Receptor-Mediated Endocytosis of Nanoparticles: Roles of Shapes

Article Cite This: J. Phys. Chem. B XXXX, XXX, XXX−XXX

pubs.acs.org/JPCB

Receptor-Mediated Endocytosis of Nanoparticles: Roles of Shapes, Orientations, and Rotations of Nanoparticles Huayuan Tang, Hongwu Zhang, Hongfei Ye, and Yonggang Zheng* International Research Center for Computational Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China S Supporting Information *

ABSTRACT: A complete understanding of the interactions between nanoparticles (NPs) and the cell membrane is essential for the potential biomedical applications of NPs. The rotation of the NP during the cellular wrapping process is of great biological significance and has been widely observed in experiments and simulations. However, the underlying mechanisms of the rotation and their potential influences on the wrapping behavior are far from being fully understood. Here, by coupling the rotation of the NP with the diffusion of the receptors, we set up a model to theoretically investigate the wrapping pathway and the internalization rate of the rotatable NP in the receptor-mediated endocytosis. Based on this model, it is found that the endocytosis proceeds through the symmetric−asymmetric or asymmetric−symmetric−asymmetric wrapping pathway due to the bending and membrane tension competition induced rotation of NP. In addition, we show that the wrapping rate in the direction that the wrapping proceeds can be largely accelerated by the rotation. Moreover, the time to fully wrap the NP depends not only on the size and shape of the NP but also on its rotation and initial orientation. These results reveal the roles of the shape, rotation, and initial orientation of the NP on the receptor-mediated endocytosis and may provide guidelines for the design of NP-based drug delivery systems. rotation sequence.35 Moreover, it has been recognized that the rotation and orientation of the NP are of great biological significance. The blood stage malaria parasite has to rotate from the initial random orientation to the tip-wrapped state to achieve the invasion of the human erythrocyte.36 Cellular uptake of Janus particles, whose surfaces are coated with two or more kinds of ligands with distinct properties, strongly depends on the initial orientations with respect to the cell membrane.37−39 Based on the examination of Ebolavirus-infected cells, it is suggested that mature Ebola virions containing nucleocapsids primarily emerge horizontally from the cell surface, whereas empty virions mainly bud vertically.40 Despite the existing evidence, the rotation of the NP with an irregular shape during the wrapping process has rarely been investigated theoretically. Although there are published theoretical works that tried to explore the influences of the shape and orientation of the NP, these works often adopted the NP of fixed orientation without rotation movement, ignoring the fact that the influences of the shape, orientation, and rotation of the NP are often coupled.18−23 For instance, by directly minimizing the total free energy of the NP−membrane system at a given wrapping

1. INTRODUCTION Understanding the cellular uptake of nanoparticles (NPs) plays a fundamental role in the promising applications of NPs in drug delivery and also assessing their potential cytotoxicity.1−5 NPs can enter cells through various mechanisms, including the direct penetration, phagocytosis, caveolae-mediated endocytosis, and receptor-mediated endocytosis,6,7 among which the receptormediated endocytosis is one of the most effective internalization mechanisms. Until now, extensive researches have been conducted to explore the receptor-mediated endocytosis of NPs and have revealed that the sizes,8−11 geometries,12−25 elastic stiffnesses,26−30 and surface properties31−33 of NPs have profound influences on the wrapping behavior. Especially, the geometries of the NPs have attracted much attention because the cell membrane might interact with the NPs with various shapes, such as nanocapsules, nanorods, and nanosheets. It has been widely observed in experiments and simulations that the wrappings of NPs with irregular shapes often accompany the rotations of the NPs.12−17 For example, the “tip recognition and rotation” mode has been proposed for the cell entry of onedimensional nanomaterials, where the uptake involves the rotation of the NP to the near-vertical configuration at relatively low receptor densities.34 For the spherocylindrical NP, coarsegrained molecular dynamics simulations showed that the endocytosis proceeds through a laying-down-then-standing-up © XXXX American Chemical Society

Received: September 27, 2017 Revised: November 30, 2017 Published: December 4, 2017 A

DOI: 10.1021/acs.jpcb.7b09619 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Figure 1. Schematic plot of a NP with the elliptical cross section wrapped by the cell membrane. (a) The initial contact state where the contact point is marked by a triangle. (b) The partial wrapping state where the outer membrane region and the adhesion region are plotted as black and red lines, respectively. (c) Schematic plot of the receptors on the membrane and the ligands on the NP. (d) The distributions of receptor density along the membrane.

