Phase combination

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. 319-321 | 1 | 2 |

Section 15.1.4. Phase combination

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. Phase combination

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Phase combination is used to filter the noise in the modified phases and eliminate the incorrect component of the modified phases through a statistical process. The observed structure-factor amplitudes are used to estimate the reliability of the phases after density modification. The estimated probability of the modified phases is combined with the probability of observed phases to produce a more reliable phase estimate, $[P_{\rm new} [\varphi ({\bf h})] = P_{\rm obs} [\varphi ({\bf h})] P_{\rm mod} [\varphi ({\bf h})]. \eqno(15.1.4.1)]$

Once a modified map has been obtained, modified phases and amplitudes may be derived from an inverse Fourier transform. The modified phases are normally combined with the initial phases by multiplication of their probability distributions. The probability distribution for the experimentally observed phases is usually described in terms of a best phase and figure of merit (Blow & Rossmann, 1961) or by Hendrickson–Lattman coefficients (Hendrickson & Lattman, 1970). In order to estimate a unimodal probability distribution for the modified phase, some estimate of the associated error must be made; this is usually achieved using the Sim weighting scheme (Sim, 1959).

Recombination with the initial phases assumes independence between the initial and modified phases and is a source of difficulties. However, in the absence of some form of phase constraint, most density-modification constraints are too weak to guarantee convergence to a reasonable solution. The exception is when high-order NCS is present; in this case, the combination of NCS and observed amplitudes is sufficient to determine the phases (Chapman et al., 1992; Tsao et al., 1992), and phase combination may be omitted; however, weighting of the phases is still necessary. In this case, it is also possible to restore missing reflections in both amplitude and phase.

15.1.4.1. Sim and $[\sigma_{a}]$ weighting

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The phase probability distribution for the density-modified phase is conventionally generated under assumptions that were made for the combination of a partial atomic model with experimental data. It assumes that the calculated amplitudes and phases arise from a density map in which some atoms are present and correctly positioned, and the remainder are completely absent (Sim, 1959). Thus, the difference between the true structure factor and the calculated value must be the effective structure factor due to the missing density alone. If the phase of this quantity is random and the amplitude is drawn from a Wilson distribution (Wilson, 1949), the following expression is obtained: $[P_{\rm mod} (\varphi ) = \exp [A \cos \varphi + B \sin \varphi], \eqno(15.1.4.2)]$ where $[\eqalign{A &= X \cos \varphi_{\exp}\cr B &= X \sin \varphi_{\exp}} \eqno(15.1.4.3)]$ and $[X = 2|F_{\exp}\|F_{\rm mod}| / \Sigma_{Q}, \eqno(15.1.4.4)]$ where $[\Sigma_{Q}]$ is the variance parameter in the Wilson distribution for the missing part of the structure. The figure of merit, w, can be derived from $[w = I_{1} (X) / I_{0} (X), \eqno(15.1.4.5)]$ where $[I_{0}]$ and $[I_{1}]$ are zero- and first-order modified Bessel functions. A similar argument follows for centric reflections.

The error estimate for the phase depends on the effective amount of missing structure that is estimated on the basis of the agreement of the modified amplitudes with their measured values, where $[\Sigma_{Q}]$ may be estimated by a number of means, for example (Bricogne, 1976), $[\Sigma_{Q} = \langle |F_{\rm obs}|^{2} - |F_{\rm mod}|^{2} \rangle, \eqno(15.1.4.6)]$ where the average is normally taken over all reflections at a particular resolution. A more sophisticated approach is the $[\sigma_{a}]$ method of Read (1986), which allows for errors in the atomic model and has also been used in density modification (Chapter 15.2 ).

Although these approaches have been applied with some success, the assumption in equation (15.1.4.1) that the density-modified amplitudes and phases are independent of the initial values is invalid. Since the density constraints are typically under-determined, it is possible to achieve an arbitrarily good agreement between the model amplitudes and their observed values without improving the phases. As a result, phase weights from density modification are typically overestimated.

This problem has traditionally been addressed by limiting the number of cycles of density modification in which weakly phased reflections are included. Typically, density modification is started with only some subset of the data, such as those reflections well phased from MIR data. Only these reflections are included in the phase recombination, with other reflections set to zero. As the calculation progresses, more reflections are introduced until all the data are included. The figures of merit of reflections that undergo fewer cycles of phase recombination will be correspondingly smaller (e.g. Leslie, 1987; Zhang & Main, 1990a). In averaging calculations where considerable phase information is available from high-order NCS, it is still typically necessary to perform phase extension over hundreds of cycles and to add a very thin resolution shell of new reflections at each cycle.

