It is crucial for hydrophobic drugs to be dissolved and stabilized by carriers in aqueous systems and then to be delivered into target cells. An amphiphilic self-assembling peptide EAK16-I (Ac-AEAKAEAKAEAKAEAK-NH2) is reported here to be able to stabilize a model hydrophobic compound, pyrene, in aqueous solution, resulting in the formation of colloidal suspensions. Egg phosphatidylcholine (EPC) vesicles are used as plasma membranes mimic. Fluorescence data shows that the pyrene is presented in the crystalline form when stabilized by EAK16-I and molecularly migrates from its peptide encapsulations into the membrane bilayers of EPC vesicles when the suspension is mixed with EPC vesicles. Furthermore, the release rate can be controlled by changing peptide-to-pyrene ratio, and the higher ratios lead to the slower release rates due to a thicker encapsulation on the pyrene microcrystals. This demonstrates that EAK16-I, as a promising nanobiomaterial, has the potential to be a hydrophobic compounds carrier.
1. Introduction
Low water solubility limits the
clinical practice of many potent hydrophobic drugs [1]. Therefore, it is
crucial in pharmaceutical industry for hydrophobic compounds to be dissolved or
stabilized by carriers in aqueous systems and to deliver them into target
cells. With good desirable biocompatibility, self-assembling peptides are
emerging as promising nanobiomaterials for hydrophobic drug delivery.
The
serendipitous discovery of self-assembling peptides has changed our view of
peptides as emerging nanobiomaterials [2]. By (re-)designing the amino acid
sequences, self-assembling peptides (8–16 residues, 2.5–5 nm in length)
[3], with good biocompatibility, have already been used in versatile biomedical
applications, such as cell cultures, tissue engineering, and surface
engineering [4–6], while only a
few studies had been reported to investigate the applicability of
self-assembling peptides as drug carriers [7–9]. To explore
this field further, it seems that more studies should be carried out to
investigate the potentials of self-assembling peptides as drug carriers. EAK16-I
is a peptide made of 16 amino acids with alternating positive and negative
charges separated by a hydrophobic amino acid residue (Ala) [2, 10, 11]. At neutral pH, Glu and Lys are negatively and positively charged, respectively. They
are believed to form complementary ion pairs when EAK16-I self-assembles into
its β-sheet microstructure which exhibits a hydrophobic and a hydrophilic surfaces.
It is expected that the hydrophobic region of EAK16-I can interact with hydrophobic compounds while
the charged residues stabilize the complex in aqueous solution.
The purpose of this study is to examine whether EAK16-I can
stabilize microcrystals of pyrene, a hydrophobic compound, in aqueous solution
and deliver them into the membrane of a cell mimic made of the egg
phosphatidylcholine (EPC) vesicles. The rate of pyrene migrates from the
colloidal crystals stabilized by EAK16-I into membrane bilayer was also studied.
We expect to reveal the potential of EAK16-I as carrier of hydrophobic
compounds and to give some clues to exploit this peptide as hydrophobic drug
carrier.
2. Materials and Methods2.1. Materials
The peptide, EAK16-I (1657 g/mol, N-terminal acetylation and
C-terminal amidation), was synthesized by Shanghai Sangon Biological
Engineering Technology & Services Co., Ltd.,
Shanghai, China. Egg Phosphatidylcholine
(EPC) was purchased from Sinopharm Chemical Reagents Co., Ltd., Shenyang, China. Pyrene
(99%) was obtained from Sigma-Aldrich, Shanghai, China, and
was recrystallized twice from ethanol before experiment. The rest chemicals
used in the experiment were acquired from Chengdu Kelong Chemical Reagents Co.,
Chengdu, China.
Deionized water (Elix Water Purification System, Millipore, Mass, USA) was used to prepare all
aqueous solutions.
2.2. Preparation of Colloidal Suspensions of Pyrene Crystals
Weighed amounts of pyrene were
added into freshly prepared EAK16-I solutions in a 10 mL vial. Two
peptide-pyrene solutions (referred to as EAKI-PY) with different
peptide-to-pyrene ratio were prepared to obtain the concentrations of 0.5 mg/mL
(3.02 × 10−4 M) for the
peptide and 1.00 mg/mL (4.94 × 10−3 M) and 0.267 mg/mL (1.32 × 10−3 M) for pyrene to obtain E05P10 and
E05P02 solutions, respectively. Then, these two samples were kept on stirring
until equilibrium was reached in about 120 hours. These solutions were deemed
at equilibrium when their fluorescence spectrums did not change in 24 hours.
