# Fractal and complex network analyses of protein molecular dynamics

Yuan-Wu Zhou1,2, Jin-Long Liu1, Zu-Guo Yu1,3, Zhi-Qin Zhao1 and Vo Anh3
1Hunan Key Laboratory for Computation and Simulation in Science and
Engineering and
Key Laboratory of Intelligent Computing and Information Processing of Ministry
of Education,
Xiangtan University, Xiangtan, Hunan 411105, China.
2Civil Construction Engineering Department, Guangxi University of Science and
Technology,
Liuzhou 545000, China.
3School of Mathematical Sciences, Queensland University of Technology, GPO Box
2434,
Brisbane, Q4001, Australia. Corresponding to<EMAIL_ADDRESS>
###### Abstract

Based on protein molecular dynamics, we investigate the fractal properties of
energy, pressure and volume time series using the multifractal detrended
fluctuations analysis (MF-DFA) and the topological and fractal properties of
their converted horizontal visibility graphs (HVGs). The energy parameters of
protein dynamics we considered are bonded potential, angle potential, dihedral
potential, improper potential, kinetic energy, Van der Waals potential,
electrostatic potential, total energy and potential energy. The shape of the
$h(q)$ curves from MF-DFA indicates that these time series are multifractal.
The numerical values of the exponent $h(2)$ of MF-DFA show that the series of
total energy and potential energy are non-stationary and anti-persistent; the
other time series are stationary and persistent apart from series of pressure
(with $H\approx 0.5$ indicating the absence of long-range correlation). The
degree distribution of their converted HVGs show that these networks are
exponential. The results of fractal analysis show that fractality exists in
these converted HVGs. For each energy, pressure or volume parameter, it is
found that the values of $h(2)$ of MF-DFA on the time series, exponent
$\lambda$ of the exponential degree distribution and fractal dimension $d_{B}$
of their converted HVGs do not change much for different proteins (indicating
some universality). We also found that after taking average over all proteins,
there is a linear relationship between $\langle h(2)\rangle$ (from MF-DFA on
time series) and $\langle d_{B}\rangle$ of the converted HVGs for different
energy, pressure and volume.

Key words: protein molecular dynamics; multifractal detrended fluctuation
analysis; horizontal visibility graph; fractal analysis; degree distribution.

##  1 Introduction

Proteins are among the most important biomacromolecules because they can
perform biological functions, which are usually determined by their
structures. To date, structures of 97362 proteins (updated in Protein Data
Bank [1] on 2014-01-28) have been obtained by experimental methods, such as
X-ray $(88.4\%)$, solution NMR $(10.6\%)$ and electron microscopy $(0.7\%)$.
On the other hand, molecular dynamics simulation packages such as NAMD (Not
(just) Another Molecular Dynamics program) [2] has been developed to learn
about different aspects of proteins. Protein molecule energy includes kinetic
energy and potential energy, and potential energy can be calculated by quasi-
empirical way (such as CHARMM [3] and AMBER [4]). In fact, Rueda et al. [5]
used NAMD to analyze the molecular dynamics of 30 real proteins. This paper
uses NAMD to simulate protein molecular dynamics, then derive the time series
for some corresponding parameters whose features reflect the aspects of
protein structures.

Apart from being characterized by an enormous number of degrees of freedom,
proteins have multidimensional potential energy surfaces [6]. It is well known
that the self-similarity exhibiting in the distributions of their biophysical
and biochemical properties can serve as an effective tool to extract the
inherently inhomogeneous and nonlinear behaviors of protein structures.
Fractal dimension (FD) has been widely used to characterize the self-
similarity of fractal objects [7, 8]. Many FD-based methods have been proposed
to investigate protein structures [6]. Fractal methods can also be used to
characterize the scaling properties of time series and then to reveal the
self-similarity of the original system [7]. A multifractal system is a
generalization of a fractal system in which a single exponent (the fractal
dimension) is not enough to describe its dynamics; instead, a continuous
spectrum of exponents (the so-called singularity spectrum) is needed [9]. The
concept of multifractal phenomena describes that different regions of an
object have different fractal properties, and multifractal scaling provides a
quantitative description of a broad range of heterogeneous phenomena [10].
Multifractal analysis was initially proposed to treat turbulence data and is a
useful way to characterize the spatial heterogeneity of both theoretical and
experimental fractal patterns [11]. The multifractal detrended fluctuations
analysis (MF-DFA) [12] is an extension of the standard detrended fluctuation
analysis (DFA) introduced by Peng et al. [13, 14]. DFA can be employed to
detect long-range correlations in stationary and nonstationary time series.
Hence MF-DFA is a suitable tool to characterize the multifractal property and
long-range correlation in time series. Multifractal analysis has been used to
study genomes [15-17], protein structures and functions (e.g. [18-22]). In
this paper, we investigate by MF-DFA [12] the scaling property of the
associated time series of energy, pressure and volume derived from simulations
of protein molecular dynamics.

In last decade, the study of complex networks has gained prominence across
many disciplines of science. Studies have shown that complex network theory
has become a powerful tool to analyze protein complex structures [23-25].
Single proteins in 3D space can also be considered as biological complex
systems emerged through the interactions of their constituent amino acids.
These interactions among the amino acids within a protein can be presented as
residues interaction network (RIN) (also called residues interaction graphs
(RIGs), protein contact network (PCN), protein structure network (PSN),
protein contact map (PCM), amino acid network (AAN)), which can be constructed
with varying definitions of nodes and edges [23-25].

