This lecture describes qualitatively the importance of flexible membranes within cells and looks at the physical properties of bilayers (Slide 1).

Slide 1 Biomembrane deformation, governed by curvature elasticity, is important in many biological processes.

There are important examples of all of the shapes described in Lectures 3 and 4 for the self-assembly of amphiphiles. For example, spherical micelles are a fundamental structure in milk; and cylindrical micelles can exist in a dense form that is very important technologically, known as entangled worm-like micelles. Inside a biological cell (Slide 2), the bilayer is certainly the most common motif present either as extended flat sheets or as curved, closed-up vesicles.

*5.1 Biological membrane deformation*

In Lecture 4, we described equilibrium shapes for micelles, which are aggregates of surfactants (e.g. lipids). In nature, and in experiments, bilayers are often observed in the form of vesicles, which are lipid bilayers rolled up to form small sacs or pouches. Typically they are much larger than micelles. A cell membrane thickness is typically a few nanometres, whereas the overall size of a vesicle can be anything between around 50 nm and 100 μm.

A key property of lipid membranes is their ability to deform. Many different types of shape changes are seen to occur due to thermal fluctuations. Slide 3 shows some examples. Shape changes occur with constant surface area and volume.

In laboratory experiments, vesicles can be used to study various aspects of membrane physics and biochemistry, including membrane mechanics and transport processes across bilayers. Biomembrane deformation is important in many biological processes, including the formation of vesicles within cells (Slide 4), where they are part of the processes of endocytosis and exocytosis (intake and expulsion of large molecules), and for intracellular transport over both short and long distances (up to about a 1 m in nerve axons).

Slide 5 shows the membranes of the Golgi apparatus. This is an intracellular membrane organelle, present in both animals and plants, that has the function of sorting and packaging different macromolecules within membrane vesicles, which are then carried within the cell or out of the cell. It is clear that in this as well as other cell biological processes, a great number of membrane vesicles are constantly being created, moved and fused into other membranes. Phospholipid membranes also make up the external boundary of cells. In the case of animals, this is called the plasma membrane. Plants have an additional cell wall outside the phospholipid bilayer (Slide 6).

Membrane deformation occurs in order for cells to take up micron-sized objects. For example, immune systems cells (e.g. macrophages) can envelop bacteria (about 1 μm). To achieve this process, known as phagocytosis, the cell’s external membrane is pushed out to envelop the object, by cytoskeletal growth. In Slide 7, an electron micrograph has captured a macrophage cell engulfing two red blood cells via the process of phagocytosis. The red arrows point to the leading edge of a membrane lamella that is wrapping around the blood cells.

Slide 7 Electron micrograph of phagocytosis. Copyright © 1994 From Molecular Biology of the Cell by Bruce Alberts, et al. Reproduced by permission of Garland Science/Taylor & Francis Books, Inc.

Deformation may also play an important role in signalling processes (Slide 8).

Slides 9 and 10 illustrate biological examples where the mechanics of membranes and cytoskeletal filaments, as well as the process by which they are formed, play an important role. Cells themselves also deform bodily, one example being red blood cells as they pass through thin blood capillaries, as are found in the peripheral circulation. Their deformation can be studied and measured in the laboratory (Slide 9).

Cell deformation also occurs during the growth of neurons (Slide 10), a process driven by the polymerisation of the protein actin into filaments. The relatively stiff actin filaments push the cell membrane. Various physical aspects, to take account of the diffusion of actin, mechanical stiffness and intracellular adhesion forces, need to be considered for a full account of such a growth process.

For more biological detail of real cell membranes, see Alberts *et al.* (2008).

*5.2 Curvature elasticity*

This discussion is based on lecture notes by Markus Deserno, which are made freely available on his website and have also been published (2009).

Deformation of a membrane costs energy. There is typically a resistance to stretching, related to a tension in the membrane, and a resistance to bending. Often the tension in a membrane is negligible (if the bilayer is in a floppy state or has a free boundary), and then the bending elastic energy is dominant in determining the membrane shape.

