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### Abstract

The attachment of macromolecules to the surface of a lipidvesicle may cause its deformations such as budding or creation of cylindrical protrusions. Diffusion of the macromolecules in the membranes may cause its shape transformations. The process of shrinking the protrusions due to diffusion of the macromolecules is investigated. It is assumed that macromolecules modify locally the spontaneous curvature and bending rigidity of the lipidmembrane. Both spontaneous curvature and bending rigidities depend on the concentration of membrane components. It has been shown that cylindrical protrusions are created when the macromolecules which induce large spontaneous curvature are accumulated at a piece of the vesiclesurface. It has been observed that here the elastic constants influence very little the evolution of the vesicle shape caused by diffusing macromolecules and the most important is the value the spontaneous curvature imposed by the macromolecules.

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

I. MODEL AND PARAMETERIZATION

II. RESULTS AND DISCUSSION

A. Formation of cylindrical protrusions

B. Diffusion for the same bending constants for all components

C. Diffusion for different bending constants

D. Shrinking of the protrusion

III. SUMMARY AND CONCLUSIONS

## Figures

Schematic illustration of the parameterization of the vesicle shape. *a* is the surface area of the vesicle, measured from the pole to some tangent point. θ(*a*) is the angle of the tangent to the shape profile with the line parallel to r-axis.

Schematic illustration of the parameterization of the vesicle shape. *a* is the surface area of the vesicle, measured from the pole to some tangent point. θ(*a*) is the angle of the tangent to the shape profile with the line parallel to r-axis.

Cylindrical protrusions created in the vesicles of the reduced volume *v* = 0.3 (the five profiles on the left side) and *v* = 0.4 (the five profiles on the right side). The black color shows the region occupied by the macromolecules. The first profile for each volume represents vesicle with the spontaneous curvature . The remaining four profiles represent the vesicles with nonuniform distribution of the spontaneous curvature given by Eq. (9) with parameters ξ = 50, and –(a), (e); –(b), (f); –(c), (g); –(d), (h), respectively. The vesicles have up-down symmetry. Only the upper halves of the profiles are presented. The bending rigidity and Gaussian rigidity are the same for each component , .

Cylindrical protrusions created in the vesicles of the reduced volume *v* = 0.3 (the five profiles on the left side) and *v* = 0.4 (the five profiles on the right side). The black color shows the region occupied by the macromolecules. The first profile for each volume represents vesicle with the spontaneous curvature . The remaining four profiles represent the vesicles with nonuniform distribution of the spontaneous curvature given by Eq. (9) with parameters ξ = 50, and –(a), (e); –(b), (f); –(c), (g); –(d), (h), respectively. The vesicles have up-down symmetry. Only the upper halves of the profiles are presented. The bending rigidity and Gaussian rigidity are the same for each component , .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components according to the colormap shown as a vertical cylinder above the vesicles. The total concentration of the macromolecules . The time step *dt* = 0.01. The elastic constants are the same for all components of the membrane , .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components according to the colormap shown as a vertical cylinder above the vesicles. The total concentration of the macromolecules . The time step *dt* = 0.01. The elastic constants are the same for all components of the membrane , .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components. The total concentration of the macromolecules . The time step *dt* = 0.01. The elastic constants are the same for all components of the membrane , .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components. The total concentration of the macromolecules . The time step *dt* = 0.01. The elastic constants are the same for all components of the membrane , .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components. The total concentration of the macromolecules . The time step *dt* = 0.01. The ratio of elastic constants .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components. The total concentration of the macromolecules . The time step *dt* = 0.01. The ratio of elastic constants .

The shape profiles and corresponding distribution of the macromolecules ϕ(*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components. The total concentration of the macromolecules . The time step *dt* = 0.01. The ratio of elastic constants .

*t*, *a*) after (a) 1, (b) 11, (c) 51, (d) 121, (e) 181, (f) 241, (g) 400 time steps for the vesicle with the reduced volume *v* = 0.3. The gray color shows the concentration of the components. The total concentration of the macromolecules . The time step *dt* = 0.01. The ratio of elastic constants .

The change of the vesicle height *h*(0) − *h*(*t*) in time for the vesicles with the reduced volume *v* = 0.3 and different values of the elastic constants: (a) —solid line, (b) —dashed line, (c) —dashed-dotted line. The total concentration of the macromolecules . The time step *dt* = 0.01.

The change of the vesicle height *h*(0) − *h*(*t*) in time for the vesicles with the reduced volume *v* = 0.3 and different values of the elastic constants: (a) —solid line, (b) —dashed line, (c) —dashed-dotted line. The total concentration of the macromolecules . The time step *dt* = 0.01.

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