Investigations into the Bending Constant and Edge Energy of Bilayers

Article pubs.acs.org/Langmuir

Investigations into the Bending Constant and Edge Energy of Bilayers of Salt-Free Catanionic Vesicles Jingfei Chen,†,§ Panfeng Long,‡ Hongguang Li,† and Jingcheng Hao*,†,‡ †

State Key Laboratory of Solid Lubrication, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China ‡ Key Laboratory of Colloid and Interface Chemistry of Ministry of Education, Shandong University, Jinan 250100, People’s Republic of China § Graduate School of the Chinese Academy of Sciences, Beijing 100080, People’s Republic of China ABSTRACT: Using molecular dynamics simulation, we performed theoretical calculations on the curvature constant and edge energy of bilayers of salt-free, zerocharged, cationic and anionic (catanionic) surfactant vesicles composed of alkylammonium cations (Cm+) and fatty acid anions (Cn−). Both the minimum size and edge energy of vesicles were calculated to examine the relation between the length of the surfactant molecules and the mechanical properties of the catanionic bilayers. Our simulation results clearly demonstrate that, when the chain lengths of the cationic and anionic surfactants are equal, both the edge energy and the rigidity of the catanionic bilayers increase dramatically, changing from around 0.36 to 2.77 kBT·nm−1 and around 0.86 to 6.51 kBT·nm−1, respectively. For the smallest catanionic vesicles, the curvature is not uniform and the surfactant molecules adopt a multicurvature arrangement in the vesicle bilayers. We suspect that the multicurvature bending of bilayers of catanionic vesicles is a common phenomenon in rigid bilayer systems, which could aid understanding of ion transport through bilayer membranes.

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To quantitatively describe the bilayer bending, Helfrich,12 in an early theoretical consideration, proposed a bilayer model using a harmonic approximation to describe the curvature energy per unit area of the lipid bilayer. In this model, the bilayers with spontaneous curvature are considered to have the lowest bending energy and any deviation from spontaneous curvature can induce an increase of bending potential energy. Meanwhile, the microenvironment in the inner and outer bilayer leaflets changes correspondingly to support the altered bilayer curvature. Based on this theoretical consideration, Jung et al.6 measured the bending constants of typical CH−CF catanionic mixture bilayers and revealed two distinct mechanisms of vesicle formation: (i) When the bending constant K ≈ kBT, the larger interlayer potential derived from Helfrich undulations results in unilamellar vesicles with a relatively wider size distribution and (ii) when K ≫ kBT, the unilamellar vesicles are mainly stabilized by the spontaneous curvature and the vesicle size distribution is in a narrower range. The bilayer edge refers to the region surrounding a bilayer fragment, in which the amphiphilic molecules reorganize to avoid contact between the solvent and the hydrophobic tails. Compared to the bulk region, the edge region with higher curvature is unfavorable to the arrangement of amphiphilic molecules and has a higher edge energy which is defined as the work needed to increase the unit length of the bilayer edge. It is

INTRODUCTION Salt-free, zero-charged, catanionic surfactant pairs,Cm+Cn−, are mixtures of cationic and anionic surfactants in an equimolar ratio, where m and n denote the chain length of the cationic and anionic surfactants, respectively. Cm+Cn− mixtures in aqueous solution can usually be prepared by eliminating the inorganic counterions such as Na+ and Br− from the aqueous solutions. In equimolar mixtures of long-chain alkyltrimethylammonium hydroxide (CmTA+OH−) and long-chain fatty acid (CnA−H+), an acid−base reaction generates the ion pairs CmTA+−−ACn (CmTA+OH− + CnA−H+ → CmTA+−−ACn + H2O), which have been widely studied as typical catanionic surfactant model systems by different groups.1−5 By varying the components in the catanionic mixtures, especially the lengths of the hydrophobic tails and their stoichiometric ratios, complex thermotropic phase behavior, i.e., a series of aggregates with a unique morphology, can be observed, including disk-like micelles4a and hollow dodecahedrons4b formed with a small excess of either the cationic or anionic surfactant. The salt-free, zero-charged mixtures produced multilamellar (or onion-like) vesicles at an equimolar mixing ratio.5b In fact, we have demonstrated that uni- and multilamellar vesicles with small sizes were obtained at equimolar mixing ratios from catanionic CH−CH or CH−CF systems.5 Theoretical models have been developed to explain the fascinating self-assembly phenomena in catanionic vesicle systems.4c,f Many studies6−11 have demonstrated that the morphology evolution and the size distribution of aggregates in bilayer systems are tightly linked to the elastic properties and edge energy of the bilayers. © 2012 American Chemical Society

