[phenixbb] measuring the angle between two DNA duplexes

Tim Gruene tg at shelx.uni-ac.gwdg.de
Tue Jan 21 01:25:15 PST 2014

```Hi Pavel,

that's the method described in
http://journals.iucr.org/a/issues/2011/01/00/sc5036/index.html ;-) based
on the moments of inertia (a computer scientist might name it
differently). I am not sure, though, you would get the desired result
for short helices. E.g. a helix defined by three atoms the eigenvalue
would point roughly in the direction of the external phosphates, which
is far from parallel with the helix axis.

Best,
Tim

On 01/21/2014 04:20 AM, Pavel Afonine wrote:
> Hi Ed,
>
> interesting idea! Although I was thinking to have a tool that is a
> little more general and a little less context dependent. Say you have
> two clouds of points that are (thinking in terms of macromolecules) two
> alpha helices (for instance), and you want to know the angle between the
> axes of the two helices. How would I approach this?..
>
> First, for each helix I would compute a symmetric 3x3 matrix like this:
>
> sum(xn-xc)**2             sum(xn-xc)*(yn-xc) sum(xn-xc)*(zn-zc)
> sum(xn-xc)*(yn-xc)     sum(yn-yc)**2 sum(yn-yc)*(yz-zc)
> sum(xn-xc)*(zn-zc)     sum(yn-yc)*(yz-zc)        sum(zn-zc)**2
>
> where (xn,yn,zn) is the coordinate of nth atom, the sum is taken over
> all atoms, and (xc,yc,zc) is the coordinate of the center of mass.
>
> Second, for each of the two matrices I would find its eigen-values and
> eigen-vectors, and select eigen-vectors corresponding to largest
> eigenvalues.
>
> Finally, the desired angle is the angle between the two eigen-vectors
> found above, which is computed trivially.
> I think this a little simpler than finding the best fit for a 3D line.
>
> What you think?
>
> Pavel
>
>
> On 1/20/14, 2:14 PM, Edward A. Berry wrote:
>>
>>
>> Pavel Afonine wrote:
>> . .
>>
>>> The underlying procedure would do the following:
>>>    - extract two sets of coordinates of atoms corresponding to two
>>> provided atom selections;
>>>    - draw two optimal lines (LS fit) passing through the above sets
>>> of coordinates;
>>>    - compute and report angle between those two lines?
>>>
>>
>> This could be innacurate for very short helices (admittedly not the
>> case one usually would be looking for angles), or determining the axis
>> of  a short portion of a curved helix. A more accurate way to
>> determine the axis- have a long canonical duplex constructed with its
>> axis along Z (0,0,1). Superimpose as many residues of that as required
>> on the duplex being tested, using only backbone atoms or even only
>> phosphates. Operate on (0,0,1) with the resulting operator (i.e. take
>> the third column of the rotation matrix) and use that as a vector
>> parallel to the axis of the duplex being tested.
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>
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--
Dr Tim Gruene
Institut fuer anorganische Chemie
Tammannstr. 4
D-37077 Goettingen

GPG Key ID = A46BEE1A

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