3DNA: a suite of software programs for the analysis, rebuilding and visualization of three-dimensional nucleic acid structures

A new paper titled Analyzing and Building Nucleic Acid Structures with 3DNA has been published in JoVE (Journal of Visualized Experiments). Specifically, the article illustrates 3DNA’s unique capability to characterize and modify DNA structures at the level of the constituent base-pair steps, and highlights a new feature in v2.1 to analyze and align an ensemble of related structures determined with NMR or generated by MD simulations.

Here is the abstract:

The 3DNA software package is a popular and versatile bioinformatics tool with capabilities to analyze, construct, and visualize three-dimensional nucleic acid structures. This article presents detailed protocols for a subset of new and popular features available in 3DNA, applicable to both individual structures and ensembles of related structures. Protocol 1 lists the set of instructions needed to download and install the software. This is followed, in Protocol 2, by the analysis of a nucleic acid structure, including the assignment of base pairs and the determination of rigid-body parameters that describe the structure and, in Protocol 3, by a description of the reconstruction of an atomic model of a structure from its rigid-body parameters. The most recent version of 3DNA, version 2.1, has new features for the analysis and manipulation of ensembles of structures, such as those deduced from nuclear magnetic resonance (NMR) measurements and molecular dynamic (MD) simulations; these features are presented in Protocols 4 and 5. In addition to the 3DNA stand-alone software package, the w3DNA web server, located at http://w3dna.rutgers.edu, provides a user-friendly interface to selected features of the software. Protocol 6 demonstrates a novel feature of the site for building models of long DNA molecules decorated with bound proteins at user-specified locations.

A new section dedicated to the JoVE paper will be set up on the 3DNA Forum soon. It will contain all the data files and scripts so our published results can be strictly reproduced. The section should also serve as a platform for open discussions of related protocols.

Early on when I started on DNA structures, I read Saenger’s book Principles of Nucleic Acid Structure and became familiar with his classification of the 28 possible base-pairs (bps) for A, G, U(T), and C involving at least two (cyclic) hydrogen bonds (see figure below).

Later on, I read from the 2nd edition of The RNA World book a list of 29 bps compiled by Burkard, Turner & Tinoco. While the one bp discrepancy (28 vs 29) has been in my mind for quite a long while, I had never paid much attention to the issue until recently while adding classifications of RNA bps (among many other functionalities) to 3DNA. A Google search did not help solve the puzzle, so I decided to dig it out by comparing the two lists.

The Burkard et al. list is titled Structures of Base Pairs Involving at Least Two Hydrogen Bonds and it mentions specifically Saenger’s list:

The structures of 29 possible base pairs that involve at least two hydrogen bonds are given in Figures 1–5 (for further descriptions, see Saenger, in Principles of nucleic acid structure, p. 120. Springer-Verlag [1984]).

However, in the five figures, Burkard et al. do not provide the corresponding Saenger numbers (I to XXVIII, 1—28) for the 28 common bps; thus it is not immediately obvious which one (i.e., the new addition by Burkard et al.) is missing from Saenger’s list. Under careful scrutiny, the absent bp turns out to be the “G•C N3-amino, amino-N3” pair in Figure 3: “Six possible flipped purine-pyrimidine mismatches.” One example of such G+C pair is found in the 5S ribosomal RNA (chain 9, G3022—C3026) of Haloarcula marismortui in PDB entry 1vq8.

The above figure shows clearly that the G+C bp does indeed have two canonical H-bonds between base atoms, and it is difficult to speculate how it escaped Saenger’s selection criteria. In the upcoming new 3DNA component, I am listing this bp as number XXIX (29), along with the other 28 base pairs.

