^{1}W. M. Keck Science Department, Scripps College, Claremont, CA, United States^{2}W. M. Keck Science Department, Pitzer College, Claremont, CA, United States^{3}Physics and Biophysics Department, University of San Diego, San Diego, CA, United States

The cytoskeleton–a composite network of biopolymers, molecular motors, and associated binding proteins–is a paradigmatic example of active matter. Particle transport through the cytoskeleton can range from anomalous and heterogeneous subdiffusion to superdiffusion and advection. Yet, recapitulating and understanding these properties–ubiquitous to the cytoskeleton and other out-of-equilibrium soft matter systems–remains challenging. Here, we combine light sheet microscopy with differential dynamic microscopy and single-particle tracking to elucidate anomalous and advective transport in actomyosin-microtubule composites. We show that particles exhibit multi-mode transport that transitions from pronounced subdiffusion to superdiffusion at tunable crossover timescales. Surprisingly, while higher actomyosin content increases the range of timescales over which transport is superdiffusive, it also markedly increases the degree of subdiffusion at short timescales and generally slows transport. Corresponding displacement distributions display unique combinations of non-Gaussianity, asymmetry, and non-zero modes, indicative of directed advection coupled with caged diffusion and hopping. At larger spatiotemporal scales, particles in active composites exhibit superdiffusive dynamics with scaling exponents that are robust to changing actomyosin fractions, in contrast to normal, yet faster, diffusion in networks without actomyosin. Our specific results shed important new light on the interplay between non-equilibrium processes, crowding and heterogeneity in active cytoskeletal systems. More generally, our approach is broadly applicable to active matter systems to elucidate transport and dynamics across scales.

## 1 Introduction

The cytoplasm is a crowded, heterogeneous, out-of-equilibrium material through which macromolecules and vesicles traverse to perform critical cellular processes such as mitosis, endocytosis, migration, and regeneration [1–4]. Macromolecules and particles diffusing through the cytoplasm and other similar materials have been shown to exhibit widely varying and poorly understood anomalous transport properties that deviate significantly from normal Brownian diffusion. In particular, the mean-squared displacement,

In steady-state, the thermal transport of particles through *in vitro* cytoskeletal systems exhibit varying degrees of subdiffusion and non-Gaussianity depending on the types and concentrations of filaments and crosslinking proteins [5, 6, 12, 13]. For example, single-particle tracking (SPT) of particles in composites of entangled actin filaments and microtubules, has revealed increasing degrees of subdiffusion (

Differential dynamic microscopy (DDM), which uses Fourier-space analysis to measure the timescales over which particle density fluctuations decay, has also been used to measure transport and quantify anomalous characteristics over larger spatiotemporal scales compared to SPT [5, 14, 15]. DDM analysis of a time-series of images provides a characteristic decay time *via* *via* SPT and DDM,

**FIGURE 1**. Combining light sheet microscopy with real-space single-particle tracking (SPT) and reciprocal-space differential dynamic microscopy (DDM) to characterize particle transport in active cytoskeletal composites. **(A)** We create composites of co-entangled microtubules (blue) and actin filaments (purple) driven out-of-equilibrium by myosin II minifilaments (green). We track the motion of embedded 1 **(B)** Schematic of the light-sheet microscope we use for data collection, which provides the necessary optical sectioning to capture dynamics in dense three-dimensional samples. **(C)** Example frame from time-series of 1 **(D)** Cartoon of expected mean-squared displacements (*versus* lag time *via* single-particle tracking (SPT) and fit to a power law **(E)** Cartoon van Hove distribution **(F)** Cartoon of expected characteristic decorrelation times

Similar SPT and DDM experiments demonstrated that crosslinking of actin and/or microtubules introduced bi-phasic transport with the subdiffusive scaling exponents dropping from

