How to quantify the relative geometry between two base-pairs? — Part 1

In the field of double helical DNA (and RNA) structures, the following two sets of parameters to normally used to quantify the relative geometry of the two base pairs in a dinucleotide step:

• shift, slide, rise, tilt, roll, and twist — which I call them stacking or simple step parameters
• x-displacement, y-displacement, helical rise, inclination, tip, and helical twist — which I call them helical parameters

3DNA calculates all of these parameters. Over the years, I have been approached quite a few times on the local helical parameters interpretation question. In literature, I've noticed numerous times of confusions the community still has over the two set of parameters, especially with regard to the issue of "twist vs helical twist" and "rise vs. helical rise". This series of blog posts is aimed to clarify the problem by providing some background information and step-by-step worked examples illustrating how the parameters are actually calculated in 3DNA (which follows SCHNAaP/SCHNArP). In my experience, knowing how the parameters are derived is the key to understand what they mean, and thus to avoid misinterpretations.

Part 1 — background information

1. The Calladine and El-Hassan Scheme (CEHS) calculates only the six step parameters, i.e., shift, slide, rise, tilt, roll, and twist. The original CEHS implementation was a few FORTRAN subroutines taking advantage the code base of Dickerson's well-known NewHelix program. In developing SCHNAaP/SCHNArP, a collaborative project between the then Sheffield and Cambridge groups, I also went through the source code of NewHelix to get a thorough knowledge of its internals in order to integrate its nice features into SCHNAaP.
   

   One thing I noticed was x-displacement, a parameter characterizing the "hole" of A-DNA in top view. The global helical axis from NewHelix was (still is) appealing to me, especially for a short, relatively straight fragment. Even for a clearly non-straight duplex, e.g., in super helical nucleosome core particle DNA, or a severely kinked DNA, the deviation from regular linear helix serves as a good parameter to quantify its overall non-linearity. As a side note, a single so-called bending angle is often misleading — for example, the bending angle as commonly cited in literature (mostly calculated from Curves), is strongly influenced by the two terminal base-pairs.

   So I devised a parallel set of global helical parameters following the CEHS scheme of angle combination: here instead of roll-tilt, I used tip-inclination. The definition of global helical axis and the point the helix passes through are based on my simplified implementation in ANSI C of the algorithms used by NewHelix.
   

   Thus SCHNAaP calculates both a set of local CEHS step parameters, and a set of global helical parameters — a unique combination of CEHS and NewHelix, made possible only after a thorough understanding of both methods. SCHNArP was developed to complete the circle, i.e., to rebuild a structure given a set of parameters, either the local step parameters: shift, slide, rise, tilt, roll, and twist, or the global helical parameters: x-displacement, y-displacement, helical rise, inclination, tip, and helical twist.

2. The RNA (Running Nucleic Acids) program by Babcock et al. calculates a set of local helical parameters. While working at Rutgers in Dr. Olson's group, I was interested in knowing how the local helical axis and the point its passes through were defined in the RNA program, since it would naturally substitute for the global one in SCHNAaP/SCHNArP to make two sets of purely local parameters. As it turned out, however, the algorithm was finally implemented in the 3DNA software package, as summarized in the email I sent out before the 13th Conversation at Albany:

   The way to calculate the local helical axis and helical parameters in 3DNA is *essentially the same* as in RNA, but expressed in a much simpler way. First, the local helical axis is calculated as dx-times-dy, following Bansal, which gives the *same* result as the "single rotation axis" detailed in the RNA paper. I still could not figure out WHY, but verified this numerically. The location where the helix passes through is based directly on the RNA paper (i.e., following Chasles's theorem). The procedure to calculate helical parameters, i.e., tip/inclination/x-disp/y-disp/etc, following the SCHNAaP paper (i.e, with tip-inclination combination, which is consistent with the roll-tilt, propeller-buckle combinations in 3DNA). What amazed us is that this much simpler tip-inclination implementation in 3DNA gives *exactly* the same numerical values as the original RNA algorithm.

3. In summary, the two sets of local parameters as implemented in 3DNA are based on a unique combination of nice features from several programs well-known in the community, including SCHNAaP/SCHNArP (CEHS), NewHelix, RNA, and NUPARM. However, the 3DNA calculated parameters are numerically different from any of them, partially because 3DNA adopts the standard base reference frame. To the best of my knowledge, 3DNA is the only software that allows for rigorous conversion between the two sets of local parameters — a utility program in 3DNA called step_hel illustrates this simple fact. Again, as I wrote before the 13th Conversation at Albany:

   Take each base-pair as a rigid block, only 6 parameters are required to relate one bp to the other. Two sets are commonly used: one set is "shift, slide, rise, tilt, roll and twist", and the other set is "x-displacement, y-displacement, helical rise, inclination, tip, and helical twist". Obviously, these two sets of parameters should be directly convertible, as demonstrated by Calladine and Drew in their 1984 JMB B-to-A transition paper, and illustrated in 3DNA manuscript.

4. Links to other programs referred to in this section:
   • SCHNAaP/SCHNArP by Lu et al.
     
   • The RNA program by Babcock et al.
     
   • FreeHelix98 by Dickerson
   • NUPARM from Banal et al.
     
   • Standards, especially the email exchanges before the 13th Conversation at Albany