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# Interplay of structure, elasticity, and dynamics in actin-based nematic materials

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved November 27, 2017 (received for review August 4, 2017)

## Significance

Thermotropic liquid crystals (LCs) are central to a wide range of commercial and emerging technologies. Lyotropic, aqueous-based LCs are common in nature, but applications have been scarce. The current understanding of their behavior is limited, and manipulating their mechanical and dynamic characteristics has been challenging. Here we show that the elasticity and temporal evolution of biopolymer-based nematic systems can be inferred from simple optical microscopy measurements, and that their mechanical properties can be manipulated by adjusting certain molecular characteristics, such as the product of length and concentration. It is also shown that the dynamic evolution of the resulting materials can be understood and predicted on the basis of a free energy functional originally developed for the study of thermotropic systems.

## Abstract

Achieving control and tunability of lyotropic materials has been a long-standing goal of liquid crystal research. Here we show that the elasticity of a liquid crystal system consisting of a dense suspension of semiflexible biopolymers can be manipulated over a relatively wide range of elastic moduli. Specifically, thin films of actin filaments are assembled at an oil–water interface. At sufficiently high concentrations, one observes the formation of a nematic phase riddled with

Recent advances have extended applications of nematic liquid crystals (LCs) onto realms that go beyond display technologies (1) and elastomers (2, 3) to colloidal/molecular self-assembly (4⇓–6), pathogen sensing (7, 8), photonic devices (9), drug delivery (10), and microfluidics (11). In LCs, topological defects correspond to locally disordered regions where the orientation of the mesogens (12) (the units that form an LC mesophase) changes abruptly. Defects exhibit unique optical and other physicochemical characteristics, and serve as the basis for many of the applications of these materials. The microstructure of an LC, the so-called “director field,” is determined by a delicate interplay between elastic forces, geometrical constraints, and the influence of applied external fields (13, 14). That microstructure dictates optical and mechanical response of the LC to external cues. Thus, the ability to control and optimize the LC elasticity is essential for emerging applications. The most widely used low-molecular-weight LCs are thermotropic (LC phases emerge at a certain temperature), and exhibit a nematic phase within a certain, relatively narrow temperature range. In such LCs, the defect size is small (

Lyotropic materials in which LC phases emerge at certain solute concentrations circumvent some of these shortcomings. In addition, certain lyotropic LCs are not toxic to many microbial species (16) and neutral to antibody–antigen binding (17), providing a possibility for biological applications (18). Here we rely on suspensions of biopolymers to form lyotropic LCs, where long and semiflexible mesogens lead to the formation of nematic phases at sufficiently high concentrations. More specifically, we use filamentous actin (F-actin), with a width of

Our 2D nematic system provides an ideal platform for detailed studies of the dynamics of defects in lyotropic LCs. To engineer lyotropic systems containing topological defects for emerging applications, it is essential that a quantitative formalism be advanced with which to predict the dynamics of lyotropic LCs. Here we combine a Landau–de Gennes free energy model and the underlying momentum conservation equations and show that such an approach is capable of describing dynamic processes in polydisperse, semiflexible biopolymer systems, paving the way for elucidating its nonequilibrium dynamics.

## Results and Discussion

The local average orientation of the mesogens is denoted by **n**, which is a position-dependent (unit) vector field that obeys nematic (head–tail) symmetry, i.e., **n** and **n** are undistinguishable. When an LC transitions from an isotropic phase to a nematic phase, topological defects emerge and annihilate over time as the material undergoes thermal equilibration. Those defects can be categorized by their winding number or topological charge (26). Fig. 1*B* provides an illustration of **1** reduces to (using Einstein notation summing all repeating indices) *A*. By minimizing the system’s total elastic energy over an area **n**), an analytical expression for the director field near a defect is obtained. Introducing an elastic distortion energy density according to *B*,

To better account for nematic symmetry, one can also construct a second-order, symmetric, traceless tensor **Q** based on **n**. The **Q** tensor (under a uniaxial assumption) is defined by **Q** tensor is introduced. Eq. **1** can be written in terms of the **Q** tensor and incorporated into the Landau–de Gennes free energy model, and a Ginzburg–Landau approach is implemented to arrive at the LC’s equilibrium structure (see *Materials and Methods*). To explore the fine structure of the topological defect, we go beyond the one-constant approximation and consider the bend-to-splay ratio **Q**-tensor simulations (see Fig. S1). To quantify the director field near a *C*) in a polar coordinate system *D*,

