Wrapping anisotropic microgel particles in lipid membranes: Effects of particle shape and membrane rigidity

Significance The cellular uptake of colloidal-sized particles of biological or synthetic origin has important implications for cellular function, and for the design of particles for diagnostic and therapeutic applications in nanomedicine. Here, we present experimental data combined with theoretical modeling showing how anisotropic microgels wrap at the lipid membrane depending on the physicochemical properties of the particles and the membrane. Important properties are the bending rigidity of the membrane, the particle shape, and the adhesion energy between the particles and the membrane. Accounting for the possibility offered by microgel systems to be custom-designed, it further opens up opportunities for future fundamental studies, therapeutic applications, and self-assembly strategies which involve nanoparticle–membrane interactions.

DT is in this case simply derived as DT = kBT /fp. RH was determined for the core-shell spherical particles at 465 and 462 60 nm at 20 and 28 • C, respectively. DT was found to decrease with ρ. Assuming an isochore transformation we can estimate 2aH 61 using the hydrodynamic radius of the core-shell and the aspect ratio from the CLSM analysis, such that bH = 2(R 3 H /(bs/as) 2 ) 1 3 .

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The full lines in Fig. S4C present the expected evolution of DT for an isochore transformation considering the experimental 63 values of RH and the dashed lines a fit with RH as fit parameter. Our values are systematically lower than expected for the 64 prolate core-shell particles, which is surprising as the deformation of the particles would be expected to reduce the swelling of the shell. It does not appear to be related to the quality of the redispersion, which was confirmed by optical microscopy but rather to some variations of the particles properties during their post-processing into prolates.   Summary of the microgel characterization. γ refer to the applied deformation during the post processing of the core-shell 82 microgels into ellipsoids. 2bc, 2ac refer to the long and short axis of the particles measured by TEM in the dried state and ρc 83 to their corresponding aspect ratio. 2b and 2a are the dimensions of the long and short axis measured in aqueous solution at 20 84 amd 28°C. DT refers to the translational diffusion coefficient and µ to the electrophoretic mobility of the core-shell microgels.

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The effective charge Q ef f is estimated from DT and µ.

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2.00 ± 0.08 6.10 ± 0.63 in Movie S1-S3. The MG2 microgel particles may be oriented in any direction when approaching the membrane. However, as 91 soon as the particles are adsorbed to the membrane they orient with their long axis parallel to the membrane surface. These 92 movies thus imply that irrespectively of the orientation at which the particles approach the membrane, they adsorb with its 93 long axis parallel to the membrane. After adsorption, the particles may further reorient and become deeply wrapped as the hexagonal structure on the DOPC membrane, the average center-to-center distance between spherical microgels is 1.19 ± 0.05 107 µm. This distance is much larger than the hydrodynamic diameter of the particles pointing that the particles significantly  The influence of the membrane tension σ was investigated for the height of the barrier for the shallow-wrapped to deep-wrapped 120 transition for ellipsoidal particles, see Fig.S12. At low tensions, the barrier for the ellipsoidal particles is higher than for 121 the spherical particles with the same volume due to the high curvature of the particle tip. At high tensions, the barrier for 122 complete-wrapping a spherical particle due to the membrane tension exceeds the barrier that we predict for deep-wrapping the 123 ellipsoidal particle.

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Role of the free membrane for wrapping ellipsoidal particles at planar lipid bilayers 125 The influence of the membrane tension σ was investigated in addition to w as shown in the main text in the discussion of 126 Fig. 6. Figure S13 further illustrates the deformation of the free membrane around the particle for various σ-values.  Energy barrier for wrapping an ellipsoidal particle with aspect ratio 2 equal volume and a spherical particle (data for the spherical particle reused from Ref. (11)).
Similarly, the radius of the mother vesicle is is the bending energy of the membrane adhered to one particle, which is a piecewise function for the spherical caps and the is the sum of the bending energy and the adhesion energy.

Tip-wrapping elongated nanoparticles: membrane holes vs. blisters
Particle wrapping at lipid-bilayer membranes is favored by the energy gain through the particle-membrane adhesion, and 154 hindered by the energy costs for deforming the membrane. These two processes are characterized by the particle-membrane 155 adhesion strength w and the membrane bending rigidity κ. Wrapping of highly elongated nanoparticles is particularly disfavored 156 by the need to strongly deform the membrane next to the tips (11). However, there are two possibilities to avoid such 157 unfavourable shapes: (i) a hole in the membrane at the tip location, which generates an open membrane boundary with a 158 penalty due to the line tension of the boundary, or (ii) the formation of a membrane "blister", which encloses the tip. In order 159 to see which of the two options is favorable, we calculated and compare the corresponding energies. We consider wrapping of 160 prolate ellipsoidal nanoparticles with aspect ratio b/a and cones that mimic a particle with a sharp tip, see Fig. S15. For a blister at the tip, the bending energy is at least a cap formed by a half sphere, and at most an entire sphere, 162 4πκ ≤ Ec < 8πκ .
[9] 163 Here, only the bending energy of the blister has to be taken into account for because the neck, which connects the blister 164 to the adhered part of the membrane, can be assumed having a catenoidal shape with vanishing bending energy (13). For 165 simplicity, we will work with the bending energy for a half-spherical blister, Ec = 4πκ, in the following. The total energy for 166 the formation of a blister at the tip is the sum of the bending energy E b of the membrane attached to the particle, the blister occurs. The total energy for the formation of a hole at the tip is therefore Wrapping energies for ellipsoidal nanoparticles. The surface of the ellipsoid with the short axis a and long axis b is parametrized 174 as r(z, ϕ) = (a(b 2 − z 2 ) 1/2 cos ϕ, a(b 2 − z 2 ) 1/2 sin ϕ, z); bending energy costs can be calculated analytically (15). The mean 175 curvature is 177 and the bending energy for an ellipsoid wrapped up to detachment length ϵ is [13]

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For a hole at the tip, the line energy is [15] 183 Figure S16 shows the total energy and the energy contributions for the formation of a hole and of a blister for particles Wrapping energies for half ellipsoids with aspect ratios b/a = 2 and b/a = 10 as function of the detachment length ϵ from the tip for (a-b) formation of a hole, and (c-d) formation of a blister. The figure shows the total energy Etot, as well as the bending energy E b , the line tension for the hole E l , the blister/cap energy Ec, and the adhesion energy E ad for the parameters in Tab. S2. The bending energy of a half-spherical cap is assumed as energy cost for the formation of a blister.
Wrapping phases for ellipsoids. For the tip-wrapping of ellipsoidal particles, we predict non-wrapped and complete-wrapped 191 states, as well as pore formation and-for sufficiently high aspect ratios-blister formation at the tips. Figure S17     Wrapping energies for cones. The surface of a cone with an opening angle γ is parametrized as r(z, ϕ) = (γz, ϕ). The adhesion energy is 208 E ad (ϵ) = −wπγ 1 + γ 2 (z 2 max − ϵ 2 ) . [19] 209 Figure S18 shows the relevant energies of the system. Unlike for ellipsoids, the bending energy diverges at the conical tip, 210 such that the theory always predicts hole or a blister formation-although at different distances from the tip. Here, it has to be 211 kept in mind that the Helfrich model applies for small mean curvatures.