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2018 | OriginalPaper | Chapter

The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws

Authors : Jemal Guven, Pablo Vázquez-Montejo

Published in: The Role of Mechanics in the Study of Lipid Bilayers

Publisher: Springer International Publishing

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Abstract

The behavior of a lipid membrane on mesoscopic scales is captured unusually accurately by its geometrical degrees of freedom. Indeed, the membrane geometry is, very often, a direct reflection of the physical state of the membrane. In this chapter we will examine the intimate connection between the geometry and the physics of fluid membranes from a number of points of view. We begin with a review of the description of the surface geometry in terms of the metric and the extrinsic curvature, examining surface deformations in terms of the behavior of these two tensors. The shape equation describing membrane equilibrium is derived and the qualitative behavior of solutions described. We next look at the conservation laws implied by the Euclidean invariance of the energy, describing the remarkably simple relationship between the stress distributed in the membrane and its geometry. This relationship is used to examine membrane-mediated interactions. We show how this geometrical framework can be extended to accommodate constraints—both global and local—as well as additional material degrees of freedom coupling to the geometry. The conservation laws are applied to examine the response of an axially symmetric membrane to localized external forces and to characterize topologically nontrivial states. We wrap up by looking at the conformal invariance of the symmetric two-dimensional bending energy, and examine some of its consequences.

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previous chapter Lipid Membranes: From Self-assembly to Elasticity

next chapter On the Computational Modeling of Lipid Bilayers Using Thin-Shell Theory

We abbreviate $\partial _a = \partial _a\mathbf {X} /\partial u^a$.

We are interested specifically in surface tensors and the scalars constructed out of them. Consider a surface reparametrization $(u^1,u^2) \rightarrow (\bar{u}^1(u^1,u^2),\bar{u}^2(u^1,u^2))$. Define $J^{\bar{a}}{}_b= \partial \bar{u}^a/\partial u^b$, with inverse $J_{\bar{a}}{}^{c}: J^{\bar{a}}{}_{c}J_{\bar{b}}{}^{c} = \delta ^{\bar{a}}{}_{\bar{b}}$. Tensor fields transform under reparametrization by matrix multiplication on each index with the Jacobian matrix of the reparametrization or its inverse. In particular, the metric transforms by $\bar{g}_{\bar{a}\bar{b}} = J_{\bar{a}}{}^{c} J_{\bar{b}}{}^{d} g_{cd}$. Note that the three Cartesian embedding functions $\mathbf {X}=(X^1,X^2,X^3)$ are each scalars under reparametrization: $\bar{X}^1(\bar{u}^1,\bar{u}^2) = X^1(u^1,u^2)$, etc.

Under reparametrization, $\sqrt{\bar{g}} = \mathrm {det} J^{-1} \sqrt{g}$.

It is simple to show that $V^a$ transforms like a vector under reparametrization.

It is straightforward to confirm that the two remaining symmetric quadratics, $C_1^2+C_2^2$ and $(C_1-C_2)^2$, can be expressed as linear combinations of $K^2$ and $\mathcal {K}_G$.

As we will show below, controlling area locally is equivalent, in equilibrium, to controlling it globally.

For simplicity we will suppose that not only the surface height but also its normal vector are fixed on the boundary.

The other half of helicoid is given by $h = c (\pi + \varphi )$.

Under a deformation $h(\mathbf {r}) \rightarrow h(\mathbf {r}) + \delta h(\mathbf {r}) $, fixed on the boundary, the change in area A (6), is given by

$$\begin{aligned} \delta A = \int d\mathbf {r}\, \nabla _0\cdot \mathbf {J} \, \delta h \qquad {(13)} \end{aligned}$$

where

$$\begin{aligned} \mathbf {J} = -\frac{ \nabla _0 h}{(1 +|\nabla _0 h|^2)^{1/2}} \,, \qquad {(14)} \end{aligned}$$

so that $\nabla _0\cdot \mathbf {J}= 0$ in equilibrium. One can evaluate $|\nabla _0h|^2 = p^2/r^2$, so that $\mathbf {J} = p(-\sin \theta ,\cos \theta ) /r (1+ p^2/r^2)^{1/2}$ and $\nabla _0\cdot \mathbf {J}= 0$.

