On its face, an angle is defined by two vectors; let's call them a and b, and if each is normalized, then the angle (in degrees) between them is: acos(dot(a, b)) * 180/pi. Geometrically, after moving the tails of the vectors into the same position (e.g., origin), the heads would normally define a plane, unless a and b are strictly parallel (0°) or anti-parallel (180°).

DNA structures are three-dimensional, normally far more complicated than a single number can quantify. The concept of DNA bending angle, as I understand it, is only applicable to DNA structures with two relatively straight fragments (as in CAP-DNA complexes). Under such situations, one can fit a least-squares (ls) linear helical axis to each of the two fragments, and calculate the angle between them. Towards this end, 3DNA outputs the following section when it judges that the input structure is not strongly curved. Using 355d/bdl084, which is distributed with 3DNA, as an example:

Global linear helical axis defined by equivalent C1' and RN9/YN1 atom pairs Deviation from regular linear helix: 3.30(0.52) Helix: -0.127 -0.275 -0.953 HETATM 9998 XS X X 999 17.536 25.713 25.665 HETATM 9999 XE X X 999 12.911 15.677 -9.080 Average and standard deviation of helix radius: P: 9.42(0.82), O4': 6.37(0.85), C1': 5.85(0.86)Where the Helix: line gives the normalized vector along the "best-fit" helical axis. The two HETATM records provides the two end points on the helix, and they are directly related to the Helix: line by a simple equation. Following the above example, we have (Octave/Matlab code):

XE = [12.911 15.677 -9.080];

XS = [17.536 25.713 25.665];

dd = XE - XS

% -4.6250 -10.0360 -34.7450

Helix = dd / norm(dd)

% -0.12685 -0.27526 -0.95296 ==> [-0.127 -0.275 -0.953]

With the two HETATM records, one can easily add them into the original PDB file to display the helical axis using a molecular graphics programs (e.g., RasMol, Jmol, PyMol). Moreover, the two helix vectors can be used to reorient the original PDB structure into a view so that one helical fragment lies along the x-axis, and the other in the xy-plane. As documented in detail in recipes #4 on "Automatic identification of double-helical regions in a DNA–RNA junction" (2008 3DNA

*Nature Protocols*paper), "The chosen view allows for easy visualization and protractor measurement of the overall bending angle between the two relatively straight helices."

The following points are well worth noting:

- The ls fitting procedure used in 3DNA follows SCHNAaP, which was based on the algorithm in the well-known NewHelix program, maintained by Dr. Richard Dickerson upto the 1990s. While fitting a global linear helical axis to strongly curved DNA structures clearly makes no much sense with derived parameters (NewHelix itself has been replaced by FreeHelix, also from Dickerson), I do believe it is meaningful to fit a linear helix with relatively straight DNA fragment. That's why I have kept this functionality in SCHNArP and 3DNA; 3DNA bending angle calculation serves as an example illustrating the point – it provides an "intuitive" way for biologists to understand how the bending angle is calculated; it can actually be measured directly.
- Instead of directly ls-fitting a linear helical axis with 3DNA, one can alternatively superimpose a regular fiber model into the DNA fragment, and then derive the straight helical axis from the fitted coordinates. The two approaches normally gives slightly different numerical values, as would be expected.
- Overall, bending angle is (at most) an approximate measure of DNA curvature. In my opinion, the concept is only applicable for comparing a set of structures, each with two relatively straight fragments. Even in such cases, the relative spatial relationship between two helical fragments is more complicated than a simple (bending) angle could quantify. Specially, it is not meaningful to make a selling point in slight differences of DNA bending angles.