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# Achiral symmetry breaking and positive Gaussian modulus lead to scalloped colloidal membranes

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved March 13, 2017 (received for review October 20, 2016)

## Significance

A number of essential processes in biology and materials science, such as vesicle fusion and fission as well as pore formation, change the membrane topology and require formation of saddle surfaces. The energetic cost associated with such deformations is described by the Gaussian curvature modulus. We show that flat 2D colloidal membranes composed of achiral rods are unstable and spontaneously form scalloped edges. Quantitative analysis of such instability estimates the Gaussian curvature modulus of colloidal membranes. The measured sign and magnitude of the modulus can be explained by a simple excluded volume argument that was originally developed for polymeric surfactants.

## Abstract

In the presence of a nonadsorbing polymer, monodisperse rod-like particles assemble into colloidal membranes, which are one-rod-length–thick liquid-like monolayers of aligned rods. Unlike 3D edgeless bilayer vesicles, colloidal monolayer membranes form open structures with an exposed edge, thus presenting an opportunity to study elasticity of fluid sheets. Membranes assembled from single-component chiral rods form flat disks with uniform edge twist. In comparison, membranes composed of a mixture of rods with opposite chiralities can have the edge twist of either handedness. In this limit, disk-shaped membranes become unstable, instead forming structures with scalloped edges, where two adjacent lobes with opposite handedness are separated by a cusp-shaped point defect. Such membranes adopt a 3D configuration, with cusp defects alternatively located above and below the membrane plane. In the achiral regime, the cusp defects have repulsive interactions, but away from this limit we measure effective long-ranged attractive binding. A phenomenological model shows that the increase in the edge energy of scalloped membranes is compensated by concomitant decrease in the deformation energy due to Gaussian curvature associated with scalloped edges, demonstrating that colloidal membranes have positive Gaussian modulus. A simple excluded volume argument predicts the sign and magnitude of the Gaussian curvature modulus that is in agreement with experimental measurements. Our results provide insight into how the interplay between membrane elasticity, geometrical frustration, and achiral symmetry breaking can be used to fold colloidal membranes into 3D shapes.

The possible configurations and shapes of 2D fluid membranes can be described by a continuum energy expression that accounts for the membrane’s out-of-plane deformations as well as the line tension associated with the membrane’s exposed edge (1, 2). Because an arbitrary deformation of a thin layer can have either mean and/or Gaussian curvature, the full theoretical description of membranes, in principle, requires two parameters, the bending and Gaussian curvature moduli. However, lipid bilayers almost always appear as edgeless 3D vesicles, which further simplify theoretical modeling. In particular, integrating Gaussian curvature over any simply closed surface yields a constant (3). Thus, the shape fluctuations of a closed vesicle only depend on the membrane-bending modulus. Consequently, experiments that interrogated mechanics or shape fluctuations of vesicles provided extensive information about the membrane curvature modulus and how it depends on the structure of the constituent particles (4⇓–6). In comparison, significantly less is known about the Gaussian modulus, despite the significant role it plays in fundamental biological and technological processes such as pore formation as well as vesicle fusion and fission (7⇓⇓⇓–11).

Recent experiments have demonstrated that, in the presence of a depleting agent, monodisperse rods robustly assemble into one-rod-length–thick 2D membranes, with in-plane liquid order (12⇓⇓⇓–16). Although more than two orders of magnitude thicker than lipid membranes, the deformations of both colloidal monolayers and lipid bilayers are described by the same elastic energy (17). However, in contrast to conventional membranes that fold into 3D vesicles, colloidal membranes appear as open structures. This presents a unique opportunity to explore the elasticity of 2D fluid sheets, a geometry for which both the Gaussian modulus and edge energy play an important role. Here, we explore the possible shapes of colloidal membranes and demonstrate an unexpected connection between the membrane’s edge structure, Gaussian curvature, and the chirality of the constituent rods.

The semicircular edge profile requires twisting of the rods at the edge, and this twist penetrates into the membrane interior over a characteristic length scale (16, 18, 19). For membranes composed of single-component chiral rods, the handedness of the edge twist along the entire circumference is uniform and dictated by the microscopic chirality of the constituent rods. With decreasing chirality, which is accomplished by mixing rods of opposite handedness, flat 2D circular membranes become unstable, and instead develop complex scalloped edges. In this limit, edge-bound rods exhibit achiral symmetry breaking, forming domains of opposite twist that are separated by cusp-like point defects, where the membrane escapes into the third dimension. The exact structure of the scalloped edge is determined by the competition between the line tension and the Gaussian curvature modulus. Line tension favors circular flat membrane that minimizes the exposed edge. In comparison, an undulating scalloped edge creates excess Gaussian curvature and is thus favored by membranes that have positive Gaussian moduli. Thus, observations of scalloped edges demonstrate that Gaussian modulus of colloidal membranes is positive. Tuning the membrane’s chiral composition effectively controls the interactions between cusp defects that can be either attractive or repulsive. Measurements of these interactions leads to an estimate of the Gaussian curvature modulus that is in agreement with the predictions of a simple theoretical model.