Numerous studies have also investigated transport in non-equilibrium cytoskeleton networks, in which activity is introduced *via* motor proteins, such as actin-associated myosin II and microtubule-associated kinesin [2, 10, 18–21]. These studies have shown evidence of vesicle movement strongly tracking with actin movement, microtubule-dependent flow, and the simultaneous presence of subdiffusive and ballistic transport dynamics. While the majority of these active matter studies have been on systems of either actin or microtubules, recent studies have used DDM and optical tweezers microrheology to characterize the dynamics of actin-microtubule composites pushed out-of-equilibrium by myosin II minifilaments straining actin filaments [14, 22, 23]. These studies showed that active actin-microtubule composites exhibited ballistic-like (

However, particle transport through active actin-microtubule composites–likely dictated by the complex combination of active processes, crowding, and interactions between mechanically distinct filaments—has remained largely unexplored. The majority of studies that have examined the combined effect of activity and crowding have been *in vivo* [1, 24–29], where a large number of conflating variables that are difficult to tune make teasing out the effect of each contribution and mechanism highly non-trivial.

Here, we take advantage of the tunability of *in vitro* reconstituted cytoskeleton composites to systematically investigate the coupled effects of non-equilibrium activity, crowding, and heterogeneity on particle transport. We combine fluorescence light sheet microscopy (fLSM) with single-particle tracking (SPT) and differential dynamic microscopy (DDM) to examine the anomalous transport of micron-sized particles within active composites of myosin II minifilaments, actin filaments, and microtubules with varying molar fractions of actin and tubulin (Figure 1A). We leverage the optical sectioning and low excitation power of fLSM (Figure 1B) to capture particle trajectories with a temporal resolution of 0.1s for durations up to 400 s (Figure 1C). Using both SPT and DDM provides transport characterization over a broad spatiotemporal range that spans ∼10^{–1} -10^{2} s and ∼10^{–1} - 10

## 2 Materials and methods

### 2.1 Protein preparation

We reconstitute rabbit skeletal actin (Cytoskeleton, Inc. AKL99) to 2 mg/ml in 5 mM Tris–HCl (pH 8.0), 0.2 mM CaCl_{2}, 0.2 mM ATP, 5% (w/v) sucrose, and 1% (w/v) dextran; porcine brain tubulin (Cytoskeleton T240) to 5 mg/ml in 80 mM PIPES (pH 6.9), 2 mM MgCl_{2}, 0.5 mM EGTA, and 1 mM GTP; and rabbit skeletal myosin II (Cytoskeleton MY02) to 10 mg/ml in 25 mM PIPES (pH 7.0), 1.25 M KCl, 2.5% sucrose, 0.5% dextran, and 1 mM DTT. We flash freeze all proteins in single-use aliquots and store at -80˚C. We reconstitute the UV-sensitive myosin II inhibitor, (-)-blebbistatin (Sigma B0560) in anhydrous DMSO and store at -20˚C for up to 6 months. Immediately prior to experiments, we remove enzymatically dead myosin II from aliquots using centrifugation clarification, as previously described [14, 22].

### 2.2 Composite network assembly

We prepare actin-microtubule composites by mixing actin monomers, tubulin dimers and a trace amount of 1 µm diameter carboxylated microspheres (Polysciences) in PEM-100 (100 mM PIPES, 2 mM MgCl_{2}, and 2 mM EGTA), 0.1% Tween-20, 1 mM ATP, and 1 mM GTP, and incubating at 37°C for 30 min to allow for polymerization of actin filaments and microtubules. We coat microspheres (beads) with AlexaFluor594 BSA (Invitrogen) to visualize the particles and prevent nonspecific interactions with the composite [30, 31]. We fix the combined molar concentration of actin and tubulin to *R*=0.08, and vary the molar fraction of actin in the composite

### 2.3 Sample preparation and imaging

We pipet prepared composites into capillary tubing with an inner diameter of *x*=800 μm and length of *y*=10 mm, then seal with epoxy. Microspheres are imaged using a custom-built fLSM with a 10*x* and *y* axes of each FOV are aligned with the cross-section and length of the capillary sample chamber, respectively.

