Torsion angles of nucleic acid structures

From the very beginning, 3DNA calculates a set of nucleic acid backbone parameters, including the six main chain torsion angles (α, β, γ, δ, ε, and ζ) around the covalent bonds, χ about the glycosidic bond, and the sugar pucker (see figure below). For double helical structures, the standard analyze output (.out file) has a section for “Main chain and chi torsion angles,” and another dedicated to “Sugar conformational parameters”. Based on my experience/understanding, these two parts are well recognized and utlizied by 3DNA users. What has receive little attention (in spite of the several posts I’ve written on the topic), though, is 3DNA’s applicability to single-stranded (ss) RNA structures for the backbone torsions, among other parameters. Using the fully refined crystal structure of the Haloarcula marismortui large ribosomal subunit (PDB entry 1jj2) as an example, the procedure is below:

find_pair -s 1jj2.pdb 1jj2.nts
analyze 1jj2.nts
    # or the above two steps can be combined:
find_pair -s 1jj2.pdb stdout | analyze stdin
    # see output file '1jj2.outs'

In retrospect, the fact that 3DNA has been little used for RNA backbone conformational analysis is of no surprise:

  • While base-pair parameters have different (oftentimes confusing) definitions, these backbone parameters are pretty “standard” — thus, for example, any program for DNA/RNA structural analysis would give the same numerical values for α or χ torsion angles.
  • The two-step process as illustrated above is a bit awkward, and the torsions are “buried” among many other parameters.
  • 3DNA is more directly “linked” (conceivably) to DNA base pairs than to RNA backbone.

So while adapting the Zp parameter for ss DNA/RNA structures in 3DNA v2.1, I also take this opportunity to add the -torsion option to analyze with the following handy features:

  • Streamline the calculation by starting directly from a PDB file and output only backbone parameters. So the above example can be shortened to analyze -t=1jj2.tor 1jj2.pdb; the output file is named 1jj2.tor.
  • Classify backbone into BI/BII conformation, and base χ into syn / anti.
  • Add pseudo-torsions, and Zp and Dp as defined by Richardson et al.
  • Handle pseudouridine sensibly, and work also for nucleic acid structure with only backbone atoms.
  • Be easy to use, efficient and robust — it takes ~1 second to process the large ribosomal subunit 1jj2 (with 2876 nucleotides consisting of 23S rRNA and 5S rRNA) on my MacBook Air.

Overall, analyze -torsion is designed to be pragmatic and allows for automatic processing of all NDB entries or molecular dynamics trajectories. Given below is an excerpt of the three sections from an analyze -torsion run on 1jj2:

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Main chain and chi torsion angles: 

Note: alpha:   O3'(i-1)-P-O5'-C5'
      beta:    P-O5'-C5'-C4'
      gamma:   O5'-C5'-C4'-C3'
      delta:   C5'-C4'-C3'-O3'
      epsilon: C4'-C3'-O3'-P(i+1)
      zeta:    C3'-O3'-P(i+1)-O5'(i+1)

      chi for pyrimidines(Y): O4'-C1'-N1-C2
          chi for purines(R): O4'-C1'-N9-C4

          If chi is in range [-90, +90], syn conformation
              otherwise, it is in anti conformation

          e-z: epsilon - zeta
              BI:  e-z = [-160, +20]
              BII: e-z = [+20, +200]

              base      chi         alpha    beta   gamma   delta  epsilon   zeta     e-z
   1 0:..10_:[..U]U   -62.5(syn)     ---     ---     56.2    74.0   142.2   -87.8  -130.1(BI) 
   2 0:..11_:[..A]A   171.5(anti)   173.2  -161.0   168.5    84.0  -112.1   -65.4   -46.7(BI) 
   3 0:..12_:[..U]U  -167.7(anti)   -70.7   168.4    53.0    78.5  -128.5   -46.4   -82.1(BI) 
   4 0:..13_:[..G]G  -172.5(anti)   -61.8   170.4    67.7    73.5  -166.7   -79.6   -87.1(BI) 
   5 0:..14_:[..C]C  -166.0(anti)   -73.0  -172.5    55.1    83.2  -143.3   -77.7   -65.6(BI) 
   6 0:..15_:[..C]C  -155.5(anti)   -60.9   174.1    47.3    80.3  -154.4   -71.2   -83.2(BI) 

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Pseudo (virtual) eta/theta torsion angles:

Note: eta:    C4'(i-1)-P(i)-C4'(i)-P(i+1)
      theta:  P(i)-C4'(i)-P(i+1)-C4'(i+1)

      eta':   C1'(i-1)-P(i)-C1'(i)-P(i+1)
      theta': P(i)-C1'(i)-P(i+1)-C1'(i+1)

      eta":   Borg(i-1)-P(i)-Borg(i)-P(i+1)
      theta": P(i)-Borg(i)-P(i+1)-Borg(i+1)

              base      eta   theta    eta'  theta'    eta"  theta"
   1 0:..10_:[..U]U    ---     ---     ---     ---     ---     --- 
   2 0:..11_:[..A]A  -174.6  -129.7   177.0  -127.7  -157.5   -75.5
   3 0:..12_:[..U]U   149.1  -105.1   174.1  -101.2  -111.0   -69.4
   4 0:..13_:[..G]G   169.0  -172.5  -156.6  -169.2   -93.3  -137.1
   5 0:..14_:[..C]C   176.2  -143.4   179.6  -140.6  -144.6  -120.6
   6 0:..15_:[..C]C   165.0  -147.7   177.4  -146.8  -149.2  -121.7

****************************************************************************
Sugar conformational parameters: 

Note: v0: C4'-O4'-C1'-C2'
      v1: O4'-C1'-C2'-C3'
      v2: C1'-C2'-C3'-C4'
      v3: C2'-C3'-C4'-O4'
      v4: C3'-C4'-O4'-C1'

      tm: the amplitude of pucker
      P:  the phase angle of pseudorotation

      Zp: z-coordinate of the 3' phosphorus atom (P) expressed in the
            standard base reference frame; it's POSITIVE when P is on
            the +z-axis side (base in anti conformation); NEGATIVE if
            P is on the -z-axis side (base in syn conformation)
      Dp: perpendicular distance of the 3' P atom to the glycosydic bond
            [as per the MolProbity paper of Richardson et al. (2010)]

              base       v0      v1      v2      v3      v4     tm       P    Puckering    Zp      Dp
   1 0:..10_:[..U]U   -11.3   -15.4    34.5   -41.8    33.5    41.6    33.8    C3'-endo   -0.13    3.53
   2 0:..11_:[..A]A    11.4   -30.2    36.9   -31.5    12.6    36.9     1.2    C3'-endo    4.74    4.78
   3 0:..12_:[..U]U     3.6   -29.3    42.4   -41.3    23.8    43.6    13.9    C3'-endo    4.67    4.82
   4 0:..13_:[..G]G   -13.0   -17.8    39.8   -47.9    38.5    47.8    33.7    C3'-endo    4.45    4.46
   5 0:..14_:[..C]C     6.0   -28.4    38.9   -36.5    19.2    39.5    10.1    C3'-endo    4.57    4.70
   6 0:..15_:[..C]C     1.9   -26.4    39.6   -39.6    23.7    41.2    16.0    C3'-endo    4.32    4.61
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