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Hierarchical organization of chiral rafts in colloidal membranes

Liquid-liquid phase separation is ubiquitous in suspensions of nanoparticles, proteins and colloids. With a few notable exceptions, surface-tension-minimizing liquid droplets in bulk suspensions continuously coalesce, increasing in size without bound until achieving macroscale phase separation. In comparison, the phase behavior of colloids, nanoparticles or proteins confined to interfaces, surfaces or membranes is significantly more complex. Inclusions distort the local interface structure leading to interactions that are fundamentally different from the well-studied interactions mediated by isotropic solvents. Here, we investigate liquid-liquid phase separation in monolayer membranes composed of dissimilar chiral colloidal rods. We demonstrate that colloidal rafts are a ubiquitous feature of binary colloidal membranes. We measure the raft free energy landscape by visualizing its assembly kinetics. Subsequently, we quantify repulsive raft-raft interactions and relate them to directly imaged raft-induced membrane distortions, demonstrating that particle chirality plays a key role in this microphase separation. At high densities, rafts assemble into cluster crystals which constantly exchange monomeric rods with the background reservoir to maintain a self-limited size. Lastly, we demonstrate that rafts can form bonds to assemble into higher-order supra-structures. Our work demonstrates that membrane-mediated liquid-liquid phase separation can be fundamentally different from the well-characterized behavior of bulk liquids. It outlines a robust membrane-based pathway for assembly of monodisperse liquid clusters which is complementary to existing methods which take place in bulk suspensions. Finally, it reveals that chiral inclusions in membranes acquire long-ranged repulsive interactions, which might play a role in stabilizing assemblages of finite size.

demonstrate that rafts can form bonds to assemble into higher-order suprastructures. Our work demonstrates that membrane-mediated liquid-liquid phase separation can be fundamentally different from the well-characterized behavior of bulk liquids. It outlines a robust membrane-based pathway for assembly of monodisperse liquid clusters which is complementary to existing methods which take place in bulk suspensions [14][15][16] . Finally, it reveals that chiral inclusions in membranes acquire long-ranged repulsive interactions, which might play a role in stabilizing assemblages of finite size 11, 17 .
In the presence of non-adsorbing polymer, mono-disperse rod-like viruses experience effective depletion attractions that drive their lateral association. These interactions can lead to assembly of one-rod-length-thick colloidal monolayer membranes that are held together by the osmotic pressure of the enveloping polymer suspension 18 . We allow membranes to sediment to the bottom of the sample chambers in which case the constituent rods point in the z direction, while all images are taken in the x-y plane.
Although they differ on molecular scales, the long-wavelength fluctuations of colloidal monolayers and lipid bilayers are described by the same free energy. In this work, we investigated the behavior of colloidal membranes containing a mixture of two rods: 880 nm long rod-like fd-Y21M virus and 1200 nm long M13KO7 virus 19 . Membranes were prepared by adding a depletant to a dilute isotropic fd-Y21M/M13KO7 mixture. For all parameters investigated, both rods co-assembled into binary membranes. At low depletant concentration the rods remained homogeneously mixed throughout the membrane (Fig. 1b,d), while at high depletant concentration they bulk phase separated ( Fig. 1b,f). Surprisingly, at intermediate concentrations we observed the formation of highly-monodisperse micron-sized droplets (colloidal rafts) enriched in short rods and floating in the background of long rods (Fig. 1e). Colloidal rafts did not coarsen with time suggesting that they are equilibrium structures.
To test this hypothesis, we brought two membranes, each containing only one type of virus, into close proximity (Fig. 1a). Upon coalescence, the interface separating dissimilar rods became unstable as short rods invaginated into the long-rod fluid, giving rise to an interconnected network of rafts held together by thin metastable bridges (Fig.   1c, Supplementary Movie 1). Over time the conjoining liquid bridges disintegrated, leading to the formation of isolated finite-sized rafts thus demonstrating that colloidal rafts have a lower free-energy than a bulk separated phase. We have explored the behavior of six phase-separating mixtures comprised of viruses of varying physical properties. In each case, we observed micro-separated phases at intermediate polymer concentrations, suggesting that rafts are a ubiquitous feature of colloidal membranes.
