SEMINVARIANT DENSITY DECOMPOSITION AND CONNECTIVITY ANALYSIS IN THE LOW RESOLUTION MACROMOLECULAR PHASING
A.G. Urzhumtsev1,2 , V.Y. Lunin2 & N.L. Lunina2
1 LCM3B, Faculté des Sciences, Université
Nancy 1, 54506, Vandoeuvre-les-Nancy, France
2 IMPB, RAS,
Pushchino, Moscow Region, 142292, Russia
Keywords : very-low-resolution,
macromolecules, phase problem, connectivity, seminvariants
When analysing very low resolution (VLR) electron density maps for biological macromolecules, it is expected to find one globular image per molecule. The connectivity analysis used earlier to improve mean-resolution density maps [1] is proposed to be used as the selection criterion in low-resolution multivariant search methods like the histogram [2] or the FAM [3] ones by comparing the number of connected components with the number of molecules. Unfortunately, a straightforward search for the combination of VLR phases which results in the map of the desired number of connectivity components fails in a number of applications. Seminvariant density decomposition (SDD) gives a light for possible reasons and suggests a way to overcome the difficulties.
Let G ={uj}j=0,…,m-1 , u0=(0,0,0) be a subgroup of the group of admissible crystallographic origin shifts. An electron density distribution (r) can be represented in the form
r(r) = roi(r) + rov(r),
where
roi(r) = [ Sj=0,…,m-1 r(r - uj) ] / m,
rov(r) = [(m-1)/m] r(r) - [S j=1,…,m-1 (r - uj) ] / m .
We call this representation seminvariant density decomposition (SDD) corresponding to the subgroup G. The Fourier series for origin-invariant roi(r) contains all structure factors, and only them, which phases are G-seminvariant, i.e. they do not change their values for all origin shifts of the G-group. As a consequence, the structure factors which phases do change their values for one of the origin shifts of the G-group are presented in the Fourier series for origin-variable part ov(r) in SDD. The function roi(r), which is a sum of m shifted copies of the same image, will dominate in the Fourier synthesis image if seminvariants dominate in the used set of structure factors, which is usually the case at a VLR.
The non-invariant part rov(r) is somehow similar to a difference synthesis and represents one positive globular image per molecule. This image is distorted by some of its m-1 flipped and shifted copies, but nevertheless it may be suitable for the connectivity analysis.
The distortion of the molecular shape by negative copies in rov(r) may be reduced when combining SDD maps corresponding to different subgroups of the group G. This kind of procedures, similar to minimal functions for Patterson analysis, can be also applied to the series of the roi(r) functions.
The possibility of an enantiomer image may be also taken into
consideration.