Membrane-Mediated Interactions Between Nonspherical Elastic Particles

The transport of particles across lipid-bilayer membranes is important for biological cells to exchange information and material with their environment. Large particles often get wrapped by membranes, a process which has been intensively investigated in the case of hard particles. However, many particles in vivo and in vitro are deformable, e.g., vesicles, filamentous viruses, macromolecular condensates, polymer-grafted nanoparticles, and microgels. Vesicles may serve as a generic model system for deformable particles. Here, we study nonspherical vesicles with various sizes, shapes, and elastic properties at initially planar lipid-bilayer membranes. Using the Helfrich Hamiltonian, triangulated membranes, and energy minimization, we predict the interplay of vesicle shapes and wrapping states. Increasing particle softness enhances the stability of shallow-wrapped and deep-wrapped states over nonwrapped and complete-wrapped states. The free membrane mediates an interaction between partial-wrapped vesicles. For the pair interaction between deep-wrapped vesicles, we predict repulsion. For shallow-wrapped vesicles, we predict attraction for tip-to-tip orientation and repulsion for side-by-side orientation. Our predictions may guide the design and fabrication of deformable particles for efficient use in medical applications, such as targeted drug delivery.

of deformable particles for efficient use in medical applications, such as targeted drug delivery.
Wrapping processes of mesoscopic particles by cell membranes are abundant in nature, such as entry and exit of viruses and parasites into host cells. 1,2Furthermore, extracellular vesicles can be taken up via endocytosis. 3Therefore, particle wrapping by lipid-bilayer membranes is also important for the design diagnostic and therapeutic agents.Particles for medical applications include lipid particles for targeted drug delivery, 4,5 and magnetic nanoparticles that serve as heat sources for cancer therapy. 6For hard particles the desired wrapping states can be achieved by controlling their shape, size, and particle-membrane adhesion strength. 7For soft particles, in addition, particle deformability plays an important role in the wrapping process.For example, an increased stability of partial-wrapped states has been reported for initially spherical vesicles with low bending rigidities of their membrane. 8Vesicles flatten upon binding to planar substrates, 9 the increased stability of partial-wrapped states is therefore similar to the increased stability reported for hard oblate ellipsoidal compared with spherical particles. 10Furthermore, soft particles can adjust to constraints and react to external stimuli.Prominent examples are filamentous viruses at plasma membranes that bend, 11 parasites that squeeze through narrow constrictions when invading host cells through the tight junction, 12 and SARS-CoV-2 virions, which are bounded by a lipid envelope, and assume non-spherical shapes near membranes. 13gineered soft particles encompass a wide variety of architectures with tunable elastic properties: microgels, 14 star polymers, 15 polymer-grafted nanoparticles, 16 polymeric shells, 17 particles made from dendrimers as building blocks, 18 vesicles, 19 and biomolecular condensates. 20For example, the elastic properties of microgels can be controlled by crosslinker density and electric charge, 21,22 of star polymers by functionality and chain length, 23 of polymer-grafted nanoparticles by grafting density and chain length, 16 and of unilamellar fluid vesicles by membrane bending rigidity and osmotic concentrations. 24Vesicles, in particular, are a versatile and well-established biomimetic system and have also been used as generic model system for soft particles. 25A zoo of vesicle shapes can be obtained by changing the membrane curvature-elastic parameters and osmotic concentrations, which includes spherical, prolate, oblate, stomatocyte, pear-shaped, and starfish shapes. 197][28] Therefore, long-ranged membrane-mediated interactions can drive self-assembly of partial-wrapped particles at lipid-bilayer membranes. 