Mesoscopic models of lipid bilayers and bilayers with embedded ...

40 Structural characterization of lipid bilayers
the same as in equation 4.1, but the angle φ is now defined as the angle between the
orientation of the vector along two beads in the chain and the normal to the bilayer
plane:
cos φ = rij · ^n
rij
= zij
rij
(4.2)
where ^n is a unit vector normal to the bilayer, and rij = ri − rj is the vector between
beads i and j (rij = |rij|). The order parameter has value 1 if this vector is on average
parallel to the bilayer normal, 0 if the orientation is random, and −0.5 if the bond is
on average parallel to the bilayer plane. With this definition of the angle φ, we can
compute the order parameter for a vector between any two beads in the lipid. In particular
we are interested in characterizing the overall order of the chains and the local
order. For the first quantity we define the indexes of the vector rij in equation 4.2 as:
i = tn and j = t1, where tn is the last bead in the lipid tail and t1 is the first one. We
call S tail the corresponding order parameter. For the local order we define i = tm+1
and j = tm with the index m increasing going toward the tail end, and call the corresponding
order parameter Sm. If m is taken progressively from the headgroup to the
tail-end of the molecule, a plot of the corresponding order parameters, Sm, gives an
indication of the persistence of order from the interfacial region to the bilayer core.
Area per lipid and bilayer thickness
The area per lipid can be experimentally estimated from X-ray or neutron scattering
[84, 85], or from the just described lipid order parameter profiles [86]. We compute
the area per lipid, AL, by dividing the total bilayer projected area by half the number
of lipids in the bilayer, since we find that, on average, there is an equal number of
lipids in each monolayer.
Experimentally the bilayer (total) thickness is computed from the peak-to-peak
headgroup distance measured by X-ray diffraction. In simulations the same approach
can also be used, and the thickness can be computed as the distance between the
peaks in the density profile.
To compute the thickness of the bilayer hydrophobic core, Dc, we consider the average
distance along the bilayer normal (which we assume to be the z-axis) between
the tail bead (or beads in case of double-tail lipids) connected to the headgroup of
the lipids in one monolayer and the ones in the opposite monolayer:
where zi t1
(i = 1, 2).
Dc = z 1 t1
− z 2 t1
(4.3)
is the average z position of the first tail beads of the lipids in monolayer i
It is also useful to consider the lipid end-to-end distance, Lee, defined as the distance
between the positions of the first bead(s), rt1 , and the last bead(s) rtn , of the