fraction, it has been found that the wrapping starts on the flat side of the ellipsoidal NP, whereas the tip-orientation configuration is more energetically favorable when the NP is deeply wrapped.21−23,27 However, the dynamic process of the evolution from the side-orientation to the tip-orientation is not clear. Moreover, there is still a lack of full understanding of the mechanism of the rotation of the NP and its interrelationship with the shape and orientation effects. In this article, we set up a model to theoretically investigate the wrapping pathway and the internalization rate of the rotatable NP, which are two important issues in the NP−membrane interaction, by the cell membrane in two-dimensional cases (Figure 1). Moreover, the dynamic wrapping behaviors of the rotatable NP are investigated by coupling the rotation of the NP with the diffusion of the receptors. The influences of the shape, rotation, and initial orientation of the NP on the wrapping behaviors are also discussed.

by the cell membrane. According to the Canham−Helfrich model,41,42 the total deformation energy of the system E is E=

Bm ∑ 2 i = 1,2

∫ Mi2 dsi + σ ∑ ∫ (1 − cos ϕi) dsi i = 1,2

(1)

where Bm and σ are the bending stiffness and surface tension of the membrane, respectively. In eq 1, the first term accounts for the bending energy Eb, whereas the last term describes the tension energy Et of the membrane. si are the arc-lengths, which are measured from the initial NP−membrane contact point and defined along the membrane, where the subscripts i = 1, 2 are used to identify quantities associated with the left and right membrane regions, respectively. Mi are the mean curvatures and ϕi are the tangent angles along the membrane. The membrane profiles in the adhesion region are assumed to conform tightly to the NP surface, whereas those in the outer free region are determined by energy minimization. To simplify the energy calculation, the tangent angles ϕi(si) of the outer free membrane are expanded into Fourier series as27,30,43,44

2. MODEL AND METHODS 2.1. Geometry of the NP−Membrane System. The schematic plot of a NP wrapped by the cell membrane is shown in Figure 1. The aspect ratio of the NP with the elliptical cross section is r = R2/R1, where R1 and R2 are the major and minor semiaxes of the NP (Figure 1a), respectively. At the initial contact stage, the NP contacts with the flat cell membrane at the initial contact point. Driven by the formation of the receptor−ligand bonds between the NP and the cell membrane, the adhesion region grows and the NP becomes partially wrapped by the membrane with the wrapping lengths a1 and a2 in the left and right sides of the initial contact point, respectively (Figure 1b). The wrapping fraction is defined as f = a/ac, where a is the total wrapping length (a = a1 + a2) and ac is the circumference of the cross section of the NP. The rotation angle of the NP is θ. The wrapping pathway of the NP is determined progressively based on the Canham−Helfrich model,41,42 and the dynamic wrapping behaviors are studied based on a coupled deformation and diffusion model (Figure 1c,d). 2.2. Wrapping Pathway of the NP. We first establish a twodimensional model to investigate the evolutions of the wrapping configurations of the rotatable NP with the elliptical cross section

ϕi(si) = ϕ0 + (ϕl − ϕ0)

si − ai + Li − ai

N

∑ Fj sin j=1

πj(si − ai) , si > ai Li − ai (2)

where Fj are the Fourier coefficients, N is the number of Fourier modes, and ϕ0 and ϕl are the tangent angles at si = ai (ai are the wrapping lengths) and si = Li (Li are the total arc-lengths of each region), respectively. Then, the total deformation energy at a given wrapping fraction can be expressed as a function of the Fourier coefficients and the rotation angle. The adhesive wrapping between the NP and the cell membrane is achieved by the formation of the receptor−ligand bond.45,46 In this article, to simplify the model, it is assumed that the receptor−ligand bonds survive throughout the wrapping process, which corresponds to a strong receptor−ligand binding under a low loading rate and has been widely adopted in previous models.8,18−20,47,48 Thus, the wrapping pathway of the rotatable NP can be determined in a progressive way: (1) Starting from the initial contact stage (a = 0), minimize the total deformation energy with respect to the Fourier series, the rotation angle and the wrapping length increments B


