Wrapping nonspherical vesicles at bio-membranes

Abstract

The wrapping of particles and vesicles by lipid bilayer membranes is a fundamental process in cellular transport and targeted drug delivery. Here, we investigate the wrapping behavior of nonspherical vesicles, such as ellipsoidal, prolate, oblate, and stomatocytes, by systematically varying the bending rigidity of the vesicle membrane and the tension of the initially planar membrane. Using the Helfrich Hamiltonian, triangulated membrane models, and energy minimization techniques, we predict multiple stable-wrapped states and identify the conditions for their coexistence. Our results demonstrate that softer vesicles bind more easily to initially planar membranes; however, complete wrapping requires significantly higher adhesion strength than rigid vesicles. As membrane tension increases, deep-wrapped states disappear at a triple point where shallow-wrapped, deep-wrapped, and complete-wrapped states coexist. The coordinates of the triple point are highly sensitive to the vesicle shape and stiffness. For stomatocytes, increasing stiffness shifts the triple point to higher adhesion strengths and membrane tensions, while for oblates, it shifts to lower values, influenced by shape changes during wrapping. Oblate shapes are preferred in shallow-wrapped states and stomatocytes in deep-wrapped states. In contrast to hard particles, where optimal adhesion strength for complete wrapping occurs at tensionless membranes, complete wrapping of soft vesicles requires finite membrane tension for optimal adhesion strength. These findings provide insights into the interplay between vesicle deformability, shape, and membrane properties, advancing our understanding of endocytosis and the design of advanced biomimetic delivery systems.

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1 Introduction

Cellular transport of information and materials, a fundamental process that ensures the survival and functionality of cells, is typically facilitated by the exchange of particles across lipid membranes.^1,2 The translocation across lipid-bilayer membranes depends on the size, elasticity, shape, and surface functionalization of the particles.³ Particles smaller than the thickness of the membrane can translocate, causing minimal membrane deformation,^4,5 while particles larger than a few nanometers translocate through endocytosis.⁶ Passive endocytosis is a multistep process that begins with the attachment of the particle to the membrane, followed by wrapping and finally detachment from the membrane.⁷ The particle wrapping process, a key aspect of endocytosis, is ubiquitous for both living and nonliving particles, such as drug-carrying nanoparticles,⁸ viruses,⁹ and parasite invasion into host cells.¹⁰ Membrane tension, regulated by cells through remodeling, plays a critical role in endocytosis: high tension slows the internalization process, while low tension facilitates membrane invagination and efficient particle uptake.^11–13 Understanding the particle wrapping process can help us optimizing biomedical applications, such as targeted drug delivery using engineered nanoparticles.^14–16

The wrapping of hard spherical particles at lipid membranes has been extensively investigated in experimental and theoretical studies.^17–21 In general, for the wrapping of hard particles, the size and shape of the particle, the bending rigidity and tension of the membrane, and the adhesion strength of the particle–membrane have been shown to primarily control the process.^22–29 In addition, the initial orientation of the anisotropic hard nanoparticles relative to the adhering membranes influences their wrapping behavior. For example, the wrapping of ellipsoidal^23,30 and spherocylindrical³¹ nanoparticles starts with an orientation in which the major axis of the particle is parallel to the membrane surface. After half-wrapping, the orientation of the particle changes, and the major axis becomes perpendicular to the membrane surface. Unlike hard particles, soft particles can deform depending on external constraints;^32,33 the deformability plays an important role in the wrapping process, in addition to the factors that apply to hard particles.

Many biologically relevant particles have anisotropic or nonspherical shapes.^34,35 For example, malaria parasites are typically egg-shaped,¹⁰ mature virions can be ellipsoidal or brick-shaped,^36,37 and vesicular stomatitis viruses (VSV) have bullet shapes.³⁸ Simulations and experiments show that for ellipsoidal particles the stability of the partial-wrapped over non-wrapped and complete-wrapped states decreases when the bending rigidity of the membrane increases.³⁹ Living particles also often have the ability to change shape and deformability throughout their life cycles. For example, murine leukemia virus (MLV) and human immunodeficiency virus (HIV) adjust their deformability through internal structural changes, significantly affecting their entry into host cells.^40,41 The mature HIV viral particles are less stiff compared to the immature HIV viral particles, which hinders their complete wrapping at the host cells.⁴¹

A wide range of synthetic deformable particles with various architectures and customizable mechanical properties can be engineered for the targeted delivery of drugs, including star polymers,^42,43 microgels,^44,45 dendrimers,^46,47 polymer-grafted nanoparticles,^48,49 and vesicles.^8,27,50 The deformability of these particles can be tuned through control parameters that change their molecular architecture or building blocks. For example, the deformability of microgels can be adjusted by changing the cross-link density and electric charge,^51,52 of polymer-grafted nanoparticles by altering the density and length of grafted polymers,⁴⁸ and of unilamellar fluid vesicles by modifying the bending rigidity and the osmotic pressure difference.⁵³ A particularly versatile and well-characterized class of deformable particles is unilamellar vesicles.²⁷ Free vesicles with fixed membrane areas, variable volumes, and symmetric lipid bilayer membranes assume a spherical shape and may change volumes significantly during wrapping. Nonspherical vesicles with volumes smaller than the maximal volume that can be enclosed by the vesicle membrane and potentially even asymmetric monolayer composition can have a complete zoo of shapes.⁵⁴

