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%\bibliographystyle{aps} |
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\title{} |
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\title{Dipolar Ordering of the Ripple Phase in Lipid Membranes} |
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\author{Xiuquan Sun and J. Daniel Gezelter} |
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\email[E-mail:]{gezelter@nd.edu} |
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\affiliation{Department of Chemistry and Biochemistry,\\ |
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\pacs{} |
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\maketitle |
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|
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\section{Introduction} |
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\label{sec:Int} |
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|
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As one of the most important components in the formation of the |
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biomembrane, lipid molecules attracted numerous studies in the past |
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several decades. Due to their amphiphilic structure, when dispersed in |
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water, lipids can self-assemble to construct a bilayer structure. The |
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phase behavior of lipid membrane is well understood. The gel-fluid |
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phase transition is known as main phase transition. However, there is |
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an intermediate phase between gel and fluid phase for some lipid (like |
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phosphatidycholine (PC)) membranes. This intermediate phase |
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distinguish itself from other phases by its corrugated membrane |
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surface, therefore this phase is termed ``ripple'' ($P_{\beta '}$) |
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phase. The phase transition between gel-fluid and ripple phase is |
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called pretransition. Since the pretransition usually occurs in room |
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temperature, there might be some important biofuntions carried by the |
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ripple phase for the living organism. |
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|
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The ripple phase is observed experimentally by x-ray diffraction |
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~\cite{Sun96,Katsaras00}, freeze-fracture electron microscopy |
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(FFEM)~\cite{Copeland80,Meyer96}, and atomic force microscopy (AFM) |
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recently~\cite{Kaasgaard03}. The experimental studies suggest two |
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kinds of ripple structures: asymmetric (sawtooth like) and symmetric |
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(sinusoidal like) ripple phases. Substantial number of theoretical |
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explaination applied on the formation of the ripple |
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phases~\cite{Marder84,Carlson87,Goldstein88,McCullough90,Lubensky93,Misbah98,Heimburg00,Kubica02,Bannerjee02}. |
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In contrast, few molecular modelling have been done due to the large |
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size of the resulting structures and the time required for the phases |
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of interest to develop. One of the interesting molecular simulations |
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was carried out by De Vries and Marrink {\it et |
| 57 |
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al.}~\cite{deVries05}. According to their dynamic simulation results, |
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the ripple consists of two domains, one is gel bilayer, and in the |
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other domain, the upper and lower leaves of the bilayer are fully |
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interdigitated. The mechanism of the formation of the ripple phase in |
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their work suggests the theory that the packing competition between |
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head group and tail of lipid molecules is the driving force for the |
| 63 |
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formation of the ripple phase~\cite{Carlson87}. Recently, the ripple |
| 64 |
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phase is also studied by using monte carlo simulation~\cite{Lenz07}, |