Just as $\ln |\mathbf {r}-\mathbf {r}'|$ is proportional to the Green’s function for the Laplacian, $-|\mathbf {r}-\mathbf {r}'|^2 \ln |\mathbf {r}-\mathbf {r}'|$ is its counterpart for the bilaplacian.

The Green’s function of the Helmholtz operator is proportional to $K_0(|\mathbf {r}-\mathbf {r}'|)$.

$\nabla _a \Phi ^a =\partial _a (\sqrt{g} \Phi ^a)/\sqrt{g}$.

Note that the Ricci identify (4) implies $R_{abcd}=-R_{bacd}$; whereas its application to the metric tensor implies $R_{abcd}=-R_{abdc}$:

$$\begin{aligned} 0= [\nabla _a,\nabla _b] g_{cd} = R_{abcd}+ R_{abdc}\,. \qquad {(34)} \end{aligned}$$

These account for all the independent constraints on $R_{abcd}$ on a two-dimensional surface.

The identity $\mathrm {det}\, K^a{}_b = (K^2 - K_{ab} K^{ab})/2$ is true for the determinant of any two-dimensional symmetric matrix.

$V_\perp ^a= \epsilon ^{ab} V_b$ is orthogonal to $V^a$.

A later derivation accommodating the finite thickness of the membrane is presented in Lomholt and Miao (2006).

In this approach, the deformation vector $\delta \mathbf {X}$ is never disassembled into normal and tangential parts, so that its reassembly is never necessary.

If $\mathbf {t}=t^a\mathbf {e}_a$ is the unit tangent vector to the curve, $\mathbf {l}\cdot \mathbf {t}=0$ or $g_{ab} l^a t^b=0$ or $l_a t^a=0$.

The local parametrization is fixed.

On a surface with boundary, this identity yields the volume of the cone standing on the surface patch, with its apex located at the origin.

Intriguingly, the quadratic contribution to $T^B_{ij}$ is trace-free in this approximation, a property we would associate with scale invariance. Yet the area itself is clearly not scale invariant. The source of this peculiarity is that, in the quadratic approximation in gradients of h, the area is represented by a massless two-dimensional scalar field on the plane which is scale invariant if the plane is scaled, but not if h is. On the other hand, $T^B_{ji}$ is not trace-free but should not have been expected to be.

Unlike the prolate, this geometry is stable with respect to membrane slippage under the ring.

This is well known in the context of global constraints. In a symmetric closed fluid membrane subject to area and volume constraints, the identity $2\sigma A - 3 P V=0$ is a consequence of the scale invariance of the bending energy, Svetina and Žekž (1989).

$K_{ab}\rightarrow K_{ab} = -|\mathbf { X}|^{2} \left( K_{ab} - 2 \,(\mathbf { X}\cdot \mathbf { n}) g_{ab} / |\mathbf { X}|^{-2}\right) $.

Fixing the discocyte area at $4 \pi r_0^2$ determines $R_S = 1.089 \, r_0$.

The bilateral symmetry (not necessarily in the original XZ plane) is preserved if the point strays off this axis; however, the up-down symmetry is broken, just as it was in the axially symmetric family.

This is the shortest distance between the two points on the surface. They are, of course, in contact in space.

An unexpected duality between the weak field behavior in one geometry and the strong field behavior in the other is evident: asymptotically, the catenoid is accurately described by the height function $h \sim \ln r$, $r\gg r_0$; this asymptotic region is mapped into the neighborhood of the poles described by $h\sim -r^2 \ln r$, $r\ll s$. Inversion provides a connection between the harmonic behavior in the former and the biharmonic behavior in the latter, Guven and Vázquez-Montejo (2013b). To understand this duality between harmonic and biharmonic function, look at the inversion in the origin $\mathbf { x}\rightarrow \mathbf { x}/|\mathbf { x}|^2$ (for transparency set the scale to one), described in the height function representation by $(r, h) \rightarrow (r, h)/ (({r^2 +h^2})$, so that $h \approx \ln r$ $\rightarrow $ $\frac{h}{r^2 + h^2}= \ln [{r}/({r^2 +h^2})]$. Now, if $h\ll r$, then $\frac{h}{r^2} \approx - \ln r $, and as claimed the Green function of the Laplacian is mapped to its biharmonic counterpart.