Recently, I came across the so-called Calcutta U-U base pair (bp) [see figure below] while reading articles on C-H…O contacts in nucleic acid structures. Not familiar with this named pair before, I was curious to find out what it’s about. After some searching, I traced the origin of the Calcutta U-U bp to the following two papers published by Sundaralingam’s group during the middle 1990s:

• Wahl et al. (1996). The structure of r(UUCGCG) has a 5’-UU-overhang exhibiting Hoogsteen-like trans UU base pairs. Nat. Struct. Biol., 3(1), 24-31. In Note added in proof, the author wrote:

We have called the novel U•U base pair, where the Hoogsteen face of one of the pyrimidines is involved in a C5-H—O4 hydrogen bond, the ‘Calcutta Base Pair’, since it was announced at the International Seminar-cum-School on Macromolecular Crystallographic Data held in Calcutta, November 16-20, 1995.

• Wahl & Sundaralingam (1997). C-H…O hydrogen bonding in biology. Trends Biochem Sci., 22(3), 97-102. In this review article, the authors noted:

We recently discovered a novel U•U base pair, referred to as the Calcutta base pair, in the crystal structure of an RNA hexamer UUCGCG (Ref. 18). The two uracil bases form a conventional N(3)-H…O(4) and an unconventional C(5)-H…O(2) hydrogen bond (Fig. 3a). The C-H…O interaction is entirely ‘voluntary’ and not ‘forced’, underlining its importance in base mispairing.

3DNA has no problem to identify the Calcutta U-U bps (or any pair for that matter); an example is shown below based on the RNA hexamer UUCGCG structure (PDB entry: 1osu) solved by Sundaralingam and colleagues.

In the new 3DNA component I’ve been working on (and to be released soon), the Calcutta U-U pair is characterized as below:

1/A.U1 3/A.U2 [U-U] Calcutta 00-n/a tHW -MW
  anti C3'-endo 8.9 --- anti C3'-endo 30.3
  dcc=11.18  dnn=8.48  dmm=7.58  tor=-174.1
  H-bonds[2]: "O4(carbonyl)-N3(imino)[2.76]; C5-O4(carbonyl)[3.27]"

  Shear=-3.67   Stretch=-0.52     Stagger=-0.89
  Buckle=-1.41  Propeller=-16.03  Opening=-90.67

The Calcutta pair is explicitly named, along with other named base pairs (e.g., Watson-Crick [WC], Wobble, and Hoogsteen bps). It is classified as type tHW (trans with Hoogsteen/WC interacting edges), following the commonly used Leontis-Westhof nomenclature. It does not belong to any of the 28 bps (00-n/a) with at least two conventional H-bonds, as categorized by Saenger. In 3DNA, the Calcutta U-U pair is of M-N type, designated as -MW.

Among the well-known named base pairs, some are after the scientists who discovered them (e.g., WC and Hoogsteen bps), while others are based on chemical/geometrical features (e.g., Wobble and Sheared G-A bps), or a combination of both (e.g., reversed WC/Hoogsteen bps). The Calcutta U-U pair is unique in that it is named after a place in India:

Kolkata, or Calcutta, is the capital of the Indian state of West Bengal. … While the city’s name has always been pronounced Kolkata or Kolikata in Bengali, the anglicized form Calcutta was the official name until 2001, when it was changed to Kolkata in order to match Bengali pronunciation.

Prior to v2.1, 3DNA does not provide any direct support for the analysis of molecular dynamics (MD) simulations trajectories of nucleic acid structures. Nevertheless, over the years, I noticed some significant applications of 3DNA in the active MD field; see my blog post (December 6, 2009) titled 3DNA in the PCCP nucleic acid simulations themed issue. In January 2011, I released a set of two Ruby scripts specifically aimed to facilitate the analysis of MD simulations trajectories. Thereafter (as of 3DNA v2.1), I have significantly refined and expanded the Ruby scripts, and consolidated the functionality under one umbrella, x3dna_ensemble with multiple sub-commands (analyze, block_image, extract, and reorient). I believe x3dna_ensemble would make it straightforward to analyze ensembles (NMR or MD simulations trajectories) of nucleic acid structures.