A physical description of a membrane requires us to know how its energy changes when we do something to it. Stretching, bending and shear are the main types of deformation, although bilayers can also sustain more subtle deformations, such as deformations of thickness. The shear modulus of membranes is very hard to measure directly, and it is usually measured indirectly, in combination with other properties, by performing deformations that involve both bending and shearing. Phospholipid bilayers typically found in biological cells usually have a completely negligible elastic shear modulus, and only a shear viscosity, that is, these phospholipid bilayers are two-dimensional fluids. Here, "two-dimensional" is used not in the strict mathematical sense but in the physical sense where motion and structure are confined to a surface.

The stretching modulus, *K _{stretch}*, can be measured experimentally, such as in micropipette experiments. Here, a capillary of a few micrometres in diameter is attached to a patch of membrane, and the membrane is aspirated by a controlled pressure. The amount by which the membrane bulges into the capillary is then observed (see Boal 2002 for further details.) Membranes can exist tension-free or under tension. A typical source of tension can arise if the membrane is enclosing a volume of fluid with a higher osmotic pressure than the outside. If the membrane is under tension, the force required to hold it and pull it is proportional to the tension. The tension and the stretching modulus act in the same way and have the same units, but while the modulus is a property of the material itself, the tension is a property that depends on the system conditions.

A membrane also has an important mode of deformation, bending, which is independent of shear and stretching. In bending a bilayer, the average area per molecule does not change (averaged over both internal and external surfaces), and the lipids don’t have to rearrange their positions in the layer. The effect of bending is to induce tilt and splay between the phospholipid molecules. This is at the origin of the bending energy. We will consider here only a simple case, of a membrane with a planar average shape, and only small deformations, as shown in Slide 11. These deformations will clearly increase the membrane area relative to the plane (i.e. stretch), and also induce a curvature.

Slide 11 Diagram of a membrane with a planar average shape, and only small bending deformations, subject to a tension σ.

We still have not defined what the strain is, for the case of bending. Being a two-dimensional surface, there are various types of bending that could be going on at a single point, and we need to look at this in a bit more detail. The mathematical language in which this is properly discussed is differential geometry.

We will limit the treatment to very simple cases, but you should be aware that this is a very well developed part of soft matter physics (see, for example, Boal 2002 or Deserno 2009).

*5.2.1 Curvature*

Following the work of W Helfrich, the study of the elasticity of cell membranes is approached via lipid bilayers. The Helfrich model works in terms of curvature energy (free energy) per unit area of a bilayer.

We first need a mathematical description of "curvature". We will describe a general section of the membrane as a surface (which is valid given the factor of about 10^{4} differences in dimensions of the membrane and the whole vesicle) and represent it by a two-dimensional surface embedded in three-dimensional space.

For a plane curve, the curvature, *c*, at a given point has a magnitude equal to the reciprocal of the radius of an osculating circle, that is a circle that closely touches the curve around a given point. Curvature is sometimes considered as a vector, pointing in the direction of that circle’s center. The smaller the radius, *r*, of the osculating circle, the larger the magnitude of the curvature will be: a nearly straight curve has a curvature close to zero; a circle of radius *r* has curvature 1/*r* everywhere.

Equation 1

When considering a curve drawn on a surface (Slide 12), one cannot readily identify the curvature of the curve with the curvature of the surface. First one has to disentangle these two different contributions. The trick is to look at two (unit) vectors: one is the local normal vector

of the surface, and the other is the principal normal

of the curve, that is the direction in which the curve locally curves. The local curvature of the curve multiplied by the scalar product between the two normal vectors,

is a curvature that no longer depends on any property of the curve, except for its direction. This resulting curvature is called the **directional curvature**. At every point, a surface has a directional curvature in each direction.

Since there are infinitely many directions, there may also be infinitely many curvatures. It turns out that there are always two directions, and they are even orthogonal (but not necessarily unique), in which the directional curvatures take their maximum and minimum values. These directions are called **principal directions**, and the corresponding curvatures are called **principal curvatures**: c_{1} and c_{2}. It is useful to define the following parameters to denote curvature,

Equation 2

extrinsic curvature * K=2H*

Equation 3

Equation 4

Note the possibility of confusion by a factor of 2. Equation 2 makes this explicit – take care whether using *K* or *H*. Also note that **Gaussian curvature** (also known as **total curvature**) has dimensions of [1/length^{2}], unlike the "linear" curvature parameters with dimension [1/length].

For a single layer, convention defines the sense of curvature. Curvature is positive if surfactant tails point towards the centre of curvature, and negative if the tails point away from the centre. (Note, however, that some authors, e.g. Israelachvili, use a different convention.)