Recent studies showed that complex network theory (such as recurrence
networks) may also be an effective method to analyze time series [26-33].
Lacasa et al. [26] proposed the visibility graph (VG) algorithm to convert
time series into complex networks. Then Luque et al. [27] proposed the
horizontal visibility algorithm to convert time series into complex networks.
It has been shown that these converted networks inherit several properties of
time series in the structure of networks [34-36]. Therefore, we can understand
time series from a new point of view using the converted networks. In this
paper, we hope to reveal some meaningful information in the associated time
series of energy, pressure and volume for real proteins from the perspective
of the horizontal visibility graphs (HVGs) [27]. This prompts us to further
study the fundamental topological and fractal properties of the converted HVGs
from different energy, pressure and volume time series of real proteins.

The fractal and self-similarity properties of complex network have also been
focused and studied widely in a variety of fields [37-39]. It is found that
many complex networks, including the world-wide web (WWW), social networks,
protein-protein interaction (PPI) networks and cellular networks, are self-
similar under a certain length-scale. Some numerical algorithms have been
proposed to calculate the fractal dimensions of complex networks. Song et al.
[40] proposed a box-counting algorithm to calculate their fractal dimension.
Kim et al. [41] studied the skeleton and fractal scaling in complex networks
via an improvement method. Later on, Zhou et al. [42] introduced an
alternative algorithm to detect self-similarity of cellular networks.
Recently, Li et al. [43] studied the fractal properties of a family of fractal
networks using the random sequential box-covering algorithm proposed by Kim et
al. [41]. Liu et al. [33] adopted this algorithm to calculate the fractal
dimensions of the recurrence networks constructed from fractional Brownian
motions. In this paper, we also adopting the random sequential box-covering
algorithm to calculate the fractal dimension of the converted HVGs of the time
series of energy, pressure and volume for real proteins.

##  2  Molecular dynamics for protein

The behavior of biomolecular systems can be modelled by molecular dynamics,
which is helpful to understand protein functions. A common procedure of
molecular dynamics [44] for proteins is as follows:

(a) Initialize the positions and velocities of all atoms. The initial
positions of atoms come from Protein Data Bank (PDB [1]). Add water and ions
to a box with protein, minimize the box system energy. Then, the initial
velocities of atoms are assigned from Maxwell’s distribution.

(b) Calculate forces for all atoms using potential energy. The protein
potential energy can be defined as

$\displaystyle V=$ $\displaystyle
V_{bonds}+V_{angles}+V_{dihedrals}+V_{impropers}$ (1)
$\displaystyle+V_{Vdw}+V_{electrostatic},$

where $V$ is protein potential energy, $V_{bonds}$ is bonded potential,
$V_{angles}$ is angle potential, $V_{dihedrals}$ is dihedral potential,
$V_{impropers}$ is improper potential, $V_{Vdw}$ is Van der Waals potential,
and $V_{electrostatic}$ is electrostatic potential. In this paper, we analyze
the potential energy and kinetic energy time series. Then, forces that act on
an atom can be written as

$\displaystyle\overrightarrow{F}=\frac{\partial V(\vec{r})}{\partial\vec{r}},$
(2)

where $\vec{r}$ is the position vector of atoms.

(c) Apply thermostat and volume changes. Update positions and velocities.
According to Newton’s law, the kinetic equation can be written as

$\displaystyle\overrightarrow{F}=m\frac{\partial^{2}\vec{r}}{\partial t^{2}}.$
(3)

Using an algorithm, such as the Verlet algorithm or LeapFrog algorithm, new
positions and velocities can be calculated.

(d) Repeat steps (b) and (c) till a termination condition is met.

(e). Analyze data. NAMD is a software for biomolecular molecular dynamics [2].
Some researchers used NAMD to achieve some important results [5]. In the
paper, we use it to simulate protein dynamics.

##  3 Multifractal detrended fluctuation analysis for time series

Detrended fluctuations analysis (DFA) introduced by Peng et al. [13, 14] can
be employed to detect long-range correlations in stationary and nonstationary
time series. The multifractal detrended fluctuations analysis (MF-DFA) [12] is
an extension of DFA. This approach can be described as follows. Given a time
series $x_{k}$, $k=1,\cdots,N$, where $N$ is the length of the series, the
profile $Y(i)$ is defined as

$\displaystyle Y(i)\equiv\sum_{k=1}^{i}[x_{k}-\langle x\rangle].\ \ \
i=1,\cdots,N,$ (4)

where $\langle x\rangle$ is the mean of sequence $x_{k}$. Then $Y(i)$ is
divided into $N_{s}\equiv int(N/s)$ non-overlapping and continuous segments of
equal length $s$. Then we estimate the variance of $y(i)$ by

$\displaystyle
F^{2}(s,v)\equiv\frac{1}{s}\sum_{i=1}^{s}\\{Y[(v-1)s+i]-y_{v}(i)\\}^{2},$ (5)

for each segment $v$, $v=1,\cdots,N$ and

$\displaystyle
F^{2}(s,v)\equiv\frac{1}{s}\sum_{i=1}^{s}\\{Y[N-(v-N_{s})s+i]-y_{v}(i)\\}^{2},$
(6)

for $v=N_{s}+1,\cdots,2N_{s}$, where $y_{v}(i)$ is a fitting polynomial in
segment $v$ (linear, quadratic, cubic, or higher order polynomial). Last we
take the average over $2N_{s}$ segments to achieve the $qth$ order fluctuation
function

