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# Selective buckling via states of self-stress in topological metamaterials

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved May 11, 2015 (received for review February 13, 2015)

## Significance

An ongoing challenge in modern materials science is to create artificial structures, termed metamaterials, with an unconventional response that can be programmed by suitable design of their geometry or topology. We demonstrate a novel metamaterial in which small structural variations single out regions that buckle selectively under external stresses. Surprisingly, these regions are at first glance indistinguishable from the rest of the structure. Our samples are examples of topological metamaterials that are the mechanical analogues of topologically protected electronic systems. As a result, the buckling response is robust against structural perturbations yet tunable without affecting secondary physical properties in the buckling region. This feature is desirable for optomechanical or thermomechanical metamaterials in which multiple characteristics need to be tuned simultaneously.

## Abstract

States of self-stress—tensions and compressions of structural elements that result in zero net forces—play an important role in determining the load-bearing ability of structures ranging from bridges to metamaterials with tunable mechanical properties. We exploit a class of recently introduced states of self-stress analogous to topological quantum states to sculpt localized buckling regions in the interior of periodic cellular metamaterials. Although the topological states of self-stress arise in the linear response of an idealized mechanical frame of harmonic springs connected by freely hinged joints, they leave a distinct signature in the nonlinear buckling behavior of a cellular material built out of elastic beams with rigid joints. The salient feature of these localized buckling regions is that they are indistinguishable from their surroundings as far as material parameters or connectivity of their constituent elements are concerned. Furthermore, they are robust against a wide range of structural perturbations. We demonstrate the effectiveness of this topological design through analytical and numerical calculations as well as buckling experiments performed on two- and three-dimensional metamaterials built out of stacked kagome lattices.

Mechanical metamaterials are artificial structures with unusual properties that originate in the geometry of their constituents, rather than the specific material they are made of. Such structures can be designed to achieve a specific linear elastic response, like auxetic (negative Poisson ratio) (1) or pentamode (zero-shear modulus) (2) elasticity. However, it is often their nonlinear behavior that is exploited to engineer highly responsive materials, the properties of which change drastically under applied stress or confinement (3⇓⇓⇓–7). Coordinated buckling of the building blocks of a metamaterial is a classic example of nonlinear behavior that can be used to drive the auxetic response (4, 8), modify the phononic properties (5), or generate 3D micro/nanomaterials from 2D templates (9).

Buckling-like shape transitions in porous and cellular metamaterials involve large deformations from the initial shape, typically studied through finite element simulations. However, many aspects of the buckling behavior can be successfully captured in an approximate description of the structure that is easier to analyze (5, 6, 10, 11). Here, we connect the mechanics of a cellular metamaterial, a foamlike structure made out of slender flexible elements (12, 13), to that of a frame—a simpler, idealized assembly of rigid beams connected by free hinges—with the same beam geometry. We exploit the linear response of a recently introduced class of periodic frames (14), inspired by topologically protected quantum materials, to induce a robust nonlinear buckling response in selected regions of two- and three-dimensional cellular metamaterials.

Frames, also known as trusses, are ubiquitous minimal models of mechanical structures in civil engineering and materials science. Their static response to an external load is obtained by balancing the forces exerted on the freely hinged nodes against internal stresses (tensions or compressions) of the beams. However, a unique set of equilibrium stresses may not always be found for any load (15). First, the structure may have loads that cannot be carried because they excite hinge motions called zero modes that leave all beams unstressed. Second, the structure may support states of self-stress—combinations of tensions and compressions on the beams that result in zero net forces on each hinge. An arbitrary linear combination of states of self-stress can be added to an internal stress configuration without disrupting static equilibrium, implying that degenerate stress solutions exist for any load that can be carried. The respective counts *d*-dimensional frame with **1** shows, states of self-stress count the excess constraints imposed by the beams on the

States of self-stress play a special role in the mechanical response of repetitive frames analyzed under periodic boundary conditions (18). Macroscopic stresses in such systems correspond to boundary loads, which can only be balanced by states of self-stress involving beams that cross the boundaries. If these states span the entire system, boundary loads are borne by tensions and compressions of beams throughout the structure. Conversely, by localizing states of self-stress to a small portion of a frame, load-bearing ability is conferred only to that region. Our approach consists of piling up states of self-stress in a specific region of a repetitive frame so that the beams participating in these states of self-stress are singled out to be compressed under a uniform load at the boundary. In a cellular material with the same beam geometry, these beams buckle when the compression exceeds their buckling threshold.

