Phase Separation and Domain Formation in Multi-Component Membranes : Finsler Geometry Modeling and Monte Carlo Simulations

In this paper, we study a surface model for membranes of three components such as DPPC, DOPC, and Cholesterol. This membrane is experimentally well known to undergo the phase separation and to form the domain structure such as the liquid ordered ( o L ) phase and the liquid disordered phase ( d L ). It is also well known that this multicomponent membrane has a lot of domain pattern transitions between the circular domains and the striped domain etc. Using the new surface model constructed on the basis of Finsler geometry, we study why those morphological changes appear on the spherical vesicles. In our model, we introduce a new variable ) ( 2 Z ∈ σ to represent the domains o L and d L , and using the value of σ we define a metric function on the surface. As a consequence, the origin of the line tension energy, which has been used to explain the domain pattern transition in the multicomponent membranes, is naturally understood in our model.


Introduction
T Artificial membranes such as vesicles are used to study the mechanism of biological cell membranes [1].The vesicles, which are composed of saturated phospholipids, unsaturated phospholipids, and cholesterols, undergo a phase separation and a domain pattern transition [2].After the phase separation, two types of domains appear: one is the liquid ordered ( o L ) phase, which is rich in the saturated phospholipids and cholesterols, and the other is the liquid disordered ( d L ) phase, which is rich in the unsaturated phospholipids.
These two domains cause the domain pattern transition, which accompanies the shape deformation [2].For example, the oblate shape membranes with two Circular o L Domains (2CD) such as Fig. 1 (a), and the prolate shape membranes with one Striped o L Domain (1SD) such as Fig. 1 (b) are well known.It is also well known that the o L phase forms many round domains, and this domain structure is very similar to the so called lipid raft.However, the mechanism of shape deformation is still unclear, although a lot of theoretical studies have been conducted.For example, the mechanism of shape deformation to 2CD and 1SD can be explained by using the models with the line tension on the boundary between o L and d L phases [3, 4].However, the origin of the line tension is unknown.The problem that should be asked is the microscopic origin of the line tension.
The purpose of this study is to find the origin of the line tension and clarify the mechanism of shape deformation, which is produced by the phase separation.For this purpose, we define a Finsler geometry surface model [5] by extending the conventional Helfrich-Polyakov model for membranes [6,7].In the new model, The metric tensor is given by 0 , 0 where ρ is a function on the surface.The inverse metric and the determinant are The metric ab g in Eq.( 2) comes from the most general metric such as

Discrete model
The discrete Hamiltonian is obtained by a discretization of the continuous Hamiltonian in (1) on the triangulated lattices such as Fig. 2. The discrete model is considered as a Finsler geometry model [5].We should note that the  Assuming that the vertex position 1 r is the local coordinate origin, we replace the symbols in (1) as follows : . , , , , From ( 1) and ( 3), we get ( ) ( ).
Not only 1 r , but also 2 r and 3 r can be chosen as the local coordinate origin.Therefore, by the cyclic permutations for the expressions in Eq. ( 4) and with the factor 1/3, we have ( ) In the expressions of Eq. ( 5), we assume that the function ρ is independent of the chioce the local coordinate, where ρ is originallly dependent on the local coordinate.
By replacing the sum over triangles with the sum over bonds, and replacing the numerical factor 2/3 with 1/4, we get The triangles are separated into two groups o L and d L , which are labeled by σ (see Fig. 3). (Don't confuse the letters for domains and the one for Finsler function.)The discrete Hamiltonian can be written as where 0 S is the aggregation energy, and σ (Fig. 3) is the function on the triangle such that S is the Gaussian bond potential, where the symbol ij l denotes the length of the bond ij (Fig. 2). 2 S is the bending energy, where the symbol i n denotes the unit normal vector of the triangle i (Fig. 3).The quantities ij γ and ij κκ are respectively called effective surface tension and effective bending rigidity.Both quantities are defined by using a function ρ on the triangle, which is defined such that

Simulation technique
We use the Monte Carlo (MC) technique for the membrane simulation [8,9].In this MC technique, the vertex position r is randomly moved to r r r δ + = ′ (Fig. 5) with the probability

Simulation results
In this study, we perform the simulations for two types of models: model 1and model 2. Model 1 is defined by , where the Finsler metric is introduced only in 2 S . 1 S in model 1 is just the same as that of the ordinary model.In model 2, the Finsler metric is introduced in both 1 S and 2 S .