2.3. EPC Vesicle Preparation
EPC (0.8 g) dissolved in chloroform was
added to a 1 L round-bottom flask. The organic solvent was evaporated by rotary evaporator at
room temperature to produce an EPC thin film inside the round-bottom flask [12]. Then, the EPC film was exposed to high vacuum for at least 2 hours to
remove residual traces of chloroform at room temperature and was resuspended in 300 mL of buffer (pH
7.4, containing 25 mM Tris-HCl acid and 0.2 mM EDTA) solution. After being bubbled nitrogen, the mixture was sonicated for 30
minutes at 0°C in a
bath type sonicator at 200 W output. This was followed by centrifugation at
12000 × g for 1.5 hours. The supernatant was filtered by 0.44μm and 0.22μm
membrane filters in turn. Liposome size was measured by Malvern Zetasizer Nano
ZS analyzer. The lipid concentration of EPC vesicles was determined by
comparing the difference between mass of the solute of buffer and that of
vesicle solution.
2.4. Steady-State Fluorescence Measurements
Fluorescence spectra were
recorded on Hitachi F-4500 spectrofluorophotometer at room temperature. Solution
samples were operated in a quartz fluorescence cuvette of 1 cm2 cross-section, while solid samples
were carried out by using a solid accessory. The following parameters were used
in experiments except special indication. Excitation and emission slits were
set to 10 nm and 2.5 nm, respectively. By setting excitation wavelength at 336
nm, the emission fluorescence spectra were scanned from 350 nm to 650 nm, with
scan speed of 240 nm/min. Excitation spectra were recorded at the selected
emission wavelengths (373 nm and 470 nm).
Solutions of different pyrene concentration (ranging from 2 × 10−6 M to 1.2 × 10−4 M) in liposome were prepared
for the calibration curve, and the solutions were referred to as EPC-PY
solutions. The Im (fluorescence intensity of the pyrene monomer) for
the EPC-PY samples was obtained by averaging the mission intensities of the
pyrene monomer for the EPC-PY solutions monitored at 373 nm for 120 seconds. To
avoid the influence of xenon lamp fluctuations, the intensity of pyrene monomer
standard (Is) is needed. A degassed and sealed solution of pyrene in
ethanol ([PY] = 3.96 × 10−5 M) was monitored at 373 nm over 120 seconds, and the measured
intensity was then averaged to yield Is after each spectrum of the
EPC-PY solutions. The corrected value of the monomer intensity (Im/Is)
for the EPC-PY solutions was gotten by dividing Im by Is.
For the experiments of the migration of pyrene from the
EAKI-PY solutions into the liposome, the appropriate volume solutions of the
EAKI-PY and EPC vesicles were mixed in quartz fluorescence cuvette. The
mixtures (referred as EPC-EAKI-PY) were constantly stirred with a small
magnetic stirrer during fluorescence measurement. All samples were prepared
less than 20 seconds before the fluorescence measurements initiated. To get the
kinetic data for pyrene transfer, a
time-dependent fluorescence measurement was required by recording
Im at 373 nm for 4 hours at 0.2-second intervals. Each Im-versus-time
profile was corrected by dividing Im by Is to avoid lamp
fluctuation. Also liposome sizes of solutions of EPC-PY and EPC-EAKI-PY were
measured by Malvern Zetasizer Nano ZS analyzer.
2.5. Scanning Electronic Microscopy (SEM)
SEM (JSM-5900LV, JEOL Ltd., Tokyo, Japan) was used in the study of the
microstructure and dimension of the peptide-pyrene complexes. An aliquot of 20μL EAKI-PY solution was placed on a freshly cleaved mica surface. After 10 minutes,
the mica surface was rinsed twice with pure water (each time 200μL) and air-dried
overnight. After gilded, the
complexes were imaged using the secondary electron (SE2) mode at 20 kV.
2.6. X-Ray Photoelectron Spectroscopy (XPS)
An aliquot of EAKI-PY solution was placed to cover a freshly
cleaved mica surface (15 × 10 mm). After air-dried overnight, the XPS spectra
of the samples were measured with a XSAM800 (Kratos Ltd., Manchester, UK) by using Al Kα (1486.6 eV) radiation at 12 kV acceleration voltage and an emission current of 15 mA, and
the analyzer mode was fixed analyzer transmission (FAT).