## Structure of Colloidal Membranes

Our experimental model system is a colloidal membrane that spontaneously assembles in a mixture of dilute monodisperse rod-like viruses and nonadsorbing polymer dextran. The viruses alone interact through repulsive screened electrostatic repulsions (20). Addition of nonadsorbing polymer induces attractive depletion interactions that lead to assembly of colloidal membranes, equilibrium structures consisting of a one-rod-length–thick monolayer of aligned rods with a fluid-like internal structure (12). For our experiments, we use wild-type filamentous virus (*fd-wt*) and *fd*-Y21M that differs from its wild-type counterpart by a point mutation in the major coat protein (21). Both viruses have comparable contour length (22); however, studies of bulk cholesteric phase demonstrate that *fd*-*wt* forms a left-handed cholesteric structure, whereas *fd*-Y21M forms a right-handed one (Fig. 1*A*) (23⇓–25). *fd-wt*/*fd*-Y12M mixture forms a homogeneous cholesteric phase with a pitch that depends on the ratio, *x*_{fd} *n*_{fd}/(*n*_{fd} + *n*_{fdY21M}), where *n*_{fd} and *n*_{fdY21M} are the concentration of *fd*-*wt* and *fd*-Y21M rods, respectively (24). The associated twist wave-number varies monotonically and smoothly from positive (right-handed) to negative (left-handed). It changes sign at *x*_{fd} = 0.26, the ratio at which the virus mixture is effectively achiral.

The structure of the colloidal membrane’s edge is determined by the balance of the surface energy associated with the rod-depletion polymer interface and the elastic distortion energy originating from the nonuniform packing of rods within the membrane. The surface energy favors a curved edge profile, whereas elastic distortions favor a squared edge (15, 16, 19, 26). For *fd*-virus–based colloidal membranes, the surface energy dominates; consequently, the membrane’s edge is curved and the edge-bound rods have to twist away from the membrane normal to fit the rounded profile imposed by the surface tension. Furthermore, the structure of the edge profile, and in particular the twist penetration depth *λ*_{t}, is independent of the chirality of the viruses; however, the chirality of the viruses does influence the effective edge tension of the membrane (19).

The tilting of edge-bound rods away from the membrane normal results in structural and optical anisotropy in the *x–y* plane (Fig. 1 *B* and *C*) (15, 16, 19). The optical anisotropy can be quantified by 2D-LC-PolScope that yields images where each pixel’s intensity is proportional to the 2D projection in the *x–y* plane of the retardance, *R* (27). The resulting twist at the edge penetrates into the membrane interior over a characteristic length scale (18). A radial retardance profile yields a twist penetration length that is significantly different between *fd*-Y21M and *fd-wt* (Fig. 1*F*). However, the 2D projection of the retardance map does not reveal the handedness of the edge twist. To extract this information, we use 3D-LC-PolScope (28). Briefly, a microlens array is introduced into the back focal plane of the objective of the 2D-LC-PolScope, producing a grid of conoscopic images on the CCD camera. Each conoscopic image determines the local orientation of rods. An azimuthally symmetric retardance profile with a dark spot in the center indicates that rods at that locality are oriented along the *z* axis. A shift of the zero-retardance spot away from the center of a conoscopic image yields the magnitude of the local virus tilting, whereas its radial position indicates the 3D direction of the birefringence vector. 3D-LC-PolScope images show that *fd-wt* membranes composed of *fd-wt* and *fd*-Y21M viruses are right- and left-handed, respectively (Fig. 1 *D* and *E*).

## Weakly Chiral Rod Mixtures Lead to Scalloped Membranes

Next, we examine the structure of colloidal membranes assembled from a mixture of *fd-wt* and *fd*-Y21M. The difference in their contour length of less than a few percent is not sufficient to induce lateral phase separation (29); instead, we observe uniformly mixed membranes throughout the entire range of parameters studied here (Fig. 2). The membranes are stable for a wide range of depletant concentrations and for all ratios *x*_{fd} (Fig. 2*A*). The twist at the membrane edge is right-handed at low *x*_{fd} and left-handed at high *x*_{fd}. Surprisingly, for intermediate x_{fd} (0.04 < *x*_{fd} < 0.45), we no longer observe flat circular membranes; instead, the membrane’s entire edge becomes decorated with a series of outward protrusions that are terminated by cusp-like defects (Fig. 2*B*). Furthermore, *z* scans indicate that such scalloped membranes are not flat but have a distinct 3D structure where a cusp point defect located below the membrane is always followed by a defect located above the same plane (Fig. 3).

2D-LC-Polscope images of the scalloped membranes demonstrate that rods at the edge of each outward protrusion have the same twist penetration length (Fig. 3 *A* and *B* and Fig. S1*A*). However, 3D-LC-Polscope reveals that adjacent protrusions have alternating left- and right-handed twist (Fig. 3*C*). The regions of opposite twist are separated by cusp-like point defects. Because each protrusion is always accompanied with two adjacent point defects alternating above and below the monolayer plane, there can only be an even number of cusp defects along the membrane circumference. The combined *z*-stack and 3D-LC-PolScope images allow us to schematize the edge structure of the scalloped membranes, which is more intricate compared with the edge structure of chiral colloidal monolayers studied previously (Fig. 3 *D–F*). The formation of the scalloped membranes is the direct consequence of molecular chirality, because scalloped membranes appear in the limit of weak chirality, that is, between 0.04 < *x*_{fd} < 0.4 (Fig. 2*A*).

In principle, there could be localized demixing of the two rods species, where *fd-wt* would preferentially localize at the edges with a left-handed twist, and *fd*-Y21M at the edges with opposite twist. This was not observed experimentally. We labeled all *fd-wt* rods with a fluorescent dye (Alexa 488) and *fd*-Y21M rods with fluorescent dye (Dylight 550). Using dual-view fluorescence, a technique that allows us to simultaneously image *fd-wt* and *fd*-Y21M fluorescent rods, we observe that, within experimental error, membranes in bulk and at the edges remain homogeneously mixed for all *x*_{fd}, even within the outward protrusion (Fig. 2*B* and Movies S1 and S2). Furthermore, using previously described techniques (18, 19), we have measured how the twist penetration length *λ*_{t}, the interfacial tension *γ*, and the edge bending rigidity *k*_{b} depend on *x*_{fd}. For scalloped membranes, we find that outward protrusions with either handedness had the same *λ*_{t} (Fig. S2) and *γ* and *k*_{b} that could not be distinguished within experimental error. Additionally, these quantities varied continuously from *x*_{fd} = 0 to *x*_{fd} = 1, which also indicates mixture homogeneity (Fig. S1).

## Membrane Coalescence Generates Cusp-Like Deformations

Lateral coalescence of colloidal membranes can lead to the formation of unconventional defect structures. For example, two laterally coalescing membranes of the same handedness can trap 180° of twist, resulting in a π-wall line defect (26). To elucidate a possible mechanism that leads to the formation of cusps in scalloped membranes, we observed membrane coarsening by using an angled-light 2D-LC-PolScope. This technique differs from conventional 2D-LC-PolScope; instead of having the light source aligned with the *z* axis, the almost closed aperture associated with the back focal plane is translated away from the optical center, resulting in the plane waves illuminating the sample at an angle. This in turn reveals the handedness of the local rod twisting. We define a coordinate system in which the optical axis lies along the *z* direction, and the membrane lies in the *x–y* plane (Fig. 4*A*). The aperture of the condenser back focal plane is placed so that the incident illumination is tilted in the *x–z* plane. It follows that the rods within a membrane along the *y* axis (dashed line in Fig. 4*B*) exhibit a variable tilt with the respect to the illumination plane. The regions where rods are perpendicular to the plane of the illuminating wave will exhibit no optical retardance, whereas retardance will increase with an increasing tilt of rods away from the angle of the incident light. As a result, the lower edge of a right-handed membrane exhibits reduced retardance (relative to the rods in the bulk) as the viruses tilt toward the light source, whereas the upper edge exhibits increased retardance due to the rods tilting away from the light source (Fig. 4*B*). By the same reasoning, a left-handed membrane will exhibit the opposite behavior, with darker and brighter regions at the top and bottom of the membrane along the *y* axis, respectively (Fig. 4*C*). This technique allows us to distinguish between left- and right-handed membrane edges with a higher spatial resolution than 3D-LC-PolScope.

For 0.04 < *x*_{fd} < 0.45, in the early stages of the sample maturation, we observe circular membranes of either edge handedness, indicating spontaneously broken achiral symmetry. Over time, the intermediate-sized membranes with mixed edge twist continue to coalesce. When two membranes with the same handedness merge, we observe the formation of either a π-wall or an array of pores at the coalescence junction, as was discussed previously (Fig. 4*B* and Movie S3). By contrast, as the two proximal edges of a membrane pair with the opposite twist rupture, the adjoining neck widens and the twist of the edge-bound rods is expelled by aligning constituent rods with the membrane normal (Fig. 4*C* and Movie S4). This coalescence process leads to a daughter membrane that has two outward protrusions and two cusp defects at which the twist of edge-bound rods switches handedness. Once formed, the cusp defects remain stable indefinitely. An outward protrusion with a pair of defects can also be imprinted into the edge using optical tweezers (Fig. 4*D* and Movie S5). The method, which allows for robust engineering of cusp defects, consists of pulling the twisted ribbons out of the membrane. The first few steps of this procedure are similar to the previously studied disk-to-ribbons transition (19). Subsequently, reversing the direction of the optical trap and dragging it toward the membrane leads to the formation of defect pairs.

## Effective Interactions Between Adjacent Point Defects

The structure of the scalloped edges greatly depends on the number fraction *x*_{fd} and how close the membrane is to the achiral limit (*x*_{fd} = 0.26). At the boundary of stability of scalloped membranes, near *x*_{fd} = 0.04 or 0.45, a pair of point defects remains bound to each other at a well-defined distance (Fig. 5*A*). In comparison, close to the achiral limit the defect pair freely moves along the edge and the total circumference of each outward protrusion exhibits significant fluctuations (Movie S6). These observations can be explained by the chiral control of membrane line tension (19). Increasing the rod chirality raises the free energy of the untwisted interior rods, whereas lowering the free energy of edge bound twisted rods, thus leading to the chiral control of the line tension. Likewise, chirality can also raise the line tension if the twist at the membrane’s edge is the opposite of the natural twist preferred by the constituent molecules.

For achiral membranes, the line tension associated with the exposed edge of left-handed and right-handed outward protrusions is roughly equal. The overall free energy does not significantly change as one outward protrusion extends its length at the expense of another one, by translating the cusp defect. In this limit, the point defects freely diffuse and the lengths of outward protrusions with either handedness exhibit significant fluctuations in agreement with experimental observations. However, away from the achiral limit, there is a finite difference in line tension between the left-handed and right-handed outward protrusions, and the free energy is minimized by reducing the length of the outward protrusions with unfavorable twist.

To quantitatively test these ideas, we have measured the effective interaction between a pair of point defects that are connected by a single protrusion. We used phase contrast microscopy to track the positions, *s*_{i}, of two adjoining defects along the membrane contour (Fig. 5*A*). For an achiral sample (*x*_{fd} = 0.26), the separation between two adjoining defects, *δs = s*_{i+1} − *s*_{i}, fluctuates by many microns over a timescale of minutes (Fig. 5*B*). However, away from achiral limit, we observe that the relative separation between these defect pairs remains well defined on experimental timescales. We measured the probability distribution function, *P*(*δs*), of the defects being separated by distance *δs*. The measured distributions are described by a Gaussian: *P*(*δs*) = exp(−*α*(*δs* − *δs*_{0})^{2}/2*k*_{B}*T*), indicating that the defects are bound by a harmonic potential centered around the equilibrium separation, *δs*_{0} (Fig. 5*C*). The equilibrium defect separation as well as the strength of the effective binding potential, *α*, depends on *x*_{fd}, the ratio of left- and right-handed rods. By varying membrane composition, we extracted how *α*, as well as the equilibrium separation, *δs*_{0} = *<δs>*, depends on *x*_{fd} (Fig. 5 *D* and *E*). Approaching the achiral mixture limit (*x*_{fd} = 0.26) leads to the increase of the mean separation *δs*_{0} and a vanishing *α*. In this limit, the adjacent defects effectively decouple from each other. Increasing chirality away from the achiral limit decreases equilibrium separation and increases the coupling strength, indicating tighter defect binding. The existence of a finite equilibrium separation indicates a competition between short-range repulsion, due to elastic distortions, and long-range attraction caused by the asymmetry of the line tension associated with the edges of the opposite twist.

The passive fluctuation analysis only maps the binding potential within a few *k*_{B}*T* around its minimum. To measure the entire binding potential, we performed active experiments where we moved one defect by *δs* using an optical trap, while simultaneously measuring the force *F* exerted on the other defect (Fig. 6 *A* and *B*). For this purpose, we embedded 1.5-μm-diameter colloidal beads into two adjoining cusp defects. Once placed there, beads remained attached to a defect for the entire duration of the experiment. To ensure that the beads do not alter the defect structure, we measured thermal fluctuations of a defect pair with and without embedded beads and found them to be identical within experimental error (Fig. S3). We then calibrated the trap to measure the zero force at the equilibrium distance *δs*_{0} (Fig. S4) and determined the optimal laser power to measure the force *F* (Fig. S5). We extracted the force as a function of *δs*, *F*(*δs*), which is averaged over 10 identical experiments (Fig. 6 *A* and *B*). In the vicinity of the equilibrium separation, *δs*_{0}, the force measurements quantitatively agree with the fluctuation experiments described above. As expected, the force is negative below *δs*_{0}, and positive above *δs*_{0}, confirming that *δs*_{0} is the stable equilibrium position. The force steeply increases for small separations and saturates at large separations, indicating that a defect pair is permanently bound. The magnitude of the force plateau and the slope of the force-increasing region depend on the number fraction x_{fd}. By moving farther away from the achiral limit, we find that the equilibrium distance between bound defects *δs*_{0} decreases. These experiments also demonstrate that the pairwise defect interactions are governed by a balance between short-range repulsion and long-range attraction.

## Modeling the Interactions Between Two Adjacent Point Defects

The theoretical model of scalloped membranes has been studied previously (30). Here, we provide a quantitative comparison of this theoretical model to experimental data. To summarize, our model reduces the overall 3D geometry of the membrane to an isolated configuration around a point defect. The outward protrusions between two neighboring cusps must form via the interplay between the line tension *γ*, the interfacial bending rigidity *k*_{b}, and the geometrical variables associated with the overall membrane deformation. For an isolated defect, the free energy is then given by the following:*σ* denotes the surface tension of the membrane. The two edge profiles (enumerated by the index *i*) with opposite handedness meet at the point defect and are in general different, because their curvature *k*_{b} is the bending modulus of each edge. The bulk terms are integrated over the membrane surface with an area element *dS*, whereas the interfacial terms are integrated along the arc length with elements *ds*_{i}. We note that the mean curvature *H* of the scalloped membrane is absent in Eq. **1**, because it can only contribute to the membrane free energy when there is a finite pressure difference of the surrounding aqueous solution above and below the membrane surface. However, in the presence of the free edges, the pressure difference vanishes in equilibrium, resulting in *H* = 0, that is, a minimal surface (*Theoretical Methods*). The stability of the scalloped membrane with respect to a flat membrane is determined by the free-energy difference *S*, the surface tension *σ* cancels out in **1** and its minimization procedure by a variational analysis, which yields the spring constant (Fig. 5*D*), the equilibrium defect separation (Fig. 5*E*), and the phase diagram (Fig. 6*C*), are discussed in detail in *Theoretical Methods*.

The structure of the scalloped edge is determined by the balance between two contributions to the free energy, the line energy and the surface energy. On the one hand, the line energy suppresses the formation of outward protrusions and cusp defects, because they increase the total membrane circumference. On the other hand, each cusp defect generates negative Gaussian curvature, which lowers the free energy of elastic deformations if the Gaussian modulus is positive and sufficiently large (30). Based on the interplay between these two contributions, our model predicts regions where scalloped membranes are more stable than flat circular membranes as a function of *x*_{fd} (Fig. 6*C*). To calculate this phase diagram, we have used theoretical fits to the experimental values for *k*_{b} (Fig. S1 and *Theoretical Methods*). When the number fraction of the virus mixture deviates from *x*_{fd} = 0.26, an increasing magnitude of *C*). The reason is that, away from the achiral limit, one of the edges (e.g., associated with *γ*_{2}) has incompatible chirality with the preferred overall handedness along the membrane boundary. Consequently, the rods along that edge tilt into a high-energy configuration, as opposed to the molecules in the adjacent outward protrusion that has lower energy (with

Theoretical predictions for the defects separation length, *δs*_{0}, and their coupling strength, *α* were fit to experimental measurements (Fig. 5 *D* and *E*; *Theoretical Methods*). The line tension *k*_{b} were taken from experiments (Fig. S1), whereas we took Δ*γ* and *k*_{b} within ∼10% error. However, because *γ* remains a free parameter, and we assume the polynomial form *x*_{fd} = 0.26 and becomes 30 *k*_{B}*T*/μm at *x*_{fd} = 0, that is, Δ*γ* stays within the bounds of experimental uncertainty. Likewise, theoretical fits to experimental curves yield the magnitude of the Gaussian modulus,

The theory quantitatively reproduces how the effective defect interactions (coupling strength *α* and equilibrium separation *δs*_{0}) depend on *x*_{fd}, thus confirming their origin: a long-range attraction due to Δ*γ* > 0, and a short-range repulsion associated with membrane Gaussian curvature (Fig. 5 *D* and *E*). The modulus *α* as a function of *x*_{fd}, extracted from a linear approximation to the theoretical force-extension curves at the point where the force vanishes, qualitatively agrees with the experimental profiles (Fig. 5*D*). We note that, without the Gaussian curvature contribution, a simpler 2D theory modeling a flat and thermodynamically unstable scalloped membrane consistently yields smaller *δs*_{0} values than those from the 3D model presented here (Fig. S6). Hence, the Gaussian curvature term with a positive modulus explains the nature of in-plane and out-of-plane deformations as well as the overall stability of the scalloped membranes. Furthermore, the model with the same parameters also reproduces the optical tweezer measurements of the effective defect interactions over a much larger range of separations (Fig. 6*B*).