In this context, note also that the symmetric saddle with $h\sim r^2 \cos 2\theta $ maps to the biharmonic dipole $h\sim \cos 2\theta $.

L. Amoasii, K. Hnia, G. Chicanne, A. Brech, B.S. Cowling, M.M. Müller, Y. Schwab, P. Koebel, A. Ferry, B. Payrastre, J. Laporte, Myotubularin and ptdins3p remodel the sarcoplasmic reticulum in muscle in vivo. J. Cell Sci. 126(8), 1806–1819 (2013). doi:10.1242/jcs.118505 CrossRef

R. Arnowitt, S. Deser, C.W. Misner, Dynamical structure and definition of energy in general relativity. Phys. Rev. 116(5), 1322–1330 (1959). doi:10.1103/PhysRev.116.1322

G. Arreaga, R. Capovilla, J. Guven, Noether currents for bosonic branes. Ann. Phys. 279(1), 126–158 (2000). doi:10.1006/aphy.1999.5979 MathSciNetCrossRefMATH

M. Arroyo, A. DeSimone, Relaxation dynamics of fluid membranes. Phys. Rev. E 79(3), 031915 (2009). doi:10.1103/PhysRevE.79.031915

P. Bassereau, B. Sorre, A. Lévy, Bending lipid membranes: experiments after w. helfrich’s model. Adv. Colloid Interface Sci. 208, 47–57 (2014). doi:10.1016/j.cis.2014.02.002. Special issue in honour of Wolfgang Helfrich

Y. Bernard, Noether’s theorem and the willmore functional. Adv. Calc. Var. (2015). doi:10.1515/acv-2014-0033 MATH

L. Bouzar, F. Menas, M.M. Müller, Toroidal membrane vesicles in spherical confinement. Phys. Rev. E 92, 032721 (2015). doi:10.1103/PhysRevE.92.032721 CrossRef

B. Božič, J. Guven, P. Vázquez-Montejo, S. Svetina, Direct and remote constriction of membrane necks. Phys. Rev. E 89, 052701 (2014). doi:10.1103/PhysRevE.89.052701

B. Božič, S.L. Das, S. Svetina, Sorting of integral membrane proteins mediated by curvature-dependent protein-lipid bilayer interaction. Soft Matter 11, 2479–2487 (2015). doi:10.1039/C4SM02289K CrossRef

P.B. Canham, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theor. Biol. 26(1), 61–81 (1970). doi:10.1016/S0022-5193(70)80032-7 CrossRef

R. Capovilla, J. Guven, Geometry of lipid vesicle adhesion. Phys. Rev. E 66, 041604 (2002a). doi:10.1103/PhysRevE.66.041604 CrossRef

R. Capovilla, J. Guven, Stresses in lipid membranes. J. Phys. A Math. Gen. 35(30), 6233 (2002b). doi:10.1088/0305-4470/35/30/302

R. Capovilla, J. Guven, Stress and geometry of lipid vesicles. J. Phys.-Condens. Mat. 16, S2187–S2191 (2004a). doi:10.1088/0953-8984/16/22/018

R. Capovilla, J. Guven, Second variation of the Helfrich-Canham Hamiltonian and reparametrization invariance. J. Phys. A Math. Gen. 37(23), 5983 (2004b). doi:10.1088/0305-4470/37/23/003 MathSciNetCrossRefMATH

R. Capovilla, J. Guven, J.A. Santiago, Lipid membranes with an edge. Phys. Rev. E 66, 021607 (2002). doi:10.1103/PhysRevE.66.021607 CrossRef

R. Capovilla, J. Guven, J.A. Santiago, Deformations of the geometry of lipid vesicles. J. Phys. A Math. Gen. 36(23), 6281 (2003). doi:10.1088/0305-4470/36/23/301

