trunk/mdRipple/mdripple.tex

%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4}
\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4}
\usepackage{graphicx}

\begin{document}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}

%\bibliographystyle{aps}

\title{}
\author{Xiuquan Sun and J. Daniel Gezelter}
\email[E-mail:]{gezelter@nd.edu}
\affiliation{Department of Chemistry and Biochemistry,\\
University of Notre Dame, \\
Notre Dame, Indiana 46556}

\date{\today} 

\begin{abstract}

\end{abstract}

\pacs{}
\maketitle

Our idea for developing a simple and reasonable lipid model to study
the ripple pahse of lipid bilayers is based on two facts: one is that
the most essential feature of lipid molecules is their amphiphilic
structure with polar head groups and non-polar tails. Another fact is
that dominant numbers of lipid molecules are very rigid in ripple
phase which allows the details of the lipid molecules neglectable. In
our model, lipid molecules are represented by rigid bodies made of one
head sphere with a point dipole sitting on it and one ellipsoid tail,
the direction of the dipole is fixed to be perpendicular to the
tail. The breadth and length of tail are $\sigma_0$, $3\sigma_0$. The
diameter of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$.  The
model of the solvent in our simulations is inspired by the idea of
``DPD'' water. Every four water molecules are reprsented by one
sphere.


Spheres interact each other with Lennard-Jones potential, ellipsoids
interact each other with Gay-Berne potential, dipoles interact each
other with typical dipole potential, spheres interact ellipsoids with
LJ-GB potential. All potentials are truncated at $25$ \AA and shifted
at $22$ \AA.


To make the simulations less expensive and to observe long-time range
behavior of the lipid membranes, all simulaitons were started from two
sepetated monolayers in the vaccum with $x-y$ anisotropic pressure
coupling, length of $z$ axis of the simulations was fixed to prevent
the shrinkage of the simulation boxes due to the free volume outside
of the bilayer, and a constant surface tension was applied to enable
the fluctuation of the surface. Periodic boundaries were used. There
were $480-720$ lipid molecules in simulations according to different
size of the heads. All the simulations were stablized for $100$ ns at
$300$ K. The resulted structures were solvated in the water (about
$6$ DPD water/lipid molecule) as the initial configurations for another
$30$ ns relaxation. All simulations with water were carried out at
constant pressure ($P=1$bar) by $3$D anisotropic coupling, and
constant surface tension ($\gamma=0.015$). Time step was
$50$ fs. Simulations were performed by using OOPSE package.


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.

\bibliography{mdripple}
\end{document}
Revision:	3147
Committed:	Mon Jun 25 21:16:17 2007 UTC (18 years, 5 months ago) by xsun
Content type:	application/x-tex
File size:	9544 byte(s)
Log Message:	create mdRipple
#	Content
1	%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4}
2	\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4}
3	\usepackage{graphicx}
4
5	\begin{document}
6	\renewcommand{\thefootnote}{\fnsymbol{footnote}}
7	\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
8
9	%\bibliographystyle{aps}
10
11	\title{}
12	\author{Xiuquan Sun and J. Daniel Gezelter}
13	\email[E-mail:]{gezelter@nd.edu}
14	\affiliation{Department of Chemistry and Biochemistry,\\
15	University of Notre Dame, \\
16	Notre Dame, Indiana 46556}
17
18	\date{\today}
19
20	\begin{abstract}
21
22	\end{abstract}
23
24	\pacs{}
25	\maketitle
26
27	Our idea for developing a simple and reasonable lipid model to study
28	the ripple pahse of lipid bilayers is based on two facts: one is that
29	the most essential feature of lipid molecules is their amphiphilic
30	structure with polar head groups and non-polar tails. Another fact is
31	that dominant numbers of lipid molecules are very rigid in ripple
32	phase which allows the details of the lipid molecules neglectable. In
33	our model, lipid molecules are represented by rigid bodies made of one
34	head sphere with a point dipole sitting on it and one ellipsoid tail,
35	the direction of the dipole is fixed to be perpendicular to the
36	tail. The breadth and length of tail are $\sigma_0$, $3\sigma_0$. The
37	diameter of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The
38	model of the solvent in our simulations is inspired by the idea of
39	``DPD'' water. Every four water molecules are reprsented by one
40	sphere.
41
42
43	Spheres interact each other with Lennard-Jones potential, ellipsoids
44	interact each other with Gay-Berne potential, dipoles interact each
45	other with typical dipole potential, spheres interact ellipsoids with
46	LJ-GB potential. All potentials are truncated at $25$ \AA and shifted
47	at $22$ \AA.
48
49
50	To make the simulations less expensive and to observe long-time range
51	behavior of the lipid membranes, all simulaitons were started from two
52	sepetated monolayers in the vaccum with $x-y$ anisotropic pressure
53	coupling, length of $z$ axis of the simulations was fixed to prevent
54	the shrinkage of the simulation boxes due to the free volume outside
55	of the bilayer, and a constant surface tension was applied to enable
56	the fluctuation of the surface. Periodic boundaries were used. There
57	were $480-720$ lipid molecules in simulations according to different
58	size of the heads. All the simulations were stablized for $100$ ns at
59	$300$ K. The resulted structures were solvated in the water (about
60	$6$ DPD water/lipid molecule) as the initial configurations for another
61	$30$ ns relaxation. All simulations with water were carried out at
62	constant pressure ($P=1$bar) by $3$D anisotropic coupling, and
63	constant surface tension ($\gamma=0.015$). Time step was
64	$50$ fs. Simulations were performed by using OOPSE package.
65
66
67	Snap shots show that the membrane is more corrugated with increasing
68	the size of the head groups. The surface is nearly perfect flat when
69	$\sigma_h$ is $1.20\sigma_0$. At $1.28\sigma_0$, although the surface
70	is still flat, the bilayer starts to splay inward, the upper leaf of
71	the bilayer is connected to the lower leaf with a interdigitated line
72	defect. Two periodicities with $100$\AA width were observed in the
73	simulation. This structure is very similiar to OTHER PAPER. The same
74	structure was also observed when $\sigma_h=1.41\sigma_0$. However, the
75	surface of the membrane is corrugated, and the periodicity of the
76	connection between upper and lower leaf membrane is shorter. From the
77	undulation spectrum of the surface (the exact form is in OUR PREVIOUS
78	PAPER), the corrugation is non-thermal fluctuation, and we are
79	confident to identify it as the ripple phase. The width of one ripple
80	is about $71$ \AA, and amplitude is about $7$ \AA. When
81	$\sigma_h=1.35\sigma_0$, we observed another corrugated surface with
82	$79$ \AA width and $10$ \AA amplitude. This structure is different to
83	the previous rippled surface, there is no connection between upper and
84	lower leaf of the bilayer. Each leaf of the bilayer is broken to
85	several curved pieces, the broken position is mounted into the center
86	of opposite piece in another leaf. Unlike another corrugated surface
87	in which the upper leaf of the surface is always connected to the
88	lower leaf from one direction, this ripple of this surface is
89	isotropic. Therefore, we claim this is a symmetric ripple phase.
90
91
92	The $P_2$ order paramter is calculated to understand the phase
93	behavior quantatively. $P_2=1$ means a perfect ordered structure, and
94	$P_2=0$ means a random structure. The method can be found in OUR
95	PAPER. Fig. shows $P_2$ order paramter of the dipoles on head group
96	raises with increasing the size of the head group. When head of lipid