Certain precautions need to be taken when interpreting the extracted magnitude of the Gaussian modulus *A* (Fig. 3*F*) and the line tension (30). This is because bigger protrusions would make the edge longer at constant *γ*, necessitating a higher *x*_{fd} = 0.26, when the protrusion size is 2 μm (Fig. 3*A*) and the line tension is *D*), the Gaussian modulus from this relation is found as *D* and *E*). The resulting discrepancy between two estimates of the Gaussian modulus may be due to the fact that our model relies on a simple geometrical assumption, an axially symmetric catenoidal surface, which likely accumulates more negative Gaussian curvature than the experimental shape of the membrane surface. Therefore, our analysis underestimates the Gaussian modulus that stabilizes the scalloped membrane over a flat configuration. Theoretically, compromising axial symmetry or the smoothness of the surface around the cusp could yield a minimal saddle surface with a lower amount of the total Gaussian curvature. On the one hand, this would restore

There is a discrepancy between the experimental (Fig. 2*A*) and theoretical (Fig. 6*C*) phase diagrams because theory implies that at *x*_{fd} = 0 and *x*_{fd} = 1. There may be two main reasons underlying the difference between the theoretical and experimental stability of the scalloped membranes. First, the theoretical phase diagram is calculated by quantifying only a point defect and the membrane deformations in its neighborhood, whereas the experimental stability of the scalloped membranes is governed by the overall membrane conformation associated with multiple defects (Fig. 2*B* and Fig. 3*A*). If the number of defect pairs is less than the overall membrane circumference can support, the scalloped membrane is still stable with respect to a flat membrane but would be in a metastable state. Experimentally, this might be the case, as is evident from the defect pair evolution as a function of chirality, leaving long flat sections between two successive pairs (Fig. 5*A*). Second, we assumed that Δ*γ* = 30 *k*_{B}*T*/µm at *x*_{fd} = 0 to fit the spring constant (Fig. 5*D*); however, Δ*γ* could be larger and could be a strongly decreasing function of *x*_{fd}. This would result in a steeper phase boundary away from *x*_{fd} = 0, and an overall stabilization of flat monolayers away from the achiral limit.

## Theoretical Methods

### The Scalloped Membrane Free Energy.

Scalloped membranes are observed in the assemblages of rod mixtures with opposite handedness. When the formation of out-of-plane cusps at the circumference of a flat colloidal membrane give rise to in-plane protrusions, the free energy of the membrane changes due to the saddle configuration of the surface around each point defect and the increase of the membrane–solution interfacial line between adjacent defects. Therefore, the scalloped membranes are only stable with respect to the flat membranes if there is a free-energy gain associated with the elastic out-of-plane deformation that compensates for the free-energy cost of longer edge along each protrusion. For an isolated point defect, we quantify the interplay between the two contributions by the membrane energy functional *Modeling the Interactions Between Two Adjacent Point Defects* in the main text and Fig. S7 for the definitions of the other unknowns in Eq. **S1**). Then, the free-energy difference between the scalloped membrane and flat membrane phases is given by the following:**S2**, **S2** yields the phase diagram as a function of *C* in the main text), which will be explained below in detail. The mean curvature modulus

### Derivation of Vanishing Mean Curvature Condition.

Locally, colloidal membranes must adopt a minimal surface configuration (*u* and *v*. Then, small deformations orthogonal to the surface are quantified by the following:*Theoretical Estimate of Gaussian Curvature Modulus* in the main text), which causes a change in the area element, is then proportional to **S4**, we can calculate the variation of **S5** has no Gaussian curvature term because of the Gaussian–Bonnet theorem, which integrates the Gaussian curvature contribution in Eq. **S1** to a constant (equal to 2π) for a closed surface. Then, the variation of **S6** apart from surface terms (see ref. 54 for the derivation of this relation; Δ: Laplace–Beltrami operator). When the surface has free edges, Eq. **S7** will still hold because **S1** will yield a line integral in terms of the geodesic curvature times the arc length of the line bounding the surface, so that its variation accompanies Eq. **S7** as a free boundary condition.

Next, we assume that the mean curvature is uniform over the saddle membrane surface: For an open surface **S7** suggests that a nonuniform mean curvature can either acquire an wave-like profile roughly with a length scale **S7**. In order for

### Evaluation of the Scalloped Membrane Free Energy.

We evaluate Eq. **S2** for a minimal surface of revolution (namely axially symmetric) with negative Gaussian curvature. A surface of revolution of radius *R*, which is embedded in Euclidean 3D space, is represented by the position vector as follows:*r*). The two principal curvatures via the second fundamental form are obtained as *a* is given by

The surface depicted in Fig. S8 is terminated by two space curves embedded in the membrane, corresponding to two free edges meeting at the point defect. Each curve is represented by the position vector *s* is the coordinate along the arc length of the curve. The curvature of these curves is obtained in terms of *i*th curve and

Next, we determine the height of the surface **S8**). The condition **S10** yields a catenoid, where

