J. Biomedical Science and Engineering, 2009, 2, 644-650
doi: 10.4236/jbise.2009.28094 Published Online December 2009 (http://www.SciRP.org/journal/jbise/
JBiSE
).
Published Online December 2009 in SciRes.http://www.scirp.org/journal/jbise
Folding rate prediction using complex network analysis for
proteins with two- and three-state folding kinetics
Hai-Yan Li1, Ji-Hua Wang1
1Key Lab of Biophysics in Universities of Shandong, Dezhou University, Shandong, China.
Email: tianwaifeixian78@163.com , jhwyh@yahoo.com.cn
Received 5 September 2009; revised 9 October 2009; accepted 10 October 2009.
ABSTRACT
It is a challenging task to investigate the different in-
fluence of long-range and short-range interactions on
two-state and three-state folding kinetics of protein.
The networks of the 30 two-state proteins and 15
three-state proteins were constructed by complex
networks analysis at three length scales: Protein Con-
tact Networks, Long-range Interaction Networks and
Short-range Interaction Networks. To uncover the
relationship between structural properties and fold-
ing kinetics of the proteins, the correlations of protein
network parameters with protein folding rate and
topology parameters contact order were analyzed.
The results show that Protein Contact Networks and
Short-range Interaction Networks (for both two-state
and three-state proteins) exhibit the “small-world”
property and Long-range Interaction networks indi-
cate “scale-free” behavior. Our results further indi-
cate that all Protein Contact Networks and Short-
range Interaction networks are assortative type.
While some of Long-range Interaction Networks are
of assortative type, the others are of disassortative
type. For two-state proteins, the clustering coeffi-
cients of Short-range Interaction Networks show
prominent correlation with folding rate and contact
order. The assortativity coefficients of Short-range
Interaction Networks also show remarkable correla-
tion with folding rate and contact order. Similar cor-
relations exist in Protein Contact Networks of
three-state proteins. For two-state proteins, the cor-
relation between contact order and folding rate is
determined by the numbers of local contacts. Short-
range interactions play a key role in determining the
connecting trend among amino acids and they impact
the folding rate of two-state proteins directly. For
three-state proteins, the folding rate is determined by
short-range and long-range interactions among resi-
dues together.
Keywords: Protein Contact Networks; Small-World;
Scale-Free; Assortative Type; Folding Rate
1. INTRODUCTION
The network concept is increasingly used to describe the
topology and dynamics of complex systems. As the es-
sential matter of life, proteins are biological macro-
molecules made up of a linear chain of amino acids and
fold into unique three-dimensional structures (native
states). Despite the large degrees of freedom, proteins
fold into their native states in a very short time. It is im-
portant to understand how proteins consistently fold into
their native-state structures and the relationship between
structures and function. A protein molecule can be
treated as a complex network with each amino acid sim-
plified as a node and the interaction between them as a
link. Efforts have been made to model proteins as net-
works for studying protein topology, small world prop-
erties and examining the nucleation in protein folding
[1-10]. Bagler and Sinha [11], in their recent protein
network analysis, constructed Protein Contact Networks
and Long-range Interaction Networks to analyze the
assortative mixing of networks and folding kinetics of
two-state proteins.
