Lee C.-Y., Lee J.C., and Gutell R.R. (2007). Networks of interactions in the secondary and tertiary structure of ribosomal RNA. Physica A, 386(1):564-572.

1. Physica A 386 (2007) 564–572
Networks of interactions in the secondary and tertiary structure
of ribosomal RNA
Chang-Yong Leea,Ã, Jung C. Leeb
, Robin R. Gutellb
a
The Department of Industrial Information, Kongju National University, Chungnam 340-702, South Korea
b
The Institute for Cellular and Molecular Biology, The University of Texas at Austin, 1 University Station A4800, Austin, TX 78712, USA
Received 13 March 2007; received in revised form 12 July 2007
Available online 28 August 2007
Abstract
We construct four different structural networks for both the secondary and tertiary structures of the 16S and 23S
ribosomal RNAs (rRNAs) in the high-resolution crystal structures of the Thermus thermophilus 30S and Haloarcula
marismortui 50S ribosomal subunits, and investigate topological characteristics of the rRNA structures by determining
relevant measures, such as the characteristic path length, the clustering coefficient, and the helix betweenness. This study
reveals that the 23S rRNA network is more compact than the 16S rRNA networks, reflecting the more globular overall
structure of the 23S rRNA relative to the 16S rRNA. In particular, the large number of tertiary interactions in the 23S
rRNA tends to cluster, accounting for its small-world network properties. In addition, although the rRNA networks are
not the scale-free network, their helix betweenness has a power-law distribution and is correlated with the phylogenetic
conservation of helices. The higher the helix betweenness, the more conserved the helix. These results suggest a potential
role of the rRNA network as a new quantitative approach in rRNA research.
r 2007 Elsevier B.V. All rights reserved.
PACS: 87.14.Gg; 87.15.Àv; 89.75.Hc
Keywords: Ribosomal RNA; Complex networks; rRNA structure; Nucleotide conservation; Small-world
1. Introduction
The network (or graph) theory [1], originated from the Ko¨ nigsberg’s seven bridges problem formulated by
Euler, was systematically studied in terms of the random network theory developed by Erdo¨ s and Re´ nyi [2].
Significant advance in the network theory was recently made by the discovery of some distinctive features that
many complex networks have in common, including the small-world [3] and the scale-free [4] properties. These
uncovered characteristics distinguish complex networks from the random and the regular networks.
Subsequent researches on the complex networks of various systems have made considerable progress in the
understanding of these systems, and studies on the complex networks have become more active across many
disciplines.
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doi:10.1016/j.physa.2007.08.030
ÃCorresponding author. Tel.: +82 041 330 1423; fax: +82 041 330 1429.
E-mail address: clee@kongju.ac.kr (C.-Y. Lee).

2. Complex networks are often classified, according to research fields, as social [5,6], technological [7–10],
and biological networks [3,10–14], to name just a few. Recently, the research community has begun to
study various types of biological networks: neural networks [3], food networks [10], metabolic networks [11],
genetic regulatory networks [12], and protein interaction networks [13,14]. Most biological networks
are composed of molecules (or substrates) and their interactions that are represented as vertices and
edges, respectively. This research on biological networks is mainly focused on the investigation of their
connectivity by determining such statistical measures as the degree distribution, the characteristic path length,
and the clustering coefficient; many biological networks are found to be a scale-free and/or small-world
network.
Besides these biological networks made of independent molecules interacting with one another, a large
biological molecule itself can be represented as a network. Proteins and RNA molecules, which are composed
of a long chain of amino acids or nucleotides with multiple interactions between them, can be constructed into
appropriate networks which represent structural features. Protein structures, which have been traditionally
viewed as molecules that catalyze essential functions in the cell, have been studied as the network of amino
acids [15–17]. Furthermore, it was shown that protein structures can be characterized as the small-world
network with which key residues for their folding process can be identified. In addition to the conventional
analysis methods, the quantitative network-based methods can reveal hidden functional as well as structural
characteristics of proteins.
