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We studied biological membranes of spherical topology within the framework of the spontaneous curvature model. Both Monte Carlo simulations and the numerical minimization of the curvature energy were used to obtain the shapes of the vesicles. The shapes of the vesicles and their energy were calculated for different values of the reduced volume. The vesicles which exhibit in-plane ordering were also studied. Minimal models have been developed in order to study the orientational ordering in colloids coated with a thin sheet of nematic liquid crystal (nematic shells). The topological defects are always present on the surfaces with the topology of a sphere. The location of the topological defects depends strongly on the curvature of the surface. We studied the nematic ordering and the formation of topological defects on vesicles obtained by the minimization of the spontaneous curvature energy.

Human red blood cells are one of the most intriguing systems in nature. The shapes of the red blood cells have been studied by many theoretical methods. The spontaneous curvature model, developed by Deuling and Helfrich [

The biological membrane forms a wall, which surrounds the cell and intercellular organelles [

The vesicles which exhibit in-plane ordering are of particular interest. An example of such a vesicle is a colloidal particle coated with a thin sheet of nematic liquid crystal, called nematic shell [

The paper is organized as follows. In Section

The vesicle shapes have been studied within the framework of Helfrich spontaneous curvature model [

Many models were formulated in order to study the shape transformations of biological membranes. The model developed by Helfrich [

We assumed that the vesicle surface is the surface of revolution, with rotational symmetry about the

Vesicle profile representation on

The fluid vesicle is represented by a set of

The randomly triangulated network acquires its lateral fluidity from a bond flip mechanism [

The microstates of the vesicle are sampled according to the Metropolis algorithm, with the energy for a given microstate

The evolution of the system is reported in

The investigated vesicle consists of

We study the nematic ordering on smooth, closed, axial-symmetric surfaces, which we have calculated within the spontaneous curvature model. We use the minimal model, developed to study nematic shells [

If we know the values of

Vesicles of any shape can be coated with a thin layer of nematic liquid crystal to get the previously described nematic shells. In order to calculate the total free energy, we integrate (

The result of minimizing the bending energy

Bending energy of closed vesicles in units

In Figure

First row: snapshots of equilibrium configurations obtained by Monte Carlo simulations as described in Section

We can see in Figure

Monte Carlo evolution of the vesicle with pressure difference

Near the boundary between prolate and stomatocyte equilibrium states, calculated with the Monte Carlo method, the fluctuations of the vesicle increase (as can be seen in the increased standard deviation of the reduced volume). Due to thermal fluctuations (i.e., the stochastic nature of the Monte Carlo method), the boundary between the prolate and stomatocyte vesicles cannot be determined with a high accuracy. For example, in Figure

We have investigated the equilibrium configuration of the nematic liquid crystal on a prolate vesicle with the reduced volume

Vector field of molecules and contour plot of

We have studied the vesicles of spherical topology within the framework of the spontaneous curvature model. The shapes of the vesicles were calculated both by Monte Carlo simulations and by the minimization of the curvature energy functional. The vesicles shapes calculated by both methods are in very good agreement. In Monte Carlo simulations, the calculations were performed for a fixed value of the pressure difference. Therefore, we were not able to obtain the vesicles of the given reduced volume,

The nematic ordering on a prolate vesicle was also studied. The shape of the vesicle was calculated in the spontaneous curvature model. The net topological charge on the surfaces with the topology of a sphere equals

In the future research, one could calculate the nematic ordering on oblate and stomatocyte vesicles. It would be possible to use the Monte Carlo simulations in order to calculate the vesicle shapes for different value of the spontaneous curvature

The authors declare that there is no conflict of interests regarding the publication of this paper.

Luka Mesarec, Miha Fošnarič, and Samo Penič contributed equally to this work.

WTG would like to acknowledge the support from NCN Grant no. 2012/05/B/ST3/03302. The authors wish to acknowledge the ARRS Grants J1-4109, J1-4136, J3-4108, and P2-0232.