Engulfment of ellipsoidal nanoparticles by membranes: full description of orientational changes

We study the engulfment of ellipsoidal nanoparticles by membranes. It has been previously predicted that wrapping by the membrane can induce reorientation of the particle, however, previous studies only considered the wrapping process constrained to either side-oriented or tip-oriented particles. In contrast, we consider here the full two-dimensional energy landscape for engulfment, where the two degrees of freedom represent (i) the amount of wrapping and (ii) the particle orientation. In this way, we obtain access to the stability limits of the differently-oriented states, as well as to the energy barriers between them. We find that prolate and oblate particles undergo qualitatively different engulfment transitions, and show that the initial orientation of the particle at first contact with the membrane influences its fate.


Introduction
The interaction between nanoparticles and biological membranes is of crucial importance in many fields such as drug delivery [1,2], biomedical imaging [3,4], nanotoxicity [5,6], or viral infection [7,8]. Indeed, in order to enter a cell, nanoparticles must first cross the cellular membrane. This is usually achieved through the process of endocytosis, by which the membrane first adheres to the particle, and then gradually wraps around it, until the particle is completely engulfed by the membrane.
The engulfment process can generally be understood as a competition between energetically favourable membrane-particle adhesion and energetically unfavourable membrane bending. This process may occur spontaneously, as observed experimentally in systems composed of particles in contact with biomimetic lipid or polymer vesicles [9][10][11][12].
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Many efforts have been made in recent years trying to understand the engulfment of nanoparticles by membranes from a theoretical perspective, and the effects of particle size and adhesiveness [13,14], membrane bilayer asymmetry [14], local curvature of the membrane [15,16], membrane tension [17], and particle rigidity [18,19] have all been investigated to a certain extent.
One particular aspect that has attracted a great deal of attention is particle shape [20][21][22][23][24][25][26][27][28][29][30][31][32]. In a number of studies, it has been found that non-spherical particles typically require higher adhesion energies to be engulfed than non-spherical particles and, most interestingly, that the engulfment process can cause spontaneous reorientation of non-spherical particles [23, 25-27, 30, 32]. The latter studies, however, either (i) consisted of dynamical simulations that did not explore the parameter space of the system in detail [25,26,32], or (ii) considered only the engulfment of the particles constrained to specific orientations, without allowing the particle to freely reorient itself [23,27,30]. The latter studies found that, in the case of ellipsoidal nanoparticles, there must be an orientational transition with increasing degree of wrapping. For a prolate nanoparticle, the transition would be from a side-oriented state at low degree (a) Wrapping of an ellipsoid by a planar membrane: there are two reaction coordinates for the process, namely the wrapped area fraction q ≡ A bo /A pa , where A bo and A pa are respectively the membrane-bound and total surface area of the particle; and the orientation angle θ, which locates the point on the surface of the ellipsoid that would make first contact with the membrane for a given orientation. These coordinates can vary in the ranges 0 q 1 and 0 θ π/2 (the range π/2 < θ < 2π can be mapped to the latter range by symmetry). (b) At low wrapping (q < 1/2), it is energetically preferred for prolate and oblate ellipsoids to be in side (θ = π/2) and tip (θ = 0) orientations, respectively. The opposite is true at high wrapping (q > 1/2). (c) Parameters used in the detailed description of the ellipsoid geometry, see section 2.2.
of wrapping, to a tip-oriented state at high degree of wrapping (the opposite would be true for an oblate nanoparticle). This was found by directly comparing the energy of the sideoriented vs the tip-oriented states at different degrees of wrapping. However, we note that, while comparing the energies of the two differently-oriented states can indicate the existence of a transition, this method does not provide any details of this transition. In particular, one cannot know whether the reorientation transition will be continuous or discontinuous, how high would the energy barrier between the two states be in the case of a discontinuous transition, or whether the differentlyoriented states remain metastable, in which case the transition may not be observed in practice.
