Chemistry Reference
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Mulet et al. (2009) has again demonstrated the presence of intermediate tran-
sient states by using both X-ray diffraction and cryogenic transmission elec-
tron microscopy. These works have settled important milestones toward the
experimental assessment of transient structures accompanying order-order
transitions.
A suitable theoretical description of transient states accompanying order-
order transitions such as the evolution of lamellar phases into inverted hex-
agonal or inverted cubic phases, has been given by Siegel in the so-called
modifi ed stalk theory (Siegel, 1999). In this approach, the stability of interme-
diate states is tested by calculating their corresponding curvature elastic free
energy [following the method proposed by Helfrich (1973)] and by inferring
that the preferred structural evolution pathway is that which maintains the
energy at its lower limit.
The Helfrich free energy consists of two terms: a term accounting for the
bending displacement from an equilibrium curvature
c
0
, and another term
associated with the bending following an imposed Gaussian curvature:
2
1
2
11
2
1
+
∫
F
=
k
+
−
c
k
dS
(1.1)
c
0
g
RR
RR
1
2
12
where
R
1
and
R
2
are the curvatures of the membrane and
k
c
and
k
g
are elastic
constants referred to as bending and saddle-splay moduli, respectively. The
term ((1/
R
1
)
(1/
R
2
)) is called the mean curvature, whereas the term 1/(
R
1
R
2
)
represents the Gaussian curvature.
Siegel was able to determine the energy of intermediate states referred to
as stalks, transmonolayer contacts (TMCs), and interlamellar attachments
(ILAs). The various steps of the structural evolutions predicted by the modi-
fi ed stalk theory are shown in Figure 1.10. In short, two lipid bilayers are
predicted to touch each other (e.g., by fl uctuation-induced contacts) and lead
to the formation of a fi rst critical step, which is a stalk. A stalk can grow in
size but at the expense of energy; a second critical step then that allows reduc-
ing energy is the transformation of a stalk into a TMC. The evolution of TMCs
into pores follows two different energetic scenarios: In the case of lamellar
→
+
cubic transition, the third critical structural change involves the transfor-
mation of a TMC into an ILA, which can then grow spontaneously in size. In
the case of lamellar
hexagonal transition, the mechanism involves the
attraction of different TMCs, which then fi nally evolve into a hexagonal
columnar lattice.
The modifi ed stalk theory has been successful in explaining a number of
experimental evidences. Nonetheless, considering the short time scales involved
in the structural changes (seconds or less) simulations are obviously to be
called up for help.
With the increasing computational power and speed, in recent years, some
specifi c features of lipid-based lyotropic liquid crystals have been correctly
captured by coarse-grained molecular dynamic simulations. One of the fi rst
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