Morphology and interaction between lipid domains
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Abstract
Cellular membranes are a heterogeneous mix of lipids, proteins and small molecules. Special groupings enriched in saturated lipids and cholesterol form liquidordered domains, known as “lipid rafts,” thought to serve as platforms for signaling, trafficking and material transport throughout the secretory pathway. Questions remain as to how the cell maintains small fluid lipid domains, through time, on a length scale consistent with the fact that no largescale phase separation is observed. Motivated by these examples, we have utilized a combination of mechanical modeling and in vitro experiments to show that membrane morphology plays a key role in maintaining small domain sizes and organizing domains in a model membrane. We demonstrate that lipid domains can adopt a flat or dimpled morphology, where the latter facilitates a repulsive interaction that slows coalescence and helps regulate domain size and tends to laterally organize domains in the membrane.
The plasma and organelle membranes of cells are composed of a host of different lipids, lipophilic molecules, and membrane proteins (1). Together, they form a heterogeneous layer capable of regulating the flow of materials and signals into and out of the cell. Lipid structure and sterol content play a key role in bilayer organization, where steric interactions and energetically costly mismatch of lipid hydrophobic thickness result in a line tension that induces lateral phase separation (2). Saturated lipids and cholesterol are sequestered into liquidordered (L _{o}) domains, often known as “lipid rafts,” distinct from an unsaturated liquiddisordered (L _{d}) phase (3 –5). Domains whose lipids include saturated sphingolipids and cholesterol, with sizes in the range of ≈50–500 nm, have been implicated in a range of biological processes from lateral protein organization and virus uptake to signaling and plasmamembrane tension regulation (6–18). In the biological setting, maintenance of small domain size is thought to arise from a combination of lipid recycling and energetic barriers to domain coalescence (19–21) [potentially provided by transmembrane proteins (22)], ostensibly resulting in a stable distribution of domain sizes. These biological examples serve as a motivation to better understand the biophysical mechanisms that maintain small lipid domains over time and pose challenges to the classical theories of phaseseparation and “domain ripening” [such as Cahn–Hilliard kinetics (23)].
A simple physical model that describes the evolution of lipid domain size and position predicts that domains diffuse and coalesce, such that the number of domains constantly decreases, whereas the average domain size constantly increases (23). Indeed, models of 2D phase separation have been studied in detail for many physical systems (24–27), and where the phase boundary is unfavorable and characterized by an energy per unit length (28), the domain size grows continuously (23, 29, 30). However, membranes can adopt 3D morphologies that affect the kinetics of phase separation (31–35). In those cases where morphology is considered as part of the phase separation model, previously uncharacterized coalescence kinetics emerge (32). Experimentally, model membranes have shown that nearly complete phase separation on the surface of a giant unilamellar vesicle can be reached in as little as 1 minute (3). With these facts in mind, our central questions are: How can model membranes that have phase separated maintain a distribution of small domain sizes on long time scales and short length scales? Are there membranemediated (i.e., elastic) forces that inhibit coalescence and spatially organize domains?
We begin to answer these questions by examining the energetics of the membrane using a linear elastic model. A phaseseparated membrane is endowed with bending stiffness, membrane tension, an energetic cost at the phase boundary, and domains of a particular size. Membrane bending and tension establish a natural length scale over which a morphological instability develops that switches domains from a flat to “dimpled” shape, similar to classical Euler buckling (36) (see Fig. 1). The dimpling instability is sizeselective and “turns on” a membranemediated interaction that inhibits domain coalescence. This transition is a precursor to budding and is distinct from transitions that require spontaneous curvature. Although variations in membrane composition may change specific parameter values, the mechanical effects we describe are generic. Thus, these systems exhibit shapedependent coarsening kinetics that are relevant for a broad class of 2D phaseseparating systems. The interaction between domains is a mechanical effect, and we use a model treating dimpled domains as curved rigid inclusions to distill the main principles governing this interaction. Experimentally, we use a model mixture of lipids and cholesterol to show that such an interaction exists between dimpled domains and is well approximated by a simple model. We hypothesize that a combination of lipid recycling (19) and elastic interactions could serve as a mechanism for the organization of domains and the maintenance of small domain sizes in cellular membranes.
