Subject: trasformation from z-matrix (or internal coordinates) to Cartesian coordinates

From: "Logean Antoine" <[email-masked]>
Date: Thu, 14 Mar 2002 17:10:33 +0100

Two or three weeks ago I posted a "geometrical problem" : 

> Taken a residue in a PDB file, how can I calculate the Cartesian 
> coordinates of CB when I know the coordinates of N, CA and C, the 
> angle N-CA-CB, C-CA-CB and the bond length CA-CB ?

I received a lot of answers (15 !) and so by solving a system of 3
equations (thank you Maple V !), 2 equations of cones (angles N-CA-CB
and C-CA-CB) and one equation of sphere (length CA-CB), I could find the
solutions : 2 points corresponding to the respective R and S amino acid
configuration. 
So thank you all (Andrew Horsfield, Daquan Gao, Ingo Brunberg, Jim
Phillips, John, Lakshmi Devi Kesavan, Michael Jakusch, Michel PETITJEAN,
Paynes, Richard and Sigismondo Boschi ) for your help.

But now I have an other similar geometrical problem concerning the
dihedral angle :

> Taken a residue in a PDB file, how can I calculate the Cartesian 
> coordinates of CG when I know the coordinates of N, CA and CB as well
as 
> the angle N-CA-CB, CA-CB-CG, the dihedral angle N-CA-CB-CG and of 
> course the bond length CB-CG ?
  
In fact this two calculations are nothing else (thank you Sigismondo !)
as a transformation of internal coordinates, represented by a Z-matrix,
to Cartesian coordinates.

Do you know how can I solve the second geometrical problem ?

Until know the only solution I have is extremely complicated : 

1) 	calculation of an Dummy atom CG' ("projection" of CG on the plan
N-
	CA-CB)
2)	translation of CA-CB-CG' to the referential, CA becoming (0,0,0)

3)	rotation around Z and Y axis to align the axis CA-CB to the
X-axis
4)	rotation around the X-axis of the value of the dihedral angle
5)	reverse transformations 2, 3, 4 and 5 to put back all the things

I am sure there is a better way to solve this problem.

I am also interested in finding a general mathematical description of
"transformation of internal coordinates to Cartesian coordinates".

Thank you all for your help ... I am waiting for answers ... 

Antoine

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
LOGEAN Antoine                              
Bioinformatics group of Dr. Rognan
UMR CNRS-ULP 7081
74, route de Rhin - B.P. 24
F-67401 Illkirch, France
                         
phone   : 0033 - (0)6 87 01 57 32
e-mail  : [email-masked]
www     : http://med-chem.u-strasbg.fr/~tony/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~