Numerical investigations of the role of curvature in strong segregation problems on a given surface
Introduction
Lipids are the bilayer structural framework for cell membranes. A lipid molecule consists of a hydrophilic headgroup and a hydrophobic tail. The headgroups point outward from the bilayer forming its boundary, while the tails form the core of the bilayer. However, lipids are more complex and are able to form thermodynamically distinct phases for different compositions and temperatures. The three phases of interest are the fluid-disordered (), fluid-ordered () and gel () phases. At low temperatures, the bilayer forms the highly-ordered gel phase. As temperature rises, the acyl chains disorder, and the bilayer ‘melts’ into the fluid-disordered phase, . When cholesterol is present a fluid-ordered phase arises. Cholesterol orders the disordered acyl chains to a certain degree and converts the phase to .
Vesicles are lipid bilayers enclosing a volume of solution in an aqueous environment and provide a simple model for a cell membrane. This model enables experimenters to control the lipid composition and phase. Fig. 1 shows two-phase vesicles formed from specific mixes of lipids. The and phases are labeled fluorescently in order to show the different phase regions; phase domains have coarsened into large areas. For this experiment, we used unsaturated lipid (DOPC) that melts from gel to fluid-disordered phase at −20 °C. This defines its low transition point (). A saturated lipid with a high is also present (DPPC, °C) along with cholesterol. The combination of high and low lipids and cholesterol has been well-characterized to form and phases in a single vesicle in a range of composition ratios (see, e.g., [2]). When the phases separate, the composition of the region contains a majority of the low lipid, DOPC. In the domain, DPPC and cholesterol are in higher concentration than DOPC. As illustrated by the images in Fig. 1, the boundaries exhibit changes in curvature, consistent with previous findings [3], [4]. In Fig. 1(a), the vesicle is roughly an elongated ellipsoid. A strongly segregated phase distribution is observed with each end of the ellipsoid covered by a different phase. While the boundary between the phases in Fig. 1(a) appears smooth, the transition in Fig. 1(b) is not. A close examination of Fig. 1(b) suggests the membrane curvature is discontinuous along the interface between regions of different phase. This is due to the fact that the equilibrium shape of a vesicle is determined primarily by minimizing bending energy and the bending energies are modulated by bending moduli of the respective phase regions. When a vesicle exhibits multiple disjoint regions of constant phase, smooth and non-smooth interfaces are possible as shown in Fig. 1(c). Images of two-component phase distributions of the type show in Fig. 1 suggest that phase distribution is coupled to membrane curvature. One of the first analytical results that coupled membrane curvature and phase was [1, Theorem 1.1] which proved that small patches of phase domains appear at places on the membrane where the Gauss curvature attains a maximum. Essentially, the boundary between the phases is energetically costly due to the packing mismatches and in a two-phase vesicle, this results in two domains, one of each phase. Recent results [3], [4] tied curvature to free energy of the system along phase boundaries. This follows the work of Helfrich, Seifert and Lipowsky in characterizing curvature in lipid systems (see, [5], [6], [7]). This work is supported by experimental outcomes of the type shown in Fig. 1. When we image two-phase vesicles, we observe that each domain coalesces into one region on the vesicle. The details of this experiment are given in [1, Appendix]. At high temperature, the two phases melt into a homogeneous fluid-disordered phase. However, when the temperature is lowered below the transition temperature, the two phases begin to separate. In the experiments shown here, the unsaturated, low lipid (DOPC) will partition into the phase while the saturated, high lipid (DPPC) and cholesterol are the major components in the phase. Depending on the ratio of these components, or may form the majority or minority phase. Thus vesicles may exhibit islands of surrounded by or vise versa. While the analytical results in [1] focused on small diameter patches, experimental work suggest similar results apply to patches that are not small and this observation is supported by the numerical results presented in this paper.
