trunk/mdRipple/mdripple.tex

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\title{Dipolar Ordering of the Ripple Phase in Lipid Membranes}
\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

\section{Introduction}
\label{sec:Int}

As one of the most important components in the formation of the
biomembrane, lipid molecules attracted numerous studies in the past
several decades. Due to their amphiphilic structure, when dispersed in
water, lipids can self-assemble to construct a bilayer structure. The
phase behavior of lipid membrane is well understood. The gel-fluid
phase transition is known as main phase transition. However, there is
an intermediate phase between gel and fluid phase for some lipid (like
phosphatidycholine (PC)) membranes. This intermediate phase
distinguish itself from other phases by its corrugated membrane
surface, therefore this phase is termed ``ripple'' ($P_{\beta '}$)
phase. The phase transition between gel-fluid and ripple phase is
called pretransition. Since the pretransition usually occurs in room
temperature, there might be some important biofuntions carried by the
ripple phase for the living organism.

The ripple phase is observed experimentally by x-ray diffraction
~\cite{Sun96,Katsaras00}, freeze-fracture electron microscopy
(FFEM)~\cite{Copeland80,Meyer96}, and atomic force microscopy (AFM)
recently~\cite{Kaasgaard03}. The experimental studies suggest two
kinds of ripple structures: asymmetric (sawtooth like) and symmetric
(sinusoidal like) ripple phases. Substantial number of theoretical
explaination applied on the formation of the ripple
phases~\cite{Marder84,Carlson87,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02}.
In contrast, few molecular modelling have been done due to the large
size of the resulting structures and the time required for the phases
of interest to develop. One of the interesting molecular simulations
was carried out by De Vries and Marrink {\it et
al.}~\cite{deVries05}. According to their dynamic simulation results,
the ripple consists of two domains, one is gel bilayer, and in the
other domain, the upper and lower leaves of the bilayer are fully
interdigitated. The mechanism of the formation of the ripple phase in
their work suggests the theory that the packing competition between
head group and tail of lipid molecules is the driving force for the
formation of the ripple phase~\cite{Carlson87}. Recently, the ripple
phase is also studied by using monte carlo simulation~\cite{Lenz07},
the ripple structure is similar to the results of Marrink except that
the connection of the upper and lower leaves of the bilayer is an
interdigitated line instead of the fully interdigitated
domain. Furthermore, the symmetric ripple phase was also observed in
their work. They claimed the mismatch between the size of the head
group and tail of the lipid molecules is the driving force for the
formation of the ripple phase.

Although the organizations of the tails of lipid molecules are
addressed by these molecular simulations, the ordering of the head
group in ripple phase is still not settlement. We developed a simple
``web of dipoles'' spin lattice model which provides some physical
insight in our previous studies~\cite{Sun2007}, we found the dipoles
on head groups of the lipid molecules are ordered in an
antiferroelectric state. The similiar phenomenon is also observed by
Tsonchev {\it et al.} when they studied the formation of the
nanotube\cite{Tsonchev04}.

In this paper, we made a more realistic coarse-grained lipid model to
understand the primary driving force for membrane corrugation and to
elucidate the organization of the anisotropic interacting head group
via molecular dynamics simulation. We will talk about our model and
methodology in section \ref{sec:method}, and details of the simulation
in section \ref{sec:experiment}. The results are shown in section
\ref{sec:results}. At last, we will discuss the results in section
\ref{sec:discussion}.

\section{Methodology and Model}
\label{sec:method}

Our idea for developing a simple and reasonable lipid model to study
the ripple phase 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. The
lipid model is shown in Figure \ref{fig:lipidMM}. Figure
\ref{fig:lipidMM}a shows the molecular strucure of a DPPC molecule. The
hydrophilic character of the head group is the effect of the strong
dipole composed by a positive charge sitting on the nitrogen and a
negative charge on the phosphate. The hydrophobic tail consists of
fatty acid chains. In our model shown in Figure \ref{fig:lipidMM}b,
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.

\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{lipidMM} 
\caption{The molecular structure of a DPPC molecule and the
coars-grained model for PC molecules.\label{fig:lipidMM}}
\end{figure}