$\displaystyle
F_{q}(s)\equiv\\{\frac{1}{2N_{s}}\sum_{v=1}^{2N_{s}}[F^{2}(s,v)]^{q/2}\\}^{1/q},\
\ \ (if\ q\neq 0).$ (7) $\displaystyle F_{q}(s)\equiv
exp\\{\frac{1}{4N_{s}}\sum_{v=1}^{2N_{s}}\ln[F^{2}(s,v)]\\},\ \ \ (if\ q=0).$
(8)

We will assume that $F_{q}(s)$ is characterized by a power law:

$F_{q}(s)\ \propto\ s^{h\left(q\right)}.$ (9)

The scaling function $h\left(q\right)$ is then determined by the regression of
$\ln F_{q}(s)$ on $\ln s$ in some range of time scale $s$.

MF-DFA is suitable for both stationary and nonstationary time series [12]. We
denote by $H$ the Hurst exponent of time series. The range $0.5<H<1$ indicates
persistence; and the range $0<H<0.5$ indicates anti-persistence; for
uncorrelated series, the scaling exponent $H$ is equal to 0.5 [45]. Assuming
the setting of fractional Brownian motion, Movahed et al. [46] proved the
relation $H=h(2)-1$ between $H$ and the exponent $h(2)$ for small scales. In
the case of fractional Gaussian noise, it was shown that $H=h(2)$ [46]. Hence
we can use the value of $H$ calculated from $h(2)$ to detect the nature of
long-range correlation (LRC) in time series under the assumption of fractional
Gaussian noise or fractional Brownian motion.

##  4 Complex networks: horizontal visibility graph for time series

A graph (or network) is a collection of nodes, which denote the elements of a
system, and links or edges, which identify the relations or interactions among
these elements.

Inspired by the concept of visibility [47], Lacasa et al. [26] proposed a
simple computational method to convert a time series into a network, known as
visibility graph (VG). It has been shown that time series structures are
inherited in the associated graphs, such as periodic, random, and fractal
series map into regular, random, and scale-free networks respectively [26]. In
addition, the degree distributions of the VGs constructed from random series
by a uniform distribution in $[0,1]$ have exponential tails $P(k)\sim
exp(-\lambda k)$ [26]. For this case, the greater the value of $\lambda$ is,
the faster attenuation of $P(k)$ is along with $k$. In contrast, the degree
distributions of the VGs converted from fractional Brownian motions have
power-law tails $P(k)\sim k^{-\alpha}$ [34].

Then a simplification of VG algorithm was proposed to map a time series into a
horizontal visibility graph (HVG) [27]. A HVG is obtained from the mapping of
a time series into a network according to the following horizontal visibility
criterion: Given a time series $\\{x_{1},x_{2},...,x_{N}\\}$, two arbitrary
data points $x_{i}$ and $x_{j}$ in the time series have horizontal visibility,
and consequently become two connected nodes in the associated graph, if any
other data point $x_{n}$ such that $i<n<j$ fulfils

$x_{i},x_{j}>x_{n}.$ (10)

Thus a connected, unweighted network can be constructed from a time series and
is called its horizontal visibility graph (HVG). In fact, the HVG of a given
time series is always a subgraph of its associated VG. Luque et al. [27] have
shown that the degree distribution of a HVG constructed from any random series
has an exponential form $P(k)=(3/4)exp(-k\ln(3/2))$. Then Lacasa et al. [35]
used the horizontal visibility algorithm to characterize and distinguish
between correlated, uncorrelated and chaotic processes. They showed that in
every case the series maps into a graph with exponential degree distribution
$P(k)\sim exp(-\lambda k)$, where the value of $\lambda$ characterizes the
specific process [35]. In this paper, we adopt the horizontal visibility
algorithm to convert the time series of energy, pressure and volume into HVGs.

##  5 Fractal analysis of horizontal visibility graph (complex networks)

The random sequential box-covering algorithm proposed by Kim et al. [41] is an
efficient algorithm for fractal analysis of complex networks. Recently, our
group used this algorithm to study the fractal properties of a family of
fractal networks and recurrence networks constructed from fractional Brownian
motions [33, 43].

For a given HVG or network, we let $N_{B}(r)$ be the minimum number of boxes
with size $r$ which are needed to cover the entire network. If the power-law
relation $N_{B}(r)\sim r^{-d_{B}}$ holds for some scaling-range of $r$ and
constant $d_{B}$, we say that the fractality exists in this network, and
$d_{B}$ is called its fractal dimension. So, in practice, we often obtain the
fractal dimension $d_{B}$ by fitting the linear relationship between
$N_{B}(r)$ and $r$ in a log-log plot.

Before using the random sequential box-covering algorithm to calculate the
fractal dimension of a network, we need to use Floyd’s algorithm [48] or
Dijkstra’s algorithm of MatLab toolbox to calculate the shortest-path distance
matrix $D$ of this network according to its adjacency matrix $A$. The random
sequential box-covering algorithm [41] can be summarized as follows.

(I) Ensure that all nodes in the network are not covered, and no node has been
selected as the center of a box.

(II) Denote the size of the network as $N$. We set $t=1,2,\ldots,T$; here we
take $T=1000$. Then we rearrange the nodes into $T=1000$ different random
orders. That is to ensure that the nodes of a network are randomly chosen as
center nodes.

