1 |
\chapter{\label{chap:md}DIPOLAR ORDERING IN THE RIPPLE PHASES OF |
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MOLECULAR-SCALE MODELS OF LIPID MEMBRANES} |
3 |
|
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\section{Introduction} |
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\label{mdsec:Int} |
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|
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A number of theoretical models have been presented to explain the |
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formation of the ripple phase. Marder {\it et al.} used a |
9 |
curvature-dependent Landau-de~Gennes free-energy functional to predict |
10 |
a rippled phase.~\cite{Marder84} This model and other related |
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continuum models predict higher fluidity in convex regions and that |
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concave portions of the membrane correspond to more solid-like |
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regions. Carlson and Sethna used a packing-competition model (in |
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which head groups and chains have competing packing energetics) to |
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predict the formation of a ripple-like phase~\cite{Carlson87}. Their |
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model predicted that the high-curvature portions have lower-chain |
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packing and correspond to more fluid-like regions. Goldstein and |
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Leibler used a mean-field approach with a planar model for {\em |
19 |
inter-lamellar} interactions to predict rippling in multilamellar |
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phases.~\cite{Goldstein88} McCullough and Scott proposed that the {\em |
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anisotropy of the nearest-neighbor interactions} coupled to |
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hydrophobic constraining forces which restrict height differences |
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between nearest neighbors is the origin of the ripple |
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phase.~\cite{McCullough90} Lubensky and MacKintosh introduced a Landau |
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theory for tilt order and curvature of a single membrane and concluded |
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that {\em coupling of molecular tilt to membrane curvature} is |
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responsible for the production of ripples.~\cite{Lubensky93} Misbah, |
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Duplat and Houchmandzadeh proposed that {\em inter-layer dipolar |
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interactions} can lead to ripple instabilities.~\cite{Misbah98} |
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Heimburg presented a {\em coexistence model} for ripple formation in |
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which he postulates that fluid-phase line defects cause sharp |
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curvature between relatively flat gel-phase regions.~\cite{Heimburg00} |
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Kubica has suggested that a lattice model of polar head groups could |
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be valuable in trying to understand bilayer phase |
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formation.~\cite{Kubica02} Bannerjee used Monte Carlo simulations of |
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lamellar stacks of hexagonal lattices to show that large head groups |
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and molecular tilt with respect to the membrane normal vector can |
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cause bulk rippling.~\cite{Bannerjee02} Recently, Kranenburg and Smit |
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described the formation of symmetric ripple-like structures using a |
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coarse grained solvent-head-tail bead model.\cite{Kranenburg2005} |
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Their lipids consisted of a short chain of head beads tied to the two |
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longer ``chains''. |
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|
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In contrast, few large-scale molecular modeling studies have been |
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done due to the large size of the resulting structures and the time |
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required for the phases of interest to develop. With all-atom (and |
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even unified-atom) simulations, only one period of the ripple can be |
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observed and only for time scales in the range of 10-100 ns. One of |
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the most interesting molecular simulations was carried out by de~Vries |
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{\it et al.}~\cite{deVries05}. According to their simulation results, |
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the ripple consists of two domains, one resembling the gel bilayer, |
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while in the other, the two leaves of the bilayer are fully |
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interdigitated. The mechanism for the formation of the ripple phase |
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suggested by their work is a packing competition between the head |
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groups and the tails of the lipid molecules.\cite{Carlson87} Recently, |
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the ripple phase has also been studied by Lenz and Schmid using Monte |
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Carlo simulations.\cite{Lenz07} Their structures are similar to the De |
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Vries {\it et al.} structures except that the connection between the |
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two leaves of the bilayer is a narrow interdigitated line instead of |
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the fully interdigitated domain. The symmetric ripple phase was also |
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observed by Lenz {\it et al.}, and their work supports other claims |
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that the mismatch between the size of the head group and tail of the |
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lipid molecules is the driving force for the formation of the ripple |
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phase. Ayton and Voth have found significant undulations in |
