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)

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][2][3][4][5][6][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 C 1 = 1/R 1 and C 2 = 1/R 2 , where R 1 and R 2 are the corresponding curvature radii. The key quantities dominating energetics of membranes are the mean curvature H = (C 1 + C 2 )/2 and the Gaussian curvature K = C 1 C 2. In general, membranes tend to minimize H 2 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 m tot is determined topologically. Namely, it holds [14]. (1) where the integral is carried over the closed membrane and da stands for an in initesimally small surface area. For example, for the spherical (toroidal) topology it holds m tot = 2 (m tot = 0). Furthermore, in "normal" (relatively weak curvatures) conditions "elementary" TDs tend to be formed. In case of i) vector, ii) rod-like, or iii) hexatic order these TDs carry winding numbers i) m 0 = ±1, ii) m 0 = ±1/2, and iii) m 0 = ±1/6. Therefore, a membrane of spherical topology exhibits at least i) two, ii) four, and iii) twelve TDs. One could intuitively understand this tendency by inspecting igure 2.
Let us assume that an in-plane ordering exists, which tends to be locally parallel. In the case of toroidal topology, the ield could stream along the parallels (lines parallel to the equatorial line). Such topology does not impose frustration on orientational order and does not require any TDs igure 2a. However, for a spherical topology TDs are unavoidable because the geometry enforces frustration to the orientational ield. In igure 2b one sees that two m=1 defects are formed at the poles of the structure. Eq.(1) also suggests that positive (negative) Gaussian curvature attracts TDs bearing positive (negative) values of m [13,15], (i.e. dm tot = Kda/2π). This is ef iciently realized in regions with K ≠ 0 where a difference between the principal curvatures is relatively small. However, if this is not the case, the difference imposes a kind of local ield, referred to as the deviatoric ield [16]. Consequently, TDs tend to be expelled from such regions because their spatially non-uniform structure is incompatible with the ield imposed uniform ordering. Note that TDs introduce local inhomogeneities in membranes, which could serve as nucleating regions for various biological (e.g., membrane ission or budding) processes [3,4]. To conclude, membranes in general host of TDs and their positions and also the number is strongly affected by curvature ields. Furthermore, TDs in membranes could play an important role in their functionality.
Related problems are also of strong interest for other branches of condensed matter physics, and even for cosmology and particle physics. Namely, TDs are an unavoidable consequence of symmetry-breaking phase transitions [12], which are ubiquitous in nature.
Furthermore, their main properties are determined topologically, which are independent of systems' microscopic details, endowing them with several universal features. For example, in general, the impact of curvature on TDs in twodimensional systems is relatively weakly understood in condensed matter systems. Namely, most theoretical studies [14,15] use covariant derivatives in expressing the elastic penalty of distorted ordering ields. Such approaches by default discard the so-called extrinsic curvature contributions [14] and take into account only intrinsic contributions. However, there are no reasonable justi ications [17,18] for this omission. Simple analysis [17] even suggests that the extrinsic and intrinsic contributions should be comparable and in several geometries, they enforce contradicting behaviour [17,18]. Therefore, by omitting extrinsic contributions several extrinsic curvature-driven phenomena could not be observed. Note that the extrinsic curvature is in biological systems commonly referred to as the deviatoric curvature and its impact on various membrane properties has been relatively well explored [9]. Therefore, other branches of physics could transfer some of this knowledge to their realm. Furthermore, the extrinsic curvature exhibits the impact of the higher dimensional space in which the lower-dimensional system of interest is embedded (i.e., effectively two-dimensional curved membranes are embedded in three dimensions). Therefore, extrinsic curvaturedriven phenomena could reveal the impacts of potentially existing higher dimensions, which is of interest in cosmology. Furthermore, the irst theory of coarsening dynamics (the so-called Kibble-Zurek mechanism [19]) of TDs following a sudden phase transition was developed in cosmology to study the coarsening of TDs in the Higgs ield of the early universe [20]. Therefore, some of this knowledge might be transferred also to membranes. Note that interactions between curvature and TDs might even resolve the origin of mysterious dark energy. Namely, the mainstream description of nature is based on the assumption that the universe is essentially lat. However, recent numerical studies [21] reveal that the current universe could exhibit inite curvature. By taking into account the impact of curvature one could reproduce effects, which are now attributed to the mysterious dark energy. Moreover, if relevant ields represent basic entities of nature [22], then TDs [23]