Instability of carbon nanoparticles interacting with lipid bilayers

Abstract

As a first step in the study of the toxicity of nanoparticles, we investigate here the energy behaviour of three distinct carbon nanoparticles interacting with a lipid bilayer; namely fullerenes, nanotubes and nanocones, using the Lennard-Jones potential together with the continuous approximation. For an assumed circular hole in the lipid bilayer, a relation for the molecular interaction energy is determined, involving the circular hole radius and the perpendicular distance of the nanoparticle from the hole. For each nanoparticle, the relation between the minimum energy location and the hole radius b is found, and for example, for a fullerene of radius 15 Å, for b > 19.03 Å, the nanoparticle relocates from the surface of the bilayer to the interior, and as the hole radius increases further it moves to the centre of the bilayer, remaining there for increasing hole radii. When the system has no external forces, the nanoparticle will not penetrate through the lipid bilayer but rather remains enclosed between the two layers.

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1 Introduction

Due to the large number of possible applications of nanoparticles in cosmetic and medical products, the possible hazards of nanoparticles in the human body are a major concern. A worst-case scenario is that nanoparticles might act as a centre of high activity and cause health issues such as skin damage or even induce diseases such as cancer. To understand the toxicity of nanoparticles, the interaction behaviour between the particles and the cell membrane is examined in order to understand the translocation of molecules into cells.

Yang and Ma¹ determine the effect of the nanoparticle size and shape on the relocation procedure across the lipid bilayer as a guide to designing the architecture of nano-cargos. They conclude that particle geometry is the key to successful nanoparticle design. The navigation of different shaped nanoparticles through biological processes is extensively reviewed by Toy et al.² Further, it has been shown that an external force is required to transport an alien particle into cells.^1,3–6 Here, we use mathematical modelling to investigate the van der Waals interaction between carbon nanostructures and a lipid bilayer. Three geometries of carbon nanostructures are considered, namely fullerenes, nanotubes and nanocones which are assumed to be the foreign particle penetrating through the dipalmitoylphosphatidylcholine (DPPC) lipid.

During the past two decades, various atomistic molecular mechanics force fields have been parameterized enabling the dynamic simulation of lipid bilayers.^7–9 Recently, several coarse grained models have been used for lipid membranes^10–14 to deal with the large dimensions of membrane compartments. Marrink et al.¹¹ describe the parametrization of a coarse grained model for a DPPC lipid system. Furthermore, Shelley et al.¹² use a coarse grained model to study the structure and self-assembly of phospholipids bilayers. Another coarse-grained force field for zwitterionic lipids and based on fitted thermodynamic and structural properties has been developed by Shinoda et al.¹⁴ Besides force field descriptions for the energy of molecular conformations that use either atomistic or coarse grained descriptions, such systems can also be studied using a continuous approach. In the continuous approximation, one assumes that discrete atomic arrangements can be replaced by a uniform atomic distribution, so that the total interaction energy between two molecules can be evaluated using an integral technique. The continuous approach has been successfully applied by a number of authors to determine the molecular interaction energy of nanostructures.^15–20

The instability of C₆₀ fullerene interaction with DPPC lipid bilayer has been studied by the present authors¹⁹ utilizing the Lennard-Jones potential function and the continuous approximation, and the relocation of the fullerene from the surface to the interior of the lipid has been reported. Further, Baowan et al.²⁰ use the same technique to determine the interaction energy of silica nanoparticle encapsulated inside a liposome, spherical lipid bilayer. Both van der Waals and electrostatic interactions are taken into account but in terms of the stability of the system the van der Waals energy plays the important role.

In this paper, we follow^19,20 and use the Lennard-Jones potential function together with the continuous approximation to determine the molecular interaction energy between certain carbon nanostructures and a DPPC lipid bilayer. Using different particle geometries, we investigate the penetration behaviour of the particles through an assumed circular hole in the lipid bilayer. The model formulations for the lipid bilayer and the carbon nanoparticles are detailed in the following section. In Section 3, the mathematical derivations for the three particle geometries interacting with the lipid are presented. Further, numerical results are given in Section 4, and finally, a brief summary of the work is presented in Section 5.