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the ripple structure is similar to the results of Marrink except that |
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the connection of the upper and lower leaves of the bilayer is an |
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interdigitated line instead of the fully interdigitated |
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domain. Furthermore, the symmetric ripple phase was also observed in |
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their work. They claimed the mismatch between the size of the head |
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group and tail of the lipid molecules is the driving force for the |
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formation of the ripple phase. |
| 72 |
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|
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Although the organizations of the tails of lipid molecules are |
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addressed by these molecular simulations, the ordering of the head |
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group in ripple phase is still not settlement. We developed a simple |
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``web of dipoles'' spin lattice model which provides some physical |
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insight in our previous studies~\cite{Sun2007}, we found the dipoles |
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on head groups of the lipid molecules are ordered in an |
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antiferroelectric state. The similiar phenomenon is also observed by |
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Tsonchev {\it et al.} when they studied the formation of the |
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nanotube\cite{Tsonchev04}. |
| 82 |
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|
| 83 |
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In this paper, we made a more realistic coarse-grained lipid model to |
| 84 |
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understand the primary driving force for membrane corrugation and to |
| 85 |
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elucidate the organization of the anisotropic interacting head group |
| 86 |
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via molecular dynamics simulation. We will talk about our model and |
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methodology in section \ref{sec:method}, and details of the simulation |
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in section \ref{sec:experiment}. The results are shown in section |
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\ref{sec:results}. At last, we will discuss the results in section |
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\ref{sec:discussion}. |
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|
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\section{Methodology and Model} |
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\label{sec:method} |
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|
| 95 |
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Our idea for developing a simple and reasonable lipid model to study |
| 96 |
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the ripple pahse of lipid bilayers is based on two facts: one is that |
| 96 |
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the ripple phase of lipid bilayers is based on two facts: one is that |
| 97 |
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the most essential feature of lipid molecules is their amphiphilic |
| 98 |
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structure with polar head groups and non-polar tails. Another fact is |
| 99 |
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that dominant numbers of lipid molecules are very rigid in ripple |
| 100 |
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phase which allows the details of the lipid molecules neglectable. In |
| 101 |
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our model, lipid molecules are represented by rigid bodies made of one |
| 102 |
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head sphere with a point dipole sitting on it and one ellipsoid tail, |
| 103 |
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the direction of the dipole is fixed to be perpendicular to the |
| 104 |
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tail. The breadth and length of tail are $\sigma_0$, $3\sigma_0$. The |
| 105 |
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diameter of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The |
| 106 |
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model of the solvent in our simulations is inspired by the idea of |
| 107 |
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``DPD'' water. Every four water molecules are reprsented by one |
| 108 |
