Motor processivity and speed determine structure and dynamics of microtubulemotor assemblies
Abstract
Active matter systems can generate highly ordered structures, avoiding equilibrium through the consumption of energy by individual constituents. How the microscopic parameters that characterize the active agents are translated to the observed mesoscopic properties of the assembly has remained an open question. These active systems are prevalent in living matter; for example, in cells, the cytoskeleton is organized into structures such as the mitotic spindle through the coordinated activity of many motor proteins walking along microtubules. Here, we investigate how the microscopic motormicrotubule interactions affect the coherent structures formed in a reconstituted motormicrotubule system. This question is of deeper evolutionary significance as we suspect motor and microtubule type contribute to the shape and size of resulting structures. We explore key parameters experimentally and theoretically, using a variety of motors with different speeds, processivities, and directionalities. We demonstrate that aster size depends on the motor used to create the aster, and develop a model for the distribution of motors and microtubules in steadystate asters that depends on parameters related to motor speed and processivity. Further, we show that network contraction rates scale linearly with the singlemotor speed in quasionedimensional contraction experiments. In all, this theoretical and experimental work helps elucidate how microscopic motor properties are translated to the much larger scale of collective motormicrotubule assemblies.
Editor's evaluation
Banks et al. demonstrate that the organization and dynamics of microtubule/kinesin asters depend upon the speed and processivity of motors. By combining in vitro reconstitutions with theory, they are able to extract parameters that relate to the dynamics of the motors. This study is of interest to readers working on microtubules, motors, and in the active matter physics field.
https://doi.org/10.7554/eLife.79402.sa0Introduction
A signature feature of living organisms is their ability to create beautiful, complex patterns of activity, as exemplified in settings as diverse as the famed flocks of starlings in Rome or the symmetrical and dazzling microtubule arrays that separate chromosomes in dividing cells (Popkin, 2016). While such organization in nature has long captured the attention of artists and scientists alike, many questions remain about how the patterns and structures created by living organisms arise. In active systems such as bird flocks or microtubulemotor arrays, energy is consumed at the local level of the individual actors, and constituents move based on interactions with their neighbors. These local actions create patterns at scales hundreds to thousands of times larger than the individual constituent. How the specific microscopic activity of each individual leads to the final largescale assembly formed remains an open question in these systems from the molecular to organismal level.
The motormicrotubule system is an excellent system in which to test this question, as many motor proteins with a variety of properties, such as speeds, stall and detachment forces, processivities, and directionalities, exist in nature. These motors play a variety of roles in cells; some transport cargo while others localize to distinct regions of the mitotic spindle (Wickstead and Gull, 2006; Mann and Wadsworth, 2019; Endow et al., 1994; Pavin and Tolić, 2021; AnjurDietrich et al., 2021). Studies have investigated how the microscopic properties of these motors makes them uniquely suited to their cellular role. For example, kinesin1s high speed and processivity make it excellent at transporting cargo (Grover et al., 2016; Hirokawa et al., 1991). However, in in vitro systems, kinesin1 tetramers are able to form asters, extensile networks, and contractile networks (Surrey et al., 2001; Sanchez et al., 2012; DeCamp et al., 2015; Ross et al., 2019). Ncd (kinesin14) and Kif11 (kinesin5) have similarly been shown to form asters in vitro, yet it remains unclear how the properties of these motors affect the structure and dynamics of the assemblies created (Surrey et al., 2001; Sanchez et al., 2012; DeCamp et al., 2015; Ross et al., 2019; Roostalu et al., 2018).
In this work, we create motormicrotubule structures with a variety of motors and develop theoretical models to connect the motor properties to the organization and dynamics of the assemblies. Our recently developed optogenetic in vitro motormicrotubule system demonstrated the formation of asters and other contractile networks with kinesin1 (K401) upon light activation (Ross et al., 2019). Briefly, we fused K401 motors to an optogenetic pair of lightdimerizable proteins, such that in the presence of light the optogenetic pair bind, acting as a crosslink between microtubules that the motor heads are walking along. Previously, we showed that this scheme enabled us to form microtubule structures with spatiotemporal control by illuminating regions of the sample at will. We now show how this system can be repurposed to ask a new set of questions with kinesin5 (Kif11) and kinesin14 (Ncd), and form asters of varying sizes with each motor, demonstrating lightcontrolled aster formation with these motors for the first time. Our controlled structure formation with these various motors enabled us to develop a theoretical model connecting the distribution of motors and microtubules in asters that depends on microscopic motor properties. We find that calculated motor distributions in an aster depend on the motor properties and fit with our experimental data. Further, by using motors with different speeds, we find that contraction rates in quasionedimensional microtubule networks directly depend on the singlemotor velocity. This theoretical and experimental work sheds light on the ways that microscopic motor properties are reflected in the 1000fold larger length scale of motormicrotubule assemblies.
Results
Aster size depends on motor used
We build on the foundational work that demonstrated the ability to control motormicrotubule systems with light (Ross et al., 2019) to consider a new set of motors with different fundamental properties. In brief, kinesin motors are fused to the lightdimerizable pair iLid and micro. In the absence of light, motor dimers walk along microtubules but do not organize them; upon activation with light, the motor dimers couple together to form tetramers, crosslinking the microtubules they are walking along as shown in Figure 1A. The optogenetic bond lasts for about 20 s before reverting to the undimerized state, thus in our experiments, we repeatedly illuminate the sample every 20 s (Guntas et al., 2015). As demonstrated by Ross et al., projecting a cylinder of light on the sample results in the formation of an aster, and different structures can be formed and manipulated by illumination with different light patterns. For the purposes of this study, we were careful to remain in a regime of motor and microtubule concentrations that produced a single aster upon illumination. However, by varying concentrations of the motors and microtubules, it is possible to form multiple smaller asters within the region, a few examples of which are shown in Figure 1—figure supplement 1. How varying the composition of the reaction mixture impacts the resulting structures warrants further investigation.
In this work, we aim to determine how the properties of the motor affect the resulting structures. While experiments with this system were previously performed with Drosophila melanogaster kinesin1 motors (K401) (Ross et al., 2019), in the present work, we investigate if other kinesin motor species with different intrinsic properties such as speed and processivity would lead to lightinducded microtubule organization. Toward this end, we use the same lightdimerizable scheme to form microtubule structures with two other motors: Ncd (D. melanogaster kinesin14) and Kif11 (Homo sapiens kinesin5). The singlemolecule properties of all three motors we use are summarized in Table 1. We measure the speed of each motor species by gliding assays (SI section ‘Gliding assay’); the processivities are based on literature values. Further, we fluorescently label the motors using mVenus or mCherry to visualize the motors and microtubules in separate imaging channels within the same assay (see Supplementary file 1).
As seen in Figure 1B and Figure 1—figure supplement 2, each of these motors is able to form asters of varying sizes in our system. It was previously unclear whether there were limits to speed, processivity, or stall force that might prevent any of these motors from forming asters in our lightcontrolled system, although Ncd has previously been shown to form asters as constitutive oligomers (Belmonte et al., 2017; Hentrich and Surrey, 2010). We found that all were able to form asters upon illumination by various excitation diameters ranging from 50 to 600 μm. Interestingly, the dynamics of aster formation by these motors seemed to roughly scale with the motor speeds – K401 formed asters the quickest, followed by Ncd236, and Kif11 took the most time to form an aster.
We sought to determine if there are discernible differences between the asters formed with the various motors. First, we measured the size of the asters using the distribution of fluorescently labeled microtubules, which peaks in the center of the aster and generally decreases moving outward, as shown in Figure 1C. We tend to observe a shoulder in the microtubule distribution (around 20 in the example in Figure 1C). This is around the size of the disordered aster core, which is discussed in SI section ‘Disordered aster core’. We defined the outer radius of the aster as the radius at which the microtubule fluorescence is twice the background microtubule concentration (see Figure 1C for an example aster outer radius determination). We found that this method agreed well with a visual inspection of the asters (Figure 1—figure supplement 2).