Although our strategy is of general applicability, it is particularly suited to isostatic lattices (19, 20), with *d* rigid body motions under periodic boundary conditions. According to Eq. **1**, these lattices only have *d* states of self-stress, insufficient to bear the *A* (14, 22), and lattice defects (23) can harbor localized states of self-stress, which can be used to drive localized buckling. Unlike their trivial counterparts, the existence of these topological states of self-stress cannot be discerned from a local count of the degrees of freedom or constraints in the region. An attractive feature for potential applications is their topological protection from perturbations of the lattice or changes in material parameters that do not close the acoustic gap of the structure (14, 24⇓⇓–27).

In the remainder of this article, we use robust states of self-stress to design buckling regions in topological metamaterials composed of flexible beams rigidly connected to each other at junctions. As shown in Fig. 1, the states of self-stress are localized to a quasi-2D domain wall obtained by stacking multiple layers of a pattern based on a deformed kagome lattice (14). The domain wall separates regions with different orientations of the same repeating unit (and hence of the topological polarization

## Linear Response of the Frame

The distinguishing feature of the isostatic periodic frame used in our design is the existence of a topological characterization of the underlying phonon band structure (14). If the only zero modes available to the structure are the *d* rigid-body translations, its phonon spectrum has an acoustic gap; i.e., all phonon modes have nonzero frequencies except for the translational modes at zero wavevector. A gapped isostatic spectrum is characterized by *d* topological indices

The count of topological mechanical states at an edge or domain wall is obtained via an electrostatic analogy. The lattice vector *A*) can be interpreted as a polarization of net degrees of freedom in the unit cell. Just as Gauss’s law yields the net charge enclosed in a region from the flux of the electric polarization through its boundary, the net number of states of self-stress (minus the zero modes) in an arbitrary portion of an isostatic lattice is given by the flux of the topological polarization through its boundary (14). In Fig. 1*A*, the left domain wall has a net outflux of polarization and harbors topological states of self-stress (the right domain wall, with a polarization influx, harbors zero modes). Although only one of each mechanical state is shown, the number of states of self-stress and zero modes is proportional to the length of the domain wall. Similar results can be obtained by using other isostatic lattices with an acoustic gap and a topological polarization, such as the deformed square lattice in ref. 23.

The linear response of a frame can be calculated from its equilibrium matrix *q* identifies independent normalized states of self-stress that span the null space of

Because we are interested in triggering buckling through uniform loads that do not pick out any specific region of the lattice, we focus on the response to affine strains, where affine beam extensions *k*). Therefore,

Eq. **5** shows that the loading of beams under affine strains is completely determined by the states of self-stress. In a frame consisting of a single repeating unit cell, loads are borne uniformly across the structure. However, if the structure also has additional states of self-stress with nonzero entries in *A*, which has a nonzero overlap with affine extensions *A* and *B*, respectively, do not single out any particular region. Although they have a nonzero overlap with both *y*, the frame can expand in the perpendicular direction to keep the tensions and compressions low in the majority of the sample. In contrast, the localized states of self-stress shown in Fig. 2 *C* and *D* have a significant overlap with **5**, a uniform compression applied to the lattice along the vertical direction, with the horizontal direction free to respond by expanding, will significantly stretch or compress only the beams participating in the localized states of self-stress.

Whereas the topological polarization guarantees the presence of states of self-stress localized to the left domain wall, their overlap with one of the three independent affine strains is determined by the specific geometry of the hinges and bars. For the frame in Fig. 2, the states of self-stress visualized in Fig. 2 *C* and *D* are crucial in triggering buckling response, which may be predicted to occur under compression along the *y* direction (or extension along the *x* direction, which would also lead to *y* compression because the lattice has a positive Poisson ratio set by *β*). In other domain wall geometries, or other orientations of the domain wall relative to the lattice, the localized states of self-stress could have small overlap with affine strains, which would make them inconsequential to the buckling behavior. Alternatively, states of self-stress that overlap significantly with shear deformations could also be realized that would enable buckling to be triggered via shear.

