Acta Crystallographica Section A

Foundations of Crystallography

Volume 58, Part 4 (July 2002)

research papers

Acta Cryst. (2002). A58, 361-369 [ doi:10.1107/S0108767302005597 ]

Algebraic direct methods for few-atoms structure models

H. A. Hauptman, D. Y. Guo, H. Xu and R. H. Blessing

Abstract: As a basis for direct-methods phasing at very low resolution for macromolecular crystal structures, normalized structure-factor algebra is presented for few-atoms structure models with N = 1, 2, 3, ... equal atoms or polyatomic globs per unit cell. Main results include: For ,

$\quad\quad\quad\,({\varphi _{\bf h} + \varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } ) \bmod 2\pi = 0.\hfill$

For

$({\varphi _{\bf h} + \varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } ) \bmod 2\pi = \left\{ \matrix{ 0\quad{\rm if }\quad | {E_{\bf h} } |{}^2 + | {E_{\bf k} } |{}^2 + | {E_{ - {\bf h} - {\bf k}} } |{}^2 - 2 \,\gt\, 0, \hfill \cr \pi\quad {\rm if }\quad | {E_{\bf h} } |{}^2 + | {E_{\bf k} } |{}^2 + | {E_{ - {\bf h} - {\bf k}} } |{}^2 - 2 \,\lt\, 0\,. \hfill } \right.$

For

, $\cos({\varphi _{\bf h} + \varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } )$ is obtained in an exact closed form as a rational function of nine magnitudes $| {E_{\bf h} } |$ , $| {E_{\bf k} }|$ , $| {E_{{\bf h} + {\bf k}} } |$ , $| {E_{{\bf h} - {\bf k}} } |$ , $| {E_{{\bf h} + 2{\bf k}} } |$ , $| {E_{2{\bf h} + {\bf k}} } |$ , $| {E_{2{\bf h}} }|$ , $| {E_{2{\bf k}} }|$ , $| {E_{2{\bf h} + 2{\bf k}} } |$ . For $N = 1,2,3,\ldots$ ,

$\displaylines{ \quad\tan ({\varphi _{\bf h} - \alpha _{\bf h} } ) \approx {{ - \textstyle\sum\limits_{\bf k} {W_{{\bf hk}} } \sin ({\varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } )} \Big/ {\textstyle\sum\limits_{\bf k} {W_{{\bf hk}} } \cos ({\varphi _{\bf k} + \varphi _{ - {\bf h} - {\bf k}} } ),}} \hfill\cr\quad \alpha _{\bf h} = \left\{ \matrix{ 0\quad{\rm if }\quad\textstyle\sum\limits_{\bf k} {\Delta _{{\bf hk}} | {\Delta _{{\bf hk}} } |} \,\gt\, 0 ,\hfill \cr \pi \quad{\rm if }\quad\textstyle\sum\limits_{\bf k} {\Delta _{{\bf hk}} | {\Delta _{{\bf hk}} } |}\, \lt\, 0 ,\hfill \cr} \right. \quad \Delta _{{\bf hk}} = | {E_{\bf h} }|{}^2 +| {E_{\bf k} } |{}^2 + | {E_{ - {\bf h} - {\bf k}} } |{}^2 - ({a /N}), \hfill\cr \quad W_{{\bf hk}} = | {\Delta _{{\bf hk}} E_{\bf h} E_{\bf k} E_{ - {\bf h} - {\bf k}} } | [{1 - ({1 - | {\Delta _{{\bf hk}} } |{}^{ - 1} } )\tanh ({N - b} )} ]. \hfill}$

Triplet discriminant $\Delta _{{\bf hk}}$ and triplet weight $W_{{\bf hk}}$ parameters, $a \approx 4.0$ and $b \approx 3.0$ , respectively, were determined empirically in numerical error analyses. Tests with phases calculated for few-atoms `super-glob' models of the protein apo-D-glyceraldehyde-3-phosphate dehydrogenase (~10000 non-H atoms) showed that low-resolution phases from the new few-atoms tangent formula were much better than conventional tangent formula phases for

and 3; phases from the two formulae were essentially the same for $N \ge 4$ .

Keywords: direct methods; few-atoms structure.