Refinement of heavy-atom parameters

Vijayan, M.; Ramaseshan, S.

doi:10.1107/97809553602060000557

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 2.4, pp. 270-271 | 1 | 2 |

Section 2.4.4.3. Refinement of heavy-atom parameters

M. Vijayan^a ^* and S. Ramaseshan^b ^‡

^a Molecular Biophysics Unit, Indian Institute of Science, Bangalore 560 012, India, and ^bRaman Research Institute, Bangalore 560 080, India
Correspondence e-mail: mv@mbu.iisc.ernet.in

2.4.4.3. Refinement of heavy-atom parameters

| top | pdf |

The least-squares method with different types of minimization functions is used for refining the heavy-atom parameters, including the occupancy factors. The most widely used method (Dickerson et al., 1961; Muirhead et al., 1967; Dickerson et al., 1968) involves the minimization of the function $[\varphi = \textstyle\sum w(F_{NH} - |{\bf F}_{N} + {\bf F}_{H}|)^{2}, \eqno(2.4.4.11)]$ where the summation is over all the reflections and w is the weight factor associated with each reflection. Here $[F_{NH}]$ is the observed magnitude of the structure factor for the particular derivative and $[{\bf F}_{N} + {\bf F}_{H}]$ is the calculated structure factor. The latter obviously depends upon the protein phase angle $[\alpha_{N}]$ , and the magnitude and the phase angle of $[{\bf F}_{H}]$ which are in turn dependent on the heavy-atom parameters. Let us assume that we have three derivatives A, B and C, and that we have already determined the heavy-atom parameters $[HA_{i}]$ , $[HB_{i}]$ and $[HC_{i}]$ . Then, $[\eqalignno{ {\bf F}_{HA} &= {\bf F}_{HA} (HA_{i})\cr {\bf F}_{HB} &= {\bf F}_{HB} (HB_{i}) &(2.4.4.12)\cr {\bf F}_{HC} &= {\bf F}_{HC} (HC_{i}). }]$ A set of approximate protein phase angles is first calculated, employing methods described later, making use of the unrefined heavy-atom parameters. These phase angles are used to construct $[{\bf F}_{N} + {\bf F}_{H}]$ for each derivative. (2.4.4.11) is then minimized, separately for each derivative, by varying $[HA_{i}]$ for derivative A, $[HB_{i}]$ for derivative B, and $[HC_{i}]$ for derivative C. The refined values of $[HA_{i}]$ , $[HB_{i}]$ and $[HC_{i}]$ are subsequently used to calculate a new set of protein phase angles. Alternate cycles of parameter refinement and phase-angle calculation are carried out until convergence is reached. The progress of refinement may be monitored by computing an R factor defined as (Kraut et al., 1962) $[R_{K} = {\sum |F_{NH} - |{\bf F}_{N} + {\bf F}_{H}||\over F_{NH}}. \eqno(2.4.4.13)]$

The above method has been successfully used for the refinement of heavy-atom parameters in the X-ray analysis of many proteins. However, it has one major drawback in that the refined parameters in one derivative are dependent on those in other derivatives through the calculation of protein phase angles. Therefore, it is important to ensure that the derivative, the heavy-atom parameters of which are being refined, is omitted from the phase-angle calculation (Blow & Matthews, 1973). Even when this is done, serious problems might arise when different derivatives are related by common sites. In practice, the occupancy factors of the common sites tend to be overestimated compared to those of the others (Vijayan, 1981; Dodson & Vijayan, 1971). Yet another factor which affects the occupancy factors is the accuracy of the phase angles. The inclusion of poorly phased reflections tends to result in the underestimation of occupancy factors. It is therefore advisable to omit from refinement cycles reflections with figures of merit less than a minimum threshold value or to assign a weight proportional to the figure of merit (as defined later) to each term in the minimization function (Dodson & Vijayan, 1971; Blow & Matthews, 1973).

If anomalous-scattering data from derivative crystals are available, the values of $[F_{H}]$ can be estimated using (2.4.4.7) or (2.4.4.9) and these can be used as the `observed' magnitudes of the heavy-atom contributions for the refinement of heavy-atom parameters, as has been done by many workers (Watenpaugh et al., 1975; Vijayan, 1981; Kartha, 1965). If (2.4.4.9) is used for estimating $[F_{H}]$ , the minimization function has the form $[\varphi = \textstyle\sum w(F_{HLE} - F_{H})^{2}. \eqno(2.4.4.14)]$ The progress of refinement may be monitored using a reliability index defined as $[R = {\sum |F_{HLE} - F_{H}|\over \sum F_{HLE}}. \eqno(2.4.4.15)]$

The major advantage of using $[F_{HLE}]$ 's in refinement is that the heavy-atom parameters in each derivative can now be refined independently of all other derivatives. Care should, however, be taken to omit from calculations all reflections for which $[F_{HUE}]$ is likely to be the correct estimate of $[F_{H}]$ . This can be achieved in practice by excluding from least-squares calculations all reflections for which $[F_{HUE}]$ has a value less than the maximum expected value of $[F_{H}]$ for the given derivative (Vijayan, 1981; Dodson & Vijayan, 1971).