In this work, we investigate the wrapping of single nonspherical vesicles with initial ellipsoidal, prolate, oblate, and stomatocytic shapes at initially planar membranes by systematically varying the relative stiffness of the vesicle membranes and the tension of the planar membranes. Using triangulated membranes and energy minimization techniques, we predict energetically stable wrapping states. The systematic increase in adhesion strength leads to transitions of the vesicles from non-wrapped to complete-wrapped states through intermediate shallow-wrapped and deep-wrapped states. Softer vesicles undergo both shape and orientation changes during wrapping, whereas stiffer vesicles exhibit only orientation changes. We calculate wrapping diagrams showing the energetically stable states for various membrane bending rigidities and tensions. As membrane tension increases, the deep-wrapped state becomes destabilized and eventually vanishes at a triple point, where shallow-, deep-, and complete-wrapped states coexist. Beyond the triple point, vesicles exhibit a discontinuous transition from the shallow-wrapped state to the complete-wrapped state. For initial stomatocytes, the triple point shifts to higher membrane tensions and adhesion strengths as the stiffness of the vesicles increases, while the trend is the opposite for initially oblate ones. In addition, we show that the optimal adhesion strength for the complete wrapping of soft oblate, prolate, and ellipsoidal vesicles occurs at a non-zero membrane tension, unlike rigid particles, which require zero membrane tension. This behavior is related to the shape change of the vesicles during wrapping. Our findings indicate that the initially stomatocyte, oblate and prolate vesicles preferred to be oblate in the shallow-wrapped states and stomatocyte in the deep-wrapped states.

The rest of the manuscript is organized as follows. In Section 2, we introduce the model and the numerical calculation techniques. In Sections 3.1 and 3.2, we show the deformation energy landscapes along with the typical vesicle shapes and characterize the vesicle shapes at various wrapping stages. In Section 3.3, we study the effect of vesicle deformability and in Section 3.4 of membrane tension on the stability of the wrapping states. Finally, in Section 4, we summarize and conclude our study.

2 Model and methods

To study particle wrapping in passive endocytosis, we employ a continuum membrane model, which is applicable to systems where the particle size is at least a few times larger than the thickness of the lipid bilayer membrane.^55,56 In a continuum membrane model, the total energy of the system is calculated using the Helfrich Hamiltonian,^57,58Here, H = (c₁ + c₂)/2 is the mean curvature, c₁ and c₂ the principal curvatures, A_v the total area of the vesicle, and A_p the area of the initially planar membrane. The adhered area A_ad is the area over which the vesicle and the initially planar membrane are attached to each other with adhesion strength w, see Fig. 1(a). The bending rigidities of the vesicle and planar membranes are κ_v and κ_p, respectively, and the tension of the initially planar membrane is σ. The membrane area A_v and the volume V_v of the vesicle are conserved using the Lagrange multipliers σ_v and p_v, respectively. The tension energy of the vesicle is omitted because its membrane area remains constant throughout the wrapping process.

Fig. 1 (a) Schematic representation of a partial-wrapped prolate vesicle at an initially planar membrane. (b) Bending energy of vesicles as a function of reduced volume.

The nonspherical shapes of the vesicles are characterized by reduced volume v = V_v/V_s, see Fig. 1(b), and Fig. S1 in the ESI.† Here, V_v is the actual volume of the vesicle with area A_v, and V_s is the volume of a spherical vesicle with the same membrane area. The maximal value, v = 1, corresponds to a spherical vesicle. The values of Lagrange multipliers σ_v and p_v depend on the vesicle shape; sudden jumps in their values indicate shape transitions; see Fig. S2 in the ESI.† The stable states for free vesicles with 0.6515 ≤ v < 1 are prolate, with 0.5915 ≤ v ≤ 0.6515 are oblate, and with v ≤ 0.5915 are stomatocytes. We refer to vesicles with v = 0.95 as ellipsoidal vesicles.⁵⁹

The wrapping fraction f_w = A_ad/A_v represents the fraction of the vesicle area that is attached to the initially planar membrane. The value of f_w varies in the range 0 to 1, where f_w = 0 corresponds to the non-wrapped (NW) state and f_w = 1 to the complete-wrapped (CW) state. The shallow-wrapped (SW) state is defined in the range 0 < f_w ≤ 0.5, and the deep-wrapped (DW) state in the range 0.5 < f_w < 1. In this work, we are focusing in particular on the wrapping behavior of four vesicle shapes: stomatocyte (v = 0.55), oblate (v = 0.63), prolate (v = 0.66) and ellipsoidal (v = 0.95). The selection of vesicle shapes at the phase boundaries between the stomatocyte, oblate, and prolate can result in complex wrapping behaviors compared to prolate vesicles with reduced volumes v > 0.7, as previously reported in the literature.³²

All integrals in eqn (1) are discretized using triangulated surfaces composed of vertices, edges, and facets.^60,61 We initialize the energy and shape calculations with a few large triangles and then iteratively minimize the energy and refine the triangulation until the desired accuracy in energy is achieved; see Fig. S3 in the ESI.† The iterative procedure is important to avoid the need for major shape changes of finely discretized membranes, which would require a high number of minimization steps. Small triangles are used where the membrane is highly curved, and large triangles are used where it is almost planar. In addition, the edges are regularly swapped to maintain the fluidity of the membranes,⁶² and vertex-averaging steps ensure that neighboring triangles have similar areas. The numerical calculations are performed with the help of the freely available software Surface Evolver.⁶³

For each of our simulations, the membrane area A_v and the volume V_v of the vesicle are fixed, while the area A_p of the initially planar membrane is determined through energy minimization, which is appropriate for an ensemble with constant membrane tension. The simulations are performed by setting w = 0 and using f_w as a constraint via a Lagrange multiplier . The total energy of the system can be expressed in terms of reduced units as

where [small sigma, Greek, tilde] = σA_v/(2πκ_p) is the reduced membrane tension. The bending rigidity ratio κ_r = κ_v/κ_p characterizes the relative deformability of the vesicle with respect to the initially planar membrane.