## Buckling in Topological Cellular Metamaterials

The cellular metamaterial differs from the ideal frame in two important ways. First, real structures terminate at a boundary, and loads on the boundary are no longer equilibrated by states of self-stress, but rather by tension configurations that are in equilibrium with forces on the boundary nodes (15). However, these tension states are closely related to system-traversing states of self-stress in the periodic case, with the forces on the boundary nodes in the finite system playing the role of the tensions exerted by the boundary-crossing beams in the periodic system. Therefore, the states of self-stress also provide information about the load-bearing regions in the finite structure away from the boundary.

In addition to having boundaries, the cellular block probed in Fig. 1 also departs from the limit of an ideal frame, as it is made of flexible beams rigidly connected at the nodes and can support external loads through shear and bending of the beams in addition to axial stretching or compression. Nevertheless, the states of self-stress and tension states of the corresponding frame (with the same beam geometry) determine the relative importance of bending to stretching in the load-bearing ability of the cellular structure (21). To verify that the localized states of self-stress in the underlying frame influence the response of the finite cellular structure, we numerically calculate the in-plane response of each layer treated as an independent 2D cellular structure with loading confined to the 2D plane. Each beam provides not just axial tension/compression resistance but also resistance to shear and bending. A beam of length *L*, cross-sectional area *A*, and area moment of inertia *I* resists (*i*) axial extensions *e* with a tension *ii*) transverse deformations *iii*) angular deflections of the end nodes θ with restoring moment *w*, the relative contribution of the bending, shear and torsional components to the total stiffness is set by the aspect ratio

The 2D linear response of such a structure is calculated by augmenting the equilibrium matrix *Materials and Methods*. The resulting equilibrium matrix is of size

We expect a similar localized compression-dominated response in each layer of the stacked structure (Fig. 1 *B* and *C*). The enhanced compressions along the left domain wall trigger buckling when the compression exceeds the Euler beam buckling threshold, *c* is a positive numerical factor determined by the clamping conditions as well as cooperative buckling effects. Buckling is signified by a loss of ability to bear axial loads, as the beam releases its compression by bending out of plane. Upon compressing the 3D sample between two plates as shown in Fig. 4*A*, we see a significant out-of-plane deflection for beams along the left domain wall (Fig. 4 *B* and *C*), consistent with buckling of the maximally stressed beams in Fig. 3*A*. The deflection in different layers is coordinated by the vertical beams connecting equivalent points, so that beams within the same column buckle either upwards (column 1) or downward (column 2) to produce a distinctive visual signature when viewed along the compression axis. Beams connecting different planes in the stacked pattern create additional states of self-stress that traverse the sample vertically, but these do not single out any region of the material, and do not couple to the specific in-plane loading of Fig. 4*A*.

We emphasize that having as many states of self-stress as there are unit cells along the domain wall is crucial for the buckling to occur throughout the domain wall. When a beam buckles, its contribution to the constraints of distances between nodes essentially disappears, reducing the number of load-bearing configurations by one. If there were only a single localized state of self-stress in the system, the buckling of a single beam would eliminate this state, and the compressions on the other beams would be relaxed, preventing further buckling events. However, the presence of multiple states of self-stress, guaranteed in this case by the topological origin of the states, allows many buckling events along the domain wall. In the *SI Text*, we use an adaptive simulation that sequentially removes highly compressed beams to show that each repeating unit along the *y* direction experiences a unique buckling event even when the loss of constraints due to other buckling events is taken into account (Fig. S1).

## Robustness of the Buckling Region

Finally, we show that the robustness of the topological states of self-stress predicted within linear elastic theory carries over to the buckling response in the nonlinear regime. There is a wide range of distortions of the deformed kagome unit cell which do not close the acoustic gap, leaving the topological invariants *A* is minimally distorted away from the regular kagome lattice, but this barely noticeable distortion (Fig. 5*A*, *Inset*) is sufficient to induce the same topological polarization

