# On the Role of Electrostatic Repulsion in Topological Defect-Driven Membrane Fission

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Topological Defects/Antidefects and Membrane Fission

## 3. Electrostatic Interaction between Charged Membrane Surfaces

#### 3.1. Modified Langevin Poisson–Boltzmann Model

^{−30}C/m), and ${n}_{w}/{N}_{A}$ = 55 mol/l, Equation (5) gives ${\epsilon}_{\mathit{r},b}$ = 78.5 for the bulk solution. The value ${p}_{0}=3.1D$ is smaller than the corresponding value in previous similar models of electric double layers considering also orientational ordering of water dipoles. For example, in the model of Abrashkin et al. [106], where the cavity field and electronic polarisability of the water molecules are not taken into account, the value of ${p}_{0}=4.86D$. The model [106] also incorrectly predicts the increase in the relative permittivity of the electrolyte solution in the direction towards the charged surface, which is in contradiction to the experimental results and defies common principles in physics [65,101,104,105]. On the contrary, Equations (1)–(3) of the described modified LPB model predicts the decrease in relative permittivity in the electrolyte solution near the charged surface [2,55,100], in agreement with the experimental observations [107,108].

#### 3.2. Osmotic Pressure between Two Charged Surfaces within a Modified Langevin Poisson–Boltzmann Model

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Singer, S.J.; Nicolson, G.L. The fluid mosaic model of the structure of cell membranes. Science
**1972**, 175, 720–731. [Google Scholar] [CrossRef] [PubMed] - Iglič, A.; Drobne, D.; Kralj-Iglic, V. Nanostructures in Biological Systems: Theory and Applications, 1st ed.; Pan/Jenny Stanford Publishing, Taylor and Francis: New York, NY, USA, 2015. [Google Scholar]
- Fošnarič, M.; Iglič, A.; May, S. Influence of rigid inclusions on the bending elasticity of a lipid membrane. Phys. Rev. E
**2006**, 74, 051503. [Google Scholar] [CrossRef] [Green Version] - Kralj-Iglič, V.; Pocsfalvi, G.; Mesarec, L.; Šuštar, V.; Hägerstrand, H.; Iglič, A. Minimizing isotropic and deviatoric membrane energy–An unifying formation mechanism of different cellular membrane nanovesicle types. PLoS ONE
**2020**, 15, e0244796. [Google Scholar] [CrossRef] - Kralj-Iglič, V.; Heinrich, V.; Svetina, S.; Žekš, B. Free energy of closed membrane with anisotropic inclusions. Eur. Phys. J. B-Cond. Matter Complex Syst.
**1999**, 10, 5–8. [Google Scholar] [CrossRef] [Green Version] - Kralj-Iglič, V.; Svetina, S.; Žekš, B. Shapes of bilayer vesicles with membrane embedded molecules. Eur. Biophys. J.
**1996**, 24, 311–321. [Google Scholar] [CrossRef] - Markin, V.S. Lateral organization of membranes and cell shapes. Biophys. J.
**1981**, 36, 1–19. [Google Scholar] [CrossRef] [Green Version] - Walani, N.; Torres, J.; Agrawal, A. Endocytic proteins drive vesicle growth via instability in high membrane tension environment. Proc. Natl. Acad. Sci. USA
**2015**, 112, E1423–E1432. [Google Scholar] [CrossRef] [Green Version] - Gov, N.S. Guided by curvature: Shaping cells by coupling curved membrane proteins and cytoskeletal forces. Philos. Trans. R. Soc. B Biol. Sci.