(III) Set the radius $r$ of boxes which will be used to cover the nodes in the
range $[1,d]$, where $d$ is the diameter of the network (i.e. the longest
distance between nodes in the network).

(IV) Treat the nodes of the $t$th kind of random orders that we have obtained
in (II) as the center of a box successively, then search all the other nodes.
If a node has a distance within $r$ to the center node and has not been
covered yet, then cover it.

(V) If no more new nodes can be covered by this box, then we abandon this box.

(VI) Repeat (IV) - (VI) until all the nodes are covered by the corresponding
boxes. We denote the number of boxes in this box covering as $N(t,r)$.

(VII) Repeat steps (III) and (VI) for all the random orders to find a box
covering with minimal number of boxes $N(t,r)$. Then denote this minimal
number of boxes as $N_{B}(r/d)$.

(VIII) For different $r$, repeat (III)-(VII). Then we use the linear
regression of $-\ln(N_{B}(r/d))$ vs $\ln(r/d)$ to estimate the fractal
dimension of the HVG or network.

##  6 Results and discussion

We use a list of 29 proteins (without ligands, RNA, or DNA, greater than 100
amino acids) as in [5] to simulate their dynamics (see Table 1) by NAMD.

Using NAMD with energy step size 200$fs$, protein structures from PDB were
used as starting points for 10-$ns$ production trajectories, performed at
constant pressure $(1atm)$ and temperature $(310K)$ using standard coupling
schemes (the same in all cases). CHARMM force fields were used in production
runs. Then, we ontained time series of bonded potential, angle potential,
dihedral potential, improper potential, electrostatic potential, Van der Waals
potential, kinetic energy, total energy, potential energy, pressure, and
volume. For example, we plotted the 11 time series of protein 1A3H in Figures
1 to 3.

Then we performed MF-DFA on these time series. As an example, we show how to
estimate the exponents $h(q)$ for the bonded potential time series of protein
1A3H in Figure 4. The numerical results showed that the best fit occurs in the
range $s\in[3,22]$. So we used this scaling-range to calculate the exponents
$h(q)$. We plotted the $h(q)$ curves for the energy, pressure and volume time
series of protein 1A3H in Figure 5 as examples. From the shape of the $h(q)$
curves of these time series for all proteins, we found that all time series
considered here are multifractal. Therefore these time series are complicated
and cannot be characterized by a single fractal dimension. In fact, they
depict heterogeneous phenomena and different regions of each time series have
different fractal properties [10]. In particular, the numerical values of
$h(2)$ (which is related to the Hurst exponent $H$) are given in Table 2. The
average values and standard deviations of $h(2)$ for all 11 parameters of 29
proteins are also listed in the bottom two rows of Table 2.

For energy parameters, the energy values are in $kcal/mol$, bonded potential
$\sim 10^{3}$, angle potential $\sim 10^{3}$, dihedral potential $\sim
10^{3}$, improper potential $\sim 10^{2}$, electrostatic potential $\sim
10^{5}$, Van der Waals potential $\sim 10^{4}$, kinetic energy $\sim 10^{4}$,
total energy $\sim 10^{5}$, and potential energy $\sim 10^{5}$. From Table 2,
we can see that the series of total energy and potential energy are non-
stationary ($h(2)>1.0$) and anti-persistent ($0<H=h(2)-1<0.5$). The other time
series are stationary and persistent apart from that of pressure (with
$H\approx 0.5$ indicating the absence of long-range correlation). Among bonded
potential, angle potential, dihedral potential, and improper potential, we can
see that the LRC in the series of angle potential is the strongest and that in
those of improper potential is the weakest, and the LRC in those of dihedral
potential is stronger than that in those of bonded potential. Among the three
largest energy parameters, namely the electrostatic potential, Vdw potential
and kinetic energy, we find that the LRC in the series of electrostatic
potential is the strongest and that in the series of Vdw potential is the
weakest.

Remark 1: The NMR structures of many proteins in PDB were obtained from
different models. For example, the structures of Protein 1WUZ in PDB were
obtained from 30 NMR models. In this case, we always use the structure
obtained from the 1st NMR model in the present study. We also tested the
effect of different NMR models on our results. We simulated the 1st model,
10th model, 20th model, and 30th model of protein 1WUZ. Our numerical results
of MF-DFA showed that the $h(q)$ curves for the four models are very close.
And it should be required in PDB that the structures of any protein obtained
from different NMR models should be similar to each other. Therefore, it is
unlikely that different NMR models of proteins would affect our results.

Then, these time series were converted into HVGs. We denote $k$ the degree of
nodes, $P(k)$ the probability of degree $k$. We obtained the plots for the
relations $k\sim P(k)$, $\ln(k)\sim\ln(P(k))$, and $k\sim\ln(P(k))$, and found
that only the linear relationship in the $k\sim\ln(P(k))$ plots is reasonable.
We gave the degree distributions $P(k)$ of the HVGs for the angle potential
and pressure time series of protein 1A3H in Figure 6 as examples. So the HVGs
of these parameters are exponential networks. We gave the numerical values of
the exponent $\lambda$ in Table 3. The average values and standard deviations
of $\lambda$ for all 11 parameters of the 29 proteins are also given in Table
3. Among the three largest energy parameters, namely the electrostatic
potential, Vdw potential and kinetic energy, the attenuation of the degree
distribution for HVGs of electrostatic potential is the fastest. The degree
distribution for HVGs of total energy has the fastest attenuation among all
energy parameters. The degree distribution for HVGs of potential energy has
the faster attenuation comparing to its components.

