Chemistry and Physics of Lipids 185 (2015) 46–60 Contents lists available at ScienceDirect Chemistry and Physics of Lipids journal homepage: www.elsevier.com/locate/chemphyslip Phase-field theories for mathematical modeling of biological membranes Guillermo R. Lázaro a,∗ , Ignacio Pagonabarraga b , Aurora Hernández-Machado a a Departament d’Estructura i Constituents de la materia, Universitat de Barcelona, Av. Diagonal 645, E08028 Barcelona, Spain b Departament de Fisica Fonamental, Universitat de Barcelona, Av. Diagonal 645, E08028 Barcelona, Spain article info Article history: Available online 20 September 2014 Keywords: Biophysics Phase-field Modelling Membrane abstract Biological membranes are complex structures whose mechanics are usually described at a mesoscopic level, such as the Helfrich bending theory. In this article, we present the phase-field methods, a useful tool for studying complex membrane problems which can be applied to very different phenomena. We start with an overview of the general theory of elasticity, paying special attention to its derivation from a molecular scale. We then study the particular case of membrane elasticity, explicitly obtaining the Helfrich bending energy. Within the framework of this theory, we derive a phase-field model for biological membranes and explore its physical basis and interpretation in terms of membrane elasticity. We finally explain three examples of applications of these methods to membrane related problems. First, the case of vesicle pearling and tubulation, when lipidic vesicles are exposed to the presence of hydrophobic polymers that anchor to the membrane, inducing a shape instability. Finally, we study the behavior of red blood cells while flowing in narrow microchannels, focusing on the importance of membrane elasticity to the cell flow capabilities. © 2014 Elsevier Ireland Ltd. All rights reserved. 1. Introduction Like many other organelles at the cell scale, membranes are composite structures that exhibit a bewildering complexity. Their basic ingredient is a lipid bilayer composed by hundreds of dif- ferent lipid species. The bilayer also contains a dense population of transmembrane proteins, which could represent up to 70% of the total mass of the membrane, and in fact these molecules define the functionality of the membrane (Alberts et al., 1994). Many other proteins are anchored to both sides of the bilayer, and membrane composition is balanced by lipid reservoirs which ensure that the physiological properties of the membrane are maintained. Among others, membranes define the cell frontiers, separating the cytosol from the external environment. They also maintain ion gradients which are necessary to produce ATP, and host the proteins that control cell signaling. Focusing specifically on the structural function, membranes present a delicate inter- play with the cortex cytoskeleton, a complex mesh formed by filaments of actin preserving cell inner structure and shape, and provides strength and compactness. This picture represents, how- ever, just a rough description of the membrane composition and ∗ Corresponding author. E-mail address: grolazaro@gmail.com (G.R. Lázaro). function, included here to evidence its extreme complexity. The comprehensive understanding of this fascinating system requires of different level of approaches. At the molecular scale, the detailed running of each microstructure can be analyzed thoroughly from the electrochemical interactions between their molecular compo- nents. However, the all-encompassing response of the membrane elements invites to a more general description, and physical the- ories such as the elastic formalism of plates and surfaces offer a formidable tool to characterize biological membranes. At this point, it is convenient to remark that physicists have focused on the spe- cific study of the human erythrocyte (Sackmann, 1995). This cell is unique among the rest of cells of the organism because it lacks any organelle and inner structure, so that all its physical proper- ties are entirely determined by its membrane, representing a much simpler structure than normal cells. The membrane of the erythro- cyte is composed by a lipid bilayer with and underlaying spectrin cytoskeleton which anchors to the cytosolic side of the bilayer. This two-dimensional scaffold has a structural function, preventing the cell from vesiculation and large deformations. In order to build a physical theory of the membrane, the com- plexity of the cell membrane suggests to consider the scales of interest. Our scope is to study phenomena at the cell scale, such as cell morphological deformations or mechanical interactions with the environment. In this context, the atomic description is clearly unaffordable: the difficulty of dealing with such a vast number http://dx.doi.org/10.1016/j.chemphyslip.2014.08.001 0009-3084/© 2014 Elsevier Ireland Ltd. All rights reserved.