Solvent flattening

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, pp. 311-314 | 1 | 2 |

Section 15.1.2.1. Solvent flattening

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

15.1.2.1. Solvent flattening

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Solvent flattening exploits the fact that the electron density in the solvent region is flat at medium resolution, owing to the high thermal motion and disorder of solvent molecules. The flattening of the solvent region suppresses noise in the map and therefore improves phases.

15.1.2.1.1. Introduction

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Biological molecules are typically irregular in shape, often taking roughly globular forms. When they are packed regularly to form a crystal lattice, there are gaps left between them, and these spaces are filled with the solvent in which the crystallization was performed. This solvent is a disordered liquid, and thus the arrangement of atoms in the solvent regions varies between unit cells, except in those small regions near the surface of the protein. The X-ray image forms an average of electron density over many cells, so the electron density over much of the solvent region appears to be constant to a good approximation.

The existence of a flat solvent region in a crystal places strong constraints on the structure-factor phases. The constraint of solvent flatness is implemented by identifying the molecular boundaries and replacing the densities in the solvent region by their mean density value.

When solving a structure, the contents of the unit cell are usually known, and so an estimate can be formed of how much of the cell volume is taken up by solvent (Matthews, 1968). If the solvent region can be located in the cell, then we can improve an electron-density map by setting the electron density in this region to the expected constant solvent density. Once the resulting modified phases are combined with the experimental data, an improvement can often be seen in the protein regions of the map (Bricogne, 1974).

The solvent region of a unit cell may usually be determined even from a poor MIR map using the following features:

(1) The mean electron density in the solvent region should be lower than that in the protein region. Note that this information will come from the low-resolution data, which dictate long-range density variations over the unit cell.
(2) The variation in density in the flat solvent region should be much smaller than that in the ordered protein region containing isolated clumps of density. The `peakiness' of the protein region comes from the high-resolution data.

A good method for locating the solvent region therefore takes into account information from both low- and high-resolution structure factors. Many methods have been proposed to locate the protein–solvent boundary. The first of these were the visual identification methods. The boundary was identified by digitizing a mini-map with the aid of a graphic tablet (Hendrickson et al., 1975; Schevitz et al., 1981). The hand-digitizing procedure was very time-consuming and prone to subjective judgmental errors. Nevertheless, these methods demonstrated the potential of solvent flattening and stimulated further improvement on boundary-identification methods. An automated method using a linked, high-density approach was first proposed by Bhat & Blow (1982). Based on the fact that the densities are generally higher in the protein region than in the solvent region, they defined the molecular boundary by locating the protein as a region of linked, high-density points.

Convolution techniques were subsequently adopted as an efficient method of molecular-boundary identification. Reynolds et al. (1985) proposed a high mean absolute density value approach. The electron density within the protein region was expected to have greater excursions from the mean density value than the solvent region, which is relatively featureless. The molecular boundary was located based on the value of a smoothed `modulus' electron density, which is the sum of the absolute values of all density points within a small box.

15.1.2.1.2. The automated convolution method for molecular-boundary identification

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Wang (1985) suggested an automated convolution method for identifying the solvent region which has achieved widespread use. His method involved first calculating a truncated map: $[\rho_{\rm trunc} ({\bf x}) = \left\{\matrix{\rho{\bf x}, &\rho({\bf x}) \gt \rho_{\rm solv}\cr 0, &\rho({\bf x}) \lt \rho_{\rm solv}\cr}\right.. \eqno(15.1.2.1)]$ The electron density is simply truncated at the expected solvent value, $[\rho_{\rm solv}]$ ; however, since the variations in density in the protein region are much larger than the variations in the solvent region, it is generally only the protein region which will be affected. Thus, the mean density over the protein region is increased. Similar results may be obtained using the mean-squared difference of the density from the expected solvent value.

A smoothed map is then formed by calculating at each point in the map the mean density over a surrounding sphere of radius R. This operation can be written as a convolution of the truncated map, $[\rho_{\rm trunc}]$ , with a spherical weighting function, $[w({\bf r})]$ , $[\rho_{\rm ave} ({\bf x}) = {\textstyle\sum\limits_{\bf r}}\ \hbox{w}({\bf r}) \rho_{\rm trunc} ({\bf x} - {\bf r}), \eqno(15.1.2.2)]$ where $[w({\bf r}) = \left\{\matrix{1-|{\bf r}|/R,\hfill &|{\bf r}| \lt R\cr 0,\hfill &|{\bf r}| \gt R\cr}\right.\ . \eqno(15.1.2.3)]$

Leslie (1987) noted that the convolution operation required in equation (15.1.2.2) can be very efficiently performed in reciprocal space using fast Fourier transforms (FFTs), $[\rho_{\rm ave} ({\bf x}) = {\scr F}^{-1} \{{\scr F} [\rho_{\rm trunc}({\bf x})] {\scr F} [w({\bf r})]\}, \eqno(15.1.2.4)]$ where $[{\scr F}]$ denotes a Fourier transform, and $[{\scr F}^{-1}]$ represents an inverse Fourier transform.

