Subject: SUMMARY: Removing Translation and Rotation after MD

From: Gerald Loeffler <[email-masked]>
Date: Tue, 08 Jul 1997 12:20:40 +0200

Hi!

Quite some time ago (sorry) I asked how exactly to remove translation
and rotation of a molecule after an MD simulation, because I felt that a
simple superimposition of all the structures to the start structure such
that the rmsd deviations of respective atoms is minimal might not
allways be appropriate.

A special "Thank You" to
	Paul Soper
	Frederic A. Van-Catledge
	Konrad Hinsen
	Michael K. Gilson
	Andrew Dalke
for taking the time to answer.

The bottom line for me is as follows:
Removing translation and rotation works _exactly_ only for the rigid
rotator. Since I'm analyzing proteins which are quite far away from
being rigid, there is not _one_ way of removing translation and
rotation. So I might as well just superimpose the structures as I've
allways done it (using profit).
Furthermore, this method is guaranteed to work, whereas methods based on
the inertia tensor might run into ambiguity problems or might align the
proteins in surprising ways (wrong handedness, ...).

	Thanks to all who responded!!
	gerald

========================================================================
My original question:
I need to remove the translational and rotational motion of a protein
after a Molecular Dynamics simulation.

I know, this is a seemingly trivial problem.

A) Up to now I didn't care much about the details and just fitted all
protein structures along the MD trajectory to the start structure such
that the C-alpha atoms are optimally (i.e., minimal root-mean-square
deviation) superimposed (e.g. using the excellent profit program).

B) However, it may be that for my current problem the other alternative
that came to my mind would be better suited, namely
        1) removing translation by superimposing the centers of gravity
           of all structures
        and subsequently
        2) removing rotation by superimposing the "principal axes of
           inertia" of all structures.

Firstly, I'm not sure if the terminology of B)2) is correct.

Secondly, I'm missing the formulas to do B)1) and B)2).

Thridly, I'd like to know if there are cases where either method A) or
method B) are definitely prefered.

Fourthly, it may be that there are other methods of removing translation
and rotation?!

Fifthly, which free program implements method B) or other intersting
methods?
========================================================================
Paul Soper:
    Many molecular modeling programs include the ability to do
both the translation and rotation you describe.  The math is
simple and should be in any introductory quantum text under a
discussion of 'the rigid rotor.'

    You're welcome to the source code of program I wrote (in 'C')
which does what you want.  The input and output formats for the
molecular structure are probably not those you'd prefer, but it
would certainly give you a starting point.  Please let me know if
you'd like a copy.
========================================================================
Frederic A. Van-Catledge:
There are any number of code (fragments) floating around that will do
both steps
in your option B.  The trick is to insure that you do not generate the
optical
antipode in the process.  This can be done by, after diagonalization of
the 
moment of inertia tensor, applying the transformation to the unit
vectors of the
starting coordinate system.  One then applies a "handedness" check to
the new
unit vectors.  If the "handedness" has changed, one does a sign-flip on
one set
(x, y, or z) of the new coordinates.  selecting the correct one can get
a little 
tricky, I suspect, if you are trying to preserve continuity in an MD
trajectory.
========================================================================
Konrad Hinsen: (in German, as you will notice...)
> A) Up to now I didn't care much about the details and just fitted all
> protein structures along the MD trajectory to the start structure such
> that the C-alpha atoms are optimally (i.e., minimal root-mean-square
> deviation) superimposed (e.g. using the excellent profit program).

Und dagegen spricht auch nichts.

> B) However, it may be that for my current problem the other alternative
> that came to my mind would be better suited, namely
>       1) removing translation by superimposing the centers of gravity
>          of all structures
>       and subsequently
>       2) removing rotation by superimposing the "principal axes of
>          inertia" of all structures.

