# Dynamic cross-correlations between entangled biofilaments as they diffuse

^{a}Department of Physics, University of Illinois at Urbana–Champaign, Urbana, IL 61801;^{b}Department of Materials Science and Engineering, Jinan University, Guangzhou 510632, China;^{c}Department of Materials Science, University of Illinois at Urbana–Champaign, Urbana, IL 61801;^{d}Department of Chemistry, University of Illinois at Urbana–Champaign, Urbana, IL 61801;^{e}IBS Center for Soft and Living Matter, Ulsan National Institute of Science and Technology, Ulju-gun, Ulsan 689-789, South Korea

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Contributed by Steve Granick, February 7, 2017 (sent for review December 20, 2016; reviewed by Marina G. Guenza and Eric R. Weeks)

## Significance

Highly entangled biofilaments are ubiquitous in the cytoskeleton and present a paradigm in polymer physics and biophysics. Here, rather than conventionally seeking to understand the self-diffusion of a single tagged entangled polymer we inquire into dynamic cross-correlations between nearby filaments as they diffuse anisotropically in aqueous solution. Our combined fluorescence tracking experiments and statistical mechanical modeling show that the continuum limit is reached only at distances beyond the filament length, in this system beyond the large distance of ≈15 µm. This noncontinuum behavior at micron-scale distance may be related to the “crowding” problem in biological function.

## Abstract

Entanglement in polymer and biological physics involves a state in which linear interthreaded macromolecules in isotropic liquids diffuse in a spatially anisotropic manner beyond a characteristic mesoscopic time and length scale (tube diameter). The physical reason is that linear macromolecules become transiently localized in directions transverse to their backbone but diffuse with relative ease parallel to it. Within the resulting broad spectrum of relaxation times there is an extended period before the longest relaxation time when filaments occupy a time-averaged cylindrical space of near-constant density. Here we show its implication with experiments based on fluorescence tracking of dilutely labeled macromolecules. The entangled pairs of aqueous F-actin biofilaments diffuse with separation-dependent dynamic cross-correlations that exceed those expected from continuum hydrodynamics up to strikingly large spatial distances of ≈15 µm, which is more than 10^{4} times the size of the solvent water molecules in which they are dissolved, and is more than 50 times the dynamic tube diameter, but is almost equal to the filament length. Modeling this entangled system as a collection of rigid rods, we present a statistical mechanical theory that predicts these long-range dynamic correlations as an emergent consequence of an effective long-range interpolymer repulsion due to the de Gennes correlation hole, which is a combined consequence of chain connectivity and uncrossability. The key physical assumption needed to make theory and experiment agree is that solutions of entangled biofilaments localized in tubes that are effectively dynamically incompressible over the relevant intermediate time and length scales.

The long-standing quest to understand why the mobility of entangled linear polymers is ultraslow normally considers the diffusion of a single average macromolecule in its average surrounding environment, most prominently envisioned as a polymer reptating through the confining Edwards–de Gennes tube composed of the identical polymers that surround it (1⇓⇓⇓⇓–6). Although differing in important respects according to polymer geometry (e.g., flexible and semiflexible chains, rigid rods, and branched polymers), all share the peculiarity that because the size of the macromolecule vastly exceeds the size of individual units along it, adjoining segments on a tagged polymer become correlated over large separations simply because they are covalently bonded and cannot cross other macromolecules. In this paper, our focus is not on the familiar single-polymer problem (1⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–13) but rather on the open question of how the motion of a given reptating macromolecule is coupled in space and time with others that reptate within its pervaded volume. The fractal and strongly interpenetrating nature of linear polymers in dense liquids causes the number of “correlated neighbors” on the macromolecular length scale to grow strongly as the polymer size increases (1⇓–3, 14). The dynamical consequences of such a feature are not addressed by the classical reptation-tube model, which ignores the correlated motion of neighboring polymers, for simplicity and perceived intractability (1⇓⇓⇓⇓–6). Analysis of the latter involves crowding effects far beyond the nearest-neighbor cage scale central to understanding slow dynamics in small molecule and colloidal liquids (15⇓⇓–18).

