Reciprocal-space interpretation of density modification

Zhang, K. Y. J.; Cowtan, K. D.; Main, P.

doi:10.1107/97809553602060000687

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 15.1, p. 319 | 1 | 2 |

Section 15.1.3. Reciprocal-space interpretation of density modification

K. Y. J. Zhang,^a K. D. Cowtan^b ^* and P. Main^c

^a Division of Basic Sciences, Fred Hutchinson Cancer Research Center, 1100 Fairview Ave N., Seattle, WA 90109, USA,^bDepartment of Chemistry, University of York, York YO1 5DD, England, and ^cDepartment of Physics, University of York, York YO1 5DD, England
Correspondence e-mail: cowtan+email@ysbl.york.ac.uk

15.1.3. Reciprocal-space interpretation of density modification

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Density modification, although mostly performed in real space for ease of application, can be understood in terms of reciprocal-space constraints on structure-factor amplitudes and phases.

Main & Rossmann (1966) showed that the NCS-averaging operation in real space can be expressed in reciprocal space as the convolution of the structure factors and the Fourier transform of the molecular envelope and the NCS matrices. Similarly, the solvent-flattening operation can be considered a multiplication of the map by some mask, $[g_{\rm sf}({\bf x})]$ , where $[g_{\rm sf}({\bf x}) = 1]$ in the protein region and $[g_{\rm sf}({\bf x}) = 0]$ in the solvent region. Thus $[\rho_{\rm mod} ({\bf x}) = g_{\rm sf} ({\bf x}) \times \rho({\bf x}). \eqno(15.1.3.1)]$ This assumes that the solvent level is zero, which can be achieved by suitable adjustment of the [F(000)] term.

If we transform this equation to reciprocal space, then the product becomes a convolution; thus $[F_{\rm mod} ({\bf h}) = (1/V) {\textstyle\sum\limits_{\bf k}} G_{\rm sf} ({\bf k}) F({\bf h} - {\bf k}), \eqno(15.1.3.2)]$ where $[G_{\rm sf}({\bf k})]$ is the Fourier transform of the mask $[g_{\rm sf}({\bf x})]$ . The solvent mask $[g_{\rm sf}({\bf x})]$ shows the outline of the molecule with no internal detail, so must be a low-resolution image. Therefore, all but the lowest-resolution terms of $[G_{\rm sf}]$ will be negligible.

The convolution expresses the relationship between phases in reciprocal space from the constraint of solvent flatness in real space. Since only the terms near the origin of $[G_{\rm sf}]$ are nonzero, the convolution can only relate phases that are local to each other in reciprocal space. Thus, it can only provide phase information for structure factors near the current phasing resolution limit.

This reasoning may also be applied to other density modifications. Histogram matching applies a nonlinear rescaling to the current density in the protein region. The equivalent multiplier, $[g_{\rm hm}({\bf x})]$ , shows variations of about 1.0 that are related to the features in the initial map. The function $[G_{\rm hm}({\bf h})]$ for histogram matching is, therefore, dominated by its origin term, but shows significant features to the same resolution as the current map or further, as the density rescaling becomes more nonlinear. Histogram matching can therefore give phase indications to twice the resolution of the initial map or beyond, although phase indications will be weak and contain errors related to the level of error in the initial map. $[\rho_{\rm mod} ({\bf x}) = g_{\rm ncs} ({\bf x}) (1/N_{\rm ncs}) {\textstyle\sum\limits_{i}} \rho_{i} ({\bf x}). \eqno(15.1.3.3)]$

Averaging may be described as the summation of a number of reoriented copies of the electron density within the region of the averaging mask (Main & Rossmann, 1966), i.e. where $[\rho_{i}({\bf x})]$ is the initial density, $[\rho({\bf x})]$ , transformed by the ith NCS operator and $[g_{\rm ncs}({\bf x})]$ is the mask of the molecule to be averaged. This summation is repeated for each copy of the molecule in the whole unit cell. The reciprocal-space averaging function, $[G_{\rm ncs}({\bf h})]$ , is the Fourier transform of a mask, as for solvent flattening, but since the mask covers only a single molecule, rather than the molecular density in the whole unit cell, the extent of $[G_{\rm ncs}({\bf h})]$ in reciprocal space is greater.

Sayre's equation is already expressed as a convolution, although in this case the function $[G({\bf h})]$ is given by the structure factors $[F({\bf h})]$ themselves. It is, therefore, the most powerful method for phase extension. However, as resolution decreases, more of the reflections required to form the convolution are missing, and the error increases.

The functions $[g({\bf x})]$ and $[G({\bf h})]$ for these density modifications are illustrated in Fig. 15.1.3.1 for a simple one-dimensional structure.

Figure 15.1.3.1| top | pdf |

The functions $[g({\bf x})]$ and $[G({\bf h})]$ for solvent flattening, histogram matching and averaging.

References

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. F. ch. 15.1, p. 319