Received: December 11, 2011 Revised: March 2, 2012 Published: March 19, 2012 5927

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2.77 kBT·nm−1 and around 0.86 to 6.51 kBT·nm−1, respectively. For the smallest catanionic vesicles, the curvature of the vesicle is not uniform and the surfactant molecules adopt a multicurvature arrangement in the vesicle bilayers. We suspect that the multicurvature bending of bilayers of catanionic vesicles is a common phenomenon in rigid bilayer systems, which could aid understanding of ion transport through bilayer membranes.

clear that the edge energy and bending energy of the bilayers are also important indicators of potential morphology change in vesicle systems. In some lipid vesicle systems, the reduction of bilayer edge energy can be achieved by adding amphiphiles with specific shape7,8 or mixing lipids of different tail lengths8,9 and a series of perforated or disk-shaped aggregates appear in these systems.9 The relationship between bending energy and edge energy has been deduced based on the energy analysis of disk-like micelles.7 Experimental results13−15 and theoretical models13 confirmed that continuous coalescence and growth of discs will make the bending energy decrease in vesiculation, while their edge energy remains relatively constant. When the edge energy exceeds the bending energy, flat discs can be induced to curve into vesicles. Given that only bending energy and edge energy contribute to the free energy change and that discs and vesicles have the same areas during the deformation, the expression describing the disk-to-vesicle transition can be given as follows:7 R d,max = 4

2κ + κ̅ λ

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THEORETICAL CALCULATIONS

Detailed Steps for Atomic-Scale Simulation of Catanionic Vesicles. To accelerate the simulation of atomistic vesicles, the simulations start from a coarse-grained (CG) vesicle template instead of a random state. The CG molecular dynamics simulation is completely based on the Martini force field.22 In the model, lipids can be presented by several different interaction sites according to the four-to-one mapping rule, and 18 CG particle types were distinguished to describe the interaction between them. This low-resolution CG model can expand the range of system size and simulation time, and still retain the key features of the systems of interest. In the CG simulation part, the cationic and anionic components were coarse-grained into 4−5 interaction sites according to the standard building block:22 C3N+ (type Q0), C2O2− (type Qa), C4 (type C1), and C3 (type C2), and 4 water molecules are presented by a CG bead (type P4). To shorten the equilibrium time, 1500−2000 cations and anions (1:1) were arranged alternately into an artificial bilayer and then solved in a cube (18 nm × 13 nm × 18 nm). A 300−400 ns dynamic simulation is enough to obtain the nearly closed vesicle template. In the atomic simulation, atomic details were reintroduced to the CG vesicle by replacing the CG sites with the corresponding mass centers of the atomic groups. 100 steps of steepest descent energy minimization were then performed to avoid the overlap between atomic groups. To avoid excessive water in the simulation box, the distance between the atomistic vesicle and the box boundaries was adjusted to ∼1 nm. The explicit water model was used to fill the box, and the water molecules inserted within the bilayer interior were then removed. The atomic dynamics simulations were performed with a time step of 5 fs using the GROMACS software package 4.0.523,24 at 300 K and 1 atm in a NPT ensemble (here “NPT ensemble” means that the number of particles, N, pressure, and temperature remain constant) using Nosé-Hoover temperature25,26 and Parrinello-Rahman isotropic pressure27,28 coupling. Long-range electrostatic interactions were treated with the particle mesh Ewald (PME)29 method with a real space cutoff of 1.0 nm. Edge Energies of Catanionic Bilayer Edges. The measurement of edge energy can be conducted by examining the variation of energy of a continuous of bilayer ribbon in NPxyLzT ensemble. To prepare the simulation box illustrated in the Figure 1, first a simulation box containing about 500 prearranged catanionic surfactants and ∼30 000 water molecules were prepared. Then a disk micelle forms after 2 ns of equilibrium. Next, all the water molecules and about 100 surfactant molecules on both ends of the disk in the z direction were removed and the simulation box length in the z direction was then adjusted to ensure the disk was in contact with its mirror image. External water molecules were added back, and then the water within the disk interior was removed. There are 203 ion pairs (406 cations and anions) and 16413 water molecules in the final simulation system. Another 2 ns of dynamics simulation in the NPT (P = 1.0 bar, T = 323 K) ensemble using the GROMACS 4.0.5 software package can make edge region parallel to the z axis to form a ribbon structure. The molecular dynamics simulation to measure the edge energy is conducted with time steps of 5 fs in a NPxyLzT (P = 1.0 bar, T = 323 K) ensemble, namely, the box length in the z direction is fixed, whereas the instantaneous pressure in the x and y directions are coupled jointly to 1.0 bar. To reduce systemic errors, 5 ns parallel simulations were conducted at least three times and the final values of edge energy were determined by calculating the average value.