But there is a significant difference in the folding be-
havior of small proteins with simple two-state kinetics
and of larger proteins having a three-state folding kinet-
ics [12]. The two-state proteins have no visible interme-
diates in the course of folding, which therefore occur as
an “all-or-none” process under all experimental condi-
tions. However, the proteins with three-state folding
kinetics fold via intermediates, which accumulate during
the early stages of folding when it occurs in denatur-
ant-free water [13-16]. Based on the work by Bagler and
Sinha, two- and three-state proteins that belong to dif-
ferent structural classes were selected from protein crys-
tal structure data bank to model the native-state protein
structures as networks. To investigate various topologi-
cal properties, the network models were constructed at
three different length scales. Protein Contact Networks
(PCNs) were built by considering the contacts between
atoms in amino acid residues. There is a natural distinc-
tion of contacts into two types: long-range and short-
H. Y. Li et al. / J. Biomedical Science and Engineering 2 (2009) 644-650 645
SciRes Copyright © 2009 JBiSE
range interactions [7]. We considered the Long-range
Interaction Networks (LINs) and Short-range Interaction
Networks (SINs) of each protein, which are subsets of
the corresponding PCNs. To investigate if the general
network parameters can offer any clue to the bio-
physical properties of the existing three dimensional
structure of a protein, these networks were analyzed to
focus on their topology including clustering coeffi-
cients, shortest path length, average degree, degree
distribution and assortative mixing behavior of the
amino acid nodes. The determination of folding rate
for two- and three-state folding kinetics has a signifi-
cant difference. To uncover the relationship between
the structural properties and the folding kinetics of the
proteins, the correlation of protein network parameters
with protein folding rate (lnkf) and topology parame-
ters contact order (CO) was analyzed. The values of
lnkf and CO are available as given in Reference [12].
Through our coarse-grained complex network model
of protein structures, it was found that short-range in-
teractions play a key role in determining the connect-
ing trend among amino acids and impact directly the
folding rate of two-state proteins. For three-state pro-
teins, the folding rate is determined by short-range and
long- range interactions among residues together.
2. METHODS
2.1. Construction of PCNs, LINs, SINs and
Their Random Networks
In this paper, 30 proteins with two-state kinetics and 15
proteins with three-state kinetics were studied and the
dataset was taken from the paper [12]. The data of these
protein structures were taken from the Protein Data Bank
(PDB) to model them as Protein Contact Networks
(PCNs) by setting the Cα atoms as the nodes, and estab-
lished a link between two nodes, if the atoms were with-
in a cut-off distance (0.8nm).
The Long-range Interaction Network (LIN) of a PCN
was obtained by considering the interactions which oc-
cur between amino acids that were twelve or more
amino acids apart in the primary sequence. A LIN was a
subset of its PCN with same numbers of nodes (N) but
fewer numbers of links due to removal of the short-range
contacts. The Short-range Interaction Network (SIN) of
a PCN was built with the amino acids separated within
twelve. For compare, the random network was con-
structed with the same numbers of residues (N) and links
as those of the PCNs, SINs, LINs.
2.2. Network Parameters
The degree of any node i is represented by .
Here is the element of the adjacency matrix, whose
value is 1 if an edge connects a node “I” to another node
j” and 0 otherwise. N is the number of nodes. Average
degree <k> of a network is defined as
N
jiji ak
1
ij
a

N
ii
k
N
k1
1.
The shortest path length is related to the link number of a
pathway between two nodes and it is the least link number
of all the pathways between two nodes. The average short-
est path length is defined as



1
11
1
1N
i
N
ij ij
L
NN
L,
where is the shortest path length between nodes i
and j.
ij
L
The average clustering coefficient C is the average
over all vertices of the fraction of the number of con-
nected pairs of neighbours for each vertex. It is calcu-
lated as follows:
N
ii
C
N
C1
1, where is the clus-
tering coefficient for a node i and defined as the fraction
of links that exist among its nearest neighbours to the
maximum number of possible links among them. It
scales the cohesiveness of the neighbours of a certain
node from the view of topology.
i
C
Many networks show “assortative mixing” on their
degrees. The Assortativity Coefficient (r) measures the
tendency of degree correlation. It is the Pearson Correla-
tion Coefficient of the degrees at either ends of an edge.
Its value was calculated using the function suggested by
Newman [17] and was given as



,
1
1
2
1
2
1
2
1
2
1221
2
11





r
kjMkjM
kjMkjM
r
ii
iiii
iI
iiii
The networks having positive r values are assortative
in nature and the negative value implies that the network
is of disassortative type.