In this paper, we investigate the behavior of a two-phase lipid vesicle via numerical simulation with finite elements. Generally, below the transition temperature, the phases separate into non-connecting domains that coarsen into larger areas. The free energy of phase properties determines the length of the boundaries separating the regions. The two phases correspond to different lipid compositions, and in cells, this fluctuation in composition is a dynamic process vital to its function. The theory predicts that a small geodesic disk of the minority lipids forms at a point of the membrane where the Gauss curvature attains a maximum. This geodesic disk is the most energetically favorable configuration when compared with a set of similar size at a different location. To demonstrate these results, we focus on a vesicle that has the shape of an ellipsoid and determine the phase distribution on this domain by directly minimizing a Landau-type free energy subject to a constraint that describes the proportion of each phase. For a generic ellipsoid M with semi-major axes where , we demonstrate that a small geodesic disk will form centered at the point of maximum Gauss curvature. We investigate similar patch formation problems in the neighborhood of other critical points of the Gauss curvature, including cases where the patch is small or of moderate size. An initial moderately sized patch placed anywhere on the ellipsoid will evolve to a geodesic patch centered at a point of maximum Gauss curvature. When the initial patch is too small and located far from the point of maximum Gauss curvature, the solution process might be unable to converge, instead terminating at a false local minimum of . However, when the initial patch is small and its center is within a few diameters of the point of maximum Gauss curvature, then the solution process will converge. We discuss the sensitivity of the solution process on the grid size h and the relation between h, the diffusion coefficient , the conservation constant m and the initial phase configuration.
In general, the bending energy is the primary factor determining vesicle shape. This paper is the first step in a broader research program where we will allow both vesicle shape and phase distribution to vary simultaneously. Ultimately, we aim to apply our approach to bio-membranes where other factors play a role in shape determination. The problem formulation and solution methodology developed in this paper can be integrated into a single model where multi-phase membranes whose free energy includes phase energy, bending energy and cytoskeleton elastic strain energy can be probed. However, before undertaking such an investigation, it is important to understand fully how our numerical approach performs on a well-defined benchmark problem, before considering membranes with more complicated topology and factors that influence its shape.
Key parameters in our numerical model are the diffusion coefficient , grid size h of the finite element mesh and a parameter which measures the percent of the minority phase in a strong segregation problem. We investigate the relationships between and h. We explore the behavior of small to moderately sized patches outside the neighborhood of a point of maximum Gauss curvature. Although we do not study dynamics in this paper, we can think of a specific distribution of the minority phase as the initial state in an evolution process that seeks a state of minimum phase energy. When the minimization process terminates in a state of lower energy, we say that the lower energy state of the system evolved from the initial configuration. We can then ask what is the stability of that configuration? Is it a local or global minimum? Is it a saddle point?
In Section 2, we present background on phase transition models and their solution approaches. In Section 3, we present the Landau free energy model that describes the phase separation property of the two lipid types. Geometric properties of the ellipsoid and the discretization of the free energy model are also discussed. In Section 4, we present numerical solutions for the phase separation problem on a ellipsoid. We demonstrate that the Landau phase energy can be used to estimate the perimeter of the boundary of a small patch. We also present several numerical experiments which demonstrate the behavior predicted by Gillmor et al. [1, Theorem 2.1]. Concluding remarks are presented in Section 5.
Section snippets
Landau phase transition theory
The Landau type free energy is one of the models commonly seen in phase transition theories [8]Here, M is a compact smooth manifold, the phase field represents the volume fraction of one of the two types of lipids, is the surface gradient with respect to the metric on M and dS denotes surface area measure. The phase field satisfies a conservation constraint given bywhere is the relative number of lipids of one type, and is
Mathematical model and its discretization
In the following, we present our numerical model for a strongly segregated two-phase ellipsoidal vesicle based on a Landau-type free energy. The discretization, grid-generation and solution algorithm are also discussed.
Numerical solutions
In the following, a minimizer of is denoted by , i.e., . The kth iterate of the SQP algorithm is denoted by . The center of mass of a set E generated by phase field is defined as
Concluding remarks
In this paper, we studied the numerical simulation of a two-phase lipid vesicle in the shape of an ellipsoid. For a tri-axial or prolate ellipsoid, our numerical results show that when m is small or of moderate size, a geodesic patch E with of the minority lipids forms at a point of the ellipsoid where the Gauss curvature attains a global maximum. When m is of moderate size (say ), a patch E initialized anywhere on the ellipsoid will evolve to a geodesic disk centered at a point
Acknowledgments
The authors thank Xiaofeng Ren for his insights and discussions on phase transition problems. This work is supported by the Swiss “High Performance and High Productivity Computing” initiative HP2C. Computational resources were provided by the Swiss Supercomputing Center CSCS. This project was initiated in Fall 2012 while F. Baginski was on sabbatical leave at the Institute for Computational Science, University of Lugano. S. Gilmor has received funding from the Keck foundation. The authors thank
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