We focus on intermediate time and length scales when the system mechanically is a soft solid showing an emergent dynamic plateau modulus due to polymer localization in two transverse spatial directions, yet the individual constituents continue to display the Fickian diffusion of a viscous liquid in the direction of the polymer contour (1⇓–3, 9). The archetypal textbook static representation (1⇓–3) of a concentrated polymer solution is threadlike polymers of contour length *L* separated by the average “mesh” distance *ξ _{m}* between neighboring strands, with time and length scales summarized in Fig. 1

*A*for stiff filaments. On intermediate scales, polymers remain entangled and slowly diffuse in the longitudinal direction but are dynamically equilibrated (localized) in their tubes, resulting in a near-constant time-averaged density as indicated by the “fat cylindrical tubes” depicted in Fig. 1

*B*.

One focus of our work is to understand the coordinated motion of entangled polymers. The hydrodynamic continuum limit of the pair or relative two-polymer diffusion coefficient is known to have correlations that decay as 1/*r*, where *r* is separation (19⇓⇓⇓⇓⇓⇓–26). Our experimental measurements cross over to this scaling at large separation but earlier they deviate upward from this prediction, suggesting conclusions about the respective importance of hydrodynamic and nonhydrodynamic correlations as a function of length scale (Fig. 1*C*). This is relevant for understanding the lower bound of distance beyond which applicability of the two-point microrheology technique is predicated (20⇓⇓⇓–24). It is also relevant to cytoskeletal dynamics of stiff biofilaments such as actin and microtubules and potentially to the “crowding” problem in biological function (11⇓–13, 27⇓⇓–30). Generic for all fluids of interpenetrating, entangled macromolecules, thematically this matter can also be explored in flexible random coil polymer solutions and melts in addition to the biofilaments considered in this paper, although we do not do so here.

Biopolymer filaments of F-actin in aqueous solution present an attractive test bed for theory and experiment to address this matter (9, 23, 31, 32). They are globally isotropic yet heavily entangled at the concentrations studied here. The intermediate time-scale regime of anisotropic diffusion, which is longer than the entanglement onset time (^{−1} s) (31) to equilibrate within a tube but far less than the terminal flow or reptation time (^{3} s) (31, 32), is experimentally accessible. The time and length scales we probe, and a cartoon of the system, are summarized in Fig. 2 *A* and *B*. The experimental system is complicated by the fact that actin length (*L*) distribution from this method of sample preparation is polydisperse (7, 9), so it is reassuring that our calculations described below find insensitivity to both the magnitude of *L* and its polydispersity over the range relevant to experiment. The geometric mesh size (*ξ _{m}* ∼0.1 µm, the average distance between filament crossing points) and the tube diameter (

*d*≈

_{T}*ξ*, a dynamical scale relevant to the emergent soft elasticity) are one to two orders of magnitude less than the mean filament length but orders of magnitude larger than the solvent water molecules (

_{m}*d*∼0.3 nm) and filament thickness (

*b*∼8 nm) (7, 9). The persistence length is large, ∼17 µm, so for diffusive transport it is reasonable to consider these stiff filaments as almost rod-like (9). The theoretical calculations have used

*L*= 15 µm, the same order of magnitude as the mean filament length, except when otherwise stated.

Experimentally, this work combines fluorescence imaging techniques pioneered by Sackmann and coworkers (9), whose spatial resolution is diffraction-limited, with subdiffraction-resolution imaging of sparsely labeled elements of these same actin filaments. We find the system is globally isotropic for the F-actin concentrations (*c* is mass fraction) studied, 0.5, 1.0, and 2.0 mg/mL (Fig. 2*B*). Relative to the critical entanglement concentration, *c _{e}* ≈ 0.4 mg/mL (31, 32), these systems are highly entangled, with

*c*/

*c*in the range 1 to 5. One fluorescent dye labels the chain backbone uniformly and hence its emission is diffraction-limited, whereas the chain is also labeled in sparse abundance with a second fluorescent dye to give subdiffraction resolution. We screen the data to find filaments in close proximity that both lie within the focal plane so that the distance between segments on them can be drawn in 2D with confidence (Fig. 2

_{e}*C*). In parallel, local displacements of each filament are visualized with subdiffraction resolution by inspecting the second dye of such sparse abundance that they serve as point sources with a peak emission that can be quantified with nanometer precision (Fig. 2

*C*) (33, 34). Earlier we introduced the technique of sparse labeling, which makes this possible (7), but here we apply it to the problem of correlated dynamics. The time dependence of relative displacement between points on neighboring filaments, each of them localized with nanometer resolution, is then inspected (Fig. 2

*D*). Appearing in epifluorescence microscope images as circular spots, centroid tracking of their positions gave a precision of ±20 nm over times as short as 50 ms.

