However, the origin of the line tension energy is not well understood. In fact, it is unclear what type of internal structure is connected to the line tension energy until now. The problem that should be asked is where the line tension energy originates. Therefore, in this paper, we clarify and discuss the microscopic origin of the line tension energy.

### Relativistic Finsler geometry

Note that the general HP model can be discretized on triangulated surfaces and becomes well defined only when it is treated in the context of Finsler geometry [ 10 , 11 , 12 ]. In addition, from the viewpoint of modeling, it is very natural to extend the HP model to the general HP model for explaining the morphological changes in multi-component membranes. Indeed, the HP model is considered as a straightforward extension of the linear chain model for polymers [ 13 ]. The remainder of this paper is organized as follows.

In SubSection 2. Finally, we summarize the results in Section 5. In Appendix A , we describe the technical details of the FG modeling. In Appendix A. From this discrete model, we obtain the model for two-component membranes by imposing a constraint on the metric function. The symbol n denotes a unit normal vector of the surface. Both S 1 and S 2 are conformally invariant. These are the HP model [ 8 , 9 ] corresponding to polymerized membranes, and the HP model and the Landau—Ginzburg model [ 15 ] have been thoroughly investigated [ 16 , 17 , 18 , 19 , 20 , 21 , 22 ].

We now introduce a discrete Hamiltonian for multi-component membranes. The technical details of the discretization of the continuous Hamiltonian S 1 and S 2 introduced in SubSection 2. Using these S 1 and S 2 , we have the total Hamiltonian S , such that:. The three-dimensional vector r denotes the vertex position of the triangulated lattice.

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As described above, the second and third terms S 1 and S 2 in S are the discrete Hamiltonians corresponding to the continuous ones introduced in Section 2. The symbol n i in S 2 expresses a unit normal vector of the triangle i.

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L o , d , L o , d correspond to the bonds represented by the duplicated lines. The symbol L o , L d refers to the bond shared by the two neighboring triangles of the L o and L d phases see Figure 1. Note that only L o , L d corresponds to the bond on the domain boundary, and the other two correspond to the bonds inside the domains L o and L d. Thus, this expression is one of the most interesting outputs of the model in this paper.

The dashed lines denote the values of c assumed in some of the simulations. Due to this bond flip, the vertices can freely diffuse over the surface, where two neighboring triangles merge and split into two different ones, and the total number of triangles remains unchanged in this process. Therefore, not only the vertices, but also the triangles diffuse over the surface. More precisely, S 0 is proportional to the total number of bonds that form the domain boundary because the mean bond length is constant or non-zero finite on the boundary. First, the fact that the mean bond length becomes constant is understood from the scale-invariant property of the partition function Z in Equation 6.

Importantly, the mean bond length is expected to be finite, although it fluctuates around the mean value, and the mean value itself varies depending on the domains or the domain boundary. The remaining problem to be clarified is how the domain boundary is formed on the triangulated surfaces. During experiments, the area fraction of L o and L d is fixed [ 3 ].

The relation between the area fraction of L o and the fraction of N T o will be described in the next section. Therefore, the triangles themselves have to diffuse over the surface to form the L o and L d domains. This triangle diffusion is numerically possible on the dynamically-triangulated surfaces, which are called triangulated fluid surfaces, via the Monte Carlo MC technique with dynamical triangulation, as described above [ 23 , 24 , 25 , 26 , 27 , 28 ]. Therefore, the model in this paper is limited to membranes with two-component domains; however, the modeling technique is applicable to membranes with multi-component domains.

Here, we comment on how to extend the model to a n -component model. The Hamiltonian of the n -states Potts model, for example, can be used for S 0. The canonical Metropolis technique is used [ 30 , 31 ]. The triangulation T is updated using the bond flip technique, as described in the previous section [ 23 , 24 , 25 , 26 , 27 , 28 ]. We use the same technique used in [ 23 , 24 , 25 , 26 , 27 , 28 ], except for the following constraint. As described in the previous section, the mean triangle areas a o and a d in the domains L o and L d are constant because of the scale invariance of Z.

This random state corresponds to the two-phase coexistence configuration. The simulations at the phase boundaries are relatively time consuming in general because the domain structure and, hence, the surface shape change very slowly at these boundaries. Two types of models, which are denoted as Model 1 and Model 2, are simulated. Model 2 is the same as the one introduced in Section 2. More precisely, the mean triangle area in the L o domain is different from that in the L d domain in Model 2.

In Table 2 , we show the parameters assumed in the simulations. This result is consistent with the experimental results reported in [ 3 ], where the area fraction of L o is changed. For this reason, the L o domain is relatively smooth compared to the L d domain. The solid lines denote the phase boundaries, and the dashed lines denote the positions for the simulations for Figure 4 a—c.

The two separated domains on the surface of the striped domain and the connected domain on the surface of two circular domains correspond to the L d phase, which is DOPC rich. D 1 and D 2 , D 3 correspond to the major and minor axes, respectively. Therefore, the surfaces with the stripe and two circular domains can be distinguished by the minor axis D 2. The three axes are perpendicular to each other. We plot D 2 vs. From the plot of D 2 vs. However, note that the change of the morphology at this phase boundary is relatively smooth.

