Abstract
In aqueous solution, lipid molecules spontaneously assemble into macroscopic bilayer membranes, which have highly interesting mechanical properties. In this chapter, we first discuss some basic aspects of this self-assembly process. In the second part, we then revisit and slightly expand a well-known continuum-level theory that describes the elastic properties pertaining to membrane geometry and lipid tilt. We then illustrate in part three several conceptually different strategies for how one of the emerging elastic parameters—the bending modulus—can be obtained in computer simulations.
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Notes
- 1.
Here we assume that a flat membrane is untilted—which is true for fluid phases, but not necessarily so for membranes in the gel phase.
- 2.
During the workshop Jemal Guven pointed out that the additional required factor \(\cos \theta \) is the equivalent of what in a \(3+1\) foliation treatment of general relativity is called the “Lapse function” (Wheeler 1964).
- 3.
Notice that if the monolayer leaflet were not fluid, a deformation that starts from a flat leaflet and ends up with one that has a non-vanishing Gaussian curvature cannot be isometric by virtue of the Theorema Egregium. Hence, this approach of writing the elastic energy by looking at the stretching away from a pivotal plane relies by construction on fluidity.
- 4.
This is merely a consequence of the fact that the length \(\ell \), which pits curvature against tilt, is microscopic. On scales larger than \(\ell \), tilt therefore only enters as a minor correction to the overall bending physics of then problem.
- 5.
Since \(T^2=T_p^2+T_q^2\), the tilt modulus along the edge (the q-direction) is just the prefactor of \(T^2\) in Eq. (124).
- 6.
In simulations that can be achieved relatively easily by suitable boundary conditions.
- 7.
Recall that we must measure the undulation spectrum \(\langle |\tilde{h}_{\varvec{q}}|^2\rangle \) over a sufficiently wide q-range to plausibly fit a spectrum, and that this spectrum decays rapidly with q.
- 8.
Hu et al. (2013) discuss buckle fluctuations in a little bit more detail. They conclude that systematic fluctuation corrections exist, but that they are small for \(f_x\). They are not necessarily small for the force \(f_y\) acting along the buckle’s ridges, though.
- 9.
One efficient construction method proceeds via the analytical expressions for position and angle of a buckle—Eqs. (139) and (140)—for mapping a flat membrane (leaflet-wise) into a buckled one. If a lipid’s center of mass in the flat configuration has coordinates \((x_0,y_0,z_0)\), map it to \((x(x_0),y_0,z(x_0))\) and rotate it around the y-axis by the angle \(\psi (x_0)\).
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Acknowledgements
Many people have contributed in numerous ways to the results presented in this chapter. We would especially like to acknowledge Luca Deseri, Patrick Diggins IV, Jemal Guven, Mingyang Hu, Zach McDargh, and Pablo Vázquez-Montejo. MD would also like to thank David Steigmann for putting together this exciting summer school. Financial support from the National Science Foundation via the grants CMMI-0941690 and CHE-1464926 is also gratefully acknowledged.
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Terzi, M.M., Deserno, M. (2018). Lipid Membranes: From Self-assembly to Elasticity. In: Steigmann, D. (eds) The Role of Mechanics in the Study of Lipid Bilayers. CISM International Centre for Mechanical Sciences, vol 577. Springer, Cham. https://doi.org/10.1007/978-3-319-56348-0_3
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