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sphere. |
| 100 |
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phase which allows the details of the lipid molecules neglectable. The |
| 101 |
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lipid model is shown in Figure \ref{fig:lipidMM}. Figure |
| 102 |
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\ref{fig:lipidMM}a shows the molecular strucure of a DPPC molecule. The |
| 103 |
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hydrophilic character of the head group is the effect of the strong |
| 104 |
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dipole composed by a positive charge sitting on the nitrogen and a |
| 105 |
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negative charge on the phosphate. The hydrophobic tail consists of |
| 106 |
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fatty acid chains. In our model shown in Figure \ref{fig:lipidMM}b, |
| 107 |
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lipid molecules are represented by rigid bodies made of one head |
| 108 |
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sphere with a point dipole sitting on it and one ellipsoid tail, the |
| 109 |
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direction of the dipole is fixed to be perpendicular to the tail. The |
| 110 |
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breadth and length of tail are $\sigma_0$, $3\sigma_0$. The diameter |
| 111 |
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of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The model of |
| 112 |
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the solvent in our simulations is inspired by the idea of ``DPD'' |
| 113 |
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water. Every four water molecules are reprsented by one sphere. |
| 114 |
|
|
| 115 |
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\begin{figure}[htb] |
| 116 |
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\centering |
| 117 |
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\includegraphics[width=\linewidth]{lipidModels} |
| 118 |
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\caption{Three different representations of DPPC lipid molecules, |
| 119 |
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including the chemical structure, an atomistic model, and the |
| 120 |
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head-body ellipsoidal coarse-grained model used in this |
| 121 |
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work.\label{fig:lipidModels}} |
| 122 |
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\end{figure} |
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|
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Spheres interact each other with Lennard-Jones potential, ellipsoids |
| 125 |
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interact each other with Gay-Berne potential, dipoles interact each |
| 126 |
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other with typical dipole potential, spheres interact ellipsoids with |
| 127 |
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LJ-GB potential. All potentials are truncated at $25$ \AA and shifted |
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at $22$ \AA. |
| 124 |
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Spheres interact each other with Lennard-Jones potential |
| 125 |
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\begin{eqnarray*} |
| 126 |
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V_{ij} = 4\epsilon \left[\left(\frac{\sigma_0}{r_{ij}}\right)^{12} - |
| 127 |
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\left(\frac{\sigma_0}{r_{ij}}\right)^6\right] |
| 128 |
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\end{eqnarray*} |
| 129 |
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here, $\sigma_0$ is diameter of the spherical particle. $r_{ij}$ is |
| 130 |
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the distance between two spheres. $\epsilon$ is the well depth. |
| 131 |
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Dipoles interact each other with typical dipole potential |
| 132 |
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\begin{eqnarray*} |
| 133 |
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V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
| 134 |
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\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
| 135 |
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\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
| 136 |
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\end{eqnarray*} |
| 137 |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
| 138 |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
| 139 |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. Identical |
| 140 |
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ellipsoids interact each other with Gay-Berne potential. |
| 141 |
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\begin{eqnarray*} |