We next demonstrate experimentally that it is possible to vary and visualize the material’s elasticity by constructing LCs comprised of the biopolymer F-actin with variable filament length. The filaments are polymerized starting from a 2-*Materials and Methods*). Actin filaments are crowded onto an oil–water interface stabilized by surfactant molecules by using methylcellulose as a depletion agent. Over a duration of 30 min to 60 min, a dense film of actin filaments forms a nematic phase (see Movie S1), evidenced by the appearance of several *A* shows that, as the filament length is increased from *A*). The corresponding director field of these images is obtained through image analysis (28) (see *Materials and Methods*), and the structural changes of these defects are shown in Fig. 2*B*. We have, therefore, realized an experimental system in which a dramatic transition in defect morphology is present and optically observed, and, in fact, is captured quantitatively by the theory described above (Fig. 1*C*). According to the Onsager type model (29), the increase in mesogen length systematically increases *D* and 2 demonstrates that the nematic elasticity theory can describe polydisperse, semiflexible biopolymer suspensions. Our experiments also demonstrate that the mechanical properties of actin-based nematic LCs are highly tunable through biochemical control of semiflexible polymer length.

To quantify the LC mechanics, we plot the angle *C*). We average *C* indicate that actin LCs exhibit

When the filament length is comparable to its persistence length, the bend-to-splay ratio

To explore the possibility of accessing a larger range of LC elasticity, we created composite LCs consisting of concentrated actin suspensions interdispersed with sparse concentration of microtubules, which are more rigid biopolymers (*i*) Microtubules are well dispersed in the actin, (*ii*) microtubules are relatively short and only weakly bent in the LC, and (*iii*) microtubule density is sufficiently small to render self-interactions negligible. We observe that microtubules are aligned with the local director field (Fig. 3*B*). This substantiates our intuition that the rigid polymers should only suppress the bend mode (increase *Simple Theory on the Elasticities of Microtubule Doped Actin LC*),

Guided by the considerations outlined above, we add a small concentration of taxol-stabilized microtubules to the solution of actin filaments. Consistent with our hypothesis, we observe that the addition of microtubules has a significant effect on defect morphology. Fig. 3*A* shows optical images of a system with and without microtubules. In the absence of microtubules, the +1/2 defects’ morphology is U-shaped, as outlined by the blue dashed line, and a significant amount of bending is also visible. When microtubules are added, all defects adopt a V-shape morphology, and bending is suppressed. In Fig. 3*C*, we find that the number of microtubules, shown in cyan, has a direct influence on the shape of +1/2 defects, which undergoes a gradual transition from a U- to a V-shape morphology, as highlighted by the director field around the defect core. Eq. **2** indicates that the change in the composite LC’s elasticity should only be a function of

To test this prediction, we inspect several tens of *Materials and Methods*). We measure *A*, the transition from a splay-dominated into a bend-dominated regime that occurs at

Having validated the theory, this framework can be used to measure the absolute values of **2** to determine

Traditional LC elasticity measurements rely on external fields or calibrations (12, 32, 36), a limitation overcome by the use of this protocol. Microtubules in the system play the role of “elastic dopants,” analogous to the chiral dopants that are added to nematic materials to increase chirality (37). Our results provide a framework within which one can simultaneously visualize, tune, and measure the material’s elasticity. We emphasize here that the idea of measuring a material’s mechanical properties by observing defects can be generalized to other ordered systems. Defects are singularity regions in an otherwise ordered material, where different elastic modes compete against each other. Thus, we expect the framework developed here could be extended to characterization of the morphology and statistics of defects, disclinations, or dislocations as a means of probing the elastic properties of other types of materials.

The results presented thus far serve to establish that the theoretical framework adopted here can, in fact, describe the equilibrium structure of lyotropic biopolymers. For emerging applications involving nonequilibrium processes, it is important that such a formalism be also able to describe the dynamics of such systems. In what follows, we therefore turn our attention to the more challenging task of characterizing and predicting the coupling between structure and dynamics that arises in actin-based LCs. When a thermotropic LC is quenched from an isotropic into a nematic state, defects emerge and annihilate each other over time. The interplay between LC elasticity, viscosity, and defect structure (configurations) determines the dynamics of annihilation events. Note that recent efforts have addressed the coupling of structure and dynamics in free-standing smectic LCs (38). In contrast to that work, our actin-based LC system represents a true nematic LC which is free of confining walls (39). As such, it offers a unique 2D platform on which to explore structure-based dynamics.

