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Curvature instability of membranes near rigid inclusions

S. Alex Rautu*

  • Simons Centre for the Study of Living Machines, National Centre for Biological Sciences (TIFR), Bellary Road, Bangalore 560065, India

  • *stefanar@ncbs.res.in
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Phys. Rev. E 97, 022414 – Published 27 February, 2018

DOI: https://doi.org/10.1103/PhysRevE.97.022414

Abstract

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, exact shape solutions are found when the membrane is curvature unstable, manifested by a negative rigidity. Near the instability point (i.e., at vanishing rigidity), the membrane is stabilized by the quartic term, giving rise to a different length scale and scale exponents for the shape and its energy profile than those found for stable membranes. The contact angle induced by an applied force at the inclusion provides a method to experimentally determine the quartic curvature modulus.

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References (49)

  1. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell, 5th ed. (Garland Science, New York, 2008).
  2. S. J. Singer and G. L. Nicolson, Science 175, 720 (1972).
  3. D. M. Engelman, Nature 438, 578 (2005).
  4. W. Helfrich, Z. Naturforsch. C: J. Bio. Sci. 28, 693 (1973).
  5. P. B. Canham, J. Theor. Biol. 26, 61 (1970).
  6. E. A. Evans, Biophys. J. 14, 923 (1974).
  7. M. Deserno, Chem. Phys. Lipids 185, 11 (2015).
  8. R. Phillips, T. Ursell, P. Wiggins, and P. Sens, Nature 459, 379 (2009).
  9. N. Dan, P. Pincus, and S. A. Safran, Langmuir 9, 2768 (1993).
  10. J.-B. Fournier, Eur. Phys. J. B 11, 261 (1999).
  11. T. R. Weikl, M. M. Kozlov, and W. Helfrich, Phys. Rev. E 57, 6988 (1998).
  12. K. S. Kim, J. Neu, and G. Oster, Biophys. J. 75, 2274 (1998).
  13. M. Goulian, R. Bruinsma, and P. Pincus, Europhys. Lett. 22, 145 (1993).
  14. P. Wiggins and R. Phillips, Biophys. J. 88, 880 (2005).
  15. S. A. Rautu, G. Rowlands, and M. S. Turner, Phys. Rev. Lett. 114, 098101 (2015).
  16. L. Miao, U. Seifert, M. Wortis, and H.-G. Döbereiner, Phys. Rev. E 49, 5389 (1994).
  17. U. Seifert, Adv. Phys. 46, 13 (1997).
  18. Ou-Yang Zhong-can and W. Helfrich, Phys. Rev. Lett. 59, 2486 (1987).
  19. Ou-Yang Zhong-can, Phys. Rev. A 41, 4517 (1990).
  20. Ou-Yang Zhong-can, Phys. Rev. E 47, 747 (1993).
  21. Zhang Shao-guang and Ou-Yang Zhong-can, Phys. Rev. E 53, 4206 (1996).
  22. O.-Y. Zhong-Can, L. Ji-Xing, and X. Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (World Scientific, Singapore, 1999).
  23. W.-M. Zheng and J. Liu, Phys. Rev. E 48, 2856 (1993).
  24. Q.-H. Liu, Z. Haijun, J.-X. Liu, and O.-Y. Zhong-Can, Phys. Rev. E 60, 3227 (1999).
  25. H. Naito, M. Okuda, and Ou-Yang Zhong-can, Phys. Rev. E 48, 2304 (1993).
  26. H. Jian-Guo and Ou-Yang Zhong-Can, Phys. Rev. E 47, 461 (1993).
  27. I. Mladenov, Eur. Phys. J. B 29, 327 (2002).
  28. H. Naito, M. Okuda, and Ou-Yang Zhong-can, Phys. Rev. Lett. 74, 4345 (1995).
  29. H. J. Deuling and W. Helfrich, Biophys. J. 16, 861 (1976).
  30. H. W. G. Lim, M. Wortis, and R. Mukhopadhyay, Proc. Natl. Acad. Sci. USA 99, 16766 (2002).
  31. V. Kralj-Iglič, S. Svetina, and B. Žekš, Eur. Biophys. J. 22, 97 (1993).
  32. V. Heinrich, S. Svetina, and B. Žekš, Phys. Rev. E 48, 3112 (1993).
  33. Yan Jie, Liu Quanhui, Liu Jixing, and Ou-Yang Zhong-can, Phys. Rev. E 58, 4730 (1998).
  34. K. S. Kim, J. Neu, and G. Oster, Phys. Rev. E 61, 4281 (2000).
  35. C. A. Haselwandter and R. Phillips, Europhys. Lett. 101, 68002 (2013).
  36. M. Deserno, Phys. Rev. E 69, 031903 (2004).
  37. M. M. Müller, M. Deserno, and J. Guven, Phys. Rev. E 76, 011921 (2007).
  38. M. Hu, P. Diggins, and M. Deserno, J. Chem. Phys. 138, 214110 (2013).
  39. S. Leibler and D. Andelman, J. Phys. Fr. 48, 2013 (1987).
  40. J.-B. Fournier and P. Galatola, Europhys. Lett. 39, 225 (1997).
  41. R. Goetz and W. Helfrich, J. Phys. II 6, 215 (1996).
  42. R. Capovilla, J. Guven, and J. A. Santiago, J. Phys. A Math. Gen. 36, 6281 (2003).
  43. Similar solutions have been found by means of an arc-length parametrization of the tangent angle ψ(s) [36, 37, 38], which allows for overhangs. By integrating over sinψ and cosψ, we attain the vertical and horizontal profiles of the membrane, respectively. However, one needs to numerically resolve for these to find the shape in terms of more relevant variables, e.g., the distance from the inclusion.
  44. For an axisymmetrical rigid inclusion [15], Eq. (1) can be written as 2π∫dr[rf(r)], with f defined in Eq. (2). The prefactor r yields an extra term in the Euler-Lagrange equation, i.e., g(r)≡r−1[−∂f/∂u′+ddr∂f/∂u″], where the remaining terms are identical to those found in the quasi-one-dimensional case. As g(r) becomes negligibly small and asymptotically vanishes far from the inclusion, the leading terms yield a shape equivalent and asymptotically exact to Eq. (7). Such axisymmetrical solutions have been studied in [49] using asymptotic methods.
  45. For the inner membrane between two inclusions, this does not hold, and one has to integrate first, to attain u(v), and then apply the corresponding boundary conditions.
  46. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  47. For ξ≫λ̃, the small gradient dudx∼u2/3 that yields the algebraic solution x(u)∼u1/3. On the other hand, we have dudx∼u for λ̃≫ξ, which leads to x(u)∼logu.
  48. B. J. G. E. Pieters, M. B. van Eldijk, R. J. M. Nolte, and J. Mecinović, Chem. Soc. Rev. 45, 24 (2016).
  49. P. Biscari and G. Napoli, Biomech. Model. Mechanobiol. 6, 297 (2007).

Outline

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Phys. Rev. E 97, 022414– Published 27 February, 2018
  • Received 19 January 2017
  • Revised 24 August 2017
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