Under this background, I am glad to read recently an article titled Structure, Stiffness and Substates of the Dickerson-Drew Dodecamer in J. Chem. Theory Comput. where 3DNA was used extensively. This work represents a re-visit of the classic Dickerson−Drew B-DNA dodecamer d-[CGCGAATTCGCG]2 using state-of-the-art MD simulations with different ionic conditions and solvation models, and compares the MD trajectories with modern crystallographic and NMR data. Among the author list (Tomas Drsata, Alberto Perez, Modesto Orozco, Alexandre Morozov, Jiri Sponer, and Filip Lankas) are some well-known figures in the MD field of nucleic acid structures.

Reading through the text, I am not sure if the newly available functionality of x3dna_ensemble was used. From the excerpts of the citations given below, however, it seems obvious that 3DNA is now well-accepted by the MD community.

Snapshots taken in 10 ps intervals were analyzed using the 3DNA program.43 From 3DNA outputs, time series of conformational parameters were extracted. These included the intra-base-pair coordinates (buckle, propeller, opening, shear, stretch, and stagger), inter-base-pair or step coordinates (tilt, roll, twist, shift, slide, and rise) as well as groove widths (based on P−P distances), backbone torsions, and sugar puckers.

Contrary to the original work of Lankas et al.,31 the intra-base-pair and step coordinates used here are those defined by 3DNA.43

Here, we apply this model together with the 3DNA definitions of the intra-base-pair and step coordinates.43

However, important differences remain, and non- negligible differences are in fact observed between individual experimental structures also in the central part of DD, even though the intra-base-pair and step coordinates are computed using the same coordinate definitions64 (we consistently use the 3DNA coordinates in this work).

Glycosidic bond “is a type of covalent bond that joins a carbohydrate (sugar) molecule to another group, which may or may not be another carbohydrate.” In nucleic acid structures, the other group is a nucleobase, and the predominated type is the N-glycosidic bond where the purine (A/G) N9 or pyrimidine (C/T/U) N1 atom connects to the C1′ atom of the five-membered (deoxy) ribose sugar ring. Another well-known type is the C-glycosidic bond in pseudouridine, the most common modified base in RNA structures where the C5 atom instead of N1 is linked to the C1′ atom of the sugar ring.

Recently, I performed a survey of all nucleic-acid-containing structures in the PDB/NDB database to see how many types of glycosidic bond are there. As always, I noticed some inconsistencies in the data: nucleotides with disconnected base/sugar, a base labeled as U but with pseudoU-type C-glycosidic bond. Shown below are a few unusual types of glycosidic bond in otherwise seemingly “normal” structures:

• The residue GN7 (number 28 on chain A) in PDB entry 1gn7 contains a N7-glycosylated guanine.

• The residue UPG (number 501 on chain A) in PDB entry 1y6f has sugar C1C (instead of C1′) atom connects to N1 of U.

• The residue XAE (number 11 on chain B) in PDB entry 2icz contains a benzo-homologous adenine.

• The residue F5H (number 206 on chain B) in PDB entry 3v06 has N1 of U connects to C2′ of a six-membered sugar ring.

The unusual glycosidic bond has implications in 3DNA calculated parameters, for example the chi torsion angle. Identifying such cases would help refine 3DNA to provide sensible parameters and to avoid possible misinterpretations.

As of today (2012-09-16), the number of 3DNA forum registrations has reached 500! A quick browse of the ‘Statistics Center’ shows that over 80% of the registrations (400+) are after March 2012, when the new 3DNA homepage/forum were launched.

The sharp increase in registration is mostly due to the streamlined, web-based way to distribute the 3DNA software package. As far as I know, the number of 3DNA registrations/downloads in the past six months is significantly higher than that of 3DNA v2.0 for over three years. Equally importantly, I have been able to fixed every reported bug, addressed each feature request, and updated the 3DNA v2.1 distribution promptly.