**2018**, 373, 20170115. [Google Scholar] [CrossRef] [PubMed] - Fošnarič, M.; Penič, S.; Iglič, A.; Kralj-Iglič, V.; Drab, M.; Gov, N.S. Theoretical study of vesicle shapes driven by coupling curved proteins and active cytoskeletal forces. Soft Matter
**2019**, 15, 5319–5330. [Google Scholar] [CrossRef] - Mesarec, L.; Góźdź, W.; Kralj, S.; Fošnarič, M.; Penič, S.; Kralj-Iglič, V.; Iglič, A. On the role of external force of actin filaments in the formation of tubular protrusions of closed membrane shapes with anisotropic membrane components. Eur. Biophys. J.
**2017**, 46, 705–718. [Google Scholar] [CrossRef] - Hägerstrand, H.; Mrówczyńska, L.; Salzer, U.; Prohaska, R.; Michelsen, K.A.; Kralj-Iglič, V.; Iglič, A. Curvature-dependent lateral distribution of raft markers in the human erythrocyte membrane. Mol. Membr. Biol.
**2006**, 23, 277–288. [Google Scholar] [CrossRef] [PubMed] - Kabaso, D.; Bobrovska, N.; Góźdź, W.; Gov, N.; Kralj-Iglič, V.; Veranič, P.; Iglič, A. On the role of membrane anisotropy and BAR proteins in the stability of tubular membrane structures. J. Biomech.
**2012**, 45, 231–238. [Google Scholar] [CrossRef] - Liese, S.; Carlson, A. Membrane shape remodeling by protein crowding. Biophys. J.
**2021**. in print. [Google Scholar] [CrossRef] [PubMed] - Mesarec, L.; Drab, M.; Penič, S.; Kralj-Iglič, V.; Iglič, A. On the role of curved membrane nanodomains and passive and active skeleton forces in the determination of cell shape and membrane budding. Int. J. Mol. Sci.
**2021**, 22, 2348. [Google Scholar] [CrossRef] [PubMed] - Fischer, T.M. Bending stiffness of lipid bilayers. III. Gaussian curvature. J. Phys. II
**1992**, 2, 337–343. [Google Scholar] [CrossRef] - Fischer, T.M. Bending stiffness of lipid bilayers. V. Comparison of two formulations. J. Phys. II
**1993**, 3, 1795–1805. [Google Scholar] [CrossRef] [Green Version] - Fournier, J.B. Nontopological saddle-splay and curvature instabilities from anisotropic membrane inclusions. Phys. Rev. Lett.
**1996**, 76, 4436. [Google Scholar] [CrossRef] - Safinya, C.R. Biomolecular materials: Structure, interactions and higher order self-assembly. Coll. Surf. A Physicochem. Eng. Asp.
**1997**, 128, 183–195. [Google Scholar] [CrossRef] - Fournier, J.B.; Galatola, P. Bilayer membranes with 2D-nematic order of the surfactant polar heads. Braz. J. Phys.
**1998**, 28, 329–338. [Google Scholar] [CrossRef] - Kralj-Iglič, V.; Babnik, B.; Gauger, D.R.; May, S.; Iglič, A. Quadrupolar ordering of phospholipid molecules in narrow necks of phospholipid vesicles. J. Stat. Phys.
**2006**, 125, 727–752. [Google Scholar] [CrossRef] - Penič, S.; Mesarec, L.; Fošnarič, M.; Mrówczyńska, L.; Hägerstrand, H.; Kralj-Iglič, V.; Iglič, A. Budding and fission of membrane vesicles: A mini review. Front. Phys.
**2020**, 8, 342. [Google Scholar] [CrossRef] - Mesarec, L.; Góźdź, W.; Iglič, A.; Kralj-Iglič, V.; Virga, E.G.; Kralj, S. Normal red blood cells’ shape stabilized by membrane’s in-plane ordering. Sci. Rep.
**2019**, 9, 1–11. [Google Scholar] [CrossRef] [Green Version] - Mares, T.; Daniel, M.; Perutkova, S.; Perne, A.; Dolinar, G.; Iglic, A.; Rappolt, M.; Kralj-Iglic, V. Role of phospholipid asymmetry in the stability of inverted hexagonal mesoscopic phases. J. Phys. Chem. B
**2008**, 112, 16575–16584. [Google Scholar] [CrossRef] - Mesarec, L.; Góźdź, W.; Iglič, A.; Kralj, S. Effective topological charge cancelation mechanism. Sci. Rep.
