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\begin{document} |
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%\bibliographystyle{aps} |
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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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University of Notre Dame, \\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\begin{abstract} |
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\end{abstract} |
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\pacs{} |
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\maketitle |
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\section{Introduction} |
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\label{sec:Int} |
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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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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 |
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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 |
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formation of the ripple phase~\cite{Carlson87}. Recently, the ripple |
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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. |
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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}. |
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In this paper, we made a more realistic coarse-grained lipid model to |
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understand the primary driving force for membrane corrugation and to |
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elucidate the organization of the anisotropic interacting head group |
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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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\section{Methodology and Model} |
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\label{sec:method} |
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|
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Our idea for developing a simple and reasonable lipid model to study |
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the ripple phase of lipid bilayers is based on two facts: one is that |
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the most essential feature of lipid molecules is their amphiphilic |
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structure with polar head groups and non-polar tails. Another fact is |
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that dominant numbers of lipid molecules are very rigid in ripple |
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phase which allows the details of the lipid molecules neglectable. The |
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lipid model is shown in Figure \ref{fig:lipidMM}. Figure |
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\ref{fig:lipidMM}a shows the molecular strucure of a DPPC molecule. The |
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hydrophilic character of the head group is the effect of the strong |
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dipole composed by a positive charge sitting on the nitrogen and a |
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negative charge on the phosphate. The hydrophobic tail consists of |
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fatty acid chains. In our model shown in Figure \ref{fig:lipidMM}b, |
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lipid molecules are represented by rigid bodies made of one head |
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sphere with a point dipole sitting on it and one ellipsoid tail, the |
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direction of the dipole is fixed to be perpendicular to the tail. The |
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breadth and length of tail are $\sigma_0$, $3\sigma_0$. The diameter |
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of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The model of |
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the solvent in our simulations is inspired by the idea of ``DPD'' |
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water. Every four water molecules are reprsented by one sphere. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{lipidMM} |
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\caption{The molecular structure of a DPPC molecule and the |
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coars-grained model for PC molecules.\label{fig:lipidMM}} |
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\end{figure} |
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Spheres interact each other with Lennard-Jones potential |
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\begin{eqnarray*} |
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V_{ij} = 4\epsilon \left[\left(\frac{\sigma_0}{r_{ij}}\right)^{12} - |
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\left(\frac{\sigma_0}{r_{ij}}\right)^6\right] |
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\end{eqnarray*} |
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here, $\sigma_0$ is diameter of the spherical particle. $r_{ij}$ is |
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the distance between two spheres. $\epsilon$ is the well depth. |
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Dipoles interact each other with typical dipole potential |
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\begin{eqnarray*} |
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V_{ij} = \frac{|\mu|^2}{4 \pi \epsilon_0 r_{ij}^3} |
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\left[ \hat{\bf u}_i \cdot \hat{\bf u}_j - 3 (\hat{\bf u}_i \cdot |
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\hat{\bf r}_{ij})(\hat{\bf u}_j \cdot \hat{\bf r}_{ij}) \right] |
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\end{eqnarray*} |
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In this potential, $\mathbf{\hat u}_i$ is the unit vector pointing |
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along dipole $i$ and $\mathbf{\hat r}_{ij}$ is the unit vector |
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pointing along the inter-dipole vector $\mathbf{r}_{ij}$. Identical |
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ellipsoids interact each other with Gay-Berne potential. |
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\begin{eqnarray*} |
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V_{ij}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}}) = 4\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, |
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{\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}})+\sigma_0}\right)^{12} |
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-\left(\frac{\sigma_0}{r_{ij}-\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}})+\sigma_0}\right)^6\right] |
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\end{eqnarray*} |
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where $\sigma_0 = \sqrt 2\sigma_s$, and orietation-dependent range |
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parameter is given by |
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\begin{eqnarray*} |
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\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{({\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
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u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
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\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
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{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
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\end{eqnarray*} |
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and the strength anisotropy function is, |
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\begin{eqnarray*} |