(1)

where Rd,max is the maximum disk radius, which is also geometrically equal to twice the minimum vesicle radius;7 λ is the edge energy (also called line tension); k and κ̅ are the Gaussian modulus and the mean bending modulus, respectively. For spherical vesicles, κ and κ̅ are usually used together to describe the bilayer bending, which can be called the effective bending constant K = 2κ + κ.̅ 16,17 In eq 1, Rd, max can be determined by time-resolved smallangle X-ray scattering (SAXS),18 κ, κ,̅ and λ can also be deduced based on analysis of the vesicle size distribution.6,13 Mechanical properties of typical catanionic surfactant vesicle bilayers such as CTAB/SOS and CTAB/FC7 have been well characterized in previous reports.6,13 Although molecular dynamics simulations can reveal new insights into bilayer systems,19,20 it is still very expensive computationally to obtain a vesicle at the atomic scale from a random configuration.21 However, the continually increasing power of computers and the appearance of more efficient computing methods have made it possible to simulate the vesicle of minimum size and determine the bilayer properties. Here using molecular dynamics simulation, for the first time to our knowledge, we performed theoretical calculations on the curvature constant and edge energy of bilayers of salt-free, zerocharged, cationic and anionic (catanionic) surfactant vesicles composed of two groups of catanionic CmTA+−−ACn bilayer systems: (i) C14TA+−−ACn mixtures, where n = 12 or 14, and (ii) C16TA+−−ACn mixtures, where n = 12, 14, or 16. First, the specific steps for simulating the minimum size of vesicles are presented, followed by discussion of how the catanionic surfactants with identical headgroups but different hydrophobic tail lengths can affect the structure of vesicles. Second, we describe the principles of the simulation of edge energies together with a comprehensive summary of edge energies and bending constants in different catanionic bilayer systems. Both the minimum size and edge energy of vesicles were calculated to examine the relation between the length of the surfactant molecules and the mechanical properties of the catanionic bilayers. Our simulation results clearly demonstrate that, when the chain lengths of the cationic and anionic surfactants are equal, both the edge energy and the rigidity of the catanionic bilayers increase dramatically, changing from around 0.36 to 5928

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Figure 3. Snapshot of a smallest C16TA+−−AC16 vesicle. A pore, positioned on the back of the vesicle, is reserved to guarantee inside and outside of the vesicles to equilibrate fully.

Figure 1. Configuration of the simulation box for determining the edge energies.

