The cytoskeleton is an active composite of filamentous proteins that dictates diverse mechanical properties and processes in eukaryotic cells by generating forces and autonomously restructuring itself. Enzymatic motors that act on the comprising filaments play key roles in this activity, driving spatiotemporally heterogeneous mechanical responses that are critical to cellular multifunctionality, but also render mechanical characterization challenging. Here, we couple optical tweezers microrheology and fluorescence microscopy with simulations and mathematical modeling to robustly characterize the mechanics of active composites of actin filaments and microtubules restructured by kinesin motors. It is discovered that composites exhibit a rich ensemble of force response behaviors–elastic, yielding, and stiffening–with their propensity and properties tuned by motor concentration and strain rate. Moreover, intermediate kinesin concentrations elicit emergent mechanical stiffness and resistance while higher and lower concentrations exhibit softer, more viscous dissipation. It is further shown that composites transition from well‐mixed interpenetrating double‐networks of actin and microtubules to de‐mixed states of microtubule‐rich aggregates surrounded by relatively undisturbed actin phases. It is this de‐mixing that leads to the emergent mechanical response, offering an alternate route that composites can leverage to achieve enhanced stiffness through coupling of structure and mechanics.
]]>The cytoskeleton is an active composite network of filamentous proteins and their associated binding proteins, including energy-transducing molecular motors that pull and walk along filaments (1,2). A primary role of the cytoskeleton is to provide mechanical integrity to cells while also allowing them to stiffen, soften, change shape, and generate forces, often in response to local stimuli (3,4). These diverse mechanical responses are often spatially heterogeneous and can range from nanoscopic to cell-spanning scales. Moreover, the nature of the response is intricately linked to the time-evolving structures and interactions of the different networks of e.g., semiflexible actin and microtubules (5).
This complex active and composite nature of the cytoskeleton has rendered it a foundational model system for probing questions in active matter physics and addressing design challenges in living materials (3,6). In vitro cytoskeleton-based active matter systems (7)(8)(9)(10)(11)(12)(13)(14)(15) typically include myosin II minifilaments (16) and/or crosslinked clusters of kinesin dimers (17), which are enzymatically-active motor proteins that harness the energy of ATP hydrolysis to bind to and pull on actin filaments and/or microtubules, respectively. Actomyosin networks have been shown to undergo bulk contraction, local contraction into foci or asters, or disordered flow depending on the concentrations of the myosin, actin and crosslinkers (12,13,18). Kinesin clusters acting on bundles of microtubules have also shown varied behaviors, ranging from the formation of locally condensed asters (19)(20)(21)(22)(23) to space-spanning networks capable of extensile restructuring that results in nematic flow and organization reminiscent of liquid crystals (8,24,25).
More recently, in vitro active cytoskeletal composites that include both actin and microtubules have been engineered and examined (26)(27)(28)(29)(30)(31)(32)(33), often revealing emergent behavior and improved material properties, such as organized dynamics (26), tunable miscibility (30,33), structural memory (31), and enhanced elasticity (28). Early work explored passive composite networks lacking molecular motors. In this simplified condition, biotin-streptavidin crosslinkers induce passive and effectively permanent crosslinking of either the actin or microtubule components (34,35). Within such composites, microtubule crosslinking is essential to eliciting elastic responses to localized strains, whereas actin-crosslinked composites exhibit yielding behavior similar to that of purely entangled composites (34).
When the concentration of actin crosslinkers was varied, an emergent elasticity was revealed at intermediate crosslinker:actin ratios 𝑅𝑅 ≃ 0.2, which decreased to values comparable to those of entangled composites as this ratio was increased to 𝑅𝑅 ≃ 0.8 (35). This counter-intuitive behavior is observed only in composites, and is driven by crosslinker-mediated network coarsening and bundling that simultaneously increases the thickness (and thus stiffness) of network fibers and as well as network mesh size. The delicate interplay of actin network microstructure and fiber rigidity leads to the development of an optimal crosslinker ratio in which the network has developed sufficient rigidity to maximize elastic response, but the mesh size remains small enough to suppress the diffusive mobility of microtubules entrapped within the actin network (35).
Notably, for actin-only networks, increasing crosslinker ratios monotonically increases the network stiffness (36)(37)(38). However, within the composites, the increased microtubule mobility enables new pathways of stress relaxation, which dominate the mechanical response and soften and fluidize the composite.
When enzymatically-active motors are included, even more dramatic structural changes are observed. Embedded motors act both as transient crosslinkers and force-generating elements, leading to dynamic responses that span spatial and temporal scales. Co-entanglement of microtubule filaments with myosin-driven actin produces active composite networks in which both actin and microtubules ballistically contract at speeds that can be tuned by the concentrations of actin and myosin (26,27). Such networks display controlled motion, enhanced elasticity, and sustained structural integrity as compared to single filament networks (28). Replacing myosin with kinesin as the active agent in similar co-entangled composites results in a much higher degree of variability in dynamics and structure, with speeds that vary by 2 orders of magnitude, phases of acceleration and deceleration, and enhanced restructuring and de-mixing of actin and microtubules, depending on the composite formulation and time after motor activation (33). Typically, network restructuring occurs over a finite timespan that is determined by the motor-driven compaction of filaments into poorly-connected, kinetically trapped network structures reminiscent of asters or disordered aggregates, with kinesin-driven composites displaying shorter active lifetimes as compared to those driven by myosin or both myosin and kinesin. Similar phenomena have been observed in composite networks comprising kinesin clusters and microtubules that are bundled by osmotic crowders acting as depletants and by microtubule binding proteins that promote antiparallel bundling (30). Adding low concentrations of actin to such networks produces fluid-like extensile dynamics, similar to those of kinesin-driven microtubule networks lacking actin. However, when the actin concentration is increased, rich dynamic structural transitions are observed, leading to the formation of onion-like asters of layered actin and microtubules or bulk contractility. Further increases in actin concentration promote de-mixing of actin and microtubules with asters and contractile regions becoming increasingly microtubule-rich (30).
Motor-driven structural transitions that form disconnected, filament-dense structures interspersed within a dilute fluid phase can undermine network percolation. From a material design perspective, this phase segregation risks fluidizing the network on mesoscopic scales, thereby compromising the ability of active composites to transmit forces over large distances or to sustain significant external stresses. Despite the importance of the mechanical response of these systems to their role in cellular processes and to materials-based applications, the majority of active composite studies have focused on the evolving structure and dynamics of the materials without regard to mechanical properties.