P. Castro-Villarreal, J. Guven, Axially symmetric membranes with polar tethers. J. Phys. A Math. Theor. 40(16), 4273 (2007a). doi:10.1088/1751-8113/40/16/002 MathSciNetCrossRefMATH

P. Castro-Villarreal, J. Guven, Inverted catenoid as a fluid membrane with two points pulled together. Phys. Rev. E 76, 011922 (2007b). doi:10.1103/PhysRevE.76.011922 CrossRef

M. Deserno, Membrane elasticity and mediated interactions in continuum theory: a differential geometric approach, in Biomembrane Frontiers, ed. by R. Faller, M.L. Longo, S.H. Risbud, T. Jue. Handbook of Modern Biophysics (Humana Press, New York, 2009), pp. 41–74. doi:10.1007/978-1-60761-314-5_2

M. Deserno, Fluid lipid membranes: from differential geometry to curvature stresses. Chem. Phys. Lipids 185, 11–45 (2015). doi:10.1016/j.chemphyslip.2014.05.001. Membrane mechanochemistry: From the molecular to the cellular scale

M. Deserno, M.M. Müller, J. Guven, Contact lines for fluid surface adhesion. Phys. Rev. E 76, 011605 (2007). doi:10.1103/PhysRevE.76.011605 CrossRef

P. Diggins IV, Z.A. McDargh, M. Deserno, Curvature softening and negative compressibility of gel-phase lipid membranes. J. Am. Chem. Soc. 137(40), 12752–12755 (2015). doi:10.1021/jacs.5b06800 CrossRef

M. Do Carmo, Differential Geometry of Curves and Surface (Prentice Hall, Upper Saddle River, 1976)

M. Do Carmo. Riemannian Geometry. (Birkhauser, Basel, 1992)

P.G. Dommersnes, J.-B. Fournier, The many-body problem for anisotropic membrane inclusions and the self-assembly of saddle defects into an egg carton. Biophys. J. 83, 2898–2905 (2002). doi:10.1016/S0006-3495(02)75299-5 CrossRef

E.A. Evans, Bending resistance and chemically induced moments in membrane bilayers. Biophys. J. 14, 923–931 (1974). doi:10.1016/S0006-3495(74)85959-X CrossRef

E.A. Evans, R. Skalak, Mechanics and Thermodynamics of Biomembranes (CRC Press, Boca Raton, 1980)

J.-B. Fournier, On the stress and torque tensors in fluid membranes. Soft Matter 3, 883–888 (2007). doi:10.1039/B701952A CrossRef

J.-B. Fournier, Dynamics of the force exchanged between membrane inclusions. Phys. Rev. Lett. 112, 128101 (2014). doi:10.1103/PhysRevLett.112.128101 CrossRef

J.-B. Fournier, P. Galatola, High-order power series expansion of the elastic interaction between conical membrane inclusions. Eur. Phys. J. E 38(8) (2015). doi:10.1140/epje/i2015-15086-3

R. Goetz, W. Helfrich, The egg carton: theory of a periodic superstructure of some lipid membranes. J. Phys. II Fr. 6(2), 215–223 (1996). doi:10.1051/jp2:1996178

M. Goulian, R. Bruinsma, P. Pincus, Long-range forces in heterogeneous fluid membranes. EPL (Europhysics Letters) 22(2), 145 (1993). doi:10.1209/0295-5075/22/2/012 CrossRef

J. Guven, Membrane geometry with auxiliary variables and quadratic constraints. J. Phys. A Math. Gen. 37(28), L313 (2004). doi:10.1088/0305-4470/37/28/L02

J. Guven, Conformally invariant bending energy for hypersurfaces. J. Phys. A Math. Gen. 38(37), 7943 (2005). doi:10.1088/0305-4470/38/37/002

J. Guven, Laplace pressure as a surface stress in fluid vesicles. J. Phys. A Math. Gen. 39(14), 3771 (2006). doi:10.1088/0305-4470/39/14/019

