# [phenixbb] measuring the angle between two DNA duplexes

Phil Evans pre at mrc-lmb.cam.ac.uk
Tue Jan 21 02:21:23 PST 2014

```One way that I've used for alpha-helices is to start with an ideal model with it's axis along say z, then get the rotation required to fit the ideal helix to the model. This works even for short helices

Phil

On 21 Jan 2014, at 09:25, Tim Gruene <tg at shelx.uni-ac.gwdg.de> wrote:

> 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
>
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