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\begin{document} |
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\title{Ripple Phase of the Lipid Bilayers: A Monte Carlo Simulation} |
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\title{Symmetry breaking and the Ripple phase} |
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\author{Xiuquan Sun and J. Daniel Gezelter\footnote{Corresponding author. Email: gezelter@nd.edu} \\ |
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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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\maketitle |
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\begin{abstract} |
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The molecular explanation for the origin and properties of the ripple |
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phase is addressed in this paper. A model which contains the surface |
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tension and dipole-dipole interactions is used to describe the |
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potential for a monolayer of simple point dipoles. The simulations are |
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carried out using Monte Carlo method. It is shown asymmetry of the |
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translational freedom of the dipoles breaks the symmetry of the |
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hexagonal lattice and allow antiferroelectric ordering of the |
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dipoles. The existence of the ripples only depends on the dipolar |
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property of the system. The structure of the ripples is affected by |
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surface tension. Only close to the hexagonal lattice, can the ripple |
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phase be reached. Surface has the lowest transition temperature on |
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hexagonal lattice elucidates the reason of the existence of the ripple |
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phase in organism. A mechanism for the phase transition of the lipid |
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bilayer is proposed. |
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The ripple phase in phosphatidylcholine (PC) bilayers has never been |
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explained. Our group has developed some simple (XYZ) spin-lattice |
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models that allow spins to vary their elevation as well as their |
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orientation. The extra degree of freedom allows hexagonal lattices of |
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spins to find states that break out of the normally frustrated |
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randomized states and are stabilized by long-range anti-ferroelectric |
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ordering. To break out of the frustrated states, the spins must form |
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``rippled'' phases that make the lattices effectively non-hexagonal. Our |
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XYZ models contain surface tension and dipole-dipole interactions to |
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describe the interaction potential for monolayers and bilayers of |
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model lipid molecules. The existence of the ripples depends primarily |
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on the strength and lattice spacing of the dipoles, while the shape |
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(wavelength and amplitude) of the ripples is only weakly sensitive to |
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the applied surface tension. Additionally, the wave vector for the |
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ripple is always perpendicular to the director axis for the |
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dipoles. Non-hexagonal lattices of dipoles are not inherently |
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frustrated, and are therefore less likely to form ripple phases |
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because they can easily form low-energy anti-ferroelectric states. |
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Indeed, we see that the dipolar order-disorder transition is |
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substantially lower for hexagonal lattices and the ordered phase for |
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this lattice is clearly rippled. |
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\end{abstract} |
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\section{Introduction} |
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\label{Int} |
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\indent |
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Fully hydrated lipids will aggregate spontaneously to form bilayers |
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which exhibit a variety of phases according to temperature and |
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composition. Among these phases, a periodic rippled |
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\section{Model and calculation method} |
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\label{Mod} |
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The model used in our simulations is shown in Fig. \ref{fmod1} and Fig. \ref{fmod2}. |
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The model used in our simulations is shown in Fig. \ref{fmod1} and |
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Fig. \ref{fmod2}. |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{picture/lattice.eps} |
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potentials.} |
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\label{fmod1} |
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\end{figure} |
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\begin{figure} |
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\centering |
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\includegraphics[width=\linewidth]{picture/xyz.eps} |
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\caption{The 6 coordinates describing the state of a 2-dipole system in our extended X-Y-Z model. $z_i$ is the height of dipole $i$ from |
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the initial x-y plane, $\theta_i$ is the angle that the dipole is away |
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the dipole on x-y plane with the x axis.} |
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\label{fmod2} |
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\end{figure} |
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The lipids are represented by the simple point-dipole. During the |
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simulations, dipoles are locked (in the x-y plane) to lattice points |
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of hexagonal (or distorted) lattice. Each dipole can move freely out |