Spheres interact each other with Lennard-Jones potential
\begin{eqnarray*}
V_{ij} = 4\epsilon \left[\left(\frac{\sigma_0}{r_{ij}}\right)^{12} -
\left(\frac{\sigma_0}{r_{ij}}\right)^6\right]
\end{eqnarray*}
here, $\sigma_0$ is diameter of the spherical particle. $r_{ij}$ is
the distance between two spheres. $\epsilon$ is the well depth.
Dipoles interact each other with typical dipole potential
\begin{eqnarray*}
V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3}
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right]
\end{eqnarray*}
In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. Identical
ellipsoids interact each other with Gay-Berne potential.
\begin{eqnarray*}
V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i},
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12}
-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right]
\end{eqnarray*} 
where $\sigma_0 = \sqrt 2\sigma_s$, and orietation-dependent range
parameter is given by
\begin{eqnarray*}
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = 
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{({\mathbf{\hat r}_{ij}}
\cdot {\mathbf{\hat u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} +
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}}
\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}}
\end{eqnarray*} 
and the strength anisotropy function is,
\begin{eqnarray*}
\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = 
{\epsilon^\nu}({\mathbf{\hat u}_i}, {\mathbf{\hat
u}_j}){\epsilon'^\mu}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j},
{\mathbf{\hat r}_{ij}})
\end{eqnarray*} 
with $\nu$ and $\mu$ being adjustable exponent, and
$\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j})$,
$\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat
r}_{ij}})$ defined as
\begin{eqnarray*} 
\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}) =
\epsilon_0\left[1-\chi^2({\mathbf{\hat u}_i} \cdot {\mathbf{\hat
u}_j})^2\right]^{-\frac{1}{2}} 
\end{eqnarray*}
\begin{eqnarray*}
\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = 
1-\frac{1}{2}\chi'\left[\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat
u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat
u}_j})^2}{1+\chi'({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} +
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}}
\cdot {\mathbf{\hat u}_j})^2}{1-\chi'({\mathbf{\hat u}_i} \cdot
{\mathbf{\hat u}_j})} \right]
\end{eqnarray*}
the diameter dependent parameter $\chi$ is given by
\begin{eqnarray*} 
\chi = \frac{({\sigma_s}^2 -
{\sigma_e}^2)}{({\sigma_s}^2 + {\sigma_e}^2)} 
\end{eqnarray*}
and the well depth dependent parameter $\chi'$ is given by
\begin{eqnarray*} 
\chi' = \frac{({\epsilon_s}^{\frac{1}{\mu}} -
{\epsilon_e}^{\frac{1}{\mu}})}{({\epsilon_s}^{\frac{1}{\mu}} +
{\epsilon_e}^{\frac{1}{\mu}})} 
\end{eqnarray*}
$\sigma_s$ is the side-by-side width, and $\sigma_e$ is the end-to-end
length. $\epsilon_s$ is the side-by-side well depth, and $\epsilon_e$
is the end-to-end well depth. For the interaction between
nonequivalent uniaxial ellipsoids (in this case, between spheres and
ellipsoids), the range parameter is generalized as\cite{Cleaver96}
\begin{eqnarray*}
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = 
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}}
\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} +
\frac{(\alpha{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - \alpha^{-1}{\mathbf{\hat r}_{ij}}
\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}}
\end{eqnarray*}
where $\alpha$ is given by
\begin{eqnarray*}
\alpha^2 =
\left[\frac{\left({\sigma_e}_i^2-{\sigma_s}_i^2\right)\left({\sigma_e}_j^2+{\sigma_s}_i^2\right)}{\left({\sigma_e}_j^2-{\sigma_s}_j^2\right)\left({\sigma_e}_i^2+{\sigma_s}_j^2\right)}
\right]^{\frac{1}{2}}
\end{eqnarray*}
the strength parameter is adjusted by the suggestion of
\cite{Cleaver96}. All potentials are truncated at $25$ \AA\ and
shifted at $22$ \AA.

\section{Experiment}
\label{sec:experiment}

To make the simulations less expensive and to observe long-time
behavior of the lipid membranes, all simulations were started from two
separate monolayers in the vaccum with $x-y$ anisotropic pressure
coupling. The length of $z$ axis of the simulations was fixed and a
constant surface tension was applied to enable real fluctuations of
the bilayer. Periodic boundaries were used. There were $480-720$ lipid
molecules in the simulations depending on the size of the head
beads. All the simulations were equlibrated for $100$ ns at $300$
K. The resulting structures were solvated in water ($6$ DPD
water/lipid molecule). These configurations were relaxed for another
$30$ ns relaxation. All simulations with water were carried out at
constant pressure ($P=1$ atm) by $3$D anisotropic coupling, and
constant surface tension ($\gamma=0.015$). Given the absence of fast
degrees of freedom in this model, a timestep of $50$ fs was
utilized. Simulations were performed by using OOPSE
package\cite{Meineke05}.

\section{Results and Analysis}
\label{sec:results}

Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is
more corrugated increasing size of the head groups. The surface is
nearly flat when $\sigma_h=1.20\sigma_0$. With
$\sigma_h=1.28\sigma_0$, 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 an interdigitated line defect. Two
periodicities with $100$ \AA\ width were observed in the
simulation. This structure is very similiar to the structure observed
by de Vries and Lenz {\it et al.}. The same basic structure is also
observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the
surface corrugations depends sensitively on the size of the ``head''
beads. From the undulation spectrum, the corrugation is clearly
non-thermal.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{phaseCartoon} 
\caption{A sketch to discribe the structure of the phases observed in
our simulations.\label{fig:phaseCartoon}}
\end{figure}

