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# Morphological transitions of elastic filaments in shear flow

Edited by Howard A. Stone, Princeton University, Princeton, NJ, and approved August 3, 2018 (received for review March 31, 2018)

## Significance

Elastic filaments and semiflexible polymers occur ubiquitously in biophysical systems and are key components of many complex fluids, yet our understanding of their conformational dynamics under flow is incomplete. Here, we report on experimental observations of actin filaments in simple shear and characterize their various dynamical regimes from tumbling to buckling and snaking. Numerical simulations accounting for elastohydrodynamics as well as Brownian fluctuations show perfect agreement with measurements. Using a reduced-order theoretical model, we elucidate the unexplained mechanism for the transition to snaking. Our results pave the way for a better understanding of biophysical processes, as well as the rheology of sheared soft materials, and provide a theoretical framework for the exploration of the dynamics of dilute and semidilute suspensions.

## Abstract

The morphological dynamics, instabilities, and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are used as an experimental model system and their conformations are investigated through fluorescence microscopy in microfluidic channels. Simulations matching the experimental conditions are also performed using inextensible Euler–Bernoulli beam theory and nonlocal slender-body hydrodynamics in the presence of thermal fluctuations and agree quantitatively with observations. We demonstrate that filament dynamics in this system are primarily governed by a dimensionless elasto-viscous number comparing viscous drag forces to elastic bending forces, with thermal fluctuations playing only a secondary role. While short and rigid filaments perform quasi-periodic tumbling motions, a buckling instability arises above a critical flow strength. A second transition to strongly deformed shapes occurs at a yet larger value of the elasto-viscous number and is characterized by the appearance of localized high-curvature bends that propagate along the filaments in apparent “snaking” motions. A theoretical model for the as yet unexplored onset of snaking accurately predicts the transition and explains the observed dynamics. We present a complete characterization of filament morphologies and transitions as a function of elasto-viscous number and scaled persistence length and demonstrate excellent agreement between theory, experiments, and simulations.

The dynamics and conformational transitions of elastic filaments and semiflexible polymers in viscous fluids underlie the complex non-Newtonian behavior of their suspensions (1) and also play a role in many small-scale biophysical processes from ciliary and flagellar propulsion (2, 3) to intracellular streaming (4, 5). The striking rheological properties of polymer solutions hinge on the microscopic dynamics of individual polymers and particularly on their rotation, stretching, and deformation under flow in the presence of thermal fluctuations. Examples of these dynamics include the coil–stretch (6, 7) and stretch–coil (8, 9) transitions in pure straining flows and the quasi-periodic tumbling and stretching of elastic fibers and polymers in shear flows (10, 11). Elucidating the physics behind these microstructural instabilities and transitions is key to unraveling the mechanisms for their complex rheological behaviors (12), from shear thinning and normal stress differences (13) to viscoelastic instabilities (14) and turbulence (15).

The case of long-chain polymers such as DNA (16), for which the persistence length

On the contrary, the dynamics of shorter polymers such as actin filaments (23), for which

The classical case of a rigid rod-like particle in a linear flow has been well understood since the work of Jeffery (24), who first described the periodic tumbling now known as Jeffery orbits occurring in shear flow. When flexibility becomes significant, viscous stresses applied on the filament can overcome bending resistance and lead to structural instabilities reminiscent of Euler buckling of elastic beams (8⇓–10, 25⇓⇓–28). On the other hand, Brownian orientational diffusion has been shown to control the characteristic period of tumbling (23, 29). In shear flow, the combination of rotation and deformation leads to particularly rich dynamics (23, 30⇓⇓⇓⇓–35), which have yet to be fully characterized and understood.

In this work, we elucidate these dynamics in a simple shear flow by combining numerical simulations, theoretical modeling, and model experiments using actin filaments. The filaments we consider here have a contour length L in the range of ^{−1}, we identify and characterize transitions from Jeffery-like tumbling dynamics of stiff filaments to buckled and finally strongly bent configurations for longer filaments.

## Results and Discussion

### Governing Parameters and Filament Dynamics.

In this problem, the filament dynamics result from the interplay of three physical effects—elastic bending forces, thermal fluctuations, and viscous stresses—and are governed by three independent dimensionless groups. First, the ratio of the filament persistence length *k*_{B} is the Boltzmann constant and *T* the temperature. Third, the anisotropic drag coefficients along the filament involve a geometric parameter

The elasto-viscous number can be viewed as a dimensionless measure of flow strength and exhibits a strong dependence on contour length. By varying L and *SI Appendix*, Movies S1–S6. In relatively weak flows, the filaments are found to tumble without any significant deformation in a manner similar to that of rigid Brownian rods. On increasing the elasto-viscous number, a first transition is observed whereby compressive viscous forces overcome bending rigidity and drive a structural instability toward a characteristic C-shaped configuration during the tumbling motion. By analogy with Euler beams, we term this deformation mode “global buckling” as it occurs over the full length of the filament. In stronger flows, this instability gives way to highly bent configurations, which we call U turns, and they are akin to the snaking motions previously observed with flexible fibers (23, 32). During those turns, the filament remains roughly aligned with the flow direction while a curvature wave initiates at one end and propagates toward the other end. At yet higher values of *SI Appendix*, Movies S5 and S6). In all cases, excellent agreement is observed between experimental measurements and Brownian dynamics simulations. Our focus here is in describing and explaining the first three deformation modes and corresponding transitions.