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The Journal of Physical Chemistry B Δa1, Δa2 with a given small increment of the total wrapping length Δa. Constraints that the membrane and the NP should be continuous at the wrapping edge and that the outer free parts of the membrane approach the same height in the remote should be satisfied. It should be noted that Δa1 and Δa2 are nonnegative in accordance with the assumption that the receptor−ligand bond would not break during the wrapping process. Once the Fourier coefficients are determined, the shapes of the membrane can be represented by the geometrical relations dxi = dsi cos ϕi and dyi = dsi sin ϕi, where xi and yi are the Cartesian coordinates. (2) The equilibrium configuration of the system with the wrapping length a = 2Δa is obtained by assigning a small increment of the total wrapping length Δa to the configuration with a = Δa using the same method in step (1). (3) Repeat step (2) until the NP is fully wrapped, where the two sides of the outer free parts of the membrane are in contact with each other. Compared with directly minimizing the total deformation energy at a given wrapping fraction,21−23 the wrapping configurations determined progressively in the present work are able to capture the rotation process of the NP. To reflect the intrinsic relationship among the parameters, nondimensional parameters are defined as E̅ = ER1/Bm, E b = E bR1/Bm , Et = EtR1/Bm , and σ̅ = σR1/Bm.27,44 2.3. Time-Dependent Wrapping Behavior of the NP. In this section, we further take the receptor−ligand binding into account to investigate how a cell membrane containing diffusive mobile receptors wraps around a cylindrical NP coated with compatible ligands when considering the rotation of the NP (Figure 1c). The ligands are assumed to be immobile and distributed on the NP surface with density ξL, whereas the receptors are mobile and undergo rapid diffusive motion in the plane of the cell membrane with an initial density ξ0. The receptor−ligand relative density ξ̃ = ξ0/ξL is usually less than unity. Once the NP comes in contact with the cell membrane, the receptors would bind with the ligands to reduce the free energy. As a result, the receptor density within the contact area is raised to ξL, and the receptors in the immediate neighborhood of the adhesion region are depleted (Figure 1d). The resulting gradient in concentration induces a global diffusive motion of receptors toward the binding site and the sizes of the contact area a1(t) and a2(t) increase with time t until the NP is fully wrapped. It should be noted that the theoretical mode has been adopted in previous studies to predict the wrapping time of the NPs of fixed orientation,8,18−20 here we further extend the model to take the rotation of the NP into consideration. The distributions of receptors ξi(si, t) in the membrane are determined from the diffusion equations ∂ξi(si , t ) = D∇2 ξi(si , t ), si > ai(t ) ∂t

d [ dt

∫0

ai(t )

∞

ξL(si) dsi +

∫a (t) ξi(si , t ) dsi] = 0 i

which leads to the continuity equation (ξL − ξi+)

ξ+i

dai + ji+ = 0, on si = ai(t ) dt

ξi(a+i ,

j+i

(4)

ji(a+i ,

where = t) and = t). In this study, the total free energy of the system F(t) is given by F(t ) = kBT

⎛

∑ ⎜⎜∫ 0

i = 1,2

ai(t ) ⎛

ξ ⎞ ⎜ −ξLeRL + ξL ln L ⎟ dsi + ξ0 ⎠ ⎝

⎝

∞

ξ

∫a (t) ξi ln ξi i

0

⎞ dsi⎟⎟ ⎠

E(a) + kBT

(5)

where kBTeRL is the energy of a single receptor−ligand bond, and kBT ln ξL/ξ0 and kBT ln ξi/ξ0 are the free energy of a single receptor associated with the loss of configurational entropy of the bounded receptors and the free receptors, respectively. E(a) is the deformation energy of the equilibrium configuration with the total wrapping length a. According to the experimental results, the cell membrane can achieve the equilibrium state in less than 1 s,49 whereas tens of seconds to tens of minutes are required to complete the receptor-mediated endocytosis of NP.11 As a result, the NP−membrane system can be treated as in a static equilibrium state with respect to the time scale of receptor diffusion, thus the configurations of the NP−membrane system can be determined by minimizing the free energy at each time step of the wrapping process.48 Differentiating eq 5 with respect to time leads to ⎛ ξ F (̇ t ) 1 dE ⎞ dai = − ∑ ⎜ξLeRL − ξL ln L+ + ξL − ξi+ − ⎟ ξi kBT kBT da ⎠ dt i = 1,2 ⎝ +

∑ i = 1,2

⎛ ∂χ ⎞2 Dξi⎜ i ⎟ dsi ai(t ) ⎝ ∂si ⎠

∫

∞

(6)

where χi(si, t) = ln(ξi/ξ0) + 1 is the local chemical potential of a receptor and dE/da reflects the deformation energy change as the wrapping proceeds. The last integral term in eq 6 is identified as the rate of energy dissipation related to the receptor transport along the cell membrane. If we require that the decrease rate of free energy during the wrapping process exactly balances the rate of energy dissipation during the receptor transport, the first term on the right side of eq 6 must vanish so that ⎛ ξ 1 dE ⎞ dai =0 ⎜ξLeRL − ξL ln L+ + ξL − ξi+ − ⎟ ξi kBT da ⎠ dt i = 1,2 ⎝