The first section of the article outlines the energetic contributions to the mechanical model and predicts the conditions under which domain dimpling occurs. The second section outlines how dimpled domains facilitate an elastic interaction and compares the model interaction to our measurements made in phaseseparated giant unilamellar vesicles.
Elastic Model and Morphological Transitions
The energetics of a lipid domain are dominated by a competition—on one hand, the applied membrane tension and bending stiffness both energetically favor a flat domain; on the other hand, the phase boundary line tension prefers any domain morphology (in 3D) that reduces the boundary length. We use a continuum mechanical model that couples these effects, relating the energetics of membrane deformation to domain morphology. As we will show, this competition results in a morphological transition from a flat to dimpled domain shape, where 2 dimpled domains are then capable of interacting elastically.
Lipid domains in a liquid state naturally adopt a circular shape to minimize the phase boundary length (3), allowing us to formulate our continuum mechanical model in polar coordinates. We employ a Monge representation, where the membrane midplane is described by a height function h(r) in the limit of small membrane deformations (i.e., ∇h < 1). With this height function, we characterize how membrane tension, bending, spontaneous curvature, and line tension all contribute to domain energetics.
Changes in membrane height alter the projected area of the membrane and hence do work against the applied membrane tension, resulting in an increase in energy written as where τ is the constant membrane tension, r _{o} is the projected radius of the domain, h _{1} is the height function of the domain, and h _{2} is the height function of the surrounding membrane (37, 38). Membrane curvature is penalized by the bending stiffness with a bending energy written as (37, 39) This model allows the domain and surrounding membrane to have differing stiffnesses, κ_{b} ^{(1)} and κ_{b} ^{(2)} respectively, characterized by the parameter σ = κ_{b} ^{(1)}/κ_{b} ^{(2)}, and from this point on we drop the superscript on κ_{b} ^{(2)}. Recent experiments suggest that the bending moduli of a cholesterolrich domain and the surrounding membrane are approximately equal (5, 40), and hence for simplicity, we assume that the bending moduli of the 2 regions are equal (i.e., σ = 1), unless otherwise noted. In addition to bending stiffness, the domain may exhibit a preferred “spontaneous” curvature due, for instance, to lipid asymmetry (35, 41). The contribution of domain spontaneous curvature can be simplified to a boundary integral that couples to the overall curvature field by where c _{o} is the spontaneous curvature of the domain, and ɛ is the membrane slope at the phase boundary. Furthermore, we assume the saddlesplay curvature moduli are equal in the 2 regions, yielding no dependence on Gaussian curvature. In principle, this contribution could be accounted for with a boundary term, explored in detail in supporting information (SI) Appendix. The phase boundary line tension is applied to the projected circumference of the domain, as shown in Fig. 1, by G _{line} = 2πr _{o}γ, where γ is the energy per unit length at the phase boundary.
Finally, a constraint must be imposed that relates the actual domain area, A, to the projected domain radius r _{o}. The energetic cost to change the area per lipid molecule is high [≈50 –100 k_{B} T/nm^{2} where k _{B} = 1.38 × 10^{−23} J/K and T = 300 K (42)]; hence, we assume the domain area is conserved during any morphological change (see SI Appendix for details). We impose this constraint using a Lagrange multiplier, τ_{o}, with units of tension by This results in an effective membrane tension within the domain τ_{1} = τ + τ_{o}, which must be negative to induce dimpling. Examining the interplay between bending and membrane tension, we see that 2 natural length scales emerge—within the domain we define \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $${\lambda }_{1}=\sqrt{\sigma {\kappa }_{b}/{\tau }_{1}}$$ \end{document} , and outside the domain we define \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $${\lambda }_{2}=\sqrt{{\kappa }_{b}/\tau }$$ \end{document} . These length scales allow us to define the relevant dimensionless parameters in this system.