**2016**, 6, 1–12. [Google Scholar] - Mesarec, L.; Iglič, A.; Kralj-Iglič, V.; Góźdź, W.; Virga, E.G.; Kralj, S. Curvature Potential Unveiled Topological Defect Attractors. Crystals
**2021**, 11, 539. [Google Scholar] [CrossRef] - Vitelli, V.; Turner, A.M. Anomalous coupling between topological defects and curvature. Phys. Rev. Lett.
**2004**, 93, 215301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bowick, M.J.; Giomi, L. Two-dimensional matter: Order, curvature and defects. Adv. Phys.
**2009**, 58, 449–563. [Google Scholar] [CrossRef] - Turner, A.M.; Vitelli, V.; Nelson, D.R. Vortices on curved surfaces. Rev. Mod. Phys.
**2010**, 82, 1301. [Google Scholar] [CrossRef] [Green Version] - Kralj-Iglič, V.; Iglič, A.; Hägerstrand, H.; Peterlin, P. Stable tubular microexovesicles of the erythrocyte membrane induced by dimeric amphiphiles. Phys. Rev. E
**2000**, 61, 4230. [Google Scholar] [CrossRef] [Green Version] - Penič, S.; Fošnarič, M.; Mesarec, L.; Iglič, A.; Kralj-Iglič, V. Active forces of myosin motors may control endovesiculation of red blood cells. Acta Chim. Slov.
**2020**, 67, 674–681. [Google Scholar] [CrossRef] - Kralj, S.; Rosso, R.; Virga, E.G. Curvature control of valence on nematic shells. Soft Matter
**2011**, 7, 670–683. [Google Scholar] [CrossRef] - Mesarec, L.; Kurioz, P.; Iglič, A.; Góźdź, W.; Kralj, S. Curvature-controlled topological defects. Crystals
**2017**, 7, 153. [Google Scholar] [CrossRef] [Green Version] - Jesenek, D.; Perutková, Š.; Góźdź, W.; Kralj-Iglič, V.; Iglič, A.; Kralj, S. Vesiculation of biological membrane driven by curvature induced frustrations in membrane orientational ordering. Int. J. Nanomed.
**2013**, 8, 677–687. [Google Scholar] [CrossRef] [Green Version] - Bowick, M.; Nelson, D.R.; Travesset, A. Curvature-induced defect unbinding in toroidal geometries. Phys. Rev. E
**2004**, 69, 041102. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mermin, N.D. The topological theory of defects in ordered media. Rev. Mod. Phys.
**1979**, 51, 591. [Google Scholar] [CrossRef] - Kurik, M.V.; Lavrentovich, O.D. Defects in liquid crystals: Homotopy theory and experimental studies. SvPhU
**1988**, 31, 196. [Google Scholar] [CrossRef] - Kamien, R.D. The geometry of soft materials: A primer. Rev. Mod. Phys.
**2002**, 74, 953. [Google Scholar] [CrossRef] [Green Version] - Helfrich, W. Elastic properties of lipid bilayers: Theory and possible experiments. Z. Naturforsch. C
**1973**, 28, 693–703. [Google Scholar] [CrossRef] [PubMed] - Helfrich, W.; Prost, J. Intrinsic bending force in anisotropic membranes made of chiral molecules. Phys. Rev. A
**1988**, 38, 3065. [Google Scholar] [CrossRef] [PubMed] - Stein, D.B.; De Canio, G.; Lauga, E.; Shelley, M.J.; Goldstein, R.E. Swirling Instability of the Microtubule Cytoskeleton. Phys. Rev. Lett.
**2021**, 126, 028103. [Google Scholar] [CrossRef] [PubMed] - Liu, C.; Elvati, P.; Majumder, S.; Wang, Y.; Liu, A.P.; Violi, A. Predicting the time of entry of nanoparticles in lipid membranes. ACS Nano
**2019**, 13, 10221–10232. [Google Scholar] [CrossRef] - Kikuchi, H.; Yokota, M.; Hisakado, Y.; Yang, H.; Kajiyama, T. Polymer-stabilized liquid crystal blue phases. Nat. Mater.
**2002**, 1, 64–68. [Google Scholar] [CrossRef] - Cordoyiannis, G.; Jampani, V.S.R.; Kralj, S.; Dhara, S.; Tzitzios, V.; Basina, G.; Nounesis, G.; Kutnjak, Z.; Tripathi, C.S.P.; Losada-Pérez, P. Different modulated structures of topological defects stabilized by adaptive targeting nanoparticles. Soft Matter
**2013**, 9, 3956–3964. [Google Scholar] [CrossRef] - Pires, D.; Fleury, J.