A computational model for capturing the distinct in- and out-of-plane response of lipid membranes

Abstract

A computational framework was developed to capture the combined fluid- and solid-like behavior of lipid membranes in a unified manner. Specifically, the in-plane diffusion of lipid molecules and the associated evolution of membrane tension were explicitly taken into account in the model. In addition, the out-of-plane movement induced bending and shearing of membrane, along with its thermal undulations caused by bombardment of medium molecules, were also considered. The capability and validity of this approach were demonstrated by simulating the enforced deformation and shape fluctuations of a lipid vesicle under a variety of testing conditions as well as their comparison with corresponding theoretical predictions. Our model could serve a useful platform for investigating processes such as cell spreading and division where morphology evolution of the membrane and transport of lipids/transmembrane proteins are known to play key roles.

Graphic Abstract

In this paper, we presented a novel computational framework to capture the combined fluid- and solid-like response of lipid membranes and then used it to investigate the enforced deformation and spontaneous shape fluctuation of lipid vesicles under a variety of experimental conditions. Specifically, the in-plane diffusion of lipid molecules and the associated evolution of membrane tension were explicitly taken into account in the model. In addition, the out-of-plane movement induced bending and shearing of membrane, along with its thermal undulations caused by bombardment of medium molecules, were also considered.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

References

  1. 1.

    Singleton, P.: Bacteria in Biology. Biotechnology and Medicine. Wiley, New York (1999)

    Google Scholar 

  2. 2.

    Goni, F.M.: The basic structure and dynamics of cell membranes: an update of the Singer-Nicolson model. Biochim. Biophys. Acta 1838(6), 1467–1476 (2014)

    Article  Google Scholar 

  3. 3.

    Siegel, R.A.: Random walks in biology. J. Control. Release 32(2), 201–202 (1994)

    Article  Google Scholar 

  4. 4.

    Divecha, N., Irvine, R.F.: Phospholipid signaling. Cell 80(2), 269–278 (1995)

    Article  Google Scholar 

  5. 5.

    Derek, M.: Elastic curvature constants of lipid monolayers and bilayers. Chem. Phys. Lipids 144(2), 146–159 (2006)

    Article  Google Scholar 

  6. 6.

    Hammad, A.F., Shelli, L.F., Rumiana, D.: Bending rigidity of charged lipid bilayer membranes. Soft Matter 15(29), 6006–6013 (2019)

    Article  Google Scholar 

  7. 7.

    Jad, E., Hefez, R., Alia, J., et al.: On calculating the bending modulus of lipid bilayer membranes from buckling simulations. J. Phys. Chem. B 124(29), 6299–6311 (2020)

    Article  Google Scholar 

  8. 8.

    Tasso, I.V., Gustavo, C.: A finite element method for viscous membranes. Comput. Methods Appl. Mech. Eng. 255, 226–237 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Lin, M., William, S.: Viscous regularization and radaptive remeshing for finite element analysis of lipid membrane mechanics. J. Comput. Phys. 227(11), 5816–5835 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Roger, A., Thang, X., Kranthi, K., et al.: Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics. J. Comput. Phys. 227(11), 5816–5835 (2008)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Feng, F., William, S.: Finite element modeling of lipid bilayer membranes. J. Comput. Phys. 220, 394–408 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Singer, S.J., Nicolson, G.L.: The fluid mosaic model of the structure of cell membranes. Science 175, 720–731 (1972)

    Article  Google Scholar 

  13. 13.

    Saffman, P.G., Delbrueck, M.: Brownian motion in biological membranes. Proc. Natl. Acad. Sci. USA 72(8), 3111–3113 (1975)

    Article  Google Scholar 

  14. 14.

    Saffman, P.G.: Brownian motion in thin sheets of viscous fluid. J. Fluid Mech. 73(4), 593–602 (1976)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Emmanue, B., Adi, P., Gamze, C., et al.: Membrane fission is promoted by insertion of amphipathic helices and is restricted by crescent BAR domains. Cell 149(1), 124–136 (2012)

    Article  Google Scholar 

  16. 16.