29,30However, the nature of the membrane-mediated interactions can either be attractive or repulsive, depends on many parameters, and is thus not easy to predict.For example, both, the deformation energy of the free membrane and the particle-membrane adhesion energy depend on the distance between interacting particles. 31Experimental and theoretical studies show membrane-mediated attraction and cluster formation of hard spherical particles on vesicles. 324][35][36] In vivo, viruses are shed as multi-virion clusters in vesicles. 37,38Ellipsoidal microgels on giant unilamellar vesicles have been found to form ordered structures with spacings that suggest membrane-mediated mutual repulsion. 14 this work, we use energy minimization to calculate and predict shapes and wrapping states for single, non-spherical vesicles that get wrapped at planar membranes.We find that adhesion to a membrane can change the shape of vesicles from prolate to oblate for partialwrapped states.Wrapping transitions can be continuous and discontinuous; shape and orientation changes of vesicles are always discontinuous with an energy barrier.Increased vesicle softness, decreased reduced vesicle volume, and increased tension of the planar membrane enhance the stability of the partial-wrapped states.We also calculate the membrane-mediated pair interaction between two prolate vesicles in shallow and deep-wrapped states.We predict attraction between shallow-wrapped vesicles in tip-to-tip orientation, and repulsion between deep-wrapped vesicles, as well as between shallow-wrapped vesicles in side-by-side orientation.For shallow-wrapped vesicles, increasing particle deformability induce a shape change from prolate to oblate and an interaction change from attractive in tip-to-tip orientation to repulsive for oblate vesicles.

Results and Discussion
Wrapping energies.Our predictions for wrapping vesicles at membranes are based on a continuum membrane model where the membranes are represented by mathematical surfaces, which is applicable for particles with radii r p 20 nm.The total energy of the system is calculated using the energy functional where H = (c 1 + c 2 )/2 is the mean curvature and c 1 and c 2 are the two principle curvatures; A v is the area of the vesicle membrane, A p the area of the initially planar membrane, and A ad is the adhered area of the vesicle membrane.The bending rigidities are κ p and κ v for the planar and the vesicle membrane, respectively; σ is the tension of the planar membrane, and w is the adhesion strength between vesicle and membrane.The shape of the vesicle is characterized by its reduced volume v = V v /V sph , where V v is the actual volume of the vesicle and V sph is the volume of a spherical vesicle with the same membrane area.The deformability of the vesicle is characterized by the ratio κ v /κ p of the bending rigidities of vesicle and planar membrane.The wrapping fraction f w = A ad /A v measures the fraction of the vesicle membrane that is adhered.For convenience, we will use the reduced membrane tension σ = σA v /(2πκ p ) and the reduced adhesion strength w = wA v /(πκ p ) in the following.
Wrapping and shape transitions for a non-spherical vesicle.The wrapping of non-spherical vesicles at a planar membrane is studied by systematically varying the bending rigidity ratio κ v /κ p and reduced volume v of the vesicle, and the tension σ of the planar membrane.Figure 1 shows minimum energy shapes of a vesicle with reduced volume v = 0.8, at a planar membrane as function of the wrapping fraction f w = A ad /A v .Wrapping always starts from the lowest mean curvature region of the vesicle surface; for a prolate vesicle binding occurs in submarine orientation with its major axis parallel to the membrane, see The deformation of the vesicles reaches its maximum at f w ≈ 0.5, where the vesicle shape is oblate and the membrane touches the rim of the vesicle where the mean curvature is maximal.For f w > 0.5, the shape of the vesicle changes back from oblate to prolate-now with rocket orientation and its major axis perpendicular to the membrane, see Fig. 1(e,f).