We perform simulations for different values of f_w in the range 0 < f_w < 1. For each value of f_w, we initialized the energy minimization with different initial configurations. During this process, the vesicle can adjust its orientation and shape, while the outer patch of the initially planar membrane is constrained to remain in a plane. We calculate the reduced deformation energy as

Here, ΔẼ_b,v, ΔẼ_b,p, and ΔẼ_s,p represent changes in the bending energy of the vesicle membrane, the bending energy of the initially planar membrane, and the surface energy of the initially planar membrane, respectively. The adhesion energy term is added to eqn (3), resulting in the total deformation energy as

ΔẼ[f_w] = ΔẼ′[f_w] − w̃f_w, (4)

where w̃ = wA_v/(πκ_p) is the reduced adhesion strength. To identify stable and unstable states, we fit the deformation energy ΔẼ at w̃ = 0 using a fourth-order polynomial function. Since the energy curve is highly non-monotonic, constructing a single fitting function is challenging. Therefore, we employ a piecewise fitting approach for different ranges of f_w. We apply different adhesion strengths (w̃) to the fitted curves, and find the different global minima in the energy landscape to determine the stable states.³²

3 Results and discussion

3.1 Wrapping states and deformation energies

Fig. 2 illustrates the wrapping states of nonspherical vesicles at initially planar membranes for various wrapping fractions, f_w. The wrapping process is consistently initiated in regions of the vesicle with the lowest curvature. For a stomatocyte (v = 0.55), initial attachment occurs in the “rocket” orientation, where the major axis of the vesicle is perpendicular to the initially planar membrane. As the wrapping progresses, the vesicle deforms to an oblate at f_w ≃ 0.07 to minimize the deformation energy, and remains oblate within the range 0.07 ≤ f_w ≤ 0.5. For f_w > 0.5, the vesicle reverts to its initial stomatocytic shape, persisting in this configuration until the wrapping of the outer sphere of the stomatocyte is completed. At f_w ≳ 0.75, the vesicle transforms into a vertical oblate, avoiding the high bending energy costs of the initially planar membrane associated with the wrapping of the inner sphere of the stomatocyte. Finally, at f_w > 0.9, the vesicle returns to an inverted stomatocyte. Upon complete wrapping, both the vesicle and the initially planar membrane recover their original shapes.

Fig. 2 Cross-sectional views of partial-wrapped nonspherical vesicles at various wrapping fractions f_w for κ_r = 1 and [small sigma, Greek, tilde] = 0.5 along with the indication of stable states (SS) and unstable states (US). Red, green, and yellow colors represent the unattached and attached membrane of the vesicle and the free initially planar membrane, respectively.

For an initially oblate vesicle (v = 0.63), the attachment to the initially planar membrane begins in a ring-shaped patch with two contact lines, as shown in Fig. 2. As the wrapping fraction f_w increases, the radius of the inner contact line decreases and vanishes at f_w ≈ 0.1, forming a circular patch with a single contact line, and remains oblate in the range of wrapping fractions 0.1 < f_w ≲ 0.5. At higher f_w values, the initially planar membrane comes into contact with the rim of the oblate, resulting in an increased bending energy cost. This energy cost induces a shape change in the vesicle from oblate to stomatocyte, which persists for 0.5 < f_w ≲ 0.79. As f_w increases further, the initially planar membrane needs to bend more to wrap the inner sphere of the stomatocyte, leading to a sharp increase in bending energy. To minimize the overall energy cost, the vesicle compromises its shape and reverts to a vertical oblate for f_w > 0.79, maintaining this shape until it is completely wrapped. Similar shape and orientation changes are observed for prolate (v = 0.66) and ellipsoidal (v = 0.95) vesicles, as illustrated in Fig. 2.

The reduced deformation energy ΔẼ, calculated using eqn (1)–(4), as a function of the wrapping fraction f_w for various reduced volumes of vesicles (v = 0.55, 0.63, 0.66 and 0.95) at a fixed relative stiffness κ_r = κ_v/κ_p = 1 and membrane tension [small sigma, Greek, tilde] = 0.5 is shown in Fig. 3(a)–(d). For an initially stomatocyte (v = 0.55), ΔẼ increases with f_w in the absence of adhesion energy (w̃ = 0), as shown in Fig. 3(a). The energy minimum at f_w = 0 suggests that the non-wrapped (NW) state is the stable configuration. However, for adhesion strength w̃ ≃ 5.3, a local minimum with the same energy appears for the first time at f_w ≃ 0.21 (point ‘A’). At this adhesion strength, the vesicle undergoes a binding transition from non-wrapped (NW) to a stable shallow-wrapped (SW) state. The stable shallow-wrapped oblate states can be found for 0.21 ≲ f_w ≲ 0.32. At adhesion strength w̃ ≃ 15.1, two energy minima at f_w ≃ 0.32 (point ‘B’) and f_w ≃ 0.74 (point ‘C’) have the same height, which correspond to the discontinuous transition from shallow-wrapped to deep-wrapped W_sd where oblate and stomatocyte coexist. At even higher adhesion strength, w̃ ≃ 33.1, the deep-wrapped stomatocytic state (point ‘D’) coexists with the complete-wrapped (CW) state at f_w = 1 (point ‘E’); the envelopment transition W_dc is also discontinuous. More details about the binding transition, the shallow-to-deep wrapped transition, and the envelopment transition are discussed in the ESI.†

Fig. 3 Deformation energy landscapes of nonspherical vesicles at initially planar membrane as a function of wrapping fraction f_w for vesicles with reduced volumes (a) v = 0.55 (stomatocyte), (b) v = 0.63 (oblate), (c) v = 0.66 (prolate), and (d) v = 0.95 (ellipsoid), at bending rigidity ratio κ_r = 1 and reduced membrane tension [small sigma, Greek, tilde] = 0.5. The minimal energy states are labeled alphabetically, with A being the wrapping fraction of the shallow-wrapped state for the binding transition, B the wrapping fraction of the shallow-wrapped state, and C the wrapping fraction of the deep-wrapped state for the transition from shallow- to deep-wrapped; D indicates the wrapping fraction of the deep-wrapped state for the envelopment transition; D1 and D2 are wrapping fractions for the deep-wrapped to deep-wrapped transition, and E indicates the complete-wrapped state.

