summary dm-averaging

[Jan Abendroth <Jan.Abendroth@uni-koeln.de>]

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Dear all,
thanks a lot for all the reply concerning my question on dm-averaging.
Due to the proposals in the last week here is a short summary:

MY QUESTION:
> Within the the asymmetric unit of my crystal there are six copies of the
> protomer grouped as three dimers A (consisting of A1 and A2), B and C. 
> I could determine the following symmetry operators:
> inter-dimer: A -> B
>              B -> C
> intra-dimer: A1 -> A2 (from subunit 1 of dimer A to subunit 2 of dimer 1)
>              B1 -> B2
>              C1 -> C2
> 
> All symmetry operators are 2-folds with a transition-component. None of
> them falls on a crystallographic axis.

SORRY FOR THIS CONFUSING STATEMENT. I MENT THAT THE OPERATOR CONSISTS OF
THE 3X3 MATRIX AND A VECTOR.

> 
> For the symmetry operations A -> B, A -> B and A1 -> A2 I use A as mask,
> for B1 -> B2 i use B as mask and for C1 -> C2 C as mask.
> 
> The question:
> Are these symmetry operations enough to cover the whole symmetry? 
> Or do I have to specify the operations A -> C, A1 -> B1, A1 -> B2 ,...
> seperately?


REPRESENTATIVELY FOR ALL THE ANSWERS THE SUGGESTION FROM KEWIN COWTAN:
> The following fundamental principle should be applied...
> 
> YOU ONLY NEED ONE MASK FOR `dm'. IF YOU HAVE MORE THAN ONE MASK YOU
> ARE DOING SOMETHING WRONG (unless you are an expert and are solving a
> very specific problem, in which case you know all about it).
> 
> What you need to do is supply one mask, which covers A1, and the
> operators which map A1-A1(*), A1-A2, A1-B1, A1-B2, A1-C1, A1-C2.
> 
> If your dimer rotations are self-inverse, then you mask could equally
> well cover A instead of A1.
> 
> If all 6 ncs ops form a closed group, then you could use a haxamer mask.
> 
> (*)-the identity.
> 

HOW TO GENERATE THE SYMMETRY OPERATOR FROM A1-B2 IF YOU JUST KNOW THE
OPERATORS A-B AND B1-B2?

(A1-B2) = (B1-B2)x(A-B): multiply the matrices, take the right order!

As all symmetry operators X consist of a 3x3 matrix M AND a vector t for
the multiplication you have to multiply the augmented 4x4 matrices:

             (m11 m12 m13  t1)
X= M + t =>  (m21 m22 m23  t2)
             (m31 m32 m33  t3)
             ( 0   0   0    1)

Take maple or a lot of paper to calculate the multiplication.

THE RESULT
nicely smoothed averaged and phase extended maps!!!!!!!!

Thanks again

Jan



-- 
------------------------------
Jan Abendroth
Institut fuer Biochemie
Universitaet Koeln
Zuelpicher Strasse 47
D-50674 Koeln

Tel: +49-(0)221 470 6455 (Labor)
                    6445 (Buero)
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