∑

(7)

Equation 7 is valid at any time t throughout the wrapping process. If we multiply eq 7 with dt/da, it becomes ⎛ ξ 1 dE ⎞ dai =0 ⎜ξLeRL − ξL ln L+ + ξL − ξi+ − ⎟ ξi kBT da ⎠ da i = 1,2 ⎝

∑

(3)

(8)

where D is the diffusivity. The diffusive fluxes of receptors ji(si, t) are prescribed by ji(si, t) = −D ∂ξi(si, t)/∂si. To determine the distribution of receptors during the wrapping process, initial and boundary conditions are required. The boundary condition at the wrapping front can be derived by the conservation of the total number of receptors, that is

dai/da are the partitions of the increments of the wrapping lengths dai with a given increment of the total wrapping length da, which can be determined once the wrapping pathway of the NP is acquired. Equation 8 couples the system deformation with the diffusion of receptors. For the symmetric wrapping of the NP (a1 = a2), it reduces to C


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The Journal of Physical Chemistry B ξLeRL − ξL ln

ξL 1 dE + =0 + + ξL − ξ − ξ kBT da

wrapping fraction are shown in Figure 2b,c, respectively. As shown in the figures, the ASR mode is divided into two stages. In stage I, the NP would be symmetrically wrapped by the membrane without any rotation movement before a critical wrapping fraction fc even when considering the rotational degree of freedom of the NP. In this stage, the wrapping lengths a1 = a2. Beyond the critical wrapping fraction, the NP starts to rotate and the wrapping configurations become asymmetric (Figure 2d) in stage II. During the rotation of the NP, the wrapping length of the left side a1 remains unchanged and the wrapping is contributed solely by the growth of a2, that is, da = da2. It should be noted that irrespective of whether the wrapping proceeds in the a1-direction or the a2-direction in stage II, the deformation energy would be the same. The obtained wrapping configurations are consistent with the simulation works (see Section 4).13,15 An important feature is that the total deformation energy of the ASR mode is less than that of the SF mode in stage II (Figure 2a). The rescaled deformation energy deference between the two modes at full wrapping stage is about 1.91 for the system with r = 0.60 and σ̅ = 0.50. For the typical parameters R1 = 30−100 nm and Bm = 20−200kBT, the energy difference can be 0.38−12.73kBT. Generally, for the system with a large-size NP and a flexible membrane, the deformation energy difference is small and easy to overcome by thermal fluctuation, whereas for the system with a small-size NP and a stiff membrane, the deformation energy difference is large and cannot be overcome by thermal fluctuation. Therefore, the rotation of the NP, which occurs beyond a critical wrapping fraction and is achieved by the asymmetric wrapping lengths, is more energetically favorable than the SF mode.

which is the same with previous studies.8,18−20 Note that dE/da = (dE/df)/(ac) and dE/df is obtained by the methods described above. Therefore, the unknowns in eq 8 are only ξ+i . For the NP wrapped by the cell membrane with finite size 2L, the receptor distribution and the wrapping length are determined by solving eqs 3, 4, and 8 using the front-tracking approach with the parameters listed in Table 1 unless otherwise specified.8,48 Table 1. Physical Constants D 4

2 −1

10 nm s

L

ξL

10 μm

−3

5 × 10

−2

nm

ξ̃

Bm

eRL

0.025

20kBT

15

3. RESULTS 3.1. Evolution of the Wrapping Configuration. First, we investigate the evolutions of the energetically favorable wrapping configurations during the cellular uptake process. To elucidate the influences of the rotation of the NP, we consider two wrapping modes: one is the symmetric wrapping of the NP of fixed orientation (SF mode), which has been extensively studied previously;18−22 another is the asymmetric wrapping of the rotatable NP (ASR mode), which allows the NP to rotate during the wrapping process. The rescaled deformation energies E̅ of the SF mode and the ASR mode as a function of the wrapping fraction f with r = 0.60 and σ̅ = 0.50 are plotted in Figure 2a. The wrapping lengths and the rotation angles as functions of the

Figure 2. Asymmetric wrapping of rotatable NP. (a) The rescaled deformation energy profiles for the system with r = 0.60 and σ̅ = 0.50. The full wrapping states are marked by the diamonds. (b) The evolutions of the wrapping lengths a1 and a2. The symmetric wrapping case where a1 = a2 is plotted as the dashed line. (c) The rotation angle θ of the NP as a function of the wrapping fraction f. (d) Some selective configurations of the SF and ASR modes. D


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Figure 3. Influences of the membrane tension on the wrapping behavior. (a) The critical wrapping fraction fc for the rotation of the NP vs the rescaled membrane tension σ̅. The rescaled energy changes as functions of the rotation angle θ for the systems with σ̅ = 5.0 and (b) r = 0.80, f = 51.0%; (c) r = 0.80, f = 56.0%; (d) r = 0.90, f = 56.0%.

