Incorporation of Soft Particles into Lipid Vesicles: Effects of Particle

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Incorporation of Soft Particles into Lipid Vesicles: Effects of Particle Size and Elasticity Xin Yi, and Huajian Gao Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03184 • Publication Date (Web): 21 Nov 2016 Downloaded from http://pubs.acs.org on November 26, 2016

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Incorporation of Soft Particles into Lipid Vesicles: Effects of Particle Size and Elasticity Xin Yi and Huajian Gao∗ School of Engineering, Brown University, Providence, Rhode Island 02912, USA E-mail: [email protected]

Abstract

Introduction

The interaction between particles and lipid biomembranes plays an essential role in many fields such as endocytosis, drug delivery and intracellular traffic. Here we conduct a theoretical study on the incorporation of elastic particles of different sizes and rigidities into a lipid vesicle through adhesive wrapping. It is shown that while the incorporation of relatively small particles involves smooth shape evolution, the vesicle wrapping of large particles exhibits a discontinuous shape transition, followed by a protrusion of the vesicle membrane at infinitesimal cost of elastic deformation energy. Moreover, softer particles require stronger adhesion energy to achieve successful internalization and delay the onset of discontinuous shape transition to a higher wrapping degree. Depending on the adhesion energy, particle-vesicle size and rigidity ratios, and the spontaneous curvature of the vesicle, a rich variety of wrapping phase diagrams consisting of stable and metastable states of no-wrapping, partial-wrapping and full-wrapping are established. The underlying mechanism of the discontinuous shape transformation of the vesicle and the relation between the uptake proneness and uptake efficiency are discussed. These results shed further light on the elasticity effects in cellular uptake of elastic particles and may provide rational design guidelines for controlled endocytosis and diagnostics delivery.

Lipid vesicles, in the fields of cell mechanics and endocytosis, serve as a minimal biomimetic model of cells with reduced structural complexity. Understanding the interaction between particles and lipid vesicles is of fundamental importance not only to the unravelling of endocytic pathways and mechanisms, 1 but also to a broad range of applications concerning drug delivery, 2,3 biomedical diagnosis, 4 intracellular distribution, 5,6 virology, cell property measurement 7 and liposome stabilization. 8 Over the past decade, it has become well established through both theoretical and experimental investigations that the vesicle-particle interaction depends strongly on particle concentration, 8–15 size, 14–22 shape, 23 and surface physiochemical properties. 15,16 For example, multiple nanoparticles adsorbed on vesicles can form linear aggregation and induce large vesicle deformation. 10–12 Smaller nanoparticles require higher adhesion energy to attach onto vesicles. 17–19 An ellipsoidal nanoparticle during uptake undergoes reorientation from an initial configuration with its long axis parallel to the vesicle membrane to a late stage configuration with long axis perpendicular to the membrane. 23 Ultrafine nanoparticles functionalized with hydrophobic coatings form a stable structure embedded within the lipid bilayers. 15 Similar phenomena on particle geometry and surface properties have also been observed for particle interaction with a lipid membrane patch. 24–33,38,39 However, compared to extensive experimental and theoretical studies focusing on the interaction between vesicles and rela-

∗ To

whom correspondence should be addressed

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tively small rigid particles, there have been no systematic studies on the size and elasticity effects in the vesicle incorporation of soft particles except for a recent study focusing on the wrapping and budding of a deformable particle of few specific small size ratios with respect to the vesicle 34 and case studies on adhesive contact between vesicles. 35–37 Typical elastic particles used in the fields of drug delivery and particle incorporation into vesicles include capsules, polymersomes, polymer-coated nanoparticles, 33,38,39 and vesicular particles such as conventional liposomes, niosomes, and ethosomes. 40–48 While there have been a number of theoretical studies on the wrapping of elastic particles modeled as fluid vesicular particles or solid thin-shelled capsules by a patch of lipid membrane, 30,49–51 they become increasingly inaccurate as the particle size increases and the global deformation of the vesicles is no longer negligible. To elucidate the collective roles of particle size and elasticity in vesicle-particle interaction, here we perform the first and comprehensive theoretical study on the adhesive interaction between lipid vesicles and soft spherical elastic particles of a wide size and rigidity range. We investigate how the particle incorporation process and vesicle morphology depend on the particle size, adhesion energy, and bending rigidity ratio between the particle and vesicle membrane. Based on the energy evolution profiles during the wrapping process, phase diagrams are established to demarcate different stable and metastable states of no-wrapping, partial-wrapping and full-wrapping. The mechanism underlying a discontinuous shape transformation of vesicle wrapping around a large particle is investigated. Moreover, the effect of possible spontaneous curvature of a vesicle on particle incorporation is analyzed. The relationship between uptake proneness and uptake efficiency is also discussed. Our results provide insights into the role of elasticity in vesicle wrapping of large particles and may provide rational design guidelines for controlled endocytosis and diagnostics delivery.

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Figure 1: Axisymmetric configurations of an elastic particle wrapped by an initially spherical vesicle in a cylindrical system (r, ϕ , z). (a) The system is divided into three regions: inner free region, contact region, and free region of the vesicle. The geometry of each region is characterized by the tangent angle ψ and normalized arclength s (scaled by the vesicle radius R). For example, the normalized arclength s in the free region of the vesicle is measured from the bottom pole of the vesicle (s = 0) to the contact edge (s = lf ). (b) Schematic configurations for the states of no-wrapping with zero contact area and full-wrapping in which the wrapped particles are completely covered by the vesicle membrane except a curved membrane neck of an infinitesimal toroid radius. Fully wrapped particles can eventually pinch off from the vesicle and become internalized. Partial-wrapping corresponds to an intermediate configuration with incomplete wrapping, as shown in (a).

Theoretical Modeling Previous numerical analysis indicates that elastic vesicular and solid thin-shelled particles exhibit similar stiffness-dependent behaviors when wrapped by a patch of lipid membrane. 49,50 Here we model elastic particles also as vesicular particles (e.g., polymersomes), and consider the adhe-


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ing from the intersection between the z-axis and the contact region to the contact edge, ψ is the tangent angle, and µ and Σ are introduced to enforce the relation r˙ = cos ψ and the constraint of a fixed Ac , respectively. In our calculations, all length scales are scaled by the radius R of the vesicle. Derivations of the governing equations in the contact region, inner free region, and free region of the vesicle are listed in the Supporting Information. At a given f , the values of ψ and r corresponding to the system configuration of the lowest elastic energy are generally unknown. To obtain the information, we vary ψ and r at the contact edge in certain ranges and pinpoint the state of the minimum system energy in (ψ , r) space by solving the equations numerically together with ψ = 0 at the pole of each region, as well as the continuity of ψ and r crossing the contact edge. The numerical procedure for the governing equations is described in detail in refs 49,53,54, and the system morphology and associated Eel are obtained, The system energy is then determined as Etot = Eel − f γ A1 or Etot /κ2 = Eel /κ2 − 2π f γ¯, where γ¯ ≡ 2γ a2 /κ2 . Depending on the wrapping degree f , the system can exhibit three characteristic wrapping states: nowrapping ( f = 0), partial-wrapping (0 < f < 1), and full-wrapping ( f → 1).

sive wrapping of an initially spherical elastic particle of radius a by a unilamellar vesicle of radius R. Both particle and vesicle are of fixed surface areas with vanishing osmotic pressure, and subject to axisymmetric elastic deformation as illustrated in Figure 1a. In our model, quantities pertaining to the vesicular elastic particle and the wrapping lipid vesicle are identified by subscripts 1 and 2, respectively. The total system energy is assumed to obey the Canham-Helfrich model as Etot = Eel − γ Ac , where ∫ ∫