3. Results and Discussion
Pyrene has very low solubility in water
(about 6.0 × 10−7 M in the saturated aqueous solution [13]). In the presence of the peptide
EAK16-I, a milk-white colloidal suspension was obtained after stirring the
EAKI-PY aqueous mixture for 3 hours, while the control sample without peptide
still remained transparent with pyrene crystals floating on the top or
precipitating at the bottom (see Figure 1). The formation of colloidal
suspension can be taken as an initial proof that large amounts of pyrene (4.94 × 10−3 M) can be
stabilized in water.
Pyrene ([PY] = 4.94 × 10−3 M) in pure water (1) and
EAK16-I ([EAK16-I] = 3.02 × 10−4 M = 0.5 mg/mL) aqueous
solution (2) after stirring for 3 hours.
In a delivery system, hydrophobic cargos need not only to
be stabilized by vehicle in water, but also to be delivered into targets. A
hydrophobic compound must be delivered into plasma membrane of the living cell.
The essential components of plasma membranes is EPC, hence we utilize EPC
vesicles as cell mimic. The lipid concentration of the EPC vesicles used in
this study was 7.32 × 10−4 M. The average diameters of EPC vesicles in EPC-PY ([PY] = 6.52 × 10−5 M) and EPC-EAKI-PY ([PY] = 5.92 × 10−5 M, [EAK16-I] = 3.52 × 10−6 M) solutions
were 129 nm and 131 nm, respectively, which were found to be virtually
identical to those of the EPC vesicles (128 nm).
Figure 2 displays the steady-state
fluorescence spectra of the pyrene solutions at equilibrium. The steady-state
fluorescence spectra show that the pyrene crystals and the E05P10 solution both
exhibit a large amount of pyrene excimer without visible monomer emission (see Figures
2(a) and 2(b)), while pyrene in liposome and the
E05P10 solution mixed with the liposome are similar, with large amounts of
pyrene monomer and excimer (see Figures 2(c) and 2(d)). I1/I3, the ratio of the
intensities of the first peak (I1, 374 nm)
and the third peak (I3, 385 nm) of the pyrene monomer, was widely
used as a polarity scale to determine the polarity of the pyrene microenvironment
[14]. Pyrene in polar solvent like water has an I1/I3 ratio about 2.02 and has lower values in apolar environment. The ratios of I1/I3 for pyrene in liposome
and the E05P10 solution mixed with the liposome are both 1.05. The value variety
indicates that pyrene has transferred from the peptide encapsulation into the
hydrophobic lipid bilayer. Similar data were acquired with E05P02 solution
(data not shown).
Steady-state fluorescence emission spectra of (a) solid pyrene
crystals, (b) E05P10 solution ([PY] = 4.94 × 10−3 M,
[EAK16-I] = 3.02 × 10−4 M), (c) pyrene in EPC vesicles
([PY] = 6.53 × 10−5 M), and (d) E05P10 solution mixed with
EPC vesicles ([PY] = 6.27 × 10−5 M, [EAK] = 3.42 × 10−6 M.), λex=336 nm.
The pyrene excimer has two forms: dynamic excimer formed via
diffusional encounters between the excited pyrene and a ground-state pyrenean static excimer
formed from the direct excitation of ground-state pyrene dimmers in crystalline
form [15]. To determine whether the pyrene excimer is dynamic excimer or static
excimer, additional experiments, measurement of excitation spectra, must be
performed. The static excimer has a red-shift excitation spectrum, and the
dynamic excimer has the same excitation spectrum as that of monomer emission [16].
Compared with excitation spectrum monitored at monomer emission,
the excitation spectra of pyrene crystals and E05P10 solution monitored at
excimer emission red shifts about 45 nm from about 335 nm to 380 nm (see
Figures 3(a) and 3(b)). This indicates that the excimers
in pyrene crystals and E05P10 solution are static excimers resulting from
pyrene molecules preassociated in solid pyrene crystals and that pyrene
molecules in pyrene crystals and E05P10 solution were both in crystalline form.
Therefore, a further proof that pyrene was stabilized in water via EAK16-I
peptide adsorbing onto the surface of pyrene microcrystals was acquired.