We found that aster size increases with excitation diameter, as shown in Figure 1B, D, consistent with what was shown by Ross et al. for K401 (Ross et al., 2019). In Ross et al., it was determined that the aster size roughly scaled with the volume of the excitation area, suggesting that the number of microtubules limits the size of the aster. This hints that there may be a density limit to the microtubules in an aster. Interestingly, we find that the size of the asters also depends on the motor used. For each excitation diameter, except for the 50 μm case, K401 formed the largest asters and Ncd formed the smallest, with Kif11 producing asters of intermediate size (Figure 1D). What is it about the different motors that confers these different structural outcomes? We found that this trend correlates with motor processivity; K401 is the most processive, followed by Kif11, and then Ncd. This is similar to the findings in Surrey et al., 2001, in which intensity of aster formation was related to motor processivity. Other factors could also be contributing to aster size such as the ratio of microtubules to motors as was suggested by Surrey et al., 2001, but not investigated in the present work, or the motor stall force.
Spatial distribution of motors in asters
The nonuniform distribution of filaments and motors in an aster is a key feature of its organization and has been the subject of previous studies. In these studies, continuum models were developed for motorfilament mixtures which predicted the radial profile of motors in confined twodimensional systems (Nédélec et al., 2001; Lee and Kardar, 2001; Sankararaman et al., 2004; Aranson and Tsimring, 2006). A notable example is the powerlaw decay prediction by Ndlec et al., who obtained it for a prescribed organization of microtubules obeying a $1/r$ decay law (Nédélec et al., 2001). Measuring the motor profiles in asters formed in a quasitwodimensional geometry (with the $z$dimension of the sample being only a few microns deep) and fitting them to a powerlaw decay, the authors found a reasonable yet noisy match between the predicted and measured trends in the decay exponent.
In our work, we also develop and test a minimal model that predicts the motor profile from the microtubule distribution and the microscopic properties of the motor. Our welldefined asters of various sizes shown above, created with varying motor properties yields an opportunity for us to test how our model and assess how the microscopic motor properties are translated to the aster scale. Additionally, in contrast to the earlier study (Nédélec et al., 2001), asters formed in our experiments are threedimensional due to the much larger depth of the flow cells (roughly 100 μm), and are thus more similar to the structures observed in cells. While the largest asters we form are likely partially compressed in the $z$direction, we assume that this effect does not significantly alter the protein distributions in the central $z$slice. For modeling purposes, then, we consider our asters to be radially symmetric outside the central disordered region (which we refer to as the aster core, as depicted schematically in Figure 2A). The core has a typical radius of $\approx 15$ μm, beyond which microtubules have a predominantly polar organization (see SI section ‘Disordered aster core’ for the discussion of the two aster regions and an example PolScope image that demonstrates their distinction).
Similar to the treatment in earlier works (Nédélec et al., 2001; Sankararaman et al., 2004; Aranson and Tsimring, 2006), we introduce two states of the motor – an unbound state where the motor can freely diffuse with a diffusion constant $D$ and a bound state where the motor walks toward the aster center with a speed $v$, depicted in Figure 2B. In the steady state of the system, which we assume our asters have reached at the end of the experiment, microtubules on average have no radial movement and hence, do not contribute to motor speed. To assess the validity of this assumption, we performed fluorescence recovery after photobleaching (FRAP) experiments of steadystate asters, and observe little radial flux of the microtubules, an example is shown in Appendix 1—figure 4. They are still dynamic, as can be seen by the angular motion that leads to the recovery of fluorescence in the photobleached areas. We denote the rates of motor binding and unbinding by ${k}_{\text{on}}$ and ${k}_{\text{off}}$, respectively. When defining the firstorder rate of motor binding, namely, ${k}_{\text{on}}{\rho}_{{}_{\text{MT}}}(r)$, we explicitly account for the local microtubule concentration ${\rho}_{{}_{\text{MT}}}(r)$ extracted from fluorescence images. This is unlike the previous models which imposed specific functional forms on the microtubule distribution (e.g., a constant value [Lee and Kardar, 2001; Sankararaman et al., 2004] or a powerlaw decay [Nédélec et al., 2001]), rendering them unable to capture the specific features often seen in our measured microtubule profiles, such as the presence of an inflection point (see Figure 1C for an example).
From these assumptions, the governing equations for the bound (${m}_{\text{b}}$) and free (${m}_{\text{f}}$) motor concentrations are shown in Figure 2C. They involve binding and unbinding terms, as well as a separate flux divergence term for each population. Solving these equations at steady state, we arrive at an equation for the total local concentration of motors defined as ${m}_{\text{tot}}(r)={m}_{\text{b}}(r)+{m}_{\text{f}}(r)$. The derivation of this result can be found in SI section ‘Model formulation’. As seen in the equation for ${m}_{\text{tot}}(r)$ (Figure 2C), knowing the microtubule distribution ${\rho}_{{}_{\text{MT}}}(r)$ along with two effective microscopic parameters, namely, the effective dissociation constant ${K}_{\text{d}}={k}_{\text{off}}/{k}_{\text{on}}$ and the length scale ${\lambda}_{0}=D/v$, we can obtain the motor distribution up to a multiplicative constant ($C$ in the equation). Note that in the special case where the motors do not move ($v\to 0$ or ${\lambda}_{0}\to \mathrm{\infty}$), the exponential term becomes 1 and an equilibrium relation between the motor and microtubule distributions dependent only on ${K}_{\text{d}}$ is recovered, as we would expect for an equilibrium system.
To test this model, we extract the average radial distributions of microtubule and motor concentrations for each aster. Then, using the microtubule profile as an input, we fit our model to the motor data and infer the effective parameters ${K}_{\text{d}}$ and ${\lambda}_{0}$ (see SI sections ‘Extraction of concentration profiles from raw images’ and ‘Model fitting’ for details). A demonstration of this procedure on an example Kif11 aster is shown in Figure 2D where a good fit to the average motor data can be observed. As a validation of our inference method, we additionally extract the radial concentration profiles inside separate wedges of the aster and show that they can be accurately captured by only choosing an appropriate multiplicative constant $C$ for each wedge, while keeping the pair $({K}_{\text{d}},{\lambda}_{0})$ inferred from average profile fixed. Fits to 5 out of 16 different wedge profiles are shown in Figure 2D for clarity. The fitting error for other asters is similarly low (Figure 2E, see Figure 2—figure supplement 1 for the collection of fitted profiles).
Plotting the inferred parameters ${K}_{\text{d}}$ and ${\lambda}_{0}$ from all fits, shown in Figure 2F, we find that they are clustered around single values for each motor type and vary between the motors. Based on the singlemolecule motor properties in Table 1 and the reported motor binding rates (Valentine and Gilbert, 2007), our expectation was that the ${K}_{\text{d}}$ values for Kif11 and K401 would have a ratio of $\approx 4.6:1$ (see SI section ‘Expected ratio of K_{d} values for K401 and Kif11 motors’), while ${K}_{\text{d}}$ for Ncd would be the highest due to its nonprocessivity. The ratio of median inferred ${K}_{\text{d}}$ values for Kif11 and K401 is $\approx 5.6:1$ – close to our expectation. However, the inferred ${K}_{\text{d}}$ values for Ncd are low and comparable to those for K401.
One possible resolution of this discrepancy comes from the finding of an in vitro study suggesting a substantial increase in the processivity of Ncd motors that act collectively (Furuta et al., 2013). Specifically, a pair of Ncd motors coupled through a DNA scaffold was shown to have a processivity reaching 1 μm (or, $\approx 100$ steps) – a value close to that reported for K401 motors. A highly processive movement was similarly observed for clusters of HSET (human kinesin14) (Norris et al., 2018) and plant kinesin14 motors (Jonsson et al., 2015).
This collective effect, likely realized for Ncd tetramers clustered on microtubules in highly concentrated aster structures, is therefore a possible cause for the low inferred values of their effective ${K}_{\text{d}}$. We also note that while a similar collective effect on processivity was observed for K401 motors (Furuta et al., 2013), it is far less dramatic since their singlemotor processivity is already about the length of our microtubules, and therefore would have a small effect on the effective ${K}_{\text{d}}$.
Next, looking at the inference results for the ${\lambda}_{0}$ parameter (Figure 2F), we can see that Kif11 and Ncd motors have an average ${\lambda}_{0}$ value of $\approx 1020$ μm, while the average value for K401 motors is $\approx 40$ μm. From the measured diffusion coefficient of $D\approx 1$ μm^{2}/s for tagged kinesin motors (Grover et al., 2016) and the singlemolecule motor speeds reported in Table 1, our rough estimate for the ${\lambda}_{0}$ parameter for Kif11 and Ncd motors was $\approx 1015$ μm, and $\approx 2$ μm for K401. While the inferred values for the two slower motors are well within the orderofmagnitude of our guess, the inferred ${\lambda}_{0}$ for the faster K401 motor is much higher than what we anticipated. This suggests a significant reduction in the effective speed. One contributor to this reduction is the stalling of motors upon reaching the microtubule ends. Recall that in our model formulation (Figure 2B) we assumed an unobstructed walk for bound motors. Since the median length of microtubules ($\approx 1.6$ μm) is comparable to the processivity of K401 motors ($\approx 1$ μm), stalling events at microtubule ends will be common, leading to a reduction of their effective speed in the bound state by a factor of $\approx 1.5$ (see SI section ‘Accounting for finite MT lengths’ for details). This correction alone, however, is not sufficient to capture the factor of $\approx 25$ discrepancy between our inference and the estimate of ${\lambda}_{0}$. We hypothesize that an additional contribution may come from the jamming of K401 motors in dense aster regions. This is motivated by the experiments which showed that K401 motors would pause when encountering obstructions during their walk (Ferro et al., 2019; Schneider et al., 2015). In contrast, for motors like Ncd and Kif11 which take fewer steps before unbinding and have a larger effective K_{d}, jamming would have a lesser effect on their effective speed as they would unbind more readily upon encountering an obstacle. Overall, our study shows that the minimal model of motor distributions proposed in Figure 2 is able to capture the distinctions in aster structure through motorspecific effective parameters, although more work needs to be done to explain the emergence of higherorder effects such as motor clustering and jamming, and their contribution to these effective parameters.