Recently we reported a statistical mechanical theory to describe intermediate-scale space–time correlations between diffusing spherical colloids in dense quasi-2D suspensions based on interparticle frictional effects and effective structural forces, and its predictions compare favorably to experiments with colloids (35). Here, we qualitatively extend this approach to the more complex case of entangled biopolymers and compare its predictions to our experiments to quantify nonhydrodynamic, noncontinuum cross-correlations in long-range pair diffusion and test the proposed mechanism. Although hydrodynamics is crucial at very short and very long distances, it is screened on the intermediate length scales of primary interest here (*ξ _{m}* <

*r*<

*L*per Fig. 1) due to polymer interpenetration (25, 26). Hence, the observed correlated motion is ascribed to a nonhydrodynamic origin. For the final cross-over to continuum behavior, our experimental results are in agreement with two-point microrheology measurements (22, 23).

## Experimental Results

The geometric mesh length of F-actin solutions is *ρ*_{r} is filament number density, where *L* is the contour length and the dynamic tube diameter *A*) are longer than required for dynamic equilibration of a filament in its tube but far below the reptation time (31, 32). Fig. 2 *C* and *D* illustrates the joint detection of the filament contour and sparse point sources along it.

To validate the credibility of our experimental measurements, first we confirm the known phenomenon that these filaments diffuse preferentially parallel to their contour (i.e., by reptation). The mean-square displacements (MSD) quantified with nanometer-level accuracy using sparse labeling were measured and plotted against time on log-log scales in Fig. 3*A*. Discriminating in this way their time-dependent position fluctuations transverse (*t*^{0.2} empirically) likely reflects limited motions such as contour length fluctuation (2). In contrast, the parallel displacements are close to linear in elapsed time (MSD ∼ *t*^{0.9} empirically), from which the implied diffusion constant is *D*_{//} ≈ 0.1 µm^{2}/s for all three concentrations that were studied. Importantly, this value is almost identical to the hydrodynamics-determined polymer diffusion constant in dilute solution, in good agreement with the reptation-tube model of entangled rods (1⇓–3, 38).

Per Fig. 2*D*, the space–time relative displacement correlation function, *r*_{0}:*r* corresponds to projection along the initial separation vector of the two tagged segments. The symbol *B* plots *c* = 0.5 mg/mL (main frame) and *c* = 2.0 mg/mL (inset) at elapsed times 0.1, 0.2, and 0.4 s. The observation that these correlations are approximately proportional to the lag time regardless of concentration suggests physically that the elementary kinetic process dominating displacement correlation is filament reptation, so it is meaningful to infer a spatially dependent relative diffusion constant, *A* *D*_{rr} is insensitive to concentration over the range that we studied.

When particles interact via conservative central forces, it is natural to expect radial correlations (*D*_{rr}) to dominate over transverse (*D*_{tt}) correlations for a nonhydrodynamic mechanism (35). Testing this proposition, our experiments indeed find that *D*_{tt} << *D*_{rr} on intermediate length scales less than the filament length (Fig. S1), further supporting the idea that a nonhydrodynamic mechanism is dominant on intermediate length scales. This differs qualitatively from the continuum hydrodynamic-based expectation that the magnitudes of radial and transverse displacement correlations should be comparable (22, 23).

Consider now the extremes of small and large separation between segments on different filaments. The smaller the separation between tagged segments, the fewer other filament segments reside between them, so physically one would anticipate that on length scales less than the physical mesh the viscous drag will approach that experienced by a single filament. Consistent with this, we find experimentally that *A* shows that the heuristic proposition that *r*_{0}*D*_{rr}* ^{HD}*(

*r*

_{0}) =

*k*/(2

_{B}T*π*

*η*

_{eff}) in the

*r*

_{0}>>

*L*hydrodynamic regime describes our data with a value

^{3}/s (22, 23), which again leads to a physically reasonable value of the effective viscosity ∼50–100 times that of water. Moreover, the data in Fig. 4

*A*are consistent with a 1/

*r*

_{0}decay but only for separations larger than roughly the filament length and with a value of

^{3}/s.

This experimental puzzle that continuum-based hydrodynamics seems to apply only beyond such large separations motivated the theoretical work that follows. It is known that at distances of order the mesh size and times below the reptation time, polymers appear as effectively fixed obstacles that “scatter” or “screen” the solvent flow field, as was articulated long ago by Doi, Edwards, and others (2, 25, 26), so we explore the consequences.