We also studied the fractal properties of the converted HVGs using the random
sequential box-covering algorithm described in Section 5\. We found that there
is a scaling-range of $r$ in which the power-law relation $N_{B}(r)\sim
r^{-d_{B}}$ holds for all converted HVGs, hence the fractality exists in these
networks. We showed how to estimate the fractal dimension $d_{B}$ of HVGs
constructed from the electrostatic potential and van der Waals potential time
series of protein 1A3H in Figure 7 as examples. The fractal dimension $d_{B}$
is the slope of linear regression between $-\ln(N_{B}(r/d))$ and $\ln(r/d)$
for each case. In Table 4, we gave the fractal dimensions $d_{B}$ for all time
series we considered here. The bottom two rows of Table 4 are the average
values and standard deviations of $d_{B}$ for all 11 parameters of 29
proteins, respectively.

From Tables 2, 3 and 4, for each energy or pressure or volume parameter, one
can find that the values of $h(2)$ of MF-DFA on the time series, exponent
$\lambda$ of exponential degree distribution and fractal dimension $d_{B}$ of
their converted HVGs do not change much for different proteins (indicating
some universality). So these methods in this paper cannot be used for the
prediction of protein structures and functions.

For 29 time series (29 proteins) of each kind of energy or pressure or volume,
we calculated the average values of $h(2)$ (from MF-DFA on time series) and
$d_{B}$ the converted HVGs. Hence we obtained 11 values of $\langle
h(2)\rangle$ and 11 values of $\langle d_{B}\rangle$ respectively. It is
surprising that there is a nice linear relationship between $\langle
h(2)\rangle$ and $\langle d_{B}\rangle$ as shown in Figure 8. The relationship
can be well fitted by the linear formula:

$\langle d_{B}\rangle=-1.5210\times\langle h(2)\rangle+3.1176.$

##  7 Conclusions

Using the NAMD software, we derived the time series of bonded potential, angle
potential, dihedral potential, improper potential, electrostatic potential,
Van der Waals potential, kinetic energy, total energy, potential energy,
pressure, and volume for each protein. The shape of the $h(q)$ curves from MF-
DFA indicates that these time series are multifractal.

In particular, the numerical values of $h(2)$ of MF-DFA (which is related to
the Hurst exponent $H$) show that the series of total energy and potential
energy are non-stationary and anti-persistent; the other time series are
stationary and persistent apart from the series of pressure (with $H\approx
0.5$ indicating the absence of long-range correlation).

The degree distributions of their converted HVGs show that these networks are
exponential. Among the three largest energy parameters, namely the
electrostatic potential, Vdw potential and kinetic energy, the attenuation of
the degree distribution of electrostatic potential is the fastest. The degree
distribution for HVGs of total energy has the fastest attenuation among all
energy parameters. The degree distribution for HVGs of potential energy has
the faster attenuation comparing to its components.

We found that there is a scaling-range of $r$ in which the power-law relation
$N_{B}(r)\sim r^{-d_{B}}$ holds for all converted HVGs, hence fractality
exists in these networks.

For each energy, pressure or volume parameter, it is found that the values of
$h(2)$ of MF-DFA on the time series, exponent $\lambda$ of exponential degree
distribution and fractal dimension $d_{B}$ of their converted HVGs do not
change much for different proteins (indicating some universality). So these
methods cannot be used for the prediction of protein structures and functions.

After taking average over all proteins, we surprisingly found that there is a
linear relationship between $\langle h(2)\rangle$ (from MF-DFA on time series)
and $\langle d_{B}\rangle$ of converted HVGs for different energy, pressure
and volume.

## ACKNOWLEDGEMENTS

This project was supported by the Natural Science Foundation of China (Grant
No. 11371016); the Chinese Program for Changjiang Scholars and Innovative
Research Team in University (PCSIRT) (Grant No. IRT1179); the Research
Foundation of Education Commission of Hunan Province of China (Grant No.
11A122); the Lotus Scholars Program of Hunan province of China; postdoctoral
research fund of Xiangtan University of China.

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Table 1: Structures representative of protein dynamics

PDB ID code The number of amino acids Helical $(\%)$ Beta sheet $(\%)$ Exp
structure 1A3H 300 38 19 X-ray 1BFG 126 8 35 X-ray 1BSN 138 26 37 NMR 2DN8 100
0 37 NMR 2DO7 101 43 9 NMR 1F39 101 3 39 X-ray 1FE6 108 90 0 X-ray 1FPR 294 30
19 X-ray 1H6T 291 17 28 X-ray 1I4S 294 68 1 X-ray 1IQQ 200 29 20 X-ray 1JLI
112 58 0 NMR 1K40 126 86 0 X-ray 1KS9 291 47 19 X-ray 1KTE 105 50 19 X-ray
1KUU 202 20 38 X-ray 1KXA 159 5 47 X-ray 1MIX 206 39 18 X-ray 1MJY 350 20 33
X-ray 1N12 298 4 55 X-ray 1OO9 294 25 17 NMR 1OOI 124 66 1 X-ray 1PHN 334 76 0
X-ray 1RAL 308 36 15 X-ray 1SUR 215 48 14 X-ray 1WUZ 103 12 34 NMR 1WWB 103 2
49 X-ray 1X0M 403 47 15 X-ray 1XGO 295 30 22 X-ray

Table 2: The exponent $h(2)$ of the time series for 29 proteins. (Ave($h(2)$)
and Std($h(2)$) denote the average and the standard deviations of $h(2)$ for
all 11 parameters of 29 proteins, respectively.)