In absence of adhesion energy ( w = 0), a monotonic increase of the wrapping energy ∆ Ẽ is observed with increasing wrapping fraction f w , see Fig. 2 (a).At finite adhesion strengths, depending on the elastic properties, three or four transitions between non-wrapped, partialwrapped, and complete-wrapped states are found.For adhesion strengths below the binding transition, w < w1 , the non-wrapped state is stable.The binding transition W 1 at adhesion strength w 1 is discontinuous, see Fig. 6S in the Supporting Information, corresponding to the merging of two separate adhesion patches, see Fig.The wrapping diagrams of the vesicle-membrane system are obtained by systematic variation of the bending-rigidity ratio κ v /κ p that effectively controls the vesicle softness, the reduced volume v that controls the shape of free vesicles, see Fig. 1S in Supporting Information, and the reduced tension σ of the planar membrane.For each parameter set, we compute the deformation energy ∆ Ẽ as function of the wrapping fraction f w , see Fig. 3S-5S  in the Supporting Information.The adhesion strengths for all three transitions increase with decreasing v, accompanied by a change of the shape of the free vesicles that become more elongated with their tips more pointed.Therefore, binding as well as complete wrapping become more difficult, and the stability of the partial-wrapped states increases.
Figure 3 shows the wrapping fractions for partial-wrapped states that coexist with the free vesicles at the W 1 transition, and for shallow-wrapped and deep-wrapped states that coexist at the W 2 transition.For the W 2 transition, the wrapping fractions f SW w 2 for the SW states and f DW w 2 for the DW state slowly increase with decreasing κ v /κ p at reduced volume v = 0.8 and tension σ = 0.5, see Fig. 3(a).The stronger increase of f SW w 2 indicates the vanishing of DW states in the limit κ v /κ p → 0. For the W 1 transition, a sudden increase in wrapping fraction of the partial-wrapped states f w 1 is observed at κ v /κ p ≃ 1, which is associated with the shape change of the vesicle from prolate to oblate.
The wrapping fractions f SW w 2 and f DW w 2 remain almost unchanged when the reduced volume is varied in the range 0.7 ≤ v ≤ 0.95 at κ v /κ p = 1 and σ = 0.5, see Fig. 3(b).However, the variation of f w 1 with v is discontinuous.A sudden drop of f w 1 is observed as v increases from 0.7 to 0.75.This is associated with the shape change of the vesicle from oblate to prolate.
For v ≥ 0.75, f w 1 decreases monotonically with v, and then again a sudden drop is observed from v = 0.85 to v = 0.9.Such a behavior is expected as the shape of the free vesicle changes from peanut-like to ellipsoid-like.For 0.75 < v < 0.85, the shape of free vesicle is peanut-like (see Fig. 1S), thus, initial attachment to membrane leads to two patches, see Fig. 1(a).The vesicle becomes ellipsoid-like for v 0.9, thus, initial binding of the vesicle to the planar membrane occurs at the middle part and forms a single patch, see Fig. 2S in the Supporting Information.For both transitions, a weak increase of the wrapping fractions of the coexisting partial-wrapped states is observed with increasing membrane tension σ at reduced volume v = 0.8 and κ v /κ p = 1, see Fig. 3(c).
For all partial-wrapped states, the vesicle deforms the free membrane, see Fig. 4. For partial-wrapped vesicles in submarine orientation, the deviation of the height field describing the free membrane from the plane is strongest around the tips and weakest along the sides.
The local bending energy of the free membrane is finite at both, tips and sides, and shows lines of vanishing bending energy separating tips and sides, see Fig. 4(c).The free-membrane deformation at the sides can be approximated by calculations for wrapping infinitely long cylindrical particles, 39,40 the deformation at the tips by calculations for wrapping spherical particles. 10,41However, the free-membrane deformation along the sides rises the particle height above the optimal value for catenoid formation around the tips.The finite mean curvature along the sides and at the tips thus originate from principal curvatures with opposite signs.Therefore, the mean curvature and also the local bending energy vanish along lines where the principal curvatures have equal magnitude.For the deep-wrapped vesicles, the free membrane deformation and the bending energy distribution are cylindrical symmetric, see Fig. 4(d).Here, the origin of the finite bending energy of the free membrane is the membrane tension that prevents a catenoidal shape of the neck.
