start/mdRipple/result.tex

Snap shots show that the membrane is more corrugated with increasing
the size of the head groups. The surface is nearly perfect flat when
$\sigma_h$ is $1.20\sigma_0$. At $1.28\sigma_0$, although the surface
is still flat, the bilayer starts to splay inward, the upper leaf of
the bilayer is connected to the lower leaf with a interdigitated line
defect. Two periodicities with $100\AA$ width were observed in the
simulation. This structure is very similiar to OTHER PAPER. The same
structure was also observed when $sigma_h=1.41\sigma_0$. However, the
surface of the membrane is corrugated, and the periodicity of the
connection between upper and lower leaf membrane is shorter. From the
undulation spectrum of the surface (the exact form is in OUR PREVIOUS
PAPER), the corrugation is non-thermal fluctuation, and we are
confident to identify it as the ripple phase. The width of one ripple
is about 71\AA, and amplitude is about 7\AA. When
$\sigma_h=1.35\sigma_0$, we observed another corrugated surface with
79\AA width and 10\AA amplitude. This structure is different to the
previous rippled surface, there is no connection between upper and
lower leaf of the bilayer. Each leaf of the bilayer is broken to
several curved pieces, the broken position is mounted into the center
of opposite piece in another leaf. Unlike another corrugated surface
in which the upper leaf of the surface is always connected to the
lower leaf from one direction, this ripple of this surface is
isotropic. Therefore, we claim this is a symmetric ripple phase.


The $P_2$ order paramter is calculated to understand the phase
behavior quantatively. $P_2=1$ means a perfect ordered structure, and
$P_2=0$ means a random structure. The method can be found in OUR
PAPER. Fig. shows $P_2$ order paramter of the dipoles on head group
raises with increasing the size of the head group. When head of lipid
molecule is small, the membrane is flat and shows strong two
dimensional characters, dipoles are frustrated on orientational
ordering in this circumstance. Another reason is that the lipids can
move independently in each monolayer, it is not nessasory for the
direction of dipoles on one leaf is consistant to another layer, which
makes total order parameter is relatively low. With increasing the
size of head group, the surface is being more corrugated, dipoles are
not allowed to move freely on the surface, they are
localized. Therefore, the translational freedom of lipids in one layer
is dependent upon the position of lipids in another layer, as a
result, the symmetry of the dipoles on head group in one layer is
consistant to the symmetry in another layer. Furthermore, the membrane
tranlates from a two dimensional system to a three dimensional system
by the corrugation, the symmetry of the ordering for the two
dimensional dipoles on the head group of lipid molecules is broken, on
a distorted lattice, dipoles are ordered on a head to tail energy
state, the order parameter is increased dramaticly. However, the total
polarization of the system is close to zero, which is a strong
evidence it is a antiferroelectric state. The orientation of the
dipole ordering is alway perpendicular to the ripple vector. These
results are consistant to our previous study on similar system. The
ordering of the tails are opposite to the ordering of the dipoles on
head group, the $P_2$ order parameter decreases with increasing the
size of head. This indicates the surface is more curved with larger
head. When surface is flat, all tails are pointing to the same
direction, in this case, all tails are parallal to the normal of the
surface, which shares the same structure with L_\beta phase. For the
size of head being $1.28\sigma_0$, the surface starts to splay inward,
however, the surface is still flat, therefore, although the order
parameter is lower, it still indicates a very flat surface. Further
increasing the size of the head, the order parameter drops dramaticly,
the surface is rippled.


