The γ correction and solvent flipping

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. 320-321 | 1 | 2 |

Section 15.1.4.3. The γ correction and solvent flipping

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

15.1.4.3. The γ correction and solvent flipping

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Abrahams & Leslie (1996) have shown that solvent flipping is dramatically more effective as a density modification than solvent flattening. This may be shown to be theoretically equivalent to performing a reflection-omit calculation for each reflection individually (Abrahams, 1997).

Solvent flattening is represented in reciprocal space by convolution of the structure factors with a function, $[G({\bf h})]$ , as shown in equation (15.1.3.2). If the origin term of G is set to zero, then the modified structure factor, $[F_{\rm mod}({\bf h})]$ , will depend on the values of all the structure factors except itself; this is equivalent to performing a reflection-omit calculation with that reflection alone omitted.

Let the origin-removed G be called $[G_{\gamma}({\bf h})]$ and its Fourier transform $[g_{\gamma}({\bf x})]$ : $[G_{\gamma} ({\bf h}) = \left\{\matrix{\phantom{G}0,\ \ & {\bf h} = 0 \cr G({\bf h}), & {\bf h} \ne 0 \cr}\ , \right. \eqno(15.1.4.7)]$ then $[g_{\gamma} ({\bf x}) = g({\bf x}) - \overline{g({\bf x})}. \eqno(15.1.4.8)]$ The convolution of the reflection data with $[G_{\gamma}({\bf h})]$ is equivalent to performing a reflection-omit calculation, omitting every reflection in turn. However, the convolution may still be performed in real space; thus, the full omit calculation becomes a simple multiplication of the map by $[g_{\gamma}({\bf x})]$ : $[\rho_{\rm mod} ({\bf x}) = \hbox{g}_{\gamma} ({\bf x}) \times \rho({\bf x}). \eqno(15.1.4.9)]$ In a solvent-flattening calculation, $[g_{\gamma}({\bf x})]$ will be equal to $[g({\bf x})]$ minus the fraction of the cell that is protein. In the case of a cell with 50% solvent, $[g_{\gamma}({\bf x})]$ has a value of 0.5 in the protein and −0.5 in the solvent. Multiplication of the map by this function results in flipping of the solvent.

If the origin term of the G function, γ, can be determined, then the flipping calculation may alternatively be performed by subtracting a copy of the initial map scaled by γ from the modified map. This is the γ correction of Abrahams (1997). This approach may be generalized to arbitrary density-modification methods by use of the perturbation γ (Cowtan, 1999). In this approach, a random perturbation is applied to the starting data. Density modification is applied to both the perturbed and unperturbed maps. The relative size of the perturbation signal in the modified map gives an estimate for γ. The perturbation γ provides effective bias correction for any combination of solvent flattening, histogram matching and averaging. γ may also be estimated as a function of resolution, allowing successful application to multi-resolution modification and possibly atomization as well.

References

Abrahams, J. P. (1997). Bias reduction in phase refinement by modified interference functions: introducing the γ correction. Acta Cryst. D53, 371–376.Google Scholar

Abrahams, J. P. & Leslie, A. G. W. (1996). Methods used in the structure determination of bovine mitochondrial F₁ ATPase. Acta Cryst. D52, 30–42.Google Scholar

Cowtan, K. D. (1999). Error estimation and bias correction in phase-improvement calculations. Acta Cryst. D55, 1555–1567.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, pp. 320-321