2 Model

This study aims at computing the energy of systems comprising carbon nanoparticles and a lipid bilayer which is assumed to be large enough to represent a cell membrane. The penetration behaviour of the nanoparticles through an assumed circular hole of radius b in the bilayer is determined. Three distinct shapes of the nanoparticles are assumed as possible structures of alien particles penetrating through the lipid bilayer, which are fullerenes, nanotubes and nanocones, and these are modelled as spheres, cylinders and cones, respectively. For the nanotubes and nanocones, two distinct configurations are considered (see Fig. 1). The carbon nanotubes are assumed to be both short and thin cylinders, and the nanocones are assumed to be one of vertical down and that of vertical up, as shown in Fig. 1(b–e). In order to make an energy comparison, we assume that all the nanoparticles are carbon nanostructures and all carbon atoms are uniformly distributed over the surfaces of the molecules with the same atomic surface densities η_c = 0.3812 Å⁻². Further, the particles are assumed to be located on the z-axis a distance Z above the hole in the lipid bilayer where Z is measured from the centres of the sphere and the cylinder and it is taken to be the closest position of the cone to the bilayer as indicated in Fig. 1.

Fig. 1 Three shapes with five possible configurations of carbon nanoparticles located a distance Z above lipid bilayer.

Here, only the van der Waals interaction arising from the Lennard-Jones function is taken into account, since it has been shown that the electrostatic energy plays only a minor effect on the systems.²⁰ The 6–12 Lennard-Jones function is given by

where ρ denotes the distance between two typical points, and A and B are attractive and repulsive Lennard-Jones constants, respectively. Further, ε denotes the well depth and σ is the van der Waals diameter, and from which we may deduce A = 4εσ⁶ and B = 4εσ¹².

The continuous approach assumes that the atoms at discrete locations on the molecule are averaged over a surface or a volume and the molecular interatomic energy is obtained by integrating over the surface or the volume of each molecule, given by

where η₁ and η₂ represent the mean surface densities or the mean volume densities of atoms on each molecule. For convenience, we may define the integral I_n asso that, E = η₁η₂(−AI₃ + BI₆).

The Lennard-Jones potential function together with the continuous approach has been successfully applied by a number of authors to determine the molecular interaction energy of nanostructures, see for example Girifalco et al.,¹⁵ Hodak and Girifalco,¹⁶ Cox et al.,^17,18 and Baowan et al.^19,20 In the first two studies,^15,16 analytical expressions are derived for the potential energies for various arrangements of a carbon nanotube and a C₆₀ fullerene. Cox et al.^17,18 use elementary mechanical principles together with the continuous approach to study the oscillatory behavior of a C₆₀ fullerene inside carbon nanotubes of various sizes. The structural behaviour and oscillatory frequency obtained there are in good agreement with molecular dynamics simulations of Qian et al.²¹ and Liu et al.²²

Here, dipalmitoylphosphatidylcholine (DPPC) is adopted as the lipid model which is represented in the MARTINI force field by a head group, consisting of choline (Q₀) and phosphate (Q_a) groups, an intermediate layer of a glycerol group (N_a) and a carbon tail group (C₁).¹¹ In this paper, the spacing between the two layers of lipids is assumed to be 3.36 Å.¹⁹ The positions for choline, phosphate and glycerol groups are taken from the work of Petrache et al.,²³ and they are detailed in Fig. 2(b) where the upper head group H₁ is assumed to be located on the xy-plane, z₂ = 0, and the other five positions of the five lipid bilayer layers are measured in a negative direction of z₂-axis. The Lennard-Jones constants for the lipid bilayer are taken from the work of Marrink et al.,¹¹ where the carbon nanoparticles are assumed to be an apolar group of type C₁. The numerical values of the Lennard-Jones constants used in this model are given in Table 1.

Fig. 2 Model for (a) a hole in bilayer and (b) structural dimensions of lipid bilayer where H₁ and H₂ represent upper and lower head groups, I₁ and I₂ denote upper and lower intermediate layers and T₁ and T₂ are upper and lower tail groups.