Membrane-mediated interactions between two partial-wrapped vesicles.The deformation of the initially planar, free membrane induces membrane-mediated interactions between two partial-wrapped vesicles.When the vesicles approach each other, both the initially planar membrane and the attached vesicles deform.We predict membrane-mediated pair interactions between shallow-and deep-wrapped prolate vesicles with v = 0.8 at initially planar membranes with a small finite membrane tension σ = 0.5.The value of the adhesion strength w is chosen such that either a deep-wrapped or a shallow-wrapped state is stable.
The interaction potentials are obtained by calculating the total energies of the system for various distances between the vesicles.The change in the total energy ∆ Ẽ is measured with respect to the vesicles at infinite distance, i.e., ∆ Ẽ = Ẽ(r) − Ẽ(∞).We calculate the totalenergy difference ∆ Ẽ as function of the reduced distance dCL = d CL /a 0 , where d CL is the minimum contactline-to-contactline distance between the two vesicles, and is the radius of a spherical vesicle with the same membrane area.To understand the origin of the membrane-mediated interaction, we split the total potential energy ∆ Ẽ into its individual where ∆ Ẽb,v is the change of bending energy of the vesicles, ∆ Ẽb,p is the change of bending energy of the planar membrane, ∆ Ẽs,p is the change of tension energy of the planar membrane, and ∆ Ẽw is the change of the adhesion energy.
For two deep-wrapped vesicles, w = 15, the energy ∆ Ẽ increases monotonically with decreasing distance between the two vesicles, see Fig. 5(a); the vesicles mutually repel each other.We find a purely repulsive interaction with decreasing strength for decreasing bending rigidity of the vesicle membrane.Within the considered distance range, the interaction potential is well described by an effective power law with exponent −2, see inset of Fig. 5(a).
In Fig. 5(b)-(d), all energy components of Eq. ( 2) are plotted as a function of dCL for κ v /κ p = 0.2, 1, and 5.The bending-energy contributions, ∆ Ẽb,p and ∆ Ẽb,v , both increase with decreasing distance between the vesicles.The deformation of the planar membrane increases as the vesicles become stiffer, thus the contribution of ∆ Ẽb,p to the total energy increases with increasing κ v /κ p .Within the range of our parameters, the relative contribution of ∆ Ẽb,v increases as the vesicles become softer, because the deformation of the vesicles becomes less expensive than that of the free membrane.The contribution of the adhesion energy ∆ Ẽw is significant for all cases.We find that ∆ Ẽw becomes more negative at short distances, which is expected as the adhered area increases at short distances; the change of wrapping fraction becomes less important with decreasing κ v /κ p , see Fig. S12 in the Supporting Information.
Thus, the contribution of ∆ Ẽw becomes weaker with decreasing κ v /κ p .For all cases, the repulsive potential mainly originates from ∆ Ẽb,p and ∆ Ẽb,v .We observe a weak increase of ∆ Ẽs,p as the distance between the vesicles decreases, which is connected to the increase of adhered membrane area as the vesicles approach each other, see Fig. S12 in the Supporting Information.The contribution of ∆ Ẽs,p to the total energy is negligible because a very small membrane tension is applied.We also investigate the membrane-mediated interaction between two shallow-wrapped vesicles with reduced volume v = 0.8 and tension σ = 0.5 for the planar membrane, see Fig. 6(a).The adhesion strength is fixed at w = 10, such that the shallow-wrapped state is stable for all considered values of κ v /κ p .For side-by-side (SS) orientation, the interaction is purely repulsive for κ v /κ p = 0.2, 1, and 5, see Fig. 6(b).The interaction strength decreases as the vesicles become softer, similar as for deep-wrapped vesicles.A qualitative change in the interaction potential is associated with the shape change of the vesicles from prolate for κ v /κ p = 5 to oblate for κ v /κ p 1. The shape change also leads to higher wrapping fractions of the vesicles from f w ≈ 0.14 for κ v /κ b = 5 to f w ≈ 0.27 for κ v /κ b = 1.Similarly, the deformation energy of the planar membrane decreases more as the prolate vesicles approach each other in side-by-side orientation than for the oblate shapes.However, the deformation energy of the free planar membrane, without the adhered part of the planar membrane, increases with increasing κ v /κ p , see Fig. S11 in the Supporting Information.