## Theory and Discussion

Our vision of the physical situation is sketched in Fig. 1. The relevant length scales of the entangled F-actin solution suggest the emergence of a dynamic structure (Fig. 1*B*) corresponding to gently bending “cylindrical fat tubes” of mean thickness the tube diameter (*d*_{T}), with an actin density that is roughly uniformly smeared inside the tube region on intermediate time scales. Importantly, because the mesh size and tube diameter nearly coincide in this semiflexible F-actin system, the tubes are densely packed and in repulsive contact (3, 36, 37). These features physically suggest a model based on “dynamic incompressibility” (small collective density fluctuations) being applicable on intermediate time and distance scales.

The problem now becomes how to understand the emergent intermolecular dynamical correlations between pairs of filaments associated with their diffusive reptation motion. The ideas described below are based on modeling biopolymer filaments as rigid rods, which for the interfilament dynamics associated with longitudinal reptation should be a reliable simplification. To formulate the theoretical model for correlated two-filament diffusion, we consider the connection between effective entropic forces and collective structure of the entangled fluid on time and length scales beyond which each polymer has equilibrated within its tube in a region of space. Effective incompressibility implies the emergence of a strong interpolymer spatial correlation known as the de Gennes “correlation hole” (2, 3) and an effective long-range repulsion that can induce space–time dynamic displacement correlations. We formulate this physical picture using quantitative statistical mechanics by combining the reptation-tube ideas of single polymer motion (4⇓–6) with force-level generalized Brownian motion ideas for predicting intermolecular dynamical correlations (35, 41⇓⇓⇓⇓–46). Here we sketch the essential elements of the model and theory and present technical details in *Supporting Information*.

As successfully implemented in recent theoretical studies of Stokes–Einstein violation of nanoparticles in polymer melts (47) and dynamic displacement correlations in dense colloidal fluids (35), the total pair diffusivity is estimated as the sum of independent hydrodynamic and nonhydrodynamic contributions, *ξ _{m}* ∼

*d*<

_{T}*r*< 10

*L*. We further assume that hydrodynamics is a second-order effect (thus

*ξ*<

_{m}*r*<

*L*, for two reasons: Solvent-mediated hydrodynamic forces are exponentially screened, as is well known from prior experimental and theoretical studies (2, 25, 26), and our present experimental observation that

*D*

_{tt}<<

*D*

_{rr}. Thus, although our data at

*r*>

*L*agree well with hydrodynamic expectations consistent with measurements by others (22, 23, 39, 40), we interpret the novel displacement correlations observed in this intermediate regime as signifying a nonhydrodynamic origin.

Consider the dynamical consequences of the above picture. On the relevant intermediate length scales the correlation hole effect implies that it is increasingly unlikely to find pairs of segments on different filaments at separations less than filament length *L*. The intermolecular segment–segment pair correlation function *h*(*r*) ∼ −1/*r*^{2} on intermediate length scales. The corresponding intersegment potential of mean force (18) is *B* graphically illustrates Eq. **2** for our experimental conditions. To gain some qualitative insight as to what the dynamical consequences might be of such distinctive power law correlations, we compute a simple metric of the number of spatially correlated segments on different rods in a spherical region of radius **2** as *h*(*r*) = *g*(*r*) − 1 is an objective statistical quantification of the deviation from random arrangement in space (on the tube diameter and larger dynamically relevant length scales) of segments on different rods surrounding a tagged segment. The results of this calculation are shown in Fig. 4*B* and one sees that *N*_{corr} grows strongly with increasing separation, before saturating at *r = L*, which corresponds to the cross-over to random structure. In rough analogy with the phenomenon of critical slowing down of collective dynamics near a phase transition due to the emergence of power law intermolecular static correlations (relevant given Eq. **2**) (49, 50), these emergent structural correlations suggest that dynamic interfilament correlations persist up to distances of approximately the polymer length.

*Supporting Information* describes mathematical implementation of the above ideas (*SI Theory and Models* and *SI Theoretical Results*). It builds on the theoretical machinery developed previously to successfully relate forces and structure to diffusive dynamical cross-correlations in dense colloidal suspensions (35). The latter applies because use of a rigid-rod model implies that each segment reptates coherently along the tube axis, which allows the dynamical analysis to be performed at the polymer center-of-mass level. The nonhydrodynamic diffusivity, *SI Theoretical Results*, *Single-Filament Diffusivity*, *Numerical Results for Monodisperse Systems*, and *Polydisperse Systems*).