The structure of the rRNA is important because it is believed that the structure dictates its
biological function. Since the rRNA, a tightly packed asymmetric macromolecule, has been considered
too large for a high-resolution structural analysis, quantitative studies on the structure proved difficult
until recent progress in the high resolution X-ray crystallography has been made. The 2:4 ˚A resolution
of the 50S subunit from the H. marismortui [18] and the 3:05 ˚A resolution of the 30S subunit from
the T. thermophilus [19] provided the first detailed views of the structure at the atomic level. These
enable us to study not only the sequence and structure, but the function of the molecules in a great
detail.
Two-thirds of the mass of the ribosome [20], the site of protein synthesis of a living cell, is RNA and the
remainder is protein. While an older and conventional paradigm dictated that the functional sites in the
ribosome is composed of protein, a long series of experiments [21] that culminated recently with the high-
resolution crystal structures of the 30S and 50S ribosomal subunits [18,19] revealed and verified that RNA
(16S rRNA in the 30S subunit; 23S and 5S rRNAs in the 50S subunit) is the active participant in protein
synthesis. Based on the simple concept that different RNA sequences with evolutionary related and similar
biological functions fold into very similar secondary and tertiary structures, the secondary structures of the
rRNAs were determined with comparative sequence analysis [22–24]. In particular, approximately 97–98% of
the basepairs predicted in these structure models are present in the high-resolution crystal structures of the T.
thermophilus 30S and H. marismortui 50S subunits [25].
Early efforts to quantitatively analyze the RNA secondary structure (including rRNA) were based on
simplified representations, including the fine- and coarse-grained tree representations. In the find-grained
tree representation [26], both basepairs and unpaired nucleotides are considered as vertices, while adjacent
base-pairs and/or unpaired nucleotides are considered as edges. This approach has been used for com-
parison of two structures, including structure alignment, motif-based searches, and quantitative measurement
of the ‘‘tree distance’’ between two structures [27]. In the coarse-grained tree representation that involves
a RNA chain of up to about 100 nucleotides [28,29], double stranded helices are represented as edges,
while single stranded loops (hairpin, internal, and multi-stem loops) are represented as vertices. This
representation is used for the study of the algebraic connectivity based on the spectral decomposition
[30]. More recently, the interaction networks of RNAs is studied for the relationships between helical
domains [31]. This study uncovers not only structural similarity but the conserved pattern and distances
between motifs.
In this paper, we study the characteristics of the rRNA structure from the biological network perspective. In
particular, we construct four rRNA networks employing both the secondary and tertiary structures of the T.
thermophilus 16S and H. marismortui 23S rRNAs by identifying each nucleotide as a vertex and chemical
bonds (either the hydrogen or the covalent bond) between nucleotides as an edge [25,32].
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3. 2. Conceptualization of rRNA molecules as networks
A biologically active RNA structure is composed of a specific sequence of four nucleotides [adenine (A),
cytosine (C), guanine (G), and uracil (U)] that is folded into its secondary structure and then into its tertiary
structure. Procaryotic ribosomes are molecules of about 250 ˚A in diameter and contain the 30S (small) and 50S
(large) subunits. The 30S subunit contains about 20 proteins and the 16S rRNA which plays a crucial role in
the decoding process of mRNA; the 50S subunit contains about 30 proteins and the 23S and 5S rRNAs which
catalyze the chemical reaction of peptide bond formation [33]. Whereas the 5S rRNA contains relatively small
number of about 120 nucleotides, the 16S and 23S rRNAs are large polymers of approximately 1500 and 3000
nucleotides, respectively. Their secondary structures comprised of double-stranded helices and single-stranded
loops are divided into secondary structural domains (four domains in the 16S rRNA and six in the 23S rRNA)
[34]. These domains range in size from approximately 150–550 and 270–840 nucleotides in the 16S and 23S
rRNAs, respectively.
In unfolded state, the rRNA is a single-stranded linear polymer of nucleotides, in which the backbone of
adjacent nucleotides is connected via the covalent bond. Thus, the order of nucleotides linked by the covalent
bond determines the sequence of the rRNA. In order for the rRNA to function biologically, the linear polymer
folds onto itself to form helices of various sizes which are main components in the 3D structure. A helix is
nothing but a pair of consecutive sequence segments that form base pairs via the hydrogen bond. Due to the
formation of helices, various types of single-stranded region (or unpaired region) occur between helices.