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Table 1. Simulation Systems

RESULTS AND DISCUSSION Molecular Dynamics Simulation for Salt-Free ZeroCharged Catanionic Vesicles. Figure 2 shows the steps

system

no. of ion pairs

no. of water molecules

equilibrium time (ns)

C14TA+−−AC12 C14TA+−−AC14 C16TA+−−AC12 C16TA+−−AC14 C16TA+−−AC16

775(+)/775(-) 705(+)/705(-) 934(+)/934(-) 862(+)/862(-) 937(+)/937(-)

63 521 57 762 83 302 73 757 92 313

5 5 8 8 8

respect to the vesicle center of mass (COM) are plotted in Figure 4. The abscissa values of the two sharp peaks represent the inner (ri) and outer (ro) radiuses of the vesicles, and the difference between them (i.e., ro − ri) refers to the thickness of the bilayer. All data characterizing the smallest catanionic vesicles are listed in Table 2. From Table 2, one can see that the size of the vesicles increases with the overall chain length. The inner radius and bilayer thickness varies with the chain length mismatch as follows: the better the chain length matches, the smaller the inner radius is and the thicker the bilayer is, which could be explained via the concept of a molecular packing parameter (P).31 v P= 0 al0 (2)

Figure 2. Steps for the simulation of the catanionic vesicles with minimum size. The coarse-grained (CG) vesicle model is replaced with an atomic artificial model in step IV, prior to the final simulation step.

where v0 and l0 are the volume and length of the hydrophobic tail and a is the average area of the headgroups at the aggregate surface. The molecular packing parameter, P, is generally used to rationalize or predict the aggregate shape or monolayer bending. For salt-free, zero-charged, catanionic C14TA+−−ACn and C16TA+−−ACn vesicles, a and l0 can be considered as constants. However, v0 increases when the chain length of the cationic and anionic surfactants matches, because spatial utilization of the hydrophobic region in the bilayer is enhanced by the matching of the chain length: the repulsion among the hydrophobic chains increases, inducing a corresponding increase of volume of the hydrophobic tails. Thus, the increased packing parameter (P) could be favorable to inner monolayer bending but unfavorable to outer monolayer bending, so the C14TA+−−AC14 and C16TA+−−AC16 vesicles have the smallest inner radiuses and the thickest bilayers in the two catanionic vesicle systems. To further investigate the impact of the matching of chain length on the vesicles, a catanionic bilayer was cut out from the

adopted to fabricate the smallest vesicles, steps I−III: The CG model22 was used to simulate nearly closed CG vesicles. An open pore is preserved to allow the inside and outside of the vesicles to further equilibrate; step IV: we reintroduced atomic details into the CG vesicles again by replacing the CG sites with the corresponding atom groups; step V: a 5−8 ns of dynamics simulation was performed using a united-atom force field30 to construct the vesicles. To obtain the smallest vesicles, the number of surfactant molecules should be adjusted to avoid forming a closed vesicle (whose size is greater than or equal to the minimum vesicle size) or a vesicle with a large pore (a pore radius of more than ∼1 nm), as shown in Figure 3. The minimum numbers of catanionic surfactant molecules constituting the vesicles were estimated according to the vesicle pore. The errors are less than ±2%. All of the compositions and simulation times of the catanionic systems involved in our simulation systems are listed in Table 1. To further analyze the smallest catanionic vesicles, the density profiles of the cationic headgroup (CH3)3N+ with 5929

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Figure 4. Density profiles of surfactant headgroup (CH3)3N+ with respect to the vesicle center of mass (COM). C14TA+−−ACn system (a) and C16TA+−−ACn system (b).

Table 2. Structural Parameters of the Minimum Size Vesicles system

inner radius (nm)

outer radius (nm)

thickness (nm)

no. of ion pairs

no. of ion pairs in the inner leaflet

no. of ion pairs in the outer leaflet

C14TA+−−AC12 C14TA+−−AC14 C16TA+−−AC12 C16TA+−−AC14 C16TA+−−AC16

3.08 2.97 3.11 3.33 3.07

5.73 5.71 6.16 6.26 6.33

2.65 2.74 3.05 2.93 3.26

801(+)/801(-) 720(+)/720(-) 950(+)/950(-) 882(+)/882(-) 955(+)/955(-)

222(+)/220(-) 182(+)/183(-) 261(+)/260(-) 216(+)/220(-) 261(+)/259(-)