Eel = 2κ1

M12 dA1 +

κ2 2

(2M2 −C0 )2 dA2

is the system elastic energy with Mi , κi , and dAi (i = 1, 2) representing the mean curvature, bending rigidity, and surface element of the particle and the vesicle, respectively; 24,49,53,54 C0 is the spontaneous curvature of the vesicle membrane; γ (> 0) is the adhesion energy and Ac the surface area of the contact region. The initial √radii of the elastic particle and vesicle are a = A1 /(4π ) and R = √ A2 /(4π ), respectively. At a certain Ac or wrapping degree f = Ac /A1 , the bending energy of the system is Eel ≡ Etot + γ Ac = Etot + f γ A1 . Introducing a reduced spontaneous curvature c0 = C0 R, we can obtain Eel /κ2 as a function of a/R, κ1 /κ2 , and c0 . We assume that the particle and the vesicle form perfect contact and they can slide against each other freely, 52 so that the bending rigidity of the contact region can be approximated as κ1 + κ2 . Variation of Eel then gives rise to the governing equations for the system morphology. 24,49,53,54 For example, the shape of the contact region of a fixed Ac is given by the governing equations

Results We now investigate the effects of stiffness and size on the incorporation of an elastic particle into a lipid vesicle. Figure 2a-c plots the elastic energy 0 as a function of the wrapchange ∆Eel = Eel − Eel ping degree f in the case of zero spontaneous curvature, where Eel ≡ Etot + γ Ac (see Methods for the expression of Etot ) denotes the system elastic 0 = 8π (κ + κ ) is the reference enenergy and Eel 1 2 ergy before the particle contacts the vesicle. For a vesicle with a small reduced spontaneous curva0 = 8πκ + 2πκ (2 − c )2 and the vesiture c0 , Eel 1 2 0 cle remains spherical when free. For a relative small particle, the energy curve ∆Eel ( f ) is smooth (Figure 2a,b and Figures S1a and S2a in the Supporting Information), indicating that the wrapping configuration evolves continuously, as illustrated in Figure 2d,e. For the incorporation of rigid particles into a vesicle of c0 = 0, the energy profiles ∆Eel

[ ( ) 1 sin ψ ψ¨ = −2 ψ˙ − cos ψ 2r r ] ) ( κ1 −1 + 1+ µ sin ψ , κ2 ( )( )( ) κ1 sin ψ sin ψ µ˙ = 1 + ψ˙ + ψ˙ − κ2 r r 2 + 2c0 ψ˙ + c0 + Σ, r˙ = cos ψ ,

z˙ = sin ψ ,

where dots denote derivatives with respect to the rescaled arclength s of the contact region measur-


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Figure 2: Evolution profiles of elastic deformation energy and particle-vesicle interaction configurations for c0 = 0. (a-c) Elastic energy ∆Eel as a vesicle wraps around a particle with increasing wrapping degree f for different size ratios a/R and bending rigidity ratios κ1 /κ2 . (d-f) Selected wrapping configurations at a/R =0.1, 0.3 and 0.4; κ1 /κ2 = ∞, 1 and 0.1; and various wrapping degrees indicated in the figure. the energy curve ∆Eel ( f ) exhibits a kink at a critical wrapping degree fc , followed by an energy plateau. The kink in ∆Eel ( f ) corresponds to a discontinuous shape transformation of the vesicle from a uniconcave shallow bowl-shaped stomatocyte into an almost closed cup (Figures 2c,f, S2b and S3b), and the plateau characterizes wrapping through propagation of a membrane protrusion during which the particle (regardless of bending rigidity) remains nearly spherical in shape. As the high curvature region of the narrow membrane neck contribute negligible membrane bending energy, 55 the protrusion part of the vesicle propagates at almost no change in bending energy (see Figures 2f, S1a and S3). In the case of a rigid particle and c0 = 0, no kink is observed until a/R exceeds ρc ≈ 0.36 (Figure S1a), and as c0 increases, both ρc and fc increase (Figure S1b). Further numerical analysis indicates that softer particles correlate with larger ρc and fc (Figures 2c and S2b). In other words, the stiffer and larger the particle is, the longer the plateau.

