 1966[link]), or in matrix form [{\bf F} = [{\bi B}] {\bf F},] which is the form of the equations used by Main (1967)[link] and by Crowther (1967)[link]. Colman (1974)[link] arrived at the same conclusions by an application of Shannon's sampling theorem. It should be noted that the elements of [B] are dependent only on knowledge of the noncrystallographic symmetry and the volume within which it is valid. Substitution of approximate phases into the right-hand side of (2.3.8.11)[link] produces a set of calculated structure factors exactly analogous to those produced by back-transforming the averaged electron density in real space. The new phases can then be used in a renewed cycle of molecular replacement.

Computationally, it has been found more convenient and faster to work in real space. This may, however, change with the advent of vector processing in `supercomputers'. Obtaining improved phases by substitution of current phases on the right-hand side of the molecular-replacement equations (2.3.8.1)[link] seems less cumbersome than the repeated forward and backward Fourier transformation, intermediate sorting, and averaging required in the real-space procedure.

References

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