Introduction

The multifunctional mechanics of eukaryotic cells are largely determined by the cytoskeleton, a dynamic network of interacting biopolymers and binding proteins. Two principal cytoskeletal biopolymers are actin filaments and microtubules. Actin filaments polymerize from globular actin monomers to ~7 nm wide semiflexible filaments with a persistence length lp ≈ 10 µm1,2, while microtubules are formed by tubulin dimers polymerizing into 25 nm wide hollow rigid tubes with lp ≈ 1 mm2,3,4,5,6. Networks of semiflexible actin filaments play critical roles in cell polarity and contractile and migratory processes7,8; while microtubules mediate intracellular trafficking and transport, chromosomal dynamics, and mitotic spindle alignment during cell division7,9. Synergistic interactions between these two filaments, which are mediated by steric and chemical interactions (i.e. entanglements and crosslinking)1,8,10,11, establish essential cell asymmetries and enable proliferation, differentiation, and migration1,8,12,13,14. While crosslinking of both filaments is ubiquitous in cells, serving important roles in locomotion, membrane reinforcement, and intercellular cargo transport4,8,14, there is mounting evidence of crosslinking between actin and microtubules, mediated by proteins such as tau, MAP2, APC, profilin, and plectin8,15,16,17,18,19,20,21. Specifically, actin-microtubule co-crosslinking has been shown to be important to cortical flow, wound healing, neuronal cone growth, cell migration, and muscle contraction7,8,9,14,15,17,22,23,24,25.

Aside from the obvious physiological relevance, investigations of actin-microtubule composites with variable crosslinking motifs are motivated by materials engineering, as they serve as a model platform for designing and investigating flexible-stiff polymer composites2,11,26,27. In order to increase stiffness, strength, and toughness in composite materials, soft polymer networks are often reinforced with stiffer, load bearing fibers such as nanotubes and carbon nanofibers11,26,28,29. Further, varying types of crosslinking have been used to reinforce and increase the mechanical strength of flexible-stiff polymer double-networks30,31,32. Crosslinking has also been shown to preserve memory and reduce network deformation in composite polymer hydrogels32,33.

Here, we use optical tweezers microrheology and confocal microscopy to characterize the mechanics and mobility of actin-microtubule composites with varying crosslinking motifs. We create equimolar composites of actin and microtubules in which: only actin is crosslinked (Actin), only microtubules are crosslinked (Microtubule), both actin and microtubules are crosslinked (Both), and actin and microtubules are crosslinked to each other (Co-linked) (Fig. 1a). We characterize mechanics by optically pulling a microsphere through the composites and simultaneously measuring the induced force during strain as well as the relaxation of force following the strain. We find that the mesoscale force response separates into two distinct classes, with one class exhibiting a substantially elastic response with minimal relaxation following strain, while the other class exhibits a softer more viscous response with nearly complete dissipation. We show that these two classes, which are only readily apparent at mesoscopic scales (>µm), are dictated by the degree to which microtubules are crosslinked.

Figure 1
figure 1

Schematic of experiments to probe the role that crosslinking plays in the mechanics and mobility of actin-microtubule composites. (a) Cartoon of molecular components comprising actin-microtubule composites with varying crosslinker motifs (None, Actin, Microtubule, Both, Co-linked, Both 2x) described in the text. The calculated length between crosslinkers for each motif is listed under each cartoon, where lc,a is the length between crosslinkers along an actin filament, lc,m is the length between crosslinkers along a microtubule, and lc is the length between crosslinkers when actin and microtubules are linked to each other. (b) Two-color laser scanning confocal micrograph of 5.8 µM Co-linked actin-microtubule composite with ~3% of actin and microtubules labeled with Alexa-488 (green) and rhodamine (red), respectively. (c) Sample microrheology data for a Co-linked composite, showing the stage displacement (purple) and the force measured for an individual trial (black) before (dark grey region), during (light grey region) and after (white region) the stage displacement. Inset: Cartoon of optical tweezers microrheology measurement. An optically trapped 4.5 µm diameter microsphere (not drawn to scale) embedded in a crosslinked composite is displaced 10 µm (x, purple) at a speed of 10 µm/s while the corresponding force the composite exerts is measured (F, black).

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Despite the relevance of actin-microtubule composites to biology, physics, and engineering, most in vitro studies to date have focused on single-component networks of either actin or microtubules4,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48. In response to strain, entangled actin networks have been reported to exhibit varying degrees of stiffening, softening, and yielding depending on the actin concentration as well as the scale and rate of the strain36,38,39. Crosslinked actin networks exhibit enhanced stiffness and prolonged, more pronounced elasticity compared to entangled solutions35,37,43,44. Entangled actin networks have also been shown to relax force via an array of mechanisms including bending, stretching, retraction, reptation, and confinement-hopping39,49,50,51,52. While many of these mechanisms are suppressed by crosslinking, crosslinked networks have been reported to exhibit relaxation and plasticity due to crosslinker unbinding/rebinding, slippage, and stretching53,54,55. Compared to entangled actin networks, entangled microtubules have been reported to respond more elastically to strain with more pronounced stiffening as well4,9,42. Further, while stretching is important to actin stress relaxation, bending plays a more dominate role in the relaxation of microtubules56. While the timescales for microtubules to relax have been suggested to be on the order of several minutes or more4,42, crosslinked microtubule networks have been reported to display some degree of relaxation due to temporary force-induced crosslinker unbinding events4.