Table 1 Numerical values of Lennard-Jones constants for DPPC lipid interacting with apolar carbon nanostructures of type C₁¹¹

Interaction	ε (eV)	σ (Å)	A (eV × Å^6)	B (eV × Å^12)
Head group	2.073 × 10^−2	6.2	4.709 × 10^3	2.675 × 10^8
Intermediate layer	2.798 × 10^−2	4.7	1.207 × 10^3	1.301 × 10^7
Tail group	3.627 × 10^−2	4.7	1.564 × 10^3	1.686 × 10^7

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The coarse grained model assigns two interaction sites to the head group, one for the choline group and one for the phosphate group; two interaction sites to the intermediate layer; and eight interaction sites to the tail group.¹¹ We assume that the intermediate group of the bilayer can be represented as a flat plane, so that the mean atomic surface density for the intermediate group, η_inter, is given by 2/65 Å⁻². Here, the factor 2 reflects the two interaction sites for the intermediate layer in the MARTINI force field. Also the tail group is approximated as a box with a thickness of a tail length [small script l] = 15 Å, and the mean atomic volume density for the tail group, η_tail, is given by 8/(65[small script l]) Å⁻³. Again, the factor 8 reflects the eight interaction sites of the tail group. Similarly, the head group can be modelled as an infinite box of thickness of 4 Å. Consequently, the mean atomic volume density of the head group is η_head = 1/(130) Å⁻³. We note that the value 65 Ų represents a lipid head group area.

3 Mathematical derivation

In this study, surface and volume integration techniques are applied to evaluate the total interaction energy between a lipid bilayer and a carbon nanostructure, and the total energy of each system is assumed to comprise:

1. Interactions between the surface of a nanoparticle and the volumes of upper and lower lipid head groups at distances Z and Z + 37.36 Å, respectively.

2. Interactions between the surface of a nanoparticle and the surfaces of upper and lower lipid intermediate layers at distances Z + 4 and Z + 37.36 Å, respectively.

3. Interactions between the surface of a nanoparticle and the volumes of upper and lower lipid tail groups at distances Z + 4 and Z + 22.36 Å, respectively.

In order to make energetic comparisons, the surface areas of the carbon nanoparticles considered here are assumed to be equal, and their mathematical derivations are detailed in the following subsections.

3.1 Spherical nanoparticle

A sphere of radius a is considered with centre assumed to be located at a distance Z measured from the uppermost surface of the lipid bilayer as shown in Fig. 1(a). Further, the lipid bilayer is assumed to have a circular hole of radius b > a. The mathematical details for the interaction energy between a sphere and a flat plane, and that between a sphere and a box can be found in,¹⁹ and for the completeness the results are restated here. We comment that in the previous work,¹⁹ the head group of the bilayer is assumed to be a flat plane whereas the tail group is represented by a box of thickness [small script l] and with no intermediate layer. In order to capture all energy contributions, an improved model of the lipid bilayer is presented here such that the effect of the intermediate layer is included and the additional atoms in the head group are combined.

The interaction energy between the sphere and the flat plane or the box may be shown (see ref. 19) to be given by

where η* is a mean surface or a mean volume density of the lipid bilayer. In the case of the intermediate layer, we have
where the superscript p refers to the flat plane interaction. For the head and the tail groups of thickness [small script l], we may deduce
where the superscript b represents the box interaction, and the above integral can be found in ref. 24 (p. 153, no. 2.513.3), from which we may deducewhere is the usual binomial coefficients and m = n − 2. We note that the thickness of the head group is taken to be [small script l] = 4 Å, and that of the tail group is [small script l] = 15 Å.

3.2 Cylindrical nanoparticle

The cylindrical carbon nanostructure, or carbon nanotube, of radius a and half-length L is assumed to be located co-axially on the z-axis with its centre being a distance Z above the lipid bilayer as shown in Fig. 1(b). A typical surface element of the cylinder has coordinates given by (a cos θ₁, a sin θ₁, z₁ + Z) where 0 < θ₁ < 2π and −L < z₁ < L.