Here, we investigate the behavior of the fd-Y21M/M13KO7 mixture, utilizing the unique features of colloidal membranes that allow us to optically manipulate rafts, quantify their nucleation dynamics as well as track the dynamics of the constituent rods.
We determined the raft free-energy landscape by measuring how k on and k off , the rates at which rods associate and dissociate from a raft, depend on the raft size. We measured k off using Fluorescence Recovery After Photobleaching (FRAP) 20 . Using a focused beam, we bleached an isolated raft of predetermined size. The fluorescence signal recovered within minutes, indicating exchange of bleached raft-bound rods with unbleached background rods (Fig. 2a, Supplementary Movie 2). The fluorescence recovery curves were fit by single exponentials (Fig. 2e). If the raft size does not change during of experiment, the fluorescence recovery time constant is 1/k off (Methods). By performing FRAP measurements on rafts of varying radii we determined the dependence of k off on raft size (Fig. 2h).
To measure k on , we created a raft population with heterogeneous radii R and quantified their subsequent growth rates (Fig. 2b, c, f, Supplementary Movie 3). Rafts with R<R critical quickly evaporated into the background membrane (Fig. 2g). When R equilibrium >R>R critical rafts grew until attaining the equilibrium size. Finally rafts with R> R equilibrium shrank slowly until achieving equilibrium. The evolution of raft size is governed by the kinetic equation: , where N(t) is the number of raft-bound rods and C BG is the concentration of short fluorescently-labeled rods in the background.
is determined from size evolution experiments ( Fig. 2g), k off is determined through FRAP experiments, and C BG is proportional to mean fluorescence of the background membrane. The only unknown is k on , making it possible to determine its dependence on raft size. We found that k on increases weakly with increasing raft size, consistent with 2D diffusion-limited association kinetics. Having measured both k on and k off (Fig. 2h), , where NA rod is the raft area and ΔG(N) is the energy cost of adding a single particle to a raft (Fig. 2i). The minimum of the excess free energy satisfies ΔGk B Tln(C BG A rod )=0, and determines the equilibrium raft size, which corresponds to ~24000 rods. Integrating ΔG over N yields the total free-energy G(N). The width of the parabola ) ( determines the fluctuations in raft area, which is ~1400 particles, corresponding to 30 nm radius fluctuations. Such fluctuations are below the optical microscope resolution limit; consistent with our observation of a monodisperse raft distribution in equilibrium. To independently validate our analysis, we performed single-molecule lifetime experiments by tracking a low-fraction of fluorescently labeled short rods in an otherwise unlabeled membrane (Fig. 2d, Supplementary Movie 4). Overlaying fluorescent images onto simultaneously acquired phase contrast images revealed raft-bound rods as a function of time, (Fig. 2j). From such data we find that for equilibrium rafts k on =3.7 min -1 µm 2 and k off =0.15 min -1 , which is in agreement with FRAP measurements (Methods). To summarize colloidal membranes enabled us to directly visualize raft nucleation dynamics and measure their free-energy landscape. So far such measurements were possible in a very few experimental systems 14,21 .
With an understanding of single-raft assembly kinetics, we subsequently measured membrane-mediated raft-raft interactions by adapting a blinking optical tweezers technique 22 . To manipulate rafts, which are repelled from a trapping laser, we created an optical plow composed of multiple beams. Using two plows we pushed a raft pair into close proximity. Subsequently, we switched off the trap and quantified raft trajectories as they were pushed apart by repulsive interactions (Supplementary Movie 5). Repeating this procedure dozens of times yielded the inter-raft potential (Methods). The effective repulsive potential is described by an exponential with a characteristic length scale of 0.65µm (Fig. 3a). The measured raft-raft interactions can be quantitatively related to raftinduced membrane distortions visualized by LC-PolScope. This technique yields images ) (t N f whose intensity indicates the local retardance, which is related to the local tilt of rods away from the membrane normal vector (Fig. 3c) 23 . The interior of a homogeneous onecomponent membrane viewed from above lacks optical anisotropy and appears dark in the LC-PolScope. In contrast, rafts imaged with LC-PolScope exhibited a spatially varying retardance, indicating local twisting of raft-bound rods.