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%\end{figure} |
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From the results of the simulation at $T = 300$ for hexagonal lattice, the system is in an antiferroelectric state. Every $3$ or $4$ arrows of the dipoles form a strip whose direction is opposite to its neighbors. $P_2$ is about $0.7$. The ripple is formed clearly. The simulation results shows the ripple has equal opportunity to be formed along different directions, this is due to the isotropic property of the hexagonal lattice. |
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We use the last configuration of this simulation as the initial condition to increase the system to $T = 400$ every $10\ kelvin$, at the same time decrease the temperature to $T = 100$ every $10\ kelvin$. The trend that the order parameter varies with temperature is plotted in Fig. \ref{t-op}. |
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\begin{figure} |
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\begin{center} |
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\centering |
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\includegraphics[width=\linewidth]{picture/hexorderpara.eps} |
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\caption{ The orderparameter $P_2$ vs temperature T at hexagonal lattice.\label{t-op}} |
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\end{center} |
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\caption{ The orderparameter $P_2$ vs temperature T at hexagonal |
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lattice.} |
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\label{t-op} |
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\end{figure} |
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The $P_2 \approx 0.9$ for $T = 100$ implies that the system is in a |
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highly ordered state. As the temperature increases, the order |
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parameter is decreasing gradually before $T = 300$, from $T = 310$ the |
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the further insight into this problem. At first, we fixed the value |
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of $s = 7$, and vary $k_r$, the results are shown in |
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Fig. \ref{kr-a-hf}. |
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\begin{figure} |
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\begin{center} |
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\centering |
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\includegraphics[width=\linewidth]{picture/kr_amplitude.eps} |
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\caption{ The amplitude $A$ of the ripples at $T = 300$ vs $k_r$ for hexagonal lattice. The height fluctuation $h_f$ vs $k_r$ is shown inset for the same situation.\label{kr-a-hf}} |
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\end{center} |
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\caption{ The amplitude $A$ of the ripples at $T = 300$ vs $k_r$ for |
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hexagonal lattice. The height fluctuation $h_f$ vs $k_r$ is shown |
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inset for the same situation.} |
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\label{kr-a-hf} |
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\end{figure} |
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When $k_r < 0.1$, due to the small surface tension part, the dipoles |
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can go far away from their neighbors, lots of noise make the ripples |
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undiscernable. So, we start from $k_r = 0.1$. However, even when $k_r |
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lattice, the distorted hexagonal lattices prefer a particular director |
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axis due to their anisotropic property. The phase diagram for this |
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system is shown in Fig. \ref{phase}. |
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\begin{figure} |
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\begin{center} |
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\centering |
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\includegraphics[width=\linewidth]{picture/phase.eps} |
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\caption{ The phase diagram with temperature $T$ and lattice variable $\gamma$. The enlarged view near the hexagonal lattice is shown inset.\label{phase}} |
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\end{center} |
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\caption{ The phase diagram with temperature $T$ and lattice variable |
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$\gamma$. The enlarged view near the hexagonal lattice is shown |
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inset.} |
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\label{phase} |
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\end{figure} |
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$T_c$ is the transition temperature. The hexagonal lattice has the |
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lowest $T_c$, and $T_c$ goes up with lattice being more |
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distorted. There is only two phases in our diagram. When we do |
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$\gamma = 1.875$, $k_r = 0.1$, $T = 260$. Under this situation, the |
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system reaches the equilibrium very quickly, and the ripples are |
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fairly stable. The results are shown in Fig. \ref{samplitude}. |
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\begin{figure} |
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\begin{center} |
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\centering |
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\includegraphics[width=\linewidth]{picture/samplitude.eps} |
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\caption{ The amplitude of ripples $A$ at $T = 260$ vs strength of dipole $s$ for $\gamma = 1.875$. The orderparameter $P_2$ vs $s$ is shown inset at the same situation.\label{samplitude}} |
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\end{center} |
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\caption{ The amplitude of ripples $A$ at $T = 260$ vs strength of |
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dipole $s$ for $\gamma = 1.875$. The orderparameter $P_2$ vs $s$ is |
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shown inset at the same situation.} |
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\label{samplitude} |
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\end{figure} |
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For small $s$, there is no long range ordering in the system, so, we |
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start from $s = 7$, and we use the rippled state as the initial |
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configuration for all the simulations to reduce the noise. There is no |
