Network Properties of the Ensemble of RNA Structures - Fig 1

<p><i>(Left)</i> Network for the toy 7-mer GGGGCCC which has 8 nodes and 16 edges (hence 32 directed edges). The expected network degree is <math><mrow><mn>32</mn><mn>8</mn><mo>=</mo><mn>4</mn></mrow></math>. Red edges indicate base pair addition or removal, while blue edges indicate shift moves. <i>(Center)</i> Feynman circular representation of secondary structure of Y RNA. <i>(Right)</i> Conventional representation of secondary structure of Y RNA. According to [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref055" target="_blank">55</a>], one function of Y RNA is to bind to certain misfolded RNAs, including 5S rRNA, as part of a quality control mechanism. The secondary structure depicted is the consensus secondary structure of Y RNA with EMBL access number AAPY01489510:220–119 from Rfam family RF00195 in the Rfam database [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref056" target="_blank">56</a>]. Images produced with sofware jViz [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref057" target="_blank">57</a>].</p

Boltzmann relative frequency for the number of neighbors for the 56 nt spliced leader RNA from <i>L. collosoma</i>, where the mean ± one standard deviation is 69.87 ± 34.04 [resp. 90.46 ± 37.71] for move set MS1 [resp. MS2] using energy model C (Turner 2004 energy parameters).

<p>The length-normalized sample mean is 1.2477 ± 0.6079 for MS1 [resp. 1.6153 ± 0.6734 for MS2]. The number of neighbors, or degree, is given on the <i>x</i>-axis. RNAsubopt -d0 -e 12 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref010" target="_blank">10</a>] was used to generate 266,065 structures <i>s</i> having free energy within 12 kcal/mol of the MFE. The sum <i>Z</i>* of all Boltzmann factors exp(−<i>E</i>(<i>s</i>)/<i>RT</i>) of the sampled structures was computed, and the ratio <i>Z</i>*/<i>Z</i> of <i>Z</i>* with respect to the partition function <i>Z</i> was determined to be 0.9998812, hence values of relative frequency should be close to the corresponding values for the Boltzmann probability. For given number <i>x</i> of neighbors, the corresponding value <i>y</i> is defined to be the sum, taken over all the structures <i>s</i>, whose degree is <i>x</i>, of the Boltzmann factor exp(−<i>E</i>(<i>s</i>)/<i>RT</i>) of <i>s</i> normalized by <i>Z</i>*. Using our code, with respect to energy model C (Turner 2004 energy parameters), we have the following values for the expected number of neighbors: <math><mrow><msub><mi>Q</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><msub><mi>Z</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>70</mn><mo>.</mo><mn>03</mn></mrow></math> (Boltzmann-MS1); <math><mrow><msub><mi>Q</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><msub><mi>Z</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>92</mn><mo>.</mo><mn>96</mn></mrow></math> (Boltzmann-MS2).</p

Defect diffusion [38], where a bulge migrates stepwise to become absorbed in an hairpin loop.

<p>The move from structure (a) to structure (b) is possible by the shift (1, 12) → (1, 13), the move from (b) to (c) by shift (2, 11) → (2, 12), etc. Our algorithm properly accounts for such moves with respect to energy models A, B, C. Image adapted from figure on page 26 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref019" target="_blank">19</a>] and produced by VARNA [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref058" target="_blank">58</a>].</p

The network of all secondary structures of the 12 nt sequence ACGUACGUACGU, where edges appear between structures that differ by a shift move.

<p>There are 35 structures, 68 edges between structures that differ by a base pair shift, hence the average network degree is <math><mrow><mn>68</mn><mn>35</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>94</mn></mrow></math>. Note that the network is not connected, unlike the previous two networks.</p

Difference in Boltzmann probabilities for 56 nt spliced leader RNA from <i>L. collosoma</i> with respect to move set MS2—see text for explanation.

