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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 the |
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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'}'' |
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phase also depends on the organization of fatty groups on tails, we |
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did not observe it either. |