### 2.4 Single-particle tracking

We use the Python package Trackpy [34] to track particle trajectories and measure the *via*

Additionally, we compute van Hove probability distributions of particle displacements,

### 2.5 Differential dynamic microscopy

We obtain the image structure function

where *q* =

We also examine the stretching exponent *q*, necessary for unequivocal determination and interpretation of the scaling exponent

## 3 Results and discussion

To elucidate the combined effects of non-equilibrium activity and steric hindrance on particle transport in crowded active matter, we leverage the tunability of reconstituted cytoskeleton composites [42] and the power of coupling real-space (SPT) and reciprocal space (DDM) transport analysis, to robustly characterize particle transport as a function of active substrate content. Specifically, we tune the composition of actomyosin-microtubule composites to display a wide range of transport characteristics by varying the molar fraction of actomyosin, which we denote by the molar actin fraction

In Figure 2A, we plot the ensemble-averaged *Δt* for particles diffusing in composites of varying *versus Δt* (Figure 2B), which is a horizontal line for normal Brownian diffusion with the

**FIGURE 2**. Actomyosin activity in actin-microtubule composites endows multi-phasic particle transport marked by pronounced subdiffusion at short lag times and superdiffusion at long lag times. **(A)** Mean-squared displacements (*versus* lag time **(B)** Mean-squared displacements scaled by lag time (*versus* lag time **(A)**. Black lines indicate fits to **(C)** Data shown in B plotted *versus* actin fraction **(D)** Lag time at which each composite transitions out of initial subdiffusive transport (**(E)** Anomalous scaling exponents **(B)** for **(C)** and **(D)** error bars indicate standard error of the mean. Color-coding in all subfigures matches the legend in **(A)**.

To evaluate the *versus*

To understand this phenomenon, we consider that active ballistic transport would only be detectable at timescales in which the network motion can move a bead more than the minimum resolvable displacement:

To determine the extent to which motor-driven transport and confinement contribute to the particle dynamics, we next evaluate the anomalous scaling exponents in the short, intermediate, and long

To understand the lower

Examining the long-time regime,

To shed further light on the mechanisms underlying the anomalous transport shown in Figure 2, we compute van Hove distributions

**FIGURE 3**. Asymmetric non-Gaussian van Hove distributions reveal a combination of heterogeneous subdiffusion and advective transport of particles in active composites. **(A)** van Hove distributions *via* SPT, for lag times **(B)** The square of the full width at half-maximum *versus* lag time **(A)**. Solid lines are fits to **(C)** The scaling exponents **(D)** A sample **(E)** The fractional difference of the half-width at half maximum **(A)**. Horizontal bars through each distribution denote the mean.

While our analysis described above assumes Gaussian distributions, Figure 3A shows that nearly all distributions have distinct non-Gaussian features similar to those reported for steady-state actin-microtubule composites [5, 7]. In particular,

The distributions for active composites are even more complex, with asymmetries and peaks at *versus* leading edge:

We also note that the asymmetry shown in Figure 3A is always in the negative direction (to the left). To determine the directionality of this asymmetry we evaluate the distributions of

To determine if this directionality is a reproducible microscale bias or is an ensemble result of averaging over trials with positive and negative anisotropies, we evaluate the van Hove distributions for each trial of the

To more quantitatively characterize the rich transport phenomena revealed in Figure 3, we first fit each

**FIGURE 4**. Actomyosin activity reduces heterogeneous non-Gaussian diffusivity and endows Gaussian-like advective transport. **(A)** Comparing van Hove distributions of composites with different **(B)**. **(C)** Fractional amplitude of the Gaussian term in each fit, **(D)** Fractional amplitude of the exponential term in each fit,

As shown in Figures 4C,D, in which

Conversely, the increased contribution from the exponential term for the trailing edge, implies that displacements comprising these exponential tails are dominated by heterogeneous transport modes such as hopping between heterogeneously distributed cages [6, 7]. To better understand this effect, we recall that particle displacements comprising the trailing edge are lagging behind the bulk translational motion of the composite. As the composite moves and restructures, a fraction of the particles remain caged in the moving composite and thus move along with it, corresponding to displacements comprising the leading edge, whereas a fraction of the particles are squeezed out or hop out of composite cages and into new ‘trailing’ cages. It is also likely that as the composite contracts and forms more heterogeneous structures and larger open voids that characteristic ‘hopping’ displacements, as well as displacements withing cages, may increase and become more heterogeneous, thereby enhancing exponential tails.

To expand the range of length and time scales over which we probe the non-equilibrium transport, and provide an independent measure of the dynamics, we complement our real-space SPT analysis with Fourier-space DDM analysis, as described in the Methods and previously [5, 6, 14]. Briefly, we compute the radially-averaged image structure function *via*

**FIGURE 5**. DDM analysis reveals ballistic-like transport of particles entrained in active composites at mesoscopic spatiotemporal scales. **(A)** Sample image structure functions **(B)** Decay times **(A)**. Dashed and dotted black lines show scaling **(C)** **(B)**. Horizontal dotted line and unity-sloped dashed line correspond to scaling indicative of normal diffusion (**(E)** and **(F)**. For **(D)** Scatter plot of **(E)** DDM scaling exponents **(C)**. **(F)** Anomalous scaling exponents *via* SPT (see Figures 2, 3). Error bars indicate 95% confidence intervals of fits. **(G)** Stretching exponent

As shown in Figure 5B,

To better visualize differences in *via* SPT (Figure 2E). To directly compare *via* SPT) and tend towards free diffusion at larger timescales (measured *via* DDM), as shown in Figure 5F.

Finally, to shed light on the competing contributions from motor-driven dynamics *versus* confinement and crowding to transport at larger spatiotemporal scales, we evaluate the dependence of the stretching exponent

We note that while other active systems display a return to Gaussian dynamics at long timescales (e.g., 46), we find no evidence of this return due to the relatively slow active network dynamics compared to other active systems. However, our inactive system (

## 4 Conclusion

Here, we couple real-space SPT and Fourier-space DDM to characterize particle transport across three decades in time (∼10^{–1}–10^{2} s) and two decades in space (∼10^{–1}—10

Figure 6 summarizes and compares the key metrics we present in Figures 2–5 that characterize these complex transport properties. Importantly, as highlighted in Figure 6, while there is clear difference between the inactive and active networks for nearly all of the transport metrics we present, we emphasize that there are very few clear monotonic dependences on

**FIGURE 6**. A robust suite of metrics reveals complex scale-dependent transport resulting from competition between motor-driven active dynamics, crowding and network connectivity. The 8-variable spider plot shows how the key metrics we use to characterize transport depend on

## Data availability statement

The datasets generated and analyzed for this manuscript are available from the corresponding author on reasonable request.

## Author contributions

JS analyzed and interpreted the data and wrote the paper; JG collected and helped analyze the data; SK and BL helped analyze data and prepare figures; RM conceived and guided the project, developed analysis code, helped analyze and interpret the data, and helped write the paper; RR-A conceived and guided the project, interpreted the data, and wrote the paper.

## Funding

This work was supported by National Institutes of Health R15 Awards (R15GM123420, 2R15GM123420-02) to RMR-A and RJM, William M. Keck Foundation Research Grant to RMR-A, and NSF RUI Award (DMR-2203791) to JYS.

## Acknowledgments

We thank AX, NB, and JK for help with data analysis and visualization. We thank GL for help with composite design. We thank DA for insightful discussions.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

## Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.1055441/full#supplementary-material

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Keywords: fluorescence microscopy, myosin, single-particle tracking, differential dynamic microscopy, cytoskeleton, actin, microtubule

Citation: Sheung JY, Garamella J, Kahl SK, Lee BY, McGorty RJ and Robertson-Anderson RM (2022) Motor-driven advection competes with crowding to drive spatiotemporally heterogeneous transport in cytoskeleton composites. *Front. Phys.* 10:1055441. doi: 10.3389/fphy.2022.1055441

Received: 27 September 2022; Accepted: 31 October 2022;

Published: 18 November 2022.

Edited by:

Roberto Cerbino, University of Vienna, AustriaReviewed by:

Fabio Giavazzi, University of Milan, ItalyPaolo Malgaretti, Helmholtz Institute Erlangen-Nürnberg, Germany

Copyright © 2022 Sheung, Garamella, Kahl, Lee, McGorty and Robertson-Anderson. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Janet Y. Sheung, jsheung@scrippscollege.edu