To address these issues, here we calculate the full twodimensional energy landscapes for the engulfment of ellipsoidal nanoparticles by membranes, as a function of both the degree of wrapping and the particle orientation (which may take all intermediate values between side-oriented and tiporiented), see figure 1(a) and (b). We find that the transition between side-oriented and tip-oriented states is discontinuous, and calculate the energy barrier for the reorientation transition, which turns out to be much larger than the thermal energy, implying that typically the particle will remain in the metastable side-oriented (for prolate particles) or tiporiented (for oblate particles) state well beyond the discontinuous transition. In the case of prolate particles, the side-oriented, weakly-wrapped state eventually does become unstable with increasing adhesion towards the deeply-wrapped, tip-oriented state. For oblate particles, however, we find that the weaklywrapped tip-oriented state becomes unstable directly towards the completely engulfed state, implying that the reorientation transition might not be observed in practice in this case. Lastly, our results show that the fate of the particle depends on the initial point of contact with the membrane.

Energetics of the system
We consider the interaction of an ellipsoidal particle with a tensionless planar membrane with a symmetric bilayer, i.e. no spontaneous curvature. The only contributions to the free energy of the system then come from the bending energy of the membrane, and the particle-membrane adhesion energy, so that the total energy of the system is E = E be + E ad . The bending energy is given by [33,34] E be = 2κ dAM 2 (1) where M is the mean curvature of the membrane, and the integral in principle runs over the whole membrane surface. Indeed, the bending energy will in general have contributions from the part of the membrane that is bound to the particle, which necessarily follows the particle shape, and the free part of the membrane, which typically will be deformed due to the presence of the particle. However, it was shown in reference [24] that, for tensionless membranes, the contribution of the free part of the membrane is negligible compared to the contribution of the bound part, because the membrane can adopt a quasi-catenoidal shape with little energetic cost. This is true even at high wrapped area fractions, in which case the unbound segment forms a narrow neck that closes concomitantly with complete engulfment, while its contribution to the energy tends to zero [17,35]. We therefore will neglect the bending energy contribution of the free membrane in the following, and focus only on the contribution of the bound part. We note also that contributions from the Gaussian curvature do not need to be taken into account, because we will not consider changes in the topology of the membrane [34]. The adhesion energy is given by [36] E ad = −|W|A bo (2) where |W| is the adhesive strength of the particle-membrane interactions, and A bo is the area of the membrane segment that is bound to the particle, see figure 1(a).

Geometry of the system
In order to calculate the full energy landscapes for the engulfment of an ellipsoidal particle, we need to calculate the bending energy E be and the bound area A bo as a function of the two independent coordinates of the process, namely the wrapped area fraction q ≡ A bo /A pa , which varies from q = 0 in the free state to q = 1 in the completely engulfed state, and the orientation angle θ describing the location of the point on the surface of the particle that would make first contact with the membrane for a given orientation, which can vary from θ = 0 for a tip-first orientation to θ = π/2 for a side-first orientation; see figure 1(a) and (b). To this end, let us first consider the parametric equations for the surface of an ellipsoid {x(ω, φ) = a sin ω cos φ, y(ω, φ) = a sin ω sin φ, z(ω, φ) = b cos ω}, with parameters as defined in figure 1(c). Here, z represents the axis of symmetry of the ellipsoid, and the x-axis is directed perpendicular to the plane of the drawing. We note that a is the semi-axis perpendicular to the symmetry axis, and b is the semi-axis along the symmetry axis, so we can define the aspect ratio r ≡ b/a, which gives r > 1 for a prolate ellipsoid, r < 1 for an oblate ellipsoid, and r = 1 for a sphere. The area element of the ellipsoid is given by [28] 1 2 sin ω dω dφ (3) whereas the mean curvature is given by Note that both depend on ω but are independent of φ, reflecting the axisymmetry of the particle shape. Moreover, we note that the mean curvature of the tip and sides of an ellipsoid can be calculated as M tip = M(0) = b a 2 and M side = M π 2 = 1 2 a 2 +b 2 ab 2 . Second, we can consider the equation of a plane z = tan α · y + h representing the membrane, where α is the angle between the plane of the membrane and a plane perpendicular to the symmetry axis of the ellipsoid, and h is the signed distance from the centre of the ellipsoid to the intersection point between the symmetry axis and the plane, see figure 1(c). Note that both α and θ can be used to define the orientation of the particle with respect to the membrane, and the two are related to each other by α = arctan(r tanθ). Similarly, h can be used as a reaction coordinate describing the extent of wrapping, and varies between h = − b cos θ for the free state to h = b cos θ for the completely engulfed state.