However, for pyrene in EPC vesicles and EAKI-PY
solution mixed with EPC vesicles, no
spectra shifts were observed; this suggested that the excimers were formed
via diffusional encounter only (see Figures 3(c) and
3(d)) [17, 18]. It also suggested that pyrene was molecularly dissolved inside the EPC vesicle
membrane while very few pyrene crystals were expected to remain after the
EAKI-PY solution mixing with the EPC vesicles completely. Similar observations
were acquired with E05P02 solution (data not shown).
Normalized steady-state fluorescence excitation spectra of (a)
solid pyrene crystals, (b) E05P10 solution ([PY] = 4.94 × 10−3 M, [EAK16-I] = 3.02 × 10−4 M), (c) pyrene
in EPC vesicles ([PY] = 6.53 × 10−5 M), and (d) E05P10
solution mixed with EPC vesicles ([PY] = 6.27 × 10−5 M,
[EAK16-I] = 3.42 × 10−6 M.), Emission wavelength
(λem) was 373 nm (—) and 468 nm (· · ·).
To monitor the migration of pyrene from
EAK16-I encapsulation into the EPC vesicles, steady-state
fluorescence spectra of PY in EPC-EAKI-PY
solution were acquired at various time intervals over a total period of 4.5 hours
after mixing EAKI-PY with liposome. All the I1/I3 values of these spectra were 1.05; this demonstrated
that the pyrene monomers were always located inside the hydrophobic lipid
bilayer [14]. As shown in Figure 2, the fluorescence signal of the pyrene
excimer can arise from those formed inside the pyrene crystals and the EPC
vesicle membrane, whereas that of the pyrene monomer had only one source.
Consequently, to monitor the transfer of pyrene during experiment more
precisely, the monomer intensity was recorded to get Im-versus-time
profile over a period of 4 hours with 0.2 seconds intervals at an emission
wavelength of 373 nm. Two profiles were shown in Figure 4. Profile 1 represents
the release of pyrene from E05P10 solution into EPC vesicles with the
final
pyrene concentrations of 1.3 × 10−5 M. Profile 1 obtained at low pyrene concentration exhibits a
continuous increase of IM with time during the first half hour and
reaches a plateau after close to 1 hour. Profile 2
represents the release of pyrene from E05P10 solution into EPC vesicles with
the final pyrene concentrations of 1.0 × 10−4 M. Profile 2 obtained at higher pyrene concentration gives a
sharp increase up to a maximum of Im with time after 0.1 hour, followed by a continuous decrease down to a plateau
reached after about 3.5 hours. The fluorescence behavior in Figure 2 indicates
that the transition undergone by pyrene from microcrystals of the E05P10
solution to individual molecules dissolved inside the lipid bilayer. Inside a
crystal, pyrene molecules are very close to each other, and the probability of
pyrene molecules to absorb a photon is small, while an equal number of pyrene
molecules are dissolved inside the vesicle as a molecular state, and the probability
of pyrene to absorb a photon becomes much bigger. As the concentration of pyrene inside the vesicle membrane
increased, the pyrene absorption increases and so does the fluorescence of the
pyrene monomer, which in turn results in the formation of more excimer inside
the membrane [19]. As shown in profile 1 of Figure 4, Im reaches
a plateau when all the pyrene is molecularly dissolved inside the vesicles at a
lower final pyrene concentration. If the amount of pyrene (aliquots of the
EAKI-PY solution) added to the liposome solution is too large, the dissolution
of pyrene molecules in the lipid vesicles leads to a rapidly increasing of
pyrene absorption. After a critical point of the pyrene concentration in
liposome, the pyrene absorption becomes too large and the inner filter effect
takes place [20], which leads to the decreasing of the pyrene monomer
fluorescence until the crystals are fully dissolved and a plateau is reached.
This is why the final pyrene concentration of profile 2 in Figure 4 is 8 times larger than that of profile
1.
Fluorescence emission of the pyrene monomer at emission
wavelength 373 nm when different amounts of E05P10 solution are added
to EPC vesicle solutions. The fluorescence intensity (Im)
is divided by that of the standard (Is). Data shown in
profile 1 ([PY] = 1.3 × 10−5 M, [EAK16-I] = 8.05 × 10−7 M) and profile 2 ([PY] = 1.1 × 10−4 M,
[EAK16-I] = 5.13 × 10−6 M) were both acquired over a
4-hour time span with 0.2-second intervals.