The total free energy of an elastic domain and its surrounding membrane is then the sum of these 5 terms, G = G _{tens} + G _{bend} + G _{spont} + G _{line} + G _{area}. Details on all the terms in the free energy can be found in SI Appendix. With this free energy in hand, we examine how the morphology of a circular domain evolves as we tune domain size and the elastic properties of the membrane.
The height field and radius can be rescaled by the elastic decay lengths such that the Euler–Lagrange equation for the domain can be written in the parameterfree form ∇^{2}(∇^{2} + β^{2})η_{1} = 0, whereas the equation for the surrounding membrane is ∇^{2}(∇^{2} − 1)η_{2} = 0, where the dimensionless variables are defined by λ_{2}η_{i} = h _{i}, λ_{2}ρ = r, λ_{2}ρ_{o} = r _{o} and β = iλ_{2}/λ_{1}. Using the same dimensionless notation, the energy from line tension and spontaneous curvature can be written as G _{line} = 2πκ_{b}ρ_{o}χ and G _{spont} = −2πσκ_{b}ɛρ_{o}υ_{o}, with υ_{o} = λ_{2} c _{o} and χ = γλ_{2}/κ_{b}. The dimensionless line tension, χ, is simply a rescaled version of the line tension γ and is 1 of 2 key parameters that characterize the morphological transition; the dimensionless domain area, α = A/λ_{2} ^{2}, is the second key parameter.
The admissible solutions for η_{1}(ρ) and η_{2}(ρ) are zeroth order Bessel functions J _{0}(βρ) and K _{0}(ρ), respectively, with the boundary conditions ∇η_{1}(0) = ∇η_{2}(∞) = 0 and ∇η_{1}(ρ_{o}) = ∇η_{2}(ρ_{o}) = ɛ. The boundary slope, ɛ, is the parameter that indicates the morphology of the domain; ɛ = 0 indicates a flat domain, whereas 0 < ɛ ≲ 1 indicates a dimpled domain. The 5 contributions to membrane deformation energy yield a relatively simple expression for the total free energy, given by Mechanical equilibrium is enforced by rendering the energy stationary with respect to the unknown parameters ɛ, ρ_{o}, and β, These equilibrium equations physically correspond to torque balance at the phase boundary, lateral force balance at the phase boundary, and domain area conservation, respectively.
Analysis of the equilibrium equations reveals a secondorder transition at a critical linetension, χ_{c}, as shown in Fig. 2. For χ less than this critical value, only the flat, trivial solution with ɛ = 0 exists. At χ_{c} a nontrivial solution describing buckled or dimpled morphologies emerges. For zero spontaneous curvature, the bifurcation is defined by a transcendental characteristic equation with \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\beta =\sqrt{({\chi }_{c}/{\rho }_{o}1)/\sigma }$$ \end{document} and \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $${\rho }_{o}=\sqrt{\alpha /\pi }$$ \end{document} . For a given dimensionless domain area, α, this defines the critical line tension required to dimple the domain. In Fig. 2 A Inset, this relation is used to generate a morphological phase diagram that shows where in the space of dimensionless domain area and line tension we find the discontinuous transition (i.e., bifurcation) from a flat domain to a dimpled domain. Near the morphological transition the boundary slope grows as \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\left\varepsilon \right\propto \sqrt{\chi /{\chi }_{c}1}$$ \end{document} , indicating that a dimple rapidly deviates from the flat state. The transition is symmetric, in that both possible dimple curvatures have the same energy, and hence the domain is equally likely to dimple upwards or downwards. In the experimentally relevant limit of small dimensionless domain area, the complexity of Eq. 7 is reduced to This leads to the conclusion that the dominant parameter governing domain dimpling at zero spontaneous curvature is \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} $$\chi \sqrt{\alpha }$$ \end{document} . For a small domain, the dimpling transition is directly regulated by domain area, the bending modulus, and line tension but only weakly depends on applied membrane tension. Intuitively, domains dimple when line tension or domain size increase, as shown in Fig. 2 A Inset. Likewise, a decrease in bending stiffness can also induce dimpling. The effects of applied membrane tension are weak because the change in projected area upon dimpling does not lead to a significant energy cost relative to the cost of bending and line tension.