Our model also provides insights on the observation that the distribution of microtubules is generally broader than that of the motors. This feature can be observed by comparing the two example profiles in Figure 2D, and it also holds for the profiles extracted from other asters (Figure 2—figure supplement 1). In SI section ‘Broader spread of the tubulin profile’, we use our model to demonstrate this feature in a special analytically tractable case, and discuss its generality across asters in greater detail. We found that the relative width of the motor distribution compared to the microtubule distribution is fairly constant among asters, with their difference being the largest for Ncd motors, consistent with our model predictions. This relationship between the shapes of the distributions may be an important factor in the spatial organization of enddirected motors in the spindle where their localization to the spindle pole is of physiological importance.
Contraction rate scales with motor speed
Besides the steadystate structure of motormicrotubule assemblies, it is also of great interest to understand their dynamics. Ross et al. demonstrated the formation of quasionedimensional contractile networks by creating two asters that are initially separated by a given distance, then activating a thin rectangular region between them to form a network connecting the asters that pulls them together (Ross et al., 2019). Example images of microtubule fluorescence during one of these experiments are shown in Figure 3a. In these experiments, an increase in maximum aster merger speed with greater initial separation was observed. We aimed to confirm this behavior with our various motors and to test the relationship between aster merger speed and singlemotor speed.
First, we tested the relationship between distance and speed in our experiments. Using optical flow to measure the contraction speed throughout the network, we observe a linear increase in contractile speed with distance from the center of the network, as shown in Figure 3b. This relationship suggests that the contractile network can be thought of as a series of connected contractile units. These findings are in agreement with results from several studies of contractile rates in actomyosin networks, that suggested telescoping models of contraction, and suggest that this may be a common mechanism across cytosketetal networks (Thoresen et al., 2011; Linsmeier et al., 2016; Schuppler et al., 2016). Independent contraction of each unit would generate the observed linear increase in speed because more contractile units are added with distance from the center of the network.
Next, we investigated how contractile speeds vary by motor in aster merger experiments. We repeated aster merger experiments with each motor and with various initial separations between the asters. Figure 3c shows the results, where each point represents the maximum aster speed measured in a single experiment by tracking the aster, and the lines are linear fits to the data for each motor (see SI section ‘Merger analysis’ for details on measuring the aster speeds). Interestingly, the ratios of the slopes of these lines match the ratios of motor speeds from Table 1. For example, the slope of the best fit line for Ncd is $\approx 0.0023{\mathrm{s}}^{1}$ and the best fit slope for Kif11 is $\approx 0.0013{\mathrm{s}}^{1}$. The ratio of these (Ncd/Kif11) is $\approx 1.8$, which closely matches the ratio of their singlemotor speeds ($\approx 1.6$). Similar calculations can be done with these two motors and K401, with the same result. Thus, we conclude that the rate of contraction in the network is set by the motor speed and the increase of network speed with distance is due to adding more connected contractile units (Figure 3—figure supplement 1). It is important to note that we only measure the initial contraction rate in these experiments, over the first ≈100 s. On this time scale, we think of the network as an elastic material, since the time for motor unbinding and rearrangement should be longer than this. The time scale for the optogenetic pair to unbind is about 20 s (Guntas et al., 2015); since binding events are independent then the average time for two to unbind is about 400 s. There are likely several motors bound between any two microtubules within the network, thus the relaxation time, or the time to observe the viscous properties of the system, will be the time for multiple motors to unbind or optogenetic links to rearrange. Therefore, to account for the change in contraction rate throughout the process, one would likely need to account for these viscous effects.
Discussion
In this work, we examined how the properties of kinesin motors determine the mesoscopic properties of the structures they create. The way in which quantities such as motor speed and processivity govern the nature of the resulting motormicrotubule structures has been an open question. Previous attempts have been made to address this question, however most of these are in the context of a single motor. While models could be developed that fit the properties they measure, these models could not be systematically tested since they did not vary the speed or processivity of the motor (Nédélec et al., 2001; Lee and Kardar, 2001; Sankararaman et al., 2004; Aranson and Tsimring, 2006). By varying motor speeds, processivities, and directionalities, we were able to quantify and model how these microscopic parameters connect to the properties of mesoscopic structures.
We demonstrate lightcontrolled aster formation with three different motors. Interestingly, the final aster size from a given illumination region varied depending upon which motor was used. Our leading hypothesis is that the key control variable is the processivity of the motors. Future work needs to be done to understand this effect and build models to explain it. Early work by Surrey et al. found that processivity affected the intensity of aster formation in simulations, which may be related to our observations, but to the best of our knowledge no model of this effect has been developed (Surrey et al., 2001). Further, we assess the distribution of motors and microtubules in the asters we form and develop a model of the steadystate aster that predicts the motor distribution given the measured microtubule distribution, with parameters that relate to the motor speed and processivity. Interestingly, the parameters we infer in some cases differ from those we would expect from the singlemolecule properties of the motors, indicating that higherorder effects such as cooperativity in collections of motors increasing the processivity of the collective, are playing important roles. In addition, we measure contraction speeds in pseudoonedimensional networks and find that the speeds are related to the singlemotor speed. In all, this work takes a step toward a mechanistic understanding of motormicrotubule assemblies, translating microscopic properties of individual interactions to the observed properties of the much largerscale assemblies. This begins to open the door to understanding how different motors and tubulins interact to form cellular scale structures with varying properties, a critical question in evolutionary biology.
Materials and methods
Cloning of motor proteins
Request a detailed protocolHuman kinesin5 (Kif11/Eg5) 5–513 was PCR amplified from mCherryKinesin11N18 plasmid (gift from Michael Davidson, Addgene # 55067). This fragment was previously shown to form functional dimers (Valentine et al., 2006). Kinesin 1 1–401 (K401) was PCR amplified from pWC2 plasmid (Addgene # 15960). Ncd 236–701 was PCR amplified from a plasmid gifted by Andrea SerraMarques.
The optogenetic proteins, iLid, and Micro were PCR amplified from pQE80L iLid (Addgene # 60408, gift from Brian Kuhlman) and pQE80L MBPSspB Micro (Addgene # 60410). mCherry was PCR amplified from mCherryKinesin11N18 and mVenus was PCR amplified from mVenus plasmid (Addgene # 27793).
Constructs were assembled by Gibson assembly of the desired motor protein, optogenetic protein, and fluorophore in order to make the plasmids listed in Supplementary file 1.
Protein expression and purification
Request a detailed protocolProtein expression and purification was done in SF9 cells. Cells were seeded at a density of 1,000,000 cells per mL in a 15 mL volume and transiently transfected with the desired plasmid using Escort IV transfection reagent, then incubated for 72 hr before purification. Cells were collected for purification by centrifugation at 500 × g for 12 min, and the pellet was resuspended in lysis buffer (200 mM NaCl, 4 mM MgCl_{2}, 0.5 mM EDTA, 1.0 mM EGTA 0.5% Igepal, 7% sucrose by weight, 20 mM imidazole pH 7.5, 10 g/mL aprotinin, 10 g/mL leupeptin, 2 mM ATP, 5 mM DTT, 1 mM PMSF) and incubated on ice for 30 min. The lysate was then clarified by centrifugation at 200,000 × g for 30 min at 4°C. Clarified supernatant was incubated with 40 L antiFLAG M2 affinity gel (SigmaAldrich A2220) for 3 hr at 4°C. To wash out unbound protein, the resin (with bound protein) was collected by centrifugation at 2000 × g for 1 min, the supernatant was removed and the resin was washed with wash buffer (for Ncd and Kif11: 150 mM KCl, 5 mM MgCl_{2}, 1 mM EDTA, 1 mM EGTA, 20 mM imidazole pH 7.5, 10 g/mL aprotinin, 10 g/mL leupeptin, 3 mM DTT, 3 mM ATP; for K401: M2B with 10 g/mL aprotinin, 10 g/mL leupeptin, 3 mM DTT, 3 mM ATP). This was repeated two more times with decreasing ATP concentration (0.3 and 0.03 mM ATP) for a total of three washes. After the third wash, about 100 L supernatant was left and the bound protein was eluted by incubation with 10 L FLAG peptide (SigmaAldrich F3290) at 4°C for 3 hr. The resin was then spun down by centrifugation at 2000 × g for 1 min and the supernatant containing the purified protein was collected. Purified protein was then concentrated to a volume of 10–20 L by centrifugation in mini spin filters (Millipore 50 kDa molecular weight cutoff). Protein was kept at 4°C and used the same day as purification or stored in 50% glycerol at –20°C for longer storage. Protein concentration was determined with QuBit Protein Assay Kit (Thermo Fisher Q33212).