Quantitative analysis of Im/Is-versus-time profiles such as those shown in Figure 4 was required to obtain
detail information of the pyrene transfer velocity from peptide encapsulation
to vesicle membranes. Therefore, the calibration curve of Im-versus-concentration
of pyrene in EPC vesicles was needed (see Figure 5). A set of EPC-PY
solutions was prepared with the pyrene
concentrations ranging from 2 × 10−6 M to 1.2 × 10−4 M. To fit the pyrene transfer
experiments in different days, Im of these solutions was divided by Is to avoid xenon lamp fluctuations. The calibration curves also show the inner
filter effect obviously. Im/Is passes through a maximum,
and its value varies little for pyrene concentration between 2.2 × 10−5 M and 5.3 × 10−5 M. Consequently, there is a
gap observed in time-dependent concentration profile for pyrene located inside
the vesicles, where no data points are reported for pyrene concentrations
ranging from 2.2 × 10−5 to 5.3 × 10−5 M. Therefore, two functions of Im/Is fitted for different pyrene concentration range were get, and both of them were
fit with sigmoidal. Up to [PY] = 3.5 × 10−5 M, the monomer
intensity was fit with one sigmoidal, Im/Is=3.077−23.834/(1+e([PY]−6.460)/8.678)(R2=0.9982). Another sigmoidal, Im/Is=1.762+1.722/(1+e([PY]−63.341)/27.522)(R2=0.9867) was used to
fit with the monomer intensity for higher pyrene concentrations.
Calibration curve for different pyrene concentrations in EPC
vesicles ([EPC] = 7.32 × 10−4 M). The concentration of
pyrene ranged from 2μM to 120μM. The fluorescence intensity
(Im) is divided by that of the standard (Is).
Up to [PY] = 35μM, the monomer intensity was fit with a sigmoidal
(R2 = 0.9982), Im/Is=3.077−23.834/(1+e([PY]−6.460)/8.678).
Another sigmoidal (R2 = 0.9867 Im/Is=1.762+1.722/(1+e([PY]−63.341)/27.522),
was used to fit with the monomer intensity for higher Pyrene
concentrations.
By transforming Im/Is-versus-time
profiles with calibration curve of Im/Is-versus-concentration
of pyrene in EPC, we got a release rate
of pyrene from its peptide carrying systems into EPC vesicles (see Figure 6). The
molecular ratios of peptide-to-pyrene used in E05P02 and E05P10 solutions are
0.229 and 0.061, respectively. For similar final pyrene concentrations in the
EPC vesicles, the release of pyrene from the EAK16-I coated pyrene crystals in
E05P02 solution proceeds in a slower manner than that in E05P10 solution. To
quantify the release rate, the pyrene release constant was acquired. The trend
shown in Figure 6 can be fitted with [PYV](t)=[PYV]eq−([PYV]eq−[PYV]0)exp(−k×t). In (1), [PYV](t),[PYV]eq, and [PYV]0 represent the pyrene concentration inside the vesicles at time t, at
equilibrium (infinite time), and at time t=0 seconds, respectively. According to (1), the release rate
constants (k) of different peptide-pyrene-EPC
systems were shown in Table 1. This
suggests that the release rate can be controlled by changing peptide-cargo
ratio.
Transfer rates of molecular pyrene from EAK16-I coated pyrene microcrystals into a solution
of EPC vesicles.
Molecular ratios of EAK16-I-to-pyrene
(a)
[PYV](b)(μmol⋅L−1)
Rate constants of transfer (k) (h−1)
0.061
13.3
7.92±0.01
0.229
15.6
2.30±0.01
0.061
63.7
6.52±0.06
0.229
61.4
1.85±0.01
0.229
62.6
(d)
1.63±0.01
0.061
104.2
(c)
fast: 7.10±0.09(78.11%);
slow: 0.60±0.02(21.89%)
(a)
molecular ratios of EAK16-I-to-pyrene for E05P10 (0.061) and
E05P02 (0.229) solution.
(b)
pyrene
concentrations in the solutions of EPC vesicles at the end points of each
transfer experiment.
(c)
two exponentials were needed to fit the release profile according to [PY](t)=[PY](t=∞)×[1−pfast×exp(−kfastt)−pslow×exp(−kslowt)] where pfast and pslow represent the proportion of fast release and slow release in overall release, respectively, and pfast+pslow=100%.