Firstly, we determine the interaction energy between a flat plane for which a typical surface element has coordinates (r cos θ₂, r sin θ₂, 0) where b < r < ∞, 0 < θ₂ < 2π and b > a denotes the hole radius in the bilayer. The distance of the flat plane to the surface of the cylinder is given by

ρ² = (a cos θ₁ − r cos θ₂)² + (a sin θ₁ − r sin θ₂)² + (z₁ + Z)² = (a − r)² + (z₁ + Z)² + 4ar sin²[(θ₁ − θ₂)/2],

and the integral I_n defined by (2) may be written asand again the superscript p refers to the plane interaction. We now definewhere α = (a − r)² + (z₁ + Z)² and β = 4ar. It can be shown that K_n is independent of either θ₁ or θ₂,²⁵ and we have simply

On letting u = sin² x, we may deduce

where F(a, b; c, z) is the usual hypergeometric function. Further, we may use a Pfaff transformation²⁴ to truncate the infinite series summation of the hypergeometric function, so that K_n becomes

Consequently, there are two remaining integrals for I^p_n

The above integral may be performed numerically utilizing the algebraic package MAPLE, and the total interaction energy between the cylinder of length 2L and the intermediate layer of the lipid is given by

E^p_cylinder = η_cη_inter(−AI^p₃ + BI^p₆), (8)

where I^p_n is given by (7).

In terms of the interaction energy between a cylinder and a box of thickness [small script l], a similar procedure is undertaken with the term (z₁ + Z) replaced by (z₁ + Z − z₂) and −[small script l] < z₂ < 0, and the integral I_n defined by (2) becomes

where the superscript b denotes the box interaction. Again, we have [small script l] = 4 Å for the thickness of the lipid bilayer head group and [small script l] = 15 Å for that of the tail group. The distance in the z-direction for each layer of the lipid is detailed in Fig. 2(b), and the total interaction energy between the cylinder of length 2L and the head or the tail group of the lipid bilayer is

E^b_cylinder = η_cη*(−AI^b₃ + BI^b₆), (10)

where η* can be either η_head or η_tail, and I^b_n is given by (9).

3.3 Conical nanoparticle

In this subsection, we consider both a “vertically down” cone and a “vertically up” cone for which mathematical determinations are similar except for the limits of integration in the z-direction.

Firstly, we consider the vertically down cone as shown in Fig. 1(d). A typical point on the cone has coordinates (r₁ cos θ₁, r₁ sin θ₁, z₁ + Z) where r₁ = az₁/h and 0 < z₁ < h. Here, Z is the distance from the upper lipid bilayer surface to the cone vertex, a is the cone base radius and h is the cone height. As with the cylindrical nanoparticle, the coordinates for the flat plane are (r₂ cos θ₂, r₂ sin θ₂, 0) where b < r₂ < ∞ and 0 < θ₂ < 2π, so that the distance between the flat plane and the surface of the cone is given by

ρ² = [(a/h)z₁ − r₂]² + (z₁ + Z)² + 4(a/h)z₁r₂ sin²[(θ₁ − θ₂)/2].

The integral I_n defined by (2) becomes

where ϕ is a cone angle, tan(ϕ/2) = a/h, and p indicates the plane interaction. We then write the above integral in terms of K_n defined by (6), and carry out precisely the same derivation as described in Section 3.2 but in this case we have α = [(a/h)z₁ − r₂]² + (z₁ + Z)² and β = 4(a/h)z₁r₂, and I^p_n for the vertically down cone becomes

The interaction energy between the vertically down cone and the intermediate layer is given by

E^p_cone = η_cη_inter(−AI^p₃ + BI^p₆), (13)

where in this case I^p_n is defined by (12).

For the interaction energy between a box of thickness [small script l] and a cone, there is an extra integral with respect to z₂ that must be determined. The term (z₁ + Z) is replaced by (z₁ + Z − z₂) where −[small script l] < z₂ < 0, and we may deduce

where b denotes the box interaction. Then the total interaction energy between the vertically down cone and the head or the tail group can be written as

E^b_cone = η_cη*(−AI^b₃ + BI^b₆), (15)

where η* can be either η_head or η_tail and I^b_n is given by (14).

For the vertically up cone, both the interaction between the cone and the flat plane and that between the cone and the box can be determined in precisely the same manner as that described for the vertically down cone. However, for the vertically up cone the integration with respect to z₁ appearing in (12) and (14) is evaluated from 0 to −h.

4 Numerical results

Interaction energies for the three carbon nanoparticle geometries involving five configurations and the lipid bilayer are determined. In an attempt to make meaningful comparison, we assume that the surface areas of the particles are all equal, and we fix the spherical radius, short cylindrical radius and two cone base radii to be a. Moreover, the radius of the thin cylinder is assumed to be a/2 to represent a long thin carbon nanotube. As a result, the length of the short cylinder becomes L = a, the heights of the two cones are , and the length of the thin cylinder is L = 2a.