Microtubule polymerization
Request a detailed protocolMicrotubules were polymerized as reported previously (Ross et al., 2019) and originally from the Mitchison lab website (Georgoulia, 2012). In brief, 75 μM unlabeled tubulin (Cytoskeleton) and 5 μM tubulinAlexa Fluor 647 (Cytoskeleton) were combined with 1 mM DTT and 0.6 mM GMPCPP in M2B buffer and incubated spun at 300,000 × g to remove aggregates, then the supernatant was incubated at 37°C for 1 hr to form GMPCPP stabilized microtubules.
Microtubule length
Request a detailed protocolThe GMPCPP stabilized microtubules were imaged with TIRF microscopy to determine their length. A flow chamber was made using a KOH cleaned slide, KOH cleaned coverslip (optionally coated with polyacrylamide), and parafilm cut into chambers. The flow cell was incubated with polyLlysine for 10 min, washed with M2B, then microtubules were flown in. The chamber was sealed with Picodent and imaged with TIRF microscopy.
Microtubules were segmented using homewritten Python code and histogrammed to determine the distribution of microtubule lengths (Appendix 1—figure 1).
Sample chamber preparation
Request a detailed protocolSlides and coverslips were cleaned with Helmanex, ethanol, and KOH, silanized, and coated with polyacrylamide as in Ross et al., 2019. Just before use, slides and coverslips were rinsed with MilliQ water and dried with compressed air. Flow chambers (3 mm wide) were cut out of Parafilm M and melted using a hotplate at 65°C to seal the slide and coverglass together, forming chambers that are $\approx 70100$ μm in height and contain ≈7 μL.
Reaction mixture preparation
Request a detailed protocolThe reaction mixture consisted of kinesin motors ($\sim 250\mathrm{nM}$), microtubules (∼1 μM tubulin), and energy mix that contained ATP, an ATP recycling system, a system to reduce photobleaching, F127 pluronic to reduce interactions with the glass surfaces, and glycerol (Ross et al., 2019). To prevent preactivation of the optogenetic proteins and photobleaching of the fluorophores, the motors and microtubules were always handled in a dark room where wavelengths of light below 520 nm were blocked with a filter or a red light was used to illuminate. The reaction mixture was prepared right before loading into the flow cell and then sealed with Picodent Speed.
Microscope instrumentation
Request a detailed protocolWe performed the experiments with an automated widefield epifluorescence microscope (Nikon TE2000). We custom modified the scope to provide two additional modes of imaging: epiilluminated pattern projection and LED gated transmitted light. We imaged light patterns from a programmable DLP chip (EKB TEchnologies DLP LightCrafter E4500 MKIITM Fiber Couple) onto the sample through a usermodified epiillumination attachment (Nikon TFL). The DLP chip was illuminated by a fiber coupled 470 nm LED (ThorLabs M470L3). The epiillumination attachment had two lightpath entry ports, one for the projected pattern light path and the other for a standard widefield epifluorescence light path. The two light paths were overlapped with a dichroic mirror (Semrock BLP01488R25). The magnification of the epiilluminating system was designed so that the imaging sensor of the camera (FliR BFLYU323S6MC) was fully illuminated when the entire DLP chip was on. Experiments were run with MicroManager (Edelstein et al., 2010), running custom scripts to controlled pattern projection and stage movement.
Activation and imaging protocol
Request a detailed protocolFor the experiments in which we make asters with excitation disks of different sizes, we use five positions within the same flow cell simultaneously in order to control for variation within flow cells and over time. Each position is illuminated with a different sized excitation region: 50, 100, 200, 400, or 600 μm diameter cylinder. Each position was illuminated with the activation light for $\sim 50200$ ms and both the microtubules (Cy5 labeled) and motors (mVenus labeled) were imaged at ×10 magnification every 15 s. After an hour of activation, a zstack of the microtubule and motor fluorescence throughout the depth of the flow chamber was taken at 5 μm increments in each position. Typically, one experiment was run per flow chamber. We placed the time limitations on the sample viewing to minimize effects related to cumulative photobleaching, ATP depletion, and global activity of the lightdimerizable proteins. After several hours, inactivated dark regions of the sample begin to show bundling of microtubules.
For the aster merger experiments, two 50 μm disks are illuminated at different distances apart. Again, five positions within the same flow cell are chosen, and the separation between the two disks varies for each position: 200, 400, 600, 800, or 1000 μm apart. For K401 and Ncd experiments, the disks are illuminated for 30 frames, for Kif11 experiments, the disks are illuminated for 60 frames (frames are every 15 s). Then, a bar ≈5 μm wide connecting the disks is illuminated to merge the asters.
Gliding assay
Request a detailed protocolMotor speeds were determined by gliding assay. Glass slides and coverslips were Helmanex, ethanol, and KOH cleaned. Flow cells with ≈10 L volume were created with double sided sticky tape, and rinsed with M2B buffer. Then, antiGFP antibody was applied and incubated for 10 min. The flow cell was then rinsed with M2B and then mVenus labeled motor proteins (at $\sim 5$ nM in M2B) were flowed in and incubated for 10 min. The flow cell was rinsed with M2B to remove unbound motors and microtubules (in M2B with 3 mM ATP and 1 mM DTT) were flowed in. Microtubules were then imaged using total internal reflection fluorescence (TIRF) microscopy at a rate of one frame per second. Individual microtubules were tracked using customwritten Python code to determine their speed. The mean microtubule speed (excluding those that were not moving) was determined as the motor speed. Appendix 1—figure 2 shows the histogram of speeds obtained for K401 motors purified from SF9 cells.
Forming a single aster
Request a detailed protocolThe experiments for this study were performed in a regime in which we obtained a single aster, in order to measure properties of the structure and compare between various motors. However, by varying concentrations of components within the system such as motor and microtubule concentration, it is possible to obtain different results. Some examples include mini asters everywhere in the sample before activation, a few asters within the activation region, or many small asters within the activation region. Example images of these cases are shown in Figure 1—figure supplement 1. The various possible resulting structures, and the perturbations to the system to obtain them, warrant further study in the future.
Disordered aster core
Request a detailed protocolWe observe that the asters we create have centers that are very dense with motors and microtubules. By fluorescence microscopy, we do not observe organized aster arms in this region and hypothesized that the microtubules are disordered in this region. To assess the extent of microtubule organization in our asters, we imaged asters with a polarized light microscope (PolScope). This microscope utilizes polarized white light to image birefringent substances. Microtubules are birefringent due to their aspect ratio; they interact differently with light polarized parallel to their long axes compared to light polarized perpendicular to their long axes. Thus, the PolScope allows determination of the alignment of microtubules, but not their plus/minus end polarity (Oldenbourg and Mei, 1995). When imaged with a PolScope, the arms of our asters are bright, indicating high alignment, and their azimuthal angle confirms that they are radially symmetric around the center (Appendix 1—figure 3). The center of the aster is dark, which we interpret to mean that this region is disordered. It is possible that the microtubules in the center could be aligned pointing in the zdirection, which could also result in the center being dark. A disordered center may be a result from steric hindrance due to a high density of microtubules in that region, which prevents the motors from aligning the microtubules. Due to the disorder in the aster center, we exclude this region from our theoretical analysis.
Steadystate aster
Request a detailed protocolWhen we measure aster size and the distributions of motors and microtubules, we do so once the aster has reached a dynamic steady state. To assess this notion, we performed FRAP experiments. We photobleached the microtubules in a fully formed aster in a grid pattern and took images of the recovery, shown in Appendix 1—figure 4. From these images, it is clear that there is little to no net radial movement of the microtubules, and slow angular motion. Therefore, at this point, the aster is no longer contracting, but the microtubules are still dynamic. We attempted similar experiments with the motors, but they recover very quickly.
Aster size
Request a detailed protocolWe formed asters of various sizes using cylindrical illumination regions, ranging in diameter from 50 to 600 μm. Figure 1—figure supplement 2 shows representative images of the microtubule fluorescence of asters formed with each motor and each excitation diameter. The yellow circles are byhand determination of the outer boundary of the aster. It can be seen in these images that for the smallest asters, the aster size can be larger than the illumination region. We believe this is due to diffusion of dimerized motors outside of the activation region, where they can bind microtubules in the background region and incorporate them into the asters. For larger excitation diameters, the microtubules in the activated region get concentrated in the aster, leaving a region depleted of microtubules between the aster and the background region. In order to measure the size of asters in a more systematic way, we used the measured microtubule distribution, as shown in Figure 1C. Outside of the central core region of the aster, microtubule fluorescence decreases monotonically before rising again to the background level outside of the activation region. We chose to define the outer radius of the aster as the radius at which the median microtubule fluorescence is twice the background fluorescence. This metric agrees well with a visual inspection of the asters.
