Biomedical Engineering Reference
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diphosphate. In the case of the monophosphate, the phosphate was located only at
the reducing end of the disaccharide. From known chemical and physical data it ap-
peared that stoichiometric ratios existed between the different chemical entities of
lipid A-phosphate, but they have not yet been elucidated. However, on the basis of
the experimental X-ray diffraction, LS, and quantitative electron microscopy data the
structures were found to be lipid A-phosphate “quasicrystals” (Faunce and Paradies,
2006; Faunce et al., 2003b, 2011; Torquato and Stillinger, 2010). These quasi crystals
can also demonstrate non-crystallographic packing of non-identical lipid A-phosphate
spheres. A spatial packing of these spheres in for example cuboctahedron or icosahe-
dron represent reasonable physical models. Other possible spatial arrangements (Tor-
quato and Stillinger, 2010) may also exist. Many aperiodically ordered materials are
known to possess 5-, 7-, 8-, 10-, and 12-fold symmetry most of these were prepared
by supercooling multi component liquids or melts (Lee et al., 2010), which show a s
phase (Kasper and Frank, 1959). A BCC phase was also discovered following an in-
crease in temperature which gave rise to subsequent dodecagonal quasicrystals which
form spherical particles. It is possible that the distinct formation of discrete (self)-
assemblies, formed by large and thermodynamically stable quasicrystals may add
information on for example local icosahedral ordering. This would also be the case
for single component lipid A-phosphates (identical subunits) or for example, a two
component system where another component acts as a copolymer or a non-identical
subunit. The equilibrium structures of such systems with respect to lipid A-phosphates
built from coordinated spherical or ellipsoidal polyhedral, as described by Frank and
Kasper in 1959, may also be controlled by components of the molecular shape and
branching, and applicable to the various subunits of lipid A-phosphates.
From X-ray diffraction traces and the electron diffraction pattern obtained for for
example lipid A-diphosphate (identical subunits), the material crystallizes to form a
BCC cubic lattice (
a
= 36.1 nm). This BCC lattice was associated with the spatial
accommodation of identical spherical (or slightly ellipsoidal) particles in the first ap-
proximation. According to the form-factor scattering at high Q, there was a spherical
domain of size R = 7.31 nm. Assuming identical spherical lipid A-diphosphate mor-
phologies and a mass density of 1.02 g/cm
3
(Faunce and Paradies, 2008), there were
on the average 28-30 lipid A-diphosphate clusters present in the initial BCC lattice,
according to the Bragg scattering. Therefore, each spherical lipid A-diphosphate clus-
ter contains approximately 514.5 lipid A-diphosphate molecules or 18.4 molecules per
aggregate. The spatial domain of R = 7.31 nm can incorporate almost 32.9 hexagons
when applying the previously determined unit cell dimensions of
a
= 3.65 nm,
c
= 1.97
nm and α = 120° (space group R32), with eight lipid A-diphosphate molecules per unit
cell, or three molecules per rhombohedral unit cell with
a
rh
= 2.25 nm and α
rh
= 66.7°.
Real space electron density maps constructed form X-ray diffraction and selected area
electron diffraction data and using the Rietveld method, the space group R32 (Faunce
et al., 2003b), for a cluster domain R = 7.31 nm. This was for a set of 30 cluster in the
BCC unit cell, where 12 have the coordination number 12, (Wyckoff positions 2b and
8i), and 16 have the coordination number 14 (8i and 8j), and four with the coordina-
tion number 15 (4g). In a comparison between the simulated and experimentally I(Q)
versus Q pattern, good agreement was revealed between the modeled and experiment
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