Folding rate prediction using complex network analysis for proteins with two- and three-state folding kinetics

doi:10.4236/jbise.2009.28094

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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

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





jiji ak



 

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











ij ij

where is the shortest path length between nodes i

and j.

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: 



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.

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



1221































kjMkjM

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

4080120 160 200 240 280

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 L≥Lr. 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

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

two state

three state

P(k)

(a)

0 2 4 6 81012141618202224

PCN-two state

PCN-three state

SIN-two state

SIN-three state

P(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

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

3.0 3.5 4.0 4.5 5.0 5.5

-2

lnkf

CSIN*<K>SIN

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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