Elastic moduli of lipid membranes: Reproducibility of AFM measures

Highlights

• Atomic force microscopy (AFM) can be used to assess elastic properties of nano-sized liposomes reproducibility is current in same conditions.

• The Young modulus measured by AFM depends on the AFM tip shape, the cantilever force constant, and especially on the interpretative theory.

• AFM results are reproducible only when determined using the same experimental setup and the same interpretative model.

• The Shell theory provides reasonable agreement between AFM data and other experimental measures.

• Molecular dynamics simulations match the results of AFM experiments for some specific experimental conditions.

Abstract

Membrane elastic properties play a major role in membrane remodeling events, such as vesicle fusion and fission. They are also crucial in drug delivery by liposomes. Different experimental techniques are available to measure elastic properties. Among them, atomic force microscopy (AFM) presents the unique advantage of being directly applicable to nano-sized liposomes. Unfortunately, different AFM measures reported in the literature show little agreement among each other and are difficult to compare with measures of bending modulus obtained by other experimental techniques or by molecular simulations. In this work we determine the bending rigidity of Egg PC liposomes in terms of Young modulus via AFM measurements, using two different tip shapes and different cantilever force constants. We interpret the measures using the Hertz and Shell models, and observe a clear dependency of the Young modulus values on the tip properties and on the interpretative theory. The effect of the AFM tip shape is less important than the effect of the cantilever force constant, and the mathematical model has a major effect on the interpretation of the data. The Shell theory provides the closest agreement between AFM data and other experimental data for the membrane bending modulus. Finally, we compare the results to calculations of bending modulus from molecular dynamics simulations of membrane buckles. Simulations provide values of bending modulus consistent with literature data, but the agreement with AFM experiments is reasonable only for some specific experimental conditions.

Introduction

Liposomes are important drug delivery systems in pharmaceutical research (Zylberberg and Matosevic, 2016). They can be used as membrane models to study the properties of biological membranes and to investigate the effect of active agents on membrane properties (Habib et al., 2015). They are generally composed of phospholipids, and have spherical structure consisting of a bilayer and/or a concentric series of multiple bilayers separated by aqueous compartments (Bangham et al., 1965). Liposomes can be obtained by various methods and are able to encapsulate hydrophobic, hydrophilic, and amphiphilic compounds, increasing their stability and controlling their release (Sebaaly et al., 2015; Hammoud et al., 2019; Ephrem et al., 2018). Lipoid E80 liposomes formulations (also called Egg PC liposomes) are among the most commonly used liposomes in drug delivery, and can encapsulate a wide variety of compounds of various physico-chemical properties (Azzi et al., 2018; Laouini et al., 2013; Fathi-Azarbayjani et al., 2015).

Bending rigidity of liposomal membranes is a crucial elastic property, and affects the formation, stability, size, shape, aggregation, permeability, and loading efficiency of drugs in liposomes (Azzi et al., 2018; Briuglia et al., 2015; Takechi-Haraya et al., 2017; Anselmo and Mitragotri, 2017). It has been shown that the effectiveness of drug delivery, the release profile of encapsulated drug from liposomes, and the circulation time of liposomes in the blood, are all related to their rigidity (Anselmo and Mitragotri, 2017; Kloxin et al., 2010). Unfortunately, quantifying this intrinsic property is still a challenge: a number of techniques have been devised, such as micropipette aspiration, shape fluctuation analysis, and X-ray scattering, and they provide different values for the bending modulus of liposomes with the same lipid composition, with differences up to a factor of 2 (see (Bochicchio and Monticelli, 2016) for a review). Most importantly, all the above techniques are normally applied to giant unilamellar vesicles, with micrometer size, or require additives that may affect membrane rigidity(Bochicchio and Monticelli, 2016). Atomic force microscopy (AFM) has emerged as a technique to estimate membrane rigidity (Takechi-Haraya et al., 2016). The main advantage of AFM is the possibility to perform measures on native nano-sized liposomes (Habib et al., 2014). In addition, AFM can be used to scan the topography of liposome suspensions (Habib et al., 2014) and determine their size distribution (Allison et al., 2010), to determine cell adhesion (Puech et al., 2006) and cell viscoelasticity (Tripathy and Berger, 2009). AFM allows determining membrane rigidity in terms of Young modulus (E) (Evans, 1974), which can be related to the bending modulus using different theories. Unfortunately, the values of Young modulus reported in the literature for membranes having the same lipid composition are even more scattered than the values of bending moduli obtained with other experimental techniques. For DPPC liposomes in the gel phase (at room temperature), Park et al. reported a value of 81 MPa (Park, 2011), over 25 % lower than values reported by Delorme et al. ($110\pm 15MPa$) and over 60 % lower than the value reported by Takechi-Haraya et al. (227 MPa) (Takechi-Haraya et al., 2016). Even larger discrepancies have been reported for Egg PC liposomes in the fluid phase: Liang et al. 2004 reported a value of 1.97 $\pm 0.75$ MPa, one order of magnitude less compared to the value reported by Takechi-Haraya et al. (19.68 MPa) (Takechi-Haraya et al., 2016; Liang et al., 2004a). Such large discrepancies make it impossible to compare values for different experimental conditions and different lipid compositions. The dependence of Young modulus on experimental parameters (tip apex (Saavedra et al., 2020; Rico et al., 2005), the speed of indentation (Lamour et al., 2020), the bottom effect of the substrate (Chiodini et al., 2020), and the calibration of the deflection sensitivity (Schillers et al., 2017)) has been investigated before. It is clear that several factors affect the bending modulus values obtained by AFM, and it is paramount to understand the origins of the differences, and establish criteria to obtain reproducible measures.

In addition to the problem of reproducibility, it is also unclear how to compare values obtained from AFM for the Young modulus with values of the bending modulus obtained with other experimental techniques. Two theories are commonly used to convert the Young modulus to a bending modulus: The Hertz and Shell model (Greenwood and Tripp, 1967; Reissner, 1946). Starting from the same Young modulus, the two theories yield different values of the bending modulus. The origin of the discrepancy is obviously the difference between the theories, but the question remains: which theory can better estimate the bending modulus?

In this work, we first describe how to achieve reproducible measures of the Young modulus using a common AFM apparatus. We repeated the measures on liposomes of approximately the same size, using four different kinds of AFM probes differing by the shape of the tip and the force constant of the cantilever. We determined which parameters affect the measure, and established a set of conditions allowing reproducible measures of the Young modulus.

In the second part of the work, we compare two common theoretical treatments used to calculate the bilayer bending modulus from the Young modulus. We also compare the two theories with experimental results obtained by other techniques and by molecular simulations. Our analysis shows that only one of the two theories provides a value of bending modulus compatible with literature data and with simulations.