While there are a number of approaches to analyzing network structural rearrangements via fluorescence microscopy, the measurement of the time-evolving local mechanical properties within the dynamically-restructuring network remains technically challenging. The emergent heterogeneity arising due to motor-driven aggregation or segregation of filaments demands the precise application of forces to determine the local mechanical responses, (28) as well as relatively large numbers of measurements in order to develop an understanding of average responses and the range of variations at each condition. Thus, although a key feature of motor-driven active cytoskeletal composites is their ability to flow, coarsen and reconfigure due to internal motor-generated forces, little is known about the interplay between material mechanics and filament motion during the emergence of these new structural phases.
To begin to establish the foundational role of motor-driven restructuring in network mechanics, we designed co-entangled composites of actin and microtubules formulated to support robust connectivity, and subjected them to active stresses and restructuring by adding varying concentrations of kinesin motors. Using a comprehensive platform comprising an optical tweezers microrheometer (OTM) capable of applying large-scale strains at specified locations within the heterogeneous sample, fluorescence microscopy to assess structural rearrangements, simulations based on lattice-based advection-diffusion models, and mathematical modeling of mechanical responses, our results reveal how kinesin motors act on composites of actin and microtubules to sculpt the mechanical and structural properties across spatiotemporal scales. We identify the presence of kinesin-driven demixing via clustering, which in turn leads to emergent complexity in mechanical response and formulationdependent heterogeneity that can be captured both in vitro and in silico. These results demonstrate the importance of hierarchical structural heterogeneity to provide new avenues for enhanced stiffness and relaxation only possible in composite designs.
As a first step, we assess the complex structural properties of active composites composed of actin, microtubules, and kinesin. We judiciously chose a ratio of actin to microtubules (45:55 molar ratio of actin to tubulin dimers) that allows for active restructuring and force-generation without the large scale flow or network rupturing that has been previously reported (30,33), and examined the effect of varying concentrations of kinesin, 𝑐𝑐 𝑘𝑘 , on the network restructuring (Fig 1A). In control networks lacking kinesin, we observed, using high-resolution two-color confocal microscopy, uniform mixing of actin and microtubules, to form a homogeneous, space-spanning composite of interpenetrating networks of actin and microtubules (Fig 1A, left). Upon addition of kinesin, we observed the formation of microtubule-rich phases that appeared to generally increase in size, density and number with increasing kinesin concentration. This kinesin-driven de-mixing of microtubules from actin was robustly observed across all samples; and, upon addition of 640 nM kinesin, the highest concentration investigated here, nearly all of the microtubules condense into aster-like aggregates surrounded by actin-rich zones (Fig. 1A, right).
To better understand the molecular drivers of this behavior, we build on our previously developed two-dimensional lattice-based advection-diffusion model of filament dynamics (33). Within the model, filament motion arises from kinesin-generated active forces that can either pull or push microtubules, as well as frictional forces that occur when passive motors act as crosslinkers between microtubules (33). In the control case without kinesin, the simulations rendered a uniform, well-mixed composite of interpenetrating networks of microtubules and actin, consistent with our experimental observations (Fig 1B, left). Similarly, upon addition of kinesin motors, the composites restructure and de-mix, with increasing segregation observed for higher kinesin concentrations. Moreover, it is clear from both experiment and simulations that the kinesin-driven motions that cluster the microtubules do not significantly restructure the actin. Rather, a two-phase material is formed, with microtubule-dense regions forming distinct, well-separated aggregates within a more uniform actin-rich background (Fig 1B, right). rotation (curved arrows) and contraction (straight arrows) that it can impart on microtubules. The scale of the schematic is indicated by the black box in the upper right corner of the 𝑐𝑐 𝑘𝑘 = 80 nM snapshot.
The simulations also provide the opportunity to quantitatively compare the simulated network structures before and after restructuring through calculation of the filament pair distribution function 𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟 , 𝑇𝑇) where the subscripts 𝑖𝑖 and 𝑗𝑗 represent the filament type, either actin (A) or microtubules (M), and which gives the probability of finding a filament (actin or microtubule) a radial distance 𝑟𝑟 from any other filament. To assess the dynamic structural changes that occur within a single filament network during the simulation, we report the difference between these quantities for the initial (𝑇𝑇 = 0) state and final (𝑇𝑇 = 𝑇𝑇 𝐹𝐹 ) states: ∆𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟 ) = 𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟 , 𝑇𝑇 𝐹𝐹 ) -𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟 , 0). For static, steady state networks, we expect ∆𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟) = 0 for all 𝑟𝑟 , as we see in Fig 2A ,B for both the actin and microtubule networks within the composites lacking kinesin (𝑐𝑐 𝑘𝑘 = 0). This result also validates that our initial simulation conditions represent homogeneous, well-mixed networks that remain well-mixed in the absence of motor activity. When comparing the distribution of microtubules to other microtubules, or actin filaments to other actin filaments, we found positive values of ∆𝑔𝑔 𝑀𝑀𝑀𝑀 (𝑟𝑟) and ∆𝑔𝑔 𝐴𝐴𝐴𝐴 (𝑟𝑟) for small filament separation distances 𝑟𝑟, indicating attractive interactions that drive clustering on those length scales; and we observed an increased clustering propensity with increased kinesin concentration, estimated by the value of ∆𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟 0 ), where 𝑟𝑟 0 = 1.25 µm is the smallest radial distance between lattice points in the simulation (Fig. 2D). As the separation distance increases, ∆𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟) curves for all 𝑐𝑐 𝑘𝑘 > 0 composites decay to zero then continue to decrease, reaching local negative-valued minima before asymptoting back to zero. This behavior indicates a depletion of filaments on intermediate length scales, which we interpret as an indication of clustering. While both filament types display these general features, the microtubule network, upon which the kinesin motors directly act, exhibited much stronger clustering effects compared to actin (Fig 2A,B,D).