We studied the effects of interaction between head groups on the
structure of lipid bilayer by changing the strength of the dipole. The
fig. shows the $P_2$ order parameter changing with strength of the
dipole. Generally the dipoles on the head group are more ordered with
increasing the interaction between heads and more disordered with
decreasing the interaction between heads. When the interaction between
heads is weak enough, the bilayer structure is not persisted any more,
all lipid molecules are melted in the water. The critial value of the
strength of the dipole is various for different system. The perfect
flat surface melts at $5$ debye, the asymmetric rippled surfaces melt
at $8$ debye, the symmetric rippled surfaces melt at $10$ debye. This
indicates that the flat phase is the most stable state, the asymmetric
ripple phase is second stalbe state, and the symmetric ripple phase is
the most unstable state. The ordering of the tails is the same as the
ordering of the dipoles except for the flat phase. Since the surface
is already perfect flat, the order parameter does not change much
until the strength of the dipole is $15$ debye. However, the order
parameter decreases quickly when the strength of the dipole is further
increased. The head group of the lipid molecules are brought closer by
strenger interaction between them. For a flat surface, a mount of free
volume between head groups is available, when the head groups are
brought closer, the surface will splay outward to be a inverse
micelle. For rippled surfaces, there is few free volume available on
between the head groups, they can be closer, therefore there are
little effect on the structure of the membrane. Another interesting
fact, unlike other systems melting directly when the interaction is
weak enough, for $\sigma_h$ is $1.41\sigma_0$, part of the membrane
melts into itself first, the upper leaf of the bilayer is totally
interdigitated with the lower leaf, this is different with the
interdigitated lines in rippled phase where only one interdigitated
line connects the two leaves of bilayer.


Fig. shows the changing of the order parameter with temperature. The
behavior of the $P_2$ orderparamter is straightforword. Systems are
more ordered at low temperature, and more disordered at high
temperature. When the temperature is high enough, the membranes are
discontinuted. The structures are stable during the changing of the
temperature. Since our model lacks the detail information for tails of
lipid molecules, we did not simulate the fluid phase with a melted
fatty chains. Moreover, the formation of the tilted ``L_{\beta'}''
phase also depends on the organization of fatty groups on tails, we
did not observe it either.
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#	User	Rev	Content
1	xsun	3147	Snap shots show that the membrane is more corrugated with increasing
2			the size of the head groups. The surface is nearly perfect flat when
3			$\sigma_h$ is $1.20\sigma_0$. At $1.28\sigma_0$, although the surface
4			is still flat, the bilayer starts to splay inward, the upper leaf of
5			the bilayer is connected to the lower leaf with a interdigitated line
6			defect. Two periodicities with $100\AA$ width were observed in the
7			simulation. This structure is very similiar to OTHER PAPER. The same
8			structure was also observed when $sigma_h=1.41\sigma_0$. However, the
9			surface of the membrane is corrugated, and the periodicity of the
10			connection between upper and lower leaf membrane is shorter. From the
11			undulation spectrum of the surface (the exact form is in OUR PREVIOUS
12			PAPER), the corrugation is non-thermal fluctuation, and we are
13			confident to identify it as the ripple phase. The width of one ripple
14			is about 71\AA, and amplitude is about 7\AA. When
15			$\sigma_h=1.35\sigma_0$, we observed another corrugated surface with
16			79\AA width and 10\AA amplitude. This structure is different to the
17			previous rippled surface, there is no connection between upper and
18			lower leaf of the bilayer. Each leaf of the bilayer is broken to
19			several curved pieces, the broken position is mounted into the center
20			of opposite piece in another leaf. Unlike another corrugated surface
21			in which the upper leaf of the surface is always connected to the
22			lower leaf from one direction, this ripple of this surface is
23			isotropic. Therefore, we claim this is a symmetric ripple phase.