<p>Difference in Boltzmann probabilities for 56 nt spliced leader RNA from <i>L. collosoma</i> with respect to move set MS2—see text for explanation.</p

Relative frequency for the Boltzmann weighted number of neighbors for the 76 nt alanine transfer RNA from <i>Mycoplasma mycoides</i> with accession code RA1180 from tRNAdb 2009 [41], where the <i>sample mean</i> ± one standard deviation is 29.11 ± 4.63 [resp. 46.51 ± 8.74] for move set MS1 [resp. MS2] using energy model C (Turner 2004 energy parameters).

<p>The length-normalized sample mean is 0.3831 ± 0.0610 for MS1 [resp. 0.6120 ± 0.1150 for MS2]. The number of neighbors, or degree, is given on the <i>x</i>-axis. RNAsubopt -d0 -e 12 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref010" target="_blank">10</a>] was used to generate 537,180 structures <i>s</i> having free energy within 12 kcal/mol of the MFE. The sum <i>Z</i>* of all Boltzmann factors exp(−<i>E</i>(<i>s</i>)/<i>RT</i>) of the sampled structures was computed, and the ratio <i>Z</i>*/<i>Z</i> of <i>Z</i>* with respect to the partition function <i>Z</i> was determined to be 0.9998202. For given number <i>x</i> of neighbors, the corresponding value <i>y</i> is defined to be the sum, taken over all the structures <i>s</i>, whose degree is <i>x</i>, of the Boltzmann factor exp(−<i>E</i>(<i>s</i>)/<i>RT</i>) of <i>s</i> normalized by <i>Z</i>*. Using our code, with respect to energy model C (Turner 2004 energy parameters), we have the following values for the expected number of neighbors expected number of neighbors: <math><mrow><msub><mi>Q</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><msub><mi>Z</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>26</mn><mo>.</mo><mn>01</mn></mrow></math> (Boltzmann-MS1); <math><mrow><msub><mi>Q</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><msub><mi>Z</mi><mrow><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><mo>=</mo><mn>37</mn><mo>.</mo><mn>61</mn></mrow></math> (Boltzmann-MS2).</p

The network of all secondary structures of the 12 nt (toy) sequence ACGUACGUACGU.

<p>The minimum free energy structure is shown in green. Edges connect structures <i>s, t</i>, such that <i>t</i> is obtained by a move in MS2 from <i>s</i>, or vice versa; i.e. structures are connected by an edge if they differ by a base pair addition, removal or shift. There are 35 structures, 126 edges between structures that differ by a base pair removal or addition, and 68 edges between structures that differ by a base pair shift. Altogether, there are 194 edges. It follows that the average network degree is <math><mrow><mn>194</mn><mn>35</mn><mo>=</mo><mn>5</mn><mo>.</mo><mn>54</mn></mrow></math>.</p

Illustration of shift moves defined in Sections “Main function <i>Q</i>_<i>n</i>” and “Recursion for function <i>Q</i>_<i>i,j</i>”.

<p>Illustration of shift moves defined in Sections “Main function <i>Q</i>_<i>n</i>” and “Recursion for function <i>Q</i>_<i>i,j</i>”.</p

Example of multiloop creation which is handled by our algorithm for energy models A, B but not for Turner energy model C.

<p>To move from (a) to (b), apply the shift (3, 13) → (13, 17); to move from (b) to (c), apply the shift (4, 12) → (12, 18). Our algorithm for the Turner energy model properly treats the move from (a) to (b), but not from (b) to (c), as explained in the Remark at the end of Section “Remaining recursions for <i>Q</i>_{<i>i</i>,<i>j</i>} and <i>Z</i>_{<i>i</i>,<i>j</i>}”. Image adapted from figure on page 27 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref019" target="_blank">19</a>] and produced by VARNA [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref058" target="_blank">58</a>].</p

Example of multiloop creation which is handled by our algorithm for all energy models, including the Turner energy model.

<p>To move from (a) to (b), remove the base pair (3, 13); to move from (b) to (c), shift (4, 12) → (12, 18); to move from (c) to (d), add base pair (13, 17). Image produced by VARNA [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0139476#pone.0139476.ref058" target="_blank">58</a>].</p