Fig. 4*A* includes our theoretical predictions for **2**, which must be proportional to the tube diameter based on our physical picture (i.e., *r*_{0} < *L* ≈ 15 µm within the experimental uncertainty. However, as anticipated, it falls below the experimental data at larger separations because the calculation ignores hydrodynamic effects beyond the macromolecular size. Moreover, correlation hole physics as the origin of collective dynamics does not apply at separations significantly beyond the filament length. This reasonable description of all of the available experimental data over all separations, which employs

## Conclusion

The experimental methods presented here to perform microrheology without use of probe particles may find general use. Using them we have considered time and length scales when entangled polymer filaments are known from the literature to be mechanically soft solids showing a dynamic plateau shear modulus due to spatial localization in two transverse directions (9, 23, 31, 32, 51) yet (as we show here experimentally) the individual molecules continue to display Fickian diffusion of a viscous liquid in the longitudinal direction of the chain contour. In these entangled systems the strongly interpenetrating nature of linear polymers causes the number of correlated “segmental neighbors” on the macromolecular length scale to increase with distance from a tagged polymer out to the filament contour length scale (51). Our analysis of the correlated motion of pairs of entangled polymers, not addressed by the classical tube model, involves considering crowding effects beyond the nearest-neighbor local cages in small-molecule and colloidal liquids.

This paper introduces several physical ideas that may apply beyond the actin biofilaments studied here: (*i*) how cooperative biofilament diffusion deviates from the single-filament behavior averaged within the entanglement time; (*ii*) the notion of emergent “dynamic incompressibility” on intermediate time and length scales, which enables us to model the experimental observations by linking collective dynamics to the correlation hole intermolecular structure of interpenetrating polymers; and (*iii*) the cutoff length scale below which the system fails to behave dynamically like a continuum fluid. We emphasize that this large cutoff far exceeds the physical and entanglement mesh lengths that are traditional to anticipate when considering biofilament or polymer solutions.

Possible extensions to other entangled polymeric systems that exhibit anisotropic dynamics can be considered, all of which generically display the distinctive feature that they respond mechanically as a soft solid on intermediate time and length scales but as a liquid as concerns their anisotropic diffusion. One can anticipate at least three distinct regimes. First, entanglement tubes may be strongly nonoverlapping,

## Materials and Methods

Segments and backbones of F-actin were visualized by two-color fluorescence microscopy focused deep into the sample to avoid potential wall effects. Images were collected typically at 10 frames per second for 200 s then analyzed by MATLAB codes written in-house to give trajectories with 10-nm precision. This work treats the segment–segment separation range 0.3–50 μm and time range 0.1–1 s. Details of the experiments and theoretical modeling, and additional results, are provided in *SI Materials and Methods*.

Here we provide technical details and additional results of our experimental and theoretical studies. *SI Materials and Methods* describes the experimental materials and methods and presents our measurements of the transverse displacement correlation function compared with its radial analog. *SI Theory and Models* describes the polymer models studied and sketches the construction of the statistical mechanical theory used to predict the radial dynamic displacement correlations. *SI Theoretical Results* presents analytical and numerical theoretical calculations that establish the asymptotic predicted behavior, how polymer concentration modifies the space–time correlated dynamics, and the influence of filament semiflexibility and length polydispersity.

## SI Materials and Methods

### Materials.

Unlabeled G-actin (rabbit skeletal muscle) and Alexa-568–labeled G-actin (rabbit skeletal muscle) were purchased from Cytoskeleton and Invitrogen, respectively. Alexa-647–labeled G-actin (rabbit skeletal muscle) was a kind gift from William Brieher, University of Illinois at Urbana–Champaign. Phalloidin (Amanita phalloides) was purchased from Sigma-Aldrich. All of the other chemicals are from Sigma-Aldrich in analytical purity. Water is deionized (18.2 MΩ·cm).

### G-Actin Reconstitution and Polymerization.

G-actin was reconstituted in fresh G-buffer (5 mM Tris [Tris(hydroxymethyl)aminomethane] at pH 8.0, supplemented with 0.2 mM CaCl_{2}, 1 mM ATP, 0.2 mM DTT, and 0.01% NaN_{3}) at 4 °C and used within 7 d of reconstitution. G-actin was polymerized into F-actin by the addition of salt (100 mM KCl and 2 mM MgCl_{2}) at room temperature for 1 h.