By contrast, when evaluating the co-distribution of actin filaments with respect to the microtubule network, we find negative values of ∆𝑔𝑔 𝑀𝑀𝐴𝐴 (𝑟𝑟) at even the smallest filament separation distances (Fig 2C), indicating an exclusion of the unlike filament type. This anticorrelation demonstrates that actin is displaced from the microtubule-rich domains that form from the kinesin-driven contraction of microtubules, and is consistent with de-mixing. The strength of this effect can be approximated by ∆𝑔𝑔 𝑀𝑀𝐴𝐴 (𝑟𝑟 0 ), which is found to monotonically decrease with increasing kinesin concentration (Fig 2D). The magnitude of this exclusion effect |∆𝑔𝑔 𝑀𝑀𝐴𝐴 (𝑟𝑟 0 )| is intermediate between the values observed for clustering of the microtubule ∆𝑔𝑔 𝑀𝑀𝑀𝑀 (𝑟𝑟 0 ) and actin ∆𝑔𝑔 𝐴𝐴𝐴𝐴 (𝑟𝑟 0 ) structures. The length scales over which this phase separation was observed can be approximated by the radial distance at which ∆𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟) = 0, which we denote as 𝑙𝑙 0 , as well as the distance at which ∆𝑔𝑔 𝑖𝑖𝑖𝑖 (𝑟𝑟) is minimal for like-filament distributions or maximal for microtubule-actin co-distributions, which we denote by 𝑙𝑙 𝑚𝑚𝑖𝑖𝑚𝑚 or 𝑙𝑙 𝑚𝑚𝑚𝑚𝑚𝑚 . Specifically, 𝑙𝑙 0 can be considered a measure of cluster size while 𝑙𝑙 𝑚𝑚𝑖𝑖𝑚𝑚 and 𝑙𝑙 𝑚𝑚𝑚𝑚𝑚𝑚 are measures of spacing between clusters. In a system with mass conservation, we expect both quantities to generally track with one another, as we observe in Fig 2E . As shown in Fig 2E , we found similar length scales of de-mixing when comparing all filament types, and, in each case, we observed a monotonic decrease in the observed length scales with increasing kinesin concentrations. Moreover, the range of values (~5 -15 µm) were generally consistent with the observed sizes and spacing between clusters in both experiment and simulation (Fig 1), and reflect the increased clustering with increasing kinesin concentration. To more quantitatively compare these structural analysis results from simulations to experimentally observed restructuring, we performed spatial image autocorrelation (SIA) analysis (28,39) on epifluorescence movies of the composites captured in identically prepared samples as the force measurements that we describe below. In SIA the correlation in intensities between two pixels separated by a distance 𝑟𝑟 is examined. In this experiment, both actin and microtubules were labeled with spectrally-indistinguishable fluorescent dyes and were simultaneously imaged such that at each condition, the composite network behavior is observed (see Methods). Similar to pair distribution functions obtained from simulations, the resulting autocorrelation function 𝑔𝑔 𝐼𝐼 (𝑟𝑟), decays from a maximum value at 𝑟𝑟 0,𝐼𝐼 = 0.41 µm (set by the pixel size) to 𝑔𝑔 𝐼𝐼 = 0 as 𝑟𝑟 → ∞, passing through a local 𝑔𝑔 𝐼𝐼 < 0 minimum; and at a given distance larger 𝑔𝑔 𝐼𝐼 (𝑟𝑟) values are suggestive of increased filament clustering. To better compare the simulated distributions shown in Fig 2A -C to experimentally determined data, we subtracted the value of 𝑔𝑔 𝐼𝐼 (𝑟𝑟) obtained for the composite without kinesin (𝑐𝑐 𝑘𝑘 = 0) from 𝑔𝑔 𝐼𝐼 (𝑟𝑟) for each 𝑐𝑐 𝑘𝑘 > 0 composite, similar to subtracting the initial time distribution from the final in simulations. As shown in Fig 2F, the resulting ∆𝑔𝑔 𝐼𝐼 (𝑟𝑟) curves display similar functional features as the simulated data, with higher kinesin concentrations generally resulting in larger 𝑔𝑔 𝐼𝐼 (𝑟𝑟 0 ) values and more pronounced minima. However, a striking distinction is that for experiments, 𝑔𝑔 𝐼𝐼 (𝑟𝑟 0 ) displays a non-monotonic dependence on 𝑐𝑐 𝑘𝑘 , reaching a maximum for 𝑐𝑐 𝑘𝑘 = 160 nM. This non-monotonicity is reminiscent of similar emergent phenomenon reported for cytoskeleton composites with increasing concentrations of actin crosslinkers (35).
Evaluating the same characteristic distances as in simulations for cluster size and spacing, 𝑙𝑙 0 and 𝑙𝑙 𝑚𝑚𝑖𝑖𝑚𝑚 , namely where ∆𝑔𝑔 𝐼𝐼 (𝑟𝑟) reaches zero and a local miminum, we find that both lengthscales show a modest decrease between 𝑐𝑐 𝑘𝑘 = 40 nM and 𝑐𝑐 𝑘𝑘 = 320 nM, similar to simulations, but subsequently increases at 𝑐𝑐 𝑘𝑘 = 640 nM (Fig 2G). This increase is indicative of the large aggregates we observe in microscopy images (Fig 1A). We note that the correlation lengthscales observed in experimental data (~30 -60 µm) are generally larger than for simulations, likely due to the larger field-of-view and system size, as well as the added dimension in 3D experiments. Specifically, the 2D experimental plane is ~10 3 -fold larger than our simulation box size, so we have many more filaments able to move into the imaging field-of-view to join clusters. Conversely, once most simulated filaments end up in clusters, and there are few freely diffusing filaments left in the box, the clusters cannot grow further. The third dimension in experiments also provides another route for filaments to move and reorganize to facilitate cluster growth.
Further examining the large lengthscales accessible to experiments, we observed positive 𝑔𝑔 𝐼𝐼 (𝑟𝑟) values out to the largest analyzed distance (𝑟𝑟 ∞ = 160 µm), for the highest kinesin concentrations (𝑐𝑐 𝑘𝑘 = 320, 640 nM), indicative of the presence of largescale clustering in these conditions (Fig 2F, inset). By contrast, the 𝑐𝑐 𝑘𝑘 = 160 nM composite, which displayed the highest short-range correlation, exhibited negative long-range correlation values as 𝑟𝑟 → 𝑟𝑟 ∞ . Together, these data suggest that as the kinesin concentration increases, de-mixing initially causes dense smallscale clustering, as indicated by the peak in 𝑔𝑔 𝐼𝐼 (𝑟𝑟 0 ) at intermediate kinesin concentration, followed by large-scale phase separation that maximizes 𝑔𝑔 𝐼𝐼 (𝑟𝑟 ∞ ) for higher kinesin concentrations.
To further corroborate the physical picture of de-mixing, we also examined the distribution of pixel intensities of the same videos. Using intensity as a proxy for mass, we evaluated the distribution of pixel intensities to identify increases (higher pixel values) and decreases (lower pixel values) in filament density due to bundling and clustering (Fig 2H). We found that as the kinesin concentration was increased from 𝑐𝑐 𝑘𝑘 = 0 nM to 80 nM, the peaks of the distribution shifted to higher intensity values, indicating bundling; and the distributions became broader, indicating the increasingly heterogeneous distribution of densities. For 𝑐𝑐 𝑘𝑘 ≥ 160 nM, two peaks emerged. The higher intensity peak occurred at an intensity that was slightly larger than that of the single peak for the 𝑐𝑐 𝑘𝑘 < 160 nM conditions, and this peak shifted to higher intensity values (shifting further right) and larger probabilities (increasing height) as 𝑐𝑐 𝑘𝑘 increased. This trend is indicative of an increasing number of bundles that also become denser due to the presence of additional kinesin motors. The second peak, which occurred at lower intensity values than the single peaks observed for 𝑐𝑐 𝑘𝑘 < 160 nM, likewise shifted to lower intensity values as 𝑐𝑐 𝑘𝑘 increased, indicating the emergence of more microtubule-poor zones. Zooming-in to examine the high intensity tails of the distributions, we found that composites with 𝑐𝑐 𝑘𝑘 ≥ 160 nM exhibited pronounced extended tails that were not observed for the lower kinesin concentrations, again indicating the formation of large and dense clusters at the higher concentrations of kinesin (Fig. 2H, inset).