Firstly, we determine the equilibrium positions for the five carbon nanostructures of a = 15 Å assumed to be interacting with a perfect lipid bilayer without any hole. The energy profiles are graphically shown in Fig. 3 where the distance δ is referred to the closest distance from the upper lipid head group to the particles. This is δ = Z − a for the cases of the sphere and the short cylinder, δ = Z − 2a for the thin cylinder, and δ = Z for the cones. It can be seen that the sphere and the cylinder of the same radii have the same behaviour in terms of the energy values and the equilibrium spacings δ. The cone of vertical up also has a similar manner. This is because a number of carbon atoms near the lipid bilayer for these three cases are equivalent and the equilibrium distances δ are approximately 4.45 Å above the bilayer. A number of carbon atoms around the tube end of the thin cylinder of radius a/2 is a half of the short cylinder, so that its energy value reduces by a half but the equilibrium spacing still be 4.45 Å. In the case of the vertically down cone, there are only a few carbon atoms at the vertex giving rise to a low interaction energy. However, this makes the cone move closer to the bilayer with an equilibrium spacing of 3.58 Å. We comment that the same energy profiles for larger carbon nanoparticles are observed, and that they differ only in the magnitude of the energy.

Fig. 3 Energy profiles for five configurations of carbon nanoparticles of a = 15 Å interacting with lipid bilayer without hole where δ refers to the closest distance from the upper lipid head group to the particles and δ = Z − a for sphere and short cylinder, δ = Z − 2a for thin cylinder, and δ = Z for cones.

Next, we study the relation between the distance Z and the hole in the lipid bilayer of radius b at the equilibrium positions of the particles. A positive value of Z indicates that the particle is located above the bilayer and a negative value of Z indicates that the particle has penetrated through the bilayer. Two sizes of the carbon nanoparticles which are a = 15 and 50 Å are determined to demonstrate the different penetration behaviours between particles smaller than the bilayer thickness and larger than the bilayer thickness, respectively. We note that in this study the thickness of the lipid bilayer is taken to be 41.36 Å.

Fig. 4 shows the relation between distance Z and hole radius b for five configurations of small carbon nanostructures where surface areas are all assumed to be 4πa² and a = 15 Å. The spherical particle moves closer to the bilayer as the hole radius gets larger. Once the hole radius is larger than the critical radius b₀ > 19.03 Å, the particle jumps into the bilayer and there are two equilibrium positions near the two head groups due to the symmetry of the two layers of lipid. Moreover, the sphere remains at the midplane of the bilayer when the hole radius is grater than 32 Å. The relocation behaviour is also observed for the two cylindrical particles with the critical hole radii b₀ of 18.15 and 11.23 Å for the short and the thin cylinders, respectively, which are around 4 Å larger than their radii. Once inside the bilayer, they will remain at the midplane. Even though the two cylinders have the similar behaviour, there are some advantages of a long thin cylinder over a short one. There will be some parts of the long thin tube passing to the other side of the bilayer which might be used for drug or gene transmission into the cell. Further, a smaller tube cross-section requires a smaller hole in the bilayer which in turn makes less disorder of lipid molecules arrangement in the cell membrane.

Fig. 4 Relation between distance Z and hole radius b for five small carbon nanostructures with equivalent surface area for (upper left) sphere of a = 15 Å, (upper right) cylinder of a = L = 15 Å, (lower left) thin cylinder of 2a = L/2 = 15 Å, and (lower right) cones vertical down and vertical up of a = 15 and .