For an oblate (v = 0.63), the reduced deformation energy ΔẼ as a function of the wrapping fraction f_w is illustrated in Fig. 3(b). The first minimum in ΔẼ indicates that the binding transition W_ns occurs at the adhesion strength w̃ ≃ 0.8 and the wrapping fraction f_w ≃ 0.16. The states within the range 0 ≤ f_w ≲ 0.17 with rim patches, having two contact lines, are on the energy barrier, while the oblate states in the range 0.17 ≲ f_w ≲ 0.34 with single contact line patches are stable. The shallow-to-deep-wrapped transition W_sd occurs at adhesion strength w̃ ≃ 17.5, where the oblate vesicle at f_w ≃ 0.34 (point ‘B’) coexists with a stomatocyte at f_w ≃ 0.78 (point ‘C’). Within the deep-wrapped (DW) regime, a deep-wrapped (D₁) to deep-wrapped (D₂) transition W_dd is observed for w̃ ≃ 28.9, where stomatocytes at f_w ≃ 0.8 (point ‘D1’) coexist with vertical oblates at f_w ≃ 0.96 (point ‘D2’). Finally, a continuous envelopment transition W_dc occurs with an adhesion strength of w̃ ≃ 55.1.

For initially prolate (v = 0.66) and ellipsoidal (v = 0.95) vesicles, the deformation energy ΔẼ as a function of f_w at fixed κ_r = 1 and [small sigma, Greek, tilde] = 0.5 is shown in Fig. 3(c) and (d), respectively. The binding transition for initially prolate vesicles (v = 0.66) is discontinuous due to the energy barrier associated with the shape transition from prolate to oblate. In contrast, the initially ellipsoidal vesicles (v = 0.95) exhibit a continuous binding transition as there is no energy barrier between non-wrapped and shallow-wrapped states; the initial attachment of vesicle occurs in submarine orientation where the major axis is parallel to the initially planar membrane. For both vesicles, the shallow-to-deep wrapped transition is discontinuous due to an energy barrier that arises from changes in shape and orientation of the vesicles; see Fig. 2. However, the envelopment transition is continuous for both the initially prolate (v = 0.66) and the initially ellipsoidal (v = 0.95) vesicles, as no energy barrier exists between the deep-wrapped and completely wrapped states. Although we focus here on stable states, depending on the values of κ_r, [small sigma, Greek, tilde], and w̃ metastable states may also appear along with the stable states.⁶⁴ However, determining the vesicle shapes corresponding to these metastable states is challenging without a comprehensive understanding of the entire energy landscape.

3.2 Vesicle shapes during wrapping

The shape of a particle can be quantified using the asphericity,^65,66 calculated from the principal moments (λ_x, λ_y, λ_z) of the gyration tensor. We define the asphericity s² as⁶⁵For a perfectly spherical particle, the principal moments are equal, λ_x = λ_y = λ_z, resulting in asphericity s² = 0. In contrast, the maximal asphericity s² = 1 is reached for rod-shaped particles, where λ_x = λ_y = 0 and λ_z ≠ 0 when aligned along the z-axis. During wrapping, the soft vesicles exhibit various shapes depending on the reduced vesicle volume v and the membrane tension [small sigma, Greek, tilde]. Thus, for nonspherical vesicles, s² (>0) increases with increasing vesicle elongation.

Fig. 4(a) illustrates the asphericity s² of the vesicles with various reduced volumes v as a function of the wrapping fraction f_w for κ_r = 1 and [small sigma, Greek, tilde] = 0.5. Initially stomatocyte vesicles (v = 0.55), resembling two connected spheres, have an asphericity close to zero. In the range 0 < f_w ≲ 0.07, the vesicles retain the stomatocytic shape and s² ≈ 0. In stable SW states with 0.07 ≲ f_w ≲ 0.5, the vesicles are oblate with asphericity s² ≃ 0.23. In the DW state, the vesicles undergo a sequence of shape transitions from stomatocytes to vertical oblates, and then back to inverted stomatocytes. This shape transition leads to a variation in asphericity from s² ≈ 0 in the range 0.5 ≲ f_w ≲ 0.75, to s² ≃ 0.23 for 0.75 ≲ f_w ≲ 0.9, and finally back to s² ≈ 0 for f_w ≳ 0.9.

Fig. 4 Asphericity variation with wrapping fraction f_w. (a) Asphericity for vesicles with various v at κ_r = 1 and [small sigma, Greek, tilde] = 0.5. (b) Asphericity for various κ_r for v = 0.63 at [small sigma, Greek, tilde] = 0.5. (c) Asphericity for various [small sigma, Greek, tilde] for v = 0.63 at κ_r = 1. The dashed lines indicate the asphericities of the corresponding free vesicles.

