trunk/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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#	Content
1	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.