The phases and figure of merit generated from density modification are more suited to the calculation of weighted $[F_{o}]$ maps than $[2mF_{o} - F_{c}]$ maps. The $[2mF_{o} - F_{c}]$ map is designed to aid the structure completion from a partial model (Main, 1979). The $[2mF_{o} - F_{c}]$ map will restore features missing from the current model at full weight if the following conditions are fulfilled. First, the model phases must be close to their true values. Secondly, the difference between the model and observed amplitudes is a good indicator of the phase error and the difference between the calculated and observed amplitudes decreases as the phases approach their true values. Neither of these assumptions are necessarily true for density modification, since it may be applied to very poor maps with almost random phases, and under most density-modification schemes the structure-factor amplitudes may be over-fitted to the observed values.

15.1.4.2. Reflection omit

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The modified map may be made more independent of the original map, as was assumed when multiplying the phase probability distributions in equation (15.1.4.1), through a reciprocal-space analogue of the omit map, the reflection-omit method.

The reflections are divided into (typically 10 or 20) sets and density-modification calculations are performed, excluding each set in turn from the calculation of the starting map, in a manner similar to a free-R-value calculation (Brünger, 1992). Density modification is applied to each map in turn, and the modified reflections from each of the free sets are combined to give a new, complete data set. This data set should be less dependent on the original amplitudes; therefore, the amplitudes may be expected to give a better indication of the quality of the modified phases.

The resulting maps obtained using solvent flattening and/or histogram matching are dramatically improved using the reflection-omit method (Cowtan & Main, 1996). In the case of averaging calculations, however, the reflection-omit approach makes little difference, since omitted reflections tend to be restored through noncrystallographic symmetry relationships to other regions of reciprocal space. It is possible that further improvements may be achieved by selecting reflection sets that approximately obey the NCS relationships.

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

Blow, D. M. & Rossmann, M. G. (1961). The single isomorphous replacement method. Acta Cryst. 14, 1195–1202.Google Scholar

Bricogne, G. (1976). Methods and programs for direct-space exploitation of geometric redundancies. Acta Cryst. A32, 832–847.Google Scholar

Brünger, A. T. (1992). Free R value: a novel statistical quantity for assessing the accuracy of crystal structures. Nature (London), 355, 472–475.Google Scholar

Chapman, M. S., Tsao, J. & Rossmann, M. G. (1992). Ab initio phase determination for spherical viruses: parameter determination for spherical-shell models. Acta Cryst. A48, 301–312.Google Scholar

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

Cowtan, K. D. & Main, P. (1996). Phase combination and cross validation in iterated density-modification calculations. Acta Cryst. D52, 43–48.Google Scholar

Hendrickson, W. A. & Lattman, E. E. (1970). Representation of phase probability distributions for simplified combination of independent phase information. Acta Cryst. B26, 136–143.Google Scholar

Leslie, A. G. W. (1987). A reciprocal-space method for calculating a molecular envelope using the algorithm of B. C. Wang. Acta Cryst. A43, 134–136.Google Scholar

Main, P. (1979). A theoretical comparison of the β, γ′ and 2F_o − F_c syntheses. Acta Cryst. A35, 779–785.Google Scholar

Read, R. J. (1986). Improved Fourier coefficients for maps using phases from partial structures with errors. Acta Cryst. A42, 140–149.Google Scholar

Sim, G. A. (1959). The distribution of phase angles for structures containing heavy atoms. II. A modification of the normal heavy-atom method for non-centrosymmetrical structures. Acta Cryst. 12, 813–815.Google Scholar

Tsao, J., Chapman, M. S. & Rossmann, M. G. (1992). Ab initio phase determination for viruses with high symmetry: a feasibility study. Acta Cryst. A48, 293–301.Google Scholar

Wilson, A. J. C. (1949). The probability distribution of X-ray intensities. Acta Cryst. 2, 318–321.Google Scholar

Zhang, K. Y. J. & Main, P. (1990a). Histogram matching as a new density modification technique for phase refinement and extension of protein molecules. Acta Cryst. A46, 41–46.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 15.1, pp. 319-321

Section 15.1.4. Phase combination

15.1.4. Phase combination

15.1.4.1. Sim and weighting

15.1.4.2. Reflection omit

15.1.4.3. The γ correction and solvent flipping

References

15.1.4.1. Sim and $[\sigma_{a}]$ weighting