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%\documentclass[aps,pre,twocolumn,amssymb,showpacs,floatfix]{revtex4} |
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\documentclass[aps,pre,preprint,amssymb,showpacs]{revtex4} |
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\usepackage{graphicx} |
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\begin{document} |
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\renewcommand{\thefootnote}{\fnsymbol{footnote}} |
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\renewcommand{\theequation}{\arabic{section}.\arabic{equation}} |
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%\bibliographystyle{aps} |
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\title{} |
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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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Our idea for developing a simple and reasonable lipid model to study |
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the ripple pahse 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. In |
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our model, lipid molecules are represented by rigid bodies made of one |
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head sphere with a point dipole sitting on it and one ellipsoid tail, |
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the direction of the dipole is fixed to be perpendicular to the |
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tail. The breadth and length of tail are $\sigma_0$, $3\sigma_0$. The |
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diameter of heads varies from $1.20\sigma_0$ to $1.41\sigma_0$. The |
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model of the solvent in our simulations is inspired by the idea of |
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``DPD'' water. Every four water molecules are reprsented by one |
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sphere. |
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Spheres interact each other with Lennard-Jones potential, ellipsoids |
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interact each other with Gay-Berne potential, dipoles interact each |
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other with typical dipole potential, spheres interact ellipsoids with |
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LJ-GB potential. All potentials are truncated at $25$ \AA and shifted |
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at $22$ \AA. |
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To make the simulations less expensive and to observe long-time range |
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behavior of the lipid membranes, all simulaitons were started from two |
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sepetated monolayers in the vaccum with $x-y$ anisotropic pressure |
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coupling, length of $z$ axis of the simulations was fixed to prevent |
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the shrinkage of the simulation boxes due to the free volume outside |
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of the bilayer, and a constant surface tension was applied to enable |
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the fluctuation of the surface. Periodic boundaries were used. There |
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were $480-720$ lipid molecules in simulations according to different |
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size of the heads. All the simulations were stablized for $100$ ns at |
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$300$ K. The resulted structures were solvated in the water (about |
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$6$ DPD water/lipid molecule) as the initial configurations for another |
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$30$ ns relaxation. All simulations with water were carried out at |
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constant pressure ($P=1$bar) by $3$D anisotropic coupling, and |
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constant surface tension ($\gamma=0.015$). Time step was |
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$50$ fs. Simulations were performed by using OOPSE package. |
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Snap shots show that the membrane is more corrugated with increasing |
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the size of the head groups. The surface is nearly perfect flat when |
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$\sigma_h$ is $1.20\sigma_0$. At $1.28\sigma_0$, although the surface |
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is still flat, the bilayer starts to splay inward, the upper leaf of |
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the bilayer is connected to the lower leaf with a interdigitated line |
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defect. Two periodicities with $100$\AA width were observed in the |
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simulation. This structure is very similiar to OTHER PAPER. The same |
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structure was also observed when $\sigma_h=1.41\sigma_0$. However, the |
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surface of the membrane is corrugated, and the periodicity of the |
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connection between upper and lower leaf membrane is shorter. From the |
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undulation spectrum of the surface (the exact form is in OUR PREVIOUS |
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PAPER), the corrugation is non-thermal fluctuation, and we are |
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confident to identify it as the ripple phase. The width of one ripple |
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is about $71$ \AA, and amplitude is about $7$ \AA. When |
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$\sigma_h=1.35\sigma_0$, we observed another corrugated surface with |
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$79$ \AA width and $10$ \AA amplitude. This structure is different to |
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the previous rippled surface, there is no connection between upper and |
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lower leaf of the bilayer. Each leaf of the bilayer is broken to |
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several curved pieces, the broken position is mounted into the center |
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of opposite piece in another leaf. Unlike another corrugated surface |
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in which the upper leaf of the surface is always connected to the |
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lower leaf from one direction, this ripple of this surface is |
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isotropic. Therefore, we claim this is a symmetric ripple phase. |
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The $P_2$ order paramter is calculated to understand the phase |
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behavior quantatively. $P_2=1$ means a perfect ordered structure, and |
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$P_2=0$ means a random structure. The method can be found in OUR |
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PAPER. Fig. shows $P_2$ order paramter of the dipoles on head group |
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raises with increasing the size of the head group. When head of lipid |
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molecule is small, the membrane is flat and shows strong two |
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dimensional characters, dipoles are frustrated on orientational |
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ordering in this circumstance. Another reason is that the lipids can |
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move independently in each monolayer, it is not nessasory for the |