As a function of **S8**–**S12** enable us to evaluate the free-energy difference between the scalloped and flat membrane configurations as follows (Eq. **S2**):**S9**. The unknowns

### Minimization Procedure for Calculating the Equilibrium Free-Energy Difference.

As a function of areal fraction **S13**. In the second step, we find the global minimum of Eq. **S13** by computing the equilibrium radius of the surface of revolution **S13** and **S14** are all functions of the areal volume fraction *Modeling the Interactions Between Two Adjacent Point Defects*). We choose

The Euler–Lagrange (EL) equations are extracted by evaluating the variations **S15** and **S16**, along with the two boundary conditions fully determine **S10**–**S12**. The lengths appearing in Eqs. **S13**–**S16** are made dimensionless via the characteristic length scale **S15** and **S16** in dimensionless form is done by the *bvp4c* boundary-value problem solver in Matlab.

The only unknowns that remain in Eqs. **S13** and **S14** are *quadv* function to evaluate the numerical quadrature of Eq. **S18**, as well as of Eqs. **S13** and **S14** in dimensionless form.

The second stage of the minimization is performed numerically by using the *fminbnd* function in Matlab, to find **S13** over an interval *fminbnd* uses the golden section search algorithm and parabolic interpolation, without using the gradient information (59).

Implementing these two stages yields the equilibrium free-energy difference as a function of *C*), where

### Determination of the Equilibrium Defect Distance and the Spring Constant.

At **S13** at a given areal fraction *E* in the main text).

In the neighborhood of the equilibrium, the free energy can in principle be fit by a parabolic profile, and its derivative will yield the force *E* in the main text. Evaluating the force numerically at *B*.

## Theoretical Estimate of Gaussian Curvature Modulus

A simple argument can be used to estimate the Gaussian curvature modulus, *D*, surrounded by the depleting polymers with radius of gyration, *R*_{g}. We assume that the polymers behave as an Asakura–Osawa ideal gas of particles with effective diameter *d*, which is related to polymer radius of gyration by the following: *z*, the distance along the membrane normal away from the midplane. In particular, if the membrane has mean curvature *H* and Gaussian curvature *z* is given by *Y* is the modulus for areal compression. Because the rods are uniform along their lengths, we take *Y* to be independent of *z*. The lateral stress is isotropic because the membrane is fluid. There is a compressive stress in the membrane even when it is flat because the polymer depletants squeeze the membrane (34). The total volume excluded to the polymers for a flat membrane of area *A* is *n* is polymer concentration. To balance this tension, the rods must experience a compressive stress *V*_{0,ex}. Specifically,, for a membrane with zero-mean curvature: *D* >> *d*.

To estimate the polymer contribution to the Gaussian curvature modulus, we use *n* ∼ 40 mg/mL, *D* ∼ 880 nm, and *d* ∼ 30 nm for a dextran with molecular weight of 500,000 g/mol (36). With these numbers, we find that *k*_{B}*T*. Despite its approximate nature, our estimate yields the magnitude *H* = 0, there is less room for the surfactant chains, which therefore must stretch and incur a higher free energy (37).

## Discussions and Conclusions

Our combined theoretical and experimental work demonstrates that membranes composed of achiral rods exhibit higher structural complexity compared with flat membranes assembled from chiral rod-like viruses. In the latter case, strong chirality enforces uniform twist of rods along the entire membrane circumference, leading to the formation of flat 2D disks. By contrast, weakly chiral or achiral membranes exhibit an intriguing instability that is driven by an interplay between the Gaussian curvature of a colloidal membrane and the spontaneous achiral symmetry breaking of rods located at the membrane’s edge. The achiral symmetry breaking induces formation of cusp-like defects. These defects in turn allow the membrane to adopt a 3D shape that decreases the overall energy associated with its negative Gaussian curvature.

Despite the important role it plays in diverse processes, measuring the Gaussian modulus of conventional lipid bilayers remains a significant experimental challenge. In comparison, the properties of the colloidal membranes described here allow us to estimate their Gaussian modulus. Conventional bilayers have a negative Gaussian modulus, which means that saddle-shaped deformations increase the membrane energy (7, 10, 11, 38). On the contrary, experiments described here, as well as previous observations of diverse assemblages with excess Gaussian curvatures such as arrays of pores and twisted ribbons (19, 26), demonstrate that colloidal monolayers, in contrast to lipid bilayers, have positive Gaussian moduli.

Achiral symmetry breaking has been observed in diverse soft systems with orientational order, ranging from lipid monolayers and nematic tactoids to confined chromonic liquid crystals (39⇓⇓⇓⇓⇓–45). In particular, the measured structure and interactions of the cusp-like defects in colloidal membranes resemble studies of point defects moving along a liquid crystalline dislocation line in the presence of chiral additives (46). The main difference is that in the colloidal membranes the achiral symmetry breaking leads to out-of-plane 3D membrane distortions that couples liquid crystal physics to membrane deformations. This is not possible for inherently confined liquid crystalline films.

