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This book is the first collection of lipid-membrane research conducted by leading mechanicians and experts in continuum mechanics. It brings the overall intellectual framework afforded by modern continuum mechanics to bear on a host of challenging problems in lipid membrane physics. These include unique and authoritative treatments of differential geometry, shape elasticity, surface flow and diffusion, interleaf membrane friction, phase transitions, electroelasticity and flexoelectricity, and computational modelling.



Mechanics and Physics of Lipid Bilayers

In this chapter we review recent work by the writer and coworkers on various aspects of the mechanics and physics of lipid bilayers. A framework for lipid bilayer surface, based on a dimension reduction procedure applied to three-dimensional liquid crystal theory, is reviewed in Sect. 1. This accommodates the non-standard effects of lipid distension and tilt. A special case of the general model in which tilt is suppressed but distension, and accompanying surface dilation, are permitted, is also derived. This is further specialized, in Sect. 2, to obtain a model of the classical type, due to Canham and Helfrich. Our approach facilitates understanding of the place of the classical theory, and its logical extensions, in a larger context. Section 3 provides a further development of the theory with surface dilation—reported here for the first time—to accommodate dissipative effects, including intra-membrane viscous flow and the diffusion of trans-membrane embedded proteins. This may be viewed as a theory of generalized capillarity, accounting for various higher order gradient effects of the Cahn–Hilliard type in the constitutive equations. A simpler variant of this model is described in Sect. 4, in which non-standard gradient effects are suppressed. This furnishes the simplest thermodynamically consistent extension of the classical theory to cover diffusion and viscosity. Finally, Sect. 5 is devoted to the electromechanical theory. This is limited to the simplest extension of the classical model to accommodate surface flexo-electricity and the coupling of surface shape with a polarization field. Restrictions on the latter, consistent with the three-dimensional electromechanical theory for liquid crystals, yield a relatively simple generalization of the classical theory appropriate for analyzing membrane response to a remote applied electric field.
David J. Steigmann

Elasticity and Hereditariness

This chapter collects the lecture notes of the module “Elasticity and Hereditatiness of Lipid Bilayers” delivered at CISM in July 2016. Such material is based primarily on three papers coauthored by this lecturer, and which have been contributing to shed light on the mechanical behavior of lipid bilayers. In particular, the breakthrough from this research is that the underlying nonlinear elastic response of lipid bilayers is fully determined as long as the membrane energy is obtained. Bending and saddle splay rigidities are shown here to be directly obtainable from the membranal response, as well as the line tension, holding together domains in which lipids are in different phases. The power law hereditariness of lipid membranes strikingly shown through rheometric tests has been analyzed in this work through a suitable energetics obtained by the author and coworkers and penalizing small perturbations of ground configurations of such systems.
Luca Deseri

Lipid Membranes: From Self-assembly to Elasticity

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.
M. Mert Terzi, Markus Deserno

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

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.
Jemal Guven, Pablo Vázquez-Montejo

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

This chapter discusses the computational modeling of lipid bilayers based on the nonlinear theory of thin shells. Several computational challenges are identified and various theoretical and computational ingredients are proposed in order to counter them. In particular, \(C^1\)-continuous, NURBS-based, LBB-conforming surface finite element discretizations are discussed. The constitutive behavior of the bilayer is based on in-plane viscosity and (near) area-incompressibility combined with the Helfrich bending model. Various shear stabilization techniques are proposed for quasi-static computations. All ingredients are formulated in the curvilinear coordinate system characterizing general surface parameterizations. The consistent linearization of the formulation is presented, and several numerical examples are shown.
Roger A. Sauer

Onsager’s Variational Principle in Soft Matter: Introduction and Application to the Dynamics of Adsorption of Proteins onto Fluid Membranes

Lipid bilayers are unique soft materials operating in general in the low Reynolds limit. While their shape is predominantly dominated by curvature elasticity as in a solid shell, their in-plane behavior is that of a largely inextensible viscous fluid. Furthermore, lipid membranes are extremely responsive to chemical stimuli. Because in their biological context they are continuously brought out-of-equilibrium mechanically or chemically, it is important to understand their dynamics. Here, we introduce Onsager’s variational principle as a general and transparent modeling tool for lipid bilayer dynamics. We introduce this principle with elementary examples, and then use it to study the sorption of curved proteins on lipid membranes.
Marino Arroyo, Nikhil Walani, Alejandro Torres-Sánchez, Dimitri Kaurin
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