PDB ID Bond-p ang-p dih-p imp-p ele-p Vdw-p Kin-e Tot-e Pot-e pre vol 1A3H
0.6266 0.7543 0.6880 0.5951 0.9052 0.6054 0.8340 1.2845 1.0454 0.4683 0.8988
1BFG 0.6269 0.7531 0.7201 0.5790 0.9401 0.6087 0.8431 1.2971 1.0685 0.4891
0.8646 1BSN 0.6227 0.7597 0.7317 0.5792 0.9283 0.5989 0.8350 1.3072 1.0590
0.4952 0.8721 2DN8 0.6260 0.7589 0.7170 0.5848 0.9229 0.6227 0.8186 1.2910
1.0518 0.4960 0.8768 2DO7 0.6254 0.7615 0.7028 0.5823 0.9344 0.6150 0.8332
1.3074 1.0630 0.5117 0.8934 1F39 0.6334 0.7481 0.7116 0.5945 0.9267 0.6087
0.8316 1.3219 1.0593 0.5175 0.8887 1FE6 0.6265 0.7512 0.7289 0.5821 0.9383
0.6009 0.8388 1.3158 1.0662 0.5172 0.8729 1FPR 0.6262 0.7562 0.7283 0.5616
0.9021 0.5948 0.8256 1.3071 1.0423 0.4762 0.9019 1H6T 0.6263 0.7551 0.7101
0.5928 0.9300 0.6282 0.8200 1.3063 1.0715 0.5062 0.8668 1I4S 0.6270 0.7237
0.7135 0.5753 0.9367 0.6002 0.8127 1.2979 1.0688 0.4874 0.8956 1IQQ 0.6265
0.7334 0.7353 0.5893 0.9225 0.5953 0.8383 1.3070 1.0701 0.4848 0.8762 1JLI
0.6276 0.7509 0.7070 0.5673 0.9296 0.5996 0.8438 1.2918 1.0450 0.4924 0.8657
1K40 0.6253 0.7614 0.7295 0.5790 0.9278 0.6141 0.8236 1.2869 1.0648 0.5143
0.8899 1KS9 0.6232 0.7621 0.7154 0.5844 0.9184 0.5912 0.8471 1.3158 1.0575
0.4762 0.8790 1KTE 0.6267 0.7597 0.7236 0.5877 0.9230 0.5937 0.8328 1.3090
1.0584 0.4725 0.8860 1KUU 0.6262 0.7599 0.7169 0.5761 0.9125 0.6029 0.8372
1.3036 1.0638 0.4824 0.8720 1KXA 0.6263 0.7684 0.7087 0.5835 0.9251 0.6196
0.8213 1.3132 1.0466 0.5200 0.8979 1MIX 0.6401 0.7643 0.7317 0.5921 0.9418
0.6290 0.8328 1.3234 1.0571 0.5342 0.8744 1MJY 0.6269 0.7665 0.6834 0.5711
0.9290 0.5903 0.8309 1.3001 1.0481 0.5120 0.8732 1N12 0.6202 0.7530 0.7149
0.5872 0.9208 0.5965 0.8118 1.2997 1.0652 0.4719 0.9032 1OO9 0.6250 0.7639
0.7081 0.5768 0.9201 0.5907 0.8310 1.3138 1.0761 0.4779 0.8849 1OOI 0.6277
0.7565 0.7014 0.5894 0.9148 0.6012 0.8324 1.2887 1.0643 0.4757 0.8916 1PHN
0.6317 0.7603 0.7061 0.5759 0.9308 0.6022 0.8339 1.3158 1.0470 0.4898 0.8814
1RAL 0.6277 0.7580 0.7222 0.5731 0.9366 0.5940 0.8344 1.2934 1.0650 0.4822
0.8810 1SUR 0.6219 0.7569 0.7242 0.5680 0.9318 0.6038 0.8322 1.3156 1.0577
0.4903 0.8772 1WUZ 0.6340 0.7606 0.7163 0.5770 0.9274 0.6067 0.8294 1.3236
1.0575 0.5172 0.8824 1WWB 0.6555 0.7572 0.7164 0.5916 0.9274 0.6213 0.8339
1.3104 1.0717 0.5005 0.8710 1X0M 0.6205 0.7513 0.7102 0.5885 0.9172 0.6072
0.8195 1.2950 1.0460 0.4925 0.8712 1XGO 0.6103 0.7550 0.7043 0.5779 0.9335
0.6051 0.8274 1.2969 1.0691 0.4836 0.8958 Ave($h(2)$) 0.6272 0.7559 0.7147
0.5815 0.9260 0.6051 0.8306 1.3049 1.0595 0.4943 0.8823 Std($h(2)$) 0.0074
0.0090 0.0123 0.0086 0.0096 0.0110 0.0086 0.0111 0.0095 0.0174 0.0114

Table 3: The exponent $\lambda$ of the horizontal visibility graphs of the
time series for 29 proteins. (Ave($\lambda$) and Std($\lambda$) denote the
average and the standard deviations of $\lambda$ for all 11 parameters of 29
proteins, respectively.)

