Integration of direct methods with isomorphous replacement techniques

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 232 | 1 | 2 |

Section 2.2.10.3. Integration of direct methods with isomorphous replacement techniques

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.10.3. Integration of direct methods with isomorphous replacement techniques

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The modulus of the isomorphous difference $[\Delta F = |F_{PH}| - |F_{P}|]$ may be assumed at a first approximation as an estimate of the heavy-atom s.f. $[F_{H}]$ . Normalization of $[|\Delta F|]$ 's and application of the tangent formula may reveal the heavy-atom structure (Wilson, 1978).

The theoretical basis for integrating the techniques of direct methods and isomorphous replacement was introduced by Hauptman (1982a). According to his notation let us denote by $[f_{j}]$ and $[g_{j}]$ atomic scattering factors for the atom labelled j in a pair of isomorphous structures, and let $[E_{\bf h}]$ and $[G_{\bf h}]$ denote corresponding normalized structure factors. Then $[\eqalign{E_{\bf h} &= |E_{\bf h}| \exp (i\varphi_{\bf h}) = \alpha_{20}^{-1/2} \textstyle\sum\limits_{j=1}^{N} f_{j} \exp (2 \pi i {\bf h} \cdot {\bf r}_{j}),\cr G_{\bf h} &= |G_{\bf h}| \exp (i\psi_{\bf h}) = \alpha_{02}^{-1/2} \textstyle\sum\limits_{j=1}^{N} g_{j} \exp (2 \pi i {\bf h} \cdot {\bf r}_{j}),}]$ where $[\alpha_{mn} = \textstyle\sum\limits_{j=1}^{N} f_{j}^{m} g_{j}^{n}.]$ The conditional probability of the two-phase structure invariant $[\Phi = \varphi_{\bf h} - \psi_{\bf h}]$ given $[|E_{\bf h}|]$ and $[|G_{\bf h}|]$ is (Hauptman, 1982a) $[P(\Phi |\; |E|, |G|) \simeq [2\pi I_{0} (Q)]^{-1} \exp (Q \cos \Phi),]$ where $[\eqalign{Q &= |EG| [2\alpha / (1 - \alpha^{2})],\cr \alpha &= \alpha_{11} / (\alpha_{20}^{1/2} \alpha_{02}^{1/2}).}]$ Three-phase structure invariants were evaluated by considering that eight invariants exist for a given triple of indices h, k, l $[({\bf h} + {\bf k} + {\bf l} = 0)]$ : $[\eqalign{\Phi_{1} &= \varphi_{\bf h} + \varphi_{\bf k} + \varphi_{\bf l} \qquad \Phi_{2} = \varphi_{\bf h} + \varphi_{\bf k} + \psi_{\bf l}\cr \Phi_{3} &= \varphi_{\bf h} + \psi_{\bf k} + \varphi_{\bf l} \qquad \Phi_{4} = \psi_{\bf h} + \varphi_{\bf k} + \varphi_{\bf l}\cr \Phi_{5} &= \varphi_{\bf h} + \psi_{\bf k} + \psi_{\bf l} \qquad \Phi_{6} = \psi_{\bf h} + \varphi_{\bf k} + \psi_{\bf l}\cr \Phi_{7} &= \psi_{\bf h} + \psi_{\bf k} + \varphi_{\bf l} \qquad \Phi_{8} = \psi_{\bf h} + \psi_{\bf k} + \psi_{\bf l}.}]$ So, for the estimation of any $[\Phi_{j}]$ , the joint probability distribution $[P (E_{\bf h}, E_{\bf k}, E_{\bf l}, G_{\bf h}, G_{\bf k}, G_{\bf l})]$ has to be studied, from which eight conditional probability densities can be obtained: $[\eqalign{&P (\Phi_{i} \|E_{\bf h}|, |E_{\bf k}|, |E_{\bf l}|, |G_{\bf h}|, |G_{\bf k}|, |G_{\bf l}|)\cr &\quad \simeq [2 \pi I_{0} (Q_{j})]^{-1} \exp [Q_{j} \cos \Phi_{j}]}]$ for $[j = 1, \ldots, 8]$ .