In practice, therefore, we calculate numerically the bending energy and the bound area fraction as and in order to obtain energy landscapes of the form E(h, α). These can then be translated into energy landscapes E(q, θ), depending on the orientation angle θ and the wrapped area fraction q ≡ A bo /A pa . We note that the total surface area A pa of an ellipsoid is given, for a prolate ellipsoid with r > 1, by and, for an oblate ellipsoid with r < 1, by The gain in adhesion energy due to complete engulfment is given by −|W|A pa , and thus depends on the total area of the particle. When comparing engulfment of ellipsoids with different aspect ratios to each other, it therefore makes sense to compare ellipsoids with equal surface area. We thus choose R ≡ A pa /(4π), i.e. the radius of a sphere with area A pa , as the typical length scale of the system. As an energy scale, we choose the bending rigidity κ. The energy land-scapesẼ(q, θ) ≡ E(q, θ)/κ, and in general the behaviour of the system, then depend only on two dimensionless parameters: the aspect ratio r, and the reduced adhesive strength w ≡ |W|R 2 /κ.

Full wrapping-orientation energy landscapes
Let us begin by exploring the full energy landscapesẼ(q, θ), as a function of the adhesive strength w, for a particle of given aspect ratio r. In figure 2, we display five such landscapes corresponding to five different values of w, for a prolate with r = 2 in (a1)-(a5), and for an oblate with r = 1/2 in (b1)-(b5).
We see that, as one would intuitively expect, for low w the energy increases with q no matter the orientation, and thus the free state is always stable, see figure 2(a1) and (b1). Similarly, for very high adhesion, the energy decreases with q no matter the orientation, and thus the completely engulfed state is stable, see figure 2(a5) and (b5). At intermediate values of w, however, we find very different behaviour for prolate and for oblate particles.
For prolate particles, we find that beyond a critical value of w, free states with θ π/2 start to become unstable towards a stable weakly-wrapped (q 0), side-oriented (θ = π/2) state, see figure 2(a2). At the same time, a metastable deeply-wrapped (q 1), tip-oriented (θ = 0) state appears. Further increasing w, these two states swap (meta)stability, the deeply-wrapped state becoming the lowest energy state, see figure 2(a3). At even higher values of w, the weakly-wrapped side-oriented state becomes unstable towards the deeply-wrapped tip-oriented state, see figure 2(a4). This instability thus involves spontaneous and complete reorientation of the particle. Finally, further increasing w, the deeplywrapped state continuously transitions into a completely engulfed state with (q = 1), see figure 2(a4).
For oblate particles, the transition develops differently with increasing w. Beyond a lower critical value, free states with θ 0 start to become unstable towards a stable weaklywrapped tip-oriented (θ = 0) state, but this state is the only stable state and there is no coexistence with a deeply-wrapped Full energy landscapes as a function of orientation θ and wrapped area fraction q ≡ A bo /A pa , for (a1)-(a5) a prolate particle with aspect ratio r = 2, and (b1)-(b5) an oblate with r = 1/2. With increasing adhesive strength w, for prolates we find a stable free state (a1, w = 0), the free state then becomes unstable for the side orientations towards a weakly-wrapped side-oriented state and a metastable deeply-wrapped tip-oriented state appears (a2, w = 1.8), the deeply-wrapped tip-oriented state becomes energetically favourable over the weakly-wrapped side-oriented state (a3, w = 2.5), the metastable weakly-wrapped side-oriented state becomes unstable towards the deeply-wrapped tip-oriented state (a4, w = 4.5), and finally the deeply-wrapped tip-oriented state transitions continuously to a completely engulfed state (a5, w = 15). For oblates, we find a stable free state (b1, w = 0), the free state then becomes unstable for the tip orientations towards a weakly-wrapped tip-oriented state (b2, w = 1.2), a metastable deeply-wrapped side-oriented state appears (b3, w = 2.5), the deeply-wrapped side-oriented state becomes energetically favorable over the weakly-wrapped tip-oriented state (b4, w = 4), and finally the metastable weakly-wrapped tip-oriented state becomes unstable towards a completely engulfed state which arises from a continuous transition of the deeply-wrapped side-oriented state (b5, w = 10). The colours (see colour bars) describe the total energy of the system in units of E/κ; the solid lines are contour lines; the arrows indicate the direction of energy gradients.