Initially oblate vesicles (v = 0.63) retain constant asphericity s² ≃ 0.23 in the SW state for 0 < f_w ≲ 0.5, as they retain their shapes, see Fig. 4(a). However, they transition to deep-wrapped stomatocytes for 0.5 ≲ f_w ≲ 0.79, reducing asphericity to s² ≃ 0. Within the DW state, the vesicles undergo a transition from stomatocyte to vertical oblate, returning the asphericity to s² ≃ 0.23 for 0.79 < f_w ≤ 1. In contrast, initially prolate vesicles (v ≃ 0.66), which are more elongated, have a high asphericity s² ≃ 0.97. In SW states, 0.07 ≲ f_w ≲ 0.5, they transition to oblates with s² ≃ 0.23. In DW states, 0.5 ≲ f_w ≲ 0.81, vesicles adopt stomatocytic states with s² ≈ 0. Finally, for f_w > 0.81, the vesicles return to prolates with a rotation of the major axis by 90° and asphericities s² ≃ 0.97. For ellipsoidal vesicles (v = 0.95), the asphericity takes a value of s² ≃ 0.1 when the vesicle adopts an oblate-ellipsoid shape in SW states for 0.1 < f_w ≤ 0.5. In DW states, the vesicles maintain prolate-ellipsoidal shape for 0.5 < f_w ≤ 1 with s² ≃ 0.38.

Fig. 4(b) illustrates the effects of relative stiffness of the vesicle, κ_r, on the asphericity s² for an initially oblate vesicle (v = 0.63) at membrane tension [small sigma, Greek, tilde] = 0.5. For soft vesicles (κ_r ≲ 2.5), both shape and orientation changes occur during the wrapping process, resulting in variations in asphericity. Across all values of κ_r, the vesicles retain their oblate shape in the SW state for 0 < f_w ≲ 0.5, maintaining an asphericity s² ≃ 0.23. However, in the DW state, the soft vesicles undergo a sequence of shape transitions as the wrapping fraction increases. For example, vesicles with κ_r = 0.1 adopt stomatocytic shapes with s² ≈ 0 for 0.5 ≲ f_w ≲ 0.72, prolate shapes with s² ≈ 0.97 for 0.72 ≲ f_w ≲ 0.9, and vertical oblate shapes with s² ≃ 0.23 for f_w ≳ 0.95. In contrast, stiff vesicles (κ_r = 10) undergo only orientation changes, maintaining asphericity s² ≃ 0.23. Similarly, the effects of membrane tension [small sigma, Greek, tilde] on the asphericity of the vesicle are shown in Fig. 4(c) for an oblate (v = 0.63) with relative stiffness κ_r = 1 at various membrane tensions. Across a wide range of wrapping fractions f_w the asphericity s² remains nearly constant and is independent of [small sigma, Greek, tilde], indicating that the shape of the vesicles is not much affected by changes in membrane tension. It decreases to s² ≲ 0.1 for 0.6 ≲ f_w ≲ 0.8, where the vesicle is a stomatocyte. For initially stomatocyte, prolate, and ellipsoidal vesicles, the effects of vesicle stiffness and membrane tension on asphericities are shown in the ESI;† see Fig. S6. The height of the center of mass of the vesicles with respect to the outer patch of the initially planar membrane can similarly indicate both shape change and orientation change during wrapping; see Fig. S7 in the ESI.†

3.3 Effect of vesicle stiffness on the wrapping diagrams

Fig. 5(a)–(d) present the wrapping diagrams for various relative stiffnesses κ_r and reduced adhesion strengths w̃ for vesicles with v = 0.55, 0.63, 0.66, and 0.95 at fixed membrane tension [small sigma, Greek, tilde] = 0.5. These wrapping diagrams were extracted from the variation of the deformation energies, ΔẼ with f_w; see Fig. S4 in the ESI.† For each value of v, the binding transition W_ns shifts to higher adhesion strengths w̃ as κ_r increases, ultimately saturating at a constant value as the deformation of the vesicles is negligible for κ_r ≳ 10. This implies that softer vesicles adhere to the membrane more easily than stiffer ones. In contrast, w̃ for the envelopment transition W_dc increases with decreasing κ_r due to the development of a sharp neck at the contact line; see Fig. 2 and 4, indicating that complete wrapping is more challenging for softer vesicles than for stiffer ones. The shape and stiffness of the vesicles also influence the nature of the binding transition, which can be continuous or discontinuous. For example, the binding transition is continuous for ellipsoidal vesicles (v = 0.95) with κ_r ≳ 1 and stiff (κ_r ≳ 1) stomatocyte (v = 0.55) vesicles, while it is discontinuous for soft (κ_r ≲ 1) ellipsoidal (v = 0.95), and soft (κ_r ≲ 2.5) stomatocyte (v = 0.55) vesicles. For initially oblate (v = 0.63) and prolate (v = 0.66) vesicles, the binding transition is discontinuous for the entire range of κ_r. The envelopment transition is discontinuous for initially stomatocytes and continuous for initially oblates and prolates. The shallow-to-deep-wrapped transition W_sd is always discontinuous and shows a moderate increase in w̃ as κ_r decreases.

Fig. 5 Wrapping diagrams in the κ_r–w̃ plane at [small sigma, Greek, tilde] = 0.5 show stable states: non-wrapped (NW), shallow-wrapped (SW), deep-wrapped (DW), and complete-wrapped (CW) for vesicles with reduced volumes: (a) v = 0.55 (stomatocyte), (b) v = 0.63 (oblate), (c) v = 0.66 (prolate), and (d) v = 0.95 (ellipsoidal). Transitions W_ns, W_sd, W_dd, and W_dc indicate NW to SW, SW to DW, DW to DW, and DW to CW transitions, respectively. Solid and dashed lines represent continuous and discontinuous transitions.