Similar to the terminology devised for describing protein structures, the architecture of the rRNA is
traditionally described hierarchically by the secondary and tertiary structures. The RNA secondary structure
is a 2D diagram consisting of many secondary structure elements including double-stranded helices and
unpaired regions. The RNA tertiary structure, in contrast, is a 3D structure in which secondary structure
elements are strategically and topologically arranged with each other to make a large number of tertiary
contacts between secondary structure elements. Thus, tertiary interactions contain secondary interactions,
pseudoknot interactions, and all other hydrogen-bond-mediated contacts comprising base–base, base–
backbone, and backbone–backbone interactions.
A detailed mapping of the secondary and tertiary structure interactions in the high-resolution 16S and
23S rRNAs crystal structures [19,18] revealed that, while the secondary structure interactions of the 16S and
23S rRNAs occur within domains, their tertiary structure interactions occur both within and between domains
[25]. In particular, the 23S rRNA contains many more tertiary interactions than the 16S rRNA.
Approximately 15% and 45% of the 180 and 460 tertiary interactions in the 16S and 23S rRNAs,
respectively, occur between domains (data not shown) suggesting that the 23S rRNA is more globular in shape
compared to the 16S rRNA.
Here we introduce and conceptualize rRNA structural networks by representing nucleotides as vertices and
their covalent sugar-phosphate backbone and hydrogen-bonds interactions as edges. We construct four rRNA
networks for the secondary structure only and secondary and tertiary structure in the high-resolution
T. thermophilus 16S and H. marismortui 23S rRNA crystal structures [19,18,25]. Moreover, multiple bonds
between a pair of nucleotides are represented as a single edge while multiple edges between a pair of vertices
are, in general, not allowed in the network theory. The four networks are denoted as T16S-2D, T16S-3D,
H23S-2D, and H23S-3D, where 2D and 3D represent secondary structure only and secondary and tertiary
structure, respectively. The four constructed rRNA networks are then analyzed by such commonly adopted
measures as the shortest path length, the clustering coefficient, and the vertex betweenness to understand
topological characteristics of the rRNA structures from a network perspective.
3. Results
3.1. Compactness and small-world property
One measure to quantify the network topology is the shortest path length lij connecting a pair of vertices
i and j. This measure is simply the minimum number of edges along the shortest path between the pair. Fig. 1
shows frequencies of the normalized shortest path length ~l for the four different networks. The frequencies of
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4. the normalized shortest path length for the constructed rRNA networks have the same standard Gaussian
distribution, despite the differences in their detailed structural characteristics.
Although the four networks have the same distribution in their mean shortest paths, their means and
standard deviations differ from one another, as shown in Table 1. For a linear polymer of N nucleotides, the
mean of the shortest path length ml % N=3$OðNÞ for Nb1. Thus, if both rRNAs were to have similar
structural characteristics, ml of the 23S rRNA network should be about twice that of the 16S rRNA network
since the 23S rRNA has about twice as many nucleotides as the 16S rRNA. Table 1, however, shows that both
the 16S and 23S rRNA networks have about the same mean shortest path length considering the 2D and 3D
networks separately. In addition, the folding of the secondary structure to its tertiary structure leads to an
approximately 2- and 3-fold reduction in the 16S and 23S rRNA networks, respectively, in both the mean
shortest path and its standard deviation. This suggests quantitatively that the 23S rRNA is structurally more
compact than the 16S rRNA.
To elaborate on these findings, we utilize the mass function MðdÞ using the cumulative density function of
the frequency distribution. That is,
MðdÞ ¼ N
X
lpd
PðlÞ, (1)
where N is the number of nucleotides, and PðlÞ is the frequency of l. The mass function is a measure of the
average number of nucleotides within a distance less than or equal to d, and it is equivalent to the average
‘‘mass’’ of the network. Incidentally, it is similar to the hop plot of the Internet diameter [35].
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-2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
frequency
normalized shortest path length
Fig. 1. The frequency distribution of the normalized shortest path length ~l ¼ ðl À ^mlÞ=^sl for the four rRNA networks, where ^ml and ^sl are
the estimated mean and standard deviation of the shortest path length: T16S-2D ð&Þ, T16S-3D ðÞ, H23S-2D ðnÞ, and H23S-3D ð,Þ.