A major problem associated with this refinement method is concerned with the effect of experimental errors on refined parameters. The values of $[F_{NH}(+) - F_{NH}(-)]$ are often comparable to the experimental errors associated with $[F_{NH}(+)]$ and $[F_{NH}(-)]$ . In such a situation, even random errors in $[F_{NH}(+)]$ and $[F_{NH}(-)]$ tend to increase systematically the observed difference between them (Dodson & Vijayan, 1971). In (2.4.4.7) and (2.4.4.9), this difference is multiplied by k or [k/2] , a quantity much greater than unity, and then squared. This could lead to the systematic overestimation of $[F_{HLE}]$ 's and the consequent overestimation of occupancy factors. The situation can be improved by employing empirical values of k, evaluated using the relation (Kartha & Parthasarathy, 1965; Matthews, 1966) $[k = {2 \sum |F_{NH} - F_{N}|\over \sum |F_{NH}(+) - F_{NH}(-)|}, \eqno(2.4.4.16)]$ for estimating $[F_{HLE}]$ or by judiciously choosing the weighting factors in (2.4.4.14) (Dodson & Vijayan, 1971). The use of a modified form of $[F_{HLE}]$ , arrived at through statistical considerations, along with appropriate weighting factors, has also been advocated (Dodson et al., 1975).

When the data are centric, (2.4.4.9) reduces to $[F_{H} = F_{NH} \pm F_{N}. \eqno(2.4.4.17)]$ Here, again, the lower estimate most often corresponds to the correct value of $[F_{H}]$ . (2.4.4.17) does not involve $[F_{NH}(+) - F_{NH}(-)]$ which, as indicated earlier, is prone to substantial error. Therefore, $[F_{H}]$ 's estimated using centric data are more reliable than those estimated using acentric data. Consequently, centric reflections, when available, are extensively used for the refinement of heavy-atom parameters. It may also be noted that in conditions under which $[F_{HLE}]$ corresponds to the correct estimate of $[F_{H}]$ , minimization functions (2.4.4.11) and (2.4.4.14) are identical for centric data.

A Patterson function correlation method with a minimization function of the type $[\varphi = \textstyle\sum w[(F_{NH} - F_{N})^{2} - F^{2}_{H}]^{2} \eqno(2.4.4.18)]$ was among the earliest procedures suggested for heavy-atom-parameter refinement (Rossmann, 1960). This procedure would obviously work well when centric reflections are used. A modified version of this procedure, in which the origins of the Patterson functions are removed from the correlation, and centric and acentric data are treated separately, has been proposed (Terwilliger & Eisenberg, 1983).

References

Blow, D. M. & Matthews, B. W. (1973). Parameter refinement in the multiple isomorphous-replacement method. Acta Cryst. A29, 56–62.Google Scholar

Dickerson, R. E., Kendrew, J. C. & Strandberg, B. E. (1961). The crystal structure of myoglobin: phase determination to a resolution of 2 Å by the method of isomorphous replacement. Acta Cryst. 14, 1188–1195.Google Scholar

Dickerson, R. E., Weinzierl, J. E. & Palmer, R. A. (1968). A least-squares refinement method for isomorphous replacement. Acta Cryst. B24, 997–1003.Google Scholar

Dodson, E., Evans, P. & French, S. (1975). The use of anomalous scattering in refining heavy atom parameters in proteins. In Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 423–436. Copenhagen: Munksgaard.Google Scholar

Dodson, E. & Vijayan, M. (1971). The determination and refinement of heavy-atom parameters in protein heavy-atom derivatives. Some model calculations using acentric reflexions. Acta Cryst. B27, 2402–2411.Google Scholar

Kartha, G. (1965). Combination of multiple isomorphous replacement and anomalous dispersion data for protein structure determination. III. Refinement of heavy atom positions by the least-squares method. Acta Cryst. 19, 883–885.Google Scholar

Kartha, G. & Parthasarathy, R. (1965). Combination of multiple isomorphous replacement and anomalous dispersion data for protein structure determination. I. Determination of heavy-atom positions in protein derivatives. Acta Cryst. 18, 745–749.Google Scholar

Kraut, J., Sieker, L. C., High, D. F. & Freer, S. T. (1962). Chymotrypsin: a three-dimensional Fourier synthesis at 5 Å resolution. Proc. Natl Acad. Sci. USA, 48, 1417–1424.Google Scholar

Matthews, B. W. (1966). The determination of the position of anomalously scattering heavy atom groups in protein crystals. Acta Cryst. 20, 230–239.Google Scholar

Muirhead, H., Cox, J. M., Mazzarella, L. & Perutz, M. F. (1967). Structure and function of haemoglobin. III. A three-dimensional Fourier synthesis of human deoxyhaemoglobin at 5.5 Å resolution. J. Mol. Biol. 28, 156–177.Google Scholar

Rossmann, M. G. (1960). The accurate determination of the position and shape of heavy-atom replacement groups in proteins. Acta Cryst. 13, 221–226.Google Scholar

Terwilliger, T. C. & Eisenberg, D. (1983). Unbiased three-dimensional refinement of heavy-atom parameters by correlation of origin-removed Patterson functions. Acta Cryst. A39, 813–817.Google Scholar

Vijayan, M. (1981). X-ray analysis of 2Zn insulin: some crystallographic problems. In Structural studies on molecules of biological interest, edited by G. Dodson, J. P. Glusker & D. Sayre, pp. 260–273. Oxford: Clarendon Press.Google Scholar

Watenpaugh, K. D., Sieker, L. C. & Jensen, L. H. (1975). Anomalous scattering in protein structure analysis. In Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 393–405. Copenhagen: Munksgaard.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.4, pp. 270-271