If membrane elastic properties are fixed (i.e., fixed κ_{b}, τ and γ), the dimplinginduced interactions “turn on” only after a critical domain size is achieved. This scenario is encountered when 2 domains, too small to dimple on their own, diffusively coalesce into a larger domain capable of dimpling and hence interacting. Indeed, such a sizeselective coalescence mechanism was observed recently in model membrane vesicles (43). This constitutes a distinct class of coarsening dynamics, where classical diffusionlimited kinetics are obeyed until the domain size distribution has matured past the critical size for dimpling—then domain coalescence is a relatively slow, interactionlimited process.
For the model domain considered in Fig. 2, with area α = π/4 (r _{o} ≃ 250 nm), the critical dimensionless line tension is χ_{c} ≃ 13, corresponding to a critical line tension of γ_{c} ≃ 0.65 k _{B} T/nm (1 k _{B} T/nm = 4.14 pN). This value compares well with theoretical estimates of the line tension (28, 44) and falls squarely in the range of values from AFM measurements (2), though it is slightly higher than the value of γ ≃ 0.22 k _{B} T/nm measured via shape analysis of fully phaseseparated vesicles (5) and γ ≃ 0.40 k _{B} T/nm measured from micropipette aspiration experiments (45). In general, measured values of the line tension depend heavily on bilayer composition, spanning a range of ≈0.05 − 1.5 k _{B} T/nm (2, 5, 45).
Spontaneous curvature does not affect the Euler–Lagrange equations and, hence, will not effect the class of equilibrium membrane shapes. However, domains with zero and nonzero spontaneous curvature exhibit qualitatively different behavior. Membranes can be asymmetric with respect to leaflet composition (6, 46, 47), endowing a domain with potentially large spontaneous curvature. The energetic contribution from spontaneous curvature takes the form of an additional line tension depending linearly on the slope taken by the domain boundary, ɛ. This breaks the symmetry of the membrane, giving an energetic preference to a dimple with the same curvature as the spontaneous curvature and eliminating the trivial ɛ = 0 solution even at small line tensions. As line tension increases, a bifurcation produces a second, stable, higherenergy dimple of the opposite curvature as υ_{o}. The more energetically stable branch of this transition corresponds to a dimpled state for all values of line tension and nonzero values of domain area, as demonstrated in Fig. 2 A. This predicts that as soon as a domain with finite spontaneous curvature forms, it dimples, regardless of size, and begins to experience interactions with any nearby dimpled domains. It is reasonable to expect that domains with similar composition will have similar spontaneous curvature, and hence form dimples whose curvature has the same sign. As we will show, dimples whose curvature has the same sign tend to interact repulsively. Such a mechanism of coalescence inhibition was observed recently in simulation (35). This indicates that control of spontaneous curvature via domain composition can regulate dimpling and hence, domain interaction (47, 48). Indeed, recent theoretical (49) work shows that lipid asymmetry leads to precisely these kinds of dimpled domains.
Calculated shapes of dimpled domains induced by line tension and spontaneous curvature are shown in Fig. 3 A, alongside dimpled domains observed on giant unilamellar vesicles, shown in Fig. 3 B and D.
Elastic Interactions of Dimpled Domains
Given 2 domains that have met the criteria for dimpling, the deformation in the membrane surrounding the domains mediates an elastic interaction when they are within a few elastic decay lengths (λ_{2}) of each other. This equips us to begin addressing how small membrane domains might be achieved on short and long time scales. As previously stated, free diffusion sets the maximum rate at which a quenched membrane can evolve into a fully phaseseparated membrane (23), where this evolution can happen in as little as a minute on the surface of a giant unilamellar vesicle (GUV) (3). By comparison, recycling and, hence, homogenization of cellular membrane is a process that takes place on the time scale of an hour or more (50). Our measurements of domain interactions (detailed below and other data shown in SI Appendix) estimate the coalescence barrier between dimpled domains at ≈5 − 10 k _{B} T. Hence, given the diffusionlimited rate of coalescence, interactions slow this process by approximately e ^{−5} ≃ 0.007 to e ^{−10} ≃ 0.00005.