I also feel confident to declare that up to now, the 3DNA Forum is spam free (at least to the extent I am aware). To this end, I’ve taken the following three measures:

• Installation of the SMF “Mod Stop Spammer”; as of this writing, it shows “3920 Spammers blocked up until today”.
• By using 3DNA-related verification questions. At its current setting, a user must answer correctly three of the ‘simple’ yet effective verification questions. Early on, I decided deliberately not to use CAPTCHA as an anti-spam means, based on my past experience.
• I’ve continuously monitored (new) registrations, and taken immediate actions against any suspicious registration. Due to the effectiveness of above two steps, so far I only have to manually handle just a few spam registrations. Nevertheless, it does illustrate the fact that no automatic method is perfect, and expert inspection is required to ensure desired results.

Overall, the new simplified way to distribute the 3DNA software package is working as intended; now users can easily access all distributed versions of 3DNA, and I can focus on support and further development of the software.

From a pure structural perspective, the designation of the two strands in an anti-parallel DNA duplex is sort of arbitrary. Thus, for a given PDB file, let’s assume that the atomic coordinates of chain A (strand I) come before those of chain B (strand II). We can swap the order of the two chains as they appear in the PDB file, i.e., list first the atomic coordinates of chain B and then those of chain A.

Structurally, the two settings corresponding to exactly the same DNA molecule. As far as 3DNA goes, however, the different orderings do make a different in calculated parameters. Using the Dickerson B-DNA dodecamer CGCGAATTCGCG solved at high resolution (PDB entry 355d) as an example, running 3DNA find_pair and analyze on ‘355d.pdb’ gives the results (abbreviated) below:

find_pair 355d.pdb 355d.bps
    # contents of file '355d.bps':
------------------------------------------------------------------
355d.pdb
355d.out
    2         # duplex
   12         # number of base-pairs
    1    1    # explicit bp numbering/hetero atoms
    1   24  0 #    1 | ....>A:...1_:[.DC]C-----G[.DG]:..24_:B<....
    2   23  0 #    2 | ....>A:...2_:[.DG]G-----C[.DC]:..23_:B<....
    3   22  0 #    3 | ....>A:...3_:[.DC]C-----G[.DG]:..22_:B<....
    4   21  0 #    4 | ....>A:...4_:[.DG]G-----C[.DC]:..21_:B<....
    5   20  0 #    5 | ....>A:...5_:[.DA]A-----T[.DT]:..20_:B<....
    6   19  0 #    6 | ....>A:...6_:[.DA]A-----T[.DT]:..19_:B<....
    7   18  0 #    7 | ....>A:...7_:[.DT]T-----A[.DA]:..18_:B<....
    8   17  0 #    8 | ....>A:...8_:[.DT]T-----A[.DA]:..17_:B<....
    9   16  0 #    9 | ....>A:...9_:[.DC]C-----G[.DG]:..16_:B<....
   10   15  0 #   10 | ....>A:..10_:[.DG]G-----C[.DC]:..15_:B<....
   11   14  0 #   11 | ....>A:..11_:[.DC]C-----G[.DG]:..14_:B<....
   12   13  0 #   12 | ....>A:..12_:[.DG]G-----C[.DC]:..13_:B<....
------------------------------------------------------------------

analyze 355d.bps
    # generate output file '355d.out', with base-pair step parameters:
****************************************************************************
    step       Shift     Slide      Rise      Tilt      Roll     Twist
   1 CG/CG      0.09      0.04      3.20     -3.22      8.52     32.73
   2 GC/GC      0.50      0.67      3.69      2.85     -9.06     43.88
   3 CG/CG     -0.14      0.59      3.00      0.97     11.30     25.11
   4 GA/TC     -0.45     -0.14      3.39     -1.59      1.37     37.50
   5 AA/TT      0.17     -0.33      3.30     -0.33      0.46     37.52
   6 AT/AT     -0.01     -0.60      3.22     -0.31     -2.67     32.40
   7 TT/AA     -0.08     -0.40      3.22      1.68     -0.97     33.74
   8 TC/GA     -0.27     -0.23      3.47      0.68     -1.69     42.14
   9 CG/CG      0.70      0.78      3.07     -3.66      4.18     26.58
  10 GC/GC     -1.31      0.36      3.37     -2.85     -9.37     41.60
  11 CG/CG     -0.31      0.21      3.17     -0.68      6.69     33.31
****************************************************************************

