Pseudo-torsions to simplify the representation of DNA/RNA backbone conformation

Backbone conformation of nucleic acid structures is most characterized by a set of 6 torsion angles (α, β, γ, δ, ε, and ζ) around the consecutive chemical bonds, chi (χ) quantifying the relative base/sugar orientation, plus the sugar pucker.

This large number of DNA/RNA backbone conformational parameters is in striking contrast to the two torsion angles (φ and ψ) in protein structures, routinely employed in Ramachandran plot. Over the years, the nucleic acid community has come up with simplified ways to represent DNA/RNA backbone conformation. Thus far, the most widely used one is the pseudo-torsion angles (See figure below) η: C4′(i-1)-P(i)-C4′(i)-P(i+1) and θ: P(i)-C4′(i)-P(i+1)-C4′(i+1).

The history of the P—C4′ virtual-bond concept and its application in RNA structure analysis have recently been reviewed by Pyle et al. in A new way to see RNA [Q Rev Biophys. 2011, 44(4), 433—466], where the following three contributions are highlighted:

  1. Olson (1980). Configurational statistics of polynucleotide chains. An updated virtual bond model to treat effects of base stacking., Macromolecules 13(3), 721—728.
  2. Malathi & Yathindra (1980). A novel virtual bond scheme to probe ordered and random coil conformations of nucleic acids: Configurational statistics of polynucleotide chains. Current Science, 49, 803—807.
  3. Duarte & Pyle (1998). Stepping through an RNA structure: A novel approach to conformational analysis. Journal of Molecular Biology, 284, 1465—1478.

More recently, Pyle et al. also employed a modified version of the pseudo-torsions, η′: C1′(i-1)-P(i)-C1′(i)-P(i+1) and θ′: P(i)-C1′(i)-P(i+1)-C1′(i+1), i.e., using C1′ instead of C4′, and found that:

The η′ and θ′ torsions are more suitable when interpreting crystallographic density because the C1′ atom is covalently bound to the nucleoside base and therefore can be more easily and accurately located within a low-resolution map.

While implementing the -torsion option to analyze to make it more explicit that 3DNA readily calculates conventional backbone torsion angles, I also take this opportunity to add the pseudo-torsion angles — η/θ and η′/θ′, among other new parameters. Moreover, while I am at it, I cannot help but also compute yet another set of pseudo-torsion angles: η″/θ″. Here, instead of C1′ or C4′, the origin of the base reference frame is employed; it can be taken as a pseudo-atom more accurately defined by the base plane than any real single atom.

The usefulness of η″/θ″, especially in comparison with η/θ and η′/θ′, remains to be determined. However, only η″/θ″ uniquely takes advantage of the two most accurately determined entities in a nucleic acid structure, the heavy phosphorus atom and the rigid base plane [see discussion (p.16) in the Richardson et al. MolProbity paper, Acta Cryst. (2010). D66, 12–21] Presumably, η″/θ″ provides a new perspective in RNA structural analysis by combining the backbone and the base.

Here is the pseudo-torsions for the yeast phenylalanine transfer RNA (6tna by simply running analyze -torsion=6tna.tor 6tna.pdb):