The physical origin of domain interaction is explained by a simple model based on the assumption that the dimpled domain shape is constant during interaction, but the domains are free to tilt by an angle ϕ, as shown in Fig. 3 C. This assumption was, in part, inspired by experimental observations of domain shapes on the surface of giant unilamellar vesicles, as shown, for example, in Fig. 3 D. The interaction energy is approximately an order of magnitude less than the free energy associated with the morphological transition itself (see Fig. 2 B), thus interaction does not perturb the domain shape significantly. Only allowing domains to rotate simplifies the interaction between 2 domains to a change in the boundary conditions in the 3 regions of interest, shown in blue in Fig. 3 C. Applying the small gradient approximation, the boundary slope is given by ɛ − ϕ in the outer regions and by ɛ + ϕ in the inner region. With the single domain boundary slope, ɛ, set by the energy minimization of the previous section (i.e., eq. 6), the pairwise energy is minimized at every domain spacing, d, by ∂G/∂ϕ = 0 to find the domain tilt angle that minimizes the deformation energy (see SI Appendix for details). This results in 2 qualitatively distinct scenarios: 2 domains whose curvatures have the same sign repel each other, whereas 2 domains whose curvatures have the opposite sign attract each other. Scaling arguments can be used to show that the strength of interaction between 2 dimpled domains increases approximately linearly with their area so long as they are both larger than some critical area (see SI Appendix for details). Mathematically, the assumption of rigidly rotating dimpled domains is identical to a previous 2D model of bendingmediated interactions between intramembrane proteins, represented by rigid conical inclusions (51).
Independent of the effects of spontaneous curvature, slight osmolar imbalances and constriction due to the lipid phase boundaries create small pressure gradients across the membrane that tend to orient all dimples on a vesicle in the same direction, resulting in net repulsive interactions between all domains. Transitions between “upward” and “downward” dimples are infrequent due to a large energy barrier. In the simplest case, where the domains are the same size, the tilt angle ϕ monotonically increases as 2 domains get closer, ϕ(d) ≃ −ɛe ^{−d}. Likewise, the interaction energy increases monotonically with decreasing domain separation, V _{int}(d) ≃ 2πκ_{b}ɛ^{2}ρ_{o} ^{2} e ^{−d}. To quantitatively compare our interaction model with experiment, we analyzed the thermal motion of small domains on the surface of giant unilamellar vesicles, as described in Materials and Methods. For direct comparison, we fit both the 1D model outlined here and the 2D inclusion model (51) to the measured potential of mean force between domains, as shown in Fig. 4. The 2 models are experimentally indistinguishable, though with a slightly different elastic decay length.
In these experiments, membrane tension was regulated by balancing the internal and external osmolarity, giving us coarse control over the elastic decay length λ_{2}. Through time, the distance between every domain pair was measured, and the net results were used to construct a histogram of centertocenter distance probability, the natural logarithm of which is the potential of mean force, as shown in Fig. 4 B. We selected vesicles that had a low density of approximately equalsized domains, and thus, generally, the interactions were described by a repulsive pairwise potential. Though areal density of domains and generic data quality varied in our experiments (see SI Appendix), all datasets exhibit the repulsive core of the elastic interaction. Multibody interactions occur, though infrequently; their effect can be seen as a small variation in the baseline of Fig. 4 B, which is not captured by the pairwise interaction model. At high membrane tension, when we would not expect dimpled domains, we qualitatively verified that domains coalesce in a rapid manner as compared with our lowtension experiments (data not shown). Other recent experiments have also observed repulsive interactions and a correspondingly slower rate of coalescence between domains on low membrane tension vesicles, and a marked increase in coalescence kinetics on the surface of taut vesicles (43).
