Many cellular processes are driven by cytoskeletal assemblies. It remains unclear how cytoskeletal filaments and motor proteins organize into cellular scale structures and how molecular properties of cytoskeletal components affect the large-scale behaviors of these systems. Here, we investigate the self-organization of stabilized microtubules in Xenopus oocyte extracts and find that they can form macroscopic networks that spontaneously contract. We propose that these contractions are driven by the clustering of microtubule minus ends by dynein. Based on this idea, we construct an active fluid theory of network contractions, which predicts a dependence of the timescale of contraction on initial network geometry, a development of density inhomogeneities during contraction, a constant final network density, and a strong influence of dynein inhibition on the rate of contraction, all in quantitative agreement with experiments. These results demonstrate that the motor-driven clustering of filament ends is a generic mechanism leading to contraction.https://doi.org/10.7554/eLife.10837.001
The ability of cells to move, divide, and carry out other processes depends on networks of protein filaments and motor proteins collectively known as the cytoskeleton. The motor proteins can move along the filaments to transport molecules and larger structures around the cell, or to rearrange the filaments themselves.
The cytoskeleton of animal, plant, and other eukaryotic cells contains two main types of filaments, known as actin filaments and microtubules. Both types of filament have distinct ends, known as the plus and minus ends. Previous studies have revealed that networks of actin filaments can rapidly contract to drive the movement of muscles and other processes. However, it is not known whether networks of microtubules can also contract.
Foster et al. studied the microtubules in extracts made from the eggs of a frog called Xenopus laevis. The experiments show that these microtubules form networks that can spontaneously contract. Foster et al. propose that this contraction is caused by the minus ends of the microtubules clustering together due to the activities of a motor protein called dynein.
To test this idea, Foster et al. developed a mathematical model based on an 'active fluid' theory. This model makes predictions that agree very well with the experimental data. The next step in this work is to find out if this model of microtubule contraction applies to other networks of microtubules.https://doi.org/10.7554/eLife.10837.002
The mechanics, motions, and internal organization of eukaryotic cells are largely determined by the cytoskeleton. The cytoskeleton consists of filaments, such as actin and microtubules, and molecular motors, which consume chemical energy to exert forces on and arrange the filaments into large-scale networks. Motor proteins, including dynein and roughly 14 different families of kinesin (Wordeman, 2010), organize microtubules to form the spindle, which segregates chromosomes during cell division. The motor protein myosin organizes actin filaments into networks which drive cell motility, polarity, cytokinesis, and left-right symmetry breakage (Mitchinson and Cramer, 1996; Mayer et al., 2010; Naganathan et al., 2014). The non-equilibrium nature of motor activity is essential for the organization of the cytoskeleton into these diverse sub-cellular structures, but it remains unclear how the interactions between filaments, different motor proteins, and other biomolecules influence the behaviors of the networks they form. In particular, it is difficult to extrapolate from the biochemical properties of motors characterized in reconstituted systems to the biological function of those motors in vivo. To address this question, we study self-organization of cytoskeletal filaments in Xenopus extracts, which recapitulate the biochemical complexity of the in vivo system.
The self-organization of cytoskeletal filaments has been extensively studied in cell extracts and in reconstituted systems of purified components. Actin can form macroscopic networks that exhibit a myosin-dependent bulk contraction (Murrell and Gardel, 2012; Bendix et al., 2008; Köhler and Bausch, 2012; Alvarado et al., 2013; Szent-Györgyi, 1943). Microtubule networks purified from neuronal extracts have also been observed to undergo bulk contraction (Weisenberg and Cianci, 1984), while microtubules in mitotic and meiotic extracts are found to assemble into asters (Gaglio et al., 1995; Mountain et al., 1999; Verde et al., 1991). Aster formation in meiotic Xenopus egg extracts is dynein-dependent, and has been proposed to be driven by the clustering of microtubule minus ends by dynein (Verde et al., 1991). It has also been suggested that dynein binds to the minus ends of microtubules in spindles and clusters the minus ends of microtubules to form spindle poles (Heald et al., 1996; Burbank et al., 2007; Khodjakov et al., 2003; Goshima et al., 2005; Elting et al., 2014) and dynein has been shown to accumulate on microtubule minus ends in a purified system (McKenney et al., 2014). Purified solutions of microtubules and kinesin can also form asters (Nédélec et al., 1997; Hentrich and Surrey, 2010; Urrutia et al., 1991), or under other conditions, dynamic liquid crystalline networks (Sanchez et al., 2012). Hydrodynamic theories have been proposed to describe the behaviors of cytoskeletal networks on length scales that are much greater than the size of individual filaments and motor proteins (Prost et al., 2015, Marchetti et al., 2013). These phenomenological theories are based on symmetries and general principles of non-equilibrium physics, with the details of the microscopic process captured by a small number of effective parameters. As hydrodynamic theories are formulated at the continuum level, they cannot be used to derive the values of their associated parameters, which must be obtained from more microscopic theories (Prost et al., 2015, Marchetti et al., 2013) or by comparison to experiments (Mayer et al., 2010; Brugués and Needleman, 2014).
