^{1}, N. Bobrovska

^{1}and A. Ciach

^{1}

### Abstract

Vesicles composed of a two component membrane with each component characterized by different spontaneous curvature are investigated by minimization of the free energy consisting of Helfrich elastic energy and entropy of mixing. The results show that mixing and demixing of membrane components can be induced by elongating a vesicle or changing its volume, if one of the components forms a complex with macromolecules on the outer monolayer. The influence of elastic coefficients on the separation of components is also examined.

The authors would like to acknowledge the support from the Polish Ministry of Science and Education, Grant No. N N 204 240534. The work of N. Bobrovska was realized within the International Ph.D. Projects Programme of the Foundation for Polish Science, cofinanced from European Regional Development Fund within Innovative Economy Operational Programme “Grants for innovation.”

I. INTRODUCTION

II. MODEL

III. RESULTS

A. Separation of components on a stiff membrane

B. The effect of entropy of mixing on the separation of components

C. Separation of components induced by elongation of the vesicle

IV. SUMMARY AND CONCLUSIONS

## Figures

The free energy as a function of the reduced volume v (Eq. (9) ). The dashed lines show the free energy obtained for constant spontaneous curvature C 0 = 4 (uniform distribution of components) for the following configurations: (i) a sphere, (ii) two, (iii) three, (iv) four, and (v) five beads, respectively. The solid curves show the free energy for (a) partially budded configurations and configurations with: (b) one, (c) two, (d) three, (e) four, (f) five small beads, and for tubular shapes (g). The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }. Configurations corresponding to the lowest value of the free energy for given v are thermodynamically stable, the remaining ones are metastable.

The free energy as a function of the reduced volume v (Eq. (9) ). The dashed lines show the free energy obtained for constant spontaneous curvature C 0 = 4 (uniform distribution of components) for the following configurations: (i) a sphere, (ii) two, (iii) three, (iv) four, and (v) five beads, respectively. The solid curves show the free energy for (a) partially budded configurations and configurations with: (b) one, (c) two, (d) three, (e) four, (f) five small beads, and for tubular shapes (g). The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }. Configurations corresponding to the lowest value of the free energy for given v are thermodynamically stable, the remaining ones are metastable.

Budding induced by the change of the reduced volume v (Eq. (9) ). From the left to the right vesicle v = 0.94, 0.95, 0.955, 0.96, 0.97, 0.975, 0.99, respectively. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }. The colormap shows mapping of concentration values to colors.

Budding induced by the change of the reduced volume v (Eq. (9) ). From the left to the right vesicle v = 0.94, 0.95, 0.955, 0.96, 0.97, 0.975, 0.99, respectively. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }. The colormap shows mapping of concentration values to colors.

The shapes of the vesicles for the family with one small bead for different values of the reduced volume v (Eq. (9) ). v = 0.705, 0.745, 0.785, 0.825, 0.865, 0.91, 0.955, respectively. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

The shapes of the vesicles for the family with one small bead for different values of the reduced volume v (Eq. (9) ). v = 0.705, 0.745, 0.785, 0.825, 0.865, 0.91, 0.955, respectively. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

The shapes of the vesicles for different values of the reduced volume v (Eq. (9) ). In the upper panel (A), the configurations with three small beads for v = 0.485, 0.535, 0.585, 0.635, 0.685, 0.735, 0.785 are presented. In the lower left panel (B), the configurations with five small beads for v = 0.505, 0.545, 0.585, 0.625, 0.665 are shown. In lower right panel (C), the configurations with tubular shapes for v = 0.345, 0.375, 0.405, 0.435, 0.465 are shown. The total concentration is ϕ tot = 0.5, the spontaneous curvatures are , , and the bending rigidity is κ^{ A } = κ^{ B }. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

The shapes of the vesicles for different values of the reduced volume v (Eq. (9) ). In the upper panel (A), the configurations with three small beads for v = 0.485, 0.535, 0.585, 0.635, 0.685, 0.735, 0.785 are presented. In the lower left panel (B), the configurations with five small beads for v = 0.505, 0.545, 0.585, 0.625, 0.665 are shown. In lower right panel (C), the configurations with tubular shapes for v = 0.345, 0.375, 0.405, 0.435, 0.465 are shown. The total concentration is ϕ tot = 0.5, the spontaneous curvatures are , , and the bending rigidity is κ^{ A } = κ^{ B }. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for different shapes of vesicles for (a) configurations without the beads, and configurations with: (b) one, (c) two, (d) three, (e) four, (f) five small beads. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }.