In the SW state, soft stomatocytes (v = 0.55) with bending rigidity ratio κ_r ≲ 1 adopt oblate shapes, while stiff ones with κ_r ≳ 2.5 retain their original shape. For intermediate stiffness, 1 ≲ κ_r ≲ 2.5, a discontinuous transition from stomatocyte to oblate occurs as the adhesion strength w̃ increases; see Fig. 5(a). In contrast, oblate (v = 0.63) vesicles consistently maintain a “rocket” orientation for all bending-rigidity ratios; see Fig. 5(b). In the DW state, soft stomatocytes with κ_r < 1 transition to vertical oblates with increasing w̃. While soft oblate vesicles (v = 0.63) undergo a shape transition as w̃ increases, moving from stomatocyte to rocket prolate for κ_r ≲ 0.5 and from stomatocyte to vertical oblate for 0.5 < κ_r ≲ 2.5. Stiffer vesicles mainly experience an orientation change rather than a shape change. In complete wrapping, regardless of the bending-rigidity ratio, the vesicles return to their initial shapes.

Initially prolate vesicles (v = 0.66) with κ_r ≲ 16 transition to oblate in the SW state, while those with κ_r ≳ 20 maintain their shape, see Fig. 5(c). For intermediate stiffness values, 16 ≲ κ_r ≲ 20, the vesicles undergo a transition from prolate to oblate with increasing w̃. In the DW state, the vesicles with κ_r < 1 experience a discontinuous transition from stomatocyte to “rocket” prolate with increasing w̃. In contrast, stiff prolates (κ_r > 1) show only an orientation change, maintaining the “rocket” prolate state. Similarly, initially ellipsoidal vesicles (v = 0.95) with 1 ≲ κ_r ≲ 5 can exhibit a discontinuous transition from prolate to oblate in the SW state as w̃ increases; see Fig. 5(d). Softer shallow-wrapped vesicles (κ_r ≲ 1) remain in the oblate state throughout, while stiffer ones (κ_r ≳ 5) remain in the prolate state. In the DW state, all ellipsoidal vesicles adopt the prolate shape with a “rocket” orientation.

Fig. 6 shows the wrapping diagrams in the κ_r–f_w plane for the same parameter values as in Fig. 5. For stomatocytes (v = 0.55) and ellipsoidal (v = 0.95) vesicles with κ_r ≳ 1, initial binding to the initially planar membranes occurs at f_wns = 0, indicating that the binding transition is continuous. In contrast, soft (κ_r ≲ 1) ellipsoidal (v = 0.95), soft (κ_r ≲ 1) stomatocyte (v = 0.55), prolate (v = 0.66), and oblate (v = 0.63) vesicles show discontinuous binding transitions to f_wns > 0. For all reduced volumes, the wrapping fractions and for the coexisting SW to DW states decrease weakly with increasing κ_r and may saturate for higher κ_r. For stomatocyte (v = 0.55), the wrapping fractions for the DW states that coexist with the CW state remain constant with increasing κ_r. For initially stomatocytes (v = 0.55), the presence of the energy barrier between the DW and CW states indicates that the envelopment transition is discontinuous. In contrast, for oblate (v = 0.63), prolate (v = 0.66), and ellipsoidal (v = 0.95) vesicles, no such energy barrier is found, indicating that the envelopment transition is continuous. The range of accessible wrapping fractions for the DW states increases slightly with increasing v.

Fig. 6 Wrapping diagrams in the κ_r–f_w at [small sigma, Greek, tilde] = 0.5 indicating stable non-wrapped (NW), shallow-wrapped (SW), deep-wrapped (DW), and complete-wrapped (CW) states and energy barriers. Data is shown for nonspherical vesicles with reduced volumes (a) v = 0.55 (stomatocyte), (b) v = 0.63 (oblate), (c) v = 0.66 (prolate), (d) v = 0.95 (ellipsoidal). The red-colored regions indicate wrapping fractions f_w for states on the energy barriers.

In summary, we have studied the wrapping behavior of nonspherical vesicles with different initial shapes—stomatocyte (v = 0.55), oblate (v = 0.63), prolate (v = 0.66), and ellipsoidal (v = 0.95)—by systematically varying the relative stiffness (κ_r) of the vesicles at a fixed reduced membrane tension [small sigma, Greek, tilde] = 0.5. For all the vesicle shapes we have considered, the adhesion strength w̃ for the binding transition decreases with decreasing relative stiffness κ_r, while it increases for envelopment transition. Softer vesicles undergo a sequence of both shape and orientation changes during the wrapping process, while stiffer vesicles exhibit only orientation changes. The binding transition is discontinuous for softer stomatocyte (v = 0.55) and ellipsoidal (v = 0.95) vesicles, and continuous for the stiffer ones. For initially oblate (v = 0.63) and the prolate (v = 0.66) vesicles, the binding transition is discontinuous regardless of stiffness. The envelopment transition is continuous for initially oblate (v = 0.63), prolate (v = 0.66), and ellipsoidal (v = 0.95) vesicles, but it is discontinuous for initially stomatocyte (v = 0.55) vesicles. For all the considered vesicle shapes, the shallow-to-deep wrapped transition is always discontinuous.