3. RESULTS AND DISCUSSION
3.1. Network Parameters of PCNs, LINs, SINs
3.1.1. Average Degree of the Networks
The average degree <k> was calculated for each of the
three type networks (PCNs, SINs, LINs) of two- and
three-state proteins. Figure 1 shows the average degree
<k> as a function of network size N. Table 1 shows the
average degree of three type networks for two- and
three-state proteins. The values of <k> have no obvious
difference between two- and three-state proteins. In
other words, the average number of contacts per residue
for three-state proteins is similar equal to that of
two-state proteins. For two-state proteins, the average
number of short-range contacts is smaller than that of
646 H. Y. Li et al. / J. Biomedical Science and Engineering 2 (2009) 644-650
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4080120 160 200 240 280
2
4
6
8
10
PCN-two state
SIN-two state
LIN-two state
PCN-three state
SIN-three state
LIN-three state
<k>
Network size N
Figure 1. Average degree <k> as a function of the network
size N for 30 two-state and 15 three-state proteins.
Table 1. Average degree <k> of PCNs, SINs, LINs.
Network type <k>two-state <k>three-state
PCNs 9.31±0.664 9.41±0.582
SINs 6.23±0.668 6.56±0.724
LINs 2.99±0.999 2.85±0.938
three-state proteins and the average number of long-
range contacts is slightly higher than that of three-state
proteins. In general, for coarse-grained complex network
model of protein structures, it has been shown, for dif-
ferent folding kinetics, that the short-range interactions
and long-range interactions are consistent with each
other for a statistical equilibrium. It is observed that the
average degree <k> for LINs shows lower values than
that of SINs and PCNs regardless of their states. It indi-
cates that long- range interactions exhibit a predominant
lower average connectivity compared with short-range
interactions. Protein structure has the strongest average
connectivity by integrating both short-range and long-
range interactions. To verify whether the observed trend
depends on the network size (i.e., the number of amino
acids of the protein), the correlation coefficient between
<k> and N was calculated. Any significant relationship
between <k> and N in SINs and LINs for two- and
three-state proteins was not found. On the other hand,
Figure 1 indicates that the <k> of PCNs (both two- and
three-state proteins) show a high positive correlation
with N. The correlation coefficients of three-state pro-
teins are higher than that of two-state proteins, and their
values are 0.672 (p=0.006) and 0.511 (p=0.004), respec-
tively.
3.1.2. “Small-World” Property
To examine whether the networks have the “small-
world” property, the average clustering coefficient C
and the average shortest path length L of each of the
networks and their respective Cr and Lr for the random
networks with the same size were calculated. Accord-
ing to Watts and Strongatz [18], a network has the
“small-world” property if C>>Cr and LLr. Cr and Lr
can be calculated using the expressions NkCr/
and
kNLrln/ln . Table 2 shows the <C> and
<L> of 30 two-state proteins and 15 three-state proteins
and the corresponding values of random networks. It is
obviously found that PCNs and SINs (both two- and
three-state proteins) are characterized by large values
of <C> and <L> compared with the corresponding
random networks, which have the typical property of
small-world networks. It indicates that any two amino
acids are connected with each other via only a few
other amino acids in both two- and three-state proteins.
Whereas LINs have similar <C> with their random
networks and their <L> are smaller than those of the
corresponding random networks. It indicates that LINs
do not exhibit the “small-world” property. Table 2 also
shows that two-state proteins have similar values of
<C> with three-state proteins for three types networks
and LINs have remarkable lower <C> than those of
PCNs and SINs. It suggests that long-range interactions
have reduced congregating of amino acids, which may
facilitate communication among distant residues in the
native structure to some extent, but such a feature can
also increase the folding time as it requires distant resi-
dues in the chain to come closer during the folding
process. Table 2 also shows that <L> of three-state
proteins are more higher compared with corresponding
two state proteins. It suggests that three-state proteins
are packed more loosely than two-state proteins and it
has a low global connectivity compared with two-state
proteins.