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\epsilon ({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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{\epsilon^\nu}({\mathbf{\hat u}_i}, {\mathbf{\hat |
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u}_j}){\epsilon'^\mu}({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, |
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{\mathbf{\hat r}_{ij}}) |
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\end{eqnarray*} |
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with $\nu$ and $\mu$ being adjustable exponent, and |
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$\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j})$, |
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$\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat |
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r}_{ij}})$ defined as |
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\begin{eqnarray*} |
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\epsilon({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}) = |
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\epsilon_0\left[1-\chi^2({\mathbf{\hat u}_i} \cdot {\mathbf{\hat |
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u}_j})^2\right]^{-\frac{1}{2}} |
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\end{eqnarray*} |
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\begin{eqnarray*} |
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\epsilon'({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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1-\frac{1}{2}\chi'\left[\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
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u}_i} + {\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
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u}_j})^2}{1+\chi'({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
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\frac{({\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - {\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_j})^2}{1-\chi'({\mathbf{\hat u}_i} \cdot |
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{\mathbf{\hat u}_j})} \right] |
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\end{eqnarray*} |
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the diameter dependent parameter $\chi$ is given by |
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\begin{eqnarray*} |
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\chi = \frac{({\sigma_s}^2 - |
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{\sigma_e}^2)}{({\sigma_s}^2 + {\sigma_e}^2)} |
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\end{eqnarray*} |
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and the well depth dependent parameter $\chi'$ is given by |
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\begin{eqnarray*} |
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\chi' = \frac{({\epsilon_s}^{\frac{1}{\mu}} - |
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{\epsilon_e}^{\frac{1}{\mu}})}{({\epsilon_s}^{\frac{1}{\mu}} + |
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{\epsilon_e}^{\frac{1}{\mu}})} |
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\end{eqnarray*} |
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$\sigma_s$ is the side-by-side width, and $\sigma_e$ is the end-to-end |
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length. $\epsilon_s$ is the side-by-side well depth, and $\epsilon_e$ |
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is the end-to-end well depth. For the interaction between |
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nonequivalent uniaxial ellipsoids (in this case, between spheres and |
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ellipsoids), the range parameter is generalized as\cite{Cleaver96} |
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\begin{eqnarray*} |
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\sigma({\mathbf{\hat u}_i}, {\mathbf{\hat u}_j}, {\mathbf{\hat r}_{ij}}) = |
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{\sigma_{0}} \left(1-\frac{1}{2}\chi\left[\frac{(\alpha{\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_i} + \alpha^{-1}{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat |
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u}_j})^2}{1+\chi({\mathbf{\hat u}_i} \cdot {\mathbf{\hat u}_j})} + |
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\frac{(\alpha{\mathbf{\hat r}_{ij}} \cdot {\mathbf{\hat u}_i} - \alpha^{-1}{\mathbf{\hat r}_{ij}} |
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\cdot {\mathbf{\hat u}_j})^2}{1-\chi({\mathbf{\hat u}_i} \cdot |
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{\mathbf{\hat u}_j})} \right] \right)^{-\frac{1}{2}} |
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\end{eqnarray*} |
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where $\alpha$ is given by |
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\begin{eqnarray*} |
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\alpha^2 = |
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\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)} |
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\right]^{\frac{1}{2}} |
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\end{eqnarray*} |
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the strength parameter is adjusted by the suggestion of |
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\cite{Cleaver96}. All potentials are truncated at $25$ \AA\ and |
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shifted at $22$ \AA. |
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\section{Experiment} |
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\label{sec:experiment} |
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To make the simulations less expensive and to observe long-time |
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behavior of the lipid membranes, all simulations were started from two |
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separate monolayers in the vaccum with $x-y$ anisotropic pressure |
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coupling. The length of $z$ axis of the simulations was fixed and a |
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constant surface tension was applied to enable real fluctuations of |
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the bilayer. Periodic boundaries were used. There were $480-720$ lipid |
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molecules in the simulations depending on the size of the head |
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beads. All the simulations were equlibrated for $100$ ns at $300$ |
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K. The resulting structures were solvated in water ($6$ DPD |
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water/lipid molecule). These configurations were relaxed for another |
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$30$ ns relaxation. All simulations with water were carried out at |
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constant pressure ($P=1$ atm) by $3$D anisotropic coupling, and |
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constant surface tension ($\gamma=0.015$). Given the absence of fast |
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degrees of freedom in this model, a timestep of $50$ fs was |
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utilized. Simulations were performed by using OOPSE |
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package\cite{Meineke05}. |
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\section{Results and Analysis} |
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\label{sec:results} |
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Snapshots in Figure \ref{fig:phaseCartoon} show that the membrane is |
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more corrugated increasing size of the head groups. The surface is |
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nearly flat when $\sigma_h=1.20\sigma_0$. With |
