Vol.2, No.9, 998-1004 (2010) Natural Science
Copyright © 2010 SciRes. OPEN ACCESS
Complex tree: the basic framework of protein-protein
interaction networks
Dai-Chuan Ma1,2, Yuan-Bo Diao1,2, Yi-Zhou Li1,2, Yan-Zi Guo1,2, Jiang Wu1,2, Meng-Long Li1,2*
1College of Chemistry, Sichuan University, Chengdu, China; *Corresponding Author: liml@scu.edu.cn
2State Key Laboratory of Biotherapy, Sichuan University, Chengdu, China
Received 22 May 2010; revised 26 July 2010; accepted 2 August 2010.
In living cells, proteins are dynamically connec-
ted through biochemical reactions, so its functi-
onal features are properly encoded into protein-
protein interaction networks (PINs). Up to pres-
ent, many efforts have been devoted to explor-
ing the basic feature of PINs. However, it is still
a challenging problem to explore a universal pr-
operty of PINs. Here we employed the complex
networks theory to analyze the protein-protein
interactions from Database of Interacting Prot-
ein. Complex tree: the unique framework of PINs
was revealed by three topological properties of
the giant component of PINs (GCOP), including
right-skewed degree distributions, relatively sm-
all clustering coefficients and short characteris-
tic path lengths. Furthermore, we proposed a no-
nlinearly growth model: complex tree model to
reflect the tree framework, the simulation resu-
lts of this model showed that GCOPs were well
represented by our model, which could be help-
ful for understanding the tree-structure: basic
framework of PINs. Source code and binaries
freely available for download at http://cic.scu.
Keywords: Complex Networks; Complex Tree;
Pins; Model Simulation
Protein-protein interactions (PPIs) are crucial to most
biochemical processes in living organisms. Identification
of protein-protein interactions has been the focus of the
post-proteomic studies. Various experimental techniques
have been developed for the large-scale PPI analysis, in-
cluding yeast two-hybrid systems [1], mass spectrometry
[2,3], protein chip [4] and so on. Consequently, great
quantity of protein interaction data from several organ-
isms such as yeast and fruit fly has been produced [2,3].
In the past decade, with the explosion of available hi-
gh throughout biological data, the analysis of biological
networks has attracted significant interest in academic
community. Since the “right-skewed” degree distribution
of metabolic networks found by Jeong et al. [5], lots of
research works have been done to understand the biolo-
gical systems in terms of network. The system’s eleme-
nts are represented as vertices (such as proteins, DNA,
RNA and small molecules) and their interactions are red-
uced to links (biochemical interactions between these bi-
ological components) connecting pairs of vertices. And
the cell’s behaviors are distinct attributed to “network of
networks” [6].
In a living cell, group of proteins participate in diverse
biochemical interactions that lead to changing the effect
of protein or forming protein complexes. All these proc-
esses constitute protein-protein interaction networks (PI-
Ns) [7]. Studying the topology features of PINs is bene-
ficial to understand the cell’s higher-level organization
mechanism. Recently, notable structure properties have
been reported in several research works of PINs [8-12],
including right-skewed degree [13], short pathway
length [7], hierarchical structure [14]. Meanwhile, many
network models have been proposed to characterize the
PINs to explore the basic universal property of PINs
[15,16]. However, it is still a challenging problem to find
a universal framework of PINs for each species.
In our works, we employed the complex networks the-
ory to analyze the datasets from the PPI database DIP in-
cluding eight species: D. melanogaster (Dmela), S. cere-
visiae (Scere), E. coli (Ecoli), C. elegans (Celeg), H. sa-
piens (Hsapi), H. pylor (Hpylo), M. musculus (Mmusc),
R. norvegicus (Rnorv). Based on the analyzing results,
we constructed a complex tree model (CTM) to mimic
the giant component of PINs (GCOP) of each species.
The simulations of CTM suggest that PINs can be well
represented by this model, and the proposed CTM is
helpful to understand the basic framework of PINs.
