International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.3, pp. 272275
Section 2.3.8. Molecular replacement ^{a}Department of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA,^{b}CABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854–5638, USA, and ^{c}Department of Biological Sciences, Columbia University, New York 10027, USA 
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 . It is then possible to compute the structure factors for another cell (p) into which the same molecule has been placed N times related by the crystallographic symmetry operators . Let the electron density at a point in the first crystallographic asymmetric unit be spatially related to the point in the nth asymmetric unit of the p crystal such that where From the definition of a structure factor, 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) and since (2.3.8.2) can be substituted in equation (2.3.8.3), we have 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 which implies Furthermore, in the h cell and thus, by combining with (2.3.8.5), (2.3.8.6) and (2.3.8.7),Now using (2.3.8.4) and (2.3.8.8) it can be shown that where S is a chosen molecular origin in the h crystal and is the corresponding molecular position in the nth asymmetric unit of the p crystal.
The use of noncrystallographic symmetry for phase determination was proposed by Rossmann & Blow (1962, 1963) and subsequently explored by Crowther (1967, 1969) and Main & Rossmann (1966). These methods were developed in reciprocal space and were primarily concerned with ab initio phase determination. Realspace averaging of electron density between noncrystallographically related molecules was used in the structure determination of deoxyhaemoglobin (Muirhead et al., 1967) and of αchymotrypsin (Matthews et al., 1967). 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 glyceraldehyde3phosphate dehydrogenase (Buehner et al., 1974; Argos et al., 1975), where the tetrameric molecule of symmetry 222 occupied one crystallographic asymmetric unit. The improvement in the essentially SIR electrondensity 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 aminoacidsequence data. In contrast to the earlier reciprocalspace 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). Of particular interest is the application to the structure determination of hexokinase (Fletterick & Steitz, 1976), where the averaging occurred both between different crystal forms and within the same crystal.
The most widely used procedure for realspace averaging is the `double sorting' technique developed by Bricogne (1976) and also by Johnson (1978). An alternative method is to maintain the complete map stored in the computer (Nordman, 1980b). This avoids the sorting operation, but is only possible given a very large computer or a lowresolution map containing relatively few grid points.
Bricogne's double sorting technique involves generating realspace nonintegral points which are related to integral grid points in the cell asymmetric unit by the noncrystallographic symmetry operators. The elements of the set are then brought back to their equivalent points in the cell asymmetric unit and sorted by their proximity to two adjacent realspace sections. The set , calculated on a finer grid than and stored in the computer memory two sections at a time, is then used for linear interpolation to determine the density values at which are successively stored and summed in the related array . A count is kept of the number of densities received at each , resulting in a final averaged aggregate, when all realspace sections have been utilized. The density to be assigned outside the molecular envelope (defined with respect to the set ) is determined by averaging the density of all unused points in . The grid interval for the set should be about onesixth of the resolution to avoid serious errors from interpolation (Bricogne, 1976). The grid point separation in the set need only be sufficient for representation of electron density, or about onethird of the resolution.
Molecular replacement in real space consists of the following steps (Table 2.3.8.1): (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 bestphase 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) to a solution (Arnold et al., 1987).

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 and the magnitude of the measurement error on the structure amplitudes. Crowther (1967, 1969) and Bricogne (1974) 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) in which seven shrinkage stages constituted the sampling of the transform of a single copy.
In an analysis of how phasing errors propagate into errors in calculations of electron density, Arnold & Rossmann (1986) concluded that the `power' of phase determination could be related to the noncrystallographic redundancy, N, the ratio of the molecular envelope volume, U, to the unit cell volume, V, the fractional error of the structurefactor amplitudes, R and the fractional completeness of the data, f, by (Arnold & Rossmann, 1986)This semiquantitative result makes intuitive sense in that the noncrystallographic redundancy and solvent content terms can be directly related to oversampling of the molecular transform in reciprocal space, and, thus, are analogous in providing phasing information. The phasing power of solvent flattening/density modification was further analysed and shown to lead to Sayre's equations (Sayre, 1952) at a limit where the molecular envelope is sufficiently detailed and shrunken to cover sharpened and separated atoms (Arnold & Rossmann, 1986). This result suggests that more detailed definitions of molecular envelopes than are traditionally used could be advantageous for phase improvement and extension procedures.
Procedures for realspace averaging have been used extensively with great success. The interesting work of Wilson et al. (1981) 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; Harrison et al., 1978; AbadZapatero et al., 1980; Liljas et al., 1982). Although the procedure has been used primarily for phase improvement, apparently successful attempts have been made at phase extension (Nordman, 1980b; Gaykema et al., 1984; Rossmann et al., 1985). Ab initio phasing of glyceraldehyde3phosphate dehydrogenase (Argos et al., 1975) 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) 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; Rayment, 1983; Rayment et al., 1983) where crude initial models led to an entirely unexpected breakdown of the Caspar & Klug (1962) concept of quasisymmetry. Ab initio phasing has also been used by combining the electrondiffraction projection data of two different crystal forms of bacterial rhodopsin (Rossmann & Henderson, 1982).
Since this article was originally written, molecular replacement has been subject of a number of reviews (Rossmann, 1990), including a historical background of the subject (Rossmann, 2001). A series of chapters pertaining to molecular replacement have been published in IT Volume F (Rossmann & Arnold, 2001a), reviewing noncrystallographic symmetry (Chapter 13.1 ; Blow, 2001), rotation (Chapter 13.2 ; Navaza, 2001b) and translation (Chapter 13.3 ; Tong, 2001b) functions, and noncrystallographic symmetry averaging for phase improvement and extension (Chapter 13.4 ; Rossmann & Arnold, 2001b). Chapters on phase improvement by density modification (Chapter 15.1 ; Zhang et al., 2001), optimal weighting of Fourier terms in map calculations (Chapter 15.2 ; Read, 2001a) and refinement calculations incorporating bulk solvent correction (Chapter 18.4 ; Dauter et al., 2001) are also recommended reading.
There has been remarkable progress in the general area of density modification, involving improvement of realspace methods for averaging and reconstruction, and treatment of solvent for iterative phase improvement and refinement calculations. The use of realspace averaging between noncrystallographically related electron density within the crystallographic asymmetric unit has become an accepted mode of extending phase information to higher resolution, particularly for complex structures such as viruses [Acharya et al., 1989; Arnold & Rossmann, 1988; Gaykema et al., 1986; Hogle et al., 1985; Luo et al., 1989; Rossmann & Arnold, 2001b (IT F Chapter 13.4 ); Rossmann et al., 1985, 1992]. Ab initio phase determination based on noncrystallographic redundancy has become fairly common (Chapman et al., 1992; Lunin et al., 2000; Miller et al., 2001; Rossmann, 1990; Tsao et al., 1992). General programs in common use for noncrystallographic symmetry averaging include BUSTERTNT [Blanc et al., 2004; Roversi et al., 2000; Tronrud & Ten Eyck, 2001 (IT F Section 25.2.4 )], CNS [Brünger et al., 1998; Brunger, Adams, DeLano et al., 2001 (IT F Section 25.2.3 )], DM/DMMULTI [Cowtan & Main, 1993; Cowtan et al., 2001 (IT F Section 25.2.2 ); Schuller, 1996; Zhang, 1993], PHASES [Furey, 2001 (IT F Section 25.2.1 ); Furey & Swaminathan, 1997], RAVE/MAVE (Jones, 1992; Kleywegt, 1996) and SOLVE/RESOLVE [Terwilliger, 2002b, 2003c; Terwilliger & Berendzen, 2001 (IT F Section 14.2.2 )].
Solvent flattening has been formulated in reciprocal space for greater computational efficiency (Leslie, 1987; Terwilliger, 1999) and solvent `flipping' is a powerful extension of solvent density modification (Abrahams, 1997; Abrahams & Leslie, 1996). Bulksolvent corrections are now commonly used in crystallographic refinement, allowing for better modelling and phase determination of lowresolution data [Brünger et al., 1998; Dauter et al., 2001 (IT F Chapter 18.4 )]. The problem of phase error estimation and analysis and bias removal has been treated extensively (Cowtan, 1999; Cowtan & Main, 1996), including extension of methods to include maximumlikelihood functions and iterative bias removal procedures [Brunger, Adams & Rice, 2001 (IT F Chapter 18.2 ); Hunt & Deisenhofer, 2003; Lamzin et al., 2001 (IT F Section 25.2.5 ); Perrakis et al., 1997; Terwilliger, 2004]. Histogram matching [Cowtan & Main, 1993; Lunin, 1993; Nieh & Zhang, 1999; Refaat et al., 1996; Zhang, 1993; Zhang et al., 2001 (IT F Chapter 15.1 )] and skeletonization [Baker et al., 1993; Zhang et al., 2001 (IT F Chapter 15.1 )], and structural fragment matching procedures (Terwilliger, 2003a) have been added to the arsenal of densitymodification methods. Automated mask and molecularenvelope definition has helped to remove the tedium and increase the efficiency and quality of densitymodification and symmetryaveraging procedures. Noncrystallographic symmetry averaging among different crystal forms (Perutz, 1954) has become increasingly common, and exploitation of the unitcell variation among flashcooled and noncooled forms of the same crystal is a broadly applicable method for phase determination (Das et al., 1996; Ding et al., 1995); soaking crystals in a series of different solvents and buffers can produce an analogous effect (Ren et al., 1995; Tong et al., 1997). Phases from noncrystallographic symmetry averaging and other `experimental' sources have been incorporated into crystallographic refinement procedures using a number of formalisms (Arnold & Rossmann, 1988; Rees & Lewis, 1983) including maximum likelihood (Pannu et al., 1998).
Let us proceed in reciprocal space doing exactly the same as is done in realspace averaging. Thus where Therefore, The next step is to perform the backtransform of the averaged electron density. Hence, where U is the volume within the averaged part of the cell. Hence, substituting for , which is readily simplified to Setting the molecular replacement equations can be written as (Main & Rossmann, 1966), or in matrix form which is the form of the equations used by Main (1967) and by Crowther (1967). Colman (1974) 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 righthand side of (2.3.8.12) produces a set of calculated structure factors exactly analogous to those produced by backtransforming the averaged electron density in real space. The new phases can then be used in a renewed cycle of molecular replacement. The reciprocalspace molecular replacement procedure has been implemented and tested in a computer program (Tong & Rossmann, 1995).
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 righthand side of the molecular replacement equations (2.3.8.1) seems less cumbersome than the repeated forward and backward Fourier transformation, intermediate sorting, and averaging required in the realspace procedure.
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