For SS orientation and κ v /κ p = 5, the high curvature of the free membrane between the particles increases the bending-energy cost for the free membrane, see Fig. S11 in the Supporting Information.This leads to partial unwrapping of the vesicles from the planar membrane, see Fig. 6(c) and Fig. S12 in the Supplementary Information.We find both bending-energy contributions, ∆ Ẽb,p and ∆ Ẽb,v , to become more negative with decreasing distance.The change of adhesion energy ∆ Ẽw is increasing with decreasing vesicle-vesicle distance, which is the dominating contribution causing the repulsion between the vesicles.
The contribution of ∆ Ẽs,p to the total energy decreases with decreasing dCL , which is expected because the total membrane area decreases as two vesicles approach each other.However, the contribution of ∆ Ẽs,p to the total energy is negligible.The total energy as function of the distance can be described by an effective power law d−2.5 CL , see the inset of Fig. 5(b).Thus, a smaller area of the vesicles is adhered to the planar membrane as they approach each other, opposite to deep-wrapped vesicles where the wrapping fraction increases.
As a result, a qualitative change is observed in the behavior of the individual components in comparison with deep-wrapped vesicles.Our shallow-wrapped vesicles in SS orientation experience stiffer membrane-mediated interaction potentials than our deep-wrapped vesicles.
For tip-to-tip (TT) orientation and κ v /κ p = 5, a cooperative deformation of the membrane by the two vesicles leads to the formation of a joint "trough" for both particles without a strong increase of the deformation-energy cost of the free membrane, see Fig. 6(a) and Fig. 7(e,f).The interaction is weakly attractive.To understand this behavior, we again look into the individual energy components, see Fig. 6(d).In contrast to the SS orientation, here we find that the wrapping fraction of the vesicles increases as the vesicles approach each other.As a result, tension-and bending-energy contributions for both the planar membrane and the vesicles increase as the distance between the vesicles decreases.The effective attraction originates from the gain in adhesion energy.For shallow-wrapped states and κ v /κ p 1 the vesicle shape changes from prolate to oblate; therefore, there is no distinction between TT and SS orientations.Now, we discuss shapes and local bending energies of the free membrane for both, deepwrapped and shallow-wrapped vesicles, see Fig. 7(a)-(f).At the contactline, where the free membrane detaches from the vesicle, the membrane height deviates most from the height of the wire frame.For two deep-wrapped vesicles with contactline-contactline distance dCL = 0.97, the free-membrane height also strongly deviates from the wire frame height in the middle between the vesicles, see Fig. 7(a).Here, the deformation is strong because the membrane aims to decrease the high local bending-energy cost, see Fig. 7(b); yet, this energy cost causes the repulsive interaction between the vesicles, compare Fig. 5(b).The repulsion leads to tilting of the vesicles, see Fig. S7 in the Supplementary Information.Interestingly, the membrane-mediated interaction induces an increase of the center-of-mass height of the vesicles with respect to the wire frame, see Fig. S8 in the Supplementary Information.
For two shallow-wrapped vesicles with contactline-contactline distance dCL = 1.36, the height of the free membrane deviates most strongly from the wire frame height at the tips of the vesicles, see Fig. 7(c).The bending energy cost between the vesicles is very high for SS orientation, see Fig. 7(d), which leads to a partial unwrapping of the vesicle sides that face each other and therefore a repulsive interaction because of the decreasing adhesion energy when the vesicles approach each other, see Fig. 6(c) and Fig. S12 in the Supplementary Information.For TT orientation, the bending energy of the membrane between the vesicles is very small although the membrane is also strongly deformed, see Fig. 7(e).The membrane height deviates strongest from the wire-frame height at the tips that face each other-and wrapping both vesicles together increases the wrapping at the sides as well.However, the overall increase of the bending energy in TT orientation is much lower than in SS orientation; the joint deformation of the membrane by both vesicles strongly increases the wrapping at their sides and thereby the adhesion energy E w , see Fig. 7(f) and Fig. 6(d).Also for shallowwrapped vesicles, the heights of the centers of mass of the vesicles decrease with decreasing distance, see Fig. S8(b) in the Supporting Information.