Figure 4. Dynamic wrapping behavior of the rotatable NP. (a) dE̅ /df vs the wrapping fraction f. (b) The rescaled receptor density ξ+/ξ0 at the wrapping front as a function of the variation of the deformation energy dE/da. (c) The evolutions of the wrapping lengths as the wrapping proceeds for the system with r = 0.60, σ̅ = 0.50, and R = 40.6 nm.

3.2. Driving Force for the Rotation. To investigate the influences of the membrane tension on the transition of the symmetric−asymmetric wrapping modes, the critical wrapping fraction fc for the NP to rotate as a function of the surface tension σ̅ is plotted in Figure 3a. Generally, the critical wrapping fraction increases as the membrane tension becomes larger, which indicates that for the system with a large membrane tension, the NP tends to adopt the symmetric wrapping configuration, where the major axis of the NP remains parallel to the membrane. This is similar to the previous results that the cellular uptake of the one-dimensional nanomaterial adopts the near-perpendicular entry mode at a small membrane tension, whereas it prefers the near-parallel mode at a large membrane tension.50 Moreover, for the NP with a more irregular shape (i.e., a small r), the asymmetric wrapping occurs at a lower wrapping fraction, the reason for which will be discussed below. To explore the underlying mechanism of the rotation of the NP, the equilibrium configurations of the systems with a prescribed rotation angle θ are calculated. The wrapping lengths

a1 and a2 are varied to minimize the deformation energy, whereas the total wrapping fraction of the system is kept unchanged. The rescaled deformation energy changes, ΔE̅ , with respect to the symmetric wrapping configuration (θ = 0°) as functions of the rotation angles θ for the systems with σ̅ = 5.0 are plotted in Figure 3b−d. For the system with r = 0.80 and f = 51.0% (Figure 3b), the bending energy decreases, whereas the surface tension energy increases with the rotation of the NP. Moreover, the increase in the surface tension energy is more dramatic than the reduction in the bending energy. As a result, the total deformation energy increases as the NP rotates, and the symmetric wrapping configuration without rotation is more energetically favorable (point A in Figure 3a). However, for the system with a larger wrapping fraction, e.g., f = 56.0% for the system with r = 0.80 as shown in Figure 3c, the reduction in the bending energy due to the rotation of the NP becomes significant and the total energy decreases as the NP rotates. Therefore, the NP tends to rotate, which leads to the asymmetric wrapping configuration (point B in Figure 3a). In contrast, for the system with the same wrapping E