-B.; Galerne, Y. Colloid particles in the interaction field of a disclination line in a nematic phase. Phys. Rev. Lett.
**2007**, 98, 247801. [Google Scholar] [CrossRef] [PubMed] - Gouy, M. Sur la constitution de la charge électrique à la surface d’un électrolyte. J. Phys. Theor. Appl.
**1910**, 9, 457–468. [Google Scholar] [CrossRef] [Green Version] - Chapman, D.L.L.I. A contribution to the theory of electrocapillarity. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1913**, 25, 475–481. [Google Scholar] [CrossRef] [Green Version] - Freise, V. Zur theorie der diffusen doppelschicht. Z. Elektrochem. Ber. Bunsenges. Phys. Chem.
**1952**, 56, 822–827. [Google Scholar] - Torrie, G.M.; Valleau, J.P. Electrical double layers. I. Monte Carlo study of a uniformly charged surface. J. Chem. Phys.
**1980**, 73, 5807–5816. [Google Scholar] [CrossRef] - Kenkel, S.W.; Macdonald, J.R. A lattice model for the electrical double layer using finite-length dipoles. J. Chem. Phys.
**1984**, 81, 3215–3221. [Google Scholar] [CrossRef] - Outhwaite, C.W.; Bhuiyan, L.B. A modified Poisson–Boltzmann equation in electric double layer theory for a primitive model electrolyte with size-asymmetric ions. J. Chem. Phys.
**1986**, 84, 3461–3471. [Google Scholar] [CrossRef] - McLaughlin, S. The electrostatic properties of membranes. Ann. Rev. Biophys. Biophys. Chem.
**1989**, 18, 113–136. [Google Scholar] [CrossRef] [PubMed] - Kralj-Iglič, V.; Iglič, A. A simple statistical mechanical approach to the free energy of the electric double layer including the excluded volume effect. J. Phys. II
**1996**, 6, 477–491. [Google Scholar] [CrossRef] [Green Version] - Bivas, I. Electrostatic and mechanical properties of a flat lipid bilayer containing ionic lipids: Possibility for formation of domains with different surface charges. Coll. Surf. A Physicochem. Eng. Asp.
**2006**, 282, 423–434. [Google Scholar] [CrossRef] - Gongadze, E.; Velikonja, A.; Perutkova, Š.; Kramar, P.; Maček-Lebar, A.; Kralj-Iglič, V.; Iglič, A. Ions and water molecules in an electrolyte solution in contact with charged and dipolar surfaces. Electrochim. Acta
**2014**, 126, 42–60. [Google Scholar] [CrossRef] - Stern, O. Zur theorie der elektrolytischen doppelschicht. ZEAPC
**1924**, 30, 508–516. [Google Scholar] - Bikerman, J.J. XXXIX. Structure and capacity of electrical double layer. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1942**, 33, 384–397. [Google Scholar] [CrossRef] - Wicke, E.; Eigen, M. Über den Einfluß des Raumbedarfs von Ionen in wäßriger Lösung auf ihre Verteilung in elektrischen Feld und ihre Aktivitätskoeffizienten. Z. Elektrochem. Ber. Bunsenges. Phys. Chem.
**1952**, 56, 551–561. [Google Scholar] - Eigen, M.; Wicke, E. The thermodynamics of electrolytes at higher concentration. J. Phys. Chem.
**1954**, 58, 702–714. [Google Scholar] [CrossRef] - Onsager, L. Electric moments of molecules in liquids. J. Am. Chem. Soc.
**1936**, 58, 1486–1493. [Google Scholar] [CrossRef] - Booth, F. The dielectric constant of water and the saturation effect. J. Chem. Phys.
**1951**, 19, 391–394. [Google Scholar] [CrossRef] - Outhwaite, C.W. A treatment of solvent effects in the potential theory of electrolyte solutions. Mol. Phys.