3.1.3. Degree Distribution
The degree distribution is an important feature which
characterizes the network topology. Figure 2 shows
the degree distribution of three types’ networks for
two- and three-state proteins. The shape of the degree
distribution of small-world network is bell-shaped,
Poisson-like. It has a pronounced peak at <k> and de-
cays exponentially for large k. Thus the topology of
the network is relatively homogeneous, all nodes hav-
ing approximately the same number of edge. The
shape of the degree distribution is Poisson distribution,
which is another typical property of “small-world”
networks. A network lacking a characteristic scale <k>
and having degree distribution of a power-law form is
known as “scale-free” network [19]. From Figure 2(a),
the long-range interaction distribution patterns (both
two- and three-state), it is noticed that a large number
of nodes with a small number of links and a small
H. Y. Li et al. / J. Biomedical Science and Engineering 2 (2009) 644-650 647
SciRes Copyright © 2009
Table 2. Values of <C> and <L> of three types networks of
two- and three-state proteins as well as those for the corre-
sponding random networks.
JBiSE
Network type <C> <Cr> <L> <Lr>
Two-state
PCNs 0.589 0.122 2.97 1.947
SINs 0.649 0.085 7.269 2.413
LINs 0.039 0.04 3.796 5.677
Three-state
PCNs 0.57 0.079 3.694 2.139
SINs 0.65 0.056 10.827 2.584
LINs 0.042 0.023 4.828 13.953
number of links with a large number of nodes in the
distribution pattern, indicating the “scale-free” behav-
iour. But from Figure 2(b), the short-range interaction
distribution patterns as well as those of total interac-
tions are of the Poisson type. This indicates again that
PCNs and SINs exhibit “small-world” behaviour, while
LINs indicate “scale-free” behaviour.
The scale-free degree distribution of LINs indicates
that proteins contain hubs, i.e. central residues, which
have a large number of long-range interactions with
other residues. The kinetic mechanism of transitions
from the denatured state to the native state is nucleation
[20]. The nucleus is composed of a set of adjacent resi-
dues, and is stabilized by long-range interactions that are
formed as the rest of the protein collapses around it. The
Poisson degree distribution means that protein structures
have a much smaller number of hubs than most self-
organized networks including most cellular or social
networks. The major reason for this deviation from the
scale-free degree distribution lies in the limited simulta-
neous binding capacity of a given amino acid side-chain
(also called as excluded volume effect). The limited
amino acid side chain binding capacity contributes to the
fact that each amino acid has a characteristic average
degree. This depends on the interaction cut-off, which
makes hydrophilic amino acids “strong hubs” (observed
at high interaction cut-off allowing low overlaps), and
hydrophobic amino acids “weak hubs” (at low interac-
tion cut-off allowing high overlaps), respectively. Hubs
are integrating various secondary structure elements, and,
therefore, it is not surprising that they increase the ther-
modynamic stability of proteins.
3.1.4. Assortative Mixing Behavior of the Nodes
The assortative mixing concept has been used in social,
technological and biological networks [17]. In social
networks assortative mixing leads to homophily, i.e.,
the tendency of individuals to associate with similar
partners. This quantity is also important to control
epidemics since assortative has a profound impact on
the percolation in networks. Contrary to social net-
works, which tend to be assortative, biological and
technological networks tend to be disassortative. Con-
cerning this aspect, the networks are classified as to
show assortative mixing, if the degree correlation is
positive, a preference for high-degree nodes to attach
to other high-degree nodes, or disassortative mixing,
otherwise. Assortativity Coefficient (r) for each of the
networks was calculated, as shown in Table 3. It indi-
cates that all the PCNs and SINs have positive r values
regardless of two-state or three-state, while the LINs
have both positive and negative r values. The ratio of
negative rL values for two-state is significantly higher
than that for three-state. The former is 17/30, while the
latter is 3/15.
0246810 12 14 16 18
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
two state
three state
P(k)
K
(a)
0 2 4 6 81012141618202224
0
5
10
15
20
25
30
35
40
45
50
55
60
65
PCN-two state
PCN-three state
SIN-two state
SIN-three state
P(K)
K
(b)
Figure 2. Degree distribution of three type networks for two- and three-state proteins. (a) LINs of two- and three-state proteins; (b)
PCNs and SINs of two- and three-state proteins.