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zero-surface-tension states of membranes simulated via dissipative |
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particle dynamics, but their results are consistent with purely |
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thermal undulations.~\cite{Ayton02} |
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|
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Although the organization of the tails of lipid molecules are |
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addressed by these molecular simulations and the packing competition |
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between head groups and tails is strongly implicated as the primary |
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driving force for ripple formation, questions about the ordering of |
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the head groups in ripple phase have not been settled. |
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|
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In Ch.~\ref{chap:mc}, we presented a simple ``web of dipoles'' spin |
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lattice model which provides some physical insight into relationship |
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between dipolar ordering and membrane buckling.\cite{sun:031602} We |
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found that dipolar elastic membranes can spontaneously buckle, forming |
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ripple-like topologies. The driving force for the buckling of dipolar |
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elastic membranes is the anti-ferroelectric ordering of the dipoles. |
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This was evident in the ordering of the dipole director axis |
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perpendicular to the wave vector of the surface ripples. A similar |
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phenomenon has also been observed by Tsonchev {\it et al.} in their |
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work on the spontaneous formation of dipolar peptide chains into |
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curved nano-structures.\cite{Tsonchev04,Tsonchev04II} |
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|
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In this chapter, we construct a somewhat more realistic molecular-scale |
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lipid model than our previous ``web of dipoles'' and use molecular |
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dynamics simulations to elucidate the role of the head group dipoles |
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in the formation and morphology of the ripple phase. We describe our |
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model and computational methodology in section \ref{mdsec:method}. |
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Details on the simulations are presented in section |
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\ref{mdsec:experiment}, with results following in section |
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\ref{mdsec:results}. A final discussion of the role of dipolar heads in |
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the ripple formation can be found in section |
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\ref{mdsec:discussion}. |
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|
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\section{Computational Model} |
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\label{mdsec:method} |
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|
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\begin{figure} |
102 |
\centering |
103 |
\includegraphics[width=\linewidth]{./figures/mdLipidModels.pdf} |
104 |
\caption{Three different representations of DPPC lipid molecules, |
105 |
including the chemical structure, an atomistic model, and the |
106 |
head-body ellipsoidal coarse-grained model used in this |
107 |
work.\label{mdfig:lipidModels}} |
108 |
\end{figure} |
109 |
|
110 |
Our simple molecular-scale lipid model for studying the ripple phase |
111 |
is based on two facts: one is that the most essential feature of lipid |
112 |
molecules is their amphiphilic structure with polar head groups and |
113 |
non-polar tails. Another fact is that the majority of lipid molecules |
114 |
in the ripple phase are relatively rigid (i.e. gel-like) which makes |
115 |
some fraction of the details of the chain dynamics negligible. Figure |
116 |
\ref{mdfig:lipidModels} shows the molecular structure of a DPPC |
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molecule, as well as atomistic and molecular-scale representations of |
118 |
a DPPC molecule. The hydrophilic character of the head group is |
119 |
largely due to the separation of charge between the nitrogen and |
120 |
phosphate groups. The zwitterionic nature of the PC headgroups leads |
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to abnormally large dipole moments (as high as 20.6 D), and this |
122 |
strongly polar head group interacts strongly with the solvating water |
123 |
layers immediately surrounding the membrane. The hydrophobic tail |
124 |
consists of fatty acid chains. In our molecular scale model, lipid |
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molecules have been reduced to these essential features; the fatty |
126 |
acid chains are represented by an ellipsoid with a dipolar ball |
127 |
perched on one end to represent the effects of the charge-separated |
128 |
head group. In real PC lipids, the direction of the dipole is |
129 |
nearly perpendicular to the tail, so we have fixed the direction of |
130 |
the point dipole rigidly in this orientation. |
131 |
|
132 |
The ellipsoidal portions of the model interact via the Gay-Berne |
133 |
potential which has seen widespread use in the liquid crystal |
134 |
community. Ayton and Voth have also used Gay-Berne ellipsoids for |
135 |
modeling large length-scale properties of lipid |
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bilayers.\cite{Ayton01} In its original form, the Gay-Berne potential |
137 |
was a single site model for the interactions of rigid ellipsoidal |
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molecules.\cite{Gay1981} It can be thought of as a modification of the |
139 |
Gaussian overlap model originally described by Berne and |
140 |
Pechukas.\cite{Berne72} The potential is constructed in the familiar |
141 |
form of the Lennard-Jones function using orientation-dependent |
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$\sigma$ and $\epsilon$ parameters, |
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\begin{multline} |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
145 |
r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
146 |
{\mathbf{\hat r}_{ij}})\left[ \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
