Mechanics of lipid bilayers

Definition of the mechanics of a lipid bilayer

Systematic studies of lipid bilayer mechanics were initiated in the 1970s by W. Helfrich (theoretical) and E. Evans (experimental), aiming at decomposing the bilayer deformation and extracting bilayer elastic parameters (102-104). In their early work, the deformation of a surface was defined with 3 independent elements: 1) area dilation or condensation, 2) in-plane extension at constant area, and 3) bending without change in rectangular shape as shown in Fig 1.2.2.

Fig 1.2.2 Illustration of surface deformation decomposed into 3 independent elements. (a) Area dilatation, when the number of molecules per unit surface area decreases as the surface area increases,

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(b) In-plane extension without surface area change, (c) Bending without change in planar shape. The three independent shape changes compose of any deformation of the surface. The image is taken from (105).

At constant temperature, the first element can be quantified by a parameter called area compressibility Ka. It describes the ratio of the fractional change in surface area at constant temperature and tension.

The second element can be quantified by a parameter called surface shear rigidity μ. The third element can be quantified by a parameter called bending rigidity Kb. According to (105), the surface shear rigidity for liquid membranes equals to 0, because the phospholipids can freely diffuse through the membrane.

For a three-dimensional isotropic material, the bending rigidity Kb is:

𝐾_𝑏= 𝐸𝑡³ 𝛼(1 − 𝜐²)

(1.2.1) where t is the thickness of the bilayer, E is Young's modulus in Pa and ν is the Poisson's ratio (103). For a material that consists of two leaflets that can slide without friction, the factor α is set to be 48, for limited friction 24 and for unlimited friction 12. We set α as 24.

For simple elastic models, when the stress is assumed to be distributed across the bilayer, Kb can be ν, the Poisson's ratio, is set to be 0.5 (106), therefore:

𝐾_𝑏=𝐸𝑡³ thickness of lipid bilayers ~5 *10^-9 m) can be understood as the effective thickness of the lipid bilayer.

The two leaflets are replaced by a thinner single sheet to maintain the same bending rigidity. This approximation is made in order to meet the criteria of equation (1.2.1), which assumes the lipid bilayer as a three-dimensional isotropic material. This assumption allows us to model the lipid bilayer as an isotropic material with finite element methods.

27 Determination methods of the mechanics of lipid bilayers

The mechanical characterization of lipid bilayers and SUVs is important. For example, SUV are known for being less stable than GUVs and planar bilayers (107) as is reflected by their lower phase transition temperature and other physical parameters. However, ideal liposome-based containers should be stable and have long lifetime in the human body (108). If the mechanical properties of SUVs can be quantified, stabilization schemes can be applied to achieve these goals. Also for this reason the choice of a lipid bilayer as a component of the genome-protecting envelope of viruses such as influenza is striking:

Influenza, a 100 nm diameter enveloped virus, was shown to be able to persist for days in rather harsh conditions (31), but unexpectedly (109) its lipid membrane is thought to be rather fluid and soft over a large range of temperatures (110). It therefore has to be determined whether the lipid envelope of the flu virus on its own can act as a barrier that is as effective as a protein capsid, or if it requires the participation of viral protein to fulfill its protective role. Studying the stability of SUV and related organelles/viruses is of great interest for both fundamental and applied purposes, as this may lead to a better understanding of biological problems such as the assembly and stability of enveloped viruses, as well as new solutions to stabilize liposomes as drug carriers. Unfortunately, almost no quantitative information on the mechanical properties of small liposomes exists so far, contrary to GUVs, which have been studied for 30 years (111).

Studies of vesicle mechanics focus on the quantification of the elasticity of the bilayer and correlating this to the biological role of the vesicle. Although the mechanics can be measured with multiple techniques, the principle is similar: to quantify the passive or active deformation of the object into the two key quantities, area compressibility Ka and bending rigidity Kb. The first active deformation technique applied micropipette aspiration to osmotically swollen red blood cells. Small reversible displacements of the cell projection in the pipette in response to the applied pressure was recorded and converted to area compressibility (102, 103). Other active deformation techniques used an electric field (112), magnetic field (113), or optical tweezers (114) to deform the object. The first passive deformation technique, observed by phase contrast microscopy, measured the thermal fluctuations ofegg lecithin bilayers caused by Brownian motion, and calculated the elasticity form the fluctuations. Many further measurements were based on this technique (115-117). Because these techniques use conventional microscope methods, the smallest size of the samples was limited to ~5 μm.

SUV are by definition liposome with a size goes up to 100 nm. Those highly curved, closed lipid bilayers are too small to be measurable by optical microscopy or micropipettes. AFM, a nanometer resolution microscopic method, images the object by direct contacting it with a very sharp tip (diameter~5-100 nm) and is also capable of measuring the mechanical properties of the sample by deforming it. In the early measurements, the mechanical properties of liposomes made from Egg PC were directly measured by AFM and were compared to that obtained by other methods. It was also shown that the liposomes can be punctured by the tip, which is reflected by the kinks shown on the force curve (118). AFM is also applied to study the pore-spanning lipid bilayers (119), which is made from GUVs and can be used as a model to study the mechanical properties of cell membranes (120). In this thesis, we have set up a precise, AFM-based force spectroscopy method to quantitatively study the mechanical properties of

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SUVs. The principle is shown by a simple schematic drawing in Fig 1.2.3. Details about the method are available in section 1.3.

Fig 1.2.3 Schematic representation of indenting a liposome with an AFM tip.