PDB ID Bond-p ang-p dih-p imp-p ele-p Vdw-p Kin-e Tot-e Pot-e pre vol 1A3H
0.4139 0.4398 0.4236 0.4129 0.4422 0.4163 0.4282 0.6238 0.4719 0.4068 0.5088
1BFG 0.4115 0.4386 0.4274 0.4092 0.4469 0.4172 0.4293 0.6341 0.4737 0.4095
0.5032 1BSN 0.4176 0.4333 0.4276 0.4149 0.4413 0.4166 0.4270 0.6273 0.4748
0.4098 0.5086 2DN8 0.4120 0.4331 0.4264 0.4198 0.4442 0.4120 0.4309 0.6177
0.4749 0.4108 0.5038 2DO7 0.4132 0.4365 0.4301 0.4145 0.4432 0.4162 0.4240
0.6131 0.4718 0.4070 0.5068 1F39 0.4109 0.4390 0.4283 0.4113 0.4453 0.4178
0.4290 0.6243 0.4776 0.4101 0.5095 1FE6 0.4068 0.4325 0.4167 0.4177 0.4442
0.4111 0.4256 0.6239 0.4776 0.4088 0.5033 1FPR 0.4188 0.4313 0.4298 0.4158
0.4387 0.4190 0.4306 0.6229 0.4752 0.3968 0.5015 1H6T 0.4188 0.4333 0.4208
0.4121 0.4428 0.4123 0.4230 0.6239 0.4749 0.4106 0.5066 1I4S 0.4102 0.4317
0.4238 0.4145 0.4403 0.4225 0.4247 0.6299 0.4745 0.4066 0.4853 1IQQ 0.4134
0.4315 0.4245 0.4176 0.4473 0.4193 0.4238 0.6247 0.4780 0.4060 0.5046 1JLI
0.4106 0.4152 0.4224 0.4157 0.4425 0.4191 0.4286 0.6330 0.4771 0.4074 0.4772
1K40 0.4161 0.4329 0.4286 0.4158 0.4485 0.4142 0.4240 0.6213 0.4705 0.4036
0.5059 1KS9 0.4128 0.4390 0.4208 0.4188 0.4427 0.4173 0.4231 0.6296 0.4717
0.4048 0.5048 1KTE 0.4188 0.4389 0.4268 0.4141 0.4486 0.4191 0.4243 0.6292
0.4763 0.4060 0.5073 1KUU 0.4132 0.4321 0.4292 0.4143 0.4428 0.4183 0.4266
0.6264 0.4792 0.4030 0.5035 1KXA 0.4111 0.4254 0.4285 0.4197 0.4352 0.4126
0.4240 0.6397 0.4733 0.4032 0.5033 1MIX 0.4163 0.4332 0.4266 0.4199 0.4490
0.4193 0.4265 0.6381 0.4794 0.4086 0.5041 1MJY 0.4176 0.4326 0.4254 0.4142
0.4430 0.4197 0.4231 0.6128 0.4793 0.4079 0.5028 1N12 0.4174 0.4236 0.4206
0.4180 0.4477 0.4197 0.4260 0.6276 0.4729 0.4022 0.5081 1OO9 0.4182 0.4371
0.4210 0.4094 0.4422 0.4170 0.4220 0.6297 0.4704 0.4058 0.5067 1OOI 0.4144
0.4315 0.4212 0.4161 0.4408 0.4162 0.4266 0.6212 0.4753 0.4085 0.5027 1PHN
0.4133 0.4255 0.4229 0.4132 0.4422 0.4186 0.4250 0.6328 0.4719 0.3952 0.5037
1RAL 0.4173 0.4303 0.4255 0.4166 0.4437 0.4103 0.4221 0.6308 0.4744 0.4003
0.4936 1SUR 0.4145 0.4350 0.4278 0.4164 0.4445 0.4189 0.4259 0.6439 0.4787
0.4076 0.4905 1WUZ 0.4152 0.4300 0.4265 0.4189 0.4423 0.4158 0.4217 0.6046
0.4747 0.4076 0.5063 1WWB 0.4103 0.4361 0.4231 0.4151 0.4460 0.4174 0.4237
0.6269 0.4799 0.4089 0.5065 1X0M 0.4155 0.4339 0.4246 0.4102 0.4426 0.4181
0.4268 0.6210 0.4778 0.4089 0.4953 1XGO 0.4169 0.4309 0.4215 0.4144 0.4447
0.4141 0.4227 0.6282 0.4712 0.4094 0.5020 Ave($\lambda$) 0.4144 0.4325 0.4249
0.4152 0.4436 0.4168 0.4255 0.6263 0.4751 0.4063 0.5023 Std($\lambda$) 0.0031
0.0052 0.0033 0.0029 0.0030 0.0029 0.0026 0.0081 0.0029 0.0038 0.0073

Table 4: The fractal dimension $d_{B}$ of the horizontal visibility graph of
the time series for 29 proteins. (Ave($d_{B}$) and Std($d_{B}$) denote the
average and the standard deviations of $d_{B}$ for all 11 parameters of 29
proteins, respectively.)