Our measurement of the pairwise potential allows us to estimate elastic properties of the membrane. The elastic decay length was fit to the 1D and 2D interaction models described above and found to be λ_{2} ^{(1D)} ≃ 240 nm and λ_{2} ^{(2D)} ≃ 270 nm, respectively. Taken with a nominal bending modulus of 25 k _{B} T, we estimate the membrane tension to be ≈4 × 10^{−4} k _{B} T/nm^{2}. From the images, we measure the size of the domains at r _{o} ≃ 350 − 400 nm, and hence ρ_{o} ≃ 1.5. We estimate the line tension, γ, using Eq. 8, based on the fact that the domains are dimpled, and find a lower bound of γ ≃ 0.49 k _{B} T/nm (1 k _{B} T/nm = 4.14 pN). This is in good agreement with theoretical estimates and values determined from experiment as discussed above. Finally, viewing the repulsive core of the interaction as an effective activation barrier to coalescence, a simple Arrhenius argument suggests a decrease in coalescence kinetics by 2–3 orders of magnitude. Indeed, such a slowing of coalescence was recently observed in a similar model membrane system (43).
Discussion
Our experiments on the surface of GUVs have 3 potentially confounding effects, all due to the spherical curvature of the vesicle. First, the surface metric is not entirely flat with respect to the image plane. Thus, measurements of distance are underestimated the farther they are made from the projected vesicle center. This problem is ameliorated by concentrating on domains that are at the bottom (or top) of the vesicle where the surface is nearly flat and demanding that our tracking software exclude domains that are out of focus; see SI Appendix for a more detailed explanation. The second potential complication is that we use a flat 2D coordinate system for our theoretical analysis; however, domains reside on a curved surface. Given that the domain deformation, and hence energy density, decays exponentially with λ_{2}, as long as λ_{2} is small with respect to the vesicle radius, the energetics that govern morphology converge on an essentially flat surface metric. The final complication is that the circular area of focus creates a fictitious confining potential for the domains, such that the effective measured potential of mean force is the sum of the elastic pairwise potential and a fictitious potential, V _{eff} = V _{int} + V _{fict}. The fictitious potential is removed by simulating noninteracting particles in a circle the same size as the radius of focus (see SI Appendix for details).
The constant tension ensemble used in our theoretical analysis has an experimental range of validity, determined by the excess area available on a thermally fluctuating membrane with conserved volume and total surface area A_{o} (i.e., a vesicle). In the limit where the morphological transitions use only a small portion (ΔA) of this excess area, defined by k _{B} T/8πκ_{b} ≫ ΔA/A_{o}, the tension is constant. Outside this regime, the tension rises exponentially with reduction in excess area, tending to stabilize dimples from fully budding (see SI Appendix for details).
In addition to the elastic mechanism of interaction, described herein, there may be other organizing forces at work in a phaseseparated membrane, for instance, those of elastic (28) or entropic (52, 53) origin. However, the putative length scale over which these effects compete with thermal fluctuations (on the order of tens of nanometers) is not accessible to the spatial and temporal resolution of our experiments. Electrostatics may also be at work, in the form of dipole–dipole repulsion due to the net difference in dipole density between the 2 phases (54 –56), although to first order, symmetry suggests there is a net zero dipole moment per unit volume of bilayer (57). In our experimental system, the modulator of repulsive interactions is membrane morphology (i.e., domain dimpling); if other interactions were a major repulsive effect, we would not expect such forces to depend markedly on largescale membrane morphology.