579(+)/581(-) 538(+)/537(-) 689(+)/690(-) 666(+)/662(-) 694(+)/696(-)

middle of each representative vesicle, as shown in Figure 5a. Surprisingly, the catanionic surfactant molecules are not

become consistent with each other. One can infer that standard spherical vesicles will be obtained as the bilayer flexibility increases. Although the smallest vesicle cannot reflect the bilayer curvature properties of the majority of the vesicles in the system, one can still deduce that a monocurvature vesicle may evolve into a multicurvature vesicle under conditions when vesicle curvature deviates far enough from spontaneous curvature. This does not occur just by varying the surfactant molecule distribution between the inner and outer bilayer leaflets, as was previously described.12 We theorize that the formation of polyhedral vesicles may occur through a multicurvature vesicles intermediate.4b,32 Calculation of Edge Energy of Catanionic Vesicles. The measurement of edge energy of lipid bilayers in experiments is usually conducted based on energy analysis of the pore opening and closing in giant unilamellar vesicles (about several micrometers),33 which are not applicable to our previous observations due to the dimension restriction of the catanionic vesicles (from several tens to hundreds nanometers).5a In these catanionic vesicle systems, the molecular dynamics approach can be adopted to measure the edge energy of a catanionic bilayer in a NPxyLzT ensemble.34 In the model, a catanionic ribbon structure is continuous in the z direction, and the catanionic surfactant molecules are rearranged into a rounded edge in the y direction. Edge energy along the two edges in the z direction can be expressed as follows:

Figure 5. Catanionic bilayer slices were cut out from the middle of minimum size vesicles. The large fragments are marked with white shadows (a). Schematic diagram of the bending materials (b).

distributed as evenly as is commonly assumed, which can be especially seen in the inner leaflets of the vesicles. These bilayer slices were seemingly made up of several bilayer fragments of varying sizes. According to the sizes of the bilayer fragments, the smaller fragments can be treated as linkers which connect the larger fragments. The inset in Figure 5a shows that, in the linker fragment site, the ratio of surfactant molecules in the inner leaflet to those in the outer one is obviously different from those in the larger fragments, and correspondingly, the curvature of the linker fragment could be greater than that of larger fragments. In multicurvature vesicles, the larger fragments with smaller curvature and small areas of linker fragments having greater curvature give an energy advantage over a vesicle with uniform curvature, which seems completely analogous to the bending of rigid materials in the macro world (Figure 5b). However, with an increase of chain length mismatch, the rigidity of the bilayer decreases and both the curvatures and the areas of the fragments tend to gradually

Λ=

1 2

⎡ Pxx + Pyy ⎤ LxLy⎢ − Pzz ⎥ 2 ⎣ ⎦

(3)

where Lx and Ly represent the lengths of the simulation box in the x and y directions, and Pxx, Pyy, and Pzz are the diagonal components of the instantaneous tensor. Edge energy (aka line tension) Λ is denoted by the product of the area of the x−y plate and the pressure derived from the elastic contraction of the bilayer in the z direction, which can be further decomposed 5930

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into bulk pressure minus instantaneous pressure in the z direction, i.e., (Pxx + Pyy)/2 − Pzz. In Figure 6, the edge energy is plotted vs chain length matching. A few reports about the edge energies of catanionic

chain lengths. Generally, a longer and more uniform tail length of surfactant molecules can enhance the bilayer rigidity, which is consistent with the experimental observations in tuning the mechanical properties of bilayers. Compared to the previous estimation, the effective bending values in this paper have three significant figures, which stems from the accurate measurement of minimum size of vesicle and edge energy. Being able to detect the small variation of energy of bilayer is a great advantage over other methods. However, a recent theory calculation of bilayer mode4f show that, the existence of additional bonds (e.g., hydrogen bonds) between the catanionic headgroups can make the bending constants increase sharply from ∼10 to ∼910 kBT within a narrow composition. Hence, we can conclude that the bending stiffness of membrane is very sensitive to the composition of bilayer and molecular structure, and the accurate measurement of the mechanical properties of bilayer is still very challengeable. In any case our results from the simulations for salt-free, zero-charged, cationic and anionic (catanionic) surfactant vesicles should open a discussion since former evaluations and experiment differ from what is shown here by 10 kBT, i.e., exp(10) in Boltzmann factors or around 25 KJ·mol−1, which an easily detected value in osmotic pressure experiments was measured from Meister et al.38 and it was discussed the instability and deformation of a spherical vesicle pressure from Zhong et al.39

