[phenixbb] measuring the angle between two DNA duplexes

Pavel Afonine pafonine at lbl.gov
Mon Jan 20 19:20:39 PST 2014

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 

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?


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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