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The Journal of Physical Chemistry B fraction f = 56.0% and r = 0.90 (Figure 3d), although both the bending energy and the tension energy are smaller compared with those of r = 0.80 at the same rotation angle, the influence of the NP shape on the bending energy is greater than on the tension energy. The reason is that the bending energy of the wrapped region is proportional to the square of the curvature, whereas the tension energy is a linear function of the NP shape. As a result, the cost of the tension energy is larger than the reduction of the bending energy as the NP rotates, which makes the rotation of the NP energetically unfavorable (point B in Figure 3a). In a word, based on the energetic analysis, it can be found that the rotation of the NP is facilitated by the reduction in the bending energy but hindered by the surface tension energy cost. Moreover, the NP with a more irregular shape switches to the asymmetric wrapping mode at a lower wrapping fraction. 3.3. Dynamic Wrapping Behavior of the Rotatable NP. Apart from the wrapping configurations of the NP−membrane system, an equally important question is how fast the NP can be internalized by the membrane. To address the question, we investigate the dynamic wrapping behavior of the NP considering its rotation movement in this section. To determine the wrapping time of the NP, the variation in the deformation energy as the wrapping proceeds dE̅/df should be obtained first (see eq 8). dE̅/df of the SF and ASR wrapping modes are shown in Figure 4a. As indicated in the figures, the rotation of the NP after the critical wrapping fraction not only reduces the total deformation energy of the system E̅ (Figure 2a) but also eliminates the variation in the deformation energy dE̅/df (Figure 4a). Based on eq 8, the receptor density at the wrapping front ξ+ increases with the deformation energy variation dE/da, as shown in Figure 4b. Note that dE/da, i.e., (Bm/R1ac) dE̅ /df, is smaller for the wrapping of the rotatable NP than that of the NP of fixed orientation after fc, thus ξ+ of the system with the rotatable NP is also smaller than that of the NP of fixed orientation. It can be predicted that the receptor density gradient is larger for the system with the rotatable NP, which would lead to a faster diffusion of the receptors and a higher wrapping speed. This can be confirmed by the evolutions of the wrapping lengths as a function of time in Figure 4c for the system with r = 0.60, σ̅ = 0.50, and R = 40.6 nm, where the size R of the NP is defined as the equivalent radius of the NP with the circular cross section, i.e., R = ac/2π. The wrapping length a2 of the system with a rotatable NP is larger than that with a NP of fixed orientation after the critical wrapping fraction. However, the wrapping in one side of the NP would stop when f > fc in the asymmetric wrapping of the rotatable NP, which makes the full wrapping time longer than the symmetric wrapping of the NP of fixed orientation, where the wrappings of the two sides proceed simultaneously all the time. Therefore, previous studies that ignored the rotation of the NP might have overestimated the wrapping speed of the NP.18−20 To further explore the shape effect of the NP on the dynamic wrapping behavior, the full wrapping time tw as a function of the NP size R with different aspect ratios r are plotted in Figure 5. There are minimum sizes Rmin for the NP to be fully wrapped, below which the full wrapping states cannot be achieved. Note that Rmin is large for the NP with a smaller r, due to which the bending energy cost is larger to wrap the curved tips of the system with a smaller r. Moreover, there are maximum sizes Rmax beyond which the NP cannot be fully wrapped due to the shortage of the receptors. For the system with a smaller r, Rmax is slightly smaller, this is because that the wrapped area of the NP with a small r is larger than that of the NP with a large r at the full

Figure 5. Full wrapping time tw of the NP as a function of the NP size R with different aspect ratios r.

wrapping state, which requires more receptors to achieve the full wrapping state. 3.4. Influences of the Initial Orientation of the NP. It is assumed that the NP interacts first with the cell membrane from the flat side in the above analyses. However, the initial contact state between the NP and the cell membrane can vary greatly due to the complex biological environment in experiments.14,51 In this section, we investigate the influences of the initial contact angles α0 (see Figure 6a) on the wrapping behaviors. The typical wrapping pathways of the NP with an initial contact angle α0 are shown in Figure 6a. The wrapping process is divided into three stages. Generally, after contacting with the NP, the membrane wraps in the direction that costs the least deformation energy. Therefore, the wrapping proceeds along the flat side of the NP in stage I until the wrapping configuration becomes symmetric (e.g., the first row in Figure 6a), where further wrapping along the either side of the NP costs equal deformation energy. Then, the wrapping switches to the symmetric mode in stage II (e.g., the second row in Figure 6a). Followed by the symmetric wrapping stage, the wrapping again turns to the asymmetric mode after a critical wrapping fraction in stage III, as discussed above. It should be noted that in stage III, depending on whether the wrapping proceeds in a1-direction (mode A) or a2-direction (mode B), different dynamic wrapping behaviors could result; although the deformation energies of the two wrapping modes are identical. The evolutions of the wrapping lengths of mode A and mode B are shown in Figure 6b. In the full wrapping stage, a large part of the NP surface is wrapped in a2-direction in mode B, whereas the wrapping lengths in the two directions are comparable in mode A. The deformation energies of the systems increase with the initial contact angle α0 in stage I due to the more curved adhesion regions (Figure 6c). The full wrapping time tw as a function of the initial contact angle α0 is plotted in Figure 6d. For the wrapping mode B, where most of the wrapping area is achieved in one direction, the full wrapping time increases as the initial contact angle becomes larger because more receptors are needed to diffuse to the wrapping region in solely one direction (Figure 6e). However, for the wrapping mode A, where the wrapping contributions of the two sides are on the same order (Figure 6e), the change in the full wrapping time is not as obvious as for mode B. Moreover, there is a minimum full wrapping time for mode A, where the wrapping lengths of the two sides are equal approximately. It should be noted that the opportunities for the wrapping to proceed in a1-direction and a2-direction are theoretically equal, and which direction the wrapping adopts actually in the experiments is influenced by other factors, such as the thermal F


