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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 94-95   | 1 | 2 |

Section 1.3.4.4.6. Derivatives for variational phasing techniques

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, 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[link]; Britten & Collins, 1982[link]; Narayan & Nityananda, 1982[link]; Bryan et al., 1983[link]; Wilkins et al., 1983[link]; Bricogne, 1984[link]; Navaza, 1985[link]; Livesey & Skilling, 1985[link]). Sayre's use of the squaring method to improve protein phases (Sayre, 1974[link]) also belongs to this category, and is amenable to the same computational strategies (Sayre, 1980[link]).

These methods differ from the density-modification procedures of Section 1.3.4.4.3.2[link] 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[link]).

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[link]; Burch et al., 1983[link]) 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[link]). In doing so, special positions and reflections must be taken into account, as in Section 1.3.4.2.2.8.[link] 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.

References

Bricogne, G. (1982). Generalised density modification methods. In Computational Crystallography, edited by D. Sayre, pp. 258–264. New York: Oxford University Press.
Bricogne, G. (1984). Maximum entropy and the foundations of direct methods. Acta Cryst. A40, 410–445.
Britten, P. L. & Collins, D. M. (1982). Information theory as a basis for the maximum determinant. Acta Cryst. A38, 129–132.
Bryan, R. K., Bansal, M., Folkhard, W., Nave, C. & Marvin, D. A. (1983). Maximum-entropy calculation of the electron density at 4 Å resolution of Pf1 filamentous bacteriophage. Proc. Natl Acad. Sci. USA, 80, 4728–4731.
Bryan, R. K. & Skilling, J. (1980). Deconvolution by maximum entropy, as illustrated by application to the jet of M87. Mon. Not. R. Astron. Soc. 191, 69–79.
Burch, S. F., Gull, S. F. & Skilling, J. (1983). Image restoration by a powerful maximum entropy method. Comput. Vision Graphics Image Process. 23, 113–128.
Livesey, A. K. & Skilling, J. (1985). Maximum entropy theory. Acta Cryst. A41, 113–122.
Narayan, R. & Nityananda, R. (1982). The maximum determinant method and the maximum entropy method. Acta Cryst. A38, 122–128.
Navaza, J. (1985). On the maximum-entropy estimate of the electron density function. Acta Cryst. A41, 232–244.
Sayre, D. (1974). Least-squares phase refinement. II. High-resolution phasing of a small protein. Acta Cryst. A30, 180–184.
Sayre, D. (1980). Phase extension and refinement using convolutional and related equation systems. In Theory and Practice of Direct Methods in Crystallography, edited by M. F. C. Ladd & R. A. Palmer, pp. 271–286. New York, London: Plenum.
Wilkins, S. W., Varghese, J. N. & Lehmann, M. S. (1983). Statistical geometry. I. A self-consistent approach to the crystallographic inversion problem based on information theory. Acta Cryst. A39, 47–60.








































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