648 H. Y. Li et al. / J. Biomedical Science and Engineering 2 (2009) 644-650
SciRes Copyright © 2009 JBiSE
Table 3. Assortativity coefficient (r) of three type networks for
two- and three-state proteins.
Network type PCNs SINs LINs
r (Two-state) 0.106~0.424 0.106~0.562 -0.566~-0.006
0~0.35
r (Three-state) 0.074~0.562 0.047~0.54 -0.349~-0.117
0.008~0.369
The r values of different networks suggest that all
PCNs and SINs are of assortative type, the LINs of
three-state proteins (except three) are also of assortative
type. While maximum of LINs of two-state have the
characteristics of disassortative mixing, few others are of
assortative type. Thus it may be said that in all of the
PCNs and SINs the residues (nodes) with high degree
have tendencies to be attached with the residues having
high degree values. The result is consistent with previ-
ous study by S. Kundu [21] and Ganesh Bagler [11]. But
in some LINs of two-state and three-state proteins hav-
ing negative r values the mixing pattern of amino acid
residues are different. Here the amino acids (nodes)
having high degree values have a tendency to be at-
tached with amino acids with smaller degree. This result
is not consistent with Ganesh Bagler, who concluded
that the assortative mixing in PCNs and LINs is a ge-
neric feature of protein structures. Recent research sug-
gests that assortative mixing by degree reduces the sta-
bility of networks [22]. In almost all biological networks
(e.g. protein interaction network, neural network etc.),
nodes of high degree tend to avoid being connected to
other highly connected nodes, i.e. these networks show
disassortative mixing. This difference of assortative
mixing between SINs and LINs may be a possible rea-
-son for the stability of native-state proteins and the re-
search of assortative mixing in LINs may give interest-
ing surprises in the future. However, the PCN is a com-
posite network of SIN and LIN. When considering the
protein structure networks, the r values had been ob-
tained, which represent a cumulative effect of either all
positive r values or a mixture of positive and negative r
values. Thus it was find that protein structure networks
always have positive r values and they are assortative.
3.2. Correlations of Protein Network Parameters
with Folding Rate (Lnkf) and Contact
Order (CO)
To uncover the relationship between the structural prop-
erties and the folding kinetics of the proteins, the corre-
lation of protein network parameters with protein folding
rate (lnkf) and contact order (CO) was studied. The cor-
relation coefficient between general network parameters
(e.g., C, L, <k>, and r) and the folding rate logarithm
(lnkf) were calculated out. And similar correlation be-
tween network parameters and CO was also discussed.
3.2.1. Correlation for Two-State Kinetics
For all the 30 two-state proteins, the clustering coeffi-
cients C of PCNs and LINs have not any significant re-
lationship with the lnkf, and the correlation coefficients
are 0.248 (p=0.186), 0.118 (p=0.534), respectively.
However, SINs have high positive correlation between C
and lnkf (correlation coefficient are 0.602, p=0.000).
From Table 2, the clustering coefficients of LINs are
significant lower than those of PCNs and SINs, which
show a low correlation with the folding rate of the pro-
teins. It indicates that clustering of amino acids that par-
ticipate in the long-range interactions, into “cliques”
slows down the folding process of two-state proteins.
SINs have the highest clustering coefficients among
them and C of SINs have significant correlation with the
folding rate, indicating that the short-range interactions
may be playing a constructive and active role in deter-
mining the rate of the two state proteins folding process.
The similar correlation occurs between r and lnkf. The
correlation coefficient between r and lnkf of SINs is
0.625 (p=0.000). For PCNs and LINs, the correlation
coefficients are 0.295 (p=0.181) and 0.121 (p=0.753),
respectively. It shows that short-range interactions play a
key role in determining the connecting trend among
amino acids and influence the folding rate of two-state
proteins directly.