Reversing the order of chains A and B in ‘355d.pdb’ as ‘355d-reversed.pdb’ and repeating the above procedure, we have the following results:

find_pair 355d-reversed.pdb 355d-reversed.bps
    # contents of file '355d-reversed.bps':
------------------------------------------------------------------
355d-reversed.pdb
355d-reversed.out
    2         # duplex
   12         # number of base-pairs
    1    1    # explicit bp numbering/hetero atoms
    1   24  0 #    1 | ....>B:..13_:[.DC]C-----G[.DG]:..12_:A<....
    2   23  0 #    2 | ....>B:..14_:[.DG]G-----C[.DC]:..11_:A<....
    3   22  0 #    3 | ....>B:..15_:[.DC]C-----G[.DG]:..10_:A<....
    4   21  0 #    4 | ....>B:..16_:[.DG]G-----C[.DC]:...9_:A<....
    5   20  0 #    5 | ....>B:..17_:[.DA]A-----T[.DT]:...8_:A<....
    6   19  0 #    6 | ....>B:..18_:[.DA]A-----T[.DT]:...7_:A<....
    7   18  0 #    7 | ....>B:..19_:[.DT]T-----A[.DA]:...6_:A<....
    8   17  0 #    8 | ....>B:..20_:[.DT]T-----A[.DA]:...5_:A<....
    9   16  0 #    9 | ....>B:..21_:[.DC]C-----G[.DG]:...4_:A<....
   10   15  0 #   10 | ....>B:..22_:[.DG]G-----C[.DC]:...3_:A<....
   11   14  0 #   11 | ....>B:..23_:[.DC]C-----G[.DG]:...2_:A<....
   12   13  0 #   12 | ....>B:..24_:[.DG]G-----C[.DC]:...1_:A<....
------------------------------------------------------------------

analyze 355d-reversed.bps
    # generate output file '355d-reversed.out', with base-pair step parameters:
****************************************************************************
    step       Shift     Slide      Rise      Tilt      Roll     Twist
   1 CG/CG      0.31      0.21      3.17      0.68      6.69     33.31
   2 GC/GC      1.31      0.36      3.37      2.85     -9.37     41.60
   3 CG/CG     -0.70      0.78      3.07      3.66      4.18     26.58
   4 GA/TC      0.27     -0.23      3.47     -0.68     -1.69     42.14
   5 AA/TT      0.08     -0.40      3.22     -1.68     -0.97     33.74
   6 AT/AT      0.01     -0.60      3.22      0.31     -2.67     32.40
   7 TT/AA     -0.17     -0.33      3.30      0.33      0.46     37.52
   8 TC/GA      0.45     -0.14      3.39      1.59      1.37     37.50
   9 CG/CG      0.14      0.59      3.00     -0.97     11.30     25.11
  10 GC/GC     -0.50      0.67      3.69     -2.85     -9.06     43.88
  11 CG/CG     -0.09      0.04      3.20      3.22      8.52     32.73
****************************************************************************

Comparing the base-pair step parameters between ‘355d.out’ and ’355d-reversed.out’, one would notice that while slide/rise/roll/twist simply switch orders, shift/tilt (the x-axis parameters) also flip their signs. On the other hand, the nucleotide serial numbers specifying base pairs (the left two columns) are identical in ‘355d.bps’ and ’355d-reversed.bps’.