We characterize the temporal shape evolution more quantitatively for each case in Fig. 2. To describe the overall shape and orientation of the filament, we introduce the gyration tensor, or the second mass moment, as

As is evident in Fig. 2, these different variables exhibit distinctive signatures in each of the three regimes and can be used to systematically differentiate between configurations. During Jeffery-like tumbling, filaments remain nearly straight with

### Order Parameters.

This descriptive understanding of the dynamics allows us to investigate transitions between deformation regimes as the elasto-viscous number increases. The dependence on *C*. In the tumbling regime, deformations are negligible beyond those induced by thermal fluctuations, as evidenced by the nearly constant values of *A* and *B* also shows a few data points for S turns at high values of

Orientational dynamics are summarized in Fig. 3*C*, showing the range *C*, where C and U turns stand apart. As

While we have not studied the tumbling frequency extensively, data based on a limited number of simulations and experiments recover the classical 2/3 scaling of frequency on flow strength (21, 23) for the explored range of parameters, with a systematic deviation toward 3/4 in strong flows in agreement with results from Lang et al. (29).

### Transitions Between Regimes and Phase Diagram.

Our experiments and simulations have uncovered three dynamical regimes with increasing values of

Thermal fluctuations do not significantly alter this threshold, but instead result in a blurred transition (9, 26, 37) with an increasingly broad transitional regime where both tumbling and C buckling can be observed for a given value of

Upon increasing *SI Appendix*, Fig. S6). This transition toward snaking dynamics has not previously been characterized. Our attempt at understanding its mechanism focuses on the onset of a U turn, which always involves the formation of a J-shaped configuration as visible in Fig. 1 and also illustrated in Fig. 5.

To elucidate the transition mechanism, we develop a theoretical model for a J configuration, which can be viewed as a precursor to the U turn. We neglect Brownian fluctuations and idealize the J shape as a semicircle of radius R connected to a straight arm forming an angle ϕ with the flow direction, with both sections undergoing a snaking motion responsible for the U turn; details of the model, which draws on analogies with the tank-treading motion of vesicles (38, 39), can be found in *SI Appendix*, *Theoretical Model*. By satisfying filament inextensibility as well as force and torque balances, and by balancing viscous dissipation in the fluid with the work of elastic forces, we are able to solve for model parameters such as R and ϕ without any fitting. A key aspect of the model is that consistent solutions for these parameters can be obtained only above a critical elasto-viscous number, and this solvability criterion thus provides a threshold

We can now discuss the initiation of the J shape, in which two possible mechanisms may be at play. On the one hand, it may be caused by the global buckling of the filament in the presence of highly compressive viscous forces, in a manner consistent with the sequence of shapes in Fig. 5*A*. Under sufficiently strong shear, compressive forces can induce a buckling instability on a filament that has not yet aligned with the compressional axis and forms only a small angle with the flow direction. Alignment of the deformed filament with the flow then results in differential tension (compression vs. tension) near its two ends, thus allowing one end to bend while the other remains straight. A second potential mechanism proposed in ref. 29 is of a local buckling occurring on the typical length scale of transverse thermal fluctuations. Our data, however, clearly show that the transition to U turns is independent of thermal fluctuations, allowing us to discard this hypothesis. Thermal fluctuations are nonetheless responsible for the existence of the transitional regime above

### Dynamics of *U* Turns.

We further characterize the dynamics during U turns, for which our theoretical model also provides predictions. The filament orientation at the onset of a turn is plotted in Fig. 6*A*, showing the tilt angle ϕ formed by the straight arm of the J shape with respect to the flow direction as a function of

After a J shape is initiated as discussed above, the curvature of the folded region remains nearly constant in time as suggested by the plateau in the bending energy (Fig. 3*C*). This provides a strong basis for approximating the bent part of the filament as a semicircle of radius R in our model. The theoretical prediction *B* (see *SI Appendix*, Fig. S5 for details). The radius of the bend is seen to decrease with

The rotation of the end-to-end vector during the U turn results primarily from tank treading of the filament along its arclength, unlike the global rotation that dominates the tumbling and C-buckling regimes. While the snaking velocity is not constant during a turn, its average value can be quantitatively measured through the time derivative of the end-to-end distance, yielding the approximation *SI Appendix*, Fig. S3). The same rescaling applied to the experimental and numerical data and using the theoretical radius *C*.

