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Warning:The following models
are all 2D. They have all the limitations of 2D, therefore may not
be appropriate for use in chemistry and biology, where the accuracy
of structure is of the fundamental importance. However,
when cautiously used, they can be used to explain certain dynamical
behavior of molecules, such as bond vibrations and conformational
transformations.
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Bilayer Membrane (e.g. lipid)
One of the goals of lipid simulations is to explain
why and how the chain amphiphiles that form the bilayer structure are
able to remain perpendicular to the membrane surface. If these molecules
intertangle with each other, ion and water channels through the bilayer
(e.g. cell membrane) might not be able to come into being, neither can
they remain open for a long time if they ever exist. By use of molecular
mechanics, structural stability of bilayer can be investigated.
Like early lipid simulations, our bilayer model is based on
the united-atom approximation, namely, the CH2 group at each kink of
the chain amphiphiles is modelled by a neutral united atom. The
head group is also represented by a united atom. The interactions of the
head group with solvent water molecules are not explicitly modelled. In
order to mimick these complicated hydrolation interactions that actually
help stabilize the lipid structure, the head group is restrained by a
harmonical potential (following van der Plog and Berendsen's pioneering
work: Molecular dynamics simulation of a bilayer membrane, J. Chem. Phys.
V. 76, 3271, 1982).
The model shown above is a bilayer membrane
made of stearic molecules, with an appropriate density of head groups on
the surface.
The upper left picture shows at low temperature
the bilayer exhibits a collective tilting behavior.
The upper right one shows
that at higher temperature the alignment of the stearic molecules become
disordered, with an average alignment orientation nearly normal to the
membrane surface. This was also the key observation of van der Plog and
Berendsen, though our Hamiltonian is distinct from theirs.
(NOTE: In the upper left picture the same tilt angle does not actually
extend all over the bilayer, the kink line on the right indicates where
the alignment starts to change its direction, but this is due to the fact
that we have used the reflectory boundary conditions instead of the periodic
boundary conditions. If using the periodic boundary condition, we shall
not observe the changing of the tilt angle.)
Another evidence for
the collective tilting motion is the kinetic energy distribution and its time
evolution when all the stearic acids start from the normal-to-surface
alignment with zero initial velocities. The kinetic energy distribution
evolution indeed has a 'waving' behavior, namely, part of the atoms have
significantly higher velocities, as is shown in the lower left picture above.
These 'hot spots' --- the atoms on which the kinetic energy concentrates,
propograte back and forth between the left and right ends, suggesting
the existence of a spatial density wave in the bilayer structure.
The lower right picture
illustrates that when the density of head groups on the membrane surface is
low (here we allow the user to change the density by adjusting the van der
Waals radii of the atoms instead of varying the volume), the inter-chain
forces become so weak that the bilayer structure cannot exist. This may
be helpful to the understanding of cell burst in cell lysis disease---
Stretching of cell membrane due to cell swelling caused by
over-osmosis weakens inter-amphiphilic
forces and finally cleaves the membrane.
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Vesicles
Vesicles are globular structures. For example, in solution
micelles can be assembled from some chain amphiphiles that have the head
groups all pointing to solution and the hydrocarbon tails inside.
In hydrophobic environment such as oil, inverted micelles can be formed ---
the head groups prefer to stay inside whereas the tails prefer to be exposed
to the medium. Like the lipid
bilayer model, the vesicle model of the Oslet does not attempt to model the
medium-head group and -tail interactions explicitly either, and the
united-atom simplification was used too (which means that an 'atom' in the
model should be viewed as a unit). Again, the emphasis
is on the inter-amphiphilic interactions and its role on stablizing the
globular structure, i.e. the molecular mechanics for the micelle structure.
Perhaps the most interesting feature of our idealized model is a
striking dynamical behavior that such a structure exhibits when no
stochastic factors are introduced to the dynamics. (NOTE: This is, of course,
unrealistic because micelles always collide with solvent atoms randomly,
however, the dynamical behavior of a system in the absence of thermal
noise, which is sometimes called intrinsic vibrational modes by theoreticians,
is believed to be more important than stochastic perturbations in determining
the thermodynamical activities of a structure.)
In the above pictures, the two in the second row are kinetic energy
distributions at different times. Since the structure is axisymmetric and
the velocities of all the atoms were set to zero at the beginning, the time
evolution of this distribution is axisymmetric. The third row show the
velocities
distributions at different times. One can observe that sometimes the
velocites of atoms at the same radial distance to the center all become
tangential,
while velocity vectors at different circles point to opposite tangential
directions,
suggesting the existence of a collective twisting mode. On the other hand,
the cases that all the velocity vectors are normal to the surface correpond
to the axial stretching mode (one of the breathing modes). The fourth and
fifth rows show the force distributions at four different times. As are
the velocity vector distributions, they are axisymmetric too. The
left picture in the fourth row shows that the contraction tendency of
the radial distance between the two outermost layers of the vesicles.
the right picture in the fourth row shows that the forces want to
whirl the vesicles around. In the left picture of the fifth row, the
resultant forces acting on the two outermost layers just become tangential.
As usual, the user can change the van der Waals parameters of the atoms
to see what happens. The left image of the last row proves chains consisting
of much smaller atoms cannot form a globular structure of the present size,
whereas the right image shows that stronger repulsions caused by increasing
the van der Waals radii of the chain atoms result in an enlargement of the
innermost circle. This is because the inter-chain repulsive forces want to
disintegrate the vesicles, but the inter-chain attractive forces want to
put them together, so the result of the compromise between attractions
and repulsions is an increment of micelle size, as a response to the
increasing of chain atom sizes. If the repulsions become great enough,
such that the medium-head group and -tail interactions cannot compete
with them, the micelle will be disrupted.
Theoretically, one can study the relationship of stability
of micelles with temperature (the thermal expansion, or the susceptibility
to thermal effects), the van der Waals radii (or equivalently, the number
of monomers), and the length of the chains (one can conjecture
that if the chains are too long spherical micelles cannot form, though they
might form nonspherical micelles).
Interestingly, when solvent molecules are explicity present, the head
groups will form with water molecules a thin hydration layer, in which
some of them may penetrate a little bit into the hydrophobic tail region.
We can imagine that it is this hydration layer that cements the architecture
of micelles from the outside.
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