The use of non-local constraints in maximum-entropy electron density reconstruction

Abstract

Different expressions of the maximum-entropy estimates of the electron density function, corresponding to different prior information are obtained. They show that no general-purpose configurational entropy of density maps exists. Some universal properties of the modellings are discussed. In particular, the meaning of super-resolution is clarified. The information of lower and upper bounds of the electron density is not in general strong enough to produce atomic maps. Atomicity is then introduced as non-local constraints and applied to the problem of phase extension using experimental data and low-resolution model phases. In all cases, the knowledge of phases up to 3.5-3 A and observed moduli up to 1.5-1 ,~ allows an estimate of the electron density of roughly the same quality as the 1 A, map obtained from a Fourier summation to be produced. IntroductionThe foundations of the theory here developed were given in a previous paper: the criterion of maximal entropy was used to obtain an estimate of the electron density function on the basis of partial information. First a maximum-entropy probability distribution of maps was obtained, its functional form being a strict consequence of the type of constraints used. Next the electron density function was estimated using this maximum-entropy probability distribution (Navaza, 1985).For the particular type of constraint considered the formulation corresponds exactly to a maximumentropy algorithm using new forms of the configurational entropy of maps and gives rise to a modelling of the maximum-entropy estimate of the sought map.In this paper a slightly different presentation is offered aiming to show that no underlying probability 0108-7673/86/040212-12501.50 distribution of maps is in fact needed in order to apply the recipes of information theory, even if we can always think in terms of frequencies in an idealized experimental situation. However, the probabilistic interpretation offers a conceptually simpler frame in which the problem of object reconstruction can be discussed.Different developments of maximum entropy have been proposed and the references can be found in the previous paper. More recent developments are those of Bricogne (1984), Livesey & Skilling (1985) and Semenovskaya, Khachaturyan & Khachaturyan (1985). It is not the aim of this paper to discuss the different formulations.Different modellings corresponding to different prior information are obtained and applied to experimental data. The results clearly showed that in the ab initio problem most maximum-entropy algorithms give uninterpretable maps. Moreover it was also found that the model phases (the 'true' phases) are not even placed in a concave region in the space of phases.Theproblem of phase extension is also considered. Good results are obtained when information on atomicity is introduced in the form of non-local constraints. From the experimentally observed moduli corresponding to 1 A resolution and the model phases up to 3.5-3 A, all the atoms were recovered for structures with differe...

Accurate structure analysis by the maximum-entropy method

“…There have been many theoretical studies (Narayan & Nityanada, 1982;Wilkins, Varghese & Lehmann, 1983;Livesey & Skilling, 1985;Navaza, 1985Navaza, , 1986Bricogne, 1988)to apply the method to problems in crystallography and, in particular, to Fourier inversion. There are many cases in crystallography which involve Fourier inversion.…”

Maxentropic Solutions to a Convex Interpolation Problem Motivated by Utility Theory

“…However, the method also admits another extension to deal with problems like (1) in which the unknown is not a density, but a function which may have any sign and not necessarily integrate to 1. Such extension seems to have been originally proposed in Navaza [22], the mathematics of which were developed by Dacunha-Castelle and Gamboa [23]. The method of maximum entropy in the mean has made its way into econometrics.…”

The use of Sayre's equation in real space

“…Although this would be possible for hypothetical structures consisting solely of identical atoms, we could expect that the exceptionally strong restriction on the densities may also be useful for more realistic structures containing heterogeneous atoms. Indeed, previous calculations based on Sayre's equation have achieved reasonable success in the field of small molecules (Krabbendam & Kroon, 1971;Navaza, 1986;Debaerdemaker, Tate & Woolfson, 1988) and macromolecules (Hoppe & Gassman, 1964;Barrett & Zwick, 1971;Sayre, 1974;Cutfield, Dodson, Dodson, Hodgkin, Isaacs, Sakabe & Sakabe, 1975;Zhang & Main, 1990;Woolfson & Yao, 1990). The present study attempts to enhance Sayre's variational method.…”

“…Sayre's equation in the form of (1) has already been used in different contexts (Navaza, 1986;Main, 1990). Some modifications have been proposed that make Sayre's equation in the form of (2) more usable (Barrett & Zwick, 1971;Sayre, 1972 Sayre, , 1975 Shiono & Woolfson, 1991).…”

“…With these modifications, the present method acquires a capability of obtaining a density map of atomic resolution even with lowresolution data. Numerical results are presented that demonstrate how the present method works at various resolutions.Sayre's equation in the form of (1) has already been used in different contexts (Navaza, 1986;Main, 1990). Some modifications have been proposed that make Sayre's equation in the form of (2) more usable (Barrett & Zwick, 1971;Sayre, 1972 Sayre, , 1975 Shiono & Woolfson, 1991).…”