Together, these results indicate that kinesin clusters drive contraction and compaction of disordered microtubules into dense, well-separated aggregates. This contraction occurs in the absence of osmotic crowding agents and does not require the presence of non-motor microtubule associated binding proteins to promote bundling. This restructuring causes modest reorganization of the actin network, as the actin filaments are squeezed out by contracting microtubules. However, the actin network remains reasonably well dispersed even at the highest kinesin concentrations.
To probe how the kinesin-driven restructuring influences the microscale mechanical properties of the composite, we applied localized but large-scale deformations within the heterogeneous material using an optical trapping-based manipulation platform (Fig 3A,B)(40-42). A single beam gradient optical trap was formed by tightly focusing a high-powered IR laser to a diffraction-limited volume within the sample chamber (43). This allowed the capture and manipulation of embedded colloidal probes (radius 𝑟𝑟 𝑝𝑝 = 2.25 µm, full details provided in Materials and Methods). The trap stiffness, which was calibrated through independent measurements, was sufficient to stably trap and hold the particle, even as the stage moved at fixed velocity, thereby dragging the particle through the sample. The displacement of the trapped particle from the trap center was simultaneously monitored in real time, and when multiplied by the known trap stiffness, provided an instantaneous readout of the force. From the known values of force and stage position, which are collected as a function of time (Fig. 3C), it is possible to construct a relationship between force and stage position. Thus, this instrument acts as a microscale mechanical testing system or microrheometer, which we use to interrogate the response to a strain (i.e., stage displacement) of 𝑠𝑠 = 20 µm, which we chose to be significantly larger than the probe size 𝑟𝑟 𝑝𝑝 and composite mesh size 𝜉𝜉 ≃ 1.2 µm (see Methods).
We performed measurements on composites with the six kinesin concentrations presented above (Figs 1,2), using 3 stage speeds (𝑣𝑣 = 6, 12, 24 µm s -1 ) for each concentration. Examining the individual force-displacement traces (Fig 3C,D), we find that all traces exhibited sharp initial increase in force at the smallest stage displacements as the particle position rapidly shifted within the trap as the stage began to move. The time constant associated with this re-equilibration is given by the ratio of local drag coefficient to the trap stiffness 𝑘𝑘 𝑂𝑂𝑂𝑂 and was typically < 50 ms. Beyond this very initial behavior, we find heterogenous responses, which can be categorized into three broad classes of responses (shown in representative traces in Fig. 3D): traces that are fully linear, suggesting elastic behavior ('elastic'), those that show an initial elastic response that softens or yields over time ('yielding'), and those that show an initially soft elastic response that significantly stiffens at large displacements ('stiffening'). The proportion of traces falling into each category varies as a function of kinesin concentration and stage speed (Fig. 3E). In general, the responses are diverse and heterogeneous, likely reflecting the structural heterogeneity of the material observed in Figures 1 and 2. The observance of a large fraction of fully linear elastic traces is notable, given the stage stroke of 20 µm, an order of magnitude larger than the size of both the probe and mesh.
To better assess the dependence of mechanical properties on composite formulation, we analyzed the force-displacement traces measured for >20 different particles in different locations and samples for each experimental condition (see Methods). For each combination of kinesin concentration 𝑐𝑐 𝑘𝑘 and speed 𝑣𝑣, we averaged together all traces that displayed elastic, yielding, or stiffening characteristics ( Figs 4A, S2). For each of these response types, we observed striking nonmonotonic behavior as the kinesin concentration was increased.
When we examined the subset of elastic traces obtained at 𝑣𝑣 = 24 µm s -1 , we found that the largest value of maximum force and maximum effective stiffness, as qualitatively assessed from the terminal force value 𝐹𝐹 𝑚𝑚𝑚𝑚𝑚𝑚 reached at the end of the strain, occurred at the intermediate value of 𝑐𝑐 𝑘𝑘 = 160 nM (Fig. 4A,B). The lowest values of 𝐹𝐹 𝑚𝑚𝑚𝑚𝑚𝑚 were surprisingly observed at the highest concentration of 𝑐𝑐 𝑘𝑘 = 640 nM. Similar general trends were observed for the elastic traces obtained at the other speeds (Fig. 4B Two notable takeaways from these results are that active composites exhibit (1) emergent mechanical resistance at intermediate kinesin concentrations and (2) viscoelastic response to the application of local strains is heterogeneous, varying from stiffening to elastic to more viscous-dominated (i.e., yielding). To shed further light on these features, and assess the ability of our simulations to capture them, we introduced spherical probes into our simulated composites and imparted oscillatory forcing on them through the composite (Fig 4C). As described in Methods and SI, we measured the probe displacement resulting from oscillatory forcing with amplitude 𝐹𝐹 0 = 100 pN and frequencies 𝜔𝜔 = 0.25, 0.5, and 1 Hz, to determine the viscoelastic stress response (Fig 4D). We chose the force amplitude to achieve particle displacements comparable to our 20 µm experimental strain (Fig 4D ), and frequencies to approximately match those in our experiments, considering 𝜔𝜔 = 𝑣𝑣/𝑠𝑠. Specifically, we measured the bead displacement amplitude and phase difference 𝜙𝜙 between the oscillation in applied force and resulting bead displacement to determine the elastic modulus 𝐺𝐺 ′ and viscous modulus 𝐺𝐺 ′′ as a function of kinesin concentration (Fig S3). To quantify the relative elasticity of the composite we evaluated the inverse loss tangent [tan 𝜙𝜙] -1 = 𝐺𝐺′/𝐺𝐺′′ which is increasingly >1 or <1 for more elastic-dominated or more viscous-dominated responses, respectively. We found that introducing kinesin into composites increased the relative elasticity for all frequencies (Fig 4E), and that peak elasticity was observed at intermediate kinesin concentrations for 0.25 Hz and 1 Hz. The 0.5 Hz data also showed a local maximum at intermediate kinesin concentrations but then increased again at 𝑐𝑐 𝑘𝑘 = 640 nM. These features align with our experimental results that display non-monotonic dependence of the force response on 𝑐𝑐 𝑘𝑘 for most formulations and speeds; and highlight the importance of both elastic and viscous contributions to the force response. Collectively, these results demonstrate the emergent elasticity and force resistance that kinesin-driven de-mixing affords, which is optimized at intermediate kinesin concentrations.