Appendix 1
Motor distributions
Model formulation
To predict the spatial distribution of motors in aster structures, we model the dynamic steady state of free (f) and bound (b) motor concentrations via
Here, ${k}_{\text{on}}$ and ${k}_{\text{off}}$ are the motor binding and unbinding rates, respectively, ${\rho}_{{}_{\text{MT}}}(\mathbf{r})$ is the spatially varying microtubule concentration at steady state (measured as μM tubulin), ${\mathbf{J}}_{v}$ is the advective flux of bound motors, and $\mathbf{J}}_{\mathrm{D}$ is the diffusive flux of free motors. Our modeling approach is similar to that used by Nédélec et al., 2001, with the main difference being in the handling of ${\rho}_{{}_{\text{MT}}}(\mathbf{r})$. Namely, they imposed a particular functional form on this distribution (${\rho}_{{}_{\text{MT}}}(\mathbf{r})=1/{\mathbf{r}}^{d1}$ with $d$ as the spatial dimension) based on an idealized representation of microtubule organization in an aster, whereas in our treatment ${\rho}_{{}_{\text{MT}}}(\mathbf{r})$ stands for the experimentally measured microtubule profiles which cannot be captured through an analogous idealization.
If the free motors have a diffusion coefficient $D$, then, in the radially symmetric setting considered in our modeling, the diffusive flux will be given by
where $\widehat{\mathbf{r}}$ is an outwardpointing unit radial vector. And if $v$ is the advection speed of bound motors, then the advective motor flux on radially organized microtubules will be
Here, we are implicitly assuming that motors constantly walk when bound, ignoring the fact that they can stall upon reaching a microtubule end. We discuss the impact of this effect later in ‘Accounting for finite MT lengths’.
At steady state, the net flux of motors at any radial distance $r$ should be zero (${\mathbf{J}}_{\mathrm{D}}(r)+{\mathbf{J}}_{v}(r)=0$), which implies a general relation between the profiles of free and bound motors, namely,
Above we introduced ${\lambda}_{0}$ as a length scale parameter that can be interpreted as the distance which is traveled by free and bound motors at similar time scales, that is, diffusion time scale (${\lambda}_{0}^{2}/D$)=advection time scale (${\lambda}_{0}/v$). Note also that the − sign at the righthand side indicates that the free motor population should necessarily have a decaying radial profile (${m}_{\text{f}}^{\mathrm{\prime}}(r)<0$) which is intuitive since at steady state the outward diffusion needs to counteract the inward advection.
To make further analytical progress, we will assume that motor binding and unbinding events are locally equilibrated (Aranson and Tsimring, 2006). This assumption is valid if motor transport is sufficiently slow compared with binding/unbinding reactions. We will justify this quasiequilibrium condition for the motors used in our study at the end of the section. It follows from this condition that
where ${K}_{\text{d}}={k}_{\text{off}}/{k}_{\text{on}}$ is the dissociation constant. Since the experimental readout reflects the total motor concentration (${m}_{\text{tot}}={m}_{\text{f}}+{m}_{\text{b}}$), we use our results (Equation 5 and Equation 6) to link ${m}_{\text{tot}}(r)$ with the microtubule profile ${\rho}_{{}_{\text{MT}}}(r)$. Specifically, using Equation 6 we find
Next, substituting the above expressions for ${m}_{\text{f}}$ and ${m}_{\text{b}}$ into Equation 5 and simplifying, we relate the motor and microtubule profiles, namely,
where ${R}_{{}_{\text{MT}}}(r)=\int {\rho}_{{}_{\text{MT}}}(r)dr$ is the integrated microtubule concentration, and $C={K}_{\text{d}}{e}^{{C}_{1}}$ is a positive constant. The presence of the multiplicative constant $C$ is a consequence of the fact that the two equations used for deriving our result (Equation 5 and Equation 6) specify the ratios of motor populations. Therefore, the result in Equation 10 predicts the relative level of the total motor concentration, given the two effective model parameters (${K}_{\text{d}}$ and ${\lambda}_{0}$), which we infer in our fitting procedure.
Note that the two variable factors on the righthand side of Equation 10 have qualitatively different structures. The first one is local and depends only on the dissociation constant (an equilibrium parameter), while the second term involves an integrated (hence, nonlocal) microtubule density term and ${\lambda}_{0}=D/v$ which depends on the advection speed $v$ (a nonequilibrium parameter). As anticipated, in the limit of vanishingly slow advection ($v\to 0$ or, ${\lambda}_{0}\to \mathrm{\infty}$) the second factor becomes 1 and an equilibrium result is recovered.
Connections to related works
Before proceeding further into our analysis, we briefly compare the expression for the motor distribution (Equation 10) with analogous results in the literature. Specifically, Nédélec et al., 2001, studied quasitwodimensional asters and in their modeling treated microtubules as very long filaments, all converging at the aster center. This setting implied $\sim 1/r$ scaling of the microtubule concentration. With this scaling, the integrated microtubule concentration in our framework becomes ${R}_{{}_{\text{MT}}}(r)=\int \frac{\alpha}{r}dr=\alpha \mathrm{ln}r$, where $\alpha $ is a constant. Substituting this form into the exponential term in Equation 10, we find $\mathrm{exp}\{{({K}_{\text{d}}{\lambda}_{0})}^{1}{R}_{{}_{\text{MT}}}(r)\}=\mathrm{exp}\{\alpha {({K}_{\text{d}}{\lambda}_{0})}^{1}\mathrm{ln}r\}\sim 1/{r}^{\beta}$, where $\beta =\alpha {({K}_{\text{d}}{\lambda}_{0})}^{1}$. It then follows from Equation 10 and the scaling ${\rho}_{{}_{\text{MT}}}(r)\sim 1/r$ that the motor concentration is a sum of two decaying power laws (namely, $\sim 1/{r}^{\beta}$ and $\sim 1/{r}^{\beta +1}$) the result obtained by Nédélec et al., 2001. A more detailed calculation can be done to demonstrate that the exponent $\beta $ matches exactly with the result derived in the earlier work, but for the purposes of our study we do not elaborate further on this comparison. We note that the experimentally measured microtubule profiles in asters (e.g., Figure 2B or Figure 3d) often have an inflection point and cannot be fitted to decaying powerlaw functions (e.g., $1/{r}^{2}$ for 3D asters), which is why the idealized setting considered by Nédélec et al., 2001, cannot be applied to our system.
Another set of works (Lee and Kardar, 2001; Sankararaman et al., 2004) also studied motor distributions in asters, but this time under the assumption of a uniform microtubule concentration (${\rho}_{{}_{\text{MT}}}(r)\sim \text{constant}$). In such a setting, our framework predicts an exponentially decaying motor profile, because ${R}_{{}_{\text{MT}}}(r)=\int {\rho}_{{}_{\text{MT}}}(r)dr\sim {\rho}_{{}_{\text{MT}}}r$ and thus, ${m}_{\text{tot}}(r)\sim {e}^{{\rho}_{{}_{\text{MT}}}r/{K}_{\text{d}}{\lambda}_{0}}$. An exponential decay was also the prediction of Lee and Kardar, 2001, although in their treatment all motors were assumed to be in the bound state. The two distinct motor states were considered in the work by Sankararaman et al., 2004, who predicted an exponential decay of motor concentration modulated by a powerlaw tail. One can show, however, that when the decay length scale of motor concentration greatly exceeds the motor processivity (as in the case of asters which we generated), the prediction of Sankararaman et al., 2004, also reduces into a pure exponential decay, matching the prediction of our model. But since the assumption of a uniform microtubule profile is clearly violated in our system, these predictions are not applicable for us.
Validity of the quasiequilibrium assumption
Earlier in the section, we assumed that motor binding and unbinding reactions were locally equilibrated, from which Equation 6 followed. Looking at the governing equation of bound motor dynamics (Equation 1), we can see that this assumption will hold true if ${k}_{\text{off}}{m}_{\text{b}}(r)\gg \mathrm{\nabla}\cdot {\mathbf{J}}_{\mathrm{v}}$. Substituting the expression of advective flux (Equation 4) and recalling that in three dimensions the divergence of a radial vector $\mathbf{A}=A\hat{\mathbf{r}}$ takes the form ${r}^{2}{\mathrm{\partial}}_{\mathrm{r}}({r}^{2}A)$, we rewrite the quasiequilibrium condition as
Now, many of the motor profiles can be approximated reasonably well by an exponentially decaying function (see Figure 2—figure supplement 1 for a collection of experimental profiles). This suggests an empirical functional form ${m}_{\text{b}}(r)\sim {e}^{r/\lambda}$ for the concentration of bound motors, where $\lambda $ is the decay length scale (note that the constant saturation level contributes to the free motor population). This functional form implies that ${m}_{\text{b}}^{\prime}(r)\approx {m}_{\text{b}}(r)/\lambda $, which, upon substituting into Equation 11, leads to
where $\lambda}_{\mathrm{v}}=v/{k}_{\text{off}$ is introduced as the motor processivity (distance traveled before unbinding). The processivities (${\lambda}_{v}$) of the three different kinesins used in our study together with the observed ranges of decay length scales ($\lambda $) of corresponding motor profiles are listed in Supplementary file 2. As can be seen, in all cases the ratio $\lambda /{\lambda}_{v}$ is much greater than one, verifying the intuitive expectation that the length scales of aster structures are much greater than the single run lengths of motors.
It is obvious from the presence of the ${r}^{1}$ term on the righthand side of Equation 13 that the condition can only be satisfied past a certain radius, since ${r}^{1}$ becomes very large when $r$ approaches zero. This threshold radius (${r}^{*}$) is set by ${r}^{*}\sim 2{\lambda}_{v}$, where the two sides of Equation 13 become comparable. The threshold radial distance that we choose to isolate the core is at least $510$ μm for the asters of our study (see the lower $x$limits in the profiles of Figure 2—figure supplement 1) which exceed ${r}^{*}$ at least a few times. This suggests that Equation 13 is valid, justifying our use of the quasiequilibrium assumption for modeling the motor distribution.
Extraction of concentration profiles from raw images
In this section, we describe our approach for extracting the radial profiles of motor and microtubule concentrations from raw fluorescence images.