### Block Labeling.

Alexa-647– (red) and Alexa-568– (green) labeled G-actin were polymerized into red and green F-actin, respectively. The red and green F-actin solutions were mixed at a molar ratio of 5:1, then the mixture was sheared rigorously by repeatedly passing them through a 26-gauge syringe needle, to fragment the filaments into ∼100-nm-long segments. The red and green segments further anneal to form ∼20-μm-long filaments in the presence of phalloidin to prevent treadmilling and depolymerization. Annealing proceeds for >12 h, giving rise to filaments with red backbones and sparsely distributed green segments. Finally, unlabeled G-actin (0.2–2 mg/mL) was polymerized in the presence of a trace amount of as-labeled F-actin and ∼30 mM vitamin C (for antiphotobleaching), giving an entangled solution with a few labeled F-actins in a “sea” of unlabeled F-actin, ready for fluorescence microscope observation.

### Two-Color Imaging and Segmental Tracking.

A two-color imaging system was built on an epifluorescence microscope (Zeiss Observer.Z1), where the signal is split to two EMCCD cameras (Andor iXon) by a Y-junction splitter (DC2; MAG Biosystems). We visualize the motion of labeled segments and backbones of F-actin in this home-built system with a 100× oil objective and focus deep into the sample (>100 μm) to avoid potential wall effects. Video images were collected typically at 10 frames per second for 200 s, which were then analyzed by MATLAB codes written in-house. The center of each segment is located in each frame to give trajectories with 10-nm precision.

This work covers a separation (*r*) range of 0.3–50 μm and a time range of 0.1–1 s. Previous two-particle microrheology was usually conducted over distance ranges ∼5–100 μm, but shorter length scales were obtained here because particle-free labeling was used. The time range was, however, mainly limited by photobleaching of labeled actin.

### F-Actin.

The mixture of labeled and unlabeled actin was deposited between a glass slide and a coverslip with a spacer (120 µm). When observed under an epifluorescence microscope with relatively small field of view (82 µm × 82 µm) and depth of focus (1.5 µm) there were no more than eight labeled actin filaments in the same view. Photobleaching was alleviated by including neutralized ascorbic acid in the actin solution. The integration and cycle time of the image acquisition was *i* is the index for segment, *j* is the associated backbone number, at *k*^{th} frame. Trajectories with erratic displacements due to tracking errors were rejected with a manual estimate of the diffusivity. At each concentration, the trajectories of ∼300 pairs of actin filaments were analyzed.

### Definition of Cross-Correlated Motion.

The trajectories of the F-actin segments are analyzed to get the segment’s vector displacement *t* with a time lag *τ*. Then the dynamic cross-correlations are defined for a given distance *r* and *τ* by** i** and

**label different segments,**

*j***’s displacement on the**

*i***and**

*i***, and**

*j**r*. This definition of

**S1**remain defined relative to the separation vector

To compute *n*, we compute the displacements and the average positions of all tagged segments observed, respectively. Then the product of projected displacement and distances between pairs

*D*_{tt} Quantification.

_{tt}

Our expectation was confirmed that displacement correlations *D _{tt}* <<

*D*on intermediate length scales less than the filament length, as shown in a plot of

_{rr}*D*/

_{rr}*D*against

_{tt}*r*(Fig. S1). These data differ from the hydrodynamic-based expectation that

*D*/

_{rr}*D*= 2 for fluids in the continuum regime, confirming that this regime is not dominated by continuum hydrodynamic-based expectations.

_{tt}## SI Theory and Models

Our focus is the dynamic displacement cross-correlation function, Eq. **1**, in the nonhydrodynamic intermediate length and time regime where the transverse MSD of a filament exhibits a (near) plateau corresponding to times longer than the time scale for equilibration in the entanglement tube but far before the long terminal reptation time (Fig. 1*B*).

### System Model and the Renormalized Cylindrical Tube Idea.

We consider a fluid of rigid rods (length, *L*) composed of bonded spherical interactions sites (diameter *ξ _{m}* (defined in the main text), which is comparable to the tube diameter,

*d*. Because

_{T}*ξ*∼

_{m}*d*, the cylindrical tubes are densely packed (nonoverlapping) and in (soft) repulsive contact (Fig.1

_{T}*B*). The system will thus have an effectively low compressibility which implies, using an argument of de Gennes (3), a direct connection between intramolecular and intermolecular site–site pair correlations on the intermediate time and length scales of interest. Specifically, the correlation hole effect of Eq.

