Computational and Mathematical Methods in Medicine

Volume 2017 (2017), Article ID 7275131, 11 pages

https://doi.org/10.1155/2017/7275131

## Curvature-Induced Spatial Ordering of Composition in Lipid Membranes

Faculty of Mechanical Engineering, Technion–Israel Institute of Technology, 32000 Haifa, Israel

Correspondence should be addressed to Sefi Givli

Received 30 January 2017; Accepted 22 March 2017; Published 4 April 2017

Academic Editor: John Mitchell

Copyright © 2017 Shimrit Katz and Sefi Givli. 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.

#### Abstract

Phase segregation of membranal components, such as proteins, lipids, and cholesterols, leads to the formation of aggregates or domains that are rich in specific constituents. This process is important in the interaction of the cell with its surroundings and in determining the cell’s behavior and fate. Motivated by published experiments on curvature-modulated phase separation in lipid membranes, we formulate a mathematical model aiming at studying the spatial ordering of composition in a two-component biomembrane that is subjected to a prescribed (imposed) geometry. Based on this model, we identified key nondimensional quantities that govern the biomembrane response and performed numerical simulations to quantitatively explore their influence. We reproduce published experimental observations and extend them to surfaces with geometric features (imposed geometry) and lipid phases beyond those used in the experiments. In addition, we demonstrate the possibility for curvature-modulated phase separation above the critical temperature and propose a systematic procedure to determine which mechanism, the difference in bending stiffness or difference in spontaneous curvatures of the two phases, dominates the coupling between shape and composition.

#### 1. Introduction

The biological lipid bilayer membrane, or in short “biomembrane,” is a fundamental building block of the cell. It forms the barrier that separates the interior of the cell from its surroundings but is also responsible for almost all interaction of the cell with its environment, including transport, adhesion, regulation, transduction, and signaling [1–5]. The diverse functionality of the biomembrane is achieved by a seemingly simple structure, two layers that are primarily made from lipid molecules and also some integral proteins, cholesterols, and other functional molecules [6, 7]. This molecular structure of the biomembrane gives rise to the so-called “fluidity” of the membrane [8]; that is, its constituent molecules are able to move relatively easy within the membrane, which resists bending and stretching but not shear [9]. Consequently, biomembranes have a dynamic structure in the sense that their molecular arrangement (local composition) can change with conditions. For example, depending on temperature (and/or osmotic pressure, acidity, etc.) the biomembrane may possess a uniform mixture of its components or it may segregate into different phases, which are rich in specific constituent and possess different mechanical and chemical properties [10–16].

The fluidity of the biomembrane combined with its spatial heterogeneity brings about a unique coupling between shape (geometry) and composition. For example, lipid phases that possess high bending stiffness highly favor regions with small (magnitude) curvature [17]. Also, the three-dimensional molecular shape of some lipids and proteins results in a nonzero spontaneous curvature that affects the geometry of the biomembrane in their neighborhood. This two-way coupling between shape and composition means that deformations exhibited by biomembranes are strongly influenced by their heterogeneous composition, while the spatial ordering of composition is modulated by the geometry of the membrane [16, 18–23].

In the last two decades, much effort has been invested into understanding the consequences of the coupling between shape and composition in biomembranes. Theoretical models have generalized uniform composition models [24–27] to account for multicomponent or multiphase membranes [28–35]. Features such as equilibrium configurations, stability [36–39], interaction with the cytoskeleton [40–42], formation of lipid rafts, anisotropy of the membrane constituents [43], and even using biomembranes as sensors or actuators [44, 45] have been investigated. The complexity of the problem has forced the usage of sophisticated numerical methods, such as advanced phase field schemes, special nonlinear finite elements, and molecular dynamics simulations [17, 46–50], while analytical derivations have commonly adopted simplifying assumptions, like small deformations, axisymmetry, and so forth. The abovementioned theoretical studies have been motivated by a large body of experimental work, for example, [10–16, 21, 51–53], that demonstrated phenomena such as phase segregation, coexistence of different phases, and formation of domains in vesicles by a variety of methods, for example, fluorescence microscopy, X-ray diffraction, proton microscopy, spin resonance, and NMR imaging.

In a recent work, Parthasarathy et al. [54] designed an elegant experiment that breaks the two-way coupling between shape and composition and enables direct investigation of the influence of the membrane geometry on the spatial ordering of its composition. To this end, they used a quartz substrate, which was topographically patterned using photolithographic microfabrication techniques. The substrate consisted of continuously alternating high and low curvature contours with one-dimensional periodicity of 2 *μ*m. In order to decouple the main membrane from the underlying substrate a double membrane system was used: first, a supported membrane of uniform composition was deposited on the substrate. Then, the “main” membrane, a giant unilamellar vesicle (GUV), was introduced on top. Parthasarathy et al. showed that beyond a critical temperature, the spatial organization of lipid phases can be directed by gradients of membrane curvature, provided that these gradients are large enough.

In the current paper we analyze this type of experiment by means of a mathematical model combined with numerical simulations. The main goal is to reproduce the experimental observations mentioned above but also to generalize them and motivate new experiments. Accordingly, the structure of the paper is as follows: Section 2 describes the main theoretical considerations, governing equations, and nondimensional quantities that govern the spatial ordering of composition in a biomembrane subjected to imposed geometry. In Section 3, we perform a numerical study aiming at understanding the role of the nondimensional quantities that were identified in Section 2, and in particular their influence on the evolution of the spatial organization of the biomembrane composition. Focus is put on the final (steady state) spatial field of the membrane composition. The main conclusions are discussed in Section 4.

#### 2. Theoretical Considerations

##### 2.1. Governing Equations

Consider a biomembrane composed of two components, for example, two different lipid molecules or two different lipid phases, that lies on a smooth nonflat surface (in their experiment, Parthasarathy et al. [54] used a double membrane system to decouple the main membrane from the underlying substrate: a “supporting” membrane of uniform composition was deposited on the substrate in order to chemically decouple the “main” membrane from the substrate, and only then the “main” membrane was introduced on top of it) that has a geometry (shape) of a continuously alternating curvature with one-dimensional periodicity; see Figure 1(b). The free energy of the biomembrane takes the form [36] where integration is performed over the entire surface area of the membrane, . Above, the first term is the (Helfrich) bending energy [25, 55] which depends on the mean curvature, , the second term, , is the specific mixing energy, and the last term describes the energetic penalty for spatial composition gradients. In addition, describes the mole fraction of the second component, which we also refer to as local composition or concentration. A few comments are in order: (i) functional (1) does not include a stretching energy term or a Gauss-curvature bending energy term. The reason is that the biomembrane lies freely on a smooth surface; thus its stretching energy vanishes. Also, the Gauss-curvature bending energy vanishes everywhere since the imposed geometry has a 1D periodicity, which results in one of the two principal curvatures being zero. (ii) The two components of the biomembrane differ in their mechanical properties; namely, they have different bending stiffness, , and spontaneous curvature, . Thus, inhomogeneity induces local spontaneous curvature and stiffness that depend on local composition, . (iii) The specific mixing energy, , combines the aggregation enthalpy and the entropy of mixing. A simple model that is often adopted for modeling the mixing energy is the so-called “gas lattice” or “regular solution” model, which takes the form [36, 42] Here, is the Boltzmann’s constant, is the critical temperature defined as the temperature at which the mixing energy changes from single-well to double-well structure, is the lipid density (number of molecules per unit area), and is the nondimensional temperature. Consequently, is convex (miscible) at temperatures but nonconvex with double-well structure at temperatures .