J. Guven, M.M. Müller, How paper folds: bending with local constraints. J. Phys. A Math. Theo. 41(5), 055203 (2008). doi:10.1088/1751-8113/41/5/055203

J. Guven, M.M. Müller, P. Vázquez-Montejo, Conical instabilities on paper. J. Phys. A Math. Theo. 45(1), 015203 (2012). doi:10.1088/1751-8113

J. Guven, P. Vázquez-Montejo, Spinor representation of surfaces and complex stresses on membranes and interfaces. Phys. Rev. E 82, 041604 (2010). doi:10.1103/PhysRevE.82.041604 MathSciNetCrossRef

J. Guven, P. Vázquez-Montejo, Constrained metric variations and emergent equilibrium surfaces. Phys. Lett. A 377(23–24), 1507–1511 (2013a). doi:10.1016/j.physleta.2013.04.031

J. Guven, P. Vázquez-Montejo, Force dipoles and stable local defects on fluid vesicles. Phys. Rev. E 87, 042710 (2013b). doi:10.1103/PhysRevE.87.042710 CrossRef

J. Guven, G. Huber, D.M. Valencia, Terasaki spiral ramps in the rough endoplasmic reticulum. Phys. Rev. Lett. 113, 188101 (2014). doi:10.1103/PhysRevLett.113.188101 CrossRef

R.C. Haussman, M. Deserno, Effective field theory of thermal casimir interactions between anisotropic particles. Phys. Rev. E 89, 062102 (2014). doi:10.1103/PhysRevE.89.062102 CrossRef

W. Helfrich, Elastic properties of lipid bilayers, theory and possible experiments. Z. Naturforsch. C 28, 693–703 (1973). http://zfn.mpdl.mpg.de/data/Reihe_C/28/ZNC-1973-28c-0693.pdf

J.H. Jellett. Sur la surface dont la courbure moyenne est constante. Journal de Mathematiques Pures et Appliquees, 163–167 (1853)

J.T. Jenkins, The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math. 32(4), 755–764 (1977). doi:10.1137/0132063 MathSciNetCrossRefMATH

F. Jülicher, The morphology of vesicles of higher topological genus: conformal degeneracy and conformal modes. J. Phys. II Fr. 6(12), 1797–1824 (1996). doi:10.1051/jp2:1996161

F. Jülicher, U. Seifert, Shape equations for axisymmetric vesicles: a clarification. Phys. Rev. E 49, 4728–4731 (1994). doi:10.1103/PhysRevE.49.4728 CrossRef

F. Jülicher, U. Seifert, R. Lipowsky, Conformal degeneracy and conformal diffusion of vesicles. Phys. Rev. Lett. 71, 452–455 (1993). doi:10.1103/PhysRevLett.71.452 CrossRef

O. Kahraman, N. Stoop, M.M. Müller, Morphogenesis of membrane invaginations in spherical confinement. EPL (Europhysics Letters) 97(6), 68008 (2012a). doi:10.1209/0295-5075/97/68008 CrossRef

O. Kahraman, N. Stoop, M.M. Müller, Fluid membrane vesicles in confinement. New J. Phys. 14(9), 095021 (2012b). doi:10.1088/1367-2630/14/9/095021 CrossRef

K.S. Kim, J. Neu, G. Oster, Curvature-mediated interactions between membrane proteins. Biophys. J. 75(5), 2274–2291 (1998). doi:10.1016/S0006-3495(98)77672-6 CrossRef

M.M. Kozlov, Fission of biological membranes: interplay between dynamin and lipids. Traffic 2(1), 51–65 (2001). doi:10.1034/j.1600-0854.2001.020107.x CrossRef

V. Kralj-Iglič, S. Svetina, B. Žekž, Shapes of bilayer vesicles with membrane embedded molecules. Eur. Biophys. J. 24(5), 311–321 (1996). doi:10.1007/BF00180372 CrossRef

V. Kralj-Iglič, V. Heinrich, S. Svetina, B. Žekž, Free energy of closed membrane with anisotropic inclusions. Eur. Phys. J. B - Condens. Matter Complex Syst. 10(1), 5–8 (1999). doi:10.1007/s100510050822