Few studies have investigated composite networks of actin and microtubules11,27,57,58 due, in part, to the incompatibility of standard polymerization conditions for each protein27. However, methods have recently been introduced to co-polymerize actin and tubulin in situ to create randomly-oriented co-entangled composites of actin and microtubules27. The first study that employed these methods used microrheology and microscopy techniques to show that a surprisingly large molar fraction of microtubules (>70%) was needed to substantially increase the response force of the composites. This intriguing result was attributed to the larger mesh size of microtubule networks compared to actin networks of the same molarity. Further, actin-rich composites softened in response to strain while microtubule-rich composites stiffened. This distinction arose from actin bending (leading to softening) that was increasingly suppressed by microtubules (leading to stiffening). This result corroborated a previous study that investigated the nonlinear macrorheology of crosslinked actin networks doped with low concentrations of microtubules, in which the high bending stiffness of microtubules served to suppress actin bending and enhance stretching in response to strain11,43.

Results and Discussion

To systematically investigate the role that crosslinking plays in composites of actin and microtubules we design actin-microtubule composites with four distinct crosslinking motifs. We use our previously established protocols27 to create equimolar co-entangled actin-microtubule composites. We then incorporate biotin-NeutrAvidin complexes into composites to selectively crosslink the filamentous proteins, keeping the crosslinker:protein ratio fixed at R = 0.0237. We create composites with: crosslinked actin (Actin), crosslinked microtubules (Microtubule), crosslinkers equally forming actin-actin and microtubule-microtubule bonds (Both), and crosslinkers binding actin to microtubules (Co-linked) (Fig. 1a). We compare these composites to a composite without crosslinkers (None) as well as one in which both filaments are crosslinked but R is doubled (Both 2x) (Fig. 1a). While all composites have the same crosslinker density R and protein concentration, the length between crosslinkers along actin filaments lc,a, along microtubules lc,m, and along both filaments lc (for Co-linked) vary for the different composites, as listed in Fig. 1a and described in Methods.

Mechanical response

To characterize the mechanical response of our designed composites we use optical tweezers microrheology to drag microspheres through each composite and measure the local force applied to each sphere by the composite during and following the strain. Figure 2a shows the measured force, F(x), as a microsphere is pulled through each type of composite. As expected, all crosslinked composites exhibit overall larger resistive forces than the composite with no crosslinkers (None). The more interesting and unexpected result that emerges is that the force curves appear to display two distinct responses dependent on the crosslinking motif: pronounced softening behavior in which the slope of the force curves approach zero at large distances, versus largely elastic behavior in which F(x) remains linear with x until the end of the strain. To better delineate between these two behaviors, we normalize F(x) by the terminal force Ft reached when x = 10 µm, which we notate as \(\widetilde{F}(x)\). As shown in Fig. 2b, \(\widetilde{F}(x)\) distinctly separates into two classes: Class 1 comprises None, Actin, and Both composites while Class 2 comprises Microtubule, Co-linked, and Both 2x composites.

Figure 2
figure 2

Mesoscale force response of crosslinked actin-microtubule composites display two distinct classes of behavior mediated by the degree of microtubule crosslinking. (a) Force F(x) that composites with varying crosslinking motifs (listed in legend) exert to resist the 10 μm strain. Dotted vertical lines separate the different regions over which F(x) and the stiffness K are averaged in (c–e). (b) The force shown in (a), normalized by the terminal force Ft reached at the end of the strain, denoted as \(\widetilde{F}(x).\) As shown, composites separate into two distinct classes – those that exhibit substantial softening and yielding (Class 1: None, Actin, Both) versus those that exhibit nearly complete elastic response (Class 2: Microtubule, Co-linked, Both 2x). (c) Force values from (a), averaged over different bead displacements xf -xi, denoted as Fxf–xi (color coded as in (a)). Data shown is for xf–xi = 1–0 µm (circles), 4–1 µm (triangles) and 10–4 µm (squares). (d) Stiffness, K = dF/dx, averaged over the same displacements as in (c), as listed in the legend. (e) The dimensionless stiffness, \(\widetilde{K}(x)\) = (10 μm) \(d\mathop{F}\limits^{ \sim }(x)\)/dx, averaged over the same displacements as in (c). Error bars in (c–e) are calculated from computing the standard error of force and stiffness values measured for each individual trial (~50–60 trials per data point).

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For a purely elastic material we expect to see a linear relationship between the resistive force and distance the microsphere is pulled (F ~ x), and once the microsphere stops the force the network exerts on the sphere F(t) should be maintained with no relaxation or dissipation (i.e. mechano-memory, F(t) ~ Ft). Conversely, a purely viscous material yields a strain-independent force response (F ~ x0) that is reached as soon as the microsphere reaches constant speed, and once the microsphere stops the network immediately and completely relaxes its force on the sphere (F(t) ~ 0). Most biopolymer networks, including entangled actin-microtubule composites, display viscoelastic behavior in between these two extremes2,27. However, crosslinking can lead to more elastic response by suppressing network rearrangements and polymer bending modes that can dissipate stress11,37. Figure 2b shows that Class 2 composites exhibit primarily elastic response while Class 1 exhibits softening and yielding to viscous behavior. The distinction between the two classes appears to be the degree to which microtubules are crosslinked, which we elaborate on further below.