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 A:...1_:[..G]G    ---   -126.6    ---   -141.5    ---   -130.4
   2 A:...2_:[..C]C   167.8  -168.3   174.6  -152.5  -151.4  -115.4
   3 A:...3_:[..G]G   160.4  -119.8  -171.9  -138.9  -123.6  -119.2
   4 A:...4_:[..G]G   148.0  -164.2   162.1  -159.2  -154.4  -124.6
   5 A:...5_:[..A]A   168.7  -137.6  -175.9  -137.8  -129.5  -115.0
   6 A:...6_:[..U]U   171.8  -145.7  -172.5  -140.5  -131.3  -124.7
   7 A:...7_:[..U]U  -151.0   -47.8  -136.0   -58.6  -117.7   -30.2
   8 A:...8_:[..U]U   160.9   159.7  -161.0  -163.6  -144.2   178.0
   9 A:...9_:[..A]A  -137.0   -48.6  -158.1  -108.9   161.5  -104.7
  10 A:..10_:[2MG]g    33.1  -135.8    93.4  -134.6   134.1  -113.0
  11 A:..11_:[..C]C   167.2  -138.3  -179.4  -137.7  -142.4  -118.7
  12 A:..12_:[..U]U   165.5  -120.7  -179.3  -128.0  -145.8  -106.7
  13 A:..13_:[..C]C   174.1  -173.6  -165.5   179.6  -120.9  -180.0
  14 A:..14_:[..A]A   173.0  -144.0   172.7  -132.4   177.6   -72.7
  15 A:..15_:[..G]G   154.7   110.6  -176.2    85.5   -97.7   -76.9
  16 A:..16_:[H2U]u    76.3    94.1    65.3   119.7  -152.8  -123.8
  17 A:..17_:[H2U]u   -36.7   -79.6   -50.7  -136.6  -142.7  -159.0
  18 A:..18_:[..G]G    -9.7  -166.8    41.7  -158.6    28.9  -120.4
  19 A:..19_:[..G]G  -131.6   -35.8  -122.9   -67.8  -104.3   -10.5
  20 A:..20_:[..G]G   160.9   -93.2  -161.6   -98.9  -174.1  -112.3
  21 A:..21_:[..A]A   -83.6   152.5   -72.8   155.7   -59.1   155.4
  22 A:..22_:[..G]G   164.1   169.4   160.0  -178.5   159.1  -157.6
  23 A:..23_:[..A]A   177.6  -148.5  -174.5  -142.7  -154.5  -114.3
  24 A:..24_:[..G]G   167.2   -98.9  -171.7  -128.6  -127.6   -99.1
  25 A:..25_:[..C]C   151.6  -153.5   167.3  -140.8  -137.7   -84.8
  26 A:..26_:[M2G]g   156.2  -137.4  -175.2  -135.2  -100.0  -104.2
  27 A:..27_:[..C]C   166.2  -145.5  -177.9  -140.4  -129.1  -116.8
  28 A:..28_:[..C]C   164.7  -140.5   175.8  -145.3  -152.7  -123.4
  29 A:..29_:[..A]A   161.2  -145.3   175.7  -144.9  -142.0  -126.0
  30 A:..30_:[..G]G  -173.5  -120.3  -158.4  -133.2  -126.6   -94.4
  31 A:..31_:[..A]A   169.8  -153.1   177.7  -140.4  -124.5   -81.5
  32 A:..32_:[OMC]c   154.4  -126.8  -178.7  -131.3  -104.1  -128.0
  33 A:..33_:[..U]U   170.0  -103.9  -179.9  -152.7  -164.6   143.6
  34 A:..34_:[OMG]g    -4.7  -123.7    41.8  -124.8    31.6   -99.6
  35 A:..35_:[..A]A   163.5  -104.3   176.9  -127.9  -137.5  -128.2
  36 A:..36_:[..A]A   175.9   173.6   180.0  -167.7  -156.4  -118.3
  37 A:..37_:[.YG]g   166.8  -131.7  -174.5  -133.0  -115.1   -82.9
  38 A:..38_:[..A]A   167.7  -121.6  -175.7  -114.3  -109.9   -79.9
  39 A:..39_:[PSU]P   168.3  -146.8  -160.2  -146.4   -98.6  -116.5
  40 A:..40_:[5MC]c   160.6  -138.7   174.0  -141.8  -139.7  -126.5
  41 A:..41_:[..U]U   164.8  -161.4   175.9  -152.3  -150.5  -117.6
  42 A:..42_:[..G]G   174.3  -140.9  -170.3  -145.4  -129.1  -121.3
  43 A:..43_:[..G]G   169.6  -159.0  -176.2  -154.9  -133.7  -133.1
  44 A:..44_:[..A]A   174.0  -121.5  -174.2  -122.0  -143.1   -74.9
  45 A:..45_:[..G]G   174.4  -132.5  -166.2  -128.1  -101.8  -128.9
  46 A:..46_:[7MG]g  -112.8  -113.4  -127.2  -138.3  -139.8  -152.1
  47 A:..47_:[..U]U   -63.2   -53.8    -1.1   -92.0    22.8  -124.7
  48 A:..48_:[..C]C   -84.7    59.6   -20.1     8.9    19.3  -104.5
  49 A:..49_:[5MC]c   -56.8  -140.1   -29.9  -143.6    98.1  -125.4
  50 A:..50_:[..U]U   173.6  -146.4  -178.3  -140.6  -147.6  -117.8
  51 A:..51_:[..G]G   160.8  -148.1  -178.6  -150.7  -140.7  -121.9
  52 A:..52_:[..U]U   164.9  -144.0   175.8  -143.5  -139.9  -114.3
  53 A:..53_:[..G]G   168.2  -140.9  -171.1  -144.0  -121.6  -117.3
  54 A:..54_:[5MU]u   167.0  -131.1   178.3  -124.9  -139.9   -77.0
  55 A:..55_:[PSU]P   167.6  -114.2  -172.8  -155.6  -113.0   146.0
  56 A:..56_:[..C]C    35.0  -121.5    52.6  -126.2    26.5   -83.8
  57 A:..57_:[..G]G   168.4  -148.1  -177.1  -131.1  -115.4  -111.7
  58 A:..58_:[1MA]a  -136.3  -133.3  -106.5  -176.7  -105.3   149.6
  59 A:..59_:[..U]U    23.0  -130.9    33.0  -115.4    48.2   -68.2
  60 A:..60_:[..C]C  -163.6   -54.3  -123.2   -76.4   -79.6   -36.4
  61 A:..61_:[..C]C   125.5  -153.3   169.7  -144.7  -153.8  -123.4
  62 A:..62_:[..A]A   172.5  -139.3  -177.0  -137.6  -150.7  -114.6
  63 A:..63_:[..C]C   165.8  -146.6  -178.5  -149.8  -139.2  -127.8
  64 A:..64_:[..A]A   164.7  -144.9   176.5  -145.8  -145.3  -118.1
  65 A:..65_:[..G]G   170.4  -152.3  -175.5  -151.5  -132.3  -122.1
  66 A:..66_:[..A]A   168.0  -152.0  -177.4  -150.2  -133.0  -118.7
  67 A:..67_:[..A]A   170.9  -141.8  -178.4  -140.4  -134.8  -123.1
  68 A:..68_:[..U]U   164.8  -135.1  -178.9  -137.9  -143.7   -95.2
  69 A:..69_:[..U]U   168.2  -154.9  -174.3  -157.1  -112.2  -144.8
  70 A:..70_:[..C]C   160.6  -153.2   170.7  -153.5  -164.4  -125.1
  71 A:..71_:[..G]G   161.8  -144.3   172.1  -143.1  -145.7  -124.2
  72 A:..72_:[..C]C   176.7  -136.4  -169.3  -134.5  -134.9   -87.1
  73 A:..73_:[..A]A   160.6  -142.8  -179.7  -139.7  -112.8  -104.4
  74 A:..74_:[..C]C  -176.9  -115.9  -163.1  -115.4  -117.2   -68.7
  75 A:..75_:[..C]C   169.8    80.9  -170.0    74.9  -108.5   -91.3
  76 A:..76_:[..A]A    ---     ---     ---     ---     ---     --- 
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