**1976**, 31, 1345–1357. [Google Scholar] [CrossRef] - Outhwaite, C.W. Towards a mean electrostatic potential treatment of an ion-dipole mixture or a dipolar system next to a plane wall. Mol. Phys.
**1983**, 48, 599–614. [Google Scholar] [CrossRef] - Iglič, A.; Gongadze, E.; Bohinc, K. Excluded volume effect and orientational ordering near charged surface in solution of ions and Langevin dipoles. Bioelectrochemistry
**2010**, 79, 223–227. [Google Scholar] [CrossRef] - Bazant, M.Z.; Kilic, M.S.; Storey, B.D.; Ajdari, A. Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions. Adv. Colloid Interface Sci.
**2009**, 152, 48–88. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gongadze, E.; Iglič, A. Asymmetric size of ions and orientational ordering of water dipoles in electric double layer model-an analytical mean-field approach. Electrochim. Acta
**2015**, 178, 541–545. [Google Scholar] [CrossRef] - Gongadze, E.; Mesarec, L.; Kralj-Iglic, V.; Iglic, A. Asymmetric finite size of ions and orientational ordering of water in electric double layer theory within lattice model. Mini Rev. Med. Chem.
**2018**, 18, 1559–1566. [Google Scholar] [CrossRef] [Green Version] - Iglič, A.; Gongadze, E.; Kralj-Iglič, V. Differential Capacitance of Electric Double Layer–Influence of Asymmetric Size of Ions, Thickness of Stern Layer and Orientational Ordering of Water Dipoles. Acta Chim. Slov.
**2019**, 66, 534–541. [Google Scholar] [CrossRef] [PubMed] - Helmholtz, H.V. Ueber einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern, mit Anwendung auf die thierisch-elektrischen Versuche (Schluss.). Ann. Phys.
**1853**, 165, 353–377. [Google Scholar] [CrossRef] [Green Version] - Helmholtz, H.V. Studien über electrische Grenzschichten. Ann. Phys.
**1879**, 243, 337–382. [Google Scholar] [CrossRef] [Green Version] - Torrie, G.M.; Valleau, J.P. Electrical double layers. 4. Limitations of the Gouy-Chapman theory. J. Phys. Chem.
**1982**, 86, 3251–3257. [Google Scholar] [CrossRef] - Nielaba, P.; Forstmann, F. Packing of ions near an electrolyte-electrode interface in the hnc/lmsa approximation to the rpm model. Chem. Phys. Lett.
**1985**, 117, 46–48. [Google Scholar] [CrossRef] - Plischke, M.; Henderson, D. Pair correlation functions and density profiles in the primitive model of the electric double layer. J. Chem. Phys.
**1988**, 88, 2712–2718. [Google Scholar] [CrossRef] - Kornyshev, A.A. Double-layer in ionic liquids: Paradigm change? J. Phys. Chem. B
**2007**, 111, 5545–5557. [Google Scholar] [CrossRef] - Mier-y-Teran, L.; Suh, S.H.; White, H.S.; Davis, H.T. A nonlocal free-energy density-functional approximation for the electrical double layer. J. Chem. Phys.
**1990**, 92, 5087–5098. [Google Scholar] [CrossRef] [Green Version] - Strating, P.; Wiegel, F.W. Effects of excluded volume on the electrolyte distribution around a charged sphere. J. Phys. A Math. Gen.
**1993**, 26, 3383. [Google Scholar] [CrossRef] - Lee, J.W.; Nilson, R.H.; Templeton, J.A.; Griffiths, S.K.; Kung, A.; Wong, B.M. Comparison of molecular dynamics with classical density functional and Poisson–Boltzmann theories of the electric double layer in nanochannels. J. Chem. Theory Comput.
**2012**, 8, 2012–2022. [Google Scholar] [CrossRef] [PubMed] - Quiroga, M.A.; Xue, K.-H.; Nguyen, T.-K.; Tułodziecki, M.; Huang, H.; Franco, A.A. A multiscale model of electrochemical double layers in energy conversion and storage devices. J. Electrochem. Soc.