Only composites in which microtubules are crosslinked (Microtubule, Co-linked, Both 2x) exhibit substantial elasticity, while Actin and None force curves display similar curvature indicative of more viscous response. This result suggests that at this crosslinking density, crosslinks do not substantially prevent semiflexible actin filaments from bending and/or rearranging to dissipate stress. Further, at a given R the length between crosslinks along an actin filament is ~4x longer than that for microtubules (see Methods), so there are comparatively fewer crosslinks along an actin filament to suppress relaxation modes. The counterintuitive result presented in Fig. 2 is that a composite in which both filaments are crosslinked (Both) exhibits soft/viscous behavior similar to None and Actin despite the fact that both actin and microtubules are crosslinked. Given the role that crosslinking plays in enhancing elasticity, one would expect that a network in which all filaments are crosslinked would be more elastic than one in which just half of the proteins are linked (Actin or Microtubule). However, because in this composite both actin and microtubules are crosslinked and the density of crosslinkers remains fixed (R = 0.02), the length between crosslinkers along each filament is doubled.

To determine if this increased lc,m and lc,a is indeed the reason for more viscous behavior for Both composites, we perform experiments in which we double the crosslinker density (Both 2x) such that the length between crosslinkers along each actin filament and microtubule is the same as in Actin and Microtubule composites, rather than being twice as long. As shown in Fig. 2, upon doubling R, the composite exhibits Class 2 elastic-like response similar to Microtubule and Co-linked networks. Thus, it appears that the critical requirement for sustained elasticity is not only that microtubules must be crosslinked but that they must be crosslinked such that the length between crosslinkers is sufficiently small. Reducing lc,m leads to a more connected network, thereby constraining filament fluctuations and rearrangements and leading to enhanced network stiffness59.

To quantify the distinct behaviors of Class 1 and Class 2 composites and examine the lengthscale dependence of the response, we average F(x) over different microsphere displacements xf – xi, which we denote as Fxf–xi (Fig. 2c). As shown, at submicron distances on the order of the mesh size (0–1 µm) there is little difference between the force response the two classes exhibit, with all crosslinked composites displaying an average force of F1–0 ≈ 3 ± 0.6 pN. This result indicates that variations in the crosslinking motif has little effect on the microscale, linear regime response of composites. Even at somewhat larger microscale displacements (1–4 µm), while crosslinked composites exhibit larger forces than None, the type of crosslinking has little effect on the measured force (with all values lying within ~1 pN of each other). Only at mesoscale distances, on the order of the length of filaments and substantially larger than the mesh size of composites, does the dramatic effect of crosslinking motif emerge, with Class 1 composites exhibiting an average force of F10–4 ≈ 9 ± 3 pN while Class 2 composites display F10–4 ≈ 15 ± 2 pN.

This scale-dependent variation is likely a result of both variable length between crosslinkers and a small fraction of crosslinker unlinking/relinking events. At the submicron level, the differences in crosslinker lengths only result in small differences in the density of crosslinkers encountered by the microsphere, and filament bending and stretching is minimal. As the distance becomes orders of magnitude larger than the crosslinker lengths, the effect of varying crosslinking densities, as well as the varying contributions of bending and stretching, become more appreciable. Further, at larger displacements (i.e. longer times), crosslinks that were initially ruptured (contributing to submicron softening) will have had time to re-link. While new un-linking events can occur, the force response at lengthscales much greater than lc reflects that of a steady-state unbinding/rebinding regime, and the response will still be dominated by stretching and/or bending modes. The varying degree to which Class 1 and Class 2 composites can undergo each mode manifests as a clear distinction between measured force values at these lengthscales.

The mechanisms described above would more clearly contribute to differences in the elasticity or stiffness of the networks rather than the absolute magnitudes of the force responses. In fact, as shown in Fig. 2b, the clear distinction between Class 1 and Class 2 is in the lengthscale-dependent curvature, which quantifies the elasticity, rather than the magnitudes of the force response. To better quantify this distinct behavior, we calculate the stiffness K of composites, analogous to the differential modulus in bulk rheology, which we quantify as K = dF/dx (Fig. 2d). We also compute the dimensionless stiffness \(\widetilde{K}(x)\) = (10 μm) \(d\mathop{F}\limits^{ \sim }(x)\)/dx, which quantifies the curvature displayed in Fig. 2b,c. Larger K and \(\widetilde{K}\) values are indicative of stiffer, more elastic networks while softer more viscous behavior results in smaller K and \(\widetilde{K}\) values.