| 142 |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 143 |
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r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 144 |
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{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
| 145 |
> |
{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
| 146 |
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-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 147 |
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{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 148 |
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\end{eqnarray*} |
| 149 |
> |
where $\sigma_0 = \sqrt 2\sigma_s$, and orietation-dependent range |
| 150 |
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parameter is given by |
| 151 |
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\begin{eqnarray*} |
| 152 |
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\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 153 |
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{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{({\mathbf{\hat r}_{ij}} |
| 154 |
> |
\cdot {\mathbf{\hat u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 155 |
> |
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 156 |
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\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
| 157 |
> |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
| 158 |
> |
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
| 159 |
> |
\end{eqnarray*} |
| 160 |
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and the strength anisotropy function is, |
| 161 |
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\begin{eqnarray*} |
| 162 |
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\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 163 |
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{\epsilon^\nu}({\mathbf{\hat u}_i}, {\mathbf{\hat |
| 164 |
> |
u}_j}){\epsilon'^\mu}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
| 165 |
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{\mathbf{\hat r}_{ij}}) |
| 166 |
> |
\end{eqnarray*} |
| 167 |
> |
with $\nu$ and $\mu$ being adjustable exponent, and |
| 168 |
> |
$\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j})$, |
| 169 |
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$\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
| 170 |
> |
r}_{ij}})$ defined as |
| 171 |
> |
\begin{eqnarray*} |
| 172 |
> |
\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}) = |
| 173 |
> |
\epsilon_0\left[1-\chi^2({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
| 174 |
> |
u}_j})^2\right]^{-\frac{1}{2}} |
| 175 |
> |
\end{eqnarray*} |
| 176 |
> |
\begin{eqnarray*} |
| 177 |
> |
\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 178 |
> |
1-\frac{1}{2}\chi'\left[\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 179 |
> |
u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 180 |
> |
u}_j})^2}{1+\chi'({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 181 |
> |
\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
| 182 |
> |
\cdot {\mathbf{\hat u}_j})^2}{1-\chi'({\mathbf{\hat u}_i} \cdot |
| 183 |
> |
{\mathbf{\hat u}_j})} \right] |
| 184 |
> |
\end{eqnarray*} |
| 185 |
> |
the diameter dependent parameter $\chi$ is given by |
| 186 |
> |
\begin{eqnarray*} |
| 187 |
> |
\chi = \frac{({\sigma_s}^2 - |
| 188 |
> |
{\sigma_e}^2)}{({\sigma_s}^2 + {\sigma_e}^2)} |
| 189 |
> |
\end{eqnarray*} |
| 190 |
> |
and the well depth dependent parameter $\chi'$ is given by |
| 191 |
> |
\begin{eqnarray*} |
| 192 |
> |
\chi' = \frac{({\epsilon_s}^{\frac{1}{\mu}} - |
| 193 |
> |
{\epsilon_e}^{\frac{1}{\mu}})}{({\epsilon_s}^{\frac{1}{\mu}} + |
| 194 |
> |
{\epsilon_e}^{\frac{1}{\mu}})} |
| 195 |
> |
\end{eqnarray*} |
| 196 |
> |
$\sigma_s$ is the side-by-side width, and $\sigma_e$ is the end-to-end |
| 197 |
> |
length. $\epsilon_s$ is the side-by-side well depth, and $\epsilon_e$ |
| 198 |
> |
is the end-to-end well depth. For the interaction between |
| 199 |
> |
nonequivalent uniaxial ellipsoids (in this case, between spheres and |
| 200 |
> |
ellipsoids), the range parameter is generalized as\cite{Cleaver96} |
| 201 |
> |
\begin{eqnarray*} |
| 202 |
> |
\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
| 203 |
> |
{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}} |
| 204 |
> |
\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
| 205 |
> |
u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
| 206 |
> |
\frac{(\alpha{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - \alpha^{-1}{\mathbf{\hat r}_{ij}} |
| 207 |
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\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
| 208 |
> |
{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
| 209 |
> |
\end{eqnarray*} |
| 210 |
> |
where $\alpha$ is given by |
| 211 |
> |
\begin{eqnarray*} |
| 212 |
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\alpha^2 = |
| 213 |
> |