For ease of visualization, we tested this design in a 2D prototype cellular metamaterial, obtained by laser-cutting voids in a 1.5-cm-thick slab of polyethylene foam (*Materials and Methods*), leaving behind beams 1–2 mm wide and 10–12 mm long (Fig. 5*A*). The aspect ratio of the beams is comparable to that of the 3D sample. Because the slab thickness is much larger than the beam width, deformations are essentially planar and uniform through the sample thickness, and can be captured by an overhead image of the selectively illuminated top surface. The sample was confined between rigid acrylic plates in free contact with the top and bottom edges, and subjected to a uniaxial in-plane compression along the vertical direction by reducing the distance between the plates. Using image analysis (*Materials and Methods*) we computed the beams’ tortuosity, defined as the ratio of the contour length of each beam to its end-to-end distance. Buckled beams have a tortuosity significantly above 1. Under a vertical compression of 4%, only beams along the left domain wall show a tortuosity above 1.05, consistent with a localized buckling response (Fig. 5*B*). However, under higher strains, the response of the lattice qualitatively changes. When beams have buckled along the entire domain wall, its response to further loading is no longer compression dominated (Fig. S1). Future buckling events, which are triggered by coupling between torsional and compressional forces on beams due to the stiff hinges, no longer single out the left domain wall and happen uniformly throughout the sample (Fig. 5*C* and Movie S2).

We have demonstrated that piling up localized states of self-stress in a small portion of an otherwise bending-dominated cellular metamaterial can induce a local propensity for buckling. Whereas this principle is of general applicability, our buckling regions exploit topological states of self-stress (14), which provide two advantages. First, they are indistinguishable from the rest of the structure in terms of node connectivity and material parameters, allowing mechanical response to be locally modified without changing the thermal, electromagnetic, or optical properties. This feature could be useful for optomechanical (30) or thermomechanical (31) metamaterial design. Second, the buckling regions are robust against structural perturbations, as long as the acoustic bulk gap of the underlying frame is maintained. This gap is a property of the unit cell geometry. Large deformations that close the gap could be induced through external actuation (32) or confinement, potentially allowing a tunable response from localized to extended buckling in the same sample, reminiscent of electrically tunable band gaps in topological insulators (33).

## Materials and Methods

### Deformed Kagome Lattice Unit Cell.

The deformed kagome lattices are obtained by decorating a regular hexagonal lattice, built from the primitive lattice vectors *a* is the lattice constant, with a three-point unit cell that results in triangles with equal sizes. The unit cells are described by a parameterization introduced in ref. 14, which uses three numbers *A*, which forms the basis for the 3D cellular structure, is reproduced by *A* parameterized by

### Construction of the Equilibrium/Compatibility Matrix.

Analysis of the linear response of a frame or a cellular material begins with the construction of the equilibrium matrix *x* axis connecting hinge 1 at *i* and the torsion angle

The forces and torque (generalized stresses) are obtained from the generalized strains *E*, the cross-sectional area *A*, the area moment of inertia *I*, and the beam length *L*:

The compatibility matrix for a beam with arbitrary orientation is obtained by projecting the displacement vectors at the end of each beam onto the axial and transverse directions using the appropriate rotation matrix, which depends on the angle made by the beam with the *x* axis. Each beam in an assembly provides three rows to the compatibility matrix, with additional columns set to zero for the degrees of freedom unassociated with that beam. In the simpler frame limit, each beam only resists axial extensions, and contributes one row (the first row of the **6**) to the compatibility matrix.

Once the equilibrium matrix is constructed, the approximate states of self-stress of the periodic frame (Fig. 2) as well as the linear response of the cellular material under loads (Fig. 3) are obtained from its singular value decomposition following the methods of ref. 15. More details of the computation are provided in the *SI Text*.

### Construction and Characterization of 2D and 3D Prototypes.

The 3D structures were printed by Materialise N.V. through laser sintering of their proprietary thermoplastic polyurethane TPU 92A-1 (tensile strength 27 MPa, density 1.2 g/cm^{3}, Young’s modulus 27 MPa).

The 2D structures were cut using a VersaLaser 3.5 laser cutter (Laser & Sign Technology) from 1.5 cm thick sheets of closed-cell cross-linked polyethylene foam EKI-1306 (EKI B.V.; tensile strength 176 kPa, density 0.03 g/cm^{3}, Young’s modulus 1.7 MPa). A characteristic load-compression curve of a 2D sample is shown in Fig. S3.

### Image Analysis of 2D Experiment.

Images of the 2D cellular prototype (Fig. 5) were obtained using a Nikon CoolPix P340 camera and stored as 3,000 × 4,000 px 24-bit JPEG images. To quantitatively identify the buckled beams in the 2D prototype under confinement, we extracted the tortuosity τ of each beam, defined as the ratio of the length of the beam to the distance between its endpoints. Tortuosity was estimated from the sample images through a series of morphological operations, as detailed in *SI Text* and Fig. S4. Straight beams have *B*), which have

## SI Text

### SVD Analysis of Equilibrium Matrix.

The linear response of the frame as well as the cellular metamaterial are obtained using the singular value decomposition (SVD) of the equilibrium matrix, as detailed in ref. 15 and summarized below. The SVD analysis simultaneously handles the spaces of node forces as well as beam stresses, and properly takes into account states of self-stress in the structure.