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for different shapes of vesicles for (a) configurations without the beads, and configurations with: (b) one, (c) two, (d) three, (e) four, (f) five small beads. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B }.

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for the different shapes of vesicles for configurations with: (a) one, (b) two, (c) three beads. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B } = 30 k b T , a 0 = 100 nm^{2} and R = 250 nm. The vesicle shapes represent minimized configurations with the best separation of the components (for the largest v for a given family of shapes), totally mixed (for the smallest v for a given family of shapes) , and intermediate. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for the different shapes of vesicles for configurations with: (a) one, (b) two, (c) three beads. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B } = 30 k b T , a 0 = 100 nm^{2} and R = 250 nm. The vesicle shapes represent minimized configurations with the best separation of the components (for the largest v for a given family of shapes), totally mixed (for the smallest v for a given family of shapes) , and intermediate. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for configurations with one bead for different sizes of nanodomains: (b) a 0 = 400 nm^{2}, (c) a 0 = 200 nm^{2}, (d) a 0 = 100 nm^{2}, (e) a 0 = 50 nm^{2}, (f) a 0 = 25 nm^{2}, (g) a 0 = 12 nm^{2}. The curve (a) refers to the calculations with neglected entropy of mixing. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B } = 30 k b T , and R = 250 nm.

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for configurations with one bead for different sizes of nanodomains: (b) a 0 = 400 nm^{2}, (c) a 0 = 200 nm^{2}, (d) a 0 = 100 nm^{2}, (e) a 0 = 50 nm^{2}, (f) a 0 = 25 nm^{2}, (g) a 0 = 12 nm^{2}. The curve (a) refers to the calculations with neglected entropy of mixing. The total concentration is ϕ tot = 0.5, , , κ^{ A } = κ^{ B } = 30 k b T , and R = 250 nm.

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for the different shapes of vesicles for configurations with: (a) one, (b) two, (c) three beads: (A) κ^{ A } = 8κ^{ B }, κ^{ B } = 30 k b T, (B) κ^{ A } = 30k b T, κ^{ B } = 8κ^{ A }. The total concentration is ϕ tot = 0.5, , , a 0 = 100 nm^{2}, and R = 250 nm. The vesicle shapes represent minimized configurations with the best separation of components (for the largest v for a given family of shapes), totally mixed (for the smallest v for a given family of shapes), and intermediate. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

Concentration in the spherical part of the vesicle (dashed curves) and in the protrusion (solid curves) as a function of the reduced volume v (Eq. (9) ) for the different shapes of vesicles for configurations with: (a) one, (b) two, (c) three beads: (A) κ^{ A } = 8κ^{ B }, κ^{ B } = 30 k b T, (B) κ^{ A } = 30k b T, κ^{ B } = 8κ^{ A }. The total concentration is ϕ tot = 0.5, , , a 0 = 100 nm^{2}, and R = 250 nm. The vesicle shapes represent minimized configurations with the best separation of components (for the largest v for a given family of shapes), totally mixed (for the smallest v for a given family of shapes), and intermediate. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 .

The shapes of vesicles with lateral segregation induced by elongation: (A) κ^{ A } = 2κ^{ B }, κ^{ B } = 30 k b T , (B) κ^{ A } = 8 κ^{ B }, κ^{ B } = 30 k b T. The total concentration is ϕ tot = 0.5, , , a 0 = 100 nm^{2}, and R = 250 nm. The height of the vesicle H = 3.1, 4.5, 5.3, 5.7, 6.3, 6.9 in R units. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 . The reduced volume v = 0.705 (Eq. (9) ).

The shapes of vesicles with lateral segregation induced by elongation: (A) κ^{ A } = 2κ^{ B }, κ^{ B } = 30 k b T , (B) κ^{ A } = 8 κ^{ B }, κ^{ B } = 30 k b T. The total concentration is ϕ tot = 0.5, , , a 0 = 100 nm^{2}, and R = 250 nm. The height of the vesicle H = 3.1, 4.5, 5.3, 5.7, 6.3, 6.9 in R units. The color of the surface reflects the value of the concentration ϕ according to the color map in Fig. 2 . The reduced volume v = 0.705 (Eq. (9) ).

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