The analytical expressions of $[Q_{j}]$ are too intricate and are not given here (the reader is referred to the original paper). We only say that $[Q_{j}]$ may be positive or negative, so that reliable triplet phase estimates near 0 or near π are possible: the larger $[|Q_{j}|]$ , the more reliable the phase estimate.

A useful interpretation of the formulae in terms of experimental parameters was suggested by Fortier et al. (1984): according to them, distributions do not depend, as in the case of the traditional three-phase invariants, on the total number of atoms per unit cell but rather on the scattering difference between the native protein and the derivative (that is, on the scattering of the heavy atoms in the derivative).

Hauptman's formulae were generalized by Giacovazzo et al. (1988): the new expressions were able to take into account the resolution effects on distribution parameters. The formulae are completely general and include as special cases native protein and heavy-atom isomorphous derivatives as well as X-ray and neutron diffraction data. Their complicated algebraic forms are easily reduced to a simple expression in the case of a native protein heavy-atom derivative: in particular, the reliability parameter for $[\Phi_{1}]$ is $[Q_{1} = 2[\sigma_{3} / \sigma_{2}^{3/2}]_{P} |E_{\bf h} E_{\bf k} E_{\bf l}| + 2[\sigma_{3} / \sigma_{2}^{3/2}]_{H} \Delta_{\bf h} \Delta_{\bf k} \Delta_{\bf l}, \eqno(2.2.10.2)]$ where indices P and H warn that parameters have to be calculated over protein atoms and over heavy atoms, respectively, and $[\Delta = (F_{PH} - F_{P}) / (\textstyle\sum f_{j}^{2})_{H}^{1/2}.]$ Δ is a pseudo-normalized difference (with respect to the heavy-atom structure) between moduli of structure factors.

Equation (2.2.10.2) may be compared with Karle's (1983) qualitative rule: if the sign of $[[(F_{\bf h})_{PH} - (F_{\bf h})_{P}] [(F_{\bf k})_{PH} - (F_{\bf k})_{P}] [(F_{\bf l})_{PH} - (F_{\bf l})_{P}]]$ is plus then the value of $[\Phi_{1}]$ is estimated to be zero; if its sign is minus then the expected value of $[\Phi_{1}]$ is close to π. In practice Karle's rule agrees with (2.2.10.2) only if the Cochran-type term in (2.2.10.2) may be neglected. Furthermore, (2.2.10.2) shows that large reliability values do not depend on the triple product of structure-factor differences, but on the triple product of pseudo-normalized differences. A series of papers (Giacovazzo, Siliqi & Ralph, 1994; Giacovazzo, Siliqi & Spagna, 1994; Giacovazzo, Siliqi & Platas, 1995; Giacovazzo, Siliqi & Zanotti, 1995; Giacovazzo et al., 1996) shows how equation (2.2.10.2) may be implemented in a direct procedure which proved to be able to estimate the protein phases correctly without any preliminary information on the heavy-atom substructure.

Combination of direct methods with the two-derivative case is also possible (Fortier et al., 1984) and leads to more accurate estimates of triplet invariants provided experimental data are of sufficient accuracy.

References

Fortier, S., Weeks, C. M. & Hauptman, H. (1984). On integrating the techniques of direct methods and isomorphous replacement. III. The three-phase invariant for the native and two-derivative case. Acta Cryst. A40, 646–651.Google Scholar

Giacovazzo, C., Cascarano, G. & Zheng, C.-D. (1988). On integrating the techniques of direct methods and isomorphous replacement. A new probabilistic formula for triplet invariants. Acta Cryst. A44, 45–51.Google Scholar

Giacovazzo, C., Siliqi, D. & Platas, J. G. (1995). The ab initio crystal structure solution of proteins by direct methods. V. A new normalizing procedure. Acta Cryst. A51, 811–820.Google Scholar

Giacovazzo, C., Siliqi, D., Platas, J. G., Hecht, H.-J., Zanotti, G. & York, B. (1996). The ab initio crystal structure solution of proteins by direct methods. VI. Complete phasing up to derivative resolution. Acta Cryst. D52, 813–825.Google Scholar

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 232