Conclusions
In this work, we have predicted the wrapping behavior and calculated membrane-mediated interactions between prolate vesicles at planar membranes.For single-vesicle systems, we systematically varied the reduced volume v of the vesicles, the bending rigidity ratio κ v /κ p , and the membrane tension σ of the planar membrane.Wrapping always starts from the minimum curvature regions of the vesicles.Thus, initial wrapping for low adhesion strengths occurs in submarine orientation where the major axis of the vesicle is parallel to the planar membrane.With increasing adhesion strength, for stiff vesicles the orientation changes from submarine to rocket, where the major axis of the vesicle is perpendicular to the planar membrane.For soft vesicles, a shape and orientation change of partial-wrapped vesicles from prolate submarine to oblate to prolate rocket can be observed.The binding transition, the shallow-to-deep-wrapped transition, and the envelopment transition are located and characterized by systematically varying the adhesion strength.We have shown that the binding and the shallow-to-deep-wrapped transition are always discontinuous, and the envelopment transition is always continuous.For fixed v and σ, we have shown that the binding transition shifts to lower adhesion strengths and the envelopment transition shifts to higher adhesion strengths with decreasing bending-rigidity ratio κ v /κ p .The softer the vesicle, the easier it attaches to the membrane, but the more difficult it gets completely wrapped-which is a generic behavior of soft, deformable particles.For fixed v and κ v /κ p , we have shown that the binding transition is independent of the membrane tension.An increase of membrane tension stabilizes shallow-wrapped states over deep-wrapped and complete-wrapped states.
The membrane-mediated interaction between two shallow-wrapped or two deep-wrapped prolate vesicles (v = 0.8) at planar membranes is studied by systematically varying the distance between two partial-wrapped vesicles.Particle systems are more complex than spherical-cap inclusions, 30,[42][43][44][45] curved membrane domains, 46 and curved proteins 47 for which the wrapping fraction does not depend on the distance.For deep-wrapped states of our prolate vesicles, we found the interaction potential to be repulsive, which originates from the deformation of both, free membrane and vesicles.The strength of the repulsive interaction decreases with the decreasing κ v /κ p , i.e., with increasing softness of the vesicles.For shallowwrapped states, a qualitative change in the interaction potential from attractive in tip-to-tip orientation to repulsive in side-by-side orientation is observed.For side-by-side orientation, the interaction potential is purely repulsive and becomes weaker as the vesicles become softer, which is associated with the shape change of the vesicles from prolate to oblate.
Although, the interaction for both, deep-wrapped vesicles and shallow-wrapped vesicles in SS orientation, is repulsive, the reason is different in both cases.For side-by-side orientation, the repulsive interaction originates from the loss of adhesion energy as the two vesicles approach each other.For tip-to-tip orientation, the interaction energy is attractive, which originates from the gain in adhesion energy as the vesicles approach each other.
Our theoretical predictions for wrapping and for pair-interactions of soft particles at membranes can be used to optimise the shapes and the elastic properties of deformable particles for efficient use in nanomedicine, such as for targeted drug delivery.It will be also interesting to investigate membrane-mediated interactions between many particles, which will help us to understand more complex problems like aggregation of virions on lipid-bilayer membranes.The understanding of the wrapping behavior of the vesicle-membrane systems is leading off to investigate the interaction of polymeric particles, e.g., star polymers, polymergrafted nanoparticles, and microgels, with lipid bilayer membranes.However, contrary to vesicles polymeric particles have bulk and shear elasticities, and their surface area increases upon binding to a membrane. 48Yet the knowledge that we gained from non-spherical vesiclemembrane systems provides a starting point to estimate the membrane-mediated interaction for anisotropic soft particles in general.