E. Kreyszig, Differential Geometry (Dover Publications, New York, 1991)MATH

R. Kusner, Geometric analysis and computer graphics, in Mathematical Sciences Research Institute Publications, vol. 17, ed. by P. Concus, R. Finn, D.A. Hoffman (Springer, New York, 1991), pp. 103–108. doi:10.1007/978-1-4613-9711-3_11

R. Lipowsky, Spontaneous tubulation of membranes and vesicles reveals membrane tension generated by spontaneous curvature. Faraday Discuss. 161, 305–331 (2013). doi:10.1039/C2FD20105D CrossRef

M.A. Lomholt, L. Miao, Descriptions of membrane mechanics from microscopic and effective two-dimensional perspectives. J. Phys. A Math. Gen. 39(33), 10323 (2006). doi:10.1088/0305-4470/39/33/005

O.V. Manyuhina, J.J. Hetzel, M.I. Katsnelson, A. Fasolino, Non-spherical shapes of capsules within a fourth-order curvature model. Eur. Phys. J. E 32(3), 223–228 (2010). doi:10.1140/epje/i2010-10631-2 CrossRef

F.C. Marques, A. Neves, Min-Max theory and the Willmore conjecture. Ann. Math. Second Series 179(2), 683–782 (2014a). doi:10.4007/annals.2014.179.2.6

F.C. Marques, A. Neves, The Willmore conjecture. Jahresbericht der Deutschen Mathematiker-Vereinigung 116(4), 201–222 (2014b). doi:10.1365/s13291-014-0104-8

Z. McDargh, P. Vázquez-Montejo, J. Guven, M. Deserno. Constriction by dynamin: Elasticity vs. adhesion. Biophy. J. 111(11), 2470–2480 (2016). doi:10.1016/j.bpj.2016.10.019

X. Michalet, D. Bensimon, Observation of stable shapes and conformal diffusion in genus 2 vesicles. Science 269(5224), 666–668 (1995). doi:10.1126/science.269.5224.666 CrossRef

S. Morlot, A. Roux, Mechanics of dynamin-mediated membrane fission. Ann. Rev. Biophys. 42(1), 629–649 (2013). doi:10.1146/annurev-biophys-050511-102247 CrossRef

M.M. Müller, Theoretical studies of fluid membrane mechanics, Ph.D. thesis, University of Mainz (Germany), 2007

M.M. Müller, M. Deserno, J. Guven, Geometry of surface-mediated interactions. Europhys. Lett. 69(3), 482 (2005a). doi:10.1209/epl/i2004-10368-1 CrossRef

M.M. Müller, M. Deserno, J. Guven, Interface-mediated interactions between particles: a geometrical approach. Phys. Rev. E 72, 061407 (2005b). doi:10.1103/PhysRevE.72.061407 MathSciNetCrossRef

M.M. Müller, M. Deserno, J. Guven, Balancing torques in membrane-mediated interactions: exact results and numerical illustrations. Phys. Rev. E 76, 011921 (2007). doi:10.1103/PhysRevE.76.011921 CrossRef

M. Mutz, D. Bensimon, Observation of toroidal vesicles. Phys. Rev. A 43, 4525–4527 (1991). doi:10.1103/PhysRevA.43.4525 CrossRef

G.-M. Nam, N.-K. Lee, H. Mohrbach, A. Johner, I.M. Kulić, Helices at interfaces. EPL (Europhysics Letters) 100(2), 28001 (2012). doi:10.1209/0295-5075/100/28001 CrossRef

H. Noguchi, Construction of nuclear envelope shape by a high-genus vesicle with pore-size constraint. Biophy. J. 111(4), 824–831 (2016a). doi:10.1016/j.bpj.2016.07.010

H. Noguchi, Membrane tubule formation by banana-shaped proteins with or without transient network structur. Sci. Rep. 6, 20935 (2016b). doi:10.1038/srep20935

A.S.H. Noguchi, M. Imai, Shape transformations of toroidal vesicles. Soft Matter 11, 193–201 (2015)CrossRef