Fluorescence normalization and calibration
When taking images with a microscope, several sources contribute to the detected pixel intensities: the camera offset, autofluorescence from the energy mix, and fluorescence coming from the tagged proteins (tubulin or motors). In addition, due to the uneven illumination of the field of view, the same protein concentration may correspond to different intensities in the raw image.
We begin the processing of raw images by first correcting for the uneven illumination. For microtubule images, we use the first movie frames as references with a uniform tubulin concentration in order to obtain an intensity normalization matrix. Each pixel intensity of the final image frame is then rescaled by the corresponding normalization factor.
Although the motor concentration is also initially uniform, the light activation region in the first frame appears photobleached, making it unsuitable for the construction of a normalization matrix. Instead, we obtain this matrix from the final frame, after masking out the neighboring region of the aster, outside of which the nonuniformity of the fluorescence serves as a proxy for uneven illumination. Intensity normalization factors inside the masked out circular region are obtained through a biquadratic interpolation scheme. The steps leading to a normalized motor image are depicted in Appendix 1—figure 5a.
After fluorescence normalization, we convert intensities into units of protein concentration using calibration factors estimated from images of samples with known protein contents. For K401 and Kif11 motors, we use the conversion 1000 intensity units $\to 815$ nM motor dimer. For Ncd dimers, which have fluorescent tags on both iLid and Micro units, we use the 1000 intensity units $\to 407$ nM conversion. In all three cases, 200 ms exposure time is used in the imaging. For tubulin, we make a rough estimate that after spinning the energy mix with tubulin, around 1 μM of tubulin remain, all of which polymerize into GMPCPP stabilized microtubules. This leads to the calibration of 360 intensity units $\to 1$ μM tubulin (100 ms exposure time).
Aster center identification
In the next step of the profile extraction pipeline, we crop out the aster region from the normalized image and identify the aster center in an automated fashion. In particular, we divide the aster into 16 equal wedges, calculate the radial profile of motors within each wedge, and define the aster center as the position that yields the minimum variability between the motor profiles extracted from the different wedges. Having identified the center, a mean radial profile for the aster is defined as the average of the 16 wedge profiles (Appendix 1—figure 5b).
Inner and outer boundary determination
Since our modeling framework applies to regions of the aster where the microtubules are ordered, we consider the concentration profiles in a limited radial range for the model fitting procedure. As we do not have a PolScope image for every aster to precisely identify the disordered core region, we prescribe a lower threshold on the radial range by identifying the position of the fastest intensity drop and adding to it a buffer interval (equal to 15% of the outer radius) to ensure that the region of transitioning from the disordered core into the ordered aster arms is not included (Appendix 1—figure 5c, top panel). As for the outer boundary, we set it as the radial position where the tubulin concentration exceeds its background value by a factor of two (Appendix 1—figure 5c, bottom panel).
Model fitting
Here, we provide the details of fitting the expression we derived for the motor distribution (Equation 10) to the profiles extracted from aster images. Since smaller asters are typically irregular and hence, do not meet the polar organization and radial symmetry assumptions of the model, we constrain the fitting procedure to larger asters formed in experiments with a minimum light illumination disk diameter of 200 μm.
The different aspects of the fitting procedure are demonstrated in Appendix 1—figure 6. Extracting the average tubulin and motor profiles, we fit our model to the motor profile and obtain the optimal values of the effective parameters ${K}_{\text{d}}$ and ${\lambda}_{0}$. With the exception of a few cases, the optimal pair $({K}_{\text{d}},{\lambda}_{0})$ corresponds to a distinct peak in the residual landscape (Appendix 1—figure 6c, note the logarithmic scale of the colorbar), suggesting that the parameters are well defined. The fit to the motor data for the example aster is shown in Appendix 1—figure 6d.
As stated in the main text, we then use the data from the separate aster wedges to assess the quality of fit for each aster. Specifically, keeping the $({K}_{\text{d}},{\lambda}_{0})$ pair inferred from the average profile fixed, we fit the 16 separate wedge profiles by optimizing over the scaling coefficient $C$ for each of the wedges, and use the model residuals to assess the fit quality. In the set shown in Appendix 1—figure 6e, with the exception of wedge 9 which contains an aggregate near the core radius, fits to all other wedge profiles are of good quality, translating into a low fitting error reported in Figure 2E of the main text.
Repeating this procedure for all other asters, we obtain the best fits to their motor profiles and the corresponding values of the optimal $({K}_{\text{d}},{\lambda}_{0})$ pairs. The collection of all average profiles, along with the best model fits and inferred parameters are shown in Figure 2—figure supplement 1.
Expected ratio of ${K}_{\text{d}}$ values for K401 and Kif11 motors
Here, we show the steps in estimating the expected ratio of ${K}_{\text{d}}$ values for the motors K401 and Kif11. Recall the definition ${K}_{\text{d}}={k}_{\text{off}}/{k}_{\text{on}}$. Knowing the motor processivities and speeds from Table 1, we calculate the offrates as ${k}_{\text{off}}=\text{speed}/\text{processivity}$. This yields a ratio ${k}_{\text{off}}^{\text{Kif11}}/{k}_{\text{off}}^{\text{K401}}\approx 1.15$. Then, using the reported onrates in Valentine and Gilbert, 2007, we find the ratio of onrates to be ${k}_{\text{on}}^{\text{Kif11}}/{k}_{\text{on}}^{\text{K401}}\approx 0.25$. Taken together, these two results lead to our estimate for the ratio of ${K}_{\text{d}}$ values reported in the main text, namely, ${K}_{\text{d}}^{\text{Kif11}}/{K}_{\text{d}}^{\text{K401}}=({k}_{\text{off}}^{\text{Kif11}}/{k}_{\text{off}}^{\text{K401}})/({k}_{\text{on}}^{\text{Kif11}}/{k}_{\text{on}}^{\text{K401}})\approx 4.6$.
Accounting for finite MT lengths
Analysis of purified microtubule images shows that the median length of microtubules is $\approx 1.6$ μm (Appendix 1—figure 1). Taking the size of a tubulin dimer to be 8 nm, this length translates into the distance traveled in $\approx 200$ motor steps, which is comparable to the processivity of K401 motors reported in Table 1 of the main text. Since motors stall when reaching microtubule ends, their effective advection speed will get reduced. Here, we account for this reduction and estimate its magnitude for the different motors used in our study.
Consider the schematic in Appendix 1—figure 3 where a motor is shown advecting on a microtubule with length $L$. If the distance $x$ between the motor and microtubule end at the moment of binding is less than the motor processivity ${\lambda}_{v}$, then the motor will reach the end and stall for a time period $\tau =1/{k}_{\text{off}}^{\mathrm{end}}$ before unbinding. On the other hand, if $x$ is greater than ${\lambda}_{v}$, the motor will not stall while bound to the microtubule and hence, its effective speed will not be reduced.
Assuming that the location of initial binding is uniformly distributed in the $[0,L]$ interval (hence, the chances of binding between $x$ and $x+\mathrm{d}x$ is $\mathrm{d}x/L$), we can calculate the effective speeds in the above two cases as
As can be seen, in both cases the effective speed is lower than the walking speed $v$. Now, if $p(L)$ is the distribution of microtubule lengths, then the mean effective motor speed evaluated over the whole microtubule population becomes
We calculate this effective speed for each motor numerically using the measured distribution $p(L)$.
The endresidence time $\tau $ was measured for rat kinesin1 motors to be $\approx 0.5\text{s}$ (Belsham and Friel, 2019). We take this estimate for our K401 motors (D. melanogaster kinesin1) and since, to our knowledge, there is no available data on endresidence times for Ncd and Kif11 motors, we use the same estimate for them (we note that in simulation studies too the endresidence time is typically guessed Surrey et al., 2001).
Using this $\tau $ estimate, the measured distribution $p(L)$, and the motor speed ($v$) and processivity ($\lambda}_{\mathrm{v}$) values from Table 1, we numerically evaluate the relative decrease in the effective speeds of the motors as
Here, we made two estimates for Ncd, first using its singlemolecule processivity ($\approx 1$ step) reported in Table 1, and then the 100fold increased processivity potentially reached due to collective effects mentioned in the main text. As a consequence of this effective speed reduction, we expect factors of $\approx 1.41$, $\approx 1.06$, and $\approx 1.01(1.41)$ increase in the inferred ${\lambda}_{0}$ values of K401, Kif11, and Ncd motors, respectively.
Broader spread of the tubulin profile
In this section, we discuss the feature of a broader tubulin distribution in greater detail. To gain analytical insights, we first consider an idealized scenario where the motor profile can be represented as an exponential decay with a constant offset for the free motor population (Appendix 1—figure 8a). Such a scenario is approximately met for many of our measured motor profiles. Using our modeling framework, we find that the microtubule distribution corresponding to such a motor profile has the shape of a truncated sigmoid (Appendix 1—figure 8b, see the second part of this section for the derivation). Indeed, microtubule distributions resembling a sigmoidal shape are observed often in our asters (Figure 2—figure supplement 1), two examples of which are shown in Appendix 1—figure 8c.