The vertically up cone behaves like the cylindrical carbon nanotube, where the critical hole radius of the cone is obtained as b₀ = 18.13 Å. However, some parts of the cone will be on the other side of the bilayer, and may be used as a conduit to transmit other molecules into the cell. Finally, for the case of the of vertically down cone, it gradually penetrates inside the bilayer as the hole radius in the bilayer increases since the circular cone radius itself increases as it movies down. The equilibrium positions for all five configurations are all at the midplane of the two layers of lipid once the hole radius is large enough, and the penetration of the nanoparticles across the bilayer is always facilitated by an external force which is in agreement with.^1,3–6

A similar penetration behaviour is observed for the large carbon nanoparticles of a = 50 Å as shown in Fig. 5. However, double minima is only occurred for the case of a spherical molecule. We also observe that the jump behaviour occurs for the sphere, two cylinders and the vertically up cone, whereas the vertically down cone gradually penetrates through the bilayer, and the external forces are required to drive the alien particles into cells.^1,3–6

Fig. 5 Relation between distance Z and hole radius b for five small carbon nanostructures with equivalent surface area for (upper left) sphere of a = 50 Å, (upper right) cylinder of a = L = 50 Å, (lower left) thin cylinder of 2a = L/2 = 50 Å, and (lower right) cones vertical down and vertical up of a = 50 and .

5 Summary

This paper presents an energetic comparison for the penetration of carbon nanostructures into a lipid bilayer. For this, the van der Waals energy is evaluated using the Lennard-Jones function and the continuous approach, which assumes that atoms in a molecule are uniformly distributed over a surface or throughout the volume of the molecule, so that an integration approach can be applied to evaluate the total energy of the system. We consider three geometries involving five distinct configurations of the nanoparticles which are a sphere, two cylinders and two cones, and we assume that all carbon atoms are distributed over their surfaces. Further, the lipid bilayer studied here is assumed to be a DPPC with two head groups, two intermediate layers and two tail groups. For this we derive analytical expressions to describe how the various energies depend on particle geometries, particle size, distance of the particle from the bilayer, and the hole size in the bilayer.

The spacing δ between the upper head group and the five particles is determined in the absence of the hole in the lipid. We find that the equilibrium location of the vertically down cone is closest to the bilayer, because of fewer carbon atoms at the vertex, inducing only a weak repulsion energy. The other four configurations have comparable values of the spacings δ. However, the energy for the thin carbon nanotube is one half of the energies of the sphere, short cylinder and vertically up cone since the number of atoms at the tube end is approximately one half of the other three shapes.

Our study shows that the particles locate inside the bilayer when the hole in the bilayer is large enough, around 4 Å greater that the particle's radius. However, they are more likely to remain between the two layers of the lipid and do not move across to be inside the cell. Therefore, external forces are required for the penetration of the nanoparticles through the cell. The penetration behaviour and the critical hole radius b₀ for the carbon nanoparticles are summarized in Table 2.

Table 2 Penetration behaviour for three geometries of five configurations of carbon nanostructures through lipid bilayer

Shapes	Penetration behaviour	Minima	b0 (Å)
a = 15 Å
Sphere	Gradually jump	Double minima	19.03
Short cylinder	Jump	Double minima	18.15
Thin cylinder	Jump	One minimum	11.23
Cone of vertical down	Gradually penetrate	One minimum	—
Cone of vertical up	Jump	One minimum	18.13
a = 50 Å
Sphere	Gradually jump	Double minima	53.21
Short cylinder	Jump	One minimum	54.31
Thin cylinder	Jump	One minimum	29.59
Cone of vertical down	Gradually penetrate	One minimum	—
Cone of vertical up	Jump	One minimum	53.09

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Nanocapsules for drug and gene delivery might be designed from these five simple structures. The spherical and the vertically down cone capsules may be utilized for chemical reactions at the surface of the cell membrane. This is because we may choose the size of the capsule for a specific lipid bilayer hole, and the capsule will stick at the upper layer of the lipid and does not move inside the cell membrane. However, if the adsorption energy is too low, it might not be a good candidate for the surface chemical reaction since the interaction between membrane and nanoparticle might be too weak. The cylindrical and vertically up cone capsules may escape a fast reaction at the surface of the cell membrane because a jump behaviour occurs for a specific lipid hole size. Further, for a tube with length longer than the thickness of the bilayer, there will be some parts of the tube getting to the other side of the cell membrane which may be used to transport the drug or the gene into cells. Using general analytical expressions for the total molecular energy of the system, our work can be viewed as a first step toward the study of transportation of nanoparticles through or inside cells.

Acknowledgements

DB gratefully thanks the Endeavour Research Scheme for the provision of an Endeavour Postdoctoral Fellowship. Financial support from the Thailand Research Fund (TRG5680072) is acknowledged.

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