(d)
from settled E05P02 solution.
Profiles for the release of molecular pyrene from EAK16-I coated
pyrene microcrystals into a solution of EPC vesicles according to
the fluorescence results. Hollow and solid symbols are for pyrene
transfer experiments carried out with E05P10 and E05P02,
respectively. The final pyrene concentrations are as follows: (°)[PY]=1.04×10−4M; (⋄)[PY]=6.37×10−5M; (□)[PY]=1.33×10−5M; (♦)[PY]=6.14×10−5M; (▼)[PY]=6.7×10−5M (from settled E05P02 solution); (▲)[PY]=1.56×10−5M.
The difference of release rates of
pyrene from pyrene crystals coated with EAK16-I can be related to a thicker
encapsulation of the pyrene crystals when a higher molecular ratio of
peptide-to-pyrene was used. As shown in Figure 7, the
sizes of the peptide-pyrene complexes in two EAKI-PY
colloidal suspensions are both several micrometers, but the surfaces of the pyrene crystals in the E05P10 (see
Figure 7(a)) and E05P02 (see Figure 7(b)) solutions exhibit different
appearances depending on the peptide-to-pyrene ratio. Pyrene crystals imaged
from the E05P10 solution display a thick encapsulation, while most of the
pyrene crystals imaged from the E05P02 solution appear to be wrapped in an even
thicker encapsulation. XPS spectra can offer elements ratios of surface layers.
The C elements of the peptide-pyrene complexes surface layers contributed by peptide and pyrene, while
the N elements of those contributed by peptide only. Thicker encapsulation meant
more peptide in the surface layers, which led to lower C-to-N ratio. The
molecular ratios of C-to-N of the E05P02 and E05P10 complexes
surface layers were 5.965 and 6.494, respectively. This indicated
that the encapsulation of the complexes from E05P02 was thicker that from E05P10. The micrographs shown in Figure 7 and the molecular
ratios of C-to-N suggested that higher peptide-to-pyrene molecular ratios lead
to the formation of a thicker peptide encapsulation which inhibits the release
of pyrene.
Scanning electron micrographs of the pyrene crystals of (a) the
E05P10 solution and (b) the E05P02 solution.
The EAKI-PY colloidal suspensions were
found to settle over time without continuous stirring. To investigate whether
the settling of the colloids has an effect on the pyrene release rate, a
release experiment was conducted with the resuspended E05P02 solution which was
settled over a period of 5 d after the equilibrium was reached and shaken vigorously before release experiment. The
results are shown in Figure 6 and Table 1. Within experimental error, no
difference was observed in the release profiles, whether the release experiment
was performed with the settled or continuously stirred E05P02 solution.
4. Conclusion
This study has demonstrated that the
mixing of a model hydrophobic compound with self-assembling EAK16-I peptide in
aqueous solution involving mechanical stirring resulted in the formation of
colloidal suspensions. EAK16-I acted as a colloidal stabilizer which interacted
with the surface of crystalline form rather than molecular state pyrene. When
the EAKI-PY solution was mixed with EPC vesicles, pyrene was transferred from
the pyrene crystals into the vesicle membrane until all pyrene crystals had
disappeared. Steady-state fluorescence showed that pyrene was molecularly
dissolved inside the vesicle membrane. The release behavior of pyrene from its
EAK16-I coating was studied by following the fluorescence of pyrene upon
exposure to EPC vesicles as a function of time. The transfer rate constants
were determined and found to be much slower when pyrene was released from the
E05P02 solution than from the E05P10 solution. Scanning electron micrographs and
C-to-N ratios of the complexes surface
layers suggested that this effect was due to thicker peptide coating which
inhibited the release of pyrene displayed in E05P02 solution. In conclusion,
our results suggested that the amphiphilic self-assembling peptide EAK16-I had
the potential to become a carrier for low molecular weight, hydrophobic
compounds whose release into target cells could be controlled by tuning the
peptide-to-cargo molecular ratio.
Acknowledgments
We thank Professor Pu Chen of the University of Waterloo for his helpful suggestions and stimulating discussions. The support of Dr. Zhaoyang Ye for preparing this manuscript was especially and gratefully acknowledged. This work was supported by the Chinese National “985 Project” of Education Ministry based in Sichuan University, and by the Analytical and Testing Center of Sichuan University for providing technique support, respectively, Chengdu, China.
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