LIKELIHOOD BASED SEARCH OF THE CENTRE OF A MACROMOLECULAR OBJECT.

Petrova T.E.¹, Lunin V.J.¹, and Podjarny A.D.²

¹IMPB RAS, 142292 Pushchino, Moscow Region, Russia,
E-mail: petrova@impb.serpukhov.su

²UPR de Biologie Structurale, IGBMC, BP 163, 67404 Illkirch CEDEX CU de Strasbourg, France,
E-mail: podjarny@igbmc.u-strasbg.fr

Keywords: ab-initio structure determination, low resolution, generalized likelihood.

The knowledge of the molecule position in the crystal cell is important in the structure determination. Two approaches have been suggested previously: the molecular replacement method and the single-sphere search. The former can be used only if an atomic model or a molecular envelope is available. The latter is based on the general hypothesis that at very low resolution a molecule can be approximated by a single sphere and that the R-factor type criteria can identify the correct position of the sphere in the unit cell. Unfortunately, this is not always the case. The goal of this work is to analyse whether the ideas of the maximal likelihood approach could resolve the problem.

In the frame of the statistical approach, a molecule is considered as one of the realisations of independent trials where every atom of the molecule is placed into the unit cell randomly and independently with some prior probability distribution function. The prior function belongs to the class of functions that are equal to a positive constant inside a sphere of some radius and some position of the centre in the unit cell, and are equal to zero outside it. For any of these models, a set of structure factors can be calculated, and therefore for every prior function the likelihood value can be defined as the probability for the calculated magnitudes of the structure factors to be equal to the observed ones. According to the maximum likelihood principle which has been widely used in the last years for choosing the prior probability function [1], one can expect that the maximal value of the likelihood function corresponds to the "best" spherical envelope.

The suggested approach was realised in the Monte-Carlo type computer procedure. The generalised likelihood [2], which is similar to the likelihood function but can be easier calculated, was employed as a criterion of choice. The experiments showed that the level of the signal for the right solution is higher than in the simple one-sphere search. However, in several cases the generalised maximum function had spurious maxima, which are usually placed on the symmetry axes. In order to eliminate these false solutions, the radius of the sphere is chosen for every position of the centre so that the summary volume of all symmetry related spheres is equal to the volume of the molecule.

The work was supported by the grant 97-04-48319 of the RFBR and through the RAS-CNRS collaboration. The search of molecular envelopes of more complicated forms is under development.

1. Bricogne, G. (1990). Acta Cryst.., A46, 284-297.
2. Lunin, V.Y., Lunina, N.L., Petrova, T.E., Urzhumtsev, A.G., Podjarny, A.D. Acta Cryst., to be published .