\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)} |
| 214 |
> |
\right]^{\frac{1}{2}} |
| 215 |
> |
\end{eqnarray*} |
| 216 |
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the strength parameter is adjusted by the suggestion of |
| 217 |
> |
\cite{Cleaver96}. All potentials are truncated at $25$ \AA\ and |
| 218 |
> |
shifted at $22$ \AA. |
| 219 |
|
|
| 220 |
+ |
\begin{figure}[htb] |
| 221 |
+ |
\centering |
| 222 |
+ |
\includegraphics[height=4in]{lipidModel} |
| 223 |
+ |
\caption{The parameters defining the behavior of the lipid |
| 224 |
+ |
models. $\sigma_h / \sigma_0$ is the ratio of the head group to body |
| 225 |
+ |
diameter. Molecular bodies all had an aspect ratio of 3.0. The |
| 226 |
+ |
dipolar strength (and the temperature and pressure) wer the only other |
| 227 |
+ |
parameters that wer varied systematically.\label{fig:lipidModel}} |
| 228 |
+ |
\end{figure} |
| 229 |
|
|
| 230 |
< |
To make the simulations less expensive and to observe long-time range |
| 231 |
< |
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. |
| 230 |
> |
\section{Experiment} |
| 231 |
> |
\label{sec:experiment} |
| 232 |
|
|
| 233 |
< |
|
| 234 |
< |
Snap shots show that the membrane is more corrugated with increasing |
| 235 |
< |
the size of the head groups. The surface is nearly perfect flat when |
| 236 |
< |
$\sigma_h$ is $1.20\sigma_0$. At $1.28\sigma_0$, although the surface |
| 237 |
< |
is still flat, the bilayer starts to splay inward, the upper leaf of |
| 238 |
< |
the bilayer is connected to the lower leaf with a interdigitated line |
| 239 |
< |
defect. Two periodicities with $100$\AA width were observed in the |
| 240 |
< |
simulation. This structure is very similiar to OTHER PAPER. The same |
| 241 |
< |
structure was also observed when $\sigma_h=1.41\sigma_0$. However, the |
| 242 |
< |
surface of the membrane is corrugated, and the periodicity of the |
| 243 |
< |
connection between upper and lower leaf membrane is shorter. From the |
| 244 |
< |
undulation spectrum of the surface (the exact form is in OUR PREVIOUS |
| 245 |
< |
PAPER), the corrugation is non-thermal fluctuation, and we are |
| 246 |
< |
confident to identify it as the ripple phase. The width of one ripple |
| 247 |
< |
is about $71$ \AA, and amplitude is about $7$ \AA. When |
| 248 |
< |
$\sigma_h=1.35\sigma_0$, 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. |
| 233 |
> |
To make the simulations less expensive and to observe long-time |
| 234 |
> |
behavior of the lipid membranes, all simulations were started from two |
| 235 |
> |
separate monolayers in the vaccum with $x-y$ anisotropic pressure |
| 236 |
> |
coupling. The length of $z$ axis of the simulations was fixed and a |
| 237 |
> |
constant surface tension was applied to enable real fluctuations of |
| 238 |
> |
the bilayer. Periodic boundaries were used. There were $480-720$ lipid |
| 239 |
> |
molecules in the simulations depending on the size of the head |
| 240 |
> |
beads. All the simulations were equlibrated for $100$ ns at $300$ |
| 241 |
> |
K. The resulting structures were solvated in water ($6$ DPD |
| 242 |
> |
water/lipid molecule). These configurations were relaxed for another |
| 243 |
> |
$30$ ns relaxation. All simulations with water were carried out at |
| 244 |
> |
constant pressure ($P=1$ atm) by $3$D anisotropic coupling, and |
| 245 |
> |
constant surface tension ($\gamma=0.015$). Given the absence of fast |
| 246 |
> |
degrees of freedom in this model, a timestep of $50$ fs was |
| 247 |
> |
utilized. Simulations were performed by using OOPSE |
| 248 |
> |
package\cite{Meineke05}. |
| 249 |
|
|
| 250 |
+ |
\section{Results and Analysis} |
| 251 |
+ |
\label{sec:results} |
| 252 |
|
|
| 253 |
< |
The $P_2$ order paramter is calculated to understand the phase |
| 254 |
< |
behavior quantatively. $P_2=1$ means a perfect ordered structure, and |
| 255 |
< |
$P_2=0$ means a random structure. The method can be found in OUR |
| 256 |
< |
PAPER. Fig. shows $P_2$ order paramter of the dipoles on head group |
| 257 |
< |
raises with increasing the size of the head group. When head of lipid |
| 258 |
< |
molecule is small, the membrane is flat and shows strong two |
| 259 |
< |
dimensional characters, dipoles are frustrated on orientational |
| 260 |
< |
ordering in this circumstance. Another reason is that the lipids can |
| 261 |
< |
move independently in each monolayer, it is not nessasory for the |
| 262 |
< |
direction of dipoles on one leaf is consistant to another layer, which |
| 263 |
< |
makes total order parameter is relatively low. With increasing the |
| 264 |
< |
size of head group, the surface is being more corrugated, dipoles are |
| 265 |
< |
not allowed to move freely on the surface, they are |
| 266 |
< |
localized. Therefore, the translational freedom of lipids in one layer |
| 253 |
> |
Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
| 254 |
> |
more corrugated increasing size of the head groups. The surface is |
| 255 |
> |
nearly flat when $\sigma_h=1.20\sigma_0$. With |
| 256 |
> |
$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
| 257 |