state, see figure 2(b2). This changes when w is further increased, as a metastable deeply-wrapped side-oriented state develops, see figure 2(b3). The two states later swap metastability, the deeply-wrapped state becoming the lowest energy state, see figure 2(b4). Finally, for even higher w we find that simultaneously the deeply-wrapped state continuously transitions into a completely engulfed state, and the weaklywrapped becomes unstable towards complete engulfment, see figure 2(b5).
The landscapes of figure 2 include arrows depicting the energy gradient, and clearly demonstrate that the wrapping process will be accompanied by extensive reorientation of the particles. In particular, the landscapes show that while initial attachment may take place for a wide range of particle orientations, the particle gets spontaneously reoriented: for prolate particles, to a side-oriented state at intermediate-low w, and to a tip-oriented state at intermediate-high w; for oblate particles, to a tip-oriented state for all intermediate w. At very high w, however, reorientation becomes less significant and one finds completely engulfed states without any particular orientation, both for prolates and for oblates.

Effect of aspect ratio: phase diagram and energy barriers
The results of the previous section show qualitatively different features in the engulfment transition with increasing adhesive strength w, depending on the aspect ratio r of the particle. We have explored systematically the wrapping-orientation energy landscapes as a function of w and r, in order to construct a phase diagram of the system, see figure 3(a).
Indeed, we find that prolates (r > 1) and oblates (r < 1) always show qualitatively different engulfment transitions; as determined by the order of the critical lines L side and L tip , describing the stability limits of the side-and tip-oriented free and completely engulfed states; the line L side , describing the stability limit of the partially-wrapped side-oriented state; and the line D, describing the discontinuous transition between tipand side-oriented states, at which both have the same energy.
For prolate particles, we find the ordering w(L side ) < w(D) < w(L side ) < w(L tip ), whereas for oblate particles, we find w(L tip ) < w(L side ) < w(D) < w(L side ). These findings align with the observations of figure 2, in particular with the fact that prolate particles initially attach at the side and the side-attached state becomes unstable towards the deeplywrapped tip-oriented state; whereas oblate particles initially attach at the tip and this tip-oriented weakly-wrapped state remains metastable up until it becomes unstable towards complete engulfment.
Whether a particle remains free, or whether it will be completely engulfed, is entirely governed by the two instability lines L side and L tip . Interestingly, in previous work [14,35] we found exact analytical expressions for the stability limits of free and completely engulfed states of adhering particles, in the presence of membranes which could, in principle, be non-planar and have non-zero spontaneous curvature. The stability limits of the two states are, in that general case, given by two different conditions. In the particular of planar membranes with zero spontaneous curvature considered here, however, the two stability limits (of the free and the completely engulfed state) become identical and given simply by |W| = 2κM 2 , where M is the curvature of the particle at the point of initial contact with the membrane (in the case (a) Phase diagram as a function of particle aspect ratio r and adhesive strength w. Prolate particles (r > 1) are free for w < w(L side ) and completely engulfed for w > w(L tip ); whereas oblate particles (r < 1) are free for w < w(L tip ) and completely engulfed for w > w(L side ), respectively. We find that prolates show coexistence between a weakly-wrapped side-oriented state and a deeply-wrapped tip-oriented state for w(L side ) < w < w(L side ), which exchange (meta)stability at the discontinuous transition D, whereas for w(L side ) < w < w(L tip ) only the deeply-wrapped tip oriented state is stable. On the other hand, for oblates we find that only a weakly-wrapped tip-oriented state is stable for w(L tip ) < w < w(L side ), which later coexists with a deeply-wrapped side-oriented state for w(L side ) < w < w(L side ), with which it switches (meta)stability at the discontinuous transition D. (b) Height of the energy barrier separating the side-oriented and tip-oriented states at the discontinuous transition D at which the two states have equal energy. of the free state) or at the location of the narrow neck that connects it to the mother membrane (in the case of the completely engulfed state). Therefore, in the present context, we obtain two critical stability limits, corresponding to that of the tip-oriented particle, given by and that of the side-oriented particle, given by where we remind that M tip = b a 2 and M side = 1 2 a 2 +b 2 ab 2 . These two analytical predictions are plotted as the black solid lines in figure 3(a), and coincide with the numerical results. One interesting feature of figure 3(a) is that there exists an optimal aspect ratio for a prolate particle (with given surface area) to bind to a membrane at a minimal value of the adhesive strength. Using equation (10), we find this optimal aspect ratio to be r ≈ 1.92, the corresponding minimal adhesive strength being w ≈ 1.33.