In Fig. 5, we show two typical defect configurations, which differ in the angle between the line connecting the defect cores and the nematic far field. For simplicity, and without loss of generality, we consider two opposite, symmetric cases. In the first, the defect line is parallel to a well-defined far field; in the second, it is perpendicular (Fig. 5 *A* and *B*). In both situations, the defects attract each other to lower the system’s total elastic free energy. Thus, in the “perpendicular” case, the *C* and *D* for the perpendicular and parallel cases, respectively, and in Movie S3. Interestingly, in the case of the parallel far field (Fig. 5*D*), the +1/2 defect deviates from a straight head-on approach to the −1/2 defect. Therefore, we have observed and analyzed two distinct types of defect annihilation events, thanks to the unique optical feature of our nematic system in which

We use our theoretical model to investigate how the flow field generated by the defect motion depends on the LC’s elastic constants and the underlying configuration. When two *Defect Annihilation*). In Fig. 5, we show the effect of defect configurations on an isolated annihilation event. We find that the perpendicular case annihilates faster than the parallel case. If one ignores hydrodynamic effects, the two cases would proceed in an identical manner (under the one-constant assumption). The inclusion of hydrodynamics breaks that symmetry. As shown in Fig. 5, the flow fields corresponding to the two configurations are markedly different. For the perpendicular case, there are two symmetric vortices on the two sides of the defect line (indicated by blue lines in Fig. 5*A* with an outward transverse flow upon annihilation), whereas, for the parallel case, upon defect annihilation, an extensional flow is produced, and the transverse flow is inward. Because the director field at the defect line in the perpendicular case is aligned with the flow, the corresponding viscosity is low, and the enhanced hydrodynamic flow accelerates the defects’ motion. In the parallel case, the flow direction is perpendicular to the local director field, and the high viscosity suppresses such flow, leading to slower dynamics. The horizontal separation of two *E*. Our experiments indicate that, on average, the perpendicular case clearly exhibits faster dynamics than the parallel case, confirming our predictions. Taken together, these results establish that defect structure influences dynamics, a feature that could be exploited to manipulate the transport of cargo in an LC by creating different flow patterns as defects annihilate.

The results presented in Fig. 5 are limited to two isolated defects, and only two ideal, extreme cases (parallel and perpendicular defect configuration) are considered. The question that emerges then is whether the simple formalism adopted here can describe dynamics in the more common case in which multiple defects with nearly arbitrary relative orientations interact with each other (41, 42). To address this question, we now consider a system of area *Materials and Methods*). We run the simulation over an extended period, and examine whether the model can capture the temporal evolution of defects that is observed in the experiments over laboratory time scales (minutes to hours). Our computer simulation can be viewed as analogous to weather forecasting: We take a pattern (director field) as the initial condition, and we rely on a simulation to predict its structure at a future time. In our quasi-2D nematic system, we show four images comparing simulations and experiments at four consecutive times, separated by a time interval of ∼200 s. As shown in Fig. 6 and in Movie S4, the model predicts the three annihilation events in correct chronological order (see the circled defects in Fig. 6), and the individual defect positions after 10 min. Shown in Fig. 7, simulations have also quantitatively captured defect separations for the three annihilating defect pairs. This agreement, which is simply based on the initial defect structure, along with our previous work using the same model to successfully describe active nematics (43), serves to underscore that the hydrodynamic formalism adopted here is able to capture key aspects of the structure and dynamics of lyotropic, polydisperse, actin-based 2D nematic systems. Despite the fact that the LC’s elastic structure and hydrodynamics are both long-ranged and nonlinearly coupled, its defect dynamics can be accurately predicted by our athermal model, implying that such defect dynamics are deterministic.

In conclusion, we have demonstrated a semiflexible biopolymer-based 2D nematic system, which exhibits highly tunable elasticity, and could be useful for future LC technologies relying on topological defects. Its elastic constants can be tuned either by varying the average filament length or by doping it with a sparse concentration of rigid polymers. We show a method to infer the ratio of splay and bend constant by visualizing the morphology of

## Materials and Methods

### Theory and Modeling.

The bulk free energy of the nematic LC, **5** can be mapped to the **1** via**n** is the eigenvector associated with the greatest eigenvalue of the **Q** tensor at each lattice point.

To obtain the static morphology of topological defects, we minimize Eq. **3** with respect to the **Q** tensor via a Ginzburg–Landau algorithm (45). To simulate the LC’s nonequilibrium dynamics, a hybrid lattice Boltzmann method is used to simultaneously solve a Beris–Edwards equation and a momentum equation which accounts for the backflow effects. By introducing a velocity gradient **H**, which drives the system toward thermodynamic equilibrium, is given by**8**, reduces to Ginzburg–Landau equation**Q** tensor at equilibrium.