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direction of dipoles on one leaf is consistant to another layer, which |
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makes total order parameter is relatively low. With increasing the |
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size of head group, the surface is being more corrugated, dipoles are |
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not allowed to move freely on the surface, they are |
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localized. 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 |
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consistant to the symmetry in another layer. Furthermore, the membrane |
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tranlates from a two dimensional system to a three dimensional system |
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by the corrugation, the symmetry of the ordering for the two |
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dimensional dipoles on the head group of lipid molecules is broken, on |
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a distorted lattice, dipoles are ordered on a head to tail energy |
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state, the order parameter is increased dramaticly. However, the total |
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polarization of the system is close to zero, which is a strong |
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evidence it is a antiferroelectric state. The orientation of the |
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dipole ordering is alway perpendicular to the ripple vector. These |
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results are consistant to our previous study on similar system. The |
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ordering of the tails are opposite to the ordering of the dipoles on |
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head group, the $P_2$ order parameter decreases with increasing the |
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size of head. This indicates the surface is more curved with larger |
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head. When surface is flat, all tails are pointing to the same |
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direction, in this case, all tails are parallal to the normal of the |
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surface, which shares the same structure with $L_{\beta}$ phase. For the |
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size of head being $1.28\sigma_0$, the surface starts to splay inward, |
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however, the surface is still flat, therefore, although the order |
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parameter is lower, it still indicates a very flat surface. Further |
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increasing the size of the head, the order parameter drops dramaticly, |
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the surface is rippled. |
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We studied the effects of interaction between head groups on the |
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structure of lipid bilayer by changing the strength of the dipole. The |
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fig. shows the $P_2$ order parameter changing with strength of the |
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dipole. Generally the dipoles on the head group are more ordered with |
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increasing the interaction between heads and more disordered with |
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decreasing the interaction between heads. When the interaction between |
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heads is weak enough, the bilayer structure is not persisted any more, |
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all lipid molecules are melted in the water. The critial value of the |
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strength of the dipole is various for different system. The perfect |
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flat surface melts at $5$ debye, the asymmetric rippled surfaces melt |
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at $8$ debye, the symmetric rippled surfaces melt at $10$ debye. This |
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indicates that the flat phase is the most stable state, the asymmetric |
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ripple phase is second stalbe state, and the symmetric ripple phase is |
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the most unstable state. The ordering of the tails is the same as the |
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ordering of the dipoles except for the flat phase. Since the surface |
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is already perfect flat, the order parameter does not change much |
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until the strength of the dipole is $15$ debye. However, the order |
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parameter decreases quickly when the strength of the dipole is further |
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increased. The head group of the lipid molecules are brought closer by |
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strenger interaction between them. For a flat surface, a mount of free |
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volume between head groups is available, when the head groups are |
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brought closer, the surface will splay outward to be a inverse |
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micelle. For rippled surfaces, there is few free volume available on |
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between the head groups, they can be closer, therefore there are |
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little effect on the structure of the membrane. Another interesting |
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fact, unlike other systems melting directly when the interaction is |
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weak enough, for $\sigma_h$ is $1.41\sigma_0$, part of the membrane |
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melts into itself first, the upper leaf of the bilayer is totally |
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interdigitated with the lower leaf, this is different with the |
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interdigitated lines in rippled phase where only one interdigitated |
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line connects the two leaves of bilayer. |
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Fig. shows the changing of the order parameter with temperature. The |
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behavior of the $P_2$ orderparamter is straightforword. Systems are |
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more ordered at low temperature, and more disordered at high |
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temperature. When the temperature is high enough, the membranes are |
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discontinuted. The structures are stable during the changing of the |
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temperature. Since our model lacks the detail information for tails of |
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lipid molecules, we did not simulate the fluid phase with a melted |
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fatty chains. Moreover, the formation of the tilted $L_{\beta'}$ phase |
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also depends on the organization of fatty groups on tails, we did not |
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observe it either. |
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\bibliography{mdripple} |
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\end{document} |