> |
bilayer starts to splay inward; the upper leaf of the bilayer is |
| 258 |
> |
connected to the lower leaf with an interdigitated line defect. Two |
| 259 |
> |
periodicities with $100$ \AA\ width were observed in the |
| 260 |
> |
simulation. This structure is very similiar to the structure observed |
| 261 |
> |
by de Vries and Lenz {\it et al.}. The same basic structure is also |
| 262 |
> |
observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
| 263 |
> |
surface corrugations depends sensitively on the size of the ``head'' |
| 264 |
> |
beads. From the undulation spectrum, the corrugation is clearly |
| 265 |
> |
non-thermal. |
| 266 |
> |
\begin{figure}[htb] |
| 267 |
> |
\centering |
| 268 |
> |
\includegraphics[width=\linewidth]{phaseCartoon} |
| 269 |
> |
\caption{A sketch to discribe the structure of the phases observed in |
| 270 |
> |
our simulations.\label{fig:phaseCartoon}} |
| 271 |
> |
\end{figure} |
| 272 |
> |
|
| 273 |
> |
When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
| 274 |
> |
morphology. This structure is different from the asymmetric rippled |
| 275 |
> |
surface; there is no interdigitation between the upper and lower |
| 276 |
> |
leaves of the bilayer. Each leaf of the bilayer is broken into several |
| 277 |
> |
hemicylinderical sections, and opposite leaves are fitted together |
| 278 |
> |
much like roof tiles. Unlike the surface in which the upper |
| 279 |
> |
hemicylinder is always interdigitated on the leading or trailing edge |
| 280 |
> |
of lower hemicylinder, the symmetric ripple has no prefered direction. |
| 281 |
> |
The corresponding cartoons are shown in Figure |
| 282 |
> |
\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
| 283 |
> |
different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
| 284 |
> |
(b) is the asymmetric ripple phase corresponding to the lipid |
| 285 |
> |
organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
| 286 |
> |
and (c) is the symmetric ripple phase observed when |
| 287 |
> |
$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
| 288 |
> |
continuous everywhere on the whole membrane, however, in asymmetric |
| 289 |
> |
ripple phase, the bilayer is intermittent domains connected by thin |
| 290 |
> |
interdigitated monolayer which consists of upper and lower leaves of |
| 291 |
> |
the bilayer. |
| 292 |
> |
\begin{table*} |
| 293 |
> |
\begin{minipage}{\linewidth} |
| 294 |
> |
\begin{center} |
| 295 |
> |
\caption{} |
| 296 |
> |
\begin{tabular}{lccc} |
| 297 |
> |
\hline |
| 298 |
> |
$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
| 299 |
> |
\hline |
| 300 |
> |
1.20 & flat & N/A & N/A \\ |
| 301 |
> |
1.28 & asymmetric flat & 21.7 & N/A \\ |
| 302 |
> |
1.35 & symmetric ripple & 17.2 & 2.2 \\ |
| 303 |
> |
1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
| 304 |
> |
\end{tabular} |
| 305 |
> |
\label{tab:property} |
| 306 |
> |
\end{center} |
| 307 |
> |
\end{minipage} |
| 308 |
> |
\end{table*} |
| 309 |
> |
|
| 310 |
> |
The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
| 311 |
> |
reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
| 312 |
> |
\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
| 313 |
> |
is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
| 314 |
> |
values are consistent to the experimental results. Note, the |
| 315 |
> |
amplitudes are underestimated without the melted tails in our |
| 316 |
> |
simulations. |
| 317 |
> |
|
| 318 |
> |
The $P_2$ order paramters (for molecular bodies and head group |
| 319 |
> |
dipoles) have been calculated to clarify the ordering in these phases |
| 320 |
> |
quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
| 321 |
> |
implies orientational randomization. Figure \ref{fig:rP2} shows the |
| 322 |
> |
$P_2$ order paramter of the dipoles on head group rising with |
| 323 |
> |
increasing head group size. When the heads of the lipid molecules are |
| 324 |
> |
small, the membrane is flat. The dipolar ordering is essentially |
| 325 |
> |
frustrated on orientational ordering in this circumstance. Another |
| 326 |
> |
reason is that the lipids can move independently in each monolayer, it |
| 327 |
> |
is not nessasory for the direction of dipoles on one leaf is |
| 328 |
> |
consistant to another layer, which makes total order parameter is |
| 329 |
> |
relatively low. With increasing head group size, the surface is |
| 330 |
> |
corrugated, and dipoles do not move as freely on the |
| 331 |
> |
surface. Therefore, the translational freedom of lipids in one layer |
| 332 |
|
is dependent upon the position of lipids in another layer, as a |
| 333 |
< |
result, the symmetry of the dipoles on head group in one layer is |
| 334 |
< |
consistant to the symmetry in another layer. Furthermore, the membrane |
| 335 |
< |
tranlates from a two dimensional system to a three dimensional system |
| 336 |
< |
by the corrugation, the symmetry of the ordering for the two |
| 337 |
< |
dimensional dipoles on the head group of lipid molecules is broken, on |
| 338 |
< |
a distorted lattice, dipoles are ordered on a head to tail energy |