Model and methods
The continuum model in Eq. ( 1) is limited to systems with particle sizes that are at least few times larger than the thickness of a lipid-bilayer membranes; bending and tension energy both contribute in the wrapping process.The total energy of the system can be expressed as a dimensionless quantity by dividing by the wrapping energy of spherical particles, πκ p , and expressing all length scales in terms of the membrane area A v of the vesicle.This gives The numerical data for the deformation energy are fitted using a piecewise function, independently for submarine and rocket orientation.In each region, we fit the energy using a fourth-order polynomial with the coefficients a 0 , ..., a 4 .For the calculation of the phase diagrams we use the sum of deformation and adhesion energy, where the latter is proportional to the wrapping fraction.
To locate different transitions, i.e. binding transitions, shallow-wrapped to deep-wrapped transitions, and envelopment transitions, we take the first derivative of the deformation energy with respect to the wrapping fraction for different adhesion strengths.The zeros of the first derivative indicate the local minima and maxima of the wrapping energies.
To calculate membrane-mediated pair interaction potentials, two vesicles are placed on the planar membrane at fixed center-of-mass distances dCC = d CC /a 0 , where a 0 = (A v /4π).
However, for the presentation of the results, we use the contactline-to-contactline distance dCL = d CL /a 0 ; we have also calculated the vesicle surface-to-surface distances dSS = d SS /a 0 , see Table S2 in the Supporting Information.For each distance, the total energy is minimized and the deformation energy ∆ Ẽ as function of the wrapping fraction f w is obtained.We exploit the mirror symmetry and calculate only a quarter or half of the actual system.For the deep-wrapped states, the contactline is stabilized by forcing it to a plane with an arbitrary tilt angle.As for the single-vesicle systems, the deformation energy is fit using fourth-order polynomials.The adhesion strength w is chosen such that the minimum of ∆ Ẽ is either a stable deep-wrapped or a stable shallow-wrapped state for all vesicle-vesicle distances.

Fig. 1 (
Fig.1 (a).As wrapping progresses, the deformation of vesicle and surrounding membrane increases, see Fig.1(b)-(d).We observe a stable oblate state of the vesicle at f w ≈ 0.3.

Figure 1 :
Figure 1: Wrapping of a non-spherical vesicle with v = 0.8 at an initially planar membrane for the wrapping fractions (a) f w = 0.05 (unstable state), (b) f w = 0.25 (stable state), (c) f w = 0.35 (stable state), (d) f w = 0.5 (unstable state), (e) f w = 0.75 (stable state), and (f) f w = 0.95 (stable state).The minimum-energy shape of the free vesicle is prolate.The bending rigidity ratio is κ v /κ p = 1 and the reduced membrane tension σ = 0.5.The left, middle, and right columns represent the side, bottom, and top views of the vesicle-membrane system, respectively.

Figure 2 :Figure 3 :
Figure 2: Wrapping of a non-spherical vesicle at a planar membrane.(a) Reduced wrapping energy ∆ Ẽ as function of the wrapping fraction f w for vesicles with reduced volume v = 0.8, bending rigidity ratio κ v /κ p = 5, and reduced membrane tension σ = 0.5.The blue circles show the numerical data for the wrapping energy in absence of adhesion strength ( w = 0) and the blue line represents the fit function.The green, red, and orange lines correspond to the adhesion strengths w associated with the wrapping transitions from non-wrapped (NW) to shallow-wrapped (SW), shallow-wrapped to deep-wrapped (DW), and deep-wrapped to complete-wrapped (CW), respectively.The energies for the (coexisting) stable states at the transitions are indicated by circles.(b) Wrapping diagram in the κ v /κ pw plane for v = 0.8 and σ = 0.5.The dashed red line of the oblate-to-prolate transition is a guide to the eye.(c) Wrapping diagram in the v-w plane for κ v /κ p = 1 and σ = 0.5.(d) Wrapping diagram in the σ-w plane for v = 0.8 and κ v /κ p = 1.All transitions and the stable wrapping states of the vesicles are labeled in the figures.