As shown in Fig. 2, both classes exhibit similar K values for submicron scales, and exhibit softening (decreasing K) from K1–0 to K4–1. Class 1 composites do soften slightly more than Class 2 composites over the first 4 µm; however, the differing degrees of softening for Class 1 and Class 2 composites is much more substantial at larger lengthscales (>4 µm). K10–4 values drop to ~0.6 ± 0.2 pN/µm for Class 1 composites, whereas for Class 2 composites K10–4 values are indistinguishable from K4–1 values (no softening). To better quantify the extent to which composites soften or remain elastic, we turn to \(\widetilde{K}(x)\). As shown, \(\widetilde{K}(x)\) values for Class 1 composites drop by ~30% from (1–4 µm) to (4–10 µm) while Class 2 composites remain nearly constant for lengthscales beyond 1 µm.

We note that the exact choice of 1 µm and 4 µm to delineate different phases of the response was motivated by visually apparent changes in the curvature of the force response as seen in Fig. 2a. However, small changes to these cutoffs did not alter the data presented in Fig. 2c–e in a statistically significant way. Further, to determine if the strain rate impacts these natural crossover values we repeated measurements using speeds of 2.5–20 µm/s. As shown in Fig S2, the variation between (0–1 µm) and (1–4 µm), as well as the magnitudes of averaged K and \(\widetilde{K}(x)\) values, are similar for all speeds.

Our results and interpretations are validated by prior measurements on actin networks crosslinked by biotin-NeutrAvidin37. In this previous work, authors found that R > 0.05 was needed for actin networks to exhibit largely elastic response, with R < 0.05 networks displaying softening and yielding to a viscous regime over a 10 µm bead displacement of comparable speed. The authors explained the transition to elastic response as arising from actin filaments stretching in the direction of the strain rather than bending non-affinely. This study also showed that force-induced crosslinker unbinding contributed to the yielding and relaxation behavior of low R networks. Such force-induced unbinding of biotin-NeutrAvidin crosslinkers has also previously been reported for crosslinked microtubule networks4.

Further, our results differ from previous actin studies in which the onset of elastic response was nearly immediate. Here, our Class 2 composites display softening behavior, similar to that of Class 1 until ~2 µm at which point a more elastic response takes over. This universal softening at small scales supports the presence of force-induced unbinding of both actin and microtubule crosslinks, as the length between crosslinkers is substantially smaller than a micron for all composites, whereas the deformation lengthscales needed to induce appreciable bending or stretching are much larger35,37,60. As the strain proceeds, the contributions from bending and stretching grow, outweighing that of crosslinker-unbinding and thus minimizing its effect on the force response. Further, unbinding events are reversible, and re-binding has been shown to contribute to sustained elastic response (coupled with partial dissipation) similar to that depicted by Class 2 composites (Fig. 2c–e)4,35,37,60,61,62.

Finally, we note that the microscale response (0–1 µm) may not solely be due to the force the composites exert but also to transient effects arising from bringing the piezoelectric stage (which moves the trapped bead) from rest to constant speed. While the response time of the stage is 0.25 ms (and accounted for in our analysis as described in Methods), the mismatch between the stage and bead velocity persists for ~25 ms (Fig S1). To test the impact of this effect on our results, we also compute the microscale quantities (F1–0, K1–0, \({\mathop{{K}}\limits^{{ \sim }}}_{{1}{-}{0}}\)) starting at 25 ms instead of 0 s. As shown (Fig S1), removing this initial period shifts the values slightly but does not change the general features or trends in the data. Further, while F1–0 and F4–1 show a weak dependence on speed (Fig S2), likely arising from this transient effect, it is much weaker than Stokes dependence (F~v) and the variation between (0–1 µm) and (1–4 µm) remains nearly constant.

Nonetheless, to remove transient effects entirely from the microscale analysis, we perform 1 µm oscillatory strain measurements for frequencies ω = 1.26–126 rad/s, from which we compute the frequency-dependent complex viscosity η(ω) (details provided in Methods). Both the magnitude and frequency dependence of η(ω) are nearly the same for all crosslinked composites, corroborating the microscale results described above. Further, the shear-thinning behavior η(ω)~ω−0.5 is indicative of a viscoelastic fluid and is comparable to previously reported scaling for entangled actin48, indicating once again that the elastic response of Class 2 composites only manifests at the mesoscale.

Relaxation dynamics

As described above, elastic and viscous systems should display starkly different relaxation behaviors following strain, with purely elastic systems exhibiting no relaxation and viscous systems exhibiting immediate and complete force relaxation. Viscoelastic systems display behavior in between these two extremes, with viscoelastic biopolymer networks exhibiting both exponential and power-law relaxation over varying timescales2,27,36,37,38,63. To characterize the relaxation dynamics of the composites, we measure the time-dependent force F(t) exerted on the bead by each composite for 15 s following the strain (Figs 1, 3).

