Asymmetric lipid bilayers from the perspective of three-dimensional liquid crystal theory
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S H O RT C O M M U N I C AT I O N
A. Agrawal · D. J. Steigmann
Asymmetric lipid bilayers from the perspective of three-dimensional liquid crystal theory
Received: 15 June 2020 / Accepted: 17 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The classical theory for asymmetric lipid bilayer surfaces is revisited from the vantage point of three-dimensional liquid crystal theory. Independent tangential motions of the leaflets comprising the bilayer are accommodated in a framework that allows for distinct leaflet properties. Keywords Lipid bilayers · Asymmetry · Spontaneous curvature 1 Introduction Our purpose in this brief exposition is to present a modern account of the classical theory of asymmetric lipid bilayers in which asymmetry, i.e., the absence of reflection symmetry with respect to the bilayer surface normal, is conferred by a property called the spontaneous curvature [1,2]. Following the seminal work of Helfrich [1], we interpret bilayers as liquid crystal films, but here we derive the associated surface theory via asymptotic expansion of three-dimensional liquid crystal theory, with film thickness—on the order of molecular dimensions—playing the role of the small parameter. In accordance with the observed phenomenology, we allow for the possibility that the two monolayer leaflets constituting the bilayer undergo relative motion in the tangent plane to the surface to which they remain congruent. Though attention is confined here to the purely mechanical theory, our formulation provides a framework in which a number of physical effects can be modeled systematically. Our motivation derives from renewed interest in asymmetric bilayers on the part of the biophysics community. For example, it has recently been demonstrated that inner (cytoplasmic) leaflets are significantly less saturated than their outer (exoplasmic) counterparts [3]. Atomistic studies indicate that this asymmetry is correlated with higher packing of lipids and inhibited diffusivity in the exoplasmic leaflet [4]. Negatively charged lipids are also present in higher concentrations in the inner leaflet. Asymmetry is thought to affect the interactions between lipids and embedded transmembrane proteins, and is known to play a key role in regulating cellular signaling, cell death and cell-to-cell interactions [5]. In Sect. 2, we present an asymptotic derivation of the two-dimensional energy of a thin film from threedimensional energies modeling the two leaflets of the bilayer as nematic liquid crystals having distinct properties. This is made explicit, in Sect. 3, for polar nematics described by energies of the Frank type [6,7]. Following a brief survey of the relevant differential geometry of surfaces in Sect. 4, in Sect. 5 we derive the differential Communicated by Andreas Öchsner. A. Agrawal Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA D. J. Steigmann (B) Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA E-mail: [email protected]
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