147 |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
148 |
\right. \\ |
149 |
\left. - \left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
150 |
{\mathbf{\hat u}_j}, {\mathbf{\hat |
151 |
r}_{ij}})+\sigma_0}\right)^6\right] |
152 |
\label{mdeq:gb} |
153 |
\end{multline} |
154 |
|
155 |
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
156 |
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
157 |
\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
158 |
are dependent on the relative orientations of the two molecules (${\bf |
159 |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
160 |
intermolecular separation (${\bf \hat{r}}_{ij}$). $\sigma$ and |
161 |
$\sigma_0$ are also governed by shape mixing and anisotropy variables, |
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\begin {eqnarray*} |
163 |
\sigma_0 & = & \sqrt{d_i^2 + d_j^2} \\ \\ |
164 |
\chi & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 - |
165 |
d_j^2 \right)}{\left( l_j^2 + d_i^2 \right) \left(l_i^2 + |
166 |
d_j^2 \right)}\right]^{1/2} \\ \\ |
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\alpha^2 & = & \left[ \frac{\left( l_i^2 - d_i^2 \right) \left(l_j^2 + |
168 |
d_i^2 \right)}{\left( l_j^2 - d_j^2 \right) \left(l_i^2 + |
169 |
d_j^2 \right)}\right]^{1/2}, |
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\end{eqnarray*} |
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where $l$ and $d$ describe the length and width of each uniaxial |
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ellipsoid. These shape anisotropy parameters can then be used to |
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calculate the range function, |
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\begin{multline} |
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\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) = \\ |
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\sigma_0 \left[ 1 - \left\{ \frac{ \chi \alpha^2 ({\bf \hat{u}}_i \cdot {\bf |
177 |
\hat{r}}_{ij} ) + \chi \alpha^{-2} ({\bf \hat{u}}_j \cdot {\bf |
178 |
\hat{r}}_{ij} ) - 2 \chi^2 ({\bf \hat{u}}_i \cdot {\bf |
179 |
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
180 |
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi^2 |
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\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\} |
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\right]^{-1/2} |
183 |
\end{multline} |
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|
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Gay-Berne ellipsoids also have an energy scaling parameter, |
186 |
$\epsilon^s$, which describes the well depth for two identical |
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ellipsoids in a {\it side-by-side} configuration. Additionally, a well |
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depth aspect ratio, $\epsilon^r = \epsilon^e / \epsilon^s$, describes |
189 |
the ratio between the well depths in the {\it end-to-end} and |
190 |
side-by-side configurations. As in the range parameter, a set of |
191 |
mixing and anisotropy variables can be used to describe the well |
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depths for dissimilar particles, |
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\begin {eqnarray*} |
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\epsilon_0 & = & \sqrt{\epsilon^s_i * \epsilon^s_j} \\ \\ |
195 |
\epsilon^r & = & \sqrt{\epsilon^r_i * \epsilon^r_j} \\ \\ |
196 |
\chi' & = & \frac{1 - (\epsilon^r)^{1/\mu}}{1 + (\epsilon^r)^{1/\mu}} |
197 |
\\ \\ |
198 |
\alpha'^2 & = & \left[1 + (\epsilon^r)^{1/\mu}\right]^{-1} |
199 |
\end{eqnarray*} |
200 |
The form of the strength function is somewhat complicated, |
201 |
\begin{eqnarray*} |
202 |
\epsilon({\bf \hat{u}}_{i}, {\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) & = & |
203 |
\epsilon_{0} \epsilon_{1}^{\nu}({\bf \hat{u}}_{i}.{\bf \hat{u}}_{j}) |
204 |
\epsilon_{2}^{\mu}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
205 |
\hat{r}}_{ij}) \\ \\ |
206 |
\epsilon_{1}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j}) & = & |
207 |
\left[1-\chi^{2}({\bf \hat{u}}_{i}.{\bf |
208 |
\hat{u}}_{j})^{2}\right]^{-1/2} |
209 |
\end{eqnarray*} |
210 |
\begin{multline*} |
211 |
\epsilon_{2}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf \hat{r}}_{ij}) |
212 |
= \\ |
213 |
1 - \left\{ \frac{ \chi' \alpha'^2 ({\bf \hat{u}}_i \cdot {\bf |
214 |
\hat{r}}_{ij} ) + \chi' \alpha'^{-2} ({\bf \hat{u}}_j \cdot {\bf |
215 |
\hat{r}}_{ij} ) - 2 \chi'^2 ({\bf \hat{u}}_i \cdot {\bf |
216 |
\hat{r}}_{ij} )({\bf \hat{u}}_j \cdot {\bf |
217 |
\hat{r}}_{ij} ) ({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j)}{1 - \chi'^2 |
218 |
\left({\bf \hat{u}}_i \cdot {\bf \hat{u}}_j\right)^2} \right\}, |
219 |
\end{multline*} |
220 |
although many of the quantities and derivatives are identical with |
221 |
those obtained for the range parameter. Ref. \citen{Luckhurst90} |
222 |
has a particularly good explanation of the choice of the Gay-Berne |
223 |
parameters $\mu$ and $\nu$ for modeling liquid crystal molecules. An |
224 |
excellent overview of the computational methods that can be used to |
225 |
efficiently compute forces and torques for this potential can be found |
226 |
in Ref. \citen{Golubkov06} |
227 |
|
228 |
The choices of parameters we have used in this study correspond to a |
229 |
shape anisotropy of 3 for the chain portion of the molecule. In |
230 |
principle, this could be varied to allow for modeling of longer or |
231 |
shorter chain lipid molecules. For these prolate ellipsoids, we have: |
232 |
\begin{equation} |
233 |
\begin{array}{rcl} |
234 |
d & < & l \\ |
235 |
\epsilon^{r} & < & 1 |
236 |
\end{array} |
237 |
\end{equation} |
238 |
A sketch of the various structural elements of our molecular-scale |
239 |
lipid / solvent model is shown in figure \ref{mdfig:lipidModel}. The |
240 |
actual parameters used in our simulations are given in table |
241 |
\ref{mdtab:parameters}. |
242 |
|
243 |
\begin{figure} |
244 |
\centering |
245 |
\includegraphics[width=\linewidth]{./figures/md2LipidModel.pdf} |
246 |
\caption{The parameters defining the behavior of the lipid |
247 |
models. $\sigma_h / d$ is the ratio of the head group to body diameter. |
248 |
Molecular bodies had a fixed aspect ratio of 3.0. The solvent model |
249 |
was a simplified 4-water bead ($\sigma_w \approx d$) that has been |
250 |
used in other coarse-grained simulations. The dipolar strength |
251 |
(and the temperature and pressure) were the only other parameters that |
252 |
were varied systematically.\label{mdfig:lipidModel}} |
253 |
\end{figure} |
254 |
|
255 |
To take into account the permanent dipolar interactions of the |
256 |
zwitterionic head groups, we have placed fixed dipole moments |
257 |
$\mu_{i}$ at one end of the Gay-Berne particles. The dipoles are |
258 |
oriented at an angle $\theta = \pi / 2$ relative to the major axis. |
259 |
These dipoles are protected by a head ``bead'' with a range parameter |
260 |
($\sigma_h$) which we have varied between $1.20 d$ and $1.41 d$. The |
261 |
head groups interact with each other using a combination of |