Das fuehrt zum selben Ergebnis mit mehr Aufwand. Noch eine
aequivalente Methode, die auch sehr effizient ist, ist die direkte
Umsetzung der Definition: man berechnet den Gesamtimpuls P, den
Gesamtdrehimpuls T, und den Traegheitstensor I, Schritt fuer Schritt
entlang der Trajektorie. Daraus berechnet man die
Schwerpunktsgeschwindigkeit P/M (M = Gesamtmasse) und die
Rotationsgeschwindigkeit (I^(-1) * T), und zieht beide von den
atomaren Geschwindigkeiten ab.

Das ist die einfachste und schnellste Methode, wenn man die
Geschwindigkeiten hat oder berechnen kann. Wenn man nur
einzelne Konfigurationen hat, bleibt nur der Fit.

> Fifthly, which free program implements method B) or other intersting
> methods?

Eigentlich sollte jedes Programm, dass MD-Trajektorien erzeugen kann,
auch eine Option zur Eliminierung der Translation und Rotation haben!
========================================================================
Michael K. Gilson:
I don't think your question is "trivial".  I think there are multiple
answers that are potentially "correct", because the question is
ambiguous. Before embarking upon a discussion of these issues, I'll
say that the bottom line of my response is that the proper way to
handle this will depend upon precisely what it is you want to
accomplish.  That is, why do you need to remove these motions?

Now for the discussion:

1) To begin, the words "translation" and "rotation" are ambiguous.
Recall that the molecular partition function involves integrals over
both momentum and position.  Therefore, in the fullest sense of the
words, "translation" and "rotation" refer not to 6 but to 12 degrees
of freedom: 6 spatial coordinates and 6 momenta. Therefore, if you
want to "remove the translational and rotational motion of a
molecule", it is not clear whether you are worrying about
  a) the translation and rotation of a molecule *in space*;
  b) the translational and rotational *momentum* of a molecule.

2) The traditional separation of translational and rotational degrees
of freedom from the internal degrees of freedom of a molecule is much
more natural for an essentially rigid molecule than for a flexible
one.  Thus, stat thermo courses and books use this separation in the
context of the rotor-harmonic oscillator approximation for a diatomic
molecule.  For such a molecule, the vibrations have very little effect
upon the rotational moments of inertia and therefore are separable
>from external rotations to a good approximation.  I would not trust
that this separation would hold to a good approximation for a flexible
molecule.  This said, people *have* made this separation when using
normal mode analysis of proteins.  This seems reasonable so long as
one is willing to live with the rather drastic harmonic approximations
required for normal mode analysis.  Given these approximations, the
rigid rotor-harmonic oscillator approximation seems okay.

3) As noted above, you might be interested in either the momenta or
the spatial coordinates.  If you are interested in spatial
coordinates, then it turns out that rotation and translation are still
ambiguous.  The reason is that one can separate internal from external
(translation/rotation) coordinates in any number of ways (Biophys
J. 35:7819,1996).  For example, one can define the "position" of the
molecule as the position of its center of coordinates; all other
atomic positions are then defined relative to this center.  But one
can also define the "position" of the molecule as the position of atom
number 1, for example.  It is just as legitimate to define the
positions of all other atoms relative to atom 1 as to the center of
coordinates.  The orientation of the molecule in space is similarly
ambiguous.  In summary, the position and orientation in space are
ambiguous for flexible molecules.

Well, this has probably made you more rather than less confused!
So I return to the bottom line: the real answer to your question
depends upon what precisely you wish to accomplish.
========================================================================
Andrew Dalke:
  I don't think the aligning of the principal axes of inertia
would work too well.  Consider uranium hexaflouride.  Its
elliposoid of inertia is a sphere but it is not rotationally
symmtetric.  Your alignment would stop saying there was a
degeneracy whereas an RMSD alignment would not have a problem.

> it may be that there are other methods of removing translation
> and rotation?!

  The one I used is Kabsch, Acta Cryst. (1978) A34, 827-828, which
does one way of doing RMSD alignment.
========================================================================
-- 

Gerald Loeffler
PostDoc in Theoretical Biophysics

EMail: [email-masked]
Phone: +43 1 79730 554
Fax:   +43 1 7987153
SMail: I.M.P. - Research Institute of Molecular Pathology
       Dr. Bohr-Gasse 7
       A-1030 Vienna
       AUSTRIA