The SVD of the equilibrium matrix *r* nonnegative values *r* being the rank of

The first *r* columns *r* and remaining

For the frame in Fig. 2 under periodic boundary conditions, the count of topological states of self-stress predicts eight localized states of self-stress if the left domain wall were isolated, in addition to the two extended states of self-stress expected under periodic boundary conditions. However, because the separation between the left and right domain walls is finite, the SVD produces only two actual states of self-stress, and six approximate states of self-stress with large *A*–*D*, respectively.

The response of the cellular network in Fig. 3 is calculated for the finite block with free edges. Unlike the frame, this is highly overconstrained and has many generalized states of self-stress (now corresponding to mixtures of tensions, shears, and torques that maintain equilibrium). An external load *A*, satisfies this condition. The generalized stresses *A*–*C*).

### Sequential Removal of Beams.

Fig. 3 showed that the axial compressions in the cellular material under uniaxial loading were concentrated along the left domain wall, matching the expectation from the analysis of the frame with similar beam geometry. However, every time a beam buckles, its axial load-bearing ability is lost. This fundamentally changes the load-bearing states of the underlying frame, and there is no guarantee that the beams with the highest compression continue to be along the left domain wall. However, the number of topological states of self-stress localized to the left domain wall in the underlying periodic frame grows linearly with the length of the domain wall (14), suggesting that the load-bearing ability of the left domain wall may not vanish completely with just a single buckling event. Here, we numerically show using a sequential analysis of the finite cellular frame that the left domain wall can support as many buckling events as there are repeating units along the *y* direction.

To recap, Fig. 3*A* shows the compressions in a finite cellular block, obtained by tiling the same pattern 3 times along the *y* direction, under forces applied to undercoordinated points at the top and bottom edge. Because the threshold for buckling under compression scales as *i* of length *F* is increased. We examine the influence of the buckling event on the compression response by removing this beam from the cellular structure and recalculating the axial tensions *A*–*C*), showing that the multiplicity of load-bearing states localized to the left domain wall is sufficient for at least one buckling event to occur for each repeating unit of the domain wall.

Furthermore, after each repeating unit along the *y* direction has experienced buckling, the beams in the rest of the lattice experience much lower compressions relative to the initial compressions along the left domain wall (Fig. S1*D*), signifying that the lattice remains entirely bending/shear dominated away from the domain wall.

By performing simulations on different system sizes, we confirmed that upon increasing the sample size along the *y* direction, the number of buckling events localized to the left domain wall increases proportionally.

### Load-Compression Curve of a 2D Sample.

A rudimentary measurement (Fig. S3) of the load-compression curve of a 2D cellular prototype has been performed to assess the effect of the localized buckling events on the mechanical response. The prototype, shown in Fig. S3*B*, uses a unit cell characterized by

The 2D cellular foam sample (460 mm × 150 mm × 15 mm; beam widths 1.5–2.5 mm; see *Materials and Methods* for more details) was positioned with its narrow edge against a supporting acrylic plate and scale (Kern PCB 10000–1). The sample was confined from above by a second acrylic plate. No lateral confinement was provided. The distance between top and bottom plates was incrementally decreased from 140 to 90 mm using a laboratory jack. At each confinement step the system was allowed to relax and the equilibrium force on the sample edge was recorded from the scale, resulting in a quasistatic measurement of the sample’s load-compression curve. A small upward drift of ca. 0.5 kN was recorded in the scale’s force response due to long relaxation times. This drift (assumed to be linear in the confinement) has been subtracted from the recorded data. The complete measurement is shown in Fig. S3*A*. Buckling of the beams along the domain wall leaves a clear signature in the form of a sudden softening of the response (region highlighted in red). Before buckling, the main resistance to confinement was provided by the compressed beams along the domain wall because the rest of the beams could easily bend to accommodate the confinement. Buckling effectively removes the compressional stiffness of the beams along the domain wall. Further confinement is only resisted by additional bending of all beams, which leads to a significantly softer response.