Such deformations can be understood by considering that fd Y21M and M13KO7 have right-and left-handed chirality, respectively 19,24 . As a consequence, the interaction energy between membrane-embedded viruses is minimized when neighboring rods are not parallel to each other, but are twisted at a small but finite angle. The radially averaged retardance profile reveals that, starting from the raft center, fd Y21M twist with a righthanded sense, leading to increasing rod tilt away from the monolayer normal. The tilt attains maximum at the raft's edge. Moving past the edge, the membrane is enriched in left-handed M13KO7. The left-handed twist induces tilting of rods back toward the membrane normal. Larger rafts attain a larger maximum edge tilt; however, the characteristic length over which twist penetrates into the background membrane is independent of raft size (Fig. 3d,e). The length scales associated with raft repulsion and raft induced twist deformation are comparable, suggesting a possible link.
Homo-chiral colloidal membranes are inherently frustrated, since the constituent rods cannot simultaneously twist locally and assemble into a monolayer globally.
Consequently, twist is expelled from the membrane interior to the edges. Increasing the rod chirality raises the free energy of interior untwisted rods while lowering the free energy of edge-bound twisted rods, leading to chiral control of edge line tension 25,26 . For uniformly mixed binary membranes, the same constraints enforce untwisting of all constituent chiral rods, thus raising their free energy. However, formation of right-handed rafts in the left-handed background allows twist to penetrate into the membrane interior, lowering raft free energy. Besides explaining raft stability, this picture also explains the origin of inter-raft repulsion. Bringing two rafts together closer than the twist penetration length requires energetically costly untwisting of chiral rods. Our hypothesis predicts that the range of raft repulsion is independent of cluster size, while its strength increases linearly with cluster size. Measured potentials between clusters of different sizes agree with these predictions (Fig. 3a, b). Furthermore, when the magnitude of the repulsive interactions is rescaled by the maximum twist at the rafts edge, all potentials collapse on top of each other, thus establishing a quantitative relationship between raft-induced chiral distortions and membrane-mediated raft interactions. These experiments describe a chiral mechanism for formation of finite-sized rafts that is complementary to others proposed mechanisms [27][28][29] .
The emergent raft repulsions can be used to assemble higher-order structures. To explore this capability, we have increased raft densities and measured the raft radial distribution function, , of resulting assemblages (Supplementary Movie 6). In the dilute limit, is zero for small separations, indicative of strong repulsive interactions, and constant elsewhere (Fig. 4a). At intermediate raft densities, exhibits liquid-like oscillations that decay after a few coordination shells (Fig. 4b). At the highest densities, rafts form a 2D crystal-like structure as evidenced by sharp peaks in (Fig. 4c). This behavior is reminiscent of conventional 2D repulsive colloids. However, there are also important differences, since colloidal crystals are assembled from immutable solid particles while the membrane-embedded crystals are assembled from highly adaptable, size-adjustable rafts, each of which constantly exchanges rods with the background.
Moreover, while conventional droplets in bulk suspensions always assume a surfaceminimizing spherical shape, the behavior of rafts embedded in a membrane is significantly more complex. For example, it is possible to assemble supra-rafts, which are liquid droplets with distinctly non-spherical shapes. At higher ionic strength and/or depletant concentrations, isolated rafts bind together to form dimers, trimers and higherorder rafts bound together by thin liquid bridges (Fig. 4d, Supplementary Movie 7).
Using such bonding rules it is possible to assemble structures of almost arbitrary complexity such as a classical bead-spring polymer (Fig. 4e, Supplementary Movie 8).