**2014**, 161, E3302. [Google Scholar] [CrossRef] - Bandopadhyay, A.; Shaik, V.A.; Chakraborty, S. Effects of finite ionic size and solvent polarization on the dynamics of electrolytes probed through harmonic disturbances. Phys. Rev. E
**2015**, 91, 042307. [Google Scholar] [CrossRef] [PubMed] - Lian, C.; Liu, K.; Van Aken, K.L.; Gogotsi, Y.; Wesolowski, D.J.; Liu, H.L.; Jiang, D.E.; Wu, J.Z. Enhancing the capacitive performance of electric double-layer capacitors with ionic liquid mixtures. ACS Energy Lett.
**2016**, 1, 21–26. [Google Scholar] [CrossRef] - Drab, M.; Kralj-Iglič, V. Diffuse electric double layer in planar nanostructures due to Fermi-Dirac statistics. Electrochim. Acta
**2016**, 204, 154–159. [Google Scholar] [CrossRef] - Budkov, Y.A.; Kolesnikov, A.L.; Goodwin, Z.A.H.; Kiselev, M.G.; Kornyshev, A.A. Theory of electrosorption of water from ionic liquids. Electrochim. Acta
**2018**, 284, 346–354. [Google Scholar] [CrossRef] - Budkov, Y.A. Nonlocal statistical field theory of dipolar particles in electrolyte solutions. J. Phys. Condens. Matter
**2018**, 30, 344001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dubtsov, A.V.; Pasechnik, S.V.; Shmeliova, D.V.; Saidgaziev, A.S.; Gongadze, E.; Iglič, A.; Kralj, S. Liquid crystalline droplets in aqueous environments: Electrostatic effects. Soft Matter
**2018**, 14, 9619–9630. [Google Scholar] [CrossRef] [PubMed] - Gavish, N.; Elad, D.; Yochelis, A. From solvent-free to dilute electrolytes: Essential components for a continuum theory. J. Phys. Chem. Lett.
**2018**, 9, 36–42. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gavish, N. Poisson–Nernst–Planck equations with steric effects—Non-convexity and multiple stationary solutions. Phys. D Nonlinear Phenom.
**2018**, 368, 50–65. [Google Scholar] [CrossRef] [Green Version] - Kruczek, J.; Chiu, S.-W.; Varma, S.; Jakobsson, E.; Pandit, S.A. Interactions of monovalent and divalent cations at palmitoyl-oleoyl-phosphatidylcholine interface. Langmuir
**2019**, 35, 10522–10532. [Google Scholar] [CrossRef] [PubMed] - Liu, X.; Tian, R.; Ding, W.; Wu, L.; Li, H. Role of ionic polarization and dielectric decrement in the estimation of surface potential of clay particles. Eur. J. Soil Sci.
**2019**, 70, 1073–1081. [Google Scholar] [CrossRef] - May, S. Differential capacitance of the electric double layer: Mean-field modeling approaches. Curr. Opin. Electrochem.
**2019**, 13, 125–131. [Google Scholar] [CrossRef] - Cruz, C.; Kondrat, S.; Lomba, E.; Ciach, A. Effect of proximity to ionic liquid-solvent demixing on electrical double layers. J. Mol. Liq.
**2019**, 294, 111368. [Google Scholar] [CrossRef] - Guardiani, C.; Gibby, W.; Barabash, M.; Luchinsky, D.; Khovanov, I.; McClintock, P. Prehistory probability distribution of ionic transitions through a graphene nanopore. In Proceedings of the 25th International Conference on Noise and Fluctuations—ICNF 2019, Neuchatel, Switzerland, 18–21 June 2019. [Google Scholar]
- Khademi, M.; Barz, D.P.J. Structure of the electrical double layer revisited: Electrode capacitance in aqueous solutions. Langmuir
**2020**, 36, 4250–4260. [Google Scholar] [CrossRef] [PubMed] - Kjellander, R.; Marčelja, S. Interaction of charged surfaces in electrolyte solutions. Chem. Phys. Lett.
**1986**, 127, 402–407. [Google Scholar] [CrossRef] - Evans, D.F.; Wennerström, H. The Colloidal Domain: Where Physics, Chemistry, Biology, and Technology Meet, 2nd ed.; Wiley-VCH: New York, NY, USA, 1999. [Google Scholar]
- Butt, H.J.; Graf, K.; Kappl, M. Physics and Chemistry of Interfaces; Wiley-VCH: Weinheim, Germany, 2003. [Google Scholar]