Harasim et al. (23) previously proposed a simplified theory of the U turn, which shares similarities with ours but assumes that the filament is aligned with the flow direction and neglects elastic stresses inside the fold. Their predictions are in partial agreement with our results in the limit of very long filaments and strong shear (*SI Appendix*, *Theoretical Model*). Their theory is unable to predict and explain the transition from buckling to U turns.

### Concluding Remarks.

Using stabilized actin filaments as a model polymer, we have systematically studied and analyzed the conformational transitions of elastic Brownian filaments in simple shear flow as the elasto-viscous number is increased. Our experimental measurements were shown to be in excellent agreement with a computational model describing the filaments as fluctuating elastic rods with slender-body hydrodynamics. By varying filament contour length and applied shear rate, we performed a broad exploration of the parameter space and confirmed the existence of a sequence of transitions, from rod-like tumbling to elastic buckling to snaking motions. While snaking motions had been previously observed in a number of experimental configurations, the existence of a C-buckling regime had not been confirmed clearly. This is due to the fact that C buckling is visible over only a limited range of elasto-viscous numbers and occurs only in simple shear flow, which is challenging to realize experimentally. We showed that both transitions are primarily governed by

While the first transition from tumbling to buckling had been previously described as a supercritical linear buckling instability (13), the transition from buckling to snaking was heretofore unexplained. Using a simple analytical model for the dynamics of the J shape that is the precursor to snaking turns, we were able to obtain a theoretical prediction for the threshold elasto-viscous number above which snake turns become possible. The model did not take thermal noise into account, but highlighted the subtle role played by tension and compression during the onset of the turn. Our analysis and model lay the groundwork for illuminating a wide range of other complex phenomena in polymer solutions, from their rheological response in flow and dynamics in semidilute solutions (40, 41) to migration under confinement and microfluidic control of filament dynamics.

## Materials and Methods

### Experimental Methods.

The protocol for assembly of the actin filaments is well controlled and reproducible. Concentrated G-actin, which is obtained from rabbit muscle and purified according to the protocol described in ref. 42, is placed into F buffer (10 mM Tris⋅HCl, pH 7.8, 0.2 mM ATP, 0.2 mM

A micro-PDMS channel is designed as a vertical Hele–Shaw cell, with length *SI Appendix*, *Microuidic Channel Geometry* for more details). To consider pure shear flow, filament and flow scales should be properly separated, and we thus chose a width (150 μm) much larger than the typical dimension of the deformed filament (

Stable flow is driven by a syringe pump (Cellix ExiGo) and particle-tracking velocimetry is used to check the agreement of the velocity profile with theoretical predictions (43). We impose flow rates Q in the range of

Images are captured by a s-CMOS camera (HAMAMATSU ORCA flash 4.0LT, 16 bits) with an exposure time of

### Modeling and Simulations.

We model the filaments as inextensible Euler–Bernoulli beams and use nonlocal slender-body hydrodynamics to capture drag anisotropy and hydrodynamic interactions (10, 25). Simulations without hydrodynamic interactions (free-draining model) were also performed but did not compare well with experiments. Brownian fluctuations are included and satisfy the fluctuation–dissipation theorem. As experiments consider only quasi-2D trajectories involving dynamics in the focal plane, we perform all simulations in 2D and indeed found better agreement compared with 3D simulations. Details of the governing equations and numerical methods are provided in *SI Appendix*, *Computational Model and Methods*. The simulation code is available upon request to the authors.

## Acknowledgments

We are grateful to Guillaume Romet-Lemonne and Antoine Jégou for providing purified actin and to Thierry Darnige for help with the programming of the microscope stage. We thank Michael Shelley, Lisa Fauci, Julien Deschamps, Andreas Bausch, Gwenn Boedec, Anupam Pandey, Harishankar Manikantan, and Lailai Zhu for useful discussions and Roberto Alonso-Matilla for checking some of our calculations. The authors acknowledge support from European Research Council Consolidator Grant 682367, from a Chinese Scholarship Council Scholarship, and from NSF Grant CBET-1532652.

## Footnotes

↵

^{1}Y.L. and B.C. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: anke.lindner{at}espci.fr.

Author contributions: Y.L., B.C., D.S., A.L., and O.d.R. designed research; Y.L., A.L., and O.d.R. performed experiments; B.C. and D.S. performed simulations; Y.L., B.C., D.S., A.L., and O.d.R. analyzed data and contributed to the theoretical model; and Y.L., B.C., D.S., A.L., and O.d.R. 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.1805399115/-/DCSupplemental.

Published under the PNAS license.

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