3.4 Effect of membrane tension on the wrapping diagrams

The effects of membrane tension on the wrapping diagrams are shown in Fig. 7 for stomatocyte (v = 0.55), oblate (v = 0.63), prolate (v = 0.66) and ellipsoidal (v = 0.95) vesicles, all with fixed relative stiffness κ_r = 1; the corresponding deformation energies are shown in Fig. S5 in the ESI.† The adhesion strength w̃ for the SW-to-DW transition increases with increasing membrane tension [small sigma, Greek, tilde], due to the increased energy cost in the neck, stabilizing SW over the DW states. As a result, the W_dc transition merges with the W_sd transition at the triple point ([small sigma, Greek, tilde]^sdc_t, w̃^sdc_t), where the SW, DW, and CW states coexist; the deformation energy landscapes at the triple points are shown in Fig. S8 in the ESI.† Beyond the triple point, the envelopment transition becomes discontinuous, with vesicles undergoing a direct transition from the SW state to the CW state. The adhesion strength w̃ for the binding transition does not change with increasing membrane tension [small sigma, Greek, tilde].

Fig. 7 Wrapping diagrams in the [small sigma, Greek, tilde]–w̃ plane at κ_r = 1. Data is shown for nonspherical vesicles with reduced volumes (a) v = 0.55 (stomatocyte), (b) v = 0.63 (oblate), (c) v = 0.66 (prolate), (d) v = 0.95 (ellipsoidal). Stable states and transitions are indicated analogously to Fig. 5; solid lines represent continuous and dashed lines discontinuous transitions, and triple points are highlighted by red diamonds.

For oblates (v = 0.63) with a bending-rigidity ratio κ_r = 1, a deep-wrapped (DW₁) to deep-wrapped (DW₂) transition occurs at membrane tensions [small sigma, Greek, tilde] ≲ 4, where stomatocytes and oblates coexist. For [small sigma, Greek, tilde] ≳ 4, vertical oblate states become unstable, and the DW₁-to-DW₂ transition vanishes at another triple point ([small sigma, Greek, tilde]^ddc_t, w̃^ddc_t), where two distinct DW states coexist with the CW state; the corresponding deformation energy landscape is shown in Fig. S9 in the ESI.† For initially stomatocytes (v = 0.55), the adhesion strength w̃ for complete wrapping gradually increases with increasing membrane tension [small sigma, Greek, tilde], since the envelope transition remains discontinuous throughout the membrane tension range [small sigma, Greek, tilde]. In contrast, for oblates (v = 0.63), the envelopment transition is continuous for membrane tensions 0 < [small sigma, Greek, tilde] ≲ [small sigma, Greek, tilde]^ddc_t and discontinuous for [small sigma, Greek, tilde] > [small sigma, Greek, tilde]^ddc_t. Similarly, for prolate and ellipsoidal vesicles, the envelopment transition is continuous for membrane tensions 0 < [small sigma, Greek, tilde] ≲ [small sigma, Greek, tilde]^sdc_t and discontinuous for [small sigma, Greek, tilde] ≳ [small sigma, Greek, tilde]^sdc_t. Consequently, the adhesion strength w̃ for the complete wrapping initially decreases with increasing [small sigma, Greek, tilde], reaches its optimal value at the triple point, and beyond the triple point w̃ increases with membrane tension [small sigma, Greek, tilde]. The wrapping diagram in the f_w–[small sigma, Greek, tilde] plane for initially stomatocyte, oblate, prolate, and ellipsoidal vesicles for the same parameter values, as mentioned in Fig. 7, is presented in Fig. S10 in the ESI.†

The coordinates of the triple points in the w̃–[small sigma, Greek, tilde] wrapping diagram are influenced by both the bending rigidity ratio κ_r and the reduced volume v of the vesicles. For stomatocytes (v = 0.55), the triple point ([small sigma, Greek, tilde]^sdc_t, w̃^sdc_t) shifts to higher values of membrane tension [small sigma, Greek, tilde] and adhesion strength w̃ as the bending rigidity ratio κ_r increases; see Fig. 8(a) and Fig. S11 in the ESI.† For high κ_r, the almost spherical shape of the outer surface reduces the cost of wrapping energy for deep-wrapped stomatocytes, and therefore, the regions of stable DW states extend to lower adhesion strengths and higher membrane tensions [small sigma, Greek, tilde] with increasing κ_r. However, as the membrane tension [small sigma, Greek, tilde] increases, independent of the vesicle shape, the energy cost associated with the neck for the DW states increases, increasing the stability of both the SW and the CW states compared to the DW state.

Fig. 8 Membrane tensions [small sigma, Greek, tilde]_t and adhesion strengths w̃_t for the triple points as functions of (a) bending rigidity ratio κ_r for initially stomatocytic (v = 0.55, circles) and oblate (v = 0.63, squares and triangles) vesicles and (b) for vesicles with various reduced volumes v at κ_r = 1.

For an oblate (v = 0.63) and κ_r ≲ 3, the vesicles transition between an oblate SW and a stomatocyte DW state with increasing adhesion strength, before further transition to oblate DW or CW states, see Fig. 8(a) and Fig. S12 in the ESI.† However, unlike initial stomatocytes, increasing κ_r leads to decreasing tensions [small sigma, Greek, tilde] and adhesion strengths w̃ for both triple points ([small sigma, Greek, tilde]^sdd_t, w̃^sdd_t) and ([small sigma, Greek, tilde]^sdc_t, w̃^sdc_t). Increasing the bending-rigidity ratio κ_r stabilizes the oblate states over the stomatocyte states and eventually leads to a direct transition between the oblate SW and the oblate DW or the CW states. The oblate DW state, due to its asymmetrical shape, lacks the advantages that a stomatocytic shape has, and stable states are only found for low tensions.

The triple point (σ^sdc_t, w̃^sdc_t) varies nonmonotonically with reduced volume v at a fixed stiffness of the vesicle κ_r = 1, as shown in Fig. 8(b). For 0.8 < v < 1, the triple point moves to higher membrane tensions and adhesion strengths as v decreases. This occurs because the vesicle becomes more elongated with decreasing v, increasing the stability of the deep-wrapped prolate states relative to the shallow-wrapped oblate states. In contrast, for 0.55 < v < 0.8, the vesicles tend to adopt an oblate shape in the SW state and a stomatocyte shape in the DW state. Consequently, the triple point shifts to lower membrane tensions and adhesion strengths.