Z.-C. Ou-Yang, Anchor ring-vesicle membranes. Phys. Rev. A 41, 4517–4520 (1990). doi:10.1103/PhysRevA.41.4517 CrossRef

Z.-C. Ou-Yang, W. Helfrich, Instability and deformation of a spherical vesicle by pressure. Phys. Rev. Lett. 59, 2486–2488 (1987). doi:10.1103/PhysRevLett.59.2486 CrossRef

Z.-C. Ou-Yang, W. Helfrich, Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39, 5280–5288 (1989). doi:10.1103/PhysRevA.39.5280 CrossRef

Z.C. Ou-Yang, J.X. Liu, Y.Z. Xie, X. Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases, Advanced series on theoretical physical science (World Scientific, Singapore, 1999)MATH

R. Phillips, T. Ursell, P. Wiggins, P. Sens, Emerging roles for lipids in shaping membrane-protein function. Nature 459, 379–385 (2009). doi:10.1038/nature08147 CrossRef

U. Pinkall, Cyclides of Dupin, in Mathematical Models from the Collections of Universities and Museums, ed. by E.G. Fischer. Advanced Lectures in Mathematics Series (Friedrick Vieweg & Son, Braunschweig, 1986), pp. 28–30. Chap. 3.3

R. Podgornik, S. Svetina, B. Žekš, Parametrization invariance and shape equations of elastic axisymmetric vesicles. Phys. Rev. E 51, 544–547 (1995). doi:10.1103/PhysRevE.51.544 CrossRef

T.R. Powers, Dynamics of filaments and membranes in a viscous fluid. Rev. Mod. Phy. 82(2), 1607–1631 (2010). doi:10.1103/RevMod-Phys.82.1607

B.J. Reynwar, G. Illya, V.A. Harmandaris, M.M. Müller, K. Kremer, M. Deserno, Aggregation and vesiculation of membrane proteins by curvature-mediated interactions. Nature 447, 461–464 (2007). doi:10.1038/nature05840 CrossRef

Y. Schweitzer, M. Kozlov, Membrane-mediated interaction between strongly anisotropic protein scaffolds. PLoS Comput. Biol. 11, 1004054 (2015). doi:10.1371/journal.pcbi.1004054 CrossRef

U. Seifert, Conformal transformations of vesicle shapes. J. Phys. A Math. Gen. 24(11), 573 (1991). doi:10.1088/0305-4470/24/11/001

U. Seifert, Vesicles of toroidal topology. Phys. Rev. Lett. 66, 2404–2407 (1991). doi:10.1103/PhysRevLett.66.2404 CrossRef

U. Seifert, Configurations of fluid membranes and vesicles. Adv. Phys. 46(1), 13–137 (1997). doi:10.1080/00018739700101488 CrossRef

U. Seifert, R. Lipowsky, Morphology of vesicles, in Structure and Dynamics of Membranes From Cells to Vesicles, ed. by R. Lipowsky, E. Sackmann. Handbook of Biological Physics, vol. 1 (North-Holland, Amsterdam, 1995), pp. 403–463. doi:10.1016/S1383-8121(06)80025-4

P. Sens, L. Johannes, P. Bassereau, Biophysical approaches to protein-induced membrane deformations in trafficking. Current Opinion Cell Biol. 20(4), 476–482 (2008). doi:10.1016/j.ceb.2008.04.004 CrossRef

H. Shiba, H. Noguchi, J.-B. Fournier, Monte carlo study of the frame, fluctuation and internal tensions of fluctuating membranes with fixed area. Soft Matter 12, 2373–2380 (2016). doi:10.1039/C5SM01900A

M. Spivak, A Comprehensive Introduction to Differential Geometry, vol. 1–5, 3rd edn. (Publish or Perish, Inc., Houston, 1999)

D.J. Steigmann, Fluid films with curvature elasticity. Arch. Rational Mech. Anal. 150(2), 127–152 (1999). doi:10.1007/s002050050183 MathSciNetCrossRefMATH