Previous studies have found that contact order (CO)
has a significant correlation with folding rate of proteins
(correlation coefficient of these 30 proteins is 0.72,
p=0.000). As an experiential parameter based on 3D
structure, though significant correlating with folding rate,
the physical meanings of CO is ambiguity. In this study,
it is found that CSIN and rSIN have a high correlation with
contact order (CO). The correlation coefficients are
0.64 (p=0.000) and 0.817 (p=0.000), respectively.
Since the clustering coefficients depend on the degree of
the node, we calculated the correlation coefficients be-
tween CSIN*<k>SIN and lnkf. It shows high positive cor-
relation (correlation coefficients are 0.733, p=0.000)
between them for these two-state proteins. A significant
high correlation also exists between CSIN*<k>SIN and
CO, the value is 0.796 (p=0.000) (see Figure 3).
CSIN measures the transitivity in the short-range inter-
action network and <k>SIN measures the average number
of short-range contacts per residue. It indicates that the
correlation between CO and lnkf is determined by the
number of local contacts for two-state proteins. It is con-
sistent with the previous study by Mirny and Shakhno-
vich [23]. It is interesting to note that despite dissimilar
quantities that CO and CSIN measure, the similar correla-
tion coefficients essentially indicate the important role of
short-range contact formation in the rate of folding for
two-state proteins.
3.2.2. Correlation for Three-State Kinetics
For three-state proteins, the clustering coefficients C of
H. Y. Li et al. / J. Biomedical Science and Engineering 2 (2009) 644-650 649
SciRes Copyright © 2009 JBiSE
3.0 3.5 4.0 4.5 5.0 5.5
-2
0
2
4
6
8
10
12
14
16
18
20
22
lnkf
CO
lnkf
CSIN*<K>SIN
CO
Figure 3. Correlation between CSIN*<k>SIN with lnkf and CO.
constructed to uncover the different influence of long-
range and short-range interactions on two- and three-
state folding kinetics. It was found that PCNs and SINs
(both two- and three-state proteins) have the typical
property of small-world networks, whereas LINs exhibit
the “scale-free” property.
the PCNs show a high positive correlation with the fold-
ing rate (correlation coefficient is 0.652, p=0.001). How-
ever, C of LINs and SINs have not significant relation-
ship with the lnkf, and the correlation coefficients be-
tween C and lnkf are 0.278 (p=0.315) and 0.405
(p=0.081), respectively. The similar correlation occurs
between r and lnkf. The correlation coefficient between r
and lnkf of PCNs is 0.603 (p=0.017). For LINs and
SINs, the correlation coefficients are 0.394 (p=0.146)
and 0.474 (p=0.075), respectively. It shows that, for
three-state proteins, the folding rate is determined by
short-range and long-range interactions among residues
together.
4. CONCLUSIONS
The network concept is increasingly used to describe the
topology and dynamics of complex systems. In this pa-
per, the three type networks (PCNs, LINs, SINs) were-
All of PCNs, SINs and nearly all LINs of three-state
proteins are of assortative type. While maximum of
LINs of two-state are of disassortative type. This differ-
ent assortative mixing behaviour of LINs may be a pos-
sible reason for the stability of native-state proteins and
the research of assortative mixing in LINs may give in-
teresting surprises in the future.
For two-state proteins, CSIN and rSIN show high corre-
lation with lnkf and CO, which indicates the correlation
between CO and lnkf is determined by the numbers of
local contacts. Short-range interactions play a key role in
determining the connecting trend among amino acids
and influence directly the folding rate of two-state pro-
teins. For three-state proteins, CPCN and rPCN also show
high correlation with lnkf and CO, which shows that the
folding rate is determined by short-range and long-range
interactions among residues together.
5. ACKNOWLEDGMENTS
This work was supported by a grant from Chinese National Key Fun-
damental Research Project (No. 30970561) and Shandong Fundamen-
tal Research Project (NO. Y2005D12).
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