To more quantitatively understand the tunable viscoelastic nature of the experimental force responses, and their underlying drivers, we use a mechano-equivalent circuit approach to model the ensembled-averaged responses. (45,46) Due to the relative infrequency of the stiffening responses, and the likelihood that that subset of traces is dominated by rare interactions of the particles with heterogeneous microstructures, we focused our analysis on the elastic and yielding responses only. We designed an equivalent circuit that consists of two Kelvin-Voigt elements in series (Fig. 5A, details in SI). The first element accounts for the composite network viscoelasticity, which is represented by a spring element with spring constant 𝜅𝜅 to represent the network stiffness, and a dashpot element with drag coefficient 𝛾𝛾 to represent viscous dissipation. A second Kelvin-Voigt element represents the effect of the optical trap stiffness 𝑘𝑘 𝑂𝑂𝑂𝑂 (which is known). We allowed the two elements to undergo relative deformation, and tracked the position of the center of the optical trap and the particle as 𝑥𝑥 1 and 𝑥𝑥 2 , respectively. Here, 𝑥𝑥 1 = 𝑣𝑣𝑡𝑡, where 𝑣𝑣 is the stage speed, 𝑡𝑡 is the elapsed time. To estimate the force response as a function of stage motion, we calculate 𝐹𝐹 = 𝑘𝑘 𝑡𝑡𝑡𝑡𝑚𝑚𝑝𝑝 (𝑥𝑥 1 -𝑥𝑥 2 ). Assuming that at 𝑡𝑡 = 0, 𝑥𝑥 1 = 𝑥𝑥 2 ≈ 0, we established:
We selected this model to capture the following phenomena: an initial elastic jump due to the re-equilibration of the particle within the optical trap as the stage begins to move, the transition to a second elastic regime as the particle engages with the composite network, and the presence of transient bonds that can dissipate stress and can be modeled via an effective viscosity (Fig. 5B). We use this approach to analyze each of the 6 kinesin concentrations at each of the 3 tested stage speeds. classes and emergent stiffness at intermediate kinesin concentrations. (A) Cartoon of mechanical circuit that models the force-displacement relationship for a bead pulled through a network by an optical trap with known trap stiffness 𝑘𝑘 𝑂𝑂𝑂𝑂 . The composite network stiffness 𝜅𝜅 and drag 𝛾𝛾 are fit parameters in the model. (B) Force 𝐹𝐹(𝑥𝑥) versus stage position 𝑥𝑥, averaged across all elastic and yielding traces for each kinesin concentration 𝑐𝑐 𝑘𝑘 , listed and color-coded according to the legend, for 𝑣𝑣 = 24 µm s -1 . Error bars denote standard error of the mean. Dashed lines are fits to the equation of motion for the mechanical circuit depicted in A. (C-F) Fit parameters 𝜅𝜅 (C,D) and 𝛾𝛾 (E,F) as a function of kinesin concentration 𝑐𝑐 𝑘𝑘 for speeds 𝑣𝑣 = 6 (green circles), 12 (purple squares) and 24 (red diamonds) µm s -1 , with error bars denoting 95% confidence intervals. Panels display (C,E) magnitudes for all kinesin concentrations on a linear scale and (D,F) values for 𝑐𝑐 𝑘𝑘 > 0 normalized by their corresponding 𝑐𝑐 𝑘𝑘 = 0 value and shown on a log scale.
For each speed, we found a nonmonotonic dependence of 𝜅𝜅 on 𝑐𝑐 𝑘𝑘 , with the highest stiffness observed at 𝑐𝑐 𝑘𝑘 = 160 nM (Fig. 5C,D), consistent with our force analysis (Fig. 4); and structural assessments that showed a higher propensity for microtubule crosslinking and small-scale bundling at intermediate kinesin concentrations (Fig 2). Additionally, we found that the highest stiffnesses occurred at the fastest stage speeds, which may reflect the reduced ability for the network to relax on the timescale over which the strain is applied. Specifically, the presence of semiflexible actin filaments in the composites allows for actin bending modes to dissipate stress (47,48). As described in SI Section S2, the predicted relaxation rate associated with actin bending in our composites is 𝜏𝜏 𝑏𝑏 -1 ≈ 25 s -1 , which is comparable to the strain rate associated with our fastest speed, 𝛾𝛾̇≃ 3𝑣𝑣 √2𝑡𝑡𝑡𝑡 ≃ 23 s -1 ,(49) but faster than the two slower rates (~5.7 s -1 , ~11 s -1 ). Thus, we may expect an increased likelihood of stress dissipation on the timescale of the slowest strain rate (i.e. at 𝑣𝑣 = 6 µm s -1 ) compared to the fastest (i.e. at 𝑣𝑣 = 24 µm s -1 ). Consistent with this understanding, we find that the viscous drag, which is a measure of stress dissipation within the composite network, is higher at slower speeds, particularly at the highest kinesin concentration (Fig. 5E,F).
When we consider the mechanical results described above (Figs 345), in the context of the composite restructuring (Fig. 12), we see that the microtubule compaction and network demixing that causes dense small-scale clustering at intermediate kinesin concentrations also provides mechanical enhancement, as observed by the increase in both stiffness and maximum force. At higher kinesin concentrations the large-scale phase separation undermines and softens the elastic response. We now aim to quantitatively understand the relationship between de-mixing and structural heterogeneity and the non-monotonic dependence of local mechanics on kinesin concentration.