It is also interesting to note that the four lines L side , L tip , L side , and D converge at the single point w = 2 and r = 1, corresponding to the case of a spherical particle. This recovers the well-known result [17] that, for spherical particles in contact with a tensionless planar membrane, particles with w < 2 remain free, whereas particles with w > 2 become completely engulfed. For w = 2, the energy landscape becomes exactly flat.
Along the discontinuous transition line D, the two (meta)stable states, one weakly-wrapped and the other deeplywrapped, one side-oriented and the other tip-oriented, have equal energies and are separated from each other by an energy barrier, corresponding to a saddle point in the energy located at q = 0.5. In figure 3(b), we plot the height of this energy barrier for wrapping-reorientation as a function of the aspect ratio. As expected, see the previous paragraph, the energy barrier vanishes for spherical particles with r = 1, and it is non-zero otherwise. We find that the energy barriers are significantly larger for oblates than for prolates. In both cases, even moderate aspect ratios lead to very high energy barriers of the order of several times κ. Given a typical value of κ 20k B T, this implies that the typical energy scale of the wrapping-reorientation barrier is of the order hundreds of k B T, and therefore it is unlikely that a particle will hop over such a barrier due to thermal fluctuations. We can therefore expect particles to be stably trapped in weakly-wrapped local minima, namely the side-oriented state for prolates in the range w(L side ) < w < w(L side ), and the tip-oriented state for oblates in the whole range w(L tip ) < w < w(L side ), see figure 3(a).

Effect of the initial particle-membrane contact point
As can be seen by examining the direction of the gradient arrows near q = 0 in the energy landscapes of figure 2, for a given value of the adhesive strength w, the free state may be stable or unstable, depending on the initial orientation of the particle or, equivalently, on the initial point of contact between the particle and the membrane. In general, we find that the free state becomes unstable at lower values of w for more weakly curved points on the particle surface. The fate of an ellipsoidal particle coming into contact with a membrane thus depends on the orientation of the particle, as well as the adhesive strength w.
The predicted fate of the particle as a function of the angle describing the initial contact point θ * and the adhesive strength w is plotted in figure 4, for a prolate particle with r = 2 in (a), and for an oblate particle with r = 1/2 in (b). Because we found in the previous section that the energy barriers between weakly-wrapped and deeply-wrapped states are typically much larger than the thermal energy, we assume that the particles will remain in the weakly-wrapped state until the latter becomes unstable.