**2**is relevant, whence long-range intrapolymer connectivity must result in long range (on the scale of

*L*) interpolymer correlations. Rotation of the cylinders is ignored because on intermediate time and length scales individual rod polymers are trapped in entanglement tubes and perform 1D reptation motion along their contours. Our physical idea is that effective repulsive forces between the dense collection of cylindrical objects can induce dynamic correlation between different reptating filaments. Because the motion of all sites on a given tagged rod is slaved via reptation, it is appropriate to use a center-of-mass (CM) model for single rod motion. The task then becomes to understand how the diffusive motions of a pair of reptating rods, each localized in its respective tube, are correlated in space and time.

### Two-Particle Dynamics and Mode Coupling Theory.

To analyze nonhydrodynamic displacement cross-correlations we extend our prior statistical dynamical theory for dense colloidal suspensions (35), which applies because for

Two coupled generalized Langevin equations describe the stochastic motion of two tagged particles. In terms of the CM, **S2** and **S3** is the frictional drag force quantified by the reptation friction constant **S3** involves the potential of mean force (PMF) (18), *g*(*r*) is the site–site pair correlation function between a pair of uncrossable rods in the dense solution, and its nonrandom part is *h*(*r*) = *g*(*r*) − 1. Viscoelastic effects associated with the space–time correlation of the forces exerted on the two-tagged sites by the surrounding fluid enter through the nonlocal in time memory term, **S4a** holds for the relative coordinate memory function **1** can then be written as

Statistical mechanical approximations are necessary to compute the memory functions and solve Eqs. **S2**–**S5**. We use a simple mode coupling theory (41⇓⇓⇓⇓–46) that has been successfully used for the nonhydrodynamic relative diffusivity of colloids in dense suspensions (35). The relevant slow dynamical variables are taken to be the single polymer and collective density fluctuations, real forces are replaced by effective forces determined by the intermolecular structural correlation function, and four point correlations are factorized into a product of two point functions. This yields the CM (*R*(*r)*, *ω*(*k*) is the single rod intermolecular structure factor,

To proceed, models for the complicated Γ(*k*,*t*) functions are necessary. The fixed separation constraint implies that dynamical decorrelation of the forces on the two-tagged sites is solely due to relaxation of the surrounding fluid, hence **S7** is sensible for intermediate times

Because the sites are effectively fixed relative to each other due to the interrod constraint, the PMF is a constant and can be dropped when solving Eq. **S3**. One then can define the renormalized CM friction constant, **S5** yields

Combining Eq. **S6** with the friction constants above Eq. **S8**, and performing the time integral, then yields the explicit result**2**. Using **S9** and performing the solid angle integrals yields**S8** and **S10** are the foundation of the nonhydrodynamic theory and the starting point to analyze the dynamic displacement correlations of rods.

### Intramolecular Structural Models.

To implement the dynamical theory a model for the intramolecular structure factor of a filament, *i*) a continuous rigid-rod model and (*ii*) a continuous Gaussian semiflexible model. Model *i* corresponds to (48)**S11**, three dimensionless quantities characterize the system: (*i*) reduced density *ii*) number of sites per rod *iii*) number of entanglements per rod **S11** and the incompressibility approximation immediately lead to Eq. **2**. Model *ii*, due to Marques and Fredrickson (48), captures a rod-like conformation inside the persistence length of a stiff filament and crosses over to coil-like behavior on larger length scales. It is given by**S11** or **S12** in the dynamic theory, the dimensionless relative diffusivity *SI Theory and Models*, *Two-Particle Dynamics and Mode Coupling Theory* and Eq. **S10** still hold.

