Equivalence of real- and reciprocal-space molecular replacement

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

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

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

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

M. G. Rossmann^a ^* and E. Arnold^b

^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

2.3.8.3. 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), or in matrix form $[{\bf F} = [{\bi B}] {\bf F},]$ 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 right-hand side of (2.3.8.11) 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) seems less cumbersome than the repeated forward and backward Fourier transformation, intermediate sorting, and averaging required in the real-space procedure.

References

Colman, P. M. (1974). Noncrystallographic symmetry and the sampling theorem. Z. Kristallogr. 140, 344–349.Google Scholar

Crowther, R. A. (1967). A linear analysis of the non-crystallographic symmetry problem. Acta Cryst. 22, 758–764.Google Scholar

Main, P. (1967). Phase determination using non-crystallographic symmetry. Acta Cryst. 23, 50–54.Google Scholar

Main, P. & Rossmann, M. G. (1966). Relationships among structure factors due to identical molecules in different crystallographic environments. Acta Cryst. 21, 67–72.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, p. 262