Figure 3
figure 3

Actin-microtubule composites display multi-mode force relaxation with varying degrees of elasticity tuned by the crosslinker motif. (a) Force relaxation after microsphere displacement as a function of time F(t) on a log-y scale for composites with varying crosslinking motifs listed in the legend in (b). Thin grey curves correspond to fits of each relaxation to the exponential sum equation displayed. Thicker grey lines guide the eye to show three distinct phases with different exponential decays (i.e different slopes). (b) F(t) normalized by the force at the beginning of relaxation, Fo. (c) The fractional force dissipation, computed as (Fo-Ff)/Fo where Ff is the force at the end of the relaxation measurement period. As shown, Class 1 composites exhibit much more force relaxation compared to Class 2 composites that retain substantial memory of imposed strain. (d) Relaxation times, τ1 (squares), τ2 (triangles), and τ3 (circles) determined from the fits shown in (a). (e) Relative relaxation coefficients Ci/(C1 + C2 + C3) for each exponential term (i = 1,2,3) determined from the fits to the data (shown in (a)). Error bars determined from fits are shown but are smaller than the size of the symbols in most cases.

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As shown in Fig. 3a, all composites exhibit some degree of force relaxation, though the extent to which the force decays depends on the crosslinking motif. This dependence becomes distinctly apparent upon normalizing F(t) by the initial force (immediately after strain) F0 (Fig. 3b). Class 1 composites exhibit faster and more complete relaxation than Class 2 composites which retain a substantial fraction of the initial force at the end of the measurement period. To quantify the varying degrees to which composites retain initial force, indicative of elastic behavior, we evaluate the fractional force dissipation (Fo–Ff)/Fo, in which Ff is the force reached at the end of the relaxation period. For reference, in a purely elastic network with no dissipation (Fo–Ff)/Fo should be 0 whereas a viscous system should exhibit a value close to 1. As shown in Fig. 3c, a clear difference between Class 1 and Class 2 composites emerges once again. Class 1 composites dissipate ~89% of initial force whereas Class 2 composites dissipate ~40–68% of the initial force.

To determine the macromolecular mechanisms underlying the observed relaxation behavior, we evaluate fits of the force curves to a range of exponential and power-law functions that have been previously used to fit relaxation data for entangled and crosslinked biopolymer networks27,36,37,38,48. In particular, crosslinked actin networks have been shown to relax via a sum of exponential decays with well-separated decay times each corresponding to different relaxation mechanisms37. Similarly, we find that the best fit to the force relaxations for all composites is a sum of three exponential decays with well-separated time constants (Fig. 3a). As described below, the agreement between the measured time constants with theoretically predicted relaxation times corroborates the choice of fits to describe the data. The function used for all composites,\(F(t)={F}_{\infty }+{C}_{1}{e}^{-t/{\tau }_{1}}+{C}_{2}{e}^{-t/{\tau }_{2}}+{C}_{3}{e}^{-t/{\tau }_{3}}\), includes an offset F∞ that is a measure of the amount of force that the composite retains indefinitely (F∞ ~ F0 and 0 for elastic and viscous systems respectively). The three distinct decay constants, with similar values for all composites, indicate that three different relaxation mechanisms with well-separated characteristic timescales contribute to the force relaxation of all composite types (Fig. 3d). The decay times, averaged over all composite types, are τ1 = 0.06 ± 0.02 s, τ2 = 0.51 ± 0.12 s and τ3 = 5.28 ± 1.83 s. However, we note that all three timescales are ~2x faster for the unlinked composite (None) compared to the crosslinked composites, which we discuss further in the next section. To evaluate the relative contribution of each mechanism to the force relaxation, we compare the corresponding fractional coefficients for each exponential term Ci /(C1 + C2 + C3) (i = 1,2,3) (Fig. 3e). For all composites, the fast timescale contributes the most to the relaxation, followed by the intermediate timescale, and finally the slowest timescale. However, Class 1 composites exhibit higher C1 values and lower C3 values than Class 2 composites, indicating that the slowest relaxation mechanism plays a more prominent role in the relaxation of Class 2 versus Class 1 composites.

To understand our fast relaxation timescale, we estimate the fastest predicted relaxation mode for entangled actin networks, which is termed the mesh time τmesh and can be thought of as the time for hydrodynamic effects to become important (i.e. for each filament to “feel” its neighbors)39,64. The mesh time for our composites, computed via τm ≈ βζξ4lp−1 where is β = 1/kBT, ζ is the translational friction coefficient and ξ is the composite mesh size (see Methods)39,63,64, is ~0.086 s, which is quite close to our fast relaxation time τ1 ≈ 0.06 s. Thus, we can interpret our fastest relaxation mechanism as arising from hydrodynamic effects of the surrounding mesh. This timescale relies on filaments being able to fluctuate and rearrange over lengthscales comparable to the mesh size. These fluctuations are hindered by rigid crosslinkers, so for smaller crosslinker lengths (Class 2) this relaxation mode should contribute less to the force relaxation. Indeed, the average C1 value for Class 1 composites is larger than for Class 2.