Figure 4 :
Figure 4: Single vesicles attached from above to initially planar membranes.The heat maps indicate (a,b) the height and (c,d) the bending energy of the free membrane around a vesicle with v = 0.8, κ v /κ p = 5, σ = 0.5 and w = 10.94 for (a,b) shallow-wrapped and (c,d) deep-wrapped states.The white areas are inside the contact line where the free membrane detaches from the vesicle.

Figure 5 :
Figure 5: Deep-wrapped vesicles at w = 15.(a) Interaction potential ∆ Ẽ as a function of distance dCL for the bending rigidity ratios κ v /κ p = 0.2, 1.0 and 5.0 at fixed v = 0.8, and σ = 0.5.Inset shows the same on log-log scale.The solid line presents a power law with an exponent −2.The individual energy contributions: the change in bending energy of the vesicle, ∆ Ẽb,v , the change in bending energy of the planar membrane, ∆ Ẽb,p , the change in surface energy of the planar membrane, ∆ Ẽs,p and the change in adhesion energy, ∆ Ẽw , as function of distance dCL are presented for (b) κ v /κ p = 5.0 (c) κ v /κ p = 1.0 and (d) κ v /κ p = 0.2.

Figure 6 :
Figure 6: Shallow-wrapped vesicles at w = 10.(a) Cross-section side view of two vesicles with side-by-side (SS) and tip-to-tip (TT) orientations at different distances for the choice of parameters v = 0.8, σ = 0.5 = 0.5, and κ v /κ p = 5.0.(b) Interaction potential ∆ Ẽ, as a function of the distance dCL between the two vesicles in shallow-wrapped states with SSand TT-orientations.For SS orientation the interaction potential is plotted for the bending rigidity ratios κ v /κ p = 0.2, 1.0 and 5.0 at v = 0.8 and σ = 0.5.The inset shows the same on a log-log scale.The solid line presents a power law with an exponent −2.5.For TT orientation the interaction potential is shown for κ v /κ p = 5.0.The individual energy contributions: the change in bending energy of the vesicle, ∆ Ẽb,v , the change in bending energy of the planar membrane, ∆ Ẽb,p , the change in surface energy of the planar membrane, ∆ Ẽs,p and the change in adhesion energy, ∆ Ẽw , as function of distance dCL are presented at κ v /κ p = 5.0 and σ = 0.5 for (c) SS-orientation and (d) TT-orientation.

Figure 7 :
Figure 7: Free-membrane deformation for two vesicles at a planar membrane.(a,c,e) Local membrane height difference ∆ h = [h − h 0 ]/a 0 with the local membrane height h and the height h 0 of the wire frame spanning the membrane patch, and (b,d,f) local bending energy of the free membrane.The values are shown in the x−y plane for vesicles with v = 0.8, reduced membrane tension σ = 0.5, and κ v /κ p = 5.0.(a, b) Deep-wrapped vesicles with their centers of mass at (x 1 , y 1 ) = (−x 2 , y 2 ) = (0.49, 0), (c,d) shallow-wrapped in side-by-side orientation with their centers of mass at (x 1 , y 1 ) = (−x 2 , y 2 ) = (0.68, 0), and (e, f) shallow-wrapped in tip-to-tip orientation with their centers of mass at (x 1 , y 1 ) = (−x 2 , y 2 ) = (0.75, 0).