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Figure 6. Rotation-dominant receptor-mediated wrapping behavior. (a) Some selective wrapping configurations of the three stages in the wrapping process. (b) The evolutions of the wrapping lengths ai for the wrapping mode A (solid lines) and mode B (dashed lines). (c) The rescaled deformation energy E̅ as a function of the wrapping fraction f for the systems with different initial contact angles α0. (d) The full wrapping time tw and (e) the wrapping lengths in the full wrapping stage vs the initial contact angle α0.

fluctuation, the nonspecific NP−membrane interaction, and the roughness of the NP surface.13,52 The above results suggest that the dynamic wrapping behaviors of the NP depend on not only the size and shape of the NP but also the rotation and initial orientation of the NP.18−20

mediated endocytosis ranges from seconds to tens of minutes,11 whereas the simulated time of these works is usually on the order of 10−5−10−4 s.13,15 To speed up the simulation, ultrahigh receptor densities are usually adopted, e.g., 50% of the lipids were chosen to be receptors,12,13 which are unrealistic compared with the typical density of the receptors 10−4 nm−2.8 In addition, one such simulation would only provide a single point in the multidimensional parameter space (e.g., the membrane tension, the aspect of the NP, and the size of NP) that we need to sample to obtain the coupling influences of the shape, rotation, and initial orientation of the NP.54 Our continuum model (i.e., the theoretical model that couples the diffusion of the receptors and the rotation of the NP proposed in the present work) provides a promising candidate to investigate the dynamic process and related mechanisms of the receptor-mediated endocytosis of rotatable NP with long time scales and large length scales. Previous works usually investigated the wrapping of elliptical NP with a prescribed initial orientation.18−20 However, due to

4. DISCUSSION Although there are several simulation works on a similar topic using the coarse-grained molecular dynamics method, the simulations are usually computationally expensive and, moreover, the time scale and length scale of those simulations are severely limited. For example, the NP simulated in those works are typical on about 10 nm.13,15 However, various types of NPs as diagnostic and therapeutic agents, e.g., viruses (10−300 nm), micelles (10−100 nm), and nanogels (10−1000 nm),53 are larger than the simulated sizes. The common coarse-grained description is thus a computationally intractable problem. Moreover, the time scale required to complete the receptorG


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Figure 7. Comparisons between the ASR mode and the side-tip mode. (a) The rescaled deformation energy profiles. The energy profile of the tip-side mode (the red line) is the minimum of the side-oriented and tip-oriented configurations, which are represented by dashed green and magenta lines, respectively. (b) The wrapping configurations of the ASR mode and the side-tip mode. In the side-tip mode, the wrapped regions at f = 50% are plotted as blue lines in the subsequent wrapping configurations.

Figure 8. Wrapping of NP by the cell membrane with a thickness hm. (a) The rescaled deformation energy as a function of the wrapping fraction for the system with a membrane thickness hm = 0.2. Other parameters are r = 0.60, σ̅ = 0.50, rm = 0.1, rc = 0.2, and ϵ = 2.0. (b) Some selective wrapping configurations.

lines) is no longer wrapped in the tip-oriented configuration. Therefore, the evolution of the wrapping configuration in the side-tip mode investigated in previous works is not continuous and not consistent with the coarse-grained molecular dynamics simulation results.12,15,35 In the present work, by considering the wrapping history of the NP, the evolution of the wrapping configuration is determined step-by-step. Thus, as illustrated by the first row in Figure 7b, there is no sudden change in the wrapped region and the continuous evolution of the wrapping configuration can be captured. Our wrapping configurations are more consistent with the coarse-grained molecular dynamics simulation results, e.g., Figure 3 in Vácha et al.12 and Figures 4 and 5 in Chen et al.15 The improved strategy proposed in the present work to determine the wrapping pathways of the rotatable NP can be further extended to the NP with a more complex shape. In this article, the free energy of the membrane is described by the classical Canham−Helfrich model, in which the thickness of the membrane is negligible compared to the size of the NP and the NP−membrane adhesive interaction is strictly local. A more sophisticated model that takes the thickness of the membrane and the finite range of the NP−membrane interaction into consideration has been proposed by Spangler et al. (referred to as the modified model, details of which can be found in the Supporting Information)55 However, as indicated in Figure 8, the deformation energy profile does not show essential differences compared with the classical Canham−Helfrich model, i.e., the deformation energy increases slowly as the wrapping proceeds in the shallow wrapping stage, but increases