- Bohinc, K.; Iglič, A.; May, S. Interaction between macroions mediated by divalent rod-like ions. EPL (Europhys. Lett.)
**2004**, 68, 494. [Google Scholar] [CrossRef] [Green Version] - Urbanija, J.; Bohinc, K.; Bellen, A.; Maset, S.; Iglič, A.; Kralj-Iglič, V.; Sunil Kumar, P.B. Attraction between negatively charged surfaces mediated by spherical counterions with quadrupolar charge distribution. J. Chem. Phys.
**2008**, 129, 09B609. [Google Scholar] [CrossRef] [Green Version] - Perutková, Š.; Frank, M.; Bohinc, K.; Bobojevič, G.; Zelko, J.; Rozman, B.; Kralj-Iglič, V.; Iglič, A. Interaction between equally charged membrane surfaces mediated by positively and negatively charged macro-ions. J. Membr. Biol.
**2010**, 236, 43–53. [Google Scholar] [CrossRef] [PubMed] - Israelachvili, J.N. Intermolecular and Surface Forces, 3rd ed.; Academic Press: London, UK, 2011. [Google Scholar]
- Velikonja, A.; Santhosh, P.B.; Gongadze, E.; Kulkarni, M.; Eleršič, K.; Perutkova, Š.; Kralj-Iglič, V.; Ulrih, N.P.; Iglič, A. Interaction between dipolar lipid headgroups and charged nanoparticles mediated by water dipoles and ions. Int. J. Mol. Sci.
**2013**, 14, 15312–15329. [Google Scholar] [CrossRef] - Prasanna Misra, R.; Das, S.; Mitra, S.K. Electric double layer force between charged surfaces: Effect of solvent polarization. J. Chem. Phys.
**2013**, 138, 114703. [Google Scholar] [CrossRef] [PubMed] - Gimsa, J.; Wysotzki, P.; Perutkova, Š.; Weihe, T.; Elter, P.; Marszałek, P.; Kralj-Iglič, V.; Müller, T.; Iglič, A. Spermidine-induced attraction of like-charged surfaces is correlated with the pH-dependent spermidine charge: Force spectroscopy characterization. Langmuir
**2018**, 34, 2725–2733. [Google Scholar] [CrossRef] - Goršak, T.; Drab, M.; Križaj, D.; Jeran, M.; Genova, J.; Kralj, S.; Lisjak, D.; Kralj-Iglič, V.; Iglič, A.; Makovec, D. Magneto-mechanical actuation of barium-hexaferrite nanoplatelets for the disruption of phospholipid membranes. J. Colloid Interface Sci.
**2020**, 579, 508–519. [Google Scholar] [CrossRef] [PubMed] - Gongadze, E.; van Rienen, U.; Kralj-Iglič, V.; Iglič, A. Langevin Poisson-Boltzmann equation: Point-like ions and water dipoles near a charged surface. Gen. Physiol. Biophys.
**2011**, 30, 130. [Google Scholar] [CrossRef] - Drab, M.; Gongadze, E.; Kralj-Iglič, V.; Iglič, A. Electric double layer and orientational ordering of water dipoles in narrow channels within a modified Langevin Poisson-Boltzmann model. Entropy
**2020**, 22, 1054. [Google Scholar] [CrossRef] - Abrashkin, A.; Andelman, D.; Orland, H. Dipolar Poisson-Boltzmann equation: Ions and dipoles close to charge interfaces. Phys. Rev. Lett.
**2007**, 99, 077801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Teschke, O.; Ceotto, G.; de Souza, E.F. Interfacial aqueous solutions dielectric constant measurements using atomic force microscopy. Chem. Phys. Lett.
**2000**, 326, 328. [Google Scholar] [CrossRef] - De Souza, E.F.; Ceotto, G.; Teschke, O. Dielectric constant measurements of interfacial aqueous solutions using atomic force microscopy. J. Mol. Catal. A Chem.
**2001**, 167, 235. [Google Scholar] [CrossRef]