D. C. Ma et al. / Natural Science 2 (2010) 998-1004
Copyright © 2010 SciRes. OPEN ACCESS
2. Methods
2.1. Data Source—Database of Interacting
Protein (DIP)
In this paper, the DIP database which collects experime-
ntally determined protein-protein interaction data was ut-
ilized as the input data. We collected the Protein-Protein
Interactions with binary relations data from DIP, version
DIP_20081014 (http://dip.doe-mbi.ucla.edu). In order to
obtain integrated topology, the “full set” subset contain-
ing PPI data identified by experiment was used. Althou-
gh false positive in “full set” might lead to false edges in
PINs and inaccurate topological features, we considered
that our analysis results are still robust because a few
local inaccuracies have less influence on global proper-
ties of PINs. Table 1 summarizes the numbers of pro-
teins and interactions for aforementioned species.
2.2. Topological Features of PINs
In this study, we constructed adjacency matrix of PIN
from the corresponding PPI pair tables, where each ele-
ment in the adjacency matrix is assigned 0 or 1 to stand
for whether it is a direct interaction, or non-interaction.
After removed loops and multiple edges, the network
was normalized into an undirected graph. All following
topological properties were calculated using the “igraph”
R package which is widely used in network analysis
2.2.1. Component Size and Giant Component
As the component size of PINs may reflect its fundam-
ental properties [17], we calculated the component size
for each species, and extracted the giant component of
PINs (GCOP) from entire network.
Measurement of component size and giant component
was listed in Table 1. For Rnorv and Mmusc, the size of
GCOP is too small for large scale analysis. Hence, in the
rest of our work we focused on the other six species.
Three main topological parameters of GCOP were
measured for these species and the details of measure-
ments are described in following section.
Table 1. Basic statistics of PINs and corresponding GCOP.
Species No. of
in PINs
PPIs in
PPIs in
Celeg 111 2651 2386 4041 3825
Dmela 72 7498 7351 22864 22602
Ecoli 362 1865 1447 6990 5879
Hpylo 19 713 686 1423 1351
Hsapi 253 1607 805 1951 1178
Mmusc 163 599 50 513 55
Rnorv 76 221 10 174 9
Scere 34 4963 4902 17570 17246
2.2.2. Node Degree Distribution
One most basic parameter of a network is its node deg-
ree distribution. In the past century, random graph is the
most important model for real systems. The degree distr-
ibution of random graph is bell-shaped and the node de-
gree clusters around the mean value. But at the begin-
ing of this century, Jeong [13] found the degree distribu-
tion of yeast PIN is far from bell-shaped distribution of
random graph. Jeong et al. deduced that the degree dis-
tri- bution follows power-law approximately and called
it “scale-free”. The phenomena so-called scale-free have
prompted many scientists to explore real networks. Up
to many present research works have been done in sys-
tem biology, showing that so many biological networks
have the “right-skewed” degree distribution [12].
In this paper, we defined the degree k of a given pro-
tein as the number of interactions with other proteins and
P(k) as the frequency distribution of degree. We collec-
ted the degree sequence datasets of GCOP and plotted
the logarithm of the cumulative degree distribution (CD-
D): Pcum(k) vs. the logarithm of k for each species. The
CCD curves were plotted in Figure 1. It is obvious that
each GCOP has the “right-skewed” degree distribution.
This means that most proteins in yeast PIN have very
few interactions and yet a few ones have many (hubs),
and the degree distribution has no well-defined peak but
is approximate to skewed line under a double logarith-
mic plot. On account of that power-low distribution is a
reasonable hypothesis for right-skewed distributions, we
made power-law parameters estimation and tested the po-
wer-law hypothesis for those distributions with the tech-
niques proposed by Aaron Clauset [18] based on maxi-
mum likelihood methods and the Kolmogorov-Smirnov
statistic. By applying this method to the degree sequen-
ces of GCOP for each species, we can not only find the
best-fit power low model for degree distributions of GC-
OPs, but also test whether the power-low distribution is
a reasonable hypothesis for those distributions. The es-
timation result is showed in Table 2.
2.2.3. Clustering Coefficient and Characteristic
Path Length
Other two basic topology properties of network are av-
erage clustering coefficient and characteristic path length.
Considering that PINs are undirected networks, we de-
fined L as the characteristic path length (also known as
the average path length) between protein pairs in PINs:
(1) ij
dij is defined as the number of links from vertex i along
the shortest path to vertex j. We measured the L of PINs
by employing the Floyd algorithm.
D. C. Ma et al. / Natural Science 2 (2010) 998-1004
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Figure 1. Cumulative component size distributions of PINs of eight species. (a) Scere; (b) Celeg;
(c) Dmela; (d) Ecoli; (e) Hpylo; (f) Hsapi; (g) Mmusc; (h) Rnorv.