the complex biological environment, the NP can be wrapped by the membrane starting from an arbitrary initial orientation. Although experiments and simulations have demonstrated that the NP with different initial orientations would exhibit very different uptake profiles,13,14 there is still a lack of the theoretical models to investigate the influences of the initial orientation on the wrapping pathway of the NP. In the present work, we show that the endocytosis of NP with an intermediate initial orientation angle would proceed through the asymmetric− symmetric−asymmetric wrapping mode and the time to fully wrap the NP also depends on the initial orientation of the NP. It should be noted that for smaller NPs, the relaxation is expected to be rather fast, compared to the time scale of membrane deformation, which would diminish the influences of the initial orientation. Based on continuum models, the orientation changes in the cellular wrapping of elliptical NP have been investigated previously by directly comparing the energy of the side-oriented (the dashed green line in Figure 7a) and the tip-oriented (the dashed magenta line in Figure 7a) configurations.21−23 A critical wrapping fraction where the tip-oriented and side-oriented configurations are of the same energy is identified, below which the NP would adopt the side-oriented configuration while taking the tip-oriented configuration when the wrapping fraction is larger than it (the red line in Figure 7a, referred to as the side-tip mode). However, as indicated by the wrapping configurations (see the second row in Figure 7b), during the transition from the side-oriented to the tip-oriented configurations, there is a sudden change in the wrapped region, i.e., part of the NP surface wrapped in the side-oriented configuration (marked by the blue H



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noticeably near f = 50%, after which the increase rate slows down again in the deep wrapping stage.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

5. CONCLUSIONS In summary, we have theoretically investigated the receptormediated endocytosis of the rotatable NP from the viewpoints of the energetically favorable configurations and the time-dependent wrapping dynamics. By considering the evolutions of the wrapping configurations, the wrapping pathway of the rotatable NP is determined progressively based on the Canham−Helfrich model. It was found that the NP with an elliptical cross section would switch from the symmetric wrapping mode to the asymmetric one accompanied by the rotation of the NP after a critical wrapping fraction. Further studies showed that the critical wrapping fraction is lower for the NP with a more irregular shape. Moreover, the energetic analysis demonstrated that the rotation is driven by the release of the bending energy, and it is suppressed by the surface tension of the membrane. By coupling the rotation of the NP with the diffusion of the receptors, the dynamic wrapping behaviors of the rotatable NP were also analyzed. The rotation of the NP was found to accelerate the wrapping speed in the direction that the wrapping proceeds. However, because the wrapping only proceeds in specific directions in the asymmetric wrapping stage, the full wrapping time is longer than the symmetric wrapping of the NP of fixed orientation, indicating that previous studies that ignored the rotation of the NP might have underestimated the full wrapping time of the NP. Moreover, for the system with an initial contact angle, the NP would experience the asymmetric− symmetric−asymmetric wrapping sequence. These results showed that apart from the size and shape of the NP, the dynamic wrapping behaviors of the NP also depend on the rotation and initial orientation of the NP. The results in the article can provide guidelines for the design of the NP-based drug delivery systems by shedding light on the fundamental principles of the rotations of the NPs in a receptormediated endocytosis. For example, the knowledge of the evolutions of wrapping configurations might provide an alternative way to identify the wrapping stages of the NP in experiments.56 Moreover, the rotation of the NPs can also be actively controlled by the external stimuli, such as the electric and magnetic fields, to accelerate or postpone the wrapping speed based on the theoretical outcomes.57,58 As a final note, it should be pointed out that further studies can be conducted to improve the analyses in the article. The roughness of NP surface, in addition to the shape irregularity discussed in the article, would make the rotational behavior of NP more complicated. Also, it will be interesting to take the breaking and rebinding of the receptor−ligand bonds into consideration.45,46 Moreover, more sophisticated models that treat the system deformations and the diffusions of receptors in three-dimensional case are required to investigate the rotation of the NP more precisely.19,59

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Article

ORCID

Yonggang Zheng: 0000-0003-0928-919X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The supports from the National Natural Science Foundation of China (Nos. 11672062, 11772082, 11232003, and 11672063), the Key Laboratory Fund of Liaoning Province of China (No. LZ2015024), the 111 Project (No. B08014), and Fundamental Research Funds for the Central Universities are gratefully acknowledged.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.7b09619. Details of the modified model, which takes the thickness of the membrane and the range of the NP−membrane adhesive interaction into consideration, and the influences of the range of the adhesive potential (PDF) I


Article


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