Another key mode of stress relaxation accessible to actin filaments and, to a lesser extent, microtubules is bending3,65. We estimate timescales for actin and microtubule bending using τB ≈ (γ/κ)[L/(3π/2)]4 where κ ≈ lpkBT is the bending rigidity, L is the filament length, γ = (4πη/ln(2ξ/r) is the perpendicular drag coefficient, η is the solvent viscosity, and r is the filament radius3,39. Using our measured lengths for actin and microtubules in composites (La ≈ 8.7 µm and Lm ≈ 18.8 µm)27 we compute bending relaxation times of τB,a ≈ 0.54 s and τB,m ≈ 0.15 s for actin and microtubules, respectively. These timescales are in close agreement with our measured τ2 values (~0.3–0.6 s), suggesting that the second relaxation mode in our experiments is due to bending of both actin and microtubules. However, Class 1 composites display larger τ2 values than Class 2, with an average value of ~0.6 s which is quite close to the predicted τB,a. Conversely, the average τ2 value for Class 2 composites is ~0.4 s, lying between τB,a and τB,m. This distinction suggests that while microtubules can freely bend in response to large strains when they are not crosslinked or minimally crosslinked (Class 1), when lc,m becomes sufficiently small (Class 2), microtubules can no longer bend to release stress. This suppression of bending modes leads to the primarily elastic response seen in Figs 2, 3.

To understand the slowest relaxation mode, which occurs on the order of ~1–10 s for all composites, we turn to previous experiments investigating the stress relaxation of crosslinked actin networks following nonlinear strain37. These studies reported the presence of two distinct relaxation mechanisms available to actin filaments, which both occur on similar timescales as our measured τ3 and are both due to force-induced crosslinker unbinding and rebinding: curvilinear diffusion (reptation) and lateral confinement-hopping37. More specifically, crosslinked filaments can evade the confinement of the surrounding crosslinked mesh by crosslinks momentarily unlinking and thereby releasing the constraint they impose on the filaments, allowing for reptation and lateral confinement hopping. Such force-induced temporary unbinding events have been previously reported for both actin and microtubule networks crosslinked with biotin-NeutrAvidin4,37. Because fast timescale fluctuations (τ1) are more suppressed in Class 2 composites, we expect this slow mode to contribute more to the relaxation of Class 2 composites compared to Class 1, which is what we see in Fig. 3e (i.e. C3 values for Class 2 composites are largely greater than for Class 1).

Steady-state mobility

To determine the steady-state mobility of actin and microtubules in each composite type, we analyze time-series of dual-color confocal images, as described in Methods and previously validated27. Briefly, for each time-series we compute the average standard deviation of the intensity over time for all pixels <δ> and normalize by the corresponding average intensity of all pixels over time <I> (Fig. 4). Higher values of this mobility metric indicate a more mobile network of filaments that fluctuates more while lower values imply a more static network.

Figure 4
figure 4

Differences between crosslinking motifs have minimal impact on the steady-state viscosity and filament mobility in actin-microtubule composites. (a) Complex viscosity η(ω) versus frequency ω for different crosslinking motifs displayed in legend. Solid line denotes power-law scaling with exponent shown. (b) For each composite, a 128 × 128 image shows the standard deviation of intensity values for each pixel over time for actin (green) and microtubules (red) in a 60 s time-series. Scale bar is 10 µm and applies to all images. (c) Box-whisker plot of the steady-state mobility, determined by computing the average standard deviation of pixel intensities <δ> normalized by the overall average pixel intensity <I> for each time-series (as described in Methods). For each composite type, <δ>/<I> is calculated separately for actin (cross-hatched) and microtubules (solid) and each data point is computed from 10–12 time-series each collected in different regions of the sample chambers of two different samples. As shown, microtubules are less mobile than actin filaments in all composites. Further, while crosslinking reduces the mobility of both filaments, the specific crosslinking motif has little effect.

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Comparing <δ>/<I> values for actin and microtubules in the same composite, we find that actin filaments in all composites fluctuate significantly more than microtubules with an average <δ>/<I> value that is ~1.5x greater than that for microtubules (p < 0.03 using KS-Test described in Methods, Fig. 4). This result, which is in line with our previous findings for composites without crosslinkers27, is not surprising given the ~100-fold lower bending rigidity of actin compared to microtubules. Comparing the mobility of the crosslinked composites to the unlinked composite (None), we find that all crosslinking motifs reduce the mobility of both filaments by ~1.4x (p ≤ 0.04 and p ≤ 0.02 for actin and microtubules, respectively). This result corroborates our relaxation data in which all three decay times for None are ~1.8x faster than for crosslinked composites (Fig. 3d). Likewise, the measured resistive force during strain for None is substantially smaller than for crosslinked composites, indicating it is more fluid and mobile (Fig. 2).

However, while we see a clear distinction in the force response of Class 1 and Class 2 composites at mesoscopic scales (and in the nonlinear regime) (Fig. 2), the mobilities of both actin and microtubules display minimal variation between the different crosslinking classes (p > 0.09 for actin, p > 0.07 for microtubules). Because the mobility we are measuring is at the submicron scale (each pixel is 0.41 μm) and in the steady-state (linear) regime, we expect our mobility results to more closely match our microrheology results at the submicron and microscopic scales (corresponding to the low-force, linear regime). As described above, at these lengths and force scales there is little difference between the different crosslinking motifs, with all composites exhibiting similar force, stiffness and viscosity values (Fig. 2). This effect is also seen in the relaxation data in which the relaxation times for all crosslinked composites are similar.