PDB ID Bond-p ang-p dih-p imp-p ele-p Vdw-p Kin-e Tot-e Pot-e pre vol 1A3H
2.2420 1.9094 2.0924 2.1066 1.6176 2.2444 1.8359 1.1711 1.4636 2.3378 1.7238
1BFG 2.1676 1.9819 2.0500 2.2011 1.6010 2.2705 1.8846 1.1743 1.4867 2.2933
1.6836 1BSN 2.2406 1.9308 2.0763 2.1631 1.6173 2.2659 1.8608 1.1663 1.4840
2.3774 1.6850 2DN8 2.1980 1.9635 2.1620 2.1677 1.6223 2.2684 1.8451 1.1764
1.4928 2.3181 1.8107 2DO7 2.1975 1.9583 2.0344 2.1446 1.6304 2.3466 1.8794
1.1816 1.5028 2.3746 1.7386 1F39 2.2468 1.9891 2.1464 2.1154 1.5909 2.2725
1.8349 1.1857 1.4706 2.2654 1.6819 1FE6 2.2630 2.0082 2.0136 2.1567 1.6369
2.3287 1.8510 1.1717 1.4698 2.2609 1.7250 1FPR 2.2390 1.9955 2.1629 2.2214
1.6071 2.3019 1.8316 1.1665 1.4861 2.3601 1.7634 1H6T 2.3016 1.9043 2.0813
2.1474 1.6370 2.2001 1.8580 1.1864 1.4765 2.3218 1.7400 1I4S 2.2240 2.0261
2.0840 2.0849 1.6208 2.2055 1.8901 1.1820 1.5158 2.3245 1.7718 1IQQ 2.2468
1.9935 2.0428 2.2282 1.6163 2.2704 1.8316 1.1674 1.4841 2.4346 1.7014 1JLI
2.2892 2.0062 2.0662 2.1833 1.6201 2.3098 1.8878 1.1816 1.4788 2.2764 1.7530
1K40 2.1872 1.9844 2.0558 2.1804 1.6435 2.2935 1.8767 1.1773 1.4626 2.4219
1.7508 1KS9 2.2384 1.9884 2.0947 2.0910 1.6336 2.3056 1.8368 1.1753 1.4988
2.3386 1.7253 1KTE 2.1812 2.0072 2.0802 2.0754 1.6411 2.2450 1.8494 1.1789
1.4744 2.3078 1.7542 1KUU 2.1128 2.0287 2.1360 2.1860 1.6510 2.2234 1.8746
1.1909 1.4877 2.3736 1.7157 1KXA 2.1854 1.9436 2.0970 2.1582 1.6303 2.3596
1.8813 1.1700 1.5108 2.2604 1.7123 1MIX 2.2149 2.0408 2.1175 2.1299 1.6383
2.3023 1.8498 1.1522 1.4852 2.3154 1.7070 1MJY 2.1973 1.9703 2.0800 2.1877
1.6672 2.2188 1.9019 1.1565 1.4943 2.3145 1.7069 1N12 2.2549 2.0646 2.0082
2.0736 1.6065 2.2561 1.8471 1.1778 1.4863 2.3750 1.7865 1OO9 2.2266 2.0059
2.0746 2.0948 1.6369 2.3035 1.8906 1.1750 1.4822 2.3019 1.7576 1OOI 2.2811
1.9514 2.1226 2.1475 1.6275 2.2415 1.8729 1.1746 1.4662 2.2810 1.7851 1PHN
2.2882 1.9190 2.0631 2.1411 1.6152 2.3023 1.8914 1.1673 1.4838 2.4080 1.7280
1RAL 2.1851 2.0142 2.0490 2.0774 1.6226 2.2371 1.8833 1.1743 1.4981 2.3651
1.7658 1SUR 2.2318 2.0036 2.0301 2.0664 1.6251 2.2950 1.9093 1.1901 1.4917
2.2883 1.7264 1WUZ 2.2801 1.9021 2.0407 2.1896 1.6406 2.3845 1.8850 1.1693
1.5043 2.4449 1.6824 1WWB 2.2323 1.9986 2.0172 2.1306 1.6187 2.3751 1.8956
1.1743 1.4875 2.2518 1.7267 1X0M 2.2126 2.0268 2.0530 2.1421 1.6553 2.2801
1.8780 1.1676 1.4813 2.3462 1.7104 1XGO 2.1929 2.0190 2.0762 2.1694 1.6323
2.3051 1.8543 1.1876 1.4716 2.3140 1.7302 Ave($d_{B}$) 2.2262 1.9840 2.0761
2.1435 1.6277 2.2832 1.8695 1.1748 1.4855 2.3328 1.7327 Std($d_{B}$) 0.0424
0.0422 0.0419 0.0456 0.0164 0.0474 0.0233 0.0090 0.0134 0.0533 0.0331

Figure 1: Bonded potential, angle potential, dihedral potential and improper
potential time series for protein 1A3H.

Figure 2: Electrostatic potential, Van der Waals potential, kinetic energy and
total energy time series for protein 1A3H.

Figure 3: Potential energy, pressure, and volume time series for protein 1A3H.

Figure 4: Linear regressions for calculating the exponents $h(q)$ for the
bonded potential time series of protein 1A3H.

Figure 5: Multifractal $h(q)$ curves for the energy, pressure and volume time
series of protein 1A3H.

Figure 6: The degree distributions $P(k)$ of the horizontal visibility graph
for the angle potential and pressure time series of protein 1A3H.

Figure 7: Linear regressions for calculating the fractal dimensions $d_{B}$ of
the horizontal visibility graphs for the electrostatic potential and van der
Waals potential time series of protein 1A3H.

Figure 8: The relationship between $\langle h(2)\rangle$ (from MF-DFA on time
series) and $\langle d_{B}\rangle$ of the constructed horizontal visibility
graphs for different energy, pressure and volume. The average is taken over 29
proteins.