for a/R < 0.35 shown in Figure S1a are similar to those reported in refs 16,22. As the particle stiffness decreases, the slope d(∆Eel )/d f decreases in the early stage and increases in the late stage of incorporation (see Figure 2a-c), which indicates that, compared to rigid particles, the incorporation of softer particles requires smaller adhesion energy in the early stage of wrapping but larger adhesion energy toward the late stage of wrapping. This is also reflected in the morphologies of the particle and the vesicle. As κ1 /κ2 decreases, the particle deforms more while the vesicle deforms less in the early stage of wrapping, and the reverse is true in the late stage of wrapping (Figure 2d,e). Similar phenomena were previously observed in the wrapping of elastic vesicular particles and solid capsules by a membrane patch, 49,50 and the underlying mechanism is attributed to a partition of elastic deformation energy between the particle and membrane patch (vesicle in the current study). As the size ratio a/R reaches a critical value ρc ,


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the case of κ1 /κ2 = 0.1 the dotted segment corresponding to the vesicle configuration of an almost closed cup are very small and almost invisible. An interesting question is whether there exists a transient state or an evolution trajectory between these two metastable configurations enforcing a continuing evolution of the vesicle volume and the incorporation depth of the particle. To explore the possibility of that transient state, we employ numerical optimization 30,56 and perform additional calculations on the vesicle wrapping of a rigid spherical particle of size a/R > ρc at different incorporation depths (see Figure S4). As shown in Figure S4, only the almost closed cup and the uniconcave stomatocyte configurations of the wrapping vesicle are observed during the particle incorporation. The theoretical determination of the most possible transient scenario between the metastable wrapping states is challenging and requires further investigation. Besides the energy profiles, the particle incorporation process could also be characterized by the evolution of vesicle volume V , as shown in Figure 3. As f increases, the vesicle volume decreases first and then increases as f exceeds a certain value till the particle becomes fully internalized. The larger the a/R, the smaller the minimum value of V /(4π R3 /3). For small particles with a/R < ρc (e.g. a/R = 0.1 in Figure S5, and a/R = 0.3 in Figure 3a), V evolves continuously and smoothly, but it exhibits a discontinuous jump at f = fc for large particles with a/R ≥ ρc (Figure 3b). Compared to the wrapping of rigid particles, a vesicle internalizing a softer particle has a larger volume V in the early and middle stages of wrapping, but a smaller V in the late stage of wrapping. At full wrapping ( f → 1), the vesicle adopts the shape of an almost closed inverted spherical cup and V /(4π R3 /3) → 1 − (a/R)3 , independent of the particle stiffness, as shown in Figures 2 and 3. Figure 4 shows typical profiles of the total system energy ∆Etot = ∆Eel − γ Ac as a function of the wrapping degree f at c0 = 0 and 9 other selected system parameters (κ1 /κ2 , a/R, γ¯). The behavior of ∆Etot ( f ) indicates that there exist five possible wrapping phases (I to V), depending on the stability of no-wrapping, partial-wrapping and full-wrapping states. In phase I, which is usually correlated with small adhesion energy γ¯, ∆Etot increases mono-

Though ∆Eel ( fc ) is higher at a larger a/R, it is insensitive to κ1 /κ2 , which is consistent with the fact that similar vesicle-particle morphologies are observed at different values of κ1 /κ2 (Figures S2b and S3b). (a)

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Figure 3: Reduced volume V /(4π R3 /3) of a lipid vesicle of radius R wrapping around an elastic particle of radius a as a function of the wrapping degree f at a/R =0.3 (a) and 0.4 (b) in the case of vesicle having zero spontaneous curvature c0 = 0. Dotted segments in (b) at f < fc and f > fc correspond to the volumes of the wrapping vesicles with configurations of the almost closed cup and the shallow bowl-shaped uniconcave stomatocyte, respectively. Compared with energetically favorable solid lines in Figure 2c, dotted segments at f < fc and f > fc correspond to metastable wrapping states in which the vesicle exhibits an almost closed cup and a uniconcave stomatocyte, respectively. Note that in