Our structural analysis shows varying degrees of clustering over a range of length scales from <10 µm to ~100 µm ( 〈𝐼𝐼〉 from the standard deviation 𝜎𝜎 𝐼𝐼 and mean 〈𝐼𝐼〉 of pixel intensities 𝐼𝐼. To quantify heterogeneity at local (< 𝑠𝑠) and global (> 𝑠𝑠) scales for a given kinesin concentration, we computed the mean and standard deviation of 𝛿𝛿 𝐼𝐼 across all tiles of all videos. The former and latter are measures of local heterogeneity, ℎ 𝐼𝐼 = 〈𝛿𝛿 𝐼𝐼 〉, and global heterogeneity, 𝐻𝐻 𝐼𝐼 = 𝜎𝜎(𝛿𝛿 𝐼𝐼 ), and their ratio 𝑝𝑝 𝐼𝐼 = 𝐻𝐻 𝐼𝐼 /ℎ 𝐼𝐼 is a measure of structural 'patchiness'. For reference, for a well-mixed system, there should be minimal global heterogeneity, i.e., all patches should be identical, so 𝑝𝑝 𝐼𝐼 should tend to zero. For fractal-like systems, the heterogeneity would be scale invariant, yielding 𝑝𝑝 𝐼𝐼 ≈ 1; and systems that are de-mixed on the scale of the patches, 𝑝𝑝 𝐼𝐼 > 1. E,F) Correlation plots that display relationships between mechanical (𝜅𝜅, 𝐹𝐹 𝑚𝑚𝑚𝑚𝑚𝑚 , 𝐺𝐺 ′ ) and structural (𝑙𝑙 0 , 𝑙𝑙 𝑚𝑚𝑖𝑖𝑚𝑚 ) parameters measured from experiments (filled symbols, black axis labels) and simulations (open symbols, grey axis labels). All experimental data shown is for 24 µm s -1 strains, 𝐹𝐹 𝑚𝑚𝑚𝑚𝑚𝑚 values are for the elastic traces, and simulated 𝐺𝐺 ′ data is for 1 Hz. We observed a non-monotonic dependence on 𝑐𝑐 𝑘𝑘 with peaks at 𝑐𝑐 𝑘𝑘 = 160 nM, consistent with our prior observations. The maximum in ℎ 𝐼𝐼 is a likely indicator of local bundling of filaments that we expect to stiffen the network by stiffening its constituents, consistent with the increased maximum force and stiffness we measured (Figs 3,4). The maximum in 𝐻𝐻 𝐼𝐼 is suggestive of larger scale de-mixing, and we found similarly high values for 𝑐𝑐 𝑘𝑘 = 640 nM, as we expected based on our structural analysis (Fig 2) and visual inspection of the videos (Fig 6A). However, at this highest kinesin concentration, local heterogeneity dropped while the patchiness remained high. Together, these features suggest that larger scale aggregation and aster formation, with patches that are largely either filled by a cluster or devoid of microtubules, dominate the force response. This broken connectivity substantially weakens the network; and the prevalence of filament-poor patches compared to clusterspanning patches tips the scales towards a softer mechanical response.
To verify this interpretation and couple structural and mechanical heterogeneity, we again turned to simulations, this time evaluating the distribution of forces exerted on filaments within 20 µm hexagonal tiles (Fig 6C), as fully described in the SI. We computed similar metrics to assess mechanical heterogeneity, replacing pixel intensity 𝐼𝐼 with force 𝑓𝑓 : ℎ 𝑓𝑓 = 〈𝛿𝛿 𝑓𝑓 = 𝜎𝜎 𝑓𝑓 /〈𝑓𝑓〉〉 and 𝐻𝐻 𝑓𝑓 = 𝜎𝜎�𝛿𝛿 𝑓𝑓 �, and 𝑝𝑝 𝐼𝐼 = 𝐻𝐻 𝐼𝐼 /ℎ 𝐼𝐼 . As shown in Fig 6D , in which we plot these metrics normalized by their values for the lowest kinesin concentration (i.e., 𝑐𝑐 𝑘𝑘 = 40 nM, metrics are not defined for 𝑐𝑐 𝑘𝑘 = 0 nM where 𝑓𝑓 = 0), we observed strong non-monotonic dependence on 𝑐𝑐 𝑘𝑘 consistent with our observations of the structural heterogeneity response shown in Fig 6B . However, and notably, all metrics were minimized for the 𝑐𝑐 𝑘𝑘 = 160 nM composite, with 𝐻𝐻 𝐼𝐼 and 𝑝𝑝 being most strongly suppressed. This increased homogeneity of forces throughout the composite is consistent with the presence of a well-connected network of stiff (bundled) fibers that can both efficiently distribute stress and provide strong elastic resistance. At higher 𝑐𝑐 𝑘𝑘 values, when demixing occurs at larger lengthscales, we observed a much higher patchiness of forces, signifying reduced mechanical connectivity, which is consistent with a weaker force response, loss of stiffness and enhanced yielding and viscous dissipation.
We have shown clear emergent elasticity of kinesin-driven composites in both experiments and simulations (Fig 6E,F). This response arises due to de-mixing of actin and microtubules and is a unique feature of the composite system (Fig S4). To summarize and provide further insight, we constructed correlation plots to depict the relationships between the structural correlation lengths obtained from SIA of experimental images with mechanical parameters of the network. Specifically, we compared the smaller structural lengthscale 𝑙𝑙 0 with stiffness 𝜅𝜅 (Fig 6E 𝑙𝑙 𝑚𝑚𝑖𝑖𝑚𝑚 values, we find that simulated and experimental data points at a given kinesin concentration loosely cluster with one another, except for the 𝑐𝑐 𝑘𝑘 = 640 nM case for reasons described above. These general features confirm that our simulations are capturing the key physics of the material system, and highlight the importance of coupling between structure and mechanics to produce the emergent behavior.
We have designed and characterized in vitro composites of actin filaments and microtubules undergoing active restructuring by kinesin motor clusters that pull on microtubules, and found them to transition from interpenetrating networks to de-mixed microtubule-rich aggregates and softer actin-rich phases. Despite this de-mixing, composites maintain structural integrity, without fracturing or completely phase-separating, and achieve steady-states that maintain viscoelastic mechanical properties. We have discovered that this restructuring, seen in both experiments and simulations, leads to rich dependence of the mechanical response on kinesin concentration, including a surprising emergence of enhanced stiffness and elasticity at intermediate kinesin concentrations. This mechanical emergence is coupled to enhanced structural heterogeneity across lengthscales. Importantly, we previously observed nonmonotonic dependence of mechanical stiffness on passive crosslinking of actin in cytoskeleton composites, suggesting this behavior may be a generalizable feature of crosslinking of one species of a composite. However, in these previous studies, there was no observable de-mixing, large-scale bundling or structural heterogeneity. Rather, the non-monotonic dependence was a result of modest microscale variations in mesh sizes and fiber stiffnesses. Here, our experiments and simulations demonstrate the importance of hierarchical structural and mechanical heterogeneity in sculpting the mechanical behavior, which we rationalize as a direct result of the internal stress generated by kinesin motors. The distribution of motorgenerated stresses measured in simulations mirrors that of the structural heterogeneity in experiments. Moreover, the stiffening behavior, a unique feature not previously reported in similar passive or active composites (28,35), also may indicate motor-generated pre-stress and densification that suppress filament bending and non-affine deformations that dissipate stress, thereby promoting a stiffening response. This interplay between structure and mechanics is likely critical to the multifunctionality of the cytoskeleton that allows for wide-ranging mechanical processes and dynamically sculpts mechanical properties in response to environmental cues and the needs of the cell. Our results and models shed important light on how to engineer and tune composite systems to exhibit emergent mechanics through independent tuning of elastic and viscous contributions of the composite constituents.
