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

International Tables for Crystallography (2012). Vol. F, ch. 13.4, pp. 359-360   | 1 | 2 |

Section 13.4.11. Convergence

M. G. Rossmanna* and E. Arnoldb

aDepartment of Biological Sciences, Purdue University, West Lafayette, IN 47907–1392, USA, and  bBiomolecular Crystallography Laboratory, CABM & Rutgers University, 679 Hoes Lane, Piscataway, NJ 08854–5638, USA
Correspondence e-mail:

13.4.11. Convergence

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Iterations consist of averaging, Fourier inversion of the average map, recombination of observed structure-factor amplitudes with calculated phases, and recalculation of a new electron-density map. Presumably, each new map is an improvement of the previous map as a consequence of using the improved phases resulting from the map-averaging procedure. However, after five or ten cycles, the procedure has usually converged so that each new map is essentially the same as the previous map. Convergence can be usefully measured by computing the correlation coefficient (CC) and R factor (R) between calculated ([F_{\rm calc}]) and observed ([F_{\rm obs}]) structure-factor amplitudes as a function of resolution (Fig.[link]). These factors are defined as [\eqalign{CC &= {{\textstyle\sum_{h}} \left(\langle F_{\rm obs} \rangle - F_{\rm obs}\right) \left(\langle F_{\rm calc} \rangle - F_{\rm calc}\right) \over \left[{\textstyle\sum_{h}} \left(\langle F_{\rm obs} \rangle - F_{\rm obs}\right)^{2} \left(\langle F_{\rm calc} \rangle - F_{\rm calc}\right)^{2}\right]^{1/2}},\cr R &= 100 \times {\textstyle\sum} \left|\left(\left|F_{\rm obs}\right| - \left|F_{\rm calc}\right|\right)\right| \Big/ {\textstyle\sum} \left|F_{\rm obs}\right|.}]Because of the lack of information immediately outside the resolution limit, these factors must necessarily be poor in the outermost resolution shell. Nevertheless, the outermost resolution shell will be the most sensitive to phase improvement as these structure factors will be the furthest from their correct values at the start of a set of iterations after a resolution extension.


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Plot of a correlation coefficient as the phases were extended from 8 to 3 Å resolution in the structure determination of Mengo virus. [Reproduced with permission from Luo et al. (1989[link]).]

Convergence of CC and R does not, however, necessarily mean that phases are no longer changing from cycle to cycle. Usually, the small-amplitude structure factors keep changing long after convergence appears to have been reached (unpublished results). However, the small-amplitude structure factors make very little difference to the electron-density maps.

The rate of convergence can be improved by suitably weighting coefficients in the computation of the next electron-density map. It can be useful to reduce the weight of those structure factors where the difference between observed and calculated amplitudes is larger than the average difference, as, presumably, error in amplitude can also imply error in phase. Various weighting schemes are generally used (Sim, 1959[link]; Rayment, 1983[link]; Arnold et al., 1987[link]; Arnold & Rossmann, 1988[link]).

As mentioned above, the rate of convergence can also be improved by inclusion of [F_{\rm calc}] values when no [F_{\rm obs}] values have been measured. However, care must be taken to use suitable weights to ensure that the [F_{\rm calc}]'s are not systematically larger or smaller than the [F_{\rm obs}] values in the same resolution range.

Monitoring the CC or R factor for different classes of reflections (e.g. [h + k + l = 2n] and [h + k + l = 2n + 1]) can be a good indicator of problems (Muckelbauer et al., 1995[link]), particularly in the presence of pseudo-symmetries. All classes of reflections should behave similarly.

The power (P) of the phase determination and, hence, the rate of convergence and error in the final phasing has been shown to be (Arnold & Rossmann, 1986[link]) proportional to [P \propto (Nf)^{1/2} / \left[R \left(U/V\right)\right],]where N is the NCS redundancy, f is the fraction of observed reflections to those theoretically possible, R is a measure of error on the measured amplitudes (e.g. [R_{\rm merge}]) and [U/V] is the ratio of the volume of the density being averaged to the volume of the unit cell. Important implications of this relationship include that the phasing power is proportional to the square root of the NCS redundancy and that it is also dependent upon solvent content and diffraction-data quality and completeness.


Arnold, E. & Rossmann, M. G. (1986). Effect of errors, redundancy, and solvent content in the molecular replacement procedure for the structure determination of biological macromolecules. Proc. Natl Acad. Sci. USA, 83, 5489–5493.Google Scholar
Arnold, E. & Rossmann, M. G. (1988). The use of molecular-replacement phases for the refinement of the human rhinovirus 14 structure. Acta Cryst. A44, 270–282.Google Scholar
Arnold, E., Vriend, G., Luo, M., Griffith, J. P., Kamer, G., Erickson, J. W., Johnson, J. E. & Rossmann, M. G. (1987). The structure determination of a common cold virus, human rhinovirus 14. Acta Cryst. A43, 346–361.Google Scholar
Muckelbauer, J. K., Kremer, M., Minor, I., Tong, L., Zlotnick, A., Johnson, J. E. & Rossmann, M. G. (1995). Structure determination of coxsackievirus B3 to 3.5 Å resolution. Acta Cryst. D51, 871–887.Google Scholar
Rayment, I. (1983). Molecular replacement method at low resolution: optimum strategy and intrinsic limitations as determined by calculations on icosahedral virus models. Acta Cryst. A39, 102–116.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

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