We find that, indeed, prolate (oblate) particles can first attach only if they make contact in the side (tip) orientation and, with increasing w, the range of orientations for which Figure 4. Predicted fate of a particle depending on the orientation θ * at initial contact with the membrane, and the adhesive strength w, for (a) a prolate with r = 2 and (b) an oblate with r = 1/2. Only the state that is reachable directly from the free state via a 'downhill' path is depicted, because the energy barriers are too large to be overcome by thermal fluctuations (see figure 3(b)). For the prolate (a), attachment and reorientation into a weakly-wrapped side-oriented state (with θ = π/2) is predicted as long as θ * is sufficiently large and w(L side ) < w < w(L side ). For higher adhesion with w(L side ) < w < w(L tip ), contact at sufficiently large θ * leads via reorientation to a tip-oriented deeply-wrapped state with θ = 0, even if initial contact in tip orientation (θ * = 0) does not lead to attachment. Finally, for w > w(L tip ), contact at any point leads to complete engulfment, with no specific orientation. For the oblate (b), within the range w(L tip ) < w < w(L side ), initial contact with sufficiently small θ * always leads to a weakly-wrapped tip-oriented state; whereas for w > w(L side ) contact at any orientation leads to complete engulfment with no specific orientation. attachment is possible widens until it comprises the whole particle surface. The adhesive strength at which attachment in the orientation given by θ * first becomes possible is again given by the instability condition for the free state [14] |W| = 2κM(θ * ) 2 (11) where M(θ * ) is the local mean curvature of the particle, as given by equation (4). For prolate particles we find that contact of an approximately side-oriented particle with the membrane will lead, with increasing w, first to a weakly-wrapped sideoriented state, then to a deeply-wrapped tip-oriented state, and finally to a completely engulfed state with no particular orientation. Tip-oriented particles, on the other hand, go directly from a free state to a deeply-wrapped tip-oriented state with increasing w. In contrast, oblate particles in any orientation will always go with increasing w first to a weakly-wrapped tiporiented state, then to a completely engulfed state, the deeplywrapped side-oriented state not being directly accessible for any value of w, in any orientation.

Relevance to experiments
We have described above our predictions for the engulfment of ellipsoidal nanoparticles by tensionless membranes. It is thus important to clarify under which conditions a membrane can be considered tensionless in an experimental setting. To this end, it is necessary to make a distinction between nanoparticle entry into actual biological cells, and nanoparticle engulfment by model biomimetic membranes such as vesicles.
In the case of cells, the cell membrane is typically connected to membrane area reservoirs, which are actively maintained by the cell at a constant tension. Depending on the type of cell under consideration, this tension can vary over a wide range, from Σ 0.003 mN m −1 for epithelial cells to Σ 0.3 mN m −1 for keratocytes [37]. Because tension contributions to the elastic energy of the membrane during particle engulfment are of the order of ΣR 2 , where R is the particle size, while bending contributions are of the order of κ, a membrane can be considered tensionless for the purpose of engulfment whenever ΣR 2 κ, which can be rewritten as R κ/Σ, thus setting an upper limit for the size of the nanoparticle [17]. Using a typical value for the membrane bending rigidity κ 20k B T, we find this upper size to range from 16 nm for keratocytes, to 165 nm for epithelial cells. Thus, whereas in the case of keratocytes tension cannot generally be neglected, epithelial cell membranes can be safely considered to be tensionless when it comes to the engulfment of nanoparticles with sizes in the range of several tens of nanometres.
Biomimetic membranes such as lipid vesicles, on the other hand, typically have a fixed membrane area and are not connected to any reservoir, and moreover have a fixed enclosed volume [34]. In this case, the membrane can be considered tensionless as long as the vesicle is sufficiently deflated, so that there is enough excess area for the particle to be engulfed and the membrane does not need to be stretched in order to accommodate the particle. Suppose a vesicle has initial volume V and surface area A, so that its volume-to-area ratio is which is equal to 1 for a spherical vesicle, and smaller than 1 if the vesicle is deflated. After completely engulfing a particle that comes from the exterior solution, the vesicle will have to enclose an increased volume V = V + V pa , while its surface area will have reduced to A = A − A pa , where V pa and A pa are the volume and area of the particle, respectively. Therefore, after engulfment, the volume-to-area ratio of the vesicle > v. A vesicle will have enough excess area for the engulfment of the particle, and its membrane can be considered tensionless, as long as v < 1. Whenever v > 1, stretching of the membrane is unavoidable, and the contribution of membrane tension needs to be taken into account [38].
We have shown above that the assumption of a tensionless membrane sets an upper limit to the range of particle sizes for which our results are applicable. A lower limit also exists, due to the fact that we are describing the membrane within a continuum-elastic theory which neglects the molecular, bilayer structure of the membrane. The lower limit is thus set by the bilayer thickness which is about 5 nm, and engulfment as described here is therefore applicable for particle sizes of the order of 10 nm and above. Particles smaller than that, of the order of the bilayer thickness or smaller, may instead adsorb within the hydrophilic lipid heads, get incorporated within the hydrophobic core of the bilayer, or they may translocate directly across the bilayer without being engulfed, depending on their surface properties [38].