A linear load compression followed by a plateau is typical of elastomeric cellular materials (12). Such materials also tend to have a secondary stiffening regime due to densification when adjacent beams contact each other along their lengths. In our measurement, however, the entire sample buckled in the lateral direction before this regime could be reached.

### Details of 2D Image Analysis.

In *Materials and Methods*, we outlined the quantitative analysis of the 2D experimental data in Fig. 5. Here, we provide more details of the image processing required to measure the tortuosity of each beam in a cellular sample under compression. The tortuosity is defined as the ratio of the length of each beam to its end-to-end distance.

The analysis starts with a raw image such as in Fig. 5*A*, a top–down view of the sample whose top surface is selectively illuminated so that the beams making up the cellular material sample are visible against a dark background. Fig. S4*A* shows a subset of a different sample with similar beam lengths and widths, on which we illustrate the image processing steps.

#### Isolating beam sections.

To quantify the tortuosity of the lattice beams, the neutral axis (center line) of each beam must be isolated. The image is first binarized through intensity thresholding (Fig. S4*B*): pixels with an intensity value above/below a certain threshold are set to 1/0. This separates the structure (1) from the background (0).

The finite-width beams are then reduced to 1-pixel-wide center lines by finding the morphological skeleton of the binarized image. To do this, we perform a repeated sequence of binary morphological operations: skeletonizing and spur removal. The former consists of removing edge **“1”** pixels without breaking up contiguously connected regions. The latter is a pruning method that removes protrusions under a certain threshold length; it reduces spurious features due to uneven lighting and resolution limitations. The resulting skeleton is shown in Fig. S4*C*.

Next, we break up the skeleton into the neutral axes of individual beams. Note that the topology of the frame, with four beams coming together at each joint, is not reproduced in the skeleton, due to the finite size of the beam joints. Instead, the skeleton has short segments connecting junctions of pairs of beams. To remove these segments, we identify the branch points of the skeleton (pixels with more than two neighbors) and clear a circular area around them. The size of the segments and of the cleared disk is of the same order as the original beam width. The resulting image, consisting of the beams’ approximate neutral axes, is shown in Fig. S4.

Once the beam axes have been separated, two steps are implemented to improve the signal-to-noise ratio. First of all, beams near the edges of the sample are ignored. They are not always separated correctly from their neighboring beams, because they have lower connectivity than beams in the bulk of the lattice; this can lead to overestimation of the tortuosity. Second, because our lattices are fairly regular, the beams have a restricted size. Beam elements above and below these length thresholds are artifacts, and we ignore them. Fig. S4*E* illustrates the processed image data after these final steps.

#### Computing the tortuosity of each neutral axis.

After the beam axes have been isolated, connected components (sets of contiguous **“1”** pixels) are found and labeled. Each connected component corresponds to an individual beam axis. Numerical data can be extracted from it by analyzing its pixels’ coordinates, *l* successfully capture the shape of the neutral axis of each beam, and smooth out pixel noise, which leads to small-wavelength features in the binary skeleton. Fig. S4*F* displays the smoothed contours superimposed on the original image, showing that the shapes of the individual beam sections are faithfully reproduced by the parametric form of Eq. **S5**.

The tortuosity τ of the beam axes is the arc length *A* along the line divided by the distance *D* between the endpoints. Both quantities can be extracted from the fitted model of the beam axes, *B*) are colored according to the corresponding tortuosity to obtain images such as Fig. 5 *B* and *C*.

## Acknowledgments

We thank Denis Bartolo, Bryan Chen, Martin van Hecke, Charlie Kane, and Tom Lubensky for useful discussions, and Jeroen Mesman at the Leiden University Fine Mechanics Department for designing and building the compression stage. This work was funded by the Foundation for Fundamental Research on Matter and by the Delta Institute for Theoretical Physics, a program of The Netherlands Organization for Scientific Research, which is funded by the Dutch Ministry of Education, Culture, and Science.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: vitelli{at}lorentz.leidenuniv.nl.

Author contributions: J.P., A.S.M., and V.V. designed research; J.P. and A.S.M. performed research; J.P., A.S.M., and V.V. contributed new reagents/analytic tools; J.P. and A.S.M. analyzed data; and J.P., A.S.M., and V.V. 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.1502939112/-/DCSupplemental.

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