262 |
Lennard-Jones, |
263 |
\begin{equation} |
264 |
V_{ij}(r_{ij}) = 4\epsilon_h \left[\left(\frac{\sigma_h}{r_{ij}}\right)^{12} - |
265 |
\left(\frac{\sigma_h}{r_{ij}}\right)^6\right], |
266 |
\end{equation} |
267 |
and dipole-dipole, |
268 |
\begin{equation} |
269 |
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
270 |
\hat{r}}_{ij})) = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
271 |
\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
272 |
\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
273 |
\end{equation} |
274 |
potentials. |
275 |
In these potentials, $\mathbf{\hat u}_i$ is the unit vector pointing |
276 |
along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
277 |
pointing along the inter-dipole vector $\mathbf{r}_{ij}$. |
278 |
|
279 |
Since the charge separation distance is so large in zwitterionic head |
280 |
groups (like the PC head groups), it would also be possible to use |
281 |
either point charges or a ``split dipole'' approximation, |
282 |
\begin{equation} |
283 |
V_{ij}({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
284 |
\hat{r}}_{ij})) = \frac{1}{{4\pi \epsilon_0 }}\left[ {\frac{{\mu _i \cdot \mu _j }}{{R_{ij}^3 }} - |
285 |
\frac{{3\left( {\mu _i \cdot r_{ij} } \right)\left( {\mu _i \cdot |
286 |
r_{ij} } \right)}}{{R_{ij}^5 }}} \right] |
287 |
\end{equation} |
288 |
where $\mu _i$ and $\mu _j$ are the dipole moments of sites $i$ and |
289 |
$j$, $r_{ij}$ is vector between the two sites, and $R_{ij}$ is given |
290 |
by, |
291 |
\begin{equation} |
292 |
R_{ij} = \sqrt {r_{ij}^2 + \frac{{d_i^2 }}{4} + \frac{{d_j^2 |
293 |
}}{4}}. |
294 |
\end{equation} |
295 |
Here, $d_i$ and $d_j$ are charge separation distances associated with |
296 |
each of the two dipolar sites. This approximation to the multipole |
297 |
expansion maintains the fast fall-off of the multipole potentials but |
298 |
lacks the normal divergences when two polar groups get close to one |
299 |
another. |
300 |
|
301 |
For the interaction between nonequivalent uniaxial ellipsoids (in this |
302 |
case, between spheres and ellipsoids), the spheres are treated as |
303 |
ellipsoids with an aspect ratio of 1 ($d = l$) and with an well depth |
304 |
ratio ($\epsilon^r$) of 1 ($\epsilon^e = \epsilon^s$). The form of |
305 |
the Gay-Berne potential we are using was generalized by Cleaver {\it |
306 |
et al.} and is appropriate for dissimilar uniaxial |
307 |
ellipsoids.\cite{Cleaver96} |
308 |
|
309 |
The solvent model in our simulations is similar to the one used by |
310 |
Marrink {\it et al.} in their coarse grained simulations of lipid |
311 |
bilayers.\cite{Marrink2004} The solvent bead is a single site that |
312 |
represents four water molecules (m = 72 amu) and has comparable |
313 |
density and diffusive behavior to liquid water. However, since there |
314 |
are no electrostatic sites on these beads, this solvent model cannot |
315 |
replicate the dielectric properties of water. Note that although we |
316 |
are using larger cutoff and switching radii than Marrink {\it et al.}, |
317 |
our solvent density at 300 K remains at 0.944 g cm$^{-3}$, and the |
318 |
solvent diffuses at 0.43 \AA$^2 ps^{-1}$ (only twice as fast as liquid |
319 |
water). |
320 |
|
321 |
\begin{table*} |
322 |
\begin{minipage}{\linewidth} |
323 |
\begin{center} |
324 |
\caption{Potential parameters used for molecular-scale coarse-grained |
325 |
lipid simulations} |
326 |
\begin{tabular}{llccc} |
327 |
\hline |
328 |
& & Head & Chain & Solvent \\ |
329 |
\hline |
330 |
$d$ (\AA) & & varied & 4.6 & 4.7 \\ |
331 |
$l$ (\AA) & & $= d$ & 13.8 & 4.7 \\ |
332 |
$\epsilon^s$ (kcal/mol) & & 0.185 & 1.0 & 0.8 \\ |
333 |
$\epsilon_r$ (well-depth aspect ratio)& & 1 & 0.2 & 1 \\ |
334 |
$m$ (amu) & & 196 & 760 & 72.06 \\ |
335 |
$\overleftrightarrow{\mathsf I}$ (amu \AA$^2$) & & & & \\ |
336 |
\multicolumn{2}c{$I_{xx}$} & 1125 & 45000 & N/A \\ |
337 |
\multicolumn{2}c{$I_{yy}$} & 1125 & 45000 & N/A \\ |
338 |
\multicolumn{2}c{$I_{zz}$} & 0 & 9000 & N/A \\ |
339 |
$\mu$ (Debye) & & varied & 0 & 0 \\ |
340 |
\end{tabular} |
341 |
\label{mdtab:parameters} |
342 |
\end{center} |
343 |
\end{minipage} |
344 |
\end{table*} |
345 |
|
346 |
\section{Experimental Methodology} |
347 |
\label{mdsec:experiment} |
348 |
|
349 |
The parameters that were systematically varied in this study were the |
350 |
size of the head group ($\sigma_h$), the strength of the dipole moment |
351 |
($\mu$), and the temperature of the system. Values for $\sigma_h$ |
352 |
ranged from 5.5 \AA\ to 6.5 \AA. If the width of the tails is taken |
353 |
to be the unit of length, these head groups correspond to a range from |
354 |
$1.2 d$ to $1.41 d$. Since the solvent beads are nearly identical in |
355 |
diameter to the tail ellipsoids, all distances that follow will be |
356 |
measured relative to this unit of distance. Because the solvent we |
357 |
are using is non-polar and has a dielectric constant of 1, values for |
358 |
$\mu$ are sampled from a range that is somewhat smaller than the 20.6 |
359 |
Debye dipole moment of the PC head groups. |
360 |
|
361 |
To create unbiased bilayers, all simulations were started from two |
362 |
perfectly flat monolayers separated by a 26 \AA\ gap between the |
363 |
molecular bodies of the upper and lower leaves. The separated |
364 |
monolayers were evolved in a vacuum with $x-y$ anisotropic pressure |
365 |
coupling. The length of $z$ axis of the simulations was fixed and a |
366 |
constant surface tension was applied to enable real fluctuations of |
367 |
the bilayer. Periodic boundary conditions were used, and $480-720$ |
368 |
lipid molecules were present in the simulations, depending on the size |
369 |
of the head beads. In all cases, the two monolayers spontaneously |
370 |
collapsed into bilayer structures within 100 ps. Following this |
371 |
collapse, all systems were equilibrated for $100$ ns at $300$ K. |
372 |
|
373 |
The resulting bilayer structures were then solvated at a ratio of $6$ |
374 |
solvent beads (24 water molecules) per lipid. These configurations |
375 |
were then equilibrated for another $30$ ns. All simulations utilizing |
376 |
the solvent were carried out at constant pressure ($P=1$ atm) with |
377 |
$3$D anisotropic coupling, and small constant surface tension |
378 |
($\gamma=0.015$ N/m). Given the absence of fast degrees of freedom in |
379 |
this model, a time step of $50$ fs was utilized with excellent energy |
380 |
conservation. Data collection for structural properties of the |
381 |
bilayers was carried out during a final 5 ns run following the solvent |
382 |
equilibration. Orientational correlation functions and diffusion |
383 |
constants were computed from 30 ns simulations in the microcanonical |
384 |
(NVE) ensemble using the average volume from the end of the constant |
385 |
pressure and surface tension runs. The timestep on these final |
386 |
molecular dynamics runs was 25 fs. No appreciable changes in phase |
387 |
structure were noticed upon switching to a microcanonical ensemble. |
388 |
All simulations were performed using the {\sc oopse} molecular |
389 |