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$\sigma_h=1.28\sigma_0$, although the surface is still flat, the |
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bilayer starts to splay inward; the upper leaf of the bilayer is |
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connected to the lower leaf with an interdigitated line defect. Two |
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periodicities with $100$ \AA\ width were observed in the |
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simulation. This structure is very similiar to the structure observed |
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by de Vries and Lenz {\it et al.}. The same basic structure is also |
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observed when $\sigma_h=1.41\sigma_0$, but the wavelength of the |
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surface corrugations depends sensitively on the size of the ``head'' |
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beads. From the undulation spectrum, the corrugation is clearly |
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non-thermal. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{phaseCartoon} |
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\caption{A sketch to discribe the structure of the phases observed in |
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our simulations.\label{fig:phaseCartoon}} |
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\end{figure} |
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When $\sigma_h=1.35\sigma_0$, we observed another corrugated surface |
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morphology. This structure is different from the asymmetric rippled |
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surface; there is no interdigitation between the upper and lower |
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leaves of the bilayer. Each leaf of the bilayer is broken into several |
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hemicylinderical sections, and opposite leaves are fitted together |
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much like roof tiles. Unlike the surface in which the upper |
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hemicylinder is always interdigitated on the leading or trailing edge |
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of lower hemicylinder, the symmetric ripple has no prefered direction. |
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The corresponding cartoons are shown in Figure |
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\ref{fig:phaseCartoon} for elucidation of the detailed structures of |
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different phases. Figure \ref{fig:phaseCartoon} (a) is the flat phase, |
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(b) is the asymmetric ripple phase corresponding to the lipid |
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organization when $\sigma_h=1.28\sigma_0$ and $\sigma_h=1.41\sigma_0$, |
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and (c) is the symmetric ripple phase observed when |
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$\sigma_h=1.35\sigma_0$. In symmetric ripple phase, the bilayer is |
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continuous everywhere on the whole membrane, however, in asymmetric |
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ripple phase, the bilayer is intermittent domains connected by thin |
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interdigitated monolayer which consists of upper and lower leaves of |
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the bilayer. |
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\begin{table*} |
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\begin{minipage}{\linewidth} |
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\begin{center} |
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\caption{} |
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\begin{tabular}{lccc} |
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\hline |
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$\sigma_h/\sigma_0$ & type of phase & $\lambda/\sigma_0$ & $A/\sigma_0$\\ |
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\hline |
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1.20 & flat & N/A & N/A \\ |
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1.28 & asymmetric flat & 21.7 & N/A \\ |
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1.35 & symmetric ripple & 17.2 & 2.2 \\ |
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1.41 & asymmetric ripple & 15.4 & 1.5 \\ |
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\end{tabular} |
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\label{tab:property} |
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\end{center} |
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\end{minipage} |
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\end{table*} |
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The membrane structures and the reduced wavelength $\lambda/\sigma_0$, |
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reduced amplitude $A/\sigma_0$ of the ripples are summerized in Table |
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\ref{tab:property}. The wavelength range is $15~21$ and the amplitude |
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is $1.5$ for asymmetric ripple and $2.2$ for symmetric ripple. These |
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values are consistent to the experimental results. Note, the |
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amplitudes are underestimated without the melted tails in our |
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simulations. |
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The $P_2$ order paramters (for molecular bodies and head group |
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dipoles) have been calculated to clarify the ordering in these phases |
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quantatively. $P_2=1$ means a perfectly ordered structure, and $P_2=0$ |
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implies orientational randomization. Figure \ref{fig:rP2} shows the |
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$P_2$ order paramter of the dipoles on head group rising with |
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increasing head group size. When the heads of the lipid molecules are |
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small, the membrane is flat. The dipolar ordering is essentially |
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frustrated on orientational ordering in this circumstance. Another |
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reason is that the lipids can move independently in each monolayer, it |
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is not nessasory for the direction of dipoles on one leaf is |
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consistant to another layer, which makes total order parameter is |
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relatively low. With increasing head group size, the surface is |
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corrugated, and dipoles do not move as freely on the |
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surface. Therefore, the translational freedom of lipids in one layer |
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is dependent upon the position of lipids in another layer, as a |
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result, the symmetry of the dipoles on head group in one layer is tied |
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to the symmetry in the other layer. Furthermore, as the membrane |
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deforms from two to three dimensions due to the corrugation, the |
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symmetry of the ordering for the dipoles embedded on each leaf is |
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broken. The dipoles then self-assemble in a head-tail configuration, |