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Figure 4: Typical profiles of the total energy change ∆Etot as a function of f at 9 selected sets of (κ1 /κ2 , a/R, γ¯) and c0 = 0. Depending on the stability of no-wrapping, partial-wrapping and full-wrapping states, the behavior of ∆Etot ( f ) can be categorized into five wrapping phases, I to V. Phase I, a stable nowrapping state with a single energy minimum at f = 0; phase II, coexistence of a stable no-wrapping state and a metastable partial- or full-wrapping state; phase III, a stable partial-wrapping state or coexistence of a stable partial- and a metastable full-wrapping states; phase IV, coexistence of a metastable state of no- or partial-wrapping and a stable full-wrapping state; phase V, a stable full-wrapping state with a single energy minimum at f = 1. Depending on the values of the set parameters, the global energy minimum in phase III-1 with two local minima could be located at a relatively either low or high f . Here in the central subfigure, only the former case is presented as an example. Depending on the values of the set parameters, the global energy minimum in phase III-1 with two local minima could be located at a relatively low or high f . Here in the central subfigure, only the former case is presented as an example. In each profile, the underlined wrapping state indicates the phase of lowest system energy. Magnitudes of ∆Etot are rescaled to a unified value. As indicated in Figure 5, some wrapping phases might not be observable at a certain κ1 /κ2 . tonically with f and the no-wrapping state prevails. As γ¯ increases, phase II arises in which a stable state of no-wrapping and a metastable state of partial- or full-wrapping coexist. For soft particles (e.g. κ1 /κ2 =1 and 0.1), increase in γ¯ could also lead to a global minimum at a state of partialwrapping and a possible energy barrier to reach a metastable state of full-wrapping (phase III). In phase IV there is a global minimum at a state of full-wrapping as well as a metastable state of no- or partial-wrapping. If γ¯ is large enough, a

stable full-wrapping state arises with a single energy minimum at f = 1 (phase V). For particles of size a/R > ρc , the wrapping vesicle exhibiting an almost closed cup f < fc and a uniconcave shallow bowl-shaped stomatocyte at f > fc is in an energetically unfavorable state, as analyzed in Figure 2c. As the slopes d(∆Eel )/d f in the metastable states of f < fc are smaller than the maximum of d(∆Eel )/d f in the energetically favorable solid lines (Figure 2c), we do not consider the effects of these metastable wrapping states on the


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Figure 5: Wrapping phase diagrams with respect to the normalized adhesion energy γ¯(≡ 2γ a2 /κ2 ) and particle-vesicle size ratio a/R at different bending rigidity ratios κ1 /κ2 = ∞, 5, 1 and 0.1 in the case of c0 = 0. Typical system energy profiles of phases I to V are exemplified in Figure 4. square of the particle radius a. If γ itself is employed, instead of γ¯, we would find that larger particle requires less adhesion energy γ to achieve full wrapping. Based on the stability of the wrapping states, larger a, γ , κ1 and lower R and κ2 would facilitate the full internalization of the particle. Since ∆Etot ( f ) for c0 = ±0.2 exhibit similar trends to those for c0 = 0 (Figure S6), the wrapping phase diagrams at relative small spontaneous curvatures c0 = ±0.2 in Figure S7 exhibit similar structures to Figure 5. As polymersomes exhibit similar features as vesicular particles in their mechanical behaviors and water permeation to some extent, much of our analysis above should be applicable to polymersomes.

wrapping phase diagrams here. With the knowledge of energy profiles ∆Etot ( f ) for c0 = 0, the wrapping phase diagrams at different κ1 /κ2 are determined in the parameter space of a/R and γ¯ (Figure 5). For simplicity, we focus on the stable wrapping states, and do not distinguish between subcategories in phases II, III and IV. For very small rigid particles, the minimum adhesion energy necessary for partial wrapping is γ¯min = 4. The same value has been obtained in the case of particle interaction with a membrane patch, 17,18,49 which is expected as γ¯min only depends on the particle size and local deformation of the (vesicle) membrane in the vicinity of the contact region. Note that there is no phase III in the wrapping phase diagram of a rigid particle. As κ1 /κ2 decreases, γ¯min decreases because the deformation of the vesicle is reduced by the flattening of the particle during adhesion, and phase III emerges and expands in the phase diagram. Note that stiffer particles require less adhesion energy γ¯ to attain full wrapping (phase V). This phenomenon is reflected in the elastic energy profiles in Figures 2a-c, S1a and S2, where softer particles exhibit larger maximum slope in the energy profiles, which has also been previously observed in the wrapping of vesicular elastic nanoparticles 49 or solid thin-walled nanocapsules 50 by a membrane patch. Compared to the wrapping phases in the case of interaction between a particle and a membrane patch, new coexistences of no- and fullwrapping states (II-2 and IV-1) are observed in the current case of wrapping by vesicles. This is attributed to the higher energy cost of vesicle deformation during particle incorporation. Note that γ¯(≡ 2γ a2 /κ2 ) in Figure 5 is proportional to the

Discussion Several methods have been proposed and developed to measure the bending rigidity κ of lipid membranes, including indentation by atomic force microscopy, optical stretching, micropipette aspiration, electrodeformation, fluctuation analysis and membrane relaxation. 57,58 As demonstrated in Figure 2d-f, the system configuration at a given wrapping degree depends on the size ratio and bending rigidity ratio between the particle and the vesicle. With the knowledge of the size ratio and the bending rigidity of either the vesicular particle or the vesicle, the κ of the other can be determined by comparing the theoretical and experimental results. This new deformation-based approach can be a valuable complement to the existing measurement techniques mentioned above, especially in the case of soft and large particles with weak to intermediate adhesion energy,


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where the morphological deformation of both particle and vesicle is striking. For example, a comparison between Figure 2e in the present study and Figure 10 in ref 59 suggests that the bending rigidities of the lipid membrane composed of sphingomyelin:cholesterol (SPM:CHOL) in ratio 2:1 and dioleoylphosphatidylcholine (DOPC) membrane are on the same order, with ratio moderately larger than 1, which is consistent with the experimental measurement. 57 Note that volumes of the vesicles in Figure 10 in ref 59 are controlled by the gain or loss of osmotically active solutes, 60 while our results are restricted to the case of free volume variation. Therefore, a more precise prediction of κ is expected when this deformation-based approach is employed with a theoretical input of the vesicle volume suggested by experimental observation. As demonstrated in Figures 2d-f and S3, a vesicle wrapping around a particle of a relatively large a/R would transform from a uniconcave shallow bowl-shaped stomatocyte to the shape of an almost closed cup and remains in that configuration till full internalization (Figure 2f). Such configurational transformation of the vesicle into a closed cup must occur beyond a certain a/R, and can be understood as follows. Theoretical analysis shows that a free-standing vesicle of relatively small c0 as considered in the present study tends to adopt a cup-shape at reduced volumes smaller than V /(4π R3 /3) ≈ 0.6 (see Figure 10 in ref 53). It can be confirmed that the observed values of V /(4π R3 /3) in the wrapping of a rigid particle of large a/R are indeed around that range and the sizes of the adhesive particle and cup-shaped vesicle match each other, we can expect that the almost closed cuplike vesicle shape as predicted in our results are energetically favorable. As the particle stiffness decreases, the elastic energy change becomes smaller. Therefore, the vesicle wrapping around a soft particle of large a/R would adopt a cup-shape at higher reduced volumes as indicated in Figure 3b. In the cases of large c0 , theoretical analysis indicates that a free-standing vesicle would adopt various shapes including prolates, pear-shapes and spherical shape with a necklacelike protrusion 61 as well as cuplike shapes at small V /(4π R3 /3). 53 Therefore, a vesicle of large c0 wrapping around the particle of large a/R might undergo shape transformation into prolates and pear-

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shapes or cuplike shapes, depending on the sign and magnitude of c0 . This deserves further detailed investigations in the future. The present and our previous studies based on the free energy show that softer particles requires stronger adhesion energy to achieve successful internalization. 30,49,50 In other words, softer particles are energetically less prone to full wrapping than stiffer ones. This stiffness-enhanced uptake proneness has recently been observed in the interaction between lipid-covered polymeric nanoparticles of radius 40 nm and HeLa as well as endothelial cells. 46 While softer liposomes merely attach onto a cell membrane after 5-h incubation, stiffer lipidcovered nanoparticles undergo successful internalization. 46 As the cell uptake of a single nanoparticle occurs usually on a time scale in the range of tens of seconds to tens of minutes, 62 these experimental results clearly demonstrate the stiffnessenhanced uptake proneness. Besides studies on stiffness-dependent uptake proneness, there have been investigations on the stiffness effect on the cell uptake rate. Cell uptake typically involves hydrodynamic effects of cell membrane, ligand diffusion, reaction of ligand-receptor binding, and protein coat assembly, each having its own time scale. Therefore, it is not surprising that softer particles in experiments exhibit higher or lower uptake rates, depending on the particle sizes, material compositions and cell types. 41,43–47 These two distinct aspects, i.e. the uptake proneness and uptake rate, may seem at the first glance to be somewhat confusing, as the free energy of wrapping might be involved in the determination of both uptake proneness and uptake rate. We emphasize that the uptake proneness characterizes the tendency of an elastic nanoparticle to be fully internalized. 30,49,50 In this sense, the uptake proneness is governed by the free energy in a conclusive and time-independent way and cannot be employed alone and straightforwardly to describe time related quantities such as the wrapping time and uptake rate. In our previous 30,49 work and the present theoretical study, an internalized elastic nanoparticle is modeled as a homogeneous vesicular nanoparticle to facilitate a minimal physical model of particle deformation in the wrapping process. This approximation is not unreasonable in the case of


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liposome composed of a single species of phospholipid. For general vesicular nanoparticles composed of a mixture of different species of lipids, proteins and/or small molecules, the composition of the vesicle plays an important role in regulating the mechanical behavior of nanoparticles during the wrapping process. For example, recent molecular dynamics simulations demonstrate that at a low surface density the ligand molecules would undergo diffusional aggregation, leading to the formation of a ligand-free domain and incomplete cell uptake. 44 Addition of cholesterol to POPC liposomes can stabilize the liposome membrane and increase its stiffness, which in turn increases the uptake proneness. For liposomes with membrane consisting of two chemically different monolayers, a non-zero spontaneous curvature might need to be introduced in the theoretical modeling to capture the asymmetry in bilayer. 19 Similar theoretical approaches can be employed to model liposomes subject to insertion or adsorption of small molecules into or onto the external liposome monolayer. Polymersomes as a relatively new class of artificial polymer vesicles exhibit similar bending property as lipid vesicles but are of orders of magnitude higher lysis tension and larger surface viscosity. 63 Therefore, polymersomes are generally more mechanically stable in comparison with liposomes, and can serve as another kind of ideal drug-delivery agents. Moreover, the membrane thickness of polymersomes can be controlled by tuning the molecular weight of building blocks. The higher the molecular weight of hydrophobic blocks, the thicker and tougher the polymersome membrane. More specific and comprehensive discussion on the mechanical properties of polymersomes can be found in ref 63. The thickness dependence of particle stiffness is also a characteristic feature for the thin-shelled solid capsules. 50 For polymer-coated nanoparticles, the mechanical properties as well as the cell-nanoparticle interaction modes strongly depend on the length, grafting density and hydrophilicity of polymer chains. 33,38,39 The water permeability of vesicles depends on the packing of the composing chains and temperature, and can also be enhanced by the incorporation of water channel proteins and peptides. 64–66 Here we have focused on the case of zero pressure difference, which is valid for high permeable

vesicles or particle incorporation at a long time scale. For vesicles and polymersomes of low permeability or in the case of fast particle incorporation, the vesicle volume can be approximated as constant as water permeation could not occur instantaneously. 10 Depending on the value of the reduced volume, a free-standing vesicle subject to volume constraint can exhibit axisymmetric shapes such as prolates, oblates and stomocytes, 53 as well as asymmetric shapes such as a starfish. 67 Particle incorporation into these vesicles might induce dramatic shape transformations of the vesicle and even break the vesicle symmetry to achieve a state of lower energy. Theoretical studies based on the continuum Helfrich membrane theory have been carried out to investigate the adsorption of a rigid spherical particle of a specific size onto a vesicle of different reduced volumes under the restriction of an axisymmetric configuration. 56 Further studies have been conducted on the size effect of particle adsorption onto an oblate-shaped vesicle, in which an axisymmetric vesicle deformation is presumably energetically favorable. 20 Recently, theoretical analysis considering the wrapping of multiple nanoparticles at nonaxisymmetric positions of prolate, oblate and stomatocyte vesicles has been performed. 68 Numerical studies have been also performed on a rigid particle interacting with a vesicle of large reduced volume 11 or zero osmotic pressure 23 and on multiple particle adsorption on a vesicle of zero osmotic pressure difference, 12 where asymmetric vesicle deformation is captured by a triangulated vesicle model. Comprehensive wrapping and morphological phase diagrams taking into account more general vesicle deformation induced by adhesive particles remain to be fully elucidated.

Conclusions We have conducted a theoretical analysis to investigate the incorporation of an elastic particle of different size and rigidity into a lipid vesicle of fixed surface area and vanishing osmotic pressure. It is shown that the vesicle volume decreases first and then increases as the wrapping proceeds beyond a certain degree till full internalization. When incorporating a small particle, the


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vesicle volume evolves continuous and the vesicle undergoes smooth shape transformation. In contrast, a vesicle wrapping a large particle exhibits a discontinuous shape transition from a uniconcave shallow bowl-shaped stomatocyte into an almost closed cup. The discontinuous shape transition occurs at a larger wrapping degree and a larger particle-vesicle size ratio for a softer particle. Due to the elastic energy partition between the elastic particle and the wrapping vesicle, a softer particle exhibits the smaller slope in its free energy profile as a function of the wrapping degree in the early wrapping stage and larger slope in the late wrapping stage. Therefore, softer particles are more prone to partial wrapping but require stronger adhesion energy to achieve successful internalization. Depending on the adhesion energy, size and rigidity ratios between the particle and vesicle, as well as the spontaneous curvature of the vesicle, a variety of wrapping phase diagrams consisting of the stable and metastable states of no-wrapping, partial-wrapping and full-wrapping have been determined. Moreover, the relation between the uptake proneness and uptake efficiency has been discussed. Since polymersomes exhibit similar mechanical features as lipid vesicles, similar elasticity effects on particle incorporation into polymersomes are expected. Our results shed light on the size and elasticity effects in cell uptake of nanoparticles and may provide rational design guidelines for controlled endocytosis and diagnostics delivery. Acknowledgement. This work was supported by the National Science Foundation (Grants CBET1344097 and CMMI-1562904). Supporting Information Available: Supplemental figures are presented on additional morphologies of particle-vesicle system in the case of zero membrane spontaneous curvature and the energy profiles, wrapping phase diagrams at finite spontaneous curvatures. Note added. During the review of our manuscript, we became aware of the theoretical study of Tang et al. 34 on a similar topic. Their work focused on the elasticity effects on both adhesive wrapping and budding of particles of few specific small size ratios with respect to a vesicle of vanishing spontaneous curvature. In our work, we investigated the effects of particle size, stiffness and spontaneous curvature of the vesicle on the adhesive wrapping,

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with results showing that the vesicle wrapping of large particles exhibits a discontinuous shape transition.

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incorporation degree

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rigid particle

soft particle


Incorporation of Soft Particles into Lipid Vesicles: Effects of Particle

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