This section describes basic mechanical and statistical-mechanical properties of lipid bilayers, which are the basic structures making up cell membranes.

Lipid molecules found in cell membranes are amiphilic molecules, with one water-soluble end, and one hydrophobic end. The water-soluble end is usually charged, while the hydrophobic end is usually one or more hydrocarbon chains. The lipids found in cell membranes are mostly molecules with two hydrocarbon chains of about 10 carbons.

The charged ends havce lowest free energy when in contact with the surrounding water. The hydrocarbon tails have lowest free energy if they are not in contact with the surrounding water. They can eliminate most of their contact with water if they organize into bilayers as shown above. The thickness of a bilayer is usually about 5 nm = 50 Å. When closed into `bubbles', bilayers provide a barrier between `inside' and `outside'; i.e. they define closed `compartments'. Large bubbles (microns in diameter) are often called `vesicles'.

Inside cells, small vesicles of » 70 nm diameter act as `packages' to carry proteins and other molecules from place to place.

Lipids are free to move around on the two-dimensional surface of biological membranes. The lipids therefore form a sort of two-dimensional `liquid'. This liquidity allows the membrane to bend, and therefore to undergo thermal fluctuations. Thermal fluctuations of membranes are sometimes called `flickering' fluctuations, due to their appearance in the microscope.

Now we forget about chemical structure, and consider the free energy associated with slight deformations of the membrane.

First let's think about stretching the membrane. If we stretch the membrane, we will force more of the interior hydrocarbon chains to contact more water, and this will cost free energy. If we double the surface area of the membrane, then the free energy cost should be on the order of about 3 k_B T per lipid.

Since each lipid only occupies about 0.4 nm² of surface, the energy per area associated with doubling the area is roughly E_{stretch-double}/Area = 7.5 k_B T/nm². This quantity has the dimensions of energy per area, or surface tension. In MKS units, we thus estimate E_{stretch-double}/Area = 0.03 J/m².

In linear elastic theory, we would expect this free energy to have the form

Note that the area-stretching elastic constant K has the units of surface tension (energy per area, or force per length), and that 1 J/m² = 1 N/m = 1000 erg/cm² = 1000 dyne/cm.

According to our previous estimate for doubling the area (so DArea = Area ), K º Y h = 2 ×0.03 J/m² = 0.06 J/m². Equivalently, the area-stretching Young modulus is Y = ^K/_h where h is the membrane thickness of about 5 nm. For our estimate above, this indicates Y = 1.2×10⁷ Pa = 12 MPa.

The modulus of the membrane is lower than that of folded DNA or protein because the origin of the free energy is mostly at the surface - the interior region of the bilayer is more or less a liquid of hydrocarbon.

Bending Elasticity

Now we apply the same calculation as for a rod (same figure) to find the energy of a simple cylindrical bend. The energy per area is just the integral across the bilayer of the strain which varies from +h/(2R) to -h/(2R).

For our estimate of Y = 1.6×10⁶ Pa and h = 5 nm, we have a bending stiffness of about k_c = 1.3 ×10^-19 J = 30 k_B T.

Note that the units of the bending elastic constant are simply energy. There is no characteristic length scale associated with bending a membrane. We will see that this gives rise to some bizarre effects.

Of course a membrane can be bent in two directions, which may have different bending radii R₁ and R₂. For a saddle shape, these two radii can even have opposite signs! The correct generalization of the simple calculation for a cylindrical bend contributed above just replaces 1/R = (1/R₁ + 1/R₂), giving us the general bending energy formula

The sum 1/R₁ + 1/R₂ is often called the mean curvature.

Some Elastic Constants of Model Membranes

Just to give an idea of the order of magnitude of the stretching and bending elastic constants of real membranes, here are some data from E. Evans and W. Rawicz taken around room temperature, for commonly studied phospholipids (from Phys. Rev. Lett. 64, 2094-2097 (1990)):

   Lipid        kc (10^-19 J)      K (N/m)
DAPC (18 C)       0.44              .135
DGDG (23 C)       0.44              .160
DMPC (29 C)       0.56              .145
SOPC (18 C)       0.90              .190

As you can see, our `pure thought' estimates of above are vaguely reasonable. However, note that they overestimate k_c while underestimating K. This inconsistency is not surprising - we have tried to base k_c and K on a single elongational modulus Y - but lipid bilayers are absolutely not isotropic elastic solids! Lipid bilayers are actually liquids in the plane, and are highly inhomogenous and anisotropic. So pure elasticity thinking can only get you so far. Reality demands that we think about separate elastic constants K and k_c.

Energy to Bend a Membrane into a Sphere

Suppose we assemble some lipids into a spherical bilayer of radius R. Everywhere on the bubble, the mean curvature is 1/R + 1/R = 2/R, so the total bending energy is just

This is a consequence of the lack of a characteristic length scale in the bending elastic constant k_c, which gives bilayers the weird property of scaling invariance of their bending energy. You can easily prove that two vesicles of the same shape, which differ just by the overall size, have the same bending energy.

The value of bending energy for real membranes is big. For DMPC from the table above, we have
E_bend = 8 ·p·0.56 ×10^-19 J = 1.4×10^-18 J = 350 k_B T

This tells us that by themselves, vesicles will be pretty stable once made! To have a `pinching' fluctuation of a large vesicle will involve an excitation energy of hundreds of k_B T, and therefore will essentially never be observed.

Energy to Bend a Membrane into a Cylinder

Now let's make a cylinder of length L and radius R. Ignoring spherical end caps (which will contribute 8pk_c to the bending energy following the above) we will have a mean curvature everywhere of 1/R, and therefore a bending energy

To pull a cylindrical tubule from a vesicle (which happens all the time inside cells) requires work done per length of tether of

If area is nearly conserved during the pulling of the tubule from the vesicle (which is the case, thanks to K) then the radius R of the tubule must be really small. In fact, the of these kinds of tubules is about R » 50 nm, with this size set by essentially the onset of strong nonlinear elastic effects. Plugging this into our force formula tells us that the characteristic force needed to pull a tubule from a DMPC vesicle is very roughly f = 3.5 pN.

These sizes of tubules are generated inside a complicated membrane structure inside eukaryote cells by the endoplasmic reticulum. To the best of our current knowledge, the tubules are being made continually by protein motors which apply forces of a few piconewtons to pull small tubules out of larger bilayers.

Proteins `floating' in cell membrane

Cytoskeleton underlying membrane: