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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 present 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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Ch.~\ref{chap:md} we developed a more realistic model for lipid |
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molecules compared to the simple point dipole one. To further address |
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the dynamics properties of the ripple phase, the simulation method is |
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switched to molecular dynamics. Symmetric and asymmetric ripple |
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phases have been observed to form in the simulations. The lipid model |
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consists of an dipolar head group and an ellipsoidal tail. Within the |
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limits of this model, an explanation for generalized membrane |
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curvature is a simple mismatch in the size of the heads with the width |
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of the molecular bodies. The persistence of a {\it bilayer} structure |
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requires strong attractive forces between the head groups. One |
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feature of this model is that an energetically favorable orientational |
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ordering of the dipoles can be achieved by out-of-plane membrane |
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corrugation. The corrugation of the surface stabilizes the long range |
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orientational ordering for the dipoles in the head groups which then |
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adopt a bulk anti-ferroelectric state. The structural properties of |
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the ripple phase we observed in the dynamics simulations are |
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consistant to that we observed in the Monte Carlo simuations of the |
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simple point dipole model. |
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To extend our simulations of lipid membranes to larger system and |
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longer time scale, an algorithm is developed in Ch.~\ref{chap:ld} for |
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carrying out Langevin dynamics simulations on complex rigid bodies by |
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incorporating the hydrodynamic resistance tensors for arbitrary shapes |
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into an 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 $9$ times larger simulation of the lipid bilayer are |
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carried out for the comparison with the molecular dynamics simulations |
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in 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 are elucidated in this dissertation. However, |
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the importance of the ripple phase in the experimental view is still a |
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mystery, hopefully, this work can contribute some flame to the |
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lighting of the experimental field. Further insights of the phase |
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behavior of the lipid membranes can be obtained by applying a atomic |
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or more detailed molecular model with information of the fatty chains |
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of the lipid molecules. |