S. Svetina, B. Žekž, Membrane bending energy and shape determination of phospholipid vesicles and red blood cells. Eur. Biophys. J. 17(2), 101–111 (1989). doi:10.1007/BF00257107 CrossRef

S. Svetina, B. Žekš, Nonlocal membrane bending: a reflection, the facts and its relevance. Adv. Colloid Interface Sci. 208, 189–196 (2014). doi:10.1016/j.cis.2014.01.010. Special issue in honour of Wolfgang Helfrich

M. Terasaki, T. Shemesh, N. Kasthuri, R.W. Klemm, R. Schalek, K.J. Hayworth, A.R. Hand, M. Yankova, G. Huber, J.W. Lichtman, T.A. Rapoport, M.M. Kozlov, Stacked endoplasmic reticulum sheets are connected by helicoidal membrane motifs. Cell 154, 285–296 (2013). doi:10.1016/j.cell.2013.06.031 CrossRef

Z.C. Tu, Z.C. Ou-Yang, Lipid membranes with free edges. Phys. Rev. E 68, 061915 (2003). doi:10.1103/PhysRevE.68.061915 CrossRef

Z.C. Tu, Z.C. Ou-Yang, A geometric theory on the elasticity of bio-membranes. J. Phys. A Math. Gen. 37(47), 11407 (2004). doi:10.1088/0305-4470/37/47/010

Z.C. Tu, Z.C. Ou-Yang, Recent theoretical advances in elasticity of membranes following helfrich’s spontaneous curvature model. Adv. Colloid Interface Sci. 208, 66–75 (2014). doi:10.1016/j.cis.2014.01.008. Special issue in honour of Wolfgang Helfrich

R.M. Wald, General Relativity (University of Chicago Press, Chicago, 2010)MATH

T.R. Weikl, M.M. Kozlov, W. Helfrich, Interaction of conical membrane inclusions: effect of lateral tension. Phys. Rev. E 57, 6988–6995 (1998). doi:10.1103/PhysRevE.57.6988 CrossRef

T.J. Willmore, Note on embedded surfaces. An. St. Univ. Iasi, sIa Mat. B 12, 493–496 (1965)

T.J. Willmore, Total Curvature in Riemannian Geometry (Ellis Horwood, Chichester, 1982)MATH

T.J. Willmore, Riemannian Geometry (Oxford University Press, Oxford, 1996)MATH

C. Yolcu, M. Deserno, Membrane-mediated interactions between rigid inclusions: an effective field theory. Phys. Rev. E 86, 031906 (2012). doi:10.1103/PhysRevE.86.031906 CrossRef

C. Yolcu, I.Z. Rothstein, M. Deserno, Effective field theory approach to casimir interactions on soft matter surfaces. EPL (Europhysics Letters) 96(2), 20003 (2011). doi:10.1209/0295-5075/96/20003 CrossRef

C. Yolcu, I.Z. Rothstein, M. Deserno, Effective field theory approach to fluctuation-induced forces between colloids at an interface. Phys. Rev. E 85, 011140 (2012). doi:10.1103/PhysRevE.85.011140 CrossRef

C. Yolcu, R.C. Haussman, M. Deserno, The effective field theory approach towards membrane-mediated interactions between particles. Adv. Colloid Interface Sci. 208, 89–109 (2014). doi:10.1016/j.cis.2014.02.017. Special issue in honour of Wolfgang Helfrich

W.-M. Zheng, J. Liu, Helfrich shape equation for axisymmetric vesicles as a first integral. Phys. Rev. E 48, 2856–2860 (1993). doi:10.1103/PhysRevE.48.2856 CrossRef

Title: The Geometry of Fluid Membranes: Variational Principles, Symmetries and Conservation Laws
Authors: Jemal Guven
Pablo Vázquez-Montejo
Publisher: Springer International Publishing
Book: The Role of Mechanics in the Study of Lipid Bilayers
Print ISBN: 978-3-319-56347-3

Electronic ISBN: 978-3-319-56348-0

Copyright Year: 2018
DOI: https://doi.org/10.1007/978-3-319-56348-0_4

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