One notable implication of this analytical connection between the two profiles is that microtubules necessarily have a broader distribution than motors, once the offset levels at the aster edge are subtracted off. To find whether this is a ubiquitous feature of our asters, we introduce radial distances ${r}_{1/2}^{(\mathrm{m})}$ and ${r}_{1/2}^{(\mathrm{t})}$ standing for the positions where the motor and tubulin distributions, respectively, are at their midconcentrations (Appendix 1—figure 8d). The ratio $\left({r}_{1/2}^{(\mathrm{t})}{r}_{\mathrm{in}}\right)/\left({r}_{1/2}^{(\mathrm{m})}{r}_{\mathrm{in}}\right)$, if greater than 1, would then be an indicator of a wider tubulin profile. Calculating this ratio for all of our asters, we find that it is always greater than 1 for all motor types (Appendix 1—figure 8e), suggesting the generality of the feature.
Idealized scenario with an exponentially decaying motor profile
Here, we first derive the analytical form for the tubulin distribution in the idealized scenario where the motor profile can be approximated as an exponential decay. We then demonstrate that, when normalized, this distribution is always broader than the motor distribution.
We start off by writing the motor distribution as
where $\lambda $ is the decay length scale, ${m}_{\mathrm{\infty}}$ is the background motor concentration corresponding to the free motor population, and $\mathrm{\Delta}m$ is the amplitude of the exponential decay. Next, using Equation 5 as well as the definition ${m}_{\mathrm{tot}}(r)={m}_{\text{b}}(r)+{m}_{\text{f}}(r)$, we set out to obtain the distributions of bound and free motor populations. From Equation 5, we have
Solving for ${m}_{\text{b}}(r)$, we find
where ${C}_{\text{b}}$ is an integration constant. Because the approximation Equation 21 applies to a finite radial interval $r\in [{r}_{\mathrm{in}},{r}_{\mathrm{out}}]$, the constant ${C}_{\text{b}}$ is generally nonzero. It specifies the relative contributions of free and bound motor populations to the total motor distribution ${m}_{\text{tot}}(r)$.
The free motor population is found by simply subtracting Equation 24 from Equation 21, that is,
Having obtained expressions for the two motor populations (bound and free), we now recall Equation 6 that relates these two populations through the local tubulin density. Using Equation 6, we find the tubulin density as
where we introduced the effective parameter $\gamma \equiv ({m}_{\mathrm{\infty}}/\mathrm{\Delta}m)(1+{\lambda}_{0}/\lambda )$. Equation 26 represents a partial sigmoid, the precise shape of which in the $r>{r}_{\mathrm{i}\mathrm{n}}$ region is defined through the parameter $\gamma $ (Appendix 1—figure 8b).
To formally demonstrate that the tubulin profiles predicted in Equation 26 are necessarily broader than the motor profile, we first normalized them after subtracting off the concentration values at the outer boundary, namely,
The local ratio of normalized tubulin and motor densities then becomes
which is always greater than 1 in the $r>{r}_{\mathrm{i}\mathrm{n}}$ region. This is indicative of the shoulder that the normalized tubulin profile often forms over normalized motor profile and demonstrates the broader spread of the tubulin distribution in this idealized setting.
Relative widths of the two distributions
We can see from Appendix 1—figure 8e that the relative widths of the motor and microtubule distributions differ most for Ncd motors (median ratio ≈ 1.55), while for K401 and Kif, the widths are more comparable (median ratio ≈ 1.35 for both motors). Here, we offer an explanation for this difference between the motors using the analytical insights developed earlier in the section.
Specifically, Equation 29 and Appendix 1—figure 8b suggest that lower values of $\gamma $ correspond to broader microtubule distributions. Substituting $r={r}_{\mathrm{in}}$ in Equation 26, we can write $\gamma $ as
where $\stackrel{~}{{\rho}_{0}}={\rho}_{MT}({r}_{\mathrm{i}\mathrm{n}})/{K}_{\text{d}}$ is the microtubule concentration near the core in units of ${K}_{\text{d}}$.
From the concentration profiles in Figure 2—figure supplement 1, we can estimate the motor decay length scale to be $\lambda \approx 20,15,10$ μm for K401, Kif11, and Ncd, respectively. Then, from our model fitting procedure, we inferred ${\lambda}_{0}\approx 40,15,20$ μm, resulting in length scale ratios ${\lambda}_{0}/\lambda \approx 2,1,2$ for the three motors. Lastly, again inspecting the profiles in Figure 2—figure supplement 1, we find the microtubule concentrations near the core in ${K}_{\text{d}}$ units to be ${\stackrel{~}{\rho}}_{0}\approx 0.5,0.2,1.3$. Note that the microtubule concentration near the core (in ${K}_{\text{d}}$ units) is the highest for Ncd. In the final step, we substitute these estimated values for ${\lambda}_{0}/\lambda $ and ${\stackrel{~}{\rho}}_{0}$ into Equation 30, and evaluate $\gamma $ for the K401, Kif11, and Ncd, respectively, to be $\gamma \approx \{3,4,0.3\}$.
This matches well with the intuition of our simple analytical study (Appendix 1—figure 8b) and the observed ratios of distribution widths (Appendix 1—figure 8e). Namely, large values of $\gamma $ for K401 and Kif11 (≈ 3 and 4, respectively) suggest a closer correspondence between the normalized motor and microtubule profiles, while the lower values of $\gamma $ for Ncd ($\approx 0.3$) suggests a wider microtubule distribution, as was observed in our asters.
Merger analysis
To measure contractile speeds in a pseudo onedimensional network, we performed aster merger experiments. In these experiments, two asters were formed with 50 μm diameter excitation disks, at varying initial separations (either 200, 400, 600, 800, or 1000 μm). The two disks were illuminated for 30 or 60 frames for experiments with Kif11 (15 s between frames). After 30 or 60 frames, a thin rectangle was illuminated between the two asters. This caused the formation of a network which contracted, pulling the two asters together. An image of two asters connected in this way is shown in Figure 3.
We used two methods to measure contractile rates. One was using optical flow to measure speeds throughout the network, as shown in Figure 3b. The open source package, Open CV, was used for the optical flow measurements. We used a dense optical flow measurement using cv.CalcOpticalFlowFarneback, which calculates optical flow speeds using the Gunnar Farneback algorithm.
To calculate the maximum contractile rate, we tracked the positions of the asters over time. The microtubule fluorescence in the region containing the asters and network (pixels 450–650 in the ydirection) was summed across the ydimension to make a onedimensional line trace of fluorescence (Appendix 1—figure 10a). This trace was smoothed using a butter filter to reduce the noise in the fluorescence intensity. An example smoothed line trace is shown in Appendix 1—figure 10a. The asters were then identified as the large peaks in fluorescence intensity using scipy.signal.find_peaks. The identified peaks are shown as red dots in Appendix 1—figure 10. The corresponding image that the line trace is from is shown in Appendix 1—figure 10b with a 50×50 pixel box around the identified aster colored in black. From the identified aster coordinates, the distance between the asters was calculated and the difference between distances in successive frames was used to calculate the merger speed. The reported speed is the speed of a single aster. Thus, the speed calculated from the change in distance between asters is divided by two, assuming the asters are moving at equal speed.
Example traces of calculated aster speeds over time after the beginning of illuminating the bar region is shown in Appendix 1—figure 10c. The speed of the asters increases while the network connecting them forms, peaks, and then decreases as the asters near each other. The peak speed is reported as the maximum aster speed in the main text. These maximum aster speeds as a function of initial separation were fit to a line using scipy.optimize.curve_fit to determine the slope of the increase in speed with separation. The best fit lines determined with this method are plotted along with the data for Ncd and Kif11 in Figure 3c. The fit slope for Ncd was 0.0024 and 0.0013 s^{−1} for Kif11. The ratio (Ncd/Kif11) of these slops is ≈ 1.8. In comparison, the ratio between measured motor speeds (115 nm/s for Ncd and 70 nm/s for Kif11) is ≈ 1.6. The slope calculated for K401 was $\approx 0.0043{\mathrm{s}}^{1}$. Thus, the ratio of slopes of aster speeds versus separation is in good agreement with the ratio of motor speeds, suggesting that the motor speed sets this slope. To illustrate this point, the best fit lines shown in Figure 3c are plotted again, with the slopes divided by the singlemotor speed. In this way, the three lines now overlap, as shown in Appendix 1—figure 9.
Model of network contraction
In our aster merger experiments, we observed a linear relationship between distance from the center of the network and aster speed. This is seen in the plot of xvelocity versus position in Figure 3b and also in the plot of maximum aster speed versus initial separation in Figure 3c. Juniper et al. observed a similar relationship between position from the center of the network and contraction speed during aster formation in Juniper et al., 2018. The speed measured at the ends of the network is a simple summation of the velocities of walking motors, the same way that a person appears to walk faster on a moving walkway than on a stationary walkway. In an ideal case, each pair of microtubules would contract relative to each other, meaning that the speed measured at the end of the network would be the number of microtubules multiplied by the walking speed of each motor. There are a number of factors that make our experiments different from this ideal case, for example, the microtubules are not perfectly aligned and there are some dead motors in the system. Thus, we consider the contractile region, of length L, to be composed of N contractile units, which are bundles of microtubules with characteristic length $\mathrm{\ell}$, that contract relative to each other. The velocity measured at the ends of the network, then is $\nu $ times the number of contractile units (L/$\mathrm{\ell}$). We summarize this in Figure 3—figure supplement 1.
where V is the network contractile rate and ${\nu}_{m}$ is the motor speed. Our observed relationship that the ratio of the slope of speed increase matched the ratio of singlemotor speeds (as discussed in the main text) supports this hypothesis. Further, we can estimate that L ≈50–100 μm, indicating that the characteristic length scale is larger than a single microtubule.
Data availability
All data associated with this study are stored on the CaltechData archive at https://doi.org/10.22002/D1.2152.

