Derivatives for variational phasing techniques

Bricogne, G.

doi:10.1107/97809553602060000551

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

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 87-88 | 1 | 2 |

Section 1.3.4.4.6. Derivatives for variational phasing techniques

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.4.6. Derivatives for variational phasing techniques

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Some methods of phase determination rely on maximizing a certain global criterion $[S[\rho\llap{$-\!$}]]$ involving the electron density, of the form $[{\textstyle\int_{{\bb R}^{3}/{\bb Z}^{3}}} K[\rho\llap{$-\!$}({\bf x})] \hbox{ d}^{3}{\bf x}]$ , under constraint of agreement with the observed structure-factor amplitudes, typically measured by a $[\chi^{2}]$ residual C. Several recently proposed methods use for $[S[\rho\llap{$-\!$}]]$ various measures of entropy defined by taking $[K(\rho\llap{$-\!$}) = - \rho\llap{$-\!$} \log (\rho\llap{$-\!$}/\mu)]$ or $[K(\rho\llap{$-\!$}) = \log \rho\llap{$-\!$}]$ (Bricogne, 1982; Britten & Collins, 1982; Narayan & Nityananda, 1982; Bryan et al., 1983; Wilkins et al., 1983; Bricogne, 1984; Navaza, 1985; Livesey & Skilling, 1985). Sayre's use of the squaring method to improve protein phases (Sayre, 1974) also belongs to this category, and is amenable to the same computational strategies (Sayre, 1980).

These methods differ from the density-modification procedures of Section 1.3.4.4.3.2 in that they seek an optimal solution by moving electron densities (or structure factors) jointly rather than pointwise, i.e. by moving along suitably chosen search directions $[v_{i}({\bf x})]$ [or $[V_{i}({\bf h})]$ ].

For computational purposes, these search directions may be handled either as column vectors of sample values $[\{v_{i}({\bf N}^{-1}{\bf m})\}_{{\bf m} \in {\bb Z}^{3}/{\bf N}{\bb Z}^{3}}]$ on a grid in real space, or as column vectors of Fourier coefficients $[\{V_{i}({\bf h})\}_{{\bf h} \in {\bb Z}^{3}/{\bf N}^{T}{\bb Z}^{3}}]$ in reciprocal space. These column vectors are the coordinates of the same vector $[{\bf V}_{i}]$ in an abstract vector space $[{\scr V} \cong L({\bb Z}^{3}/{\bf N}{\bb Z}^{3})]$ of dimension $[{\scr N} = |\hbox{det } {\bf N}|]$ over $[{\bb R}]$ , but referred to two different bases which are related by the DFT and its inverse (Section 1.3.2.7.3).

The problem of finding the optimum of S for a given value of C amounts to achieving collinearity between the gradients $[\nabla S]$ and $[\nabla C]$ of S and of C in $[{\scr V}]$ , the scalar ratio between them being a Lagrange multiplier . In order to move towards such a solution from a trial position, the dependence of $[\nabla S]$ and $[\nabla C]$ on position in $[{\scr V}]$ must be represented. This involves the $[{\scr N} \times {\scr N}]$ Hessian matrices H(S) and H(C), whose size precludes their use in the whole of $[{\scr V}]$ . Restricting the search to a smaller search subspace of dimension n spanned by $[\{{\bf V}_{i}\}_{i = 1, \ldots, n}]$ we may build local quadratic models of S and C (Bryan & Skilling, 1980; Burch et al., 1983) with respect to n coordinates X in that subspace: $[\eqalign{S({\bf X}) &= S({\bf X}_{0}) + {\bf S}_{0}^{T} ({\bf X} - {\bf X}_{0})\cr &\quad + {\textstyle{1 \over 2}}({\bf X} - {\bf X}_{0})^{T} {\bf H}_{0}(S) ({\bf X} - {\bf X}_{0})\cr C({\bf X}) &= C ({\bf X}_{0}) + {\bf C}_{0}^{T}({\bf X} - {\bf X}_{0})\cr &\quad + {\textstyle{1 \over 2}}({\bf X} - {\bf X}_{0})^{T} {\bf H}_{0}(C) ({\bf X} - {\bf X}_{0}).}]$ The coefficients of these linear models are given by scalar products: $[\eqalign{[{\bf S}_{0}]_{i} &= ({\bf V}_{i}, \nabla S)\cr [{\bf C}_{0}]_{i} &= ({\bf V}_{i}, \nabla C)\cr [{\bf H}_{0}(S)]_{ij} &= [{\bf V}_{i}, {\bf H}(S){\bf V}_{j}]\cr [{\bf H}_{0}(C)]_{ij} &= [{\bf V}_{i}, {\bf H}(C){\bf V}_{j}]}]$ which, by virtue of Parseval's theorem, may be evaluated either in real space or in reciprocal space (Bricogne, 1984). In doing so, special positions and reflections must be taken into account, as in Section 1.3.4.2.2.8. Scalar products involving S are best evaluated by real-space grid summation, because H(S) is diagonal in this representation; those involving C are best calculated by reciprocal-space summation, because H(C) is at worst $[2 \times 2]$ block-diagonal in this representation. Using these Hessian matrices in the wrong space would lead to prohibitively expensive convolutions instead of scalar (or at worst $[2 \times 2]$ matrix) multiplications.

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 87-88