Apart from explicitly swapping the two strands in PDB data file, one can simply switch around the nucleotide serial numbers generated with find_pair in order to analyze a DNA duplex based on its complementary sequence instead of the primary one. For example, starting from the same PDB file ‘355d.pdb’, we change ‘355d.bps’ to ’355d-cs.bps’ as below,

------------------------------------------------------------------
355d.pdb
355d-cs.out
    2         # duplex
   12         # number of base-pairs
    1    1    # explicit bp numbering/hetero atoms
   13   12
   14   11
   15   10
   16    9
   17    8
   18    7
   19    6
   20    5
   21    4
   22    3
   23    2
   24    1
------------------------------------------------------------------

Run analyze 355d-cs.bps, one would get exactly the same parameters in output file ’355d-cs.out’ as in ’355d-reversed.out’.

As of v2.1, I’ve switched from Perl to Ruby as the scripting language for 3DNA. Consequently, the Perl scripts in previous versions of 3DNA (v1.5 and v2.0) are now obsolete. I’ll only correct bugs in existing Perl scripts, but will not add any new features.

For back reference, the scripts are still available from a separate directory $X3DNA/perl_scripts, with the following contents:

OP_Mxyz*          dcmnfile*         nmr_strs*
README            del_ms*           pdb_frag*
block_atom*       expand_ids*       x3dna2charmm_pdb*
blocview.pl*      manalyze*         x3dna_r3d2png*
bp_mutation*      mstack2img*       x3dna_setup.pl*
cp_std*           nmr_ensemble*     x3dna_utils.pm

Among them, x3dna_setup.pl and blocview.pl have corresponding Ruby versions: x3dna_setup and blocview. Actually, the .pl file extension (for Perl) was added to avoid confusion with the new Ruby scripts.

Some of the functionalities have been incorporated into the Ruby script x3dna_utils:

------------------------------------------------------------------------
A miscellaneous collection of 3DNA utilities
    Usage: x3dna_utils [-h|-v] sub-command [-h] [options]
    where sub-command must be one of: 
        block_atom -- generate a base block schematic representation
        cp_std -- select standard PDB datasets for analyze/rebuild
        dcmnfile -- remove fixed-name files generated with 3DNA
        x3dna_r3d2png -- convert .r3d to image with Raster3D or PyMOL
------------------------------------------------------------------------
  --version, -v:   Print version and exit
     --help, -h:   Show this message

Along the same line, ensemble-related functionalities (for NMR or molecular dynamics simulations) have been consolidated and extended into the new Ruby script x3dna_ensemble:

------------------------------------------------------------------------
Utilities for the analysis and visualization of an ensemble
    Usage: x3dna_ensemble [-h|-v] sub-command [-h] [options]
    where sub-command must be one of: 
        analyze -- analyze MODEL/ENDMDL delineated ensemble (NMR or MD)
        block_image -- generate a base block schematic image
        extract -- extract structural parameters after running 'analyze'
        reorient -- reorient models to a particular frame/orientation
------------------------------------------------------------------------
  --version, -v:   Print version and exit
     --help, -h:   Show this message

Conceivably, C programs in 3DNA can also be consolidated. For backward compatibility, however, all existing C programs will be kept — and refined as necessary — in the current 3DNA v2.x series. As of v3.x, I’ll completely re-organize 3DNA incorporating my years of experience in programming languages and knowledge of macromolecular structures.