CaltechDATAImages for Motor processivity and speed determine structure and dynamics of motormicrotubule assemblies.https://doi.org/10.22002/D1.2152
References

Theory of selfassembly of microtubules and motorsPhysical Review. E, Statistical, Nonlinear, and Soft Matter Physics 74:031915.https://doi.org/10.1103/PhysRevE.74.031915

A theory that predicts behaviors of disordered cytoskeletal networksMolecular Systems Biology 13:941.https://doi.org/10.15252/msb.20177796

Orientational order of motile defects in active nematicsNature Materials 14:1110–1115.https://doi.org/10.1038/nmat4387

Computer control of microscopes using µmanagerCurrent Protocols in Molecular Biology Chapter 14:Unit14.https://doi.org/10.1002/0471142727.mb1420s92

Mutants of the Drosophila ncd microtubule motor protein cause centrosomal and spindle pole defects in mitosisJournal of Cell Science 107 (Pt 4):859–867.https://doi.org/10.1242/jcs.107.4.859

Microtubule organization by the antagonistic mitotic motors kinesin5 and kinesin14The Journal of Cell Biology 189:465–480.https://doi.org/10.1083/jcb.200910125

Kinesin associates with anterogradely transported membranous organelles in vivoThe Journal of Cell Biology 114:295–302.https://doi.org/10.1083/jcb.114.2.295

Macroscopic equations for pattern formation in mixtures of microtubules and molecular motorsPhysical Review. E, Statistical, Nonlinear, and Soft Matter Physics 64:056113.https://doi.org/10.1103/PhysRevE.64.056113

Kinesin5 regulation and function in mitosisTrends in Cell Biology 29:66–79.https://doi.org/10.1016/j.tcb.2018.08.004

Dynamic concentration of motors in microtubule arraysPhysical Review Letters 86:3192–3195.https://doi.org/10.1103/PhysRevLett.86.3192

New polarized light microscope with precision universal compensatorJournal of Microscopy 180:140–147.https://doi.org/10.1111/j.13652818.1995.tb03669.x

Mechanobiology of the mitotic spindleDevelopmental Cell 56:192–201.https://doi.org/10.1016/j.devcel.2020.11.003

SelfOrganized pattern formation in motormicrotubule mixturesPhysical Review. E, Statistical, Nonlinear, and Soft Matter Physics 70:031905.https://doi.org/10.1103/PhysRevE.70.031905

Boundaries steer the contraction of active gelsNature Communications 7:13120.https://doi.org/10.1038/ncomms13120

Reconstitution of contractile actomyosin bundlesBiophysical Journal 100:2698–2705.https://doi.org/10.1016/j.bpj.2011.04.031

To step or not to step? how biochemistry and mechanics influence processivity in kinesin and Eg5Current Opinion in Cell Biology 19:75–81.https://doi.org/10.1016/j.ceb.2006.12.011

A “ holistic” kinesin phylogeny reveals new kinesin families and predicts protein functionsMolecular Biology of the Cell 17:1734–1743.https://doi.org/10.1091/mbc.e05111090
Article and author information
Author details
Funding
Foundational Questions Institute (FQXi 1816)
 Rachel A Banks
 Matt Thomson
 Rob Phillips
 Soichi Hirokawa
 Heun Jin Lee
 Vahe Galstyan
John Templeton Foundation (51250)
 Rachel A Banks
 Vahe Galstyan
 Heun Jin Lee
 Soichi Hirokawa
 Matt Thomson
 Rob Phillips
National Institutes of Health (1R35 GM11804301)
 Rachel A Banks
 Vahe Galstyan
 Heun Jin Lee
 Soichi Hirokawa
 Rob Phillips
John Templeton Foundation (60973)
 Rachel A Banks
 Vahe Galstyan
 Heun Jin Lee
 Soichi Hirokawa
 Matt Thomson
 Rob Phillips
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
We are grateful to Dan Needleman, Madhav Mani, Peter Foster, and Ana Duarte for fruitful discussions. We acknowledge support from the NIH through grant 1R35 GM11804301; the John Templeton Foundation as part of the Boundaries of Life Initiative through grants 51250 and 60973; the Foundational Questions Institute and Fetzer Franklin Fund through FQXi 1816.
Version history
 Preprint posted: October 23, 2021 (view preprint)
 Received: April 11, 2022
 Accepted: February 7, 2023
 Accepted Manuscript published: February 8, 2023 (version 1)
 Version of Record published: March 14, 2023 (version 2)
Copyright
© 2023, Banks et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
Metrics

 1,026
 views

 175
 downloads

 3
 citations
Views, downloads and citations are aggregated across all versions of this paper published by eLife.
Download links
Downloads (link to download the article as PDF)
Open citations (links to open the citations from this article in various online reference manager services)
Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)
Further reading

 Developmental Biology
 Physics of Living Systems
Geometric criteria can be used to assess whether cell intercalation is active or passive during the convergent extension of tissue.

 Computational and Systems Biology
 Physics of Living Systems
Locomotion is a fundamental behavior of Caenorhabditis elegans (C. elegans). Previous works on kinetic simulations of animals helped researchers understand the physical mechanisms of locomotion and the musclecontrolling principles of neuronal circuits as an actuator part. It has yet to be understood how C. elegans utilizes the frictional forces caused by the tension of its muscles to perform sequenced locomotive behaviors. Here, we present a twodimensional rigid body chain model for the locomotion of C. elegans by developing Newtonian equations of motion for each body segment of C. elegans. Having accounted for frictioncoefficients of the surrounding environment, elastic constants of C. elegans, and its kymogram from experiments, our kinetic model (ElegansBot) reproduced various locomotion of C. elegans such as, but not limited to, forwardbackward(omega turn)forward locomotion constituting escaping behavior and deltaturn navigation. Additionally, ElegansBot precisely quantified the forces acting on each body segment of C. elegans to allow investigation of the force distribution. This model will facilitate our understanding of the detailed mechanism of various locomotive behaviors at any given frictioncoefficients of the surrounding environment. Furthermore, as the model ensures the performance of realistic behavior, it can be used to research actuatorcontroller interaction between muscles and neuronal circuits.