The clustering, sometimes also called transitivity, gen-
erally means the presence of a heightened number of tri-
angles (groups of three vertices each of which is con-
nected to the others) in the network [19]. In many sys
tems it has been found that if constituent A is connected
to constituent B and constituent B to constituent C, then
there is a heightened probability that constituent A will
also be connected to constituent C. And we computed
the clustering coefficient of PIN following the defini-
tions given by Watts and Strogatz [20], defining a local
D. C. Ma et al. / Natural Science 2 (2010) 998-1004
Copyright © 2010 SciRes. OPEN ACCESS
Table 2. Average node degree < k >, maximum connectivity kmax, clustering coefficient and characteris-
tic path length, and the power-law testing results of CCD for the six species.
Species dmin k
max <k> p-value C L
Celeg 2.49 7 187 1.603 0.118 0.022 4.809
Dmela 3.50 28 178 3.074 0.104 0.012 4.401
Ecoli 1.65 1 152 4.063 0 0.109 3.819
Hpylo 2.24 3 54 1.969 0 0.016 4.137
Hsapi 2.79 5 37 1.463 0.523 0.107 6.533
Scere 2.93 16 283 3.518 0.022 0.060 4.422
clustering value Ci:
number of triangles connected to vertex
number of triples centered on vertex
Then the global clustering coefficient for the whole ne-
twork is the average:
The calculation results are also showed in Table 2.
2.3. Complex Tree Model
In the past few years, notable discoveries of exploring
complex networks have redefined our understanding of
complex systems. Meanwhile, certain models have been
constructed to mimic real networks [20-24]. The BA
model is the most famous stochastic model which gener-
ates scale-free topology by the combination of network
size linear growth and preference attachment rule [25].
However the randomness in this model makes it hard to
gain a visual understanding of scale-free topology. A
deterministic model [22] was constructed by Barabasi to
solve this problem. The unique property of this model is
its hierarchical fashion. After that, Ravasz and Barabasi
uncovered the hierarchical structure in metabolic net-
works [26], they proposed a deterministic hierarchical
network model to explain the modularity and hierarchi-
cal organization in real networks [27]. Up to present,
many network models have been proposed to explain the
organization mechanism of biochemical system, but it is
still a challenging problem to construct a universal
model for PINs.
Inspired by previous study, we tried to build a model
to reflect the unique framework of PINs: complex tree.
We constructed this model in a tree-like fashion to mim-
ic the GCOP. In GCOPs, there’re many varieties of tree-
structure, it’s really difficult to build a deterministic mo-
del for it. So we made a simplification here and built this
heuristic model by adding shortcuts on a tree substrate,
we called it “complex tree model”. The details of model
construction are depicted as follow.
2.3.1. Substrate
In building small-world model, Watts made random rew-
iring test on several substrates, including tree, ring lattice,
and many other structures. Finally the one dimension ring
lattice was chosen because of its equivalence [20]. Rece-
ntly, Dong-Hee Kim et al. found thatmany real systems
have their own communication kernels and the commun-
ication kernel of scale-free network is scale-free tree that
is called the skeleton of complex networks [28]. They
deduced that scale-free networks can decompose into
(scale-free) trees and shortcuts. So in this paper we fo-
cused on building a simple model by adding shortcuts in
a tree-structure. The tree graph was chosen as substrate
to construct the CTM. Here we focused on the situation
of perfect binary tree. Its inherent hierarchical structure
and connectivity make it a proper substrate. By altering
the number of levels, n, the size of substrate N(n) can
grow nonlinearly as:
() 21
2.3.2. Shortcut Adding
In the construction of NW small-world model, Newman
and Watts added a few shortcuts between different parts
of the ring lattice. As a result, the local construction do-
esn’t change, but the distance between two remote parts
reduces dramatically by long-range connections [29]. In
our model, we added shortcuts between two nodes in
different hierarchies according to a simple rule. An arbi-
trary node in level n is linked with nodes of lower levels
(excluding the two leaf nodes already connected) with
0() n
Pn Pk
P0 is the shortcut adding probability of the main root
to all other nodes below level 0 except the two leaf
nodes it already linked. Following this simple adding
rule, the nodes in higher levels get more shortcuts than
those in lower levels, and the adding probability drop
exponentially with n. We ‘tuned’ the amount of shortcuts
by changing P0 from 0 to 1. While P0 = 0, there is no
shortcut and loop, the model is just a perfect binary tree
D. C. Ma et al. / Natural Science 2 (2010) 998-1004
Copyright © 2010 SciRes. OPEN ACCESS
with big characteristic path length; P0 = 1, the root gets
every node connected and the maximum amount of sh-
ortcuts is added to the substrate. In the region 0 < P0 < 1,
some interesting phenomena can be seen between two
extremes above. The details of model construction are
shown in Figure 2, which depicts both the substrate and
the shortcuts adding process. The model simulation is
described in the next section.