    Graber, Z.T., Shi, T., Baumgart, T.: Cations induce shape remodeling of negatively charged phospholipid membranes. Phys. Chem. Chem. Phys. 19(23), 15285–15295 (2017)

    Article  Google Scholar 

  17. 17.

    Bondar, A., Kellert, S.: Lipid membranes and reactions at lipid interfaces: theory, experiments, and applications. J. Membr. Biol. 251(3), 295–298 (2018)

    Article  Google Scholar 

  18. 18.

    Jakob, A., Fuller, A., Kathleen, W.: A tethered bilayer lipid membrane that mimics microbial membranes. Phys. Chem. Chem. Phys. 20(18), 12958–12969 (2018)

    Article  Google Scholar 

  19. 19.

    Balleza, D.: Mechanical properties of lipid bilayers and regulation of mechanosensitive function. Channels 6(4), 220–233 (2012)

    Article  Google Scholar 

  20. 20.

    Amelie, B., Romain, G., Catherine, L.J., et al.: Interdigitation between triglycerides and lipids modulates surface properties of lipid droplets. Biophys. J. 112(7), 1417–1430 (2017)

    Article  Google Scholar 

  21. 21.

    Boal, D.: Mechanics of the Cell. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  22. 22.

    Kinnere, K., Zachary, P., Greg, M.A., et al.: Mechanism of shape determination in motile cells. Nature 453(7194), 475–480 (2008)

    Article  Google Scholar 

  23. 23.

    Helfrich, W., Servuss, M.: Undulations, steric interaction and cohesion of fluid membranes. Nuovo Cimento D 3(1), 137–151 (1984)

    Article  Google Scholar 

  24. 24.

    Janke, W., Kleinert, H.: Fluctuation pressure of membrane between walls. Phys. Lett. A 117(7), 353–357 (1986)

    Article  Google Scholar 

  25. 25.

    Zarda, P.R., Chien, S., Skalak, R.: Elastic deformations of red blood cells. J. Biomech. 10(4), 211–221 (1977)

    MATH  Article  Google Scholar 

  26. 26.

    Fang, C., Zheng, F., Yao, J.: A model for bridging microtubule dynamics with nuclear envelope shape evolution during closed mitosis. J. Mech. Phys. Solids 144, 104116 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Fang, C., Hui, T.H., Wei, X.: A combined experimental and theoretical investigation on cellular blebbing. Sci. Rep. 7(1), 16666 (2017)

    Article  Google Scholar 

  28. 28.

    Pozrikidis, C.: Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  29. 29.

    Lu, C., Cao, Y., Mumford, D.: Surface evolution under curvature flows. J. Vis. Commun. Image Rep. 13(12), 65–81 (2002)

    Article  Google Scholar 

  30. 30.

    Desbrun, M., Meyer, M, Schroder, P., et al.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, vol. 1999, pp. 317–324 (1999)

  31. 31.

    Wirtz, D.: Particle-tracking microrheology of living cells: principles and applications. Ann. Rev. Biophys. 38(1), 301–326 (2009)

    Article  Google Scholar 

  32. 32.

    Reinhard, L.: The conformation of membranes. Nature 349(6309), 475–481 (1991)

    Article  Google Scholar 

  33. 33.

    Helfrich, W., Servuss, R.M.: Undulations, steric interaction and cohesion of fluid membranes. Il Nuovo Cimento D 3(1), 137–151 (1984)

    Article  Google Scholar 

  34. 34.

    Kleinert, H.: Fluctuation pressure of membrane between walls. Phys. Lett. Sect. A 257(5–6), 269–274 (1999)

    Article  Google Scholar 

  35. 35.

    Hu, B., Shenoy, V.B., Lin, Y.: Buckling and enforced stretching of bio-filaments. J. Mech. Phys. Solids 60(11), 1941–1951 (2012)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Lin, Y., Wei, X., Qian, J., et al.: A combined finite element-Langevin dynamics (FEM- LD) approach for analyzing the mechanical response of bio-polymer networks. J. Mech. Phys. Solids 62(1), 2–18 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Li, L., Wang, X., Shao, Y., et al.: Entropic pressure between fluctuating membranes in multilayer systems. Sci. China Phys. Mech. Astron. 61(12), 128711 (2018)

    Article  Google Scholar 

  38. 38.

