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\chapter{\label{chap:conclusion}CONCLUSION} |
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This dissertation has shown the efforts to the understanding of the |
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structural properties and phase behavior of lipid membranes. In |
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Ch.~\ref{chap:mc}, we presented a simple model for dipolar elastic |
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membranes that gives lattice-bound point dipoles complete |
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orientational freedom as well as translational freedom along one |
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coordinate (out of the plane of the membrane). There is an additional |
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harmonic term which binds each of the dipoles to the six nearest |
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neighbors on either triangular or distorted lattices. The |
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translational freedom of the dipoles allows triangular lattices to |
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find states that break out of the normal orientational disorder of |
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frustrated configurations and which are stabilized by long-range |
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anti-ferroelectric ordering. In order to break out of the frustrated |
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states, the dipolar membranes form corrugated or ``rippled'' phases |
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that make the lattices effectively non-triangular. We observe three |
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common features of the corrugated dipolar membranes: 1) the corrugated |
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phases develop easily when hosted on triangular lattices, 2) the wave |
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vectors for the surface ripples are always found to be perpendicular |
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to the dipole director axis, and 3) on triangular lattices, the dipole |
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director axis is found to be parallel to any of the three equivalent |
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lattice directions. |
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In Ch.~\ref{chap:md} we developed a more realistic model for lipid |
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molecules. To further address the dynamical properties of the |
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formation of the ripple phase, Molecular Dynamics was used to simulate |
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these systems. The lipid model consists of a dipolar head group and an |
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ellipsoidal tail. Within the limits of this model, an explanation for |
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generalized membrane curvature is the simple mismatch in the size of |
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the heads with the width of the molecular bodies. The persistence of |
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a {\it bilayer} structure requires strong attractive forces between |
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the head groups. One feature of this model is that an energetically |
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favorable orientational ordering of the dipoles can be achieved by |
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out-of-plane membrane corrugation. The corrugation of the surface |
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stabilizes the long range orientational ordering for the dipoles in |
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the head groups which then adopt a bulk anti-ferroelectric |
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state. Symmetric and asymmetric ripple phases were observed to form in |
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the simulations.The structural properties of the ripple phase we |
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observed in the dynamics simulations are consistant with those we |
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observed in the Monte Carlo simuations of the simple point dipole |
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model. |
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To extend our simulations of lipid membranes to larger systems and |
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longer time scales, we developed an algorithm for carrying out |
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Langevin dynamics simulations on complex rigid bodies by incorporating |
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the hydrodynamic resistance tensors for arbitrary shapes into an |
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advanced symplectic integration scheme. The integrator gives |
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quantitative agreement with both analytic and approximate hydrodynamic |
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theories for a number of model rigid bodies, and works well at |
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reproducing the solute dynamical properties (diffusion constants, and |
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orientational relaxation times) obtained from explicitly-solvated |
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simulations. A simulation of the lipid bilayer was carried out that |
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was $9$ times the size of for the molecular dynamics simulations in |
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Ch.~\ref{chap:md}, the results show the structural stability of the |
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ripple phase. |
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The structural properties and the formation mechanism for the ripple |
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phase of lipid membranes have been elucidated in this |
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dissertation. However, the biological importance of the ripple phase |
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is still a mystery. Additionally, experimental conformation of our |
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predictions (dipoles that align perpendicular to the membrane ripples) |
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is still required. Hopefully, this work can kindle some interest among |
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experimentalists. Further insights into the phase transitions of lipid |
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membranes can be obtained by applying more detailed molecular or |
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atomic scale model with information of the fatty chains of the lipid |
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molecules. |