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and the order parameter increases dramaticaly. However, the total |
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polarization of the system is still close to zero. This is strong |
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evidence that the corrugated structure is an antiferroelectric |
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state. The orientation of the dipolar is always perpendicular to the |
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ripple wave vector. These results are consistent with our previous |
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study on dipolar membranes. |
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|
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The ordering of the tails is essentially opposite to the ordering of |
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the dipoles on head group. The $P_2$ order parameter decreases with |
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increasing head size. This indicates the surface is more curved with |
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larger head groups. When the surface is flat, all tails are pointing |
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in the same direction; in this case, all tails are parallel to the |
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normal of the surface,(making this structure remindcent of the |
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$L_{\beta}$ phase. Increasing the size of the heads, results in |
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rapidly decreasing $P_2$ ordering for the molecular bodies. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{rP2} |
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\caption{The $P_2$ order parameter as a funtion of the ratio of |
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$\sigma_h$ to $\sigma_0$.\label{fig:rP2}} |
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\end{figure} |
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|
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We studied the effects of the interactions between head groups on the |
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structure of lipid bilayer by changing the strength of the dipole. |
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Figure \ref{fig:sP2} shows how the $P_2$ order parameter changes with |
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increasing strength of the dipole. Generally the dipoles on the head |
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group are more ordered by increase in the strength of the interaction |
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between heads and are more disordered by decreasing the interaction |
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stength. When the interaction between the heads is weak enough, the |
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bilayer structure does not persist; all lipid molecules are solvated |
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directly in the water. The critial value of the strength of the dipole |
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depends on the head size. The perfectly flat surface melts at $5$ |
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debye, the asymmetric rippled surfaces melt at $8$ debye, the |
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symmetric rippled surfaces melt at $10$ debye. The ordering of the |
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tails is the same as the ordering of the dipoles except for the flat |
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phase. Since the surface is already perfect flat, the order parameter |
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does not change much until the strength of the dipole is $15$ |
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debye. However, the order parameter decreases quickly when the |
364 |
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strength of the dipole is further increased. The head groups of the |
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lipid molecules are brought closer by stronger interactions between |
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them. For a flat surface, a large amount of free volume between the |
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head groups is available, but when the head groups are brought closer, |
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the tails will splay outward, forming an inverse micelle. For rippled |
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surfaces, there is less free volume available between the head |
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groups. Therefore there is little effect on the structure of the |
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membrane due to increasing dipolar strength. Unlike other systems that |
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melt directly when the interaction is weak enough, for |
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$\sigma_h=1.41\sigma_0$, part of the membrane melts into itself |
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first. The upper leaf of the bilayer becomes totally interdigitated |
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with the lower leaf. This is different behavior than what is exhibited |
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with the interdigitated lines in the rippled phase where only one |
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interdigitated line connects the two leaves of bilayer. |
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\begin{figure}[htb] |
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\centering |
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\includegraphics[width=\linewidth]{sP2} |
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\caption{The $P_2$ order parameter as a funtion of the strength of the |
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dipole.\label{fig:sP2}} |
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\end{figure} |
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|
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Figure \ref{fig:tP2} shows the dependence of the order parameter on |
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temperature. The behavior of the $P_2$ order paramter is |
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straightforward. Systems are more ordered at low temperature, and more |
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disordered at high temperatures. When the temperature is high enough, |
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the membranes are instable. Since our model lacks the detailed |
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information on lipid tails, we can not simulate the fluid phase with |
391 |
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melted fatty acid chains. Moreover, the formation of the tilted |
392 |
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$L_{\beta'}$ phase also depends on the organization of fatty groups on |
393 |
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tails. |
394 |
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\begin{figure}[htb] |
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\centering |
396 |
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\includegraphics[width=\linewidth]{tP2} |
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\caption{The $P_2$ order parameter as a funtion of |
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temperature.\label{fig:tP2}} |
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\end{figure} |
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|
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\section{Discussion} |
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\label{sec:discussion} |
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|
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\bibliography{mdripple} |
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\end{document} |