Methods

Rabbit skeletal actin, biotinylated actin and Alexa-488-labeled actin were purchased from Cytoskeleton (AKL99, AB07) and Thermofisher (A12373) and stored at −80 °C in a Ca buffer (2 mM Tris (pH 8.0), 0.2 mM ATP, 0.5 mM DTT, 0.1 mM CaCl2). Porcine brain tubulin, biotinylated tubulin, and rhodamine-labeled tubulin were purchased from Cytoskeleton (T240, T33P, TL590M) and stored at −80 °C in PEM-100 (100 mM PIPES (pH 6.8), 2 mM MgCl2, and 2 mM EGTA).

Crosslinked networks were formed using preassembled biotin-NeutrAvidin complexes optimized previously for stable, isotropic crosslinking of actin networks37. As depicted in Fig. 1a, crosslinker complexes are comprised of NeutrAvidin flanked on either side by biotinylated actin and/or tubulin, depending on the desired crosslinking motif. We incorporated crosslinkers into previously established protocols to form co-entangled actin-microtubule composites27. In short, actin monomers, tubulin dimers, and crosslinker complexes were added to PEM-100 supplemented with 1 mM ATP, 1 mM GTP, and 5 µM Taxol. To image composites, 0.13 µM of Alexa-488-labeled actin filaments and rhodamine-labeled microtubules, as well as oxygen scavenging agents [4.5 mg/ml glucose, 0.5% β-mercaptoethanol, 4.3 mg/ml glucose oxidase, 0.7 mg/ml catalase] were added (Fig. 1b). To enable microrheology measurements, carboxylated microspheres (Polysciences, Inc.), with a 4.5 µm diameter, were also added (Fig. 1c). The solution was then mixed, pipetted into a ~20 µL sample chamber, and allowed to incubate at 37 °C for 30 minutes to co-polymerize both proteins.

We prepared six different composite types that varied in their crosslinking motif as follows (Fig. 1a): (1) no crosslinkers are present (None); (2) actin filaments in the composite are crosslinked to each other (Actin); (3) microtubules in the composite are crosslinked to each other (Microtubule); (4) both actin and microtubules are crosslinked but they cannot crosslink to each other (Both); (5) actin and microtubules are crosslinked to each other but not to themselves (Co-linked); and (6) a composite similar to Both but in which the densities of actin-actin and microtubule-microtubule crosslinks are both doubled (Both 2x).

For all presented data, total protein concentration was held fixed at 5.8 µM with an equimolar ratio of actin to tubulin, and the molar ratio of crosslinker to protein was fixed at R = 0.02. The mesh size of actin and microtubule networks comprising each composite are ξa = 0.3/ca1/2 = 0.85 µm and ξm = 0.89/ct1/2 = 1.58 µm in which ca and ct are the actin and tubulin concentrations in units of mg/ml34,42,67. From these mesh sizes we calculated the effective composite mesh as ξ = (ξm−3 + ξa−3)−1/3 = 0.81 µm27. We estimated the length between crosslinkers for all three types of crosslinking used in composites (actin crosslinking, microtubule crosslinking, and actin-microtubule co-linking) as follows (Fig. 1a). The length between crosslinkers along an actin filament was determined by lc,a = ½lmon × R−1 where lmon = 2.7 nm is the length that each actin monomer adds to an actin filament. The ½ prefactor takes into account that each crosslinker is shared between two filaments. Similarly, the length between crosslinkers along a microtubule was calculated by lc,m = ½(lring/13) × R−1 where every 13 tubulin dimers adds lring = 7.8 nm in length to the microtubule. For Actin and Microtubule composites, lc,a = 67.5 nm and lc,m = 15 nm. In the composite where crosslinkers are equally distributed between actin-actin links and microtubule-microtubule links (Both), R remains fixed so there are half as many crosslinkers available to each network. Thus, lc,a and lc,m both double (i.e. lc,a = 135 nm, lc,m = 30 nm). For composites in which both networks are linked but R is doubled (Both 2x) lc,a and lc,m remain the same as in the Actin and Microtubule cases. For Co-linked composites, the length between crosslinkers that link actin to microtubules was computed as lc,a-m = (lc,a×lc,m)1/2 = 31 nm.

The optical trap used in all microrheology measurements, which has been previously described and validated27,36,37,68, was formed by a IX71 fluorescence microscope (Olympus) outfitted with a 1064 nm Nd:YAG fiber laser (Manlight) and focused with a 60× 1.4 NA objective (Olympus). A position-sensing detector (Pacific Silicon Sensor) measured the deflection of the trapping laser, which is proportional to the force acting on the trapped microsphere. Stokes drag method was used to calibrate trap stiffness69,70. A nanopositioning piezoelectric stage (Mad City Labs) was used to move the trapped microsphere relative the sample (Fig. 1c). Stage position and laser deflection were recorded at 20 kHz using custom-written Labview codes. The response time of the stage is 0.25 ms, so the data presented in Fig. 2 (strain phase) and Fig. 3 (relaxation phase) start at 2.5e-4 s when the stage has reached constant velocity (strain phase) or come to a complete stop (relaxation phase).