When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface
morphology. This structure is different from the asymmetric rippled
surface; there is no interdigitation between the upper and lower
leaves of the bilayer. Each leaf of the bilayer is broken into several
hemicylinderical sections, and opposite leaves are fitted together
much like roof tiles. Unlike the surface in which the upper
hemicylinder is always interdigitated on the leading or trailing edge
of lower hemicylinder, the symmetric ripple has no prefered direction.
The corresponding cartoons are shown in Figure
\ref{fig:phaseCartoon} for elucidation of the detailed structures of
different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase,
(b) is the asymmetric ripple phase corresponding to the lipid
organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$,
and (c) is the symmetric ripple phase observed when
$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is
continuous everywhere on the whole membrane, however, in asymmetric
ripple phase, the bilayer is intermittent domains connected by thin
interdigitated monolayer which consists of upper and lower leaves of
the bilayer.
\begin{table*}
\begin{minipage}{\linewidth}
\begin{center}
\caption{}
\begin{tabular}{lccc}
\hline 
$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\
\hline 
1.20 & flat & N/A & N/A \\
1.28 & asymmetric flat & 21.7 & N/A \\
1.35 & symmetric ripple & 17.2 & 2.2 \\
1.41 & asymmetric ripple & 15.4 & 1.5 \\
\end{tabular}
\label{tab:property}
\end{center}
\end{minipage}
\end{table*}

The membrane structures and the reduced wavelength $\lambda/\sigma_0$,
reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table
\ref{tab:property}. The wavelength range is $15~21$ and the amplitude
is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These
values are consistent to the experimental results. Note, the
amplitudes are underestimated without the melted tails in our
simulations.

The $P_2$ order paramters (for molecular bodies and head group
dipoles) have been calculated to clarify the ordering in these phases
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$
implies orientational randomization. Figure \ref{fig:rP2} shows the
$P_2$ order paramter of the dipoles on head group rising with
increasing head group size. When the heads of the lipid molecules are
small, the membrane is flat. The dipolar ordering is essentially
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 head group size, the surface is
corrugated, and dipoles do not move as freely on the
surface. 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 tied
to the symmetry in the other layer. Furthermore, as the membrane
deforms from two to three dimensions due to the corrugation, the
symmetry of the ordering for the dipoles embedded on each leaf is
broken. The dipoles then self-assemble in a head-tail configuration,
and the order parameter increases dramaticaly. However, the total
polarization of the system is still close to zero. This is strong
evidence that the corrugated structure is an antiferroelectric
state. The orientation of the dipolar is always perpendicular to the
ripple wave vector. These results are consistent with our previous
study on dipolar membranes.

The ordering of the tails is essentially opposite to the ordering of
the dipoles on head group. The $P_2$ order parameter decreases with
increasing head size. This indicates the surface is more curved with
larger head groups. When the surface is flat, all tails are pointing
in the same direction; in this case, all tails are parallel to the
normal of the surface,(making this structure remindcent of the
$L_{\beta}$ phase. Increasing the size of the heads, results in
rapidly decreasing $P_2$ ordering for the molecular bodies.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{rP2} 
\caption{The $P_2$ order parameter as a funtion of the ratio of
$\sigma_h$ to $\sigma_0$.\label{fig:rP2}}
\end{figure}

We studied the effects of the interactions between head groups on the
structure of lipid bilayer by changing the strength of the dipole.
Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with
increasing strength of the dipole. Generally the dipoles on the head
group are more ordered by increase in the strength of the interaction
between heads and are more disordered by decreasing the interaction
stength. When the interaction between the heads is weak enough, the
bilayer structure does not persist; all lipid molecules are solvated
directly in the water. The critial value of the strength of the dipole
depends on the head size. The perfectly flat surface melts at $5$
debye, the asymmetric rippled surfaces melt at $8$ debye, the
symmetric rippled surfaces melt at $10$ debye. 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 groups of the
lipid molecules are brought closer by stronger interactions between
them. For a flat surface, a large amount of free volume between the
head groups is available, but when the head groups are brought closer,
the tails will splay outward, forming an inverse micelle. For rippled
surfaces, there is less free volume available between the head
groups. Therefore there is little effect on the structure of the
membrane due to increasing dipolar strength. Unlike other systems that
melt directly when the interaction is weak enough, for
$\sigma_h=1.41\sigma_0$, part of the membrane melts into itself
first. The upper leaf of the bilayer becomes totally interdigitated
with the lower leaf. This is different behavior than what is exhibited
with the interdigitated lines in the rippled phase where only one
interdigitated line connects the two leaves of bilayer.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{sP2} 
\caption{The $P_2$ order parameter as a funtion of the strength of the
dipole.\label{fig:sP2}}
\end{figure}

Figure \ref{fig:tP2} shows the dependence of the order parameter on
temperature. The behavior of the $P_2$ order paramter is
straightforward. Systems are more ordered at low temperature, and more
disordered at high temperatures. When the temperature is high enough,
the membranes are instable. Since our model lacks the detailed
information on lipid tails, we can not simulate the fluid phase with
melted fatty acid chains. Moreover, the formation of the tilted
$L_{\beta'}$ phase also depends on the organization of fatty groups on
tails.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{tP2} 
\caption{The $P_2$ order parameter as a funtion of
temperature.\label{fig:tP2}}
\end{figure}

\section{Discussion}
\label{sec:discussion}

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