2.4. Simulations
Here we concentrated on the substrate n = 10, N = 1023
and added shortcuts on it. When altering P0 from 0 to 1;
the topological features can be altered dramatically dur-
ing this process. We calculated three main topological
parameters (CDD, C, L) of CTM to describe this sto-
chastic process. Furthermore, with the aim to represent
the topological features of GCOP, we focused on these
cases of CTM with n = 9 N = 511, n = 10 N = 1023, n =
11 N = 2047, n = 12 N = 4095, n = 13 N = 8191, which
made the CTM attained similar size with the GCOP for
five species. By tuning P0, we found out the certain P0
for each scale of CTM to fit PINs, and compared the
topological features of GCOP with fitted CTM. The
comparison result is depicted in Table 3.
Several basic statistics of the PINs are summarized in
Table 1. It is distinct that the GCOP of Dmela, Scere,
Ecoli, Celeg, Hsapi, Hpylo is remarkably big, containing
most proteins and interactions of PINs. Especially in the
PINs of Celeg, Dmela, Hpylo and Scere, the giant com-
ponents contain more than 90% proteins of whole PINs.
Figure 2. The generation process of CTM (n = 3, P0 = 0.3).
Table 3. Comparisons between PINs and CTM.
N C L Levels/P0 N C L
Hpylo 686 0.016 4.137 8/0.15 511 0.0184.153
Ecoli 1447 0.109 3.819 9/0.20 1023 0.0493.582
Celeg 2386 0.022 4.809 10/0.10 2047 0.0154.751
Scere 4902 0.073 4.146 11/0.15 4095 0.0264.010
Dmela 7351 0.012 4.401 12/0.10 8191 0.0134.605
For Rnorv and Mmusc, the size of GCOP is smaller and
not suitable for large scale analysis.
Table 2 shows the average node degree <k>, maxi-
mum node degree kmax, characteristic path length L,
clustering coefficient C and the power-law estimation
results of the CCD (scaling parameter, lower-bound and
p-value) for every GCOP. The <k> ranges from 1.60 to
4.06, which means many proteins have only a few inter-
actions. But there exist some proteins with many interac-
tions in PINs, the number of interactions can be as high
as 283 (Protein JSN1 in S. cerevisiae, dip: 1281N). The
kmax values are much larger than <k>it strongly indic-
ates that PINs cannot be well depicted by random graph.
Then we made the power-law estimation for the CDD of
GCOP, the results show that the scaling parameter lies
between 1.65 and 3.5 of GCOP for six species, and the
low-bound dmin ranges from 1 to 28. Furthermore po-
wer-law hypothesis was evaluated by goodness-of-fit
test [18]. We calculated the goodness-of-fit between the
cumulative degree distribution and the best fitted power
law model. The resulting p-value was listed in P-value
columns. Here, a relatively conservative choice was made
that the power law was ruled out if p-value 0.1. When
the p-value is greater than 0.1, power law is a plausible
model for the CDD, otherwise it is rejected. The results
show that the power-law is not a proper model for Ecoli,
Hpylo and Scere but a reasonable one for Celeg, Hsapi,
Dmela. The characteristic path length L of PINs lies be-
tween 3.81 and 6.53.
The characteristic path length L of the GCOP for six
species is around 4, which means that any two proteins
can indirectly interact via relatively short successive bio-
chemical reactions. The clustering coefficient C of the
GCOP is relatively small; the highest clustering coeffi-
cient appears in GCOP of Ecoli is 0.108 and for other
species C is around 0.03. This is important evidence for
that GCOP is a multi-scale network with tree structure.