The Fourier transform of the truncated density can be readily calculated using FFTs. The Fourier transform of the weighting function can be calculated analytically by $[\eqalignno{g(s) &= {\scr F}[w({\bf r})] = {{3[\sin (2\pi Rs) - 2\pi Rs \cos (2\pi Rs)]} \over {(2\pi Rs)^{3}}}\cr &\quad - {{3\{4\pi Rs \sin (2\pi Rs) - [(2\pi Rs)^{2} - 2]\cos (2\pi Rs) - 2\}} \over {(2\pi Rs)^{4}}},\cr& &(15.1.2.5)}]$ where $[s = 2\sin\theta /\lambda.]$

Therefore, the averaging of the truncated electron density by a spherical weighting function can be achieved by two FFTs. This greatly reduced the time required for calculating the averaged density. Other weighting functions may be implemented by the same approach.

A cutoff value, $[\rho_{\rm cut}]$ , is then calculated, which divides the unit cell into two portions occupying the correct volumes for the protein and solvent regions. All points in the map where $[\rho_{\rm ave} ({\bf x}) \lt \rho_{\rm cut}]$ can then be assumed to be in the solvent region. A typical mask obtained from an MIR map by this means, and the modified map, are shown in Fig. 15.1.2.2.

Figure 15.1.2.2| top | pdf |

Solvent mask determined from a map by Wang's method.

The radius of the sphere, R, used in equation (15.1.2.3) for the averaging of electron densities is generally around 8 Å. The molecular envelope derived from such an averaged map tends to lose details of the protein molecular surface. Paradoxically, a large averaging sphere is required for the identification of the protein–solvent boundary based on the difference between the mean density of the protein and solvent, which is very small and can only be distinguished when a sufficiently large area of the map is averaged. Abrahams & Leslie (1996) proposed an alternative method of molecular-boundary identification that uses the standard deviation of the electron density within a given radius relative to the overall mean at every grid point of a map. The local-standard-deviation map is the square root of a convolution of a sphere and the squared map, which can be calculated in reciprocal space in a similar way to the procedure described in equations (15.1.2.4) and (15.1.2.5) as proposed by Leslie (1987). By integrating the histogram of the local-standard-deviation map, the cutoff value of the local standard deviation corresponding to the solvent fraction can be calculated. Using this procedure, a molecular envelope that contains more details of the protein molecular surface can be obtained, since the radius of the averaging sphere can be as low as 4 Å (Abrahams & Leslie, 1996).

15.1.2.1.3. The solvent-flattening procedure

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Once the envelope has been determined, solvent flattening is performed by simply setting the density in the solvent region to the expected value, $[\rho_{\rm solv}]$ : $[\rho_{\rm mod} ({\bf x}) = \left\{\matrix{\rho({\bf x}), &\rho_{\rm ave} ({\bf x}) \gt \rho_{\rm cut}\cr \rho_{\rm solv}, &\rho_{\rm ave}({\bf x}) \lt \rho_{\rm cut}\cr}\right.. \eqno(15.1.2.6)]$ If the electron density has not been calculated on an absolute scale, the solvent density may be set to its mean value.

A related method is solvent flipping, developed by Abrahams & Leslie (1996). In this approach, the flattening operation is modified by the introduction of a relaxation factor, γ, where γ is positive, effectively `flipping' the density in the solvent region. $[\rho_{\rm mod} ({\bf x}) = \left\{\matrix{\rho({\bf x}),\hfill &\rho_{\rm ave} ({\bf x}) \gt \rho_{\rm cut}\cr \rho_{\rm solv} - [\gamma/(1 - \gamma)] [\rho({\bf x}) - \rho_{\rm solv}], &\rho_{\rm ave} ({\bf x}) \lt \rho_{\rm cut}\cr}\right.. \eqno(15.1.2.7)]$ The effect of this modification is to correct for the problem of independence in phase combination and is discussed in Section 15.1.4.3.

References

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International Tables for Crystallography (2006). Vol. F. ch. 15.1, pp. 311-314