Lipids in Membranes - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 37

chains, up to 8 carbons in length, such as detergents, form micelles when a critical
micelle concentration is reached.
A theoretical analysis and relevant experimental investigations [ 7 ] indicate that
the monolayer's intrinsic curvature fundamentally distinguishes the bilayer from
the non-bilayer lipids due to lipid phase properties. A quantitative shape concept
involves a determination of the free energy per lipidmoleculewhen it occupies a given
molecular volume of a given shape [ 15 ]. Here, it is assumed that the free energy per
molecule may be partioned into components arising from the elastic bending of the
lipid monolayers and hydrocarbon packing energies, besides other necessary lattice-
specific energy components which come from hydration and electrostatic potentials.
Lipid monolayers are practically flat in the lamellar L α phase (see Fig. 3.2 ) but rolled
into tight cylinders in the non-lamellar H II phase (see Fig. 3.3 ). The theory suggests
the following form for the elastic free energy:

k 1

2

1
R 0

μ E =

R −

(3.1)

Here, k is the elastic constant and R is the radius of curvature of the lipid/water inter-
face. For a cross-section of the membrane surface at a point under consideration,
using two planes that are perpendicular to the surface and oriented in two special
directions called the principal directions, the principal curvatures are the curvatures
of the two lines of intercepts between the planes and the surface which have almost
circular shapes in close proximity to the point under consideration. The radii of
these two circular fragments are called the principal radii of curvature. For simplic-
ity we have considered both to be R . R 0 is the equilibrium value or the intrinsic
radius of curvature, which is defined self-consistently as the radius of curvature that
minimizes R 0 . R 0 is a sole prop-

erty of the lipid. Widening the splay of the lipid tails accounts for a decrease in R 0
while increasing the effective head group area accounts for an increase of R 0 (see
Fig. 3.4 ). In thismodel [ 7 ] hydrocarbon-packing constraints are thought to prevent the
expression of large radii of curvature. Depending on the lipid phase, not all hydrocar-
bon lipid tails may have the same relaxed lengths. As an example, Fig. 3.3 illustrates
this in H II phases. In this case, the hydrocarbon-packing free energy may be very
large. That is why in curved structures a competition between the packing free energy
and the elastic free energy (Eq. 3.1 ) is unavoidable, and this competition appears as a
general phenomenon associated with local expression of the intrinsic curvature. It is,
therefore, clear that the lipid phase properties are lipid structure-specific, because the
energy contributions depend highly on the specific lipid head group and tail geome-
try, as well as the organization of the biological environment. The existence of lipids
in lamellar or non-lamellar phases therefore follows from the participating lipid's
physical properties. These phase preferences can be experimentally investigated by
measuring the thermodynamic transition temperatures for the lipid mixture between
lamellar/non-lamellar phases, or between different substates within both lamellar
and non-lamellar phases. A calorimetric study is the best experimental tool in this

μ E . In an elastically relaxed lipid monolayer R =