^{1,a)}, Aleš Iglič

^{1}, Daniel M. Kroll

^{2}and Sylvio May

^{2}

### Abstract

Monte Carlo simulations are employed to investigate the ability of a charged fluidlike vesicle to adhere to and encapsulate an oppositely charged spherical colloidal particle. The vesicle contains mobile charges that interact with the colloid and among themselves through a screened electrostatic potential. Both migration of charges on the vesiclesurface and elastic deformations of the vesicle contribute to the optimization of the vesicle-colloid interaction. Our Monte Carlo simulations reveal a discontinuous wrapping transition of the colloid as a function of the number of charges on the vesicle. Upon reducing the bending stiffness of the vesicle, the transition terminates in a critical point. At large electrostatic screening length we find a reentrant wrapping-unwrapping behavior upon increasing the total number of charges on the vesicle. We present a simple phenomenological model that qualitatively captures some features of the wrapping transition.

S.M. thanks the NSF for support through Grant No. DMR-0605883; D.M.K. acknowledges the support from the NSF under Grant Nos. DMR-0513393 and DMR-0706017. The authors also acknowledge the support of the Slovenian Research Agency through Grant No. BI-U.S./08-10/072.

I. INTRODUCTION

II. THEORETICAL MODEL

A. Monte Carlo simulations

B. Phenomenological model

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Vesicles
- 77.0
- Colloidal systems
- 55.0
- Electrostatics
- 38.0
- Cell membranes
- 17.0
- Monte Carlo methods
- 15.0

## Figures

Bending free energy of a vesicle that partially wraps a spherical colloid of *effective* radius plotted as a function of the wrapping parameter . The surface area of the vesicle corresponds to a sphere of radius . The method to calculate is outlined in the Appendix. The Monte Carlo simulations of the present work all correspond to the case and .

Bending free energy of a vesicle that partially wraps a spherical colloid of *effective* radius plotted as a function of the wrapping parameter . The surface area of the vesicle corresponds to a sphere of radius . The method to calculate is outlined in the Appendix. The Monte Carlo simulations of the present work all correspond to the case and .

The wrapping of a colloid (radius ) by a vesicle (which, if spherical, would have a radius of ). Left: snapshots of representative colloid-vesicle configurations obtained from simulations for different numbers of charged vertices on the membrane: , , and (from left to right). The shape of the vesicle is represented by a triangulated surface; mobile charges on the vesicle and fixed charges on the colloid are indicated by dots. Right: cross sections of colloid-vesicle complexes obtained by minimizing the bending energy of the vesicle (see Appendix) for different values of the wrapping parameter .

The wrapping of a colloid (radius ) by a vesicle (which, if spherical, would have a radius of ). Left: snapshots of representative colloid-vesicle configurations obtained from simulations for different numbers of charged vertices on the membrane: , , and (from left to right). The shape of the vesicle is represented by a triangulated surface; mobile charges on the vesicle and fixed charges on the colloid are indicated by dots. Right: cross sections of colloid-vesicle complexes obtained by minimizing the bending energy of the vesicle (see Appendix) for different values of the wrapping parameter .

Number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. Results from Monte Carlo simulations are marked by the symbols and , corresponding to fully unwrapped and fully wrapped initial states of the simulation run, respectively. At about a discontinuous wrapping transition occurs. The dashed line is the prediction of our phenomenological model. The inset shows the number of charged vertices in the wrapping region . All results correspond to and .

Number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. Results from Monte Carlo simulations are marked by the symbols and , corresponding to fully unwrapped and fully wrapped initial states of the simulation run, respectively. At about a discontinuous wrapping transition occurs. The dashed line is the prediction of our phenomenological model. The inset shows the number of charged vertices in the wrapping region . All results correspond to and .

Number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. Results from Monte Carlo simulations are marked by the symbols (connected by dashed lines) and (connected by dotted lines), corresponding to fully unwrapped and fully wrapped initial states of the simulation run, respectively. The two minimum states predicted by the phenomenological model are marked by the solid line and the dashed curve for the full and weak wrapping regimes, respectively. The inset shows the corresponding free energy . At (see the vertical dotted line in the main figure) both states have the same free energy. All results correspond to and .

Number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. Results from Monte Carlo simulations are marked by the symbols (connected by dashed lines) and (connected by dotted lines), corresponding to fully unwrapped and fully wrapped initial states of the simulation run, respectively. The two minimum states predicted by the phenomenological model are marked by the solid line and the dashed curve for the full and weak wrapping regimes, respectively. The inset shows the corresponding free energy . At (see the vertical dotted line in the main figure) both states have the same free energy. All results correspond to and .

Number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. Results from Monte Carlo simulations are shown together with the prediction (dashed line) from the phenomenological model. The inset displays a representative colloid-vesicle complex corresponding to . All results correspond to and .

Number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. Results from Monte Carlo simulations are shown together with the prediction (dashed line) from the phenomenological model. The inset displays a representative colloid-vesicle complex corresponding to . All results correspond to and .

The number of charged vertices required to induce a discontinuous wrapping transition as a function of the bending stiffness . Data points from our simulations are connected by straight lines. The inset shows the free energy corresponding to points A (for ), B (for ), and C (for ); is a reference energy and is the probability to find vertices within the wrapping region. We have measured that probability from our simulations. Note that our computational results for indicate a critical point at about . No critical point is predicted by our phenomenological model (dashed line). All results are derived from the Debye length .

The number of charged vertices required to induce a discontinuous wrapping transition as a function of the bending stiffness . Data points from our simulations are connected by straight lines. The inset shows the free energy corresponding to points A (for ), B (for ), and C (for ); is a reference energy and is the probability to find vertices within the wrapping region. We have measured that probability from our simulations. Note that our computational results for indicate a critical point at about . No critical point is predicted by our phenomenological model (dashed line). All results are derived from the Debye length .

Simulation results for the number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. The two data sets ▲ and ▼ correspond to the two spontaneous curvatures and of a fully charged vesicle segment. Results for the absence of composition-curvature coupling marked by and are replotted from Fig. 3. A representative colloid-vesicle complex for and is also shown. All results correspond to and .

Simulation results for the number of vertices in the wrapping region as a function of the total number of charged vertices in the vesicle. The two data sets ▲ and ▼ correspond to the two spontaneous curvatures and of a fully charged vesicle segment. Results for the absence of composition-curvature coupling marked by and are replotted from Fig. 3. A representative colloid-vesicle complex for and is also shown. All results correspond to and .

Shape of a rotationally symmetric vesicle that partially wraps a spherical colloid of radius . The arc length and the functions (distance from the axis of symmetry) and (angle between the vesicle normal and the axis of symmetry) are indicated. The optimal shape is calculated as outlined in the Appendix.

Shape of a rotationally symmetric vesicle that partially wraps a spherical colloid of radius . The arc length and the functions (distance from the axis of symmetry) and (angle between the vesicle normal and the axis of symmetry) are indicated. The optimal shape is calculated as outlined in the Appendix.

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