97	molecule is small, the membrane is flat and shows strong two
98	dimensional characters, dipoles are frustrated on orientational
99	ordering in this circumstance. Another reason is that the lipids can
100	move independently in each monolayer, it is not nessasory for the
101	direction of dipoles on one leaf is consistant to another layer, which
102	makes total order parameter is relatively low. With increasing the
103	size of head group, the surface is being more corrugated, dipoles are
104	not allowed to move freely on the surface, they are
105	localized. Therefore, the translational freedom of lipids in one layer
106	is dependent upon the position of lipids in another layer, as a
107	result, the symmetry of the dipoles on head group in one layer is
108	consistant to the symmetry in another layer. Furthermore, the membrane
109	tranlates from a two dimensional system to a three dimensional system
110	by the corrugation, the symmetry of the ordering for the two
111	dimensional dipoles on the head group of lipid molecules is broken, on
112	a distorted lattice, dipoles are ordered on a head to tail energy
113	state, the order parameter is increased dramaticly. However, the total
114	polarization of the system is close to zero, which is a strong
115	evidence it is a antiferroelectric state. The orientation of the
116	dipole ordering is alway perpendicular to the ripple vector. These
117	results are consistant to our previous study on similar system. The
118	ordering of the tails are opposite to the ordering of the dipoles on
119	head group, the $P_2$ order parameter decreases with increasing the
120	size of head. This indicates the surface is more curved with larger
121	head. When surface is flat, all tails are pointing to the same
122	direction, in this case, all tails are parallal to the normal of the
123	surface, which shares the same structure with $L_{\beta}$ phase. For the
124	size of head being $1.28\sigma_0$, the surface starts to splay inward,
125	however, the surface is still flat, therefore, although the order
126	parameter is lower, it still indicates a very flat surface. Further
127	increasing the size of the head, the order parameter drops dramaticly,
128	the surface is rippled.
129
130
131	We studied the effects of interaction between head groups on the
132	structure of lipid bilayer by changing the strength of the dipole. The
133	fig. shows the $P_2$ order parameter changing with strength of the
134	dipole. Generally the dipoles on the head group are more ordered with
135	increasing the interaction between heads and more disordered with
136	decreasing the interaction between heads. When the interaction between
137	heads is weak enough, the bilayer structure is not persisted any more,
138	all lipid molecules are melted in the water. The critial value of the
139	strength of the dipole is various for different system. The perfect
140	flat surface melts at $5$ debye, the asymmetric rippled surfaces melt
141	at $8$ debye, the symmetric rippled surfaces melt at $10$ debye. This
142	indicates that the flat phase is the most stable state, the asymmetric
143	ripple phase is second stalbe state, and the symmetric ripple phase is
144	the most unstable state. The ordering of the tails is the same as the
145	ordering of the dipoles except for the flat phase. Since the surface
146	is already perfect flat, the order parameter does not change much
147	until the strength of the dipole is $15$ debye. However, the order
148	parameter decreases quickly when the strength of the dipole is further
149	increased. The head group of the lipid molecules are brought closer by
150	strenger interaction between them. For a flat surface, a mount of free
151	volume between head groups is available, when the head groups are
152	brought closer, the surface will splay outward to be a inverse
153	micelle. For rippled surfaces, there is few free volume available on
154	between the head groups, they can be closer, therefore there are
155	little effect on the structure of the membrane. Another interesting
156	fact, unlike other systems melting directly when the interaction is
157	weak enough, for $\sigma_h$ is $1.41\sigma_0$, part of the membrane
158	melts into itself first, the upper leaf of the bilayer is totally
159	interdigitated with the lower leaf, this is different with the
160	interdigitated lines in rippled phase where only one interdigitated
161	line connects the two leaves of bilayer.
162
163
164	Fig. shows the changing of the order parameter with temperature. The
165	behavior of the $P_2$ orderparamter is straightforword. Systems are
166	more ordered at low temperature, and more disordered at high
167	temperature. When the temperature is high enough, the membranes are
168	discontinuted. The structures are stable during the changing of the
169	temperature. Since our model lacks the detail information for tails of
170	lipid molecules, we did not simulate the fluid phase with a melted
171	fatty chains. Moreover, the formation of the tilted $L_{\beta'}$ phase
172	also depends on the organization of fatty groups on tails, we did not
173	observe it either.
174
175	\bibliography{mdripple}
176	\end{document}