modeling program.\cite{Meineke2005} |
390 |
|
391 |
A switching function was applied to all potentials to smoothly turn |
392 |
off the interactions between a range of $22$ and $25$ \AA. The |
393 |
switching function was the standard (cubic) function, |
394 |
\begin{equation} |
395 |
s(r) = |
396 |
\begin{cases} |
397 |
1 & \text{if $r \le r_{\text{sw}}$},\\ |
398 |
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
399 |
{(r_{\text{cut}} - r_{\text{sw}})^3} |
400 |
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
401 |
0 & \text{if $r > r_{\text{cut}}$.} |
402 |
\end{cases} |
403 |
\label{mdeq:dipoleSwitching} |
404 |
\end{equation} |
405 |
|
406 |
\section{Results} |
407 |
\label{mdsec:results} |
408 |
|
409 |
The membranes in our simulations exhibit a number of interesting |
410 |
bilayer phases. The surface topology of these phases depends most |
411 |
sensitively on the ratio of the size of the head groups to the width |
412 |
of the molecular bodies. With heads only slightly larger than the |
413 |
bodies ($\sigma_h=1.20 d$) the membrane exhibits a flat bilayer. |
414 |
|
415 |
Increasing the head / body size ratio increases the local membrane |
416 |
curvature around each of the lipids. With $\sigma_h=1.28 d$, the |
417 |
surface is still essentially flat, but the bilayer starts to exhibit |
418 |
signs of instability. We have observed occasional defects where a |
419 |
line of lipid molecules on one leaf of the bilayer will dip down to |
420 |
interdigitate with the other leaf. This gives each of the two bilayer |
421 |
leaves some local convexity near the line defect. These structures, |
422 |
once developed in a simulation, are very stable and are spaced |
423 |
approximately 100 \AA\ away from each other. |
424 |
|
425 |
With larger heads ($\sigma_h = 1.35 d$) the membrane curvature |
426 |
resolves into a ``symmetric'' ripple phase. Each leaf of the bilayer |
427 |
is broken into several convex, hemicylinderical sections, and opposite |
428 |
leaves are fitted together much like roof tiles. There is no |
429 |
interdigitation between the upper and lower leaves of the bilayer. |
430 |
|
431 |
For the largest head / body ratios studied ($\sigma_h = 1.41 d$) the |
432 |
local curvature is substantially larger, and the resulting bilayer |
433 |
structure resolves into an asymmetric ripple phase. This structure is |
434 |
very similar to the structures observed by both de~Vries {\it et al.} |
435 |
and Lenz {\it et al.}. For a given ripple wave vector, there are two |
436 |
possible asymmetric ripples, which is not the case for the symmetric |
437 |
phase observed when $\sigma_h = 1.35 d$. |
438 |
|
439 |
\begin{figure} |
440 |
\centering |
441 |
\includegraphics[width=\linewidth]{./figures/mdPhaseCartoon.pdf} |
442 |
\caption{The role of the ratio between the head group size and the |
443 |
width of the molecular bodies is to increase the local membrane |
444 |
curvature. With strong attractive interactions between the head |
445 |
groups, this local curvature can be maintained in bilayer structures |
446 |
through surface corrugation. Shown above are three phases observed in |
447 |
these simulations. With $\sigma_h = 1.20 d$, the bilayer maintains a |
448 |
flat topology. For larger heads ($\sigma_h = 1.35 d$) the local |
449 |
curvature resolves into a symmetrically rippled phase with little or |
450 |
no interdigitation between the upper and lower leaves of the membrane. |
451 |
The largest heads studied ($\sigma_h = 1.41 d$) resolve into an |
452 |
asymmetric rippled phases with interdigitation between the two |
453 |
leaves.\label{mdfig:phaseCartoon}} |
454 |
\end{figure} |
455 |
|
456 |
Sample structures for the flat ($\sigma_h = 1.20 d$), symmetric |
457 |
($\sigma_h = 1.35 d$, and asymmetric ($\sigma_h = 1.41 d$) ripple |
458 |
phases are shown in Figure \ref{mdfig:phaseCartoon}. |
459 |
|
460 |
It is reasonable to ask how well the parameters we used can produce |
461 |
bilayer properties that match experimentally known values for real |
462 |
lipid bilayers. Using a value of $l = 13.8$ \AA~for the ellipsoidal |
463 |
tails and the fixed ellipsoidal aspect ratio of 3, our values for the |
464 |
area per lipid ($A$) and inter-layer spacing ($D_{HH}$) depend |
465 |
entirely on the size of the head bead relative to the molecular body. |
466 |
These values are tabulated in table \ref{mdtab:property}. Kucera {\it |
467 |
et al.} have measured values for the head group spacings for a number |
468 |
of PC lipid bilayers that range from 30.8 \AA\ (DLPC) to 37.8 \AA\ (DPPC). |
469 |
They have also measured values for the area per lipid that range from |
470 |
60.6 |
471 |
\AA$^2$ (DMPC) to 64.2 \AA$^2$ |
472 |
(DPPC).\cite{NorbertKucerka04012005,NorbertKucerka06012006} Only the |
473 |
largest of the head groups we modeled ($\sigma_h = 1.41 d$) produces |
474 |
bilayers (specifically the area per lipid) that resemble real PC |
475 |
bilayers. The smaller head beads we used are perhaps better models |
476 |
for PE head groups. |
477 |
|
478 |
\begin{table*} |
479 |
\begin{minipage}{\linewidth} |
480 |
\begin{center} |
481 |
\caption{Phase, bilayer spacing, area per lipid, ripple wavelength |
482 |
and amplitude observed as a function of the ratio between the head |
483 |
beads and the diameters of the tails. Ripple wavelengths and |
484 |
amplitudes are normalized to the diameter of the tail ellipsoids.} |
485 |
\begin{tabular}{lccccc} |
486 |
\hline |
487 |
$\sigma_h / d$ & type of phase & bilayer spacing (\AA) & area per |
488 |
lipid (\AA$^2$) & $\lambda / d$ & $A / d$\\ |
489 |
\hline |
490 |
1.20 & flat & 33.4 & 49.6 & N/A & N/A \\ |
491 |
1.28 & flat & 33.7 & 54.7 & N/A & N/A \\ |
492 |
1.35 & symmetric ripple & 42.9 & 51.7 & 17.2 & 2.2 \\ |
493 |
1.41 & asymmetric ripple & 37.1 & 63.1 & 15.4 & 1.5 \\ |
494 |
\end{tabular} |
495 |
\label{mdtab:property} |
496 |
\end{center} |
497 |
\end{minipage} |
498 |
\end{table*} |
499 |
|
500 |
The membrane structures and the reduced wavelength $\lambda / d$, |
501 |
reduced amplitude $A / d$ of the ripples are summarized in Table |
502 |
\ref{mdtab:property}. The wavelength range is $15 - 17$ molecular bodies |
503 |
and the amplitude is $1.5$ molecular bodies for asymmetric ripple and |
504 |
$2.2$ for symmetric ripple. These values are reasonably consistent |
505 |
with experimental measurements.\cite{Sun96,Katsaras00,Kaasgaard03} |
506 |
Note, that given the lack of structural freedom in the tails of our |
507 |
model lipids, the amplitudes observed from these simulations are |
508 |
likely to underestimate of the true amplitudes. |
509 |
|
510 |
\begin{figure} |
511 |
\centering |
512 |
\includegraphics[width=\linewidth]{./figures/mdTopDown.pdf} |
513 |
\caption{Top views of the flat (upper), symmetric ripple (middle), |
514 |
and asymmetric ripple (lower) phases. Note that the head-group |
515 |
dipoles have formed head-to-tail chains in all three of these phases, |
516 |
but in the two rippled phases, the dipolar chains are all aligned {\it |
517 |
perpendicular} to the direction of the ripple. Note that the flat |
518 |
membrane has multiple vortex defects in the dipolar ordering, and the |
