Dielectric properties and lamellarity of single liposomes measured by in-liquid scanning dielectric microscopy

Liposomes are widely used as drug delivery carriers and as cell model systems. Here, we measure the dielectric properties of individual liposomes adsorbed on a metal electrode by in-liquid scanning dielectric microscopy in force detection mode. From the measurements the lamellarity of the liposomes, the separation between the lamellae and the specific capacitance of the lipid bilayer can be obtained. As application we considered the case of non-extruded DOPC liposomes with radii in the range ~ 100–800 nm. Uni-, bi- and tri-lamellar liposomes have been identified, with the largest population corresponding to bi-lamellar liposomes. The interlamellar separation in the bi-lamellar liposomes is found to be below ~ 10 nm in most instances. The specific capacitance of the DOPC lipid bilayer is found to be ~ 0.75 µF/cm2 in excellent agreement with the value determined on solid supported planar lipid bilayers. The lamellarity of the DOPC liposomes shows the usual correlation with the liposome's size. No correlation is found, instead, with the shape of the adsorbed liposomes. The proposed approach offers a powerful label-free and non-invasive method to determine the lamellarity and dielectric properties of single liposomes. Supplementary Information The online version contains supplementary material available at 10.1186/s12951-021-00912-6.


S1. Comparison of the distribution of the spherical equivalent radii of the liposomes assuming a constant area and volume
The surface area, Scap, and volume, Vcap, of a spherical cap of diameter D and height h are given, respectively, by (1) The surface area, Ssphere, and volume, Vsphere, of a sphere of radius R are given, respectively, by (1) Req,S in Eq. (S3) has been used to generate the distribution of equivalent radii in Fig. 2d (reproduced in Fig. S1, red bars, for easier reference). The distribution of radii obtained by using Req,V in Eq.
(S3) is shown in Fig. S1 (blue bars). Note that the color palette of the scale is the same as in Fig. 2, but the range of values covered is different, so that we can visualize all the images in a common color scale.

S3. Equivalent complex permittivity of uni-and bi-lamellar core-shell spheroidal liposomes
in an external uniform ac electric field.
For a uni-lamellar core-shell spheroidal liposome of height h, width D and shell thickness tm, in a uniform external ac electric field, the equivalent homogeneous complex permittivity, εeq * , in the direction of the external electric field is given by (2), is the complex permittivity of the solution entrapped inside the liposome, with εlip and σlip being its permittivity and conductivity (here we do not consider polarization losses). ω is the angular frequency of the ac electric field and εm the lipid bilayer permittivity, assumed to be real (without losses or conductivity). Finally, Lz is the polarization factor given by (3) ( ) In the spherical limit, one has D=h and Lz=1/3, and one recovers the well-known relation for the equivalent homogeneous permittivity of a spherical core-shell particle (2). In the homogeneous limit, when εlip * = εm, one has εeq * = εm, as it should be. Note that the equivalent homogeneous complex permittivity of a core-shell uni-lamellar liposome depends on the size of the liposome.
For the case of a bi-lamellar liposome, one applies recursively Eq. (S6) following the scheme shown in Fig. S1 as described elsewhere (2).

Figures S3.
Recursive approach to determine the equivalent complex permittivity of a bi-lamellar spheroidal liposome.
One obtains,

S7. Geometrical model for the adsorbed liposomes.
We have modelled the cap geometry of the adsorbed liposomes by the axial revolution of a function of the form where X0 is the center of the liposome (position of the maximum) and ( ) By substituting Eqs. (S11) into Eq. (S10), one has that the shape is described by the revolution of the function This geometry is determined from three parameters of the topographic image, the height, h, the width, D, and the full width at half maximum FWHM. A spherical cap geometry also provides a good description of the measured topography, although, since it is determined from only two parameters of the topography (the height, h, and the width, D) it is slightly less accurate. Explicitly, the spherical cap geometry is given by where Rc is the radius of curvature, which for a spherical cap is given by (1) In Fig. S7 we compare the predictions of the two geometrical models with the measured topography of four adsorbed liposomes spanning the full range of sizes considered. In general, both models describe correctly the measured topography, but the phenomenological model adjusts better the profiles for larger liposomes. The relationships in Eq. (S15) together with Eq. (S12) allows generating liposome geometries of any size compatible with the shape of the adsorbed liposomes. We used this fact, for instance, to generate generic theoretical predictions, as those in Fig. 3 of the main manuscript.