| 339 |
< |
state, the order parameter is increased dramaticly. However, the total |
| 340 |
< |
polarization of the system is close to zero, which is a strong |
| 341 |
< |
evidence it is a antiferroelectric state. The orientation of the |
| 342 |
< |
dipole ordering is alway perpendicular to the ripple vector. These |
| 343 |
< |
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. |
| 333 |
> |
result, the symmetry of the dipoles on head group in one layer is tied |
| 334 |
> |
to the symmetry in the other layer. Furthermore, as the membrane |
| 335 |
> |
deforms from two to three dimensions due to the corrugation, the |
| 336 |
> |
symmetry of the ordering for the dipoles embedded on each leaf is |
| 337 |
> |
broken. The dipoles then self-assemble in a head-tail configuration, |
| 338 |
> |
and the order parameter increases dramaticaly. However, the total |
| 339 |
> |
polarization of the system is still close to zero. This is strong |
| 340 |
> |
evidence that the corrugated structure is an antiferroelectric |
| 341 |
> |
state. The orientation of the dipolar is always perpendicular to the |
| 342 |
> |
ripple wave vector. These results are consistent with our previous |
| 343 |
> |
study on dipolar membranes. |
| 344 |
|
|
| 345 |
+ |
The ordering of the tails is essentially opposite to the ordering of |
| 346 |
+ |
the dipoles on head group. The $P_2$ order parameter decreases with |
| 347 |
+ |
increasing head size. This indicates the surface is more curved with |
| 348 |
+ |
larger head groups. When the surface is flat, all tails are pointing |
| 349 |
+ |
in the same direction; in this case, all tails are parallel to the |
| 350 |
+ |
normal of the surface,(making this structure remindcent of the |
| 351 |
+ |
$L_{\beta}$ phase. Increasing the size of the heads, results in |
| 352 |
+ |
rapidly decreasing $P_2$ ordering for the molecular bodies. |
| 353 |
+ |
\begin{figure}[htb] |
| 354 |
+ |
\centering |
| 355 |
+ |
\includegraphics[width=\linewidth]{rP2} |
| 356 |
+ |
\caption{The $P_2$ order parameter as a funtion of the ratio of |
| 357 |
+ |
$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
| 358 |
+ |
\end{figure} |
| 359 |
|
|
| 360 |
< |
We studied the effects of interaction between head groups on the |
| 361 |
< |
structure of lipid bilayer by changing the strength of the dipole. The |
| 362 |
< |
fig. shows the $P_2$ order parameter changing with strength of the |
| 363 |
< |
dipole. Generally the dipoles on the head group are more ordered with |
| 364 |
< |
increasing the interaction between heads and more disordered with |
| 365 |
< |
decreasing the interaction between heads. When the interaction between |
| 366 |
< |
heads is weak enough, the bilayer structure is not persisted any more, |
| 367 |
< |
all lipid molecules are melted in the water. The critial value of the |
| 368 |
< |
strength of the dipole is various for different system. The perfect |
| 369 |
< |
flat surface melts at $5$ debye, the asymmetric rippled surfaces melt |
| 370 |
< |
at $8$ debye, the symmetric rippled surfaces melt at $10$ debye. This |
| 371 |
< |
indicates that the flat phase is the most stable state, the asymmetric |
| 372 |
< |
ripple phase is second stalbe state, and the symmetric ripple phase is |
| 373 |
< |
the most unstable state. The ordering of the tails is the same as the |
| 374 |
< |
ordering of the dipoles except for the flat phase. Since the surface |
| 375 |
< |
is already perfect flat, the order parameter does not change much |
| 376 |
< |
until the strength of the dipole is $15$ debye. However, the order |
| 377 |
< |
parameter decreases quickly when the strength of the dipole is further |
| 378 |
< |
increased. The head group of the lipid molecules are brought closer by |
| 379 |
< |
strenger interaction between them. For a flat surface, a mount of free |
| 380 |
< |
volume between head groups is available, when the head groups are |
| 381 |
< |
brought closer, the surface will splay outward to be a inverse |
| 382 |
< |
micelle. For rippled surfaces, there is few free volume available on |
| 383 |
< |
between the head groups, they can be closer, therefore there are |
| 384 |
< |
little effect on the structure of the membrane. Another interesting |
| 385 |
< |
fact, unlike other systems melting directly when the interaction is |
| 386 |
< |
weak enough, for $\sigma_h$ is $1.41\sigma_0$, part of the membrane |
| 387 |
< |
melts into itself first, the upper leaf of the bilayer is totally |
| 388 |
< |
interdigitated with the lower leaf, this is different with the |
| 389 |
< |
interdigitated lines in rippled phase where only one interdigitated |
| 390 |
< |
line connects the two leaves of bilayer. |
| 360 |
> |
We studied the effects of the interactions between head groups on the |
| 361 |
> |
structure of lipid bilayer by changing the strength of the dipole. |
| 362 |
> |
Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
| 363 |
> |
increasing strength of the dipole. Generally the dipoles on the head |
| 364 |
> |
group are more ordered by increase in the strength of the interaction |
| 365 |