A key feature of networks of cytoskeletal filaments and motor proteins that enters hydrodynamic theories, and differentiates these non-equilibrium systems from passive polymer networks, is the presence of additional, active stresses (Prost et al., 2015, Marchetti et al., 2013). These active stresses can be contractile or extensile, with profound implications for the large-scale behavior of cytoskeletal networks. Contractile stresses can result from a preferred association of motors with filament ends (Kruse and Jülicher, 2000; Hyman and Karsenti, 1996), nonlinear elasticity of the network (Liverpool et al., 2009), or the buckling of individual filaments (Murrell and Gardel, 2012; Lenz, 2014; Soares e Silva et al., 2011). Extensile active stresses can arise from polarity sorting or result from the mechanical properties of individual molecular motors (Gao et al., 2015; Blackwell et al. 2015). In networks with dynamically growing and shrinking filaments, polymerization dynamics can also contribute to the active stress. Experimentally, acto-myosin systems (Murrell and Gardel, 2012; Bendix et al., 2008; Köhler and Bausch, 2012; Alvarado et al., 2013; Szent-Györgyi, 1943) and microtubule networks from neuronal extracts (Weisenberg and Cianci, 1984) are observed to be contractile, while purified solutions of microtubules and kinesin can form extensile liquid crystalline networks (Sanchez et al., 2012). It is unclear which microscopic properties of filaments and motor proteins dictate if the active stress is contractile or extensile in these different systems.
Here, we investigate the motor-driven self-organization of stabilized microtubules in Xenopus meiotic egg extracts. These extracts are nearly undiluted cytoplasm and recapitulate a range of cell biological processes, including spindle assembly and chromosome segregation (Hannak and Heald, 2006). We have discovered that, in addition to microtubules forming asters in this system as previously reported (Verde et al., 1991), the asters assemble themselves into a macroscopic network that undergoes a bulk contraction. We quantitatively characterized these contractions and found that their detailed behavior can be well understood using a simple coarse-grained model of a microtubule network in which dynein drives the clustering of microtubule minus ends. This end clustering mechanism leads to a novel form of active stress, which drives the system to a preferred microtubule density. Our results suggest that the dynein-driven clustering of microtubule minus ends causes both aster formation and network contraction, and have strong implications for understanding the role of dynein in spindle assembly and pole formation. Furthermore, the close agreement we find between experiments and theory demonstrates that simple continuum models can accurately describe the behavior of the cytoskeleton, even in complex biological systems.
To further study the motor-induced organization of microtubules, we added 2.5 M Taxol to Xenopus egg extracts and loaded them into microfluidic channels (Figure 1A). Taxol causes microtubules to rapidly assemble and stabilize (Mitchison et al., 2013), which allowed us to decouple the effects of motor-driven self-assembly from the complicating effects of polymerization-depolymerization dynamics. In some regions of the channel, microtubules organized into asters (Figure 1B) as observed previously (Verde et al., 1991). A NUMA antibody was used to locate microtubule minus ends (Mitchison et al., 2013), and was found to localize to the aster core, confirming the polarity of the aster (Gaglio et al., 1995). Isolated asters were found to interact and coalesce (Figure 1C, Video 1). In other regions of the channel, microtubules formed networks of aster-like structures (Figure 1D), which were highly dynamic and exhibited large-scale motion that persisted for several tens of seconds (Figure 1E, Video 2). NUMA was found to localize to the interior of these structures, confirming their aster-like nature (Figure 1F,G).
To characterize these large-scale motions, we next imaged networks at lower magnification, obtaining a field of view spanning the entire channel width. The networks, which initially filled the entire channel (width = 1.4 mm), underwent a strong contraction, which was uniform along the length of the channel (Figure 2A, Video 3). The contractile behavior of these microtubule networks is highly reminiscent of the contractions of actin networks in these extracts (Bendix et al., 2008), but in our experiments actin filaments are not present due to the addition of 10 Cytochalasin D. We characterized the dynamics of microtubule network contractions by measuring the width, W(t), of the network as a function of time (Figure 2B). Occasionally, we observed networks tearing along their length (Video 4), yet these tears seemed to have little impact on the contraction dynamics far from the tearing site, arguing that the Poisson ratio of the network is 0. We then calculated the fraction contracted of the network:
The time course of (t) was found to be well fit by an exponential relaxation:
We next sought to investigate which processes determine the timescale of contraction and the extent that the network contracts. For this, we exploited the fact that different mechanisms predict different dependence of the timescale on the channel dimensions. For instance, in a viscoelastic Kelvin-Voight material driven to contract by a constant applied stress, = /E depends solely on the viscosity and the Young’s modulus E and is independent of the size of the channel (Oswald, 2009). In contrast, in a poroelastic material driven by a constant stress, (Coussy, 2004), where is the width of the channel. Thus, studying how varies with channel width provides a means to test the validity of these models.
We fabricated microfluidic channels of varying width, = 1.4, 0.9, 0.44, and 0.16 mm, all with height = 125 m, loaded the channels with extracts supplemented with 2.5 M Taxol, and imaged the networks at low magnification (Figure 3A, Video 5). Results for each channel width were averaged together to produce master curves of the width, W(t) (Figure 3B), and fraction contracted, (t) (Figure 3C), of the networks in each channel. Visual inspection of the fraction-contracted curves, (t), reveals that networks in smaller channels contract faster, but all reach a similar final fraction contracted (Figure 3C). To quantify these trends, we fit the (t) curves using (Equation 2) and extracted the characteristic time to contract, , and the final fraction contracted, , for each channel width. