Thermal fluctuations of the lipid membrane determine particle uptake into Giant Unilamellar Vesicles

Phagocytic particle uptake is crucial for the fate of both living cells and pathogens. Invading particles have to overcome fluctuating lipid membranes as the first physical barrier. However, the energy and the role of the fluctuation-based particle-membrane interactions during particle uptake are not understood. We tackle this problem by indenting the membrane of differently composed Giant Unilamellar Vesicles (GUVs) with optically trapped particles until particle uptake. By continuous 1 MHz tracking and autocorrelating the particle’s positions within 30µs delays for different indentations, the fluctuations’ amplitude, the damping, the mean forces, and the energy profiles were obtained. Remarkably, the uptake energy into a GUV becomes predictable since it increases for smaller fluctuation amplitudes and longer relaxation time. Our observations could be explained by a mathematical model based on continuous suppression of fluctuation modes. Hence, the reduced particle uptake energy for protein-ligand interactions LecA-Gb3 or Biotin-Streptavidin results also from pronounced, low-friction membrane fluctuations.


Supplementary note 1: Membrane fluctuations with and without bead
The Supplementary Figs. S1 and S2 represent an image composition from two movies of GUV fluctuations without bead indentation (Supplementary Figs. S1 a and S2 a) and with bead indentation (Supplementary Figs. S1 b and S2 b). Three horizontal and one vertical line scans represent kymographs of the fluctuating, fluorescent membrane. The total widths 2 | ( )| hq  of all fluctuation modes q as well as the mean displacements from the bead indentation can be roughly estimated from the fluorescence movies. However, the measurement of precise fluctuations in time and space is only possible through interferometric thermal noise tracking.

Supplementary note 2: fluctuation width of planar and spherical membranes
In Supplementary  of the sum of N independent coupling modes (bottom graph). Here the dependence on g R represents the radial spherical fluctuations. For the experimental relevant parameters, the difference in the fluctuation widths is negligible.

Supplementary note 3: change of relaxation times by adding or suppressing fluctuation modes
The relaxation time of summing up the modes depends on the number ranging from n 0 to N mx . The change in the slope depends on the relation between the optical trap stiffness  opt and the membrane parameters like GUV radius g R , bending rigidity K or membrane tension σ. This is illustrated for five examples in the following Supplementary Fig. S4: Supplementary Fig. S4: The relaxation times (AC times  c ) of a bead in contact with the membrane increase with increasing lower mode number n 0 , corresponding to subsequent exclusion of lower modes. This is true for most, but not all GUV / membrane parameters and trap stiffnesses (see the case for a weak optical trap, black dotted line). The maximum mode numbers are N mx1 = R g1 /R b = 15 and N mx2 = R g2 /R b = 26.

For the parallel connection between bead and membrane the Langevin equation in Fourier space is
Hence, the response function of both the viscoelastic behavior of the optically trapped bead and all membrane modes (serially connected) can be written as: The squared modulus of the total response function is (see Supplementary Fig. S5 In the high-frequency case  > opt bd opt     , i.e. on short timescales, we find: If the membrane viscous drag is much larger than that of the bead Fig. S5 c) the response function reads: Fig. S5: Comparison of response functions for a trapped bead, for the membrane and the combination of bead and membrane. a Decaying real part (red), increasing imaginary part (blue) and slowly decreasing modules (green) of the total response function from the bead connected to membrane. The case of a trapped bead only is shown in black. b The power spectral density, being proportional to |()|², reveals the same high frequency dependence for a tense and an extremely flaccid membrane (K0, 0), which is mainly determined by the viscous drag. c The imaginary parts dominate the fluctuation response in the high-frequency case, where membrane parameters (K,) become unimportant.
Hence, the total response is purely determined by friction Without bead and optical trapping forces the bead displacement is: If  mem = 0 or in the high-frequency case, the membrane response

Supplementary note 5: Model for membrane deformation by a particle
Bending and stretching the membrane costs energy (and force), which are proportional to K for bending and to  for stretching lipids relative to each other, as described by the Helfrich free energy 2,3 . Here h() is the height function describing the shape of the membrane, which can be described as a function of the polar angle relative to the axis of indentation: Using fluorescence microscopy, one can see two different types of GUV deformations 4 . As sketched in Fig. 2e and Supplementary Fig. S6, there is a global deformation from the round GUV with radius R g to an ellipsoidally deformed GUV (with half axes a and c) caused by force F opt of the trapped particle. This deformation, defined by the mean free energy () glo gl Gh, can be quantified by the global indentation distance h gl . In addition, there is a local deformation caused by the bead, which generates a toroidal membrane indentation with torus minor radius R c . Two circles with appropriate radii R c fit well to fluorescence indentation profiles of the GUV (Fig. 2e bottom).
The local indentation distance is denoted as h lo and is defined by the local free energy ()

loc lo
Ghof the membrane. The corresponding damped (non-linear) springs form a serial connection with each other, but a parallel connection with the optical trap, as outlined in Fig. 2e  , as required for a serial connection of forces.

Supplementary Fig. S6: Deformation schematic for the bead indentation into a GUV.
A global deformation results from the change of a spherical vesicle (see green dashed line) with radius R g to an elliptically deformed vesicle with half axis a and c. For this situation, the global indentation height of the bead is h gl (), which is a function of the polar angle . The additional local deformation results in a local shape, which can be described by the surface of a torus with a minor torus radius R c (indicated by the red dotted lines). It is assumed that the torus radius R c is not changing during the indentation expressed by the local height h lo ().
The Helfrich free energy 4 in Eq. (9) here, G ad (h lo ) = 2πwR b h lo is the adhesion energy with surface energy density w, which will become important for specific biological binding partners used in the study.
In the following we explain how to calculate the local bending and stretching energies and the global bending and stretching energies depending on their indentation lengths h gl and h lo , respectively.

The global deformation
It considers the change of the spherical GUV of radius R g to an oblate ellipsoid shaped GUV with half axis a, and c. From these two half-axis, the ellipticity eps(h) is defined and thereby the increase of the GUV surface area, resulting in the global stretching energy with surface tension σ   Experimental membrane parameters ( 1 , K 1 , ), (0.5 1 , K 1 , ), ( 2 , K 2 , ) are chosen as shown in Supplementary Fig. S7. Supplementary Fig. S7: Experimental values used for the simulations. For exemplary calculations in MathCad 15 we used two different trap stiffnesses  opt,1 and  opt,2 , two different membrane surface tensions  1 and  2 , and two different bending rigidities K 1 and K 2 .
In Supplementary Fig. S8 the global stretching and bending energies are compared in the upper graph and the corresponding forces in the bottom graph. It can be seen that the global stretching energy rises stronger than the global bending energy at low indentations h gl , whereas bending energies and forces dominate for larger h gl values. Supplementary Fig. S8: Global stretching energy/force and global bending energy/force. Global energies (in k B T) and forces (in pN) for stretching and bending are compared for three GUVS, with different radii (R g1 and R g2 ), bending rigidities (K 1 and K 2 ) and surface tensions ( 1 and  2 ) as a function of global indentation height h gl in µm. The sum of both global deformation energies is shown up to 3000 k B T. Supplementary Fig. S9 shows the sums of global energies and global forces. Hence, the global deformation energy rises smoothly due to initial stretching and then more steeply due to the high bending energy costs.

The local deformation
It considers the energy costs for bending and stretching the membrane when a flat circular area is deformed into a spherical surface area defined by the radius of the bead and a torus surface area defined by concave (inner) torus radius, as shown in Supplementary Fig. S10. By analyzing the fluorescence indentation movies (see Fig. 3a), we found that the torus radius is approximately 2.5 times the bead radius for the smaller, tense GUV (2 = 15µm) and is approximately independent of the indentation height, i.e. R C1  2.5R b . For the larger, flaccid GUV (2 = 26µm), the torus radius is approximately five times the bead radius i.e. R C2  5R b, once more approximately independent of the indentation height. As sketched in Supplementary Fig. S10, the local indentation height is a function of angle  according to and where .
As shown in Supplementary Figs. S11 and S12, the area of the flat circle and the torus begin to decrease for  > 90° such that () lo A h   0 at the length max lo h , which is determined numerically, but can be approximated by max The increase in cap plus torus area relative to the flat area with  can be written as: and is illustrated in Supplementary Fig. S11 at the bottom. The increase in membrane area with indentation height h lo for the sphere, the torus and the projected flat area is shown in Supplementary Fig. S12