24
25
26			The $P_2$ order paramter is calculated to understand the phase
27			behavior quantatively. $P_2=1$ means a perfect ordered structure, and
28			$P_2=0$ means a random structure. The method can be found in OUR
29			PAPER. Fig. shows $P_2$ order paramter of the dipoles on head group
30			raises with increasing the size of the head group. When head of lipid
31			molecule is small, the membrane is flat and shows strong two
32			dimensional characters, dipoles are frustrated on orientational
33			ordering in this circumstance. Another reason is that the lipids can
34			move independently in each monolayer, it is not nessasory for the
35			direction of dipoles on one leaf is consistant to another layer, which
36			makes total order parameter is relatively low. With increasing the
37			size of head group, the surface is being more corrugated, dipoles are
38			not allowed to move freely on the surface, they are
39			localized. Therefore, the translational freedom of lipids in one layer
40			is dependent upon the position of lipids in another layer, as a
41			result, the symmetry of the dipoles on head group in one layer is
42			consistant to the symmetry in another layer. Furthermore, the membrane
43			tranlates from a two dimensional system to a three dimensional system
44			by the corrugation, the symmetry of the ordering for the two
45			dimensional dipoles on the head group of lipid molecules is broken, on
46			a distorted lattice, dipoles are ordered on a head to tail energy
47			state, the order parameter is increased dramaticly. However, the total
48			polarization of the system is close to zero, which is a strong
49			evidence it is a antiferroelectric state. The orientation of the
50			dipole ordering is alway perpendicular to the ripple vector. These
51			results are consistant to our previous study on similar system. The
52			ordering of the tails are opposite to the ordering of the dipoles on
53			head group, the $P_2$ order parameter decreases with increasing the
54			size of head. This indicates the surface is more curved with larger
55			head. When surface is flat, all tails are pointing to the same
56			direction, in this case, all tails are parallal to the normal of the
57			surface, which shares the same structure with L_\beta phase. For the
58			size of head being $1.28\sigma_0$, the surface starts to splay inward,
59			however, the surface is still flat, therefore, although the order
60			parameter is lower, it still indicates a very flat surface. Further
61			increasing the size of the head, the order parameter drops dramaticly,
62			the surface is rippled.
63
64
65			We studied the effects of interaction between head groups on the
66			structure of lipid bilayer by changing the strength of the dipole. The
67			fig. shows the $P_2$ order parameter changing with strength of the
68			dipole. Generally the dipoles on the head group are more ordered with
69			increasing the interaction between heads and more disordered with
70			decreasing the interaction between heads. When the interaction between
71			heads is weak enough, the bilayer structure is not persisted any more,
72			all lipid molecules are melted in the water. The critial value of the
73			strength of the dipole is various for different system. The perfect
74			flat surface melts at $5$ debye, the asymmetric rippled surfaces melt
75			at $8$ debye, the symmetric rippled surfaces melt at $10$ debye. This
76			indicates that the flat phase is the most stable state, the asymmetric
77			ripple phase is second stalbe state, and the symmetric ripple phase is
78			the most unstable state. The ordering of the tails is the same as the
79			ordering of the dipoles except for the flat phase. Since the surface
80			is already perfect flat, the order parameter does not change much
81			until the strength of the dipole is $15$ debye. However, the order
82			parameter decreases quickly when the strength of the dipole is further
83			increased. The head group of the lipid molecules are brought closer by
84			strenger interaction between them. For a flat surface, a mount of free
85			volume between head groups is available, when the head groups are
86			brought closer, the surface will splay outward to be a inverse
87			micelle. For rippled surfaces, there is few free volume available on
88			between the head groups, they can be closer, therefore there are
89			little effect on the structure of the membrane. Another interesting
90			fact, unlike other systems melting directly when the interaction is
91			weak enough, for $\sigma_h$ is $1.41\sigma_0$, part of the membrane
92			melts into itself first, the upper leaf of the bilayer is totally
93			interdigitated with the lower leaf, this is different with the
94			interdigitated lines in rippled phase where only one interdigitated
95			line connects the two leaves of bilayer.
96
97
98			Fig. shows the changing of the order parameter with temperature. The
99			behavior of the $P_2$ orderparamter is straightforword. Systems are
100			more ordered at low temperature, and more disordered at high
101			temperature. When the temperature is high enough, the membranes are
102			discontinuted. The structures are stable during the changing of the
103			temperature. Since our model lacks the detail information for tails of
104			lipid molecules, we did not simulate the fluid phase with a melted
105			fatty chains. Moreover, the formation of the tilted ``L_{\beta'}''
106			phase also depends on the organization of fatty groups on tails, we
107			did not observe it either.