In 3DNA, each base pair (bp) is specified by the identity of its two comprising nucleotides (nts), and their interactions. Some examples are shown below based on the PDB entry 1ehz (the crystal structure of yeast phenylalanine tRNA at 1.93 Å resolution), with the shorthand form on the right:

....>A:...1_:[..G]G-----C[..C]:..72_:A<....  G-C
....>A:...4_:[..G]G-*---U[..U]:..69_:A<....  G-U
....>A:...9_:[..A]A-**+-A[..A]:..23_:A<....  A+A
....>A:..15_:[..G]G-**+-C[..C]:..48_:A<....  G+C
....>A:..26_:[M2G]g-**--A[..A]:..44_:A<....  g-A

Specification of a nucleotide

The nt specification string consists of 6 fields and follows the pattern below, with the number of characters in each field inside the parentheses:

modelNum(4)>chainId(1):ntNum(4)insCode(1):[ntName(3)]baseName(1)

1. modelNum(4) — the model number is up to 4 digits, right-justified, with each leading space replaced by a dot. If no model number is available, as is the case for 1ehz (and virtually all other x-ray crystal structures in the PDB), it is written as .... (4 dots).
2. chainId(1) — the chain id is 1-char long, with space replaced by underscore.
3. ntNum(4) — the nt residue number, handled as for the model number.
4. insCode(1) — insertion code, handled as for the chain id.
5. ntName(3) — the nt residue name is up to 3-char long, right-justified, with each leading space replaced by a dot.
6. baseName(1) — the base name is 1-char long, mapped from ntName(3) following $X3DNA/config/baselist.dat. Note that modified nucleotides are put in lower case to distinguish them from the canonical ones — for example, M2G to g.

For the complementary base in a bp, the order of the 6 fields is reversed — see examples above. To see the full list of nts in a PDB data file, run: find_pair -s 1ehz.pdb stdout (here using 1ehz as an example).

Specification of a base pair

The pattern of a bp is M-xyz-N, where M and N are 1-char base names (as in aforesaid field #6), and the three characters xyz have the following meaning:

• z — the sign of the dot product of the z-axes of the M and N base reference frames. It is positive (+) if the two z-axes point in similar directions, as in Hoogsteen or reverse Watson-Crick bps. Conversely, it is negative (-) when the two z-axes point in opposite directions, as in the canonical Watson-Crick and Wobble bps. See figure below:

• y — it is - if M and N are in a so-called Watson-Crick geometry (the two y-axes of the M and N base reference frames are anti-parallel, so are the two z-axes, whilst the two x-axes are parallel), e.g., the G-U Wobble pair; otherwise, *.
• x — it is - for Watson-Crick bps, otherwise, *.

By design, Watson-Crick bps would be of the pattern M-----N, Wobble bps M-*---N, and non-canonical bps M-**+-N or M-**--N. Thus by browsing through the 3DNA output, users can readily identify these three bp types.

The shortened form is represented as MzN; following aforementioned notation, it can be either M-N or M+N. The relative direction of the two z-axes is critical in effecting 3DNA-calculated bp (and step) parameters, as detailed in the 2003 3DNA NAR paper:

To calculate the six complementary base pair parameters of an M–N pair (Shear, Stretch, Stagger, Buckle, Propeller and Opening), where the two z‐axes run in opposite directions, the reference frame of the complementary base N is rotated about the x2‐axis by 180°, i.e. reversing the y2‐ and z2‐axes in Figure 2a. Under this convention, if the base pair is reckoned as an N–M pair, rather than an M–N pair, the x‐axis parameters (Shear and Buckle) reverse their signs. For an M+N pair, e.g. the Hoogsteen A+U in Figure 2b, the x2‐, y2‐ and z2‐axes do not change sign; thus all six parameters for an N+M pair are of opposite sign(s) from those for an M+N pair.

The M-N and M+N bp designation is unique to 3DNA. In combination with the corresponding 6 bp parameters (shear, stretch, stagger, buckle, propeller, and opening), 3DNA provides a rigorous description of all possible bps. This contrasts and complements with the conventional Saenger scheme and the 3-edge based Leontis/Westhof notation.

The 3DNA M-N vs M+N bp designation is base-centric, without concerning the sugar-phosphate backbone. The chi (χ) torsion angle, which characterizes base/sugar relative orientation, can be in either anti or syn conformation; thus similar backbone(S) can accommodate either M-N or M+N.