In fact, one circular domain surface, which is not shown as a snapshot in Figure 3 , can be observed at the boundary. This implies that the stripe domain surface and one circular domain surface have the same bending energy, or in other words, the bending energy is degenerate. Additionally, note that the phase boundary between the two circular and random domains appears to be continuous.

This means that the shape of the two circular domain surfaces continuously changes to the random domain surface. At this phase boundary, the surface shape continuously changes from pancake to sphere. These are calculated on the dashed horizontal and vertical lines in Figure 3. For this reason, the area of the triangles in the L o domain becomes considerably smaller than that in the L d domain.

The shape of the one circular surface in the one circular region is almost spherical, such as the one shown in Figure 6 , and this result is in contrast to the result in [ 4 ], where the one circular phase is separated into two phases: the prolate and oblate phases.

The solid lines on the phase diagram denote the phase boundaries, and the dashed lines denote the positions for the simulations for Figure 7 a—c. To observe the variation of the surface size at the phase boundaries, we calculate the size D 2 on the dashed lines in Figure 6 and plot them in Figure 7 a,b.

Moreover, the phase boundary is also not as clear because of the same reason as that for Model 1.

In fact, the surface shape at the phase boundary between the two circular and stripe domains is not always stable in Model 2, similar to that in Model 1. The phase boundary between the two circular and the random domains is also expected to be continuous in Model 2. The boundaries of one circular to two circular and one circular to stripe are also not as clear, and the boundary of one circular to random is continuous.

These are calculated on the dashed horizontal and vertical lines in Figure 6. The size of the surface changes almost discontinuously and smoothly at the phase boundaries, which are denoted by the dashed lines.

## Download A Brief Introduction To Finsler Geometry

Another difference between Model 1 and Model 2, other than the appearance of the stable one circular domain, is the raft-like domain and the budding domain. More precisely, the budding domain can also be seen in Model 1; however, it is more clear in Model 2. Note that the budding domain in some of the budding surfaces goes inside the surface, and some of them self-intersect because no self-avoiding interaction is assumed.

Note that S 0 has a non-zero positive value only on the boundary bonds between L o and L d , while S 2 has a non-zero value on all of the bonds. Moreover, note that the boundary length between L o and L d becomes longer shorter if the total number of circular domains increases decreases , whereas the areas of L o and L d remain constant and are independent of the total number of L o domains. The solid lines on the phase diagram denote the phase boundaries.

## Finsler Geometry Modeling of Phase Separation in Multi-Component Membranes

As described in Section 2 , the bond length is expected to be well defined in the sense that the mean bond length is constant on the surface, although this constant varies depending on the domains or the domain boundary to which the bond belongs. The data in Figure 9 a,b are obtained on the dashed lines in Figure 3 , and those in Figure 9 c,d are obtained on the lines in Figure 6. These data shown in Figure 9 indicate that the simulations including the energy discretization are successful.

The data in a and b c and d are obtained on the dashed lines in Figure 3 Figure 6. In other words, we have extended the HP model to explain the morphological changes of the three-component membranes in the context of FG modeling. The results obtained from Monte Carlo MC simulations are consistent with the experimental results that have been reported in the literature. We confirm the phase separation of the L o and L d domains on the surface and that the surface shows a variety of morphologies, such as the two circular domain, the stripe domain, the raft domain and the budding domain.

Indeed, the value of S 0 is only the total number of bonds on the boundary between L o and L d in our new model. This interaction is closely connected to the property of the new model that the surface strength, such as the surface tension and the bending rigidity, is dependent on the bond position on the surface. To obtain the discrete model from the continuous surface model introduced in Section 2. The Hamiltonian is defined on the triangulated surfaces, which are composed of three simplexes, such as vertices, bonds and triangles.

Thus, all physical quantities, including Hamiltonians and the metric function, are defined on these simplexes labeled by integers.

## Homogeneous Finsler Spaces

We start with the discrete metric g a b , such that:. For this reason, the metric g a b can be diagonalizable. This metric depends only on x and is independent of y , and hence, it simply corresponds to the Riemannian metric. By replacing:. We have three possible coordinate origins in the triangles. For arbitrary g a b , we always have the metric of the form in Equation A1 by the same procedure as described above.

Thus, we finally have:. Indeed, from the expression given in Equation A9 , we have:. In this subsection, we show that the discrete surface models constructed above are well defined only in the context of Finsler geometry modeling [ 10 ]. For this purpose, we should first remind ourselves of the fact that the symmetry properties in Equation A10 can be observed in the model only under the condition of Equation A5.

This symmetry is not present in the model of Equation A4. In fact, we have:. From this, we can see that the Gaussian bond potential energy and also the bending energy of the bond 12 of one surface configuration differs from that of the opposite orientation configuration. However, we have no reason for the difference in S 1 for two surfaces with different orientations. Thus, the model defined by Equation A4 , which is orientation asymmetric, is ill defined in the context of conventional surface modeling.

Moreover, we have to remark that the model defined by Equation A8 , which is orientation symmetric, is also ill defined. These two square lengths of the bond 12 must be the same. However, we have:. The edge length of triangles should uniquely be given as the basic requirement even in the discrete models. Therefore, in a model construction on the triangulated lattices, we always obtain an ill-defined discrete model if we start with an arbitrary Riemannian metric in which the elements are defined on the triangles.

However, these ill-defined models in Equations A4 and A8 become well defined in the context of Finsler geometry [ 10 , 11 , 12 ]. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username.

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