## SI Theoretical Results

### Single-Filament Diffusivity.

Before discussing the relative diffusivity we comment on the renormalized single-rod friction constant predicted by the dynamic theory sketched above. In principle, it is modified by the structural correlation effects we consider (41, 46). However, for both the rigid-rod and semiflexible models we have explicitly verified that the structural correlation effects do not modify the single-polymer reptation friction,

### Analytic Large Separation Limit.

The large separation limit of the relative diffusivity can be generically evaluated by taking the **S10**. Because **S10** reduces to**S13** in Eq. **S8** and expanding for large separations

### Numerical Results for Monodisperse Systems.

We first consider monodisperse filaments of length **S10** are evaluated using the rigid-rod structure of Eq. **S11** and the experimental physical mesh and tube diameter expressions given in the main text. We perform calculations in two ways: (*i*) hold the rod length fixed and vary the polymer concentration (and hence the tube diameter and mesh size) and (*ii*) hold the polymer concentration fixed and vary the rod length.

For the first case, Fig. S2*A* shows the relative diffusivity normalized by the single filament diffusivity (*A*). In Fig. S2*A* *c* = 1 mg/mL and rod length varies between **S10**], but because experiments cannot precisely enforce this constraint and are performed for **S15** shows that in the large separation limit at constant concentration one has**S16** the large separation limit behavior is expected to collapse if one uses the length scale *B*. No universal collapse of the curves on all length scales exists.

For the second case above we investigate the concentration dependence of the relative diffusivity at fixed rod length *L* = 15 μm. Fig. S3*A* shows calculations for concentrations varying from *c* = 0.5 mg/mL to 16 mg/mL (from top to bottom at large separation). The small and large separation regimes discussed above for the first case are again observed. No intermediate scaling regime is observed except at the largest concentration. This is primarily due to the fact that *c* = 0.5 mg/mL to *c* = 16 mg/mL. As the concentration increases there are modest differences in the small separation plateau, mainly as a result of the small differences in the modest quantitative corrections to the single particle friction discussed above. At large separations the relative diffusivity correlations decrease with increasing concentration, as predicted by Eq. **S16** (collapse shown in Fig. S3*B*). Whereas based on the limiting analytic analysis we expect **S13** is not quantitatively accurate at smaller concentrations. For the large concentrations we indeed find collapse when scaling by *c* (not shown), confirming this explanation. The fact that the relative diffusivity decreases modestly with increasing concentration may seem counterintuitive; however, it is a unique feature of rods that occurs primarily due to the enhanced effects of the correlation hole at higher concentrations via the effective repulsive force which enters (to leading order) as the gradient of

The role of polymer semiflexiblity is studied by using Eq. **S12** in Eq. **S10**. We find that for realistic (high) persistence lengths of F-actin all of the qualitative features of the concentration and length dependence remain the same as predicted for rigid rods. Hence, we only present the influence of *L* = 15 μm and *c* = 1 mg/mL, where the relative diffusivity as a function of separation is normalized by the single polymer diffusivity. Calculations for several persistence lengths are shown spanning semiflexible

### Polydisperse Systems.

We briefly consider the influence of contour length polydispersity on the relative diffusivity for the rigid-rod model of filaments. For illustration, an exponential distribution of filament lengths with a lower cutoff at

There are several choices of how to implement averaging over the length distribution in the dynamical theory. We choose to average the effective force, which is determined by

Fig. S5*A* shows the average effective force as a function of the dimensionless wavevector *kL*. The red and blue curves show results for the monodisperse system (**S10** via the replacement *L*/*d*. Fig. S5*B* shows the relative diffusivity normalized by the single rod diffusivity for polydisperse systems as a function of the dimensionless separation at *c* = 1 mg/mL. For all cases studied, except the most extreme one with

## Acknowledgments

We thank William Brieher (University of Illinois at Urbana–Champaign) for donating the fluorescent-labeled actin. This work was supported by Institute for Basic Science Project IBS-R020-D1 (to S.G.). The theoretical work was supported by US Department of Energy, Division of Materials Science Award DEFG02-07ER46471 (to Z.E.D. and K.S.S.) through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana–Champaign. Experiments were supported by US Department of Energy Award DEFG02-02ER46019 (to B.T., L.J., and S.G.).

## Footnotes

↵

^{1}B.T., Z.E.D., and L.J. contributed equally to this work.- ↵
^{2}To whom correspondence should be addressed. Email: sgranick{at}illinois.edu.

Author contributions: B.T., Z.E.D., L.J., K.S.S., and S.G. designed research; B.T., Z.E.D., L.J., and K.S.S. performed research; B.T., Z.E.D., L.J., K.S.S., and S.G. analyzed data; and B.T., Z.E.D., L.J., K.S.S., and S.G. wrote the paper.

Reviewers: M.G.G., University of Oregon; and E.R.W., Emory University.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1620935114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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