The simulation results of CTM include the CDD, C
and L for the family of CTM. By altering the number of
levels (n), the size of substrate N(n) can ascend nonline-
arly. Furthermore, with the increase of P0, the adding
rule can result in nodes in higher levels can obtain more
shortcuts, and the lower level nodes have less chance to
connect with other nodes. Especially the nodes in lowest
level can be only linked by upper nodes. The shift of the
CDD curve for the family of CTM is depicted in Figure
3. Just a few shortcuts between different hierarchies lead
to rapid descending of L. Meanwhile, the C rises up with
more loops formed in shortcut adding process. Charac-
teristic path length L(p) and clustering coefficient C(p)
of the CTM family are showed in Figure 4. Table 3
shows that the two main topological parameters of CTM
are really approximate to the parameters of GCOP; GC-
OP can be well presented by CTM.
D. C. Ma et al. / Natural Science 2 (2010) 998-1004
Copyright © 2010 SciRes. OPEN ACCESS
Figure 3. Degree distribution of the CTM model with different
P0 (0.2, 0.5, 0.8, 1.0), n = 9.
Figure 4. Characteristic path length L and clustering coeffi-
cient C for the family of CTM with n = 9. L0 and C0 is the
characteristic path length and clustering coefficient of CTM
with P0 = 1.
In this paper, we employed COMPLEX NETWORKS
THEORY to analyze the PPI database DIP (Oct. 2008).
Eight species (Ecoli, Hpylo, Celeg, Dmela, Hsapi, Mm-
usc, Rnorv and Scere) were considered, but the GCOP of
Mmusc and Rnorv are smaller, we focused on the large-
scale analysis of GCOP of other six species. Three glo-
bal parameters (node degree distribution, characteristic
path length L, and the clustering coefficient C) were us-
ed to characterize the GCOP. The logarithm of Pcum(k) vs.
the logarithm of node degree log k indicated that the
CCDs of PINs are a group of right-skewed curves. We
also tested the power-law behavior for CCD curves.
Thus in some cases power-law may not be more interes-
ting than any other heavy-tailed distribution. But in our
work, the goal is to infer plausible mechanisms that mi-
ght underlie the formation and evolution of PINs; it may
matter greatly whether the CDD of PINs follows a pow-
er law or not. We employed a new testing technique to
evaluate the power-law hypothesis for CDD. The estim-
ation results show that power-law is a plausible model
for the CDD of GCOP for Celeg, Dmela and Hsapi, but
for other three species: Ecoli, Hpylo, Scere, the power-
law is ruled out. Furthermore, there are only a few prote-
ins of which the degree values are larger than lower bo-
und dmin in GCOP of Celeg, Dmela and Hsapi (Celeg
227/2386 Dmela 287/7351 Hsapi 132/805), the best-
fitted part of CDD is really short. In conclusion, the po-
wer-law model might not be a proper model for the CDD
of GCOP and previous scale-free models are not proper
for PINs.
The calculations of characteristic path length L and
clustering coefficient C indicate that the GCOPs are mu-
lti-scale networks without many loops. The long-range
interactions between different local parts make the L be
close to the small-world limit given by random networks,
otherwise the long-range interactions have no significant
effect on local structure, the clustering coefficient is rel-
atively small, and those issues suggest that tree structure
be the basic framework of GCOP. With the aim to repre-
sent the framework of GCOP, we proposed a nonlinearly
growth model: CTM. In our model, network size can be
changed nonlinearly with different n (number of tree
levels) and the amount of shortcuts can be altered by
altering P0(shortcut adding probability of the main root
to all other nodes below level 0 except the two leaf
nodes it already linked). We found out the P0 and n to
approximate the GCOP after accomplishing mass com-
puter simulations of CTM. From the comparison result
between GCOP and according CTM in Table 3, it is
clear that the giant component can be well represented
by CTM.
Our study only offers a starting point for understand-
ing the simple nature of PINs: complex tree. The propo-
sed CTM offers us a new perspective in exploring the
PINs. In this model, the upper nodes of substrate have m
= 2 leaves. For further modeling, we may generalize this
situation by changing the arbitrary parameter m, such as
to 3 or 4, or connect the brotherhood leaves in the same
level. In summary, our research might be helpful to und-
erstand the basic framework of PINs: complex tree and
the CTM would be a powerful tool for correlating the
topological with functional properties of the PINs.
The authors gratefully thank Aaron Clauset for sharing the Power-law
estimation code. The work was funded by the National Natural Science
Foundation of China (20775052, 20972103).
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