Protein Preparation: Rabbit skeletal actin (Cytoskeleton, Inc. AKL99) was reconstituted to 2 mg mL -1 in 5 mM Tris-HCl (pH 8.0), 0.2 mM CaCl2, 0.2 mM ATP, 5% (w/v) sucrose, and 1% (w/v) dextran. Porcine brain tubulin (Cytoskeleton, Inc. T240) and HiLyte488-labeled porcine brain tubulin (Cytoskeleton, Inc. TL488M-A) were reconstituted to 5 mg mL -1 with 80 mM PIPES (pH 6.9), 2 mM MgCl2, 0.5 mM EGTA, and 1 mM GTP. All cytoskeleton proteins were flash frozen single-use aliquots and stored at -80ºC. Biotinylated kinesin-401 (21,50) was expressed in Rosetta (DE3) pLysS competent E. coli cells (ThermoFisher), purified, and flash-frozen into single-use aliquots, as described previously (33). To prepare force-generating kinesin clusters, kinesin-401 dimers were incubated with NeutrAvidin (ThermoFisher) at a 2:1 ratio in PEM-100 buffer (100 mM PIPES, 2 mM MgCl2, 2 mM EGTA) supplemented with 4 µM DTT for 30 min at 4°C. Clusters were prepared fresh and used within 24 hrs.
Composite Sample Preparation: Composites of actin filaments and microtubules at a 45:55 molar ratio, were polymerized by combining 1.35 µM actin monomers, 1.55 µM tubulin dimers, and 0.1 µM HiLyte488 tubulin dimers in PEM-100 (100 mM PIPES, 2 mM MgCl2, 2 mM EGTA) supplemented with 0.1% Tween, 10 mM ATP, 4 mM GTP, 5 µM Taxol, 1.08 µM phalloidin, and 0.27 µM ActiStain488 phalloidin (Cytoskeleton, Inc. PHDG1-A) and incubating for 1 hr at 37°C. The fluorophores for both microtubules and actin were chosen to be spectrally similar to allow both filaments to be visible in a single field-of-view with the same excitation/emission filter combination, a requirement due to the fact that the epifluorescence microscope outfitted with our optical tweezers can only accommodate a single fluorescence channel at a time. For optical tweezers experiments, 0.02% (v/v) of 4.5 µm diameter carboxylated microspheres (Polysciences, Inc.), coated with BSA to inhibit non-specific interactions with the network, were added (28). For two-color confocal microscopy measurements, HiLyte488 tubulin dimers were replaced with rhodamine tubulin dimers (Cytoskeleton, Inc. TL590M) to allow for separate imaging of actin and microtubules in different channels of the confocal microscope (see below for additional details).
Following polymerization, and immediately prior to experiments, an oxygen scavenging system (45 µg mL -1 glucose, 43 µg mL -1 glucose oxidase, 7 µg mL -1 catalase, 0.005% β-mercaptoethanol) was added to reduce photobleaching, followed by kinesin clusters at final kinesin concentrations of 𝑐𝑐 𝑘𝑘 = 0, 40, 80, 160, 320, and 640 nM.
For both optical tweezers and confocal experiments, the final sample was gently flowed into a sample chamber made from a glass slide and coverslip separated by ~100 µm of double-stick tape to accommodate ~10 µL. Both the glass slide and coverslip of the chamber were passivated with BSA prior to flowing in the sample. The chamber was sealed with UV curable glue to prevent sample leakage and evaporation. This process completed ~5 mins after the addition of kinesin to the sample, and the sealed sample was incubated for an additional 25 mins to allow for motor-driven restructuring prior to measurements.
The composite mesh size 𝜉𝜉 is determined from the mesh sizes for the actin and microtubule networks comprising the composite, 𝜉𝜉 𝐴𝐴 ≃ 1.46𝑐𝑐
𝐴𝐴 -1/2 ≃ 1.26 µm and 𝜉𝜉 𝑀𝑀 ≃ 2.68 𝑐𝑐 𝑂𝑂 -1/2 ≃ 2.15 µm, where 𝑐𝑐 𝐴𝐴 and 𝑐𝑐 𝑂𝑂 are the molarities of actin and tubulin, via the relation 𝜉𝜉 ≃ (𝜉𝜉 𝐴𝐴 3 + 𝜉𝜉 𝑂𝑂 3 ) -1/3 ,(48) yielding a composite mesh size of 𝜉𝜉 ≃ 1.19 µm. The ratio of kinesin clusters to tubulin 𝑅𝑅 = 𝑐𝑐 𝑘𝑘 4𝑐𝑐 𝑇𝑇
= 0, 0.006, 0.012, 0.024, 0.048 and 0.097 for 𝑐𝑐 𝑘𝑘 = 0 -640 nM, where the 4 accounts for the ~4 kinesins (two dimers) per motor cluster.
OTM experiments were performed using an optical trap formed by a 1064 nm Nd:YAG fiber laser (Manlight), focused with a 60× 1.4 NA objective (Olympus), and custom-built around an Olympus IX71 epifluorescence microscope (41,51). The force was measured using a position-sensing detector (Pacific Silicon Sensor) to record the deflection of the trapping laser, which over the operating range used within the reported experiments, is proportional to the force acting on the trapped microsphere. The proportionality constant that provides the trap stiffness 𝑘𝑘 𝑂𝑂𝑂𝑂 was determined to be 𝑘𝑘 𝑂𝑂𝑂𝑂 ≃ 68 pN/µm using the Stokes drag method (40,41,51). Imaging of the labeled filaments and probes was achieved using a broadband LED source (XCITE) with 488nm/525nm excitation/emission filters and a Hamamatsu ORCA-Flash 4.0LT CMOS camera with a 1024 x 1024 square-pixel field-of-view and frame rate of 20 s -1 .
For each force measurement, an optically trapped microsphere was dragged back and forth in the ±𝑥𝑥-direction through the sample in a sawtooth pattern using a nanopositioning piezoelectric stage (Mad City Labs) that moves the sample chamber relative to the fixed trap (Fig 3a). The distance the probe was moved in each half-cycle (i.e., the strain amplitude) and total perturbation time was fixed at 𝑠𝑠 = 20 µm and 𝑡𝑡 𝑓𝑓 = 50 s for all measurements. The stage position and laser deflection were recorded at 20 kHz, and the stage position was updated at 400 Hz using custom-written National Instruments LabVIEW programs. Measurements were performed at 3 different for each kinesin concentration: 𝑣𝑣 = 6, 12, 24 µm s -1 . Prior to each strain and force measurement, an image of the labeled filaments surrounding the probe in a 202 µm x 135 µm (900 x 600 pixel) area centered at the center of the strain path was captured. The same protocol was repeated but using a piezoelectric mirror to move the trapped bead relative to the sample (keeping the sample chamber fixed), and recording 1000-frame videos of the labeled filaments. These videos were used in the image analysis presented in Figs 2 and 6.