Figure 6. Plot of edge energies of catanionic bilayer edge vs. chain length matching of catanionic surfactants.

bilayers have appeared. The edge energies of lipids with different saturated carbon chain lengths (C12 −C18) are in the range of 1−12 pN,35 whereas the edge energies of the catanionic bilayers in this work are slightly larger, in the range of 5−16 pN, which may be attributed to the electrostatic attraction between the headgroups of the catanionic surfactants. Moreover, the reduction of chain length in both cationic and anionic hydrophobic tails, or only in the anionic component seems to have the same effect on reducing the edge energies: about a 4−6 pN decrease of edge energy is associated with a 2 carbon atom reduction in the hydrophobic tail. In view of the difference between bilayer thickness (∼3 nm) and outer radius of vesicle (∼6 nm), it is reasonable that the outer radius minus half of a bilayer thickness is considered as the vesicle radius, instead of just the outer radius. According to the geometrical relationship in the disk-to-vesicle deformation, the maximum disk radius, Rd, max, can be obtained by doubling the minimum vesicle radius, Rv,min. The effective bending constants K = 2κ + κ̅ determined from eq 1, the maximum radius of the disk Rd,max, and edge energy, Λ, are listed in Table 3. The radius of the minimum size of vesicles at ∼8−9 nm is conspicuously smaller than that measured in experiments (∼15−20 nm).18,36 In fact, the smallest vesicle is extremely rare in practical experiments, even though it exists in theory. Thus, the theoretical simulation could provide a more accurate understanding of vesicle formation. The effective bending values of the catanionic bilayer of C16TA+−−ACn range from 0.86 ± 0.51 to 6.51 ± 1.06 kBT·nm−1, which is reasonable since experimental values of K = 2κ + κ̅ are ∼0.7 to 6.0 kBT for other catanionic bilayers.6,37 The catanionic bilayer rigidity is closely associated with the overall chain length and the matching of the

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CONCLUSION In conclusion, molecular dynamics has been employed to mimic the smallest vesicles formed in catanionic surfactant mixtures with a cationic quaternary ammonium group and an anionic carboxylate group but with various alkyl chain lengths. The cross sections of catanionic vesicles demonstrate that in some of the bilayers, the vesicles adopt multicurvature instead of identical curvature, especially for rigid bilayer systems. Presumably, multicurvature bending is the optimal organization for amphiphilic molecules residing in a curved bilayer, when bilayer curvature deviates far away from the spontaneous curvature. Comprehensive understanding of the multicurvature bilayer will necessarily enrich membrane theory and have great significance for further studies of surfactant self-assembly in solution. The effective measurement of catanionic bilayers also shows that molecular dynamics is a powerful tool to accurately characterize the mechanical properties of bilayers, which are crucial for understanding the phase behavior of bilayer systems and could contribute to our understanding of ion transport through bilayer membranes.

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AUTHOR INFORMATION

Corresponding Author

*Telephone: +86-531-88366074. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

Table 3. Effective Bending Constants of Catanionic Bilayersa system −

C14TA − AC12 C14TA+−−AC14 C16TA+−−AC12 C16TA+−−AC14 C16TA+−−AC16 +

a

RV,min (nm)

Rd,max (nm)

4.41 4.34 4.64 4.79 4.70

8.82 8.68 9.28 9.58 9.40

edge energy (pN) 3.41 6.02 1.63 6.30 12.38

± ± ± ± ±

1.20 1.30 1.00 1.90 2.00

edge energy (kB·nm−1) 0.76 1.35 0.37 1.41 2.77

± ± ± ± ±

0.27 0.29 0.22 0.43 0.45

K = 2κ + κ̅ (kBT·nm−1) 1.68 ± 0.60 2.93 ± 0.63 0.86 ± 0.51 3.38 ± 1.03 6.51 ± 1.06

1 pN ≈ 0.224 kBT·nm−1, at T = 323 K. 5931

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ACKNOWLEDGMENTS This work was financially supported by the NSFC (Grant No. 21033005), the National Basic Research Program of China (973 Program, 2009CB930103), and NFS of Shandong Province (2009ZRB01876).

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