Biological membranes: The laboratory of fundamental physics

Copyright: © 2019 Kralj S, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Biological membranes present an essential constituent of living cells. Their main role is to separate the interior of a cell from its surrounding, however allowing the selective transfer of speci ic material through it. Con iguration changes of membranes are often correlated with important biological processes [1-7]. For example, they might trigger divisions of cells [4], adaptation of red blood cells [1] to temporal conditions during their transport to different parts of biological tissues, they might be involved in cancerous [5] and cell death [6] processes. Membrane structures are in general extremely complex, however, their key properties are often dominated by geometry. This was irst illustrated by Helfrich [8] who constructed a minimal model of membranes introducing curvature ields. Locally, these are represented by principal curvatures C1 = 1/R1 and C2 = 1/R2, where R1 and R2 are the corresponding curvature radii. The key quantities dominating energetics of membranes are the mean curvature H = (C1 + C2)/2 and the Gaussian curvature K = C1 C2. In general, membranes tend to minimize H2 for given boundary conditions and K plays an important role if a membrane undergoes a topological change (i.e. in membrane ission or fusion processes). Furthermore, membranes in general often exhibit some kind of in-plane order, which enormously increases the complexity of potential membrane responses to various stimuli. This ordering could be due to anisotropic membrane constituents [9], lexible hydrocarbon chains of lipids [10] or due to anisotropic proteins embedded within membranes [11]. If in-plane order exists topological defects (TDs) [12] are inevitably formed in membranes if they do not exhibit toroidal symmetry [13]. TDs in membranes correspond to points or lines where the in-plane ield is (mathematically) not uniquely de ined as illustrated in igure 1. Consequently, such regions are in general energetically costly. In practice, membranes avoid such singularities by locally “melting” [4] of in-plane order or by a local phase separation [9]. The former case corresponds to relatively strong local luctuations, via which the in-plane ordering is averaged out. In the latter case a membrane ingredient responsible for anisotropic ordering moves to “nonsingular” membrane parts. One can assign to TDs a topological charge, which is a conserved quantity. If membranes are treated as effectively two-dimensional objects, the topological charge equals [12] the winding number m. The later determines the total reorientation of the in-plane ield divided by 2π on encircling the defect center counterclockwise. Examples if igure 1 represent TDs bearing charges m=1 (Figure 1a) and m=-1 (Figure 1b). In general, TDs behave like localized electric charges, where m plays the role of an electrical charge. Furthermore, TDs are energetically costly and lat parts of the membrane tend to expel them. However, in closed membranes, their total winding number mtot is determined topologically. Namely, it holds [14]. (1)