> |
between heads and are more disordered by decreasing the interaction |
| 366 |
> |
stength. When the interaction between the heads is weak enough, the |
| 367 |
> |
bilayer structure does not persist; all lipid molecules are solvated |
| 368 |
> |
directly in the water. The critial value of the strength of the dipole |
| 369 |
> |
depends on the head size. The perfectly flat surface melts at $5$ |
| 370 |
> |
$0.03$ debye, the asymmetric rippled surfaces melt at $8$ $0.04$ |
| 371 |
> |
$0.03$ debye, the symmetric rippled surfaces melt at $10$ $0.04$ |
| 372 |
> |
debye. The ordering of the tails is the same as the ordering of the |
| 373 |
> |
dipoles except for the flat phase. Since the surface is already |
| 374 |
> |
perfect flat, the order parameter does not change much until the |
| 375 |
> |
strength of the dipole is $15$ debye. However, the order parameter |
| 376 |
> |
decreases quickly when the strength of the dipole is further |
| 377 |
> |
increased. The head groups of the lipid molecules are brought closer |
| 378 |
> |
by stronger interactions between them. For a flat surface, a large |
| 379 |
> |
amount of free volume between the head groups is available, but when |
| 380 |
> |
the head groups are brought closer, the tails will splay outward, |
| 381 |
> |
forming an inverse micelle. When $\sigma_h=1.28\sigma_0$, the $P_2$ |
| 382 |
> |
order parameter decreases slightly after the strength of the dipole is |
| 383 |
> |
increased to $16$ debye. For rippled surfaces, there is less free |
| 384 |
> |
volume available between the head groups. Therefore there is little |
| 385 |
> |
effect on the structure of the membrane due to increasing dipolar |
| 386 |
> |
strength. However, the increase of the $P_2$ order parameter implies |
| 387 |
> |
the membranes are flatten by the increase of the strength of the |
| 388 |
> |
dipole. Unlike other systems that melt directly when the interaction |
| 389 |
> |
is weak enough, for $\sigma_h=1.41\sigma_0$, part of the membrane |
| 390 |
> |
melts into itself first. The upper leaf of the bilayer becomes totally |
| 391 |
> |
interdigitated with the lower leaf. This is different behavior than |
| 392 |
> |
what is exhibited with the interdigitated lines in the rippled phase |
| 393 |
> |
where only one interdigitated line connects the two leaves of bilayer. |
| 394 |
> |
\begin{figure}[htb] |
| 395 |
> |
\centering |
| 396 |
> |
\includegraphics[width=\linewidth]{sP2} |
| 397 |
> |
\caption{The $P_2$ order parameter as a funtion of the strength of the |
| 398 |
> |
dipole.\label{fig:sP2}} |
| 399 |
> |
\end{figure} |
| 400 |
|
|
| 401 |
+ |
Figure \ref{fig:tP2} shows the dependence of the order parameter on |
| 402 |
+ |
temperature. The behavior of the $P_2$ order paramter is |
| 403 |
+ |
straightforward. Systems are more ordered at low temperature, and more |
| 404 |
+ |
disordered at high temperatures. When the temperature is high enough, |
| 405 |
+ |
the membranes are instable. For flat surfaces ($\sigma_h=1.20\sigma_0$ |
| 406 |
+ |
and $\sigma_h=1.28\sigma_0$), when the temperature is increased to |
| 407 |
+ |
$310$, the $P_2$ order parameter increases slightly instead of |
| 408 |
+ |
decreases like ripple surface. This is an evidence of the frustration |
| 409 |
+ |
of the dipolar ordering in each leaf of the lipid bilayer, at low |
| 410 |
+ |
temperature, the systems are locked in a local minimum energy state, |
| 411 |
+ |
with increase of the temperature, the system can jump out the local |
| 412 |
+ |
energy well to find the lower energy state which is the longer range |
| 413 |
+ |
orientational ordering. Like the dipolar ordering of the flat |
| 414 |
+ |
surfaces, the ordering of the tails of the lipid molecules for ripple |
| 415 |
+ |
membranes ($\sigma_h=1.35\sigma_0$ and $\sigma_h=1.41\sigma_0$) also |
| 416 |
+ |
show some nonthermal characteristic. With increase of the temperature, |
| 417 |
+ |
the $P_2$ order parameter decreases firstly, and increases afterward |
| 418 |
+ |
when the temperature is greater than $290 K$. The increase of the |
| 419 |
+ |
$P_2$ order parameter indicates a more ordered structure for the tails |
| 420 |
+ |
of the lipid molecules which corresponds to a more flat surface. Since |
| 421 |
+ |
our model lacks the detailed information on lipid tails, we can not |
| 422 |
+ |
simulate the fluid phase with melted fatty acid chains. Moreover, the |
| 423 |
+ |
formation of the tilted $L_{\beta'}$ phase also depends on the |
| 424 |
+ |
organization of fatty groups on tails. |
| 425 |
+ |
\begin{figure}[htb] |
| 426 |
+ |
\centering |
| 427 |
+ |
\includegraphics[width=\linewidth]{tP2} |
| 428 |
+ |
\caption{The $P_2$ order parameter as a funtion of |
| 429 |
+ |
temperature.\label{fig:tP2}} |
| 430 |
+ |
\end{figure} |
| 431 |
|
|
| 432 |
< |
Fig. shows the changing of the order parameter with temperature. The |
| 433 |
< |
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. |
| 432 |
> |
\section{Discussion} |
| 433 |
> |
\label{sec:discussion} |
| 434 |
|
|
| 435 |
|
\bibliography{mdripple} |
| 436 |
|
\end{document} |