519 |
ordering on the lower leaf of the bilayer can be in an entirely |
520 |
different direction from the upper leaf.\label{mdfig:topView}} |
521 |
\end{figure} |
522 |
|
523 |
The orientational ordering in the system is observed by $P_2$ order |
524 |
parameter, which is calculated from Eq.~\ref{mceq:opmatrix} |
525 |
in Ch.~\ref{chap:mc}. Here, $\hat{\bf u}_i$ can refer either to the |
526 |
principal axis of the molecular body or to the dipole on the head |
527 |
group of the molecule. Since the molecular bodies are perpendicular to |
528 |
the head group dipoles, it is possible for the director axes for the |
529 |
molecular bodies and the head groups to be completely decoupled from |
530 |
each other. |
531 |
|
532 |
Figure \ref{mdfig:topView} shows snapshots of bird's-eye views of the |
533 |
flat ($\sigma_h = 1.20 d$) and rippled ($\sigma_h = 1.35, 1.41 d$) |
534 |
bilayers. The directions of the dipoles on the head groups are |
535 |
represented with two colored half spheres: blue (phosphate) and yellow |
536 |
(amino). For flat bilayers, the system exhibits signs of |
537 |
orientational frustration; some disorder in the dipolar head-to-tail |
538 |
chains is evident with kinks visible at the edges between differently |
539 |
ordered domains. The lipids can also move independently of lipids in |
540 |
the opposing leaf, so the ordering of the dipoles on one leaf is not |
541 |
necessarily consistent with the ordering on the other. These two |
542 |
factors keep the total dipolar order parameter relatively low for the |
543 |
flat phases. |
544 |
|
545 |
With increasing head group size, the surface becomes corrugated, and |
546 |
the dipoles cannot move as freely on the surface. Therefore, the |
547 |
translational freedom of lipids in one layer is dependent upon the |
548 |
position of the lipids in the other layer. As a result, the ordering of |
549 |
the dipoles on head groups in one leaf is correlated with the ordering |
550 |
in the other leaf. Furthermore, as the membrane deforms due to the |
551 |
corrugation, the symmetry of the allowed dipolar ordering on each leaf |
552 |
is broken. The dipoles then self-assemble in a head-to-tail |
553 |
configuration, and the dipolar order parameter increases dramatically. |
554 |
However, the total polarization of the system is still close to zero. |
555 |
This is strong evidence that the corrugated structure is an |
556 |
anti-ferroelectric state. It is also notable that the head-to-tail |
557 |
arrangement of the dipoles is always observed in a direction |
558 |
perpendicular to the wave vector for the surface corrugation. This is |
559 |
a similar finding to what we observed in our earlier work on the |
560 |
elastic dipolar membranes.\cite{sun:031602} |
561 |
|
562 |
The $P_2$ order parameters (for both the molecular bodies and the head |
563 |
group dipoles) have been calculated to quantify the ordering in these |
564 |
phases. Figure \ref{mdfig:rP2} shows that the $P_2$ order parameter for |
565 |
the head-group dipoles increases with increasing head group size. When |
566 |
the heads of the lipid molecules are small, the membrane is nearly |
567 |
flat. Since the in-plane packing is essentially a close packing of the |
568 |
head groups, the head dipoles exhibit frustration in their |
569 |
orientational ordering. |
570 |
|
571 |
The ordering trends for the tails are essentially opposite to the |
572 |
ordering of the head group dipoles. The tail $P_2$ order parameter |
573 |
{\it decreases} with increasing head size. This indicates that the |
574 |
surface is more curved with larger head / tail size ratios. When the |
575 |
surface is flat, all tails are pointing in the same direction (normal |
576 |
to the bilayer surface). This simplified model appears to be |
577 |
exhibiting a smectic A fluid phase, similar to the real $L_{\beta}$ |
578 |
phase. We have not observed a smectic C gel phase ($L_{\beta'}$) for |
579 |
this model system. Increasing the size of the heads results in |
580 |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
581 |
|
582 |
\begin{figure} |
583 |
\centering |
584 |
\includegraphics[width=\linewidth]{./figures/mdRP2.pdf} |
585 |
\caption{The $P_2$ order parameters for head groups (circles) and |
586 |
molecular bodies (squares) as a function of the ratio of head group |
587 |
size ($\sigma_h$) to the width of the molecular bodies ($d$). \label{mdfig:rP2}} |
588 |
\end{figure} |
589 |
|
590 |
In addition to varying the size of the head groups, we studied the |
591 |
effects of the interactions between head groups on the structure of |
592 |
lipid bilayer by changing the strength of the dipoles. Figure |
593 |
\ref{mdfig:sP2} shows how the $P_2$ order parameter changes with |
594 |
increasing strength of the dipole. Generally, the dipoles on the head |
595 |
groups become more ordered as the strength of the interaction between |
596 |
heads is increased and become more disordered by decreasing the |
597 |
interaction strength. When the interaction between the heads becomes |
598 |
too weak, the bilayer structure does not persist; all lipid molecules |
599 |
become dispersed in the solvent (which is non-polar in this |
600 |
molecular-scale model). The critical value of the strength of the |
601 |
dipole depends on the size of the head groups. The perfectly flat |
602 |
surface becomes unstable below $5$ Debye, while the rippled |
603 |
surfaces melt at $8$ Debye (asymmetric) or $10$ Debye (symmetric). |
604 |
|
605 |
The ordering of the tails mirrors the ordering of the dipoles {\it |
606 |
except for the flat phase}. Since the surface is nearly flat in this |
607 |
phase, the order parameters are only weakly dependent on dipolar |
608 |
strength until it reaches $15$ Debye. Once it reaches this value, the |
609 |
head group interactions are strong enough to pull the head groups |
610 |
close to each other and distort the bilayer structure. For a flat |
611 |
surface, a substantial amount of free volume between the head groups |
612 |
is normally available. When the head groups are brought closer by |
613 |
dipolar interactions, the tails are forced to splay outward, first forming |
614 |
curved bilayers, and then inverted micelles. |
615 |
|
616 |
When $\sigma_h=1.28 d$, the $P_2$ order parameter decreases slightly |
617 |
when the strength of the dipole is increased above $16$ Debye. For |
618 |
rippled bilayers, there is less free volume available between the head |
619 |
groups. Therefore increasing dipolar strength weakly influences the |
620 |
structure of the membrane. However, the increase in the body $P_2$ |
621 |
order parameters implies that the membranes are being slightly |
622 |
flattened due to the effects of increasing head-group attraction. |
623 |
|
624 |
A very interesting behavior takes place when the head groups are very |
625 |
large relative to the molecular bodies ($\sigma_h = 1.41 d$) and the |
626 |
dipolar strength is relatively weak ($\mu < 9$ Debye). In this case, |
627 |
the two leaves of the bilayer become totally interdigitated with each |
628 |
other in large patches of the membrane. With higher dipolar |
629 |