Conclusion
We have shown that lipid domains are subject to a morphological dimpling transition that depends on the bilayer elastic properties and domain size. Dimpling allows 2 domains in proximity to repulsively interact due to the deformation in the surrounding membrane. Our model makes 2 key predictions: (i) at zero spontaneous curvature, the domain size distribution reaches a critical point where coalescence is arrested by repulsive interactions (43) and (ii) domains with finite spontaneous curvature are always subject to interaction and hence should always coalesce at a rate slower than the diffusionlimited rate (35). Additionally, the strength of elastic interactions is augmented by increasing line tension or domain area, with an approximately linear scaling. We proposed a simple 1D model of an elastic interaction that mediates dimpleddomain repulsion and then used a standard ternary membrane system to verify the existence of dimpled domains and their subsequent repulsive interaction. Our model offers a mechanism that works against diffusiondriven coalescence, to maintain small lipid domains over time.
Materials and Methods
GUVs were prepared from a mixture of DOPC (1,2dioleoylsnglycero3phosphocholine), DPPC (1,2dipalmitoylsnglycero3phosphocholine) and cholesterol (Avanti Polar Lipids, Inc.) (25:55:20/molar) that exhibits liquid–liquid phase coexistence (3). Fluorescence contrast between the 2 lipid phases is provided by the rhodamine headgrouplabeled lipids: DOPE (1,2dioleoylsnglycero3phosphoethanolamineN(lissamine rhodamine B sulfonyl)) or DPPE (1,2dipalmitoylsnglycero3phosphoethanolamineN(lissamine rhodamine B sulfonyl)), at a molar fraction of ≈0.005. The leaflet compositions are presumed symmetric, and, hence, υ_{o} = 0.
GUVs were formed via electroformation (3, 58). Briefly, 3–4 μg of lipid in chloroform were deposited on an indium–tin oxidecoated slide and dessicated for ≈2 h to remove excess solvent. The film was then hydrated with a 100 mM sucrose solution and heated to ≈50°C to be above the miscibility transition temperature. An alternating electric field was applied; 10 Hz for 120 m, 2 Hz for 50 m, at ≈500 V olts/m over ≈2 mm. Low membrane tensions were achieved by careful osmolar balancing with sucrose (≈100 mM) inside the vesicles, and glucose (≈100 – 108 mM) outside.
Domains were induced by a temperature quench (see SI Appendix) and imaged by using standard TRITC epifluorescence microscopy at 80× magnification with a cooled (−30°C) CCD camera (6.7 × 6.7 μm^{2} per pixel, 20 MHz digitization; Roper Scientific). Images were taken from the top or bottom of a GUV where the surface metric is approximately flat (see SI Appendix). Datasets contained ≈500 –1,500 frames collected at 10–20 Hz with a varying number of domains (usually 5–10). The frame rate was chosen to minimize exposuretime blurring of the domains while allowing sufficiently large diffusive domain motion. Software was written to track the position of each wellresolved domain and calculate the radial distribution function. The raw radial distribution function was corrected for the fictitious confining potential of the circular geometry (see SI Appendix). In the dilute interaction limit, pairwise interactions dominate, and the negative natural logarithm of the radial distribution function is the interaction potential (potential of mean force) plus a constant, as shown in Fig. 4 B.
Note Added in Proof.
Just recently, another group (59) has independently come to similar conclusions about the presence of elastically mediated interactions among dimpled domains, specifically commenting on their tendency to order domains.
Acknowledgments
We thank Patricia Bassereau, Evan Evans, Ben Freund, Kerwyn Huang, Greg Huber, Sarah Keller, and Udo Seifert for stimulating discussion and comments on the manuscript and Jenny Hsaio for help with experiments. T.U. and R.P. acknowledge the support of National Science Foundation (NSF) Award CMS0301657, NSF Award ACI0204932, Nanoscale Interdisciplinary Research Teams Award CMS0404031, and the National Institutes of Health Director's Pioneer Award, and National Institutes of Health Award RO1 GM084211. W.S.K. acknowledges support from NSF CAREER Award CMMI0748034.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: phillips{at}pboc.caltech.edu

Edited by L. B. Freund, Brown University, Providence, RI, and approved June 10, 2009

Author contributions: T.S.U. and W.S.K. performed research; T.S.U. analyzed data; and T.S.U., W.S.K., and R.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0903825106/DCSupplemental.
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