For the data presented in Figs 1–3, a trapped bead was displaced 10 µm at a constant speed of 10 µm/s relative to the sample chamber. The speed and displacement were chosen to ensure we were well within the nonlinear regime, which we previously showed to occur at ~3–5 μm/s27; and accessing mesoscopic scales, i.e. several times larger than ξ and comparable to the lengths of the filaments. We did not exceed this speed and displacement because for larger strain parameters the bead would not reliably stay trapped for the duration of the strain as the strength of the composite exceeded the strength of the trap. Oscillatory strain measurements (Fig. 4) were also performed in which the trapped sphere was sinusoidally displaced over an amplitude A = 1 µm at frequencies ω = 1.26–126 rad/s. From the amplitudes of the oscillatory force curves Fmax, as well as the phase shift Δϕ between the force and stage position curves we computed the complex viscosity, η(ω) = Fmax(ω)/3πAd where d is the bead diameter (Fig. 4), as described previously71.

Custom-written MATLAB scripts were used for post-measurement analysis. All displayed data are averages over 50–60 different trials taken between two samples. All trials were taken in different regions of the sample chamber, each separated by >100 µm.

A Nikon A1R laser scanning confocal microscope with a 60× 1.4 NA objective and QImaging QICAM CCD camera were used to collect 2D time-series of composites. The microscope is outfitted with 488 nm and 561 nm lasers and simultaneously records separate images for each laser channel (green and red) to separately visualize Alexa-488-actin (green) and rhodamine-tubulin (red) (Figs 1b, 4). Time-series of 512 × 512 images (0.41 µm/pixel) in each channel were recorded at 16 fps for 60 seconds. For each composite type, two different samples were imaged with 3–5 time-series taken for each sample.

Time-series from each channel (green and red) were analyzed separately to determine the mobility of actin and microtubules in the composites, as previously described27. In brief, every 16 frames in the time-series were averaged together to create time-series with 1 second frames27. As demonstrated and described previously27, we selected 1 second frames to average out the frame-to-frame shot noise that obscured the images of the filaments, enhancing the signal to noise without altering the trends in the data (Fig S3); and because the filaments do not significantly change position over that time (within the resolution of the measurement). From these averaged time-series, we used FIJI/ImageJ to create a single collapsed image for each channel that represented the standard deviation of each pixel over time (Fig. 4). The average standard deviation over all pixels was then calculated to give a single standard deviation value for the entire time-series <δ>. This value, which represents the variation in intensity values for each pixel over time, is a measure of the extent to which filaments in the network fluctuate over time. To account for differences in overall intensity among different time-series, we also computed the average intensity over all pixels and frames for each time-series <I>, and normalized each standard deviation by this value. The resulting metric <δ>/<I>, which quantifies the mobility of actin and microtubules in each time-series, is presented in Fig. 4. We carried out this analysis for all time-series collected for each composite type. We used the Kolmogorov-Smirnov statistical test (KS-Test, http://www.physics.csbsju.edu/stats/KS-test.html) to compare different data sets. From KS-tests, we computed probability values p for finding overlapping data and use the accepted value of p < 0.05 as the cutoff for determining when two data sets were significantly different.

Conclusion

Actin and microtubules form interacting networks within the cytoskeleton, providing cells with mechanical integrity and enabling a myriad of mechanical processes such as locomotion, morphogenesis, intracellular transportation, and division. Many of these diverse functions are mediated by crosslinking proteins that can bind actin, microtubules, or both proteins. Further, the role of crosslinking between actin and microtubules has received much recent attention and is thought to be responsible for maintaining cell shape and polarity, growth and structure of neurons, and spindle positioning in mitosis8,15,16. The design and characterization of composites of flexible and stiff polymers is also a topic of current interest in materials engineering26,66, yet how crosslinking can tune the mechanical response of these composites has remained largely unexplored.

Here, we characterize the mobility and mechanics – from submicron to mesoscopic scales – of custom-designed composites of actin and microtubules with distinct crosslinking patterns. We create composites in which actin is crosslinked (Actin), microtubules are crosslinked (Microtubule), both actin and microtubules are crosslinked (Both), and actin and microtubules are crosslinked to each other (Co-linked). We use optical tweezers microrheology and dual-color fluorescence confocal microscopy to measure the nonlinear force response and mobility of each composite type. We show that composites separate into two distinct classes which are dictated by the degree to which microtubules in each composite are crosslinked. Class 1, which comprises None, Actin, and Both composites, displays largely viscoelastic response to the imposed strain with ample softening. In contrast, Class 2, comprised of Microtubule, Co-linked and Both 2x composites, elicit a substantially elastic response to perturbation. Notably, this distinction only emerges at mesoscopic scales in response to nonlinear forcing. Conversely, the crosslinking motif has little impact on the microscopic linear regime mechanics and steady-state filament mobility. These intriguing scale-dependent dynamics likely play a significant role in the ability of the composite cytoskeleton to mediate a vast array of different mechanical processes, and could be exploited to design multifunctional soft materials with scale-dependent mechanical properties. The custom-designed composites and measurement techniques demonstrated in this study further lay the groundwork for building complexity into in vitro cytoskeleton networks to determine how each component plays a role in cellular mechanics and in the design of novel biomimetic materials.