strength, the interdigitation is limited to single lines that run |
630 |
through the bilayer in a direction perpendicular to the ripple wave |
631 |
vector. |
632 |
|
633 |
\begin{figure} |
634 |
\centering |
635 |
\includegraphics[width=\linewidth]{./figures/mdSP2.pdf} |
636 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
637 |
molecular bodies (b) as a function of the strength of the dipoles. |
638 |
These order parameters are shown for four values of the head group / |
639 |
molecular width ratio ($\sigma_h / d$). \label{mdfig:sP2}} |
640 |
\end{figure} |
641 |
|
642 |
Figure \ref{mdfig:tP2} shows the dependence of the order parameters on |
643 |
temperature. As expected, systems are more ordered at low |
644 |
temperatures, and more disordered at high temperatures. All of the |
645 |
bilayers we studied can become unstable if the temperature becomes |
646 |
high enough. The only interesting feature of the temperature |
647 |
dependence is in the flat surfaces ($\sigma_h=1.20 d$ and |
648 |
$\sigma_h=1.28 d$). Here, when the temperature is increased above |
649 |
$310$K, there is enough jostling of the head groups to allow the |
650 |
dipolar frustration to resolve into more ordered states. This results |
651 |
in a slight increase in the $P_2$ order parameter above this |
652 |
temperature. |
653 |
|
654 |
For the rippled surfaces ($\sigma_h=1.35 d$ and $\sigma_h=1.41 d$), |
655 |
there is a slightly increased orientational ordering in the molecular |
656 |
bodies above $290$K. Since our model lacks the detailed information |
657 |
about the behavior of the lipid tails, this is the closest the model |
658 |
can come to depicting the ripple ($P_{\beta'}$) to fluid |
659 |
($L_{\alpha}$) phase transition. What we are observing is a |
660 |
flattening of the rippled structures made possible by thermal |
661 |
expansion of the tightly-packed head groups. The lack of detailed |
662 |
chain configurations also makes it impossible for this model to depict |
663 |
the ripple to gel ($L_{\beta'}$) phase transition. |
664 |
|
665 |
\begin{figure} |
666 |
\centering |
667 |
\includegraphics[width=\linewidth]{./figures/mdTP2.pdf} |
668 |
\caption{The $P_2$ order parameters for head group dipoles (a) and |
669 |
molecular bodies (b) as a function of temperature. |
670 |
These order parameters are shown for four values of the head group / |
671 |
molecular width ratio ($\sigma_h / d$).\label{mdfig:tP2}} |
672 |
\end{figure} |
673 |
|
674 |
Fig. \ref{mdfig:phaseDiagram} shows a phase diagram for the model as a |
675 |
function of the head group / molecular width ratio ($\sigma_h / d$) |
676 |
and the strength of the head group dipole moment ($\mu$). Note that |
677 |
the specific form of the bilayer phase is governed almost entirely by |
678 |
the head group / molecular width ratio, while the strength of the |
679 |
dipolar interactions between the head groups governs the stability of |
680 |
the bilayer phase. Weaker dipoles result in unstable bilayer phases, |
681 |
while extremely strong dipoles can shift the equilibrium to an |
682 |
inverted micelle phase when the head groups are small. Temperature |
683 |
has little effect on the actual bilayer phase observed, although higher |
684 |
temperatures can cause the unstable region to grow into the higher |
685 |
dipole region of this diagram. |
686 |
|
687 |
\begin{figure} |
688 |
\centering |
689 |
\includegraphics[width=\linewidth]{./figures/mdPhaseDiagram.pdf} |
690 |
\caption{Phase diagram for the simple molecular model as a function |
691 |
of the head group / molecular width ratio ($\sigma_h / d$) and the |
692 |
strength of the head group dipole moment |
693 |
($\mu$).\label{mdfig:phaseDiagram}} |
694 |
\end{figure} |
695 |
|
696 |
We have computed translational diffusion constants for lipid molecules |
697 |
from the mean-square displacement, |
698 |
\begin{equation} |
699 |
D = \lim_{t \rightarrow \infty} \frac{1}{6 t} \langle {|\left({\bf |
700 |
r}_{i}(t) - {\bf r}_{i}(0) \right)|}^2 \rangle, |
701 |
\label{mdeq:msdisplacement} |
702 |
\end{equation} |
703 |
of the lipid bodies. Translational diffusion constants for the |
704 |
different head-to-tail size ratios (all at 300 K) are shown in table |
705 |
\ref{mdtab:relaxation}. We have also computed orientational correlation |
706 |
times for the head groups from fits of the second-order Legendre |
707 |
polynomial correlation function, |
708 |
\begin{equation} |
709 |
C_{\ell}(t) = \langle P_{\ell}\left({\bf \mu}_{i}(t) \cdot {\bf |
710 |
\mu}_{i}(0) \right) \rangle |
711 |
\end{equation} |
712 |
of the head group dipoles. The orientational correlation functions |
713 |
appear to have multiple components in their decay: a fast ($12 \pm 2$ |
714 |
ps) decay due to librational motion of the head groups, as well as |
715 |
moderate ($\tau^h_{\rm mid}$) and slow ($\tau^h_{\rm slow}$) |
716 |
components. The fit values for the moderate and slow correlation |
717 |
times are listed in table \ref{mdtab:relaxation}. Standard deviations |
718 |
in the fit time constants are quite large (on the order of the values |
719 |
themselves). |
720 |
|
721 |
Sparrman and Westlund used a multi-relaxation model for NMR lineshapes |
722 |
observed in gel, fluid, and ripple phases of DPPC and obtained |
723 |
estimates of a correlation time for water translational diffusion |
724 |
($\tau_c$) of 20 ns.\cite{Sparrman2003} This correlation time |
725 |
corresponds to water bound to small regions of the lipid membrane. |
726 |
They further assume that the lipids can explore only a single period |
727 |
of the ripple (essentially moving in a nearly one-dimensional path to |
728 |
do so), and the correlation time can therefore be used to estimate a |
729 |
value for the translational diffusion constant of $2.25 \times |
730 |
10^{-11} m^2 s^{-1}$. The translational diffusion constants we obtain |
731 |
are in reasonable agreement with this experimentally determined |
732 |
value. However, the $T_2$ relaxation times obtained by Sparrman and |
733 |
Westlund are consistent with P-N vector reorientation timescales of |
734 |
2-5 ms. This is substantially slower than even the slowest component |
735 |
we observe in the decay of our orientational correlation functions. |
736 |
Other than the dipole-dipole interactions, our head groups have no |
737 |
shape anisotropy which would force them to move as a unit with |
738 |
neighboring molecules. This would naturally lead to P-N reorientation |
739 |
times that are too fast when compared with experimental measurements. |
740 |
|
741 |
\begin{table*} |
742 |
\begin{minipage}{\linewidth} |
743 |
\begin{center} |
744 |
\caption{Fit values for the rotational correlation times for the head |
745 |
groups ($\tau^h$) and molecular bodies ($\tau^b$) as well as the |
746 |
translational diffusion constants for the molecule as a function of |
747 |
the head-to-body width ratio. All correlation functions and transport |
748 |
coefficients were computed from microcanonical simulations with an |
749 |
average temperture of 300 K. In all of the phases, the head group |
750 |
correlation functions decay with an fast librational contribution ($12 |