Frequencies are normalized such that, for each network, the sum of frequencies over all shortest path lengths is unity. By the scaling, all
distributions collapse to the standard Gaussian distribution, Nð0; 1Þ (solid line).
Table 1
The estimated mean (^ml) and its standard deviation (^sl) of the shortest path length for four rRNA networks
T16S-2D H23S-2D T16S-3D H23S-3D
^ml 66.2 72.0 30.5 24.4
^sl 82.6 65.4 36.5 21.0
C.-Y. Lee et al. / Physica A 386 (2007) 564–572 567

5. As shown in Fig. 2, the plotted mass functions for all rRNA networks reveal that the tertiary interactions
reduce the maximum shortest path length lmax in both the 16S and 23S rRNAs, where lmax satisfies
MðlmaxÞ % N. More importantly, the 23S rRNA has a slope more than twice the 16S rRNA in both the
secondary and tertiary structures, indicating that nucleotides in the 23S rRNA are more densely packed than
those in the 16S rRNA. Since the 23S rRNA contains about twice as many nucleotides as the 16S rRNA,
the 23S rRNA network would contain about twice as many nucleotides as the 16S rRNA within the same
distance if the packing density is comparable for both rRNAs. A similar finding was established from the mass
fractal dimension analysis of rRNA molecules. While the mass fractal dimension of the 16S rRNA molecule is
less than three, that of the 23S rRNA is close to three, implying that the 23S rRNA is a more compact 3D
object [36].
The characteristic compactness of the H23-3D network is due to an increase in the number of domain-
domain tertiary interactions in 23S rRNA, relative to 16S rRNA. These extra tertiary interactions between
different domains reduce the overall simple sequence distance between the nucleotides in different domains.
While the more compact 23S rRNA itself is responsible for the peptide bond formation [37,18], the less
compact 16S rRNA might be related to the higher degree of structural flexibility of the 30S subunit during
translocation of mRNA and tRNAs, including the rotational rigid-body motion between the head and the rest
of the 30S subunit [38]. In particular, it has been reported that the 30S subunit might undergo the ratchet-like
movement relative to the large 50S subunit [39]. In contrast to the 30S subunit, the 50S subunit might not be
associated with any significant movements in its core region during protein synthesis because of its much
compact structure except peripheral regions.
We further address the shortest path length from the perspective of the small-world network [3]. Nucleotides
in rRNA secondary structures contain only two or three interactions, including basepairing interactions with
their basepairing partners and/or covalent interactions with their neighboring nucleotides in sequence, so that
they are not clustered and hardly show an appreciable cliquishness. Thus, the rRNA networks based only on
the secondary structure interactions (2D) do not form a small-world network. In contrast, the networks based
on the secondary and tertiary structure interactions (3D) contain many tertiary interactions that result in a
perceptible cliquishness characteristic of the small-world network.
As shown in Table 2, we calculate the characteristic path length L and the clustering coefficient C [3] in both
of the T16S-3D and H23S-3D networks, and compare these values with networks of randomly rewired tertiary
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0 50 100 150 200
0
500
1000
1500
2000
2500
3000
b=100.7
b=32.6
b=33.0
b=14.9
massfunction,M(d)
distance, d
Fig. 2. Plots of the mass function MðdÞ versus the distance d for four rRNA networks: T16S-2D ðÞ, T16S-3D ðÞ, H23S-2D ðÞ, and
H23S-3D ð’Þ. Dotted lines are estimated slopes b’s which are 14.9, 33.0, 32.6, and 100.7 for T16S-2D, T16S-3D, H23-2D, and H23S-3D,
respectively.
C.-Y. Lee et al. / Physica A 386 (2007) 564–572568

6. interactions while maintaining the same number of tertiary interactions. In both of the T16S-3D and H23S-3D
networks, Lreal$Lregular and CrealbCrandom$NÀ1
, suggesting that both networks are far from the random
network and resemble the regular grid network. Interestingly, Lreal$Lregular and CrealbCrewired for the 23S
rRNA tertiary network, featuring the small-world property. The H23S-3D network is highly clustered like a
regular lattice, yet has a small characteristic path length like a random network. The same is not true for the
T16S-3D network where Creal is just an order of magnitude larger than Crewired. This finding can be accounted
by the fact that the 23S rRNA has many more tertiary structure interactions than the 16S rRNA. Experiments
have shown that a RNA sequence form its secondary structure first, and then folds into its tertiary structure
[40,41]. This hierarchical folding of RNA structure suggests that the formation of tertiary structure
interactions, especially in the 23S rRNA, results in the clustering of nucleotides.