In conclusions, we have demonstrated assembly of thermodynamically stable, monodisperse, surfactant-free liquid droplets which are driven by membrane-mediated repulsive chiral interactions that are orders of magnitude longer-ranged than attractive depletion interactions 30 . These results may be relevant for biological membranes, which contain diverse chiral inclusions. In particular they suggest chirality as a plausible mechanism that stabilizes assembly of finite-sized clusters such as lipid rafts.
Intriguingly, it is known that cholesterol, an effective chiral dopant of conventional liquid crystals, is essential for the assembly of lipid rafts. Correspondence and request for materials should be addressed to Z. D. (zdogic@brandeis.edu)

Materials and Methods
Virus growth and sample preparation: Colloidal rafts were assembled from membranes containing a mixture of filamentous bacteriophages fd-Y21M and M13K07.
The rod-like fd-Y21M virus has a 6 nm diameter, 880 nm contour length and a 9.9 µm persistence length 19 . M13KO7 has the same diameter, but a 1200 nm contour length and persistence length of 2.8 µm 31 where k off and k on are rod dissociation and association rates and C BG is the concentration of fluorescent fd-Y21M rods in the background membrane 20 . The experimentally measured total raft intensity is given by I tot (t)=αN F (t)+βN B (t) where α and β are the intensities of single fluorescent labeled and bleached viruses, respectively. Perfect bleaching would result in β=0. However, we find that this not the case as bleached rods retain finite fluorescence. Solving for I tot (t) yields: , ( where I 0 is the raft fluorescence intensity after completion of the photo-bleaching event and is the mean fluorescence intensity of the background membrane. Eq. (1) shows that the fluorescence recovery process can be approximated by a single exponential rate, in agreement with experimental findings. The exponential time constant yields the rod dissociation constant 1/k off and its dependence on raft size (Fig. 2e,h).

Evolution of polydisperse rafts yields
where N(t) is the total number of raft bound rods at time t. Cluster size evolution experiments yielded and N(t ) (Fig. 2g), while C BG was extracted from fluorescent membrane images, and k off and its size were obtained from previously described single raft FRAP measurements. It follows that k on and its dependence on raft size can be uniquely determined from Eq. 2, as all other parameters are determined from independent experiments.
Measuring the raft free energy landscape: The change in free energy, ΔG, to add a single rod to a raft containing N rods is given by: Where A rod is the effective area occupied by a single fd-Y21M virus, N is the number of rods in a raft and A rod N is the raft size. Assuming dilute suspension conditions, the chemical potential of fd-Y21M rods dissolved in the background membrane is given by: . The net free energy change associated with taking a single fd-Y21M virus from the background and inserting it into a raft containing N rods is given by Fig. 2i). The absolute raft free energy landscape up to a constant offset was obtained by numerically integrating ΔG-µ using a cumulative trapezoidal rule (Fig. 2i).

Single molecule analysis:
We prepared raft bearing membranes with non-fluorescent long and short rods. These membranes were doped with a very low volume fraction of fluorescent short rods (∼ 0.003%) enabling us to visualize the motion of single rods within the membrane background and rafts. The membranes were simultaneously visualized in phase contrast and fluorescence microscopy to determine the number of raft-bound labeled rods, N f . Standard analysis shows that the decay of the autocorrelation function of N f is given by: 37 . Therefore, k off and k on can be determined uniquely using this expression. These measurements were performed for equilibrium sized rafts only. Raft separations were measured as a function of time, once the traps were switched off, using standard video tracking methods 38 . The time lapse between successive frames was 100ms and the exposure time was 50ms.
The discretized probability of a raft pair being separated by distance r j at time t+Δt is given by: , where P is the transition probability for a raft pair initially separated by distance r i at time t to be separated by distance r j at time t+Δt later. ρ(r i ) is the probability of a raft pair being separated by a distance r i at time t 22 . Experimentally, P is determined by binning the trajectories according to the initial and final separations in each time step. The steady state solution to Eq. 1 is equal to the equilibrium probability, ρ(r), computed by calculating the eigenvector of the transition probability matrix. The inter-raft potential is given by: .