    Li, L., Song, F.: Entropic force between biomembranes. Acta Mech. Sin. 32(5), 970–975 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Michale, X., Bensimon, D., Fourcade, B.: Fluctuating vesicles of nonspherical topology. Phys. Rev. Lett. 72(1), 168–171 (1994)

    Article  Google Scholar 

  40. 40.

    Morse, D.C., Milner, S.T.: Statistical mechanics of closed fluid membranes. Phys. Rev. E 52(6), 5918–5945 (1995)

    Article  Google Scholar 

  41. 41.

    May, E.R., Narang, A., Kopelevich, D.I.: Role of molecular tilt in thermal fluctuations of lipid membranes. Phys. Rev. E 76(2), 021913 (2007)

    Article  Google Scholar 

  42. 42.

    Yi, X., Gao, H.: Cell interaction with graphene microsheets: near-orthogonal cutting versus parallel attachment. Nanoscale 7(12), 5457–5467 (2015)

    Article  Google Scholar 

  43. 43.

    Yu, M., Xu, L., Tian, F., et al.: Rapid transport of deformation-tuned nanoparticles across biological hydrogels and cellular barriers. Nat. Commun. 9(1), 2607 (2018)

    Article  Google Scholar 

  44. 44.

    Yang, Y., Yang, X., Liang, L., et al.: Large-area graphene-nanomesh/carbon-nanotube hybrid membranes for ionic and molecular nanofiltration. Science 364(6445), 1057–1062 (2019)

    Article  Google Scholar 

  45. 45.

    Lin, Y.: Mechanics model for actin-based motility. Phys. Rev. E 79(2), 021916 (2009)

    Article  Google Scholar 

  46. 46.

    Yang, L., Gong, Z., Lin, Y., et al.: Disordered topography mediates filopodial extension and morphology of cells on stiff materials. Adv. Funct. Mater. 27(38), 1702689 (2017)

    Article  Google Scholar 

  47. 47.

    Li, L., Hu, J., Xu, G., et al.: Binding constant of cell adhesion receptors and substrate-immobilized ligands depends on the distribution of ligands. Phys. Rev. E 97(1), 1702689 (2018)

    Google Scholar 

  48. 48.

    Xu, G., Feng, X., Zhao, H., et al.: Theoretical study of the competition between cell-cell and cell-matrix adhesions. Phys. Rev. E 80(1), 011921 (2009)

    Article  Google Scholar 

  49. 49.

    Lin, Y., Shenoy, V.B., Bai, L.: A microscopic formulation for the actin-driven motion of Listeria in curved paths. Biophys. J. 99(4), 1043–1052 (2010)

    Article  Google Scholar 

  50. 50.

    Lin, Y.: A model of cell motility leading to biphasic dependence of transport speed on adhesive strength. J. Mech. Phys. Solids 58(4), 502–514 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Yi, X., Shi, X., Gao, H.: Cellular uptake of elastic nanoparticles. Phys. Rev. Lett. 107(9), 98101 (2011)

    Article  Google Scholar 

  52. 52.

    Zhu, Q., Zheng, F., Liu, A.P., et al.: Shape transformation of the nuclear envelope during closed mitosis. Biophys. J. 111(10), 2309–2316 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

This research was funded by the Research Grants Council of the Hong Kong Special Administration Region (Grants GRF/17257016 and GRF/17210618), and the National Natural Science Foundation of China (Grant 11872325).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yuan Lin.

Additional information

Executive Editor: Xi-Qiao Feng.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wen, P., Wei, X. & Lin, Y. A computational model for capturing the distinct in- and out-of-plane response of lipid membranes. Acta Mech. Sin. (2021). https://doi.org/10.1007/s10409-020-01033-3

Download citation

Keywords

  • Cell membrane
  • Thermal undulations
  • Lipid diffusion
  • Finite element method