As long as the particle size is above the lower limit set by the bilayer thickness and below the upper limit set by the membrane tension just discussed, the results described here will apply. We have predicted that the transition region separating free and completely engulfed states is located between the stability lines L tip and L side as given by equations (9) and (10). This transition region thus occurs for adhesive strengths of the order of |W| 2κ/R 2 , where R is the particle size, or equivalently, for particle sizes of the order of R 2κ/|W|. The critical particle size thus depends on the strength of particle-membrane attraction, and on the bending rigidity of the membrane. In reference [14], we found that this critical size can vary over several orders of magnitude depending on the system under consideration, ranging from 20 nm for silica particles and DMPC membranes, to 3 μm for glass particles DOPC/DOPG membranes.
Throughout the paper, we have focussed on ellipsoids as a paradigmatic example of a non-spherical shape. Of course, many other nanoparticle shapes can be engineered. Ellipsoids, nevertheless, capture essential features of non-spherical particles, such as non-uniform surface curvatures, and the distinction between prolate and oblate shapes. We expect that many of the lessons learned here for ellipsoidal particles can be applied to other particle shapes too. In particular, the stability limit (11) for the free state as a function of the particle-membrane contact point, which in the absence of spontaneous curvature coincides with the stability limit of the completely engulfed state, should be applicable to any arbitrary particle shape. Particles will thus become attached at the points of their surface with low mean curvature at low adhesion strength, and at higher adhesive strength a transition towards complete engulfment will occur. The latter transition may be accompanied by particle reorientation, so that the point of highest curvature on the particle surface will be the last to be wrapped.
Our findings may be used as guidelines for the design of nanoparticle shapes tailored to specific purposes. In particular, several important aspects of figure 3 should be stressed. For fixed particle surface area, the optimal shape for complete engulfment corresponds to a sphere, in the sense that a spherical shape requires minimal adhesive strength. However, if the goal is for the particle to only be attached to the membrane, without being completely engulfed, then nonspherical particles are preferable. In fact, oblate particles with very low aspect ratio r 1 require very little adhesive strength to become attached, thanks to their very weakly curved tip. On the other hand, and somewhat surprisingly, we find an optimal aspect ratio for prolate particles to become attached in side orientation with minimal adhesive strength, given by r 1.92. Moreover, from figure 4, we learn that spontaneous formation of deeply-wrapped partially engulfed states can only be achieved for prolate particles. These states could be experimentally useful as they involve a very large binding energy, see also figure 2(a4), but at the same time the particle is still exposed to the exterior compartment, allowing e.g. for micromanipulation from the outside or interaction with reagents added to the exterior solution.

Conclusion
In summary, we have studied here the full energy landscapes for combined wrapping and reorientation of ellipsoidal particles by tensionless membranes. These two-dimensional energy landscapes show that the wrapping process depends on the orientation of the particle when it first makes contact with the membrane. Moreover, we find that wrapping can be accompanied by a reorientation of the particle. Prolate particles at low adhesive strength become attached in side orientation, whereas at higher adhesive strength this weakly-wrapped side-oriented state becomes unstable towards a deeply-wrapped tip-oriented state. Oblate particles, on the other hand, become attached in tip orientation; this weakly-wrapped tip-oriented state coexists with a deeply-wrapped side-oriented state over a range of values of the adhesive strength, but only becomes unstable towards the completely engulfed state, at higher values of adhesion.
While the existence of wrapping-induced reorientation of ellipsoidal particles had already been predicted on the basis of a comparison of the energies of side-oriented vs tip-oriented states at different degrees of wrapping [23,27], the full energy landscapes at arbitrary orientations had not been considered before. Therefore, the stability limits of the different states, as well as the energy barriers separating them, had not been accessible until now.
We stress again that the results obtained here are for tensionless membranes. For cases in which the membrane tension is very high, with Σ κ/R 2 , we expect that engulfment will always proceed in side orientation for prolate particles, and in tip orientation for oblate particles, as these orientations minimize the tension energy of the system. A non-trivial competition between bending and tension is expected to occur for intermediate tensions Σ ∼ κ/R 2 ; however, in order to study such intermediate values, the contribution of the free part of the membrane to the energy of the system will need to be taken into account.