for a few ripples to fit in the simulation box. Furthermore, as the ripple structure is well ordered, the relaxation time of the structure and of the simulation box geometry in the plane of the bilayer is larger than the maximal possible simulation time.
As a consequence, it was impossible to accurately measure the equilibrium ripple re-peat distance for the ripple structures from the simulations in the frame of this work.
Also, the exact equilibrium structure of the different domains could not be deter-mined, whether the interdigitated domain and the kink actually exist in equilibrium, or whether they are an expression of finite size effects.
However, it is possible to estimate limits for the repeat distance from the structures formed in systems of different sizes.
When cooling down a bilayer in the fluid phase Lα, single, asymmetric ripple structures were found in 12x12-lipid-systems and 15x15-lipid-systems, double rip-ples where found along the long axis of 30x12-lipid-systems.
In systems with intermediate sizes (20x20 and 24x24), no clear elongated ripples evolved. Instead, branched and interconnected structures of the thicker and thinner parts of the bilayer as shown in figure 8.10 on the facing page are formed. Still, the arrangement of the leaflet domains very much resembles the asymmetric ripple struc-ture: the bilayer exhibits thicker, gel-like, well ordered domains as well as interfacial interdigitated domains were the upper and the lower leaflets of two gel-like domains form a joint interdigitated domain, with the corresponding opposite leaflets end in disordered, fluid-like domains.
The different ripple structures formed in systems of different sizes differ mainly in the size of the interdigitated domains (labelled “B” in figure 8.3b on page 101) and the kink angle, as can be seen in the simulation snapshots of three different occurences of the ripple structure in figure 8.8 on page 105. While it is a relatively large domain in subfigure 8.8a, it is much smaller in the other subfigures, down to its complete disappearence in some cases. Figure 8.9 gives sketches of two alternatives of the asymmetric ripple structures, with either no kink and a small interdigitated region to an extended interdigitated region with a strong kink.
It can be assumed, that by varying between these two extremes, the system can compensate the mismatch between the simulation box geometry and the equilibrium ripple repeat distance to a certain amount. Unfortunately, caused by the slow relax-ation of the box geometry, it was not possible to establish the equilibrium the domain in the frame of this work.
From these facts, the equilibrium ripple repeat distance can be estimated to be between 12 and 15 lipid diameters. From the behaviour of the systems, the author assumes that it is closer to 15 lipid diameters, and that the interdigitated domain is minimal, i.e. that there is direct contact between both leaflets at the ripple.
8.3. Symmetric ripple structures in the model
When cooling down a sufficiently large patch of a model bilayer at pressurep∗ = 2.0 in the fluid phaseLα, in some cases, the structure depicted in figure 8.11 is formed. In
A A
A A
B C C
(a) Simulation snapshot of the structure. The horizontal axis lies in the bilayer plane and is parallel to the ripple direction, the vertical axis is parallel to the bilayer normal. To stress the structure, the particle coordinates are smoothed over10adjacent configurations. The colour of the lipid chains encodeslz: red lipid chains point upwards, while blue chains point downwards and green lipids are in between. The head beads are depicted as spheres. For an explanation of the domain labels, see the text.
B
A A
C C C
A A
C
(b) Sketch of the structure. The purple arrows indicate the lipid tiltθ. For an explanation of the domain names, see the text.
Figure 8.11: Symmetric ripple structure obtained when slowly cooling down a 30x12-lipid-system from theLα-phase.
(a) 12x30-lipid-system atT = 1.18, slowly cooled down fromT= 1.19(Lα)
(b) 12x30-lipid-system atT = 1.16, cooled down fromT = 1.2(Lα)
Figure 8.12: Height of the bilayer in thex-y-plane for different occurences of the sym-metric ripple structure.
8.3. Symmetric ripple structures in the model
-10 0 10 20 30
x
-40 -30 -20 -10 0 10 20 30 40 50 60
Tilt angle θ
upper leaflet lower leaflet
Figure 8.13: Average lipid tiltθagaintsx-coordinate in the symmetric ripple structure.
(a)
(b)
Figure 8.14: Snapshots of different occurences of the symmetric ripple structure. The vertical axis is parallel to the bilayer normal, the horizontal axis is parallel to the ripple direcion. The systems correspond to the systems in figure 8.12 on the facing page.
the following, this structure is identified with the prevalent structure in the metastable symmetric ripple phasePβ(mst).
The structure shares most of the characteristics of the asymmetric ripple structure described in the previous section. Firstly, the structure consists of large, well-ordered gel-like domains (labelled A), interdigitated domains of varying size (labelled B) and smaller, disordered, fluid-like domains (labelled C).
On average, the lipids in the structure are mostly ordered and reside on a hexag-onal lattice – the radial distribution functions and structure factors of the structure are virtually indistinguishable from those of the asymmetric ripple structure and are therefore not shown. Instead of being tilted towards the bilayer normal on average, the lipids in the ordered domains exhibit splay that continuously changes the tilt (see figure 8.13 on the previous page). The value of the splay is dxdθ = 2.3±0.2, as in the case of the asymmetric ripple structure.
However, there are a number of significant differences between the structures. First of all, the arrangement of the domains is different from the asymmetric ripple structure described in the previous section, leading to a symmetric height profile which has roughly double the repeat distance of the asymmetric structure. This can be seen in figure 8.12 on page 108, where the height of the bilayer, i.e. the maximalzcoordinate in the plane of the bilayer is visualised. Note that in the figure, the bilayer heighth is plotted instead of the bilayer thicknessd, as the thickness of the bilayer varies with the same repeat distance, as in the asymmetric phase, while the height has roughly the double repeat distance.
As in the asymmetric ripple structure, the lipids of two splayed domains on both sides of the ripple are interconnected via an interdigitated intermediate domain. In the symmetric ripple structure, the upper leaflet on one side interconnects with the upper leaflet on the other side. To allow for a continuous, splayed, ordered leaflet, the system has to locally tilt the plane of the bilayer, which results in the height variations of the symmetric ripple phase. In the cavity that is formed in the opposite leaf-let, fluid-like disordered domains form that end the corresponding ordered domains, where the tilt of the leaflets is reset.
Again, only limits for the equilibrium repeat distance and size of the interdigitated domain can be given, as the relaxation times of the box geometry are very slow.
The largest systems that were simulated only fit a single symmetric ripple structure.
Different occurences of the structure are shown in figure 8.14, the repeat distances of which were between28and40σ. In section 8.4 on the next page, it will be discussed, why the repeat distance is about the double repeat distance of the asymmetric phase.
In general, the amplitude of the ripple profile is larger than in the asymmetric ripple, and the bilayer has a distinct curvature.
Note, that a transition from the asymmetric ripple structure to the symmetric ripple structure or vice versa has never been observed. It can be assumed, that the energy barrier between both structures is too high for such an event to occur within feasible simulation times.