**Figure 1.**Typical vesicle shapes calculated by MC simulations [11] for the two-component membrane composed of highly curved isotropic flexible nanodomains (marked in red) and the nanodomains with zero intrinsic curvature (marked in blue) are presented in panel (

**a**). Membrane nanodomains with high intrinsic curvature (red) are accumulated in undulated membrane protrusions. Panels (

**b**,

**c**) show the orientational ordering profiles in the necks of undulated membrane buds/protrusions. Topological antidefects are accumulated in the necks. Consequently, the shape with three prominent thin necks (

**b**) is transformed into two distinct closed membrane shapes (

**c**) as a result of the rupture of one neck. The positions of antidefects in panels (

**b**,

**c**) are marked by small squares. Orientational ordering profiles with the superimposed nematic director fields in the vicinity of topological antidefects are magnified. The figure also shows an example of the vesicle budding (panel (

**d**)) and the formation of the detached daughter vesicle (panel (

**e**)) driven by the formation of topological antidefects in the neck prior to the fission process. The shape and orientational ordering profile were calculated as described in [22]. Panels (

**a**–

**c**) are adapted from [11,22].

**Figure 2.**A schematic figure of the electrolyte solution between two charged surfaces at the distance $H$. The surface charge densities of both surfaces are negative, ${\sigma}_{1}<0$ and ${\sigma}_{2}<0$.

**Figure 3.**Distribution of the electric potential and the magnitude of the electric field strength along the line passing through the midpoint between the surface of both spheres perpendicular to the line connecting the centres of the sphere (see Figure 4). The calculation were performed by solving the modified LPB eqution (Equations (1)–(3)) for two spheres with equal surface charge densities $\sigma =-$0.25 As/m

^{2}. The radii of the spheres are ${R}_{p}=R=$ 1 nm (

**a**), ${R}_{p}=R=$ 2 nm (

**b**), and ${R}_{p}=$ 2 nm and $R=$ 1 nm (

**c**) (see also Figure 4). The values of other parameters are $H=1$ nm (see Figure 4), ${p}_{0}=$ 3.1 Debye, ${n}_{0}/{N}_{A}=$ 0.15 mol/L, ${n}_{w}/{N}_{A}=$ 55 mol/L, and $T=$ 298 K.

**Figure 4.**Schematic figure of two electrically charged spheres (vesicles) at the distance $H$ with the radii ${R}_{p}$ and $R$.

**Figure 5.**The calculated osmotic pressure between two negatively charged spherical vesicles of the same radius (${R}_{p}=R=$ 10 nm) and same surface charge density as a function of the distance between their surfaces (H, see Figure 4). The values of the model parameters are ${n}_{0}/{N}_{A}=$ 0.15 mol/L, ${n}_{w}/{N}_{A}=$ 55 mol/L, and ${p}_{0}=$ 3.1 Debye.

**Figure 6.**The calculated (net) osmotic pressure between two negatively charged neighbouring spheres/vesicles embedded in a electrolyte solution as a function of the radius of the smaller sphere/vesicle ($R$). The radius of the larger sphere/vesicle ${R}_{p}=$ 10 nm (see also Figure 4). The distance between the spheres/vesicles is $H=$ 1 nm for all values of $R$. Both spheres have equal surface charge densities ($\sigma =$ −0.25 As/m

^{2}). The value of the osmotic pressure for $R=$ 10 nm corresponds to the case where both spheres/vesicles have the same radius (Figure 5). The values of other model parameters are the same as in Figure 5. Note that Equation (8) has limited validity for small values of $R$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gongadze, E.; Mesarec, L.; Kralj, S.; Kralj-Iglič, V.; Iglič, A.
On the Role of Electrostatic Repulsion in Topological Defect-Driven Membrane Fission. *Membranes* **2021**, *11*, 812.
https://doi.org/10.3390/membranes11110812

**AMA Style**

Gongadze E, Mesarec L, Kralj S, Kralj-Iglič V, Iglič A.
On the Role of Electrostatic Repulsion in Topological Defect-Driven Membrane Fission. *Membranes*. 2021; 11(11):812.
https://doi.org/10.3390/membranes11110812

**Chicago/Turabian Style**

Gongadze, Ekaterina, Luka Mesarec, Samo Kralj, Veronika Kralj-Iglič, and Aleš Iglič.
2021. "On the Role of Electrostatic Repulsion in Topological Defect-Driven Membrane Fission" *Membranes* 11, no. 11: 812.
https://doi.org/10.3390/membranes11110812