*5.2.2 Curvature and free energy*

The local bending energy per unit area, f_{0}, is directly related to the curvature by

Equation 5

where κ and κ_{G} are called the **bending rigidity** and **Gaussian bending rigidity**, respectively, and have the dimensions of energy. The bending rigidity κ is typically of the order of 1–20 *k*_{B}*T* in magnitude.

The total free energy, *F*, can be evaluated by integrating the distortions over the surface *A*,

Equation 6

In Equation 6, τ is the area coefficient and is related to the chemical potential of the surfactant. In a closed vesicle, it is a constant since the area is constant, and it can be ignored. The integral of the last term is also a constant, due to a theorem called the Gauss–Bonnet theorem (see Kleman & Lavrentovich pp84–8.)

*K*_{0} is the spontaneous curvature and is determined by the details of the membrane (e.g. is it a bilayer or just a single membrane?). The curvature of a free membrane is not *K*_{0} but the curvature that minimises *F*. For a bilayer made of identical monolayers, the spontaneous curvature is zero (Slide 13). However, in general, a flat structure is not necessarily the lowest free energy state.

For model systems with *K*_{0} = 0, one can write the following approximation for the energy (Slide 14),

Equation 7

For a sphere, this yields

* F*=8Πk

Equation 8

It can be shown (though it is beyond the scope of this lecture) that there is an equilibrium curvature, K_{0}, related to the difference between the optimal area per chain dictated by the head packing and that preferred by the chain stretching energy,

Equation 9

This reinforces the idea (from Lecture 4) that the state of aggregation is determined by the "shape" of the building block (see Slide 15). If *a*_{0} is larger than *v*_{0}/*l*_{c}, the spontaneous curvature will be positive, implying that the system prefers to pack with the headgroups on the outside.

For cell membranes, as opposed to vesicles, the tails may not be entirely fluid, and this sets limits to *l*_{c} that may be substantially less than the maximum tail length, and there is a curvature dependence of ε_{N}. In addition, in bilayers, the attractive and repulsive forces may not act in the same plane. For example, repulsive forces between lipid head groups will act in a plane that is displaced from the midplane of the bilayer.

Finally, for bilayers, although the curvature elasticity is the sum from each of the monolayers, since the layers have finite thickness, these curvatures are not simply equal.

*5.3 Real cell membranes*

The previous discussion uses a simple model for membranes. In real cell membranes, energy related to curvature, and the intrinsic curvature of bilayers, are important in defining the shape. Other important physical parameters of the membrane are its viscosity (fluidity) and its phase behavior. The viscosity of the membrane determines how fast objects confined to the membrane can move. Typical such “objects” are the membrane proteins, and assemblies of these proteins. Their function is to regulate membrane processes. For example, there are pumps that regulate the concentration of calcium, potassium, pH, etc across the membrane. There are membrane proteins that bind to specific chemicals and trigger a particular response. The sequence of responses is called a signalling pathway. While many of these processes are very specific, and the details are different for each chemical pathway, this is also an area where general physical principles are important. We have already seen what it takes to bend a membrane. This energy needs to be found in order, for example, to “bud” a vesicle from the membrane.

A real biological membrane is made not of one single phospholipid but by many hundreds of chemically different phospholipids. In biological membranes, the composition of inner and outer leaflets in the bilayer is not the same. These multiphase systems have so many components that a complete phase diagram is impossibly complicated to measure. However, there is a very important simplification. Phospholipids fall into two classes: saturated and unsaturated. Slide 16 (Slide 3 from Lecture 3) is a diagram of an unsaturated phospholipid. There is a kink in one of the fatty acid tails. A saturated phospholipid does not have that kink. Saturated lipids pack much more efficiently and with stronger order among themselves, giving bilayer phases with high viscosity. Unsaturated phospholipids pack with less order and have much higher mobility, hence the membrane has much lower viscosity. In many biological membranes, there is a ratio of roughly 1:1 of saturated:unsaturated. There is also a high concentration of cholesterol, which is a smaller lipid that is “dissolved” in the hydrophobic layer. Such systems have a remarkably simple phase diagram, separating into two coexisting fluid phases at a temperature below around 25 °C. Active research is in progress that aims to establish if and how the vicinity to the critical point is related to any biological role.