From an entirely different perspective, a number of emerging techniques have been developed to fold, wrinkle, and shape thin elastic sheets with in-plane elasticity (47⇓⇓–50). So far, these efforts were focused on studying instability of thin elastic films with finite in-plane shear modulus. The methods to achieve folding or wrinkling of thin sheets involves either engineering of in-plane heterogeneities or imposing an external force. Our work demonstrates that simpler uniform elastic sheets lacking in-plane rigidity can spontaneously assume complex 3D folding patterns that decorate its edge.

Finally, methods described here and in our previous work should be applicable to any monodisperse rod type with sufficiently large aspect ratio. Thus, they might offer a scalable method for robust assembly of photovoltaic devices composed of nanorods. Our previous investigation of chiral *fd-wt* colloidal membranes demonstrated that the twist at their edges introduces a significant energetic barrier that suppresses their lateral coalescence (26). In such samples, membranes with diameters ranging from 10 to 100 μm are commonly found. Compared with chiral colloidal membranes, we find that colloidal membranes of monodisperse virus mixtures that are close to the achiral limit coalesce much faster and can easily reach millimeter dimensions.

## Materials and Methods

### Sample Preparation.

Both viruses, *fd* and *fd*-Y21M, were grown in bacteria and purified as described elsewhere (19). *fd*-Y21M, has a single point mutation in the amino acid sequence of the major coat protein: amino acid number 21 is replaced from Y to M. *fd* and *fd*-Y21M were labeled with fluorescent dye as described elsewhere (51). The preparation of optical chambers was described elsewhere (19).

### Optical Microscopy.

Experiments were carried out on an inverted microscope (Nikon TE 2000) equipped with traditional polarization optics, a differential interference contrast (DIC) module, a fluorescence imaging module, and 2D-LC-Polscope module. For dual-view fluorescence imaging, we used DV2 from Photometrics. We used a 100× oil-immersion objective (PlanFluor, N.A. 1.3, for DIC and PlanApo, N.A. 1.4, for phase contrast). Images were recorded with cooled CCD cameras [CoolSnap HQ (Photometrics) or Retiga Exi (QImaging)]. For 3D-LC-PolScope measurements, we used a Zeiss Axiovert 200M microscope with a Plan Apochromat oil-immersion objective (63×/1.4 N.A.) and a monochrome CCD camera (Retiga 4000R; QImaging).

### Laser Tweezers.

A 1,064-nm laser (Coherent Compass) was brought into the optical path of an inverted microscope (Nikon Eclipse Te2000-u) and focused with a 100× objective onto the image plane (Nikon PlanFlour, N.A. 1.3). To simultaneously trap multiple beads, a single beam was time shared between different positions using an acousto-optic deflector (IntraAction-276HD) (52). Bead position was measured using back focal plane interferometry and a quadrant photodiode (QPD) (53). A separate 830-nm laser (Point Source Iflex-2000) was used as a detection beam. To calibrate the photodiode, we scan a bead across the detection beam in known step sizes and measure the corresponding voltage change. Trap stiffness was calibrated by analyzing the power spectrum of the bead position (53).

## Acknowledgments

We acknowledge conversations with William Irvine, Robert Pelcovits, and Leroy Jia. We also acknowledge use of Brandeis Materials Research Science and Engineering Centers (MRSEC) optical and biomaterial synthesis facility supported by National Science Foundation (NSF) Grant MRSEC-1420382. T.G. acknowledges the Agence National de la Recherche Française (Grant ANR-11-PDOC-027) for support. P.S., C.N.K., R.B.M., and Z.D. acknowledge support of NSF through Grants MRSEC-1420382 and NSF-DMR-1609742. T.R.P. acknowledges support of the NSF through Grants MRSEC-1420382 and NSF-CMMI-1634552. R.D.K. was partially supported by a Simons Investigator grant and Grant NSF-DMR-1262047.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: zdogic{at}brandeis.edu.

Author contributions: T.G., R.B.M., and Z.D. designed research; C.N.K. developed the theoretical model of defect interactions; R.D.K. and T.R.P. provided theoretical estimate of the Gaussian curvature modulus; R.B.M. contributed to the theoretical model; T.G., C.N.K., P.S., M.J.Z., and A.W. performed research; R.O. contributed new reagents/analytic tools; P.S. acquired coalescence movies; A.W. contributed optical-tweezer measurements; R.O. contributed microscopy expertise; T.G., C.N.K., and T.R.P. analyzed data; and T.G., C.N.K., T.R.P., and Z.D. 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/lookup/suppl/doi:10.1073/pnas.1617043114/-/DCSupplemental.

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