Degenerate planar anchoring is implemented through a Fournier–Galatola expression (48) that penalizes out-of-plane distortions of the **Q** tensor. The associated free energy expression is given by**P** is the projection operator associated with the surface normal **Q** field is governed by (49)

Using an Einstein summation rule, the momentum equation for the nematics can be written as (47, 51)**8**, using a finite difference method. The momentum equation, Eq. **12**, is solved simultaneously via a lattice Boltzmann method over a D3Q15 grid (53). The implementation of stress follows the approach proposed by Guo et al. (54). Our model and implementation were validated by comparing our simulation results to predictions using the Ericksen–Leslie–Parodi theory (55⇓–57). The units are chosen as follows: The unit length

### Experimental Methods.

#### Proteins.

Monomeric actin is purified from rabbit skeletal muscle acetone powder (58) (Pel-Freez Biologicals) stored at _{2}, 0.2 mM DTT, _{3}). Fluorescent labeling of actin was done with a tetramethylrhodamine-6-maleimide dye (Life Technologies). Capping protein [mouse, with a HisTag, purified from bacteria (59); gift from the Dave Kovar laboratory, The University of Chicago, Chicago] is used to regulate actin filament length. The 1:10 ratio of fluorescently labeled tubulin (cat _{4},1 mM EGTA, pH 6.8) at 5 mg/mL. The mixture was centrifuged at 4 °C at 30,000 × *g* for 10 min, and the supernatant was incubated in PEM-100 at 37 °C in the presence of 1 mM GMPCPP (cat #NU-405L; Jena Biosciences) for 20 min. Taxol was added to stabilize polymers to 50 μM final concentration. The microtubule length was shortened by shearing through Hamilton Syringe (Mfr #81030, Item #EW-07939-13) before adding 3 μL of volume to the actin polymers.

#### Experimental assay and microscopy.

The actin is polymerized in 1X F-buffer (10 mM imidazole, pH 7.5, 50 mM KCL, 0.2 mM EGTA, 1 mM MgCl_{2}, and 1 mM ATP). To prevent photobleaching, an oxygen scavenging system [4.5 mg/mL of glucose, 2.7 mg/mL of glucose oxidase (cat

The experiment is done in a cylindrical chamber which is a glass cylinder glued to a coverslip (60). Coverslips are sonicated clean with water and ethanol. The surface is treated with triethoxy(octyl)silane in isopropanol to produce a hydrophobic surface. To prepare a stable oil–water interface, PFPE-PEG-PFPE surfactant (cat

The sample is imaged using an inverted microscope (Eclipse Ti-E; Nikon) with a spinning disk confocal head (CSU-X; Yokagawa Electric), equipped with a Complementary metal-oxide semiconductor camera (Zyla-4.2 USB 3; Andor). A 40

#### Image analysis.

To extract the director field, the optical images were band-pass filtered and unsharp masked in the ImageJ software (61) to remove noise and spatial irregularities in brightness.

The resulting images were analyzed using an algorithm as described in methods of Cetera et al. (28). In short, the algorithm computes 2D fast Fourier transform of a small local square section (of side

The microtubule number density (*C*). The resulting image is then thresholded to separate all of the microtubule bundles whose

## Acknowledgments

N.K. thanks Dr. Kimberly Weirich for useful discussions and purified proteins, Dr. Samantha Stam for assisting with experiments, and Dr. Patrick Oakes for helping in director field analysis. R.Z. acknowledges helpful discussions with Dr. Shuang Zhou and Dr. Takuya Yanagimachi, and is grateful for the support of the University of Chicago Research Computing Center for assistance with the calculations carried out in this work. This work was supported primarily by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation (NSF) under Award DMR-1420709. M.L.G. and J.L.R. acknowledge support from NSF Grant MCB-1344203. J.J.d.P. acknowledges support from NSF Grant DMR-1710318. The design of tubulin–actin composites in the J.L.R. and J.J.d.P. group was supported by the US Army Research Office through the Multidisciplinary University Research Initiative (MURI Award W911NF-15-1-0568). N.K. acknowledges the Yen Fellowship of the Institute for Biophysical Dynamics, The University of Chicago.

## Footnotes

↵

^{1}R.Z. and N.K. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: depablo{at}uchicago.edu or gardel{at}uchicago.edu.

Author contributions: R.Z., N.K., M.L.G., and J.J.d.P. designed research; R.Z. and N.K. performed research; J.L.R. contributed new reagents/analytic tools; R.Z. and N.K. analyzed data; and R.Z., N.K., M.L.G., and J.J.d.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1713832115/-/DCSupplemental.

Published under the PNAS license.

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