Model 1
The parameters used in the simulations for model 1 are shown in Tables 1, 2. The input parameters by hand are λ , which is the coefficient of 0 S in ( 4), ( )  The parameters ij κ is automatically determined according to the combination of the two neighboring triangles as listed in Tables 2. ( )  φ are large, also from Fig. 8(b) that 2CD appears if λ and o φ are small.Therefore, the membrane shape changes 1SD to 2CD, if λ and o φ are reduced.We also find from Fig. 8(c) that a random pattern appears if λ is very small, where the domains are not separated.Moreover, we find from Fig. 8(d) that the budding appears.The budding is a phase with the domains which grow inside or outside the surface.( ) In this simulations, we find that the surface shape changes 1SD to 2CD, if λ and κ are reduced, just as in the cases of model 1.Moreover, we find from Fig. 9(c) that the raft appears.The raft is a phase of the multi circular domains.The budding also appears, if λ is very large.These raft and budding can be seen in the real multi-component membranes.

Conclusions
In this paper, we study a coarse grained Finsler geometric surface model for three components membrane.The purpose is to understand the origin of the line tension on the boundary between o L and d L phases in those membranes.We have peformed Monte Carlo simulations on the dynamically triangulatted spherical surfaces.
The results are summurised as follows.
These results obtained in both models 1 and 2 are consistent with the experimental results.Recalling that the aggregation energy 0 S is proportional to the length of the boundary between o L and d L phases, we find that the of σ with the surface is considered as the origin of the line tension on the boundary.This interaction between σ and the surface is introduced by the Finsler metric, and therefore, the origin of the line tension energy is understood in the context of Finsler geometry modeling of membranes.
In the model of this paper, we assume that the function ρ used in the Finsler metric is independent of the local coordinate on the trianle.For this reason, the effective surface tension and bending rigidity become symmetric . Because of this property, the Hamiltonian of the surface is identical with that of the inverted surface.This implies that the model introduced in this paper is able to describe the area difference and the spontaneous curvature, both of which separates the spherical membrane from the inverted membranes, if the assumption for ρ is neglected.Therefore, it is interesting to study the origin of spontaneous curvature and area difference in bilayer membranes in the context of Finsler geometry modeling.The problem that will be asked is the microscopic origin of spontaneous curvature and area difference in bilayer membranes.This will be the future study.

Figure 1 .
Figure 1.General form of (a) 1SD, and (b) 2CD.The blue (white) region corresponds to o L ( d L ) phase.
interaction between this σ and the membrane shape is naturally introduced by a Finsler metric, which is defined by using the value of σ .

1
Continuous modelThe Hamiltonian (=energy function) of the continuous mode is defined by a mapping r from a two-dimensional surface M to the three-dimensional Euclidean space 3 R .The continuous description of the Hamiltonian is as follows: vector along the local coordinate or bond.

Figure 2 .
Figure 2. Triangles for the descritization, the vertex positions 1 r , 2 r , 3 r , the unit normal vectors

Figure 3 .
Figure 3. Geometrical quantities on the two neighboring triangles of the bond ij (on a triangulated spherical surface).

This 0 S
corresponds to the line tension on the boundary between o L and d L , because the value of 0 S is proportional to the length of the boundary.1

.
The effective surface tension ij γ is defined by ρ of two neighboring triangles, as mentioned above.For this reason ij γ has three different values such that .4(c)).For ij κ , we also have three different values, which are exactly the same as those for ij γ .We should note that these ij γ and ij κ depend on the position and the direction on the surface, because these are defined on the bonds by using ρ 's on the two neighboring triangles.Moreover, ij γ and ij κ are symmetric under the exchange of i and j (

Figure 4 .
Figure 4. Three types of combinations of the triangles (a) o L

.
This implies that r is updated to r′ with the probability 1 (

Figure 5 .
Figure 5. MC update of the vertex position r to r r r δ + = ′ In this study, the lattice structure is changed by bond flip technique such as in Fig.6 This bond flip makes the vertices diffuse freely over the surface, and hence the surface becomes a fluid surface.After the bond flip, i σ

Figure 6 .
Figure 6.A bond flip as a MC process, which makes the surface fluid.

Figure 7 .
Figure 7.The variable σ is changed by the bond flips over the surface.

Table 1 .
are the total number of o L triangles and the total number of triangles), which is the area fraction of o L phase, the bending rigidity κ .These are listed in Tables 1. c is the parameter for the function ρ in (4).The value of c here, we assumed is 05 .0 = c .The parameters for the simulations for model 1. λ , o φ , and κ are fixed by hand as the inputs.

.
At the domain boundary, the surface stiffness is fixed to the intermediate value of these two such that ( ) We see from Fig.8(a) that 1SD appears if λ and o

Figure 8 .4.2 Model 2
Figure 8. Snapshots of the surfaces corresponding to the domains (a) 1SD, (b) 2CD, (c) random, (d) budding, which are obtained from the simulations for model 1

Figure 9 .
Figure 9. Snapshots of the surfaces corresponding to the domains (a) 1SD, (b) 2CD, (c) raft, (d) budding, which are obtained from the simulations for model 2.

Table 2 .
The parameters for the simulations for model 1.The

Table 3 .
The parameters for the simulations for model 2. λ , , and κ are fixed by hand as the inputs.

Table 4 .
The parameters for the simulations of model 2. the