751 |
\pm 1$ ps). There are additional moderate ($\tau^h_{\rm mid}$) and |
752 |
slow $\tau^h_{\rm slow}$ contributions to orientational decay that |
753 |
depend strongly on the phase exhibited by the lipids. The symmetric |
754 |
ripple phase ($\sigma_h / d = 1.35$) appears to exhibit the slowest |
755 |
molecular reorientation.} |
756 |
\begin{tabular}{lcccc} |
757 |
\hline |
758 |
$\sigma_h / d$ & $\tau^h_{\rm mid} (ns)$ & $\tau^h_{\rm |
759 |
slow} (\mu s)$ & $\tau^b (\mu s)$ & $D (\times 10^{-11} m^2 s^{-1})$ \\ |
760 |
\hline |
761 |
1.20 & $0.4$ & $9.6$ & $9.5$ & $0.43(1)$ \\ |
762 |
1.28 & $2.0$ & $13.5$ & $3.0$ & $5.91(3)$ \\ |
763 |
1.35 & $3.2$ & $4.0$ & $0.9$ & $3.42(1)$ \\ |
764 |
1.41 & $0.3$ & $23.8$ & $6.9$ & $7.16(1)$ \\ |
765 |
\end{tabular} |
766 |
\label{mdtab:relaxation} |
767 |
\end{center} |
768 |
\end{minipage} |
769 |
\end{table*} |
770 |
|
771 |
\section{Discussion} |
772 |
\label{mdsec:discussion} |
773 |
|
774 |
Symmetric and asymmetric ripple phases have been observed to form in |
775 |
our molecular dynamics simulations of a simple molecular-scale lipid |
776 |
model. The lipid model consists of an dipolar head group and an |
777 |
ellipsoidal tail. Within the limits of this model, an explanation for |
778 |
generalized membrane curvature is a simple mismatch in the size of the |
779 |
heads with the width of the molecular bodies. With heads |
780 |
substantially larger than the bodies of the molecule, this curvature |
781 |
should be convex nearly everywhere, a requirement which could be |
782 |
resolved either with micellar or cylindrical phases. |
783 |
|
784 |
The persistence of a {\it bilayer} structure therefore requires either |
785 |
strong attractive forces between the head groups or exclusionary |
786 |
forces from the solvent phase. To have a persistent bilayer structure |
787 |
with the added requirement of convex membrane curvature appears to |
788 |
result in corrugated structures like the ones pictured in |
789 |
Fig. \ref{mdfig:phaseCartoon}. In each of the sections of these |
790 |
corrugated phases, the local curvature near a most of the head groups |
791 |
is convex. These structures are held together by the extremely strong |
792 |
and directional interactions between the head groups. |
793 |
|
794 |
The attractive forces holding the bilayer together could either be |
795 |
non-directional (as in the work of Kranenburg and |
796 |
Smit),\cite{Kranenburg2005} or directional (as we have utilized in |
797 |
these simulations). The dipolar head groups are key for the |
798 |
maintaining the bilayer structures exhibited by this particular model; |
799 |
reducing the strength of the dipole has the tendency to make the |
800 |
rippled phase disappear. The dipoles are likely to form attractive |
801 |
head-to-tail configurations even in flat configurations, but the |
802 |
temperatures are high enough that vortex defects become prevalent in |
803 |
the flat phase. The flat phase we observed therefore appears to be |
804 |
substantially above the Kosterlitz-Thouless transition temperature for |
805 |
a planar system of dipoles with this set of parameters. For this |
806 |
reason, it would be interesting to observe the thermal behavior of the |
807 |
flat phase at substantially lower temperatures. |
808 |
|
809 |
One feature of this model is that an energetically favorable |
810 |
orientational ordering of the dipoles can be achieved by forming |
811 |
ripples. The corrugation of the surface breaks the symmetry of the |
812 |
plane, making vortex defects somewhat more expensive, and stabilizing |
813 |
the long range orientational ordering for the dipoles in the head |
814 |
groups. Most of the rows of the head-to-tail dipoles are parallel to |
815 |
each other and the system adopts a bulk anti-ferroelectric state. We |
816 |
believe that this is the first time the organization of the head |
817 |
groups in ripple phases has been addressed. |
818 |
|
819 |
Although the size-mismatch between the heads and molecular bodies |
820 |
appears to be the primary driving force for surface convexity, the |
821 |
persistence of the bilayer through the use of rippled structures is a |
822 |
function of the strong, attractive interactions between the heads. |
823 |
One important prediction we can make using the results from this |
824 |
simple model is that if the dipole-dipole interaction is the leading |
825 |
contributor to the head group attractions, the wave vectors for the |
826 |
ripples should always be found {\it perpendicular} to the dipole |
827 |
director axis. This echoes the prediction we made earlier for simple |
828 |
elastic dipolar membranes, and may suggest experimental designs which |
829 |
will test whether this is really the case in the phosphatidylcholine |
830 |
$P_{\beta'}$ phases. The dipole director axis should also be easily |
831 |
computable for the all-atom and coarse-grained simulations that have |
832 |
been published in the literature.\cite{deVries05} |
833 |
|
834 |
Experimental verification of our predictions of dipolar orientation |
835 |
correlating with the ripple direction would require knowing both the |
836 |
local orientation of a rippled region of the membrane (available via |
837 |
AFM studies of supported bilayers) as well as the local ordering of |
838 |
the membrane dipoles. Obtaining information about the local |
839 |
orientations of the membrane dipoles may be available from |
840 |
fluorescence detected linear dichroism (LD). Benninger {\it et al.} |
841 |
have recently used axially-specific chromophores |
842 |
2-(4,4-difluoro-5,7-dimethyl-4-bora-3a,4a-diaza-s-indacene-3-pentanoyl)-1-hexadecanoyl-sn-glycero-3-phospocholine |
843 |
($\beta$-BODIPY FL C5-HPC or BODIPY-PC) and 3,3' |
844 |
dioctadecyloxacarbocyanine perchlorate (DiO) in their |
845 |
fluorescence-detected linear dichroism (LD) studies of plasma |
846 |
membranes of living cells.\cite{Benninger:2005qy} The DiO dye aligns |
847 |
its transition moment perpendicular to the membrane normal, while the |
848 |
BODIPY-PC transition dipole is parallel with the membrane normal. |
849 |
Without a doubt, using fluorescence detection of linear dichroism in |
850 |
concert with AFM surface scanning would be difficult experiments to |
851 |
carry out. However, there is some hope of performing experiments to |
852 |
either verify or falsify the predictions of our simulations. |
853 |
|
854 |
Although our model is simple, it exhibits some rich and unexpected |
855 |
behaviors. It would clearly be a closer approximation to reality if |
856 |
we allowed bending motions between the dipoles and the molecular |
857 |
bodies, and if we replaced the rigid ellipsoids with ball-and-chain |
858 |
tails. However, the advantages of this simple model (large system |
859 |
sizes, 50 fs time steps) allow us to rapidly explore the phase diagram |
860 |
for a wide range of parameters. Our explanation of this rippling |
861 |
phenomenon will help us design more accurate molecular models for |
862 |
corrugated membranes and experiments to test whether or not |
863 |
dipole-dipole interactions exert an influence on membrane rippling. |