Clarification on Rebuilding Reference Frames from the Middle Frame

[nrawal222]

Dear Xiang-Jun,

I have been having trouble rebuilding the base reference frames from the middle frame and base pair parameters outputted by x3DNA/DSSR.

Middle Frame: [[0.825, 0.565, 0.001], [-0.565, 0.824, -0.032], [-0.018, 0.026, 1.0]]
Base Pair Parameters: [-0.176, -0.054, 0.306, -0.548, -9.232, -3.527]

I am implementing the method from the x3DNA user manual (attached below). It seems that one of the frames I am calculating (labeled F2) has its y and z axes reflected.

Reference Frames calculated by me:
F1c: [[ 0.838,  0.54   ,  0.082], [-0.539,  0.841, -0.034], [-0.087, -0.016,  0.996]]
*F2c: [[ 0.806,  0.586, -0.0802], [-0.59,  0.807, -0.025], [ 0.051 ,  0.067,  0.996]]

*y and z axes are reflected

Reference Frames outputted by DSSR:
F1: [[0.838, 0.54, 0.082], [-0.539, 0.841, -0.034], [-0.087, -0.016, 0.996]]
F2: [[0.806, 0.587, -0.081], [0.59, -0.807, 0.024], [-0.051, -0.067, -0.996]]

I can just multiply the frame by this matrix: [[ 1,  0,  0], [ 0, -1,  0], [ 0,  0, -1]], but I was wondering what might be wrong with my implementation.

I have attached the formulas I am using, as well as the pdb I am analyzing and DSSR json file output. Let me know if I should attach anything else that may be helpful.

Edit: The bases I am analyzing in the pdb file are the modified 8OG-C pair (Base numbers 2 and 5).

Thank you for your time,
Nil

[xiangjun]

Hi Nil,

You've touched a subtle yet important point in 3DNA/DSSR. The 'difference' you noticed is as expected: it simply represents a 180-degree rotation around the x-axis of the reference frame of the complementary base for an M–N type pair (as is the case for WC pairs). Please read the 2003 3DNA paper (also the 2015 DSSR paper) for further information. As you know, the 3DNA source code is available for your examination.

To better understand the point, you could try the following two things:

• Attach reference frame F2 (from DSSR) to the complementary base, and show an image in 3D. Note that you also need the origin in order to show its position and orientations.
• Derive the base-pair parameters from F1c and F2c (from your calculations), you should get the same result mentioned at the top of your post.

Working through those steps would be a great exercise for you. Please post back your results on the Forum -- that would surely benefit other 3DNA/DSSR users who care about such (essential) details.

HTH,

Xiang-Jun

[nrawal222]

Thank you for the quick response Xiang-Jun.

As described in the 2003 3DNA paper (https://www.ncbi.nlm.nih.gov/pubmed/12930962), since this is a Watson-Crick base pair (M-N), and the faces of the base pairs are opposite each other (M is up, N is down), it is convention to have the frame of N rotated 180 on its x-axis, so that the frame's z-axis faces down. Thus the scalar product between the z-axis of each frame would be negative.

I have attached a picture below to show the difference between F2c (green) and F2 (blue). F2 and F2c share the same x-axis. The yellow dots are the origins of the atoms of base M, and the red dots are the origins of the atoms of base N. Rotating F2c 180 degrees about its x-axis would make it equivalent to F2.

Calculating the base pair parameters between F1c and F2c returned the same result as DSSR. This means that to calculate the base pair parameters, F2 needs to be rotated 180 degrees about its x-axis before base pair parameters are calculated.

[xiangjun]

Hi Nil,

Thanks for your follow-up! It is a good example of what I've always encouraged 3DNA/DSSR users to follow. Presumably, a non-trivial thread would include clarifications along the way and a summary on the topic (from user's perspective).

To make the point of the thread clear, I've created an image of the G2--C23 Watson-Crick pair (PDB id 355d) with base reference frames. The G2 base is filled with green, and its z-axis is pointing upwards. The C23 base is not filled, and its z-axis is pointing downwards. A rotation of the C23 base frame around its x-axis by 180 degrees would bring its y- and z-axes (yellow) parallel with the G2 y- and z-axes (green). As is clear from the figure, the rotated C23 frame is not perfectly aligned with the G2 base frame. The base-pair parameters (shear, buckle, propeller etc.) are used to characterize the deformation. An ideal, perfectly planar WC pair would have all the six parameters to be zeros --- that's how the standard base reference frame is defined.

HTH,

Xiang-Jun