For each (𝑣𝑣, 𝑐𝑐 𝑘𝑘 ) combination, measurements were repeated using multiple beads located in different regions of the sample chamber, which were separated by ≥100 µm. This suite of measurements for each condition was then repeated for three different samples for a total of ≥24 beads per condition. While we performed cyclic straining for each measurement, in which the same probe was repeatedly pulled through the material, we found that a significant number of probes were lost on the return after the initial pull, and that for those that were not lost, the measured force ranges for subsequent pulls were considerably smaller, suggesting that the initial pull caused network damage or plastic deformation. Thus, data presented in Figs 3-6 are solely for the initial loading cycle.
Post-acquisition analysis of measured force 𝐹𝐹(𝑥𝑥, 𝑡𝑡) (Figs 3456) was performed using customwritten MATLAB scripts. Each 50 s measurement was divided into individual cycles and the second half of each cycle, when the probe is moving in the -𝑥𝑥 direction, was removed. The last 1% of the forward cycle (0.2 µm) was also removed to avoid artifacts that arise from attempting to instantaneously switch the stage motion from +𝑣𝑣 to -𝑣𝑣 (the stage response rate is 400 Hz). Average data shown for each condition represent averages over all valid trials and error bars represent standard error. Trials were considered invalid if the bead was pulled out of the trap.
To characterize the composite structure, we performed Spatial Image Autocorrelation (SIA) analysis (39,51) on each frame of each of the 1000-frame videos described above. SIA measures the correlation in intensity 𝑔𝑔 𝐼𝐼 of two pixels in an image as a function of separation distance 𝑟𝑟. Autocorrelation curves 𝑔𝑔 𝐼𝐼 (𝑟𝑟) were generated by taking the fast Fourier transform of the image 𝐹𝐹(𝐼𝐼), multiplying by its complex conjugate, applying an inverse Fourier transform 𝐹𝐹 -1 , and normalizing by the squared intensity: 𝑔𝑔 𝐼𝐼 (𝑟𝑟) = 𝐹𝐹 -1 ��𝐹𝐹�𝐼𝐼(𝑡𝑡)�� 2 �
[𝐼𝐼(𝑡𝑡)] 2
. To determine the effect of motor activity on the structure we subtract 𝑔𝑔 𝐼𝐼 (𝑟𝑟) for the no-motor case (𝑐𝑐 𝑘𝑘 =0) from 𝑔𝑔 𝐼𝐼 (𝑟𝑟) for each kinesin concentration: ∆𝑔𝑔 𝐼𝐼 (𝑟𝑟, 𝑐𝑐 𝑘𝑘 ) = 𝑔𝑔 𝐼𝐼 (𝑟𝑟, 𝑐𝑐 𝑘𝑘 ) -𝑔𝑔 𝐼𝐼 (𝑟𝑟, 0), which we show in Fig 2 . We observed no dependence on strain speed so the data shown is the average and standard error across all frames of all videos for a given kinesin concentration 𝑐𝑐 𝑘𝑘 .
To determine the unperturbed composite structure and dynamics, videos of composites with distinctly-labeled actin and microtubules were recorded using a Nikon A1R laser scanning confocal microscope with a 60× 1.4 NA oil-immersion objective (Nikon), 488 nm laser with 488/525 nm excitation/emission filters (to excite/image actin), and 561 nm laser with 565/591 nm excitation/emission filters (to excite/image microtubules) (Fig 1). For each kinesin concentration, four time-series (videos) of 512 × 512 square-pixel (213 µm × 213 µm) images were collected at 1.86 fps for a total of 1116 frames (10 mins). Videos were collected at 10, 20, 30 and 40 mins after the addition of kinesin motors, with each video collected in a different field of view separated by >500 µm. We observed no dependence of the composite structure on the time that the video was acquired, indicating the motor activity is largely halted after 10 mins. All videos include two channels that separate the actin and microtubule signals such that they can be processed separately and compared.
Computational Model: We have recently developed a simple Lattice-gas model of microfilament composites which adequately captures experimentally observable filament restructuring in the composite (see Ref 33). Here, we build on this model to predict the restructuring of the composites, and the resulting changes in composite mechanics due to varying levels of motor activity. In this model, the available space is defined as a hexagonal grid with periodic boundary conditions. Each grid point can be occupied by a single microtubule or single actin filament center or can be empty. The filaments can interact with neighbors within reach, via 1) motor-generated forces that can either pull the interacting filaments towards each other or push them away from each other; and 2) motor crosslinks that increase the friction forces on the interacting filaments and allow forces to be transmitted through crosslinked filament clusters. The movement of a filament center to a neighboring grid point within a small temporal time step is then a stochastic event whose probability can be calculated using the transition rate based on the first passage times. We purposefully choose a minimal approach to capture the dynamics to shed light on the competing factors of activity and friction. Our model assumes a single length for all filaments of 𝑙𝑙 = 5 µm while in experiments actin and microtubules assume distributions of lengths (𝑙𝑙 𝐴𝐴 ≃ 4 ± 3 µm and 𝑙𝑙 𝑀𝑀 ≃ 8 ± 4 µm) (48). We treat all filaments as rigid rods while actin in experiments is semiflexible with a persistence length of ~17 µm. Our model also assumes uniform motor density throughout the simulation space, which might locally under or overestimate motor generated forces within the composites. Our simulations are in 2D while experimental composites span 3D.
We implement our model on a 100 µm x 100 µm 2D space with a hexagonal lattice, where the lattice spacing is 1.25 µm. Each 5-µm long filament interacts with other filaments located within 4 grid points in all directions. Initially, each lattice point is either occupied by a single microtubule center, a single actin filament center, or is left empty using probabilities matching the average volume fraction occupied by these elements. The initial orientations of microtubules and actin filaments are randomly distributed. We enforce that there is a single occupancy per lattice site for filament centers, but the extended length of each filament allows for interactions with other filaments within its interaction radius. At any snapshot in time each filament has a specific orientation that is a function of the net force on the filament due to motor forces between interacting filaments.
The movement of the filaments is simulated in each iteration by calculating the likelihood of each possible movement, 𝑝𝑝 𝑖𝑖𝑖𝑖 for all grid points i and j, where at least one of them contains a filament center, and randomly picking one of these movements to occur based on these probabilities. The simulation is run for 𝑇𝑇 𝑆𝑆 = 5 minutes, which we find is sufficient to reach quasisteady state. The model calculations and simulations are coded in python and the scripts are available on GitHUB (52). A cartoon depiction of the model is shown in Figs 1 and S1 and numerical values for all model parameters are included in Table S1.
To quantify the degree of clustering and segregation of the different filaments, we compute the filament pair distribution function 𝑔𝑔 𝐴𝐴𝑀𝑀 (𝑟𝑟 , 𝑇𝑇 ) where the subscripts 𝐴𝐴 and 𝑀𝑀 represent the