In summary, we have investigated the wrapping behavior of nonspherical vesicles with different initial shapes—stomatocyte (v = 0.55), oblate (v = 0.63), prolate (v = 0.66), and ellipsoidal (v = 0.95)—by systematically varying the reduced tension ([small sigma, Greek, tilde]) of an initially planar membrane at a fixed bending rigidity ratio κ_r = 1. The adhesion strength w̃ for the binding transition remains constant with increasing membrane tension [small sigma, Greek, tilde]. The stability of the shallow-wrapped state increases with increasing [small sigma, Greek, tilde]. At high membrane tension [small sigma, Greek, tilde], the deep-wrapped state disappears at a triple point where the shallow-wrapped, deep-wrapped, and completely wrapped states coexist. For initially oblate vesicles (v = 0.63), an additional triple point is observed where two distinct deep-wrapped states (DW₁ and DW₂) coexist with the completely wrapped state. We have shown the coordinates of the triple point is influenced by both the bending rigidity ratio κ_r and the reduced volume v of the vesicles. For initially oblate (v = 0.63), prolate (v = 0.66), and ellipsoidal (v = 0.95) vesicles, the triple point marks the minimum adhesion strength w̃ for complete wrapping which occurs at a finite membrane tension.

4 Conclusions and outlook

We calculated the shapes, orientations, and energies for the wrapping of initially prolate, oblate, and stomatocytic vesicles at initially planar membranes. A systematic variation of reduced volumes, bending rigidity ratios, and membrane tensions leads to a rich phase behavior. Whereas shallow-wrapped states are often oblate, deep-wrapped states stabilize stomatocytic shapes. Transitions between shallow-wrapped and deep-wrapped states are always discontinuous, whereas the binding and envelopment transitions can be either continuous or discontinuous depending on vesicle stiffness and membrane tension. Softer vesicles undergo major shape changes during wrapping, while stiffer ones show only orientation changes, and at high tensions, the transition can occur directly from SW to CW states. Stomatocytic shapes can, because of their near-spherical shape, stabilize deep-wrapped over shallow-wrapped states and also for initially oblate and prolate vesicles. For soft vesicles, the binding transition is shifted to lower adhesion strengths, and the envelopment transition to higher adhesion strengths, in agreement with previous studies.^32,33

Increasing membrane tension destabilizes deep-wrapped states that vanish at a triple point, leading to the coexistence of shallow-, deep-, and complete-wrapped states. Vesicle asphericities and triple-point coordinates can be used to systematically characterize phase diagrams as functions of elastic parameters. With increasing stiffness, the triple points between the shallow-wrapped, deep-wrapped, and complete-wrapped states shift to lower tensions and adhesion strengths for initially oblate vesicles and to higher values for initially stomatocytic vesicles. For oblate, prolate and ellipsoidal vesicles, the optimal membrane tension for uptake is finite, indicating that the adhesion strength required for complete wrapping is not the lowest for vanishing tension, as expected for hard particles.^19,23,25 Because cells can regulate their membrane tension,^67,68 non-spherical vesicles can favor endocytosis or shallow-wrapped states; the latter may eventually facilitate fusion and direct delivery of the vesicle volume to the cytosol.⁸

Vesicles are a versatile class of elastic particles, exhibiting unique behaviors not observed in particles with 3D elasticity, such as polymeric particles. In particular, in the regime of stable deep-wrapped states, vesicles show internal discontinuous shape transitions with stomatocytic states extending to higher membrane tensions. Studying initial vesicle shapes close to the phase boundaries between stomatocytic, oblate, and prolate vesicles leads to an intriguing qualitatively novel finding compared to prolate vesicles with reduced volumes v > 0.7, which we studied earlier.³² Our findings on the wrapping of nonspherical vesicles at initially planar membranes provide valuable insights for designing tailored deformable particles, with potential applications in membrane attachment, fusion, and endocytosis.

This work highlights how vesicle shape, deformability, and membrane tension influence wrapping transitions on initially planar membranes. While our current model assumes a contact potential, many realistic systems involve long-range adhesive interactions.⁶⁹ Thus, future studies could include wrapping nonspherical vesicles at lipid membranes by systematically varying the range of adhesion interaction potential to mimic realistic systems. For vesicles with very low reduced volumes, large energy barriers may lead to metastable shapes,⁷⁰ which can make the wrapping transitions more complex. Investigating the wrapping behavior of metastable states of such vesicles could be a valuable extension of this work. Additionally, membrane deformations induce membrane-mediated interactions, which become particularly important in crowded environments. Predicting these interaction potentials between multiple vesicles could offer insights into the aggregation behavior of virions on biomembranes. However, the knowledge we gained from this work may also contribute to biomedical applications, particularly targeted drug delivery using deformable nanoparticles.

Author contributions

JM designed the research. AKS, RM, and JM conducted the research and analyzed the data. JM and AKS drafted the manuscript.

Data availability

The data used in this study are available in the main manuscript or the ESI.† Additional raw data and analysis scripts can be obtained from the corresponding author upon request.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

JM thanks Thorsten Auth (Jülich) for valuable discussions and input on the wrapping behavior of vesicles on membranes. JM acknowledges funding support from the Science and Engineering Research Board (SERB/ANRF) under the MATRICS grant (grant no. MTR/2023/001538) and IIT Bhubaneswar. AKS thanks the Council of Scientific and Industrial Research (CSIR) for funding his PhD.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5sm00150a

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