3.2. Betweenness and nucleotide conservation
Comparative sequence analysis of rRNA sequences for organisms that span across the three primary
divisions of life (Archaea, Bacteria, and Eukaryotes) revealed that many of the highly conserved nucleotides
are found clustered in a few specific regions of the rRNA structure [34,42]. The relative conservation of
nucleotides in rRNA structure is related to their prominence or importance within a network environment.
This prominence of a network is called the centrality in the network, since it measures which vertex
(nucleotide) is best connected to other vertices or the most influential in the formation of the network.
Although the relative importance of a vertex is quantified by various measures such as the vertex betweenness,
the degree, and eigenvector centrality [43], here we employ the vertex betweenness as a measure and investigate
its correlation with the nucleotide conservation.
The vertex betweenness of a vertex vi, bðviÞ, is defined as the number of shortest paths between all pairs of
vertices passing through the vertex vi [43,44] and is usually normalized by the maximum possible value, so that
it takes values between 0 and 1. Since some vertices can be visited more frequently than others in a network,
the betweenness of a vertex can be also used as a measure for the amount of ‘‘traffic’’ that runs through the
vertex [45]. In this regard, the vertex betweenness can be used as an indicator for which vertex (nucleotide in
rRNA) is the most influential to hold a network.
Similarly, the helix betweenness can be defined by treating rRNA secondary structure helices as ‘‘vertices’’
to measure for which helix in the rRNAs is the most influential helix to hold the rRNA structure networks.
The helix betweenness of a helix j, Bj, is defined as the average vertex betweenness for all vertices in a helix.
That is,
Bj ¼
1
nj
X
vi2Aj
bðviÞ, (2)
where nj is the number of vertices (or nucleotides) in a helix j and Aj is the set of vertices in the helix j.
Fig. 3 shows distributions of the helix betweenness of the H23-2D and H23-3D networks, both of which
approximately have power-law distributions of the form
PðBÞ / BÀa
, (3)
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Table 2
The characteristic path length ðLrealÞ and the clustering coefficient ðCrealÞ for T16S-3D and H23S-3D, which are compared to those with
the randomly rewired tertiary interactions (Lrewired and Crewired )
Network N Lreal Lrewired Lregular %
ffiffiffiffiffi
N
p
Creal Crewired Crandom % NÀ1
T16S-3D 1521 33.1 12.0 39.0 0.010 0.0031 0.00066
H23S-3D 2922 27.7 11.4 54.1 0.014 0.0006 0.00034
Lregular and Crandom represent the characteristic path length of a regular grid network and the clustering coefficient of a random network,
respectively.
ffiffiffiffiffi
N
p
and NÀ1
represent the typical characteristic path length of a regular grid network and clustering coefficient of a random
network, respectively, where N is the number of nucleotides (vertices) in rRNA sequence comprising the 16S and 23S rRNA networks.
C.-Y. Lee et al. / Physica A 386 (2007) 564–572 569

7. implying the absence of any characteristic scale of the helix betweenness. Similar power-law behavior are
found for the T16S-2D and T16S-3D networks (data not shown). These results indicate that the power-law
behavior of the betweenness that was first observed in the scale-free networks [45] also occurs in non-scale-free
networks. Some other non-scale-free networks also follow a power-law behavior, as reported in the
collaboration network and the neural network of Caenorhabditis elegans [46].
The power-law distribution tells that high values of helix betweenness is not just statistically forbidden or
rare, but are as equally important in the organization of a network as those with low values. The power-law
also indicates that while most of helices in the rRNA structure networks have low helix betweenness values, a
few of the helices have high helix betweenness values that are statistically significant. This implies that only a
few helices of high betweenness dictate the global structure, while most of helices are participated in clustering
local structures. Interestingly, tertiary interactions slightly alter the power-law distribution as shown in Fig. 3.
Since the tertiary structure necessarily has more edges linked by the tertiary interactions, the tertiary structure
may yield a new shortest path that is not possible in the secondary structure. Moreover, the number of vertices
along the shortest path in the tertiary structure is always less than that in the secondary structure. Therefore,
tertiary interactions reduce the extent of the betweenness, and thus the number of helices with higher helix
betweenness values are reduced.
We also introduce the term helix conservation as a measure of the average of the nucleotide conservations
for all nucleotides in a helix. The nucleotide conservation values can be computed for each nucleotide position
in a sequence alignment using the modified Shannon equation [42],
CONS ¼
X
i
Pi log2 ð4PiÞ þ PD log2 ðPDÞ, (4)
where Pi is the frequency of nucleotide i at a given position and PD is the frequency of deletions at that
position. The computed nucleotide conservation values range between À1 and 2 ðÀ1pCONSp2Þ.
The plots of the helix betweenness versus the helix conservation computed for the bacterial alignment for
the T16S-2D and H23S-2D networks are shown in Fig. 4. When a nucleotide with CONS41:3 is arbitrary
considered highly conserved, helices with high helix betweenness values are referred to as highly conserved
helices in both the T16S-2D and H23S-2D networks. The converse, however, is not true. In particular, vast
majority of helices with their betweenness greater than 0.1 is highly conserved, suggesting that the most
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1E-3 0.01 0.1
1E-4
1E-3
0.01
0.1
probability
helix betweenness
Fig. 3. Log–log plots of the helix betweenness versus its probability for H23S-2D ð’Þ and H23S-3D ðÞ. The solid and the dotted
lines represent estimated slopes, which are À1:13 Æ 0:07 and À1:11 Æ 0:08 for H23S-2D and H23S-3D, respectively. The distribution of
H23S-3D is shifted vertically for the display purpose.
C.-Y. Lee et al. / Physica A 386 (2007) 564–572570

8. important helices that maintain the rRNA structure networks have minimal variation in bacterial rRNAs.
A similar result is also obtained for the networks of tertiary structures.
4. Summary and conclusion
In this paper, the four 16S and 23S rRNA structural networks were constructed using the secondary and
tertiary interactions mapped in the high-resolution crystal structures of the T. thermophilus 30S and
H. marismortui 50S subunits, by treating nucleotides and their interactions as vertices and edges, respectively.
Subsequently, their topological characteristics were investigated quantitatively with various measures and then
related to the biological and structural implications of the rRNAs.
The mass functions for the rRNA networks revealed that the 23S rRNA is more compact than the 16S
rRNA in both the secondary-only (2D) and secondary and tertiary (3D) networks. The tertiary inter-
actions, especially in the H23-3D network, cluster nucleotides in a way that increase the cliquishness of the
network. In addition, the helix betweenness follows a power-law distribution. Only a few helices have
high centrality in the formation of the global structure of rRNA, while the rest are associated with the
clustering of local structures. Furthermore, helices with higher betweenness are usually highly conserved in
Bacteria.
These results could uncover the characteristics of the rRNA that are not discernible with other qualitative
method, and suggest a potential role of rRNA networks as a new quantitative approach for RNA research. In
particular, network analysis of the 16S and 23S rRNA structures may give some insights into RNA structure
and function by providing some useful quantitative measures to describe the topological characteristics of
RNA structure and folding.
This work was supported by the Korea Research Foundation Grant funded by the Korean Government
(MOEHRD) (KRF-2005-041-H00052 to CYL), the Welch Foundation (F-1427 to RRG), and the National
Institutes of Health (GM067317 to RRG). The main calculations were performed by using the
supercomputing resource of the Korea Institute of Science and Technology Information(KISTI).
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0.00 0.05 0.10 0.15 0.20 0.25 0.30
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
helixconservation
0.00 0.05 0.10 0.15 0.20
-0.5
0.0
0.5
1.0
1.5
2.0
helixconservation
helix betweenness
Fig. 4. Plots of the helix betweenness versus the helix conservation in the bacterial domain for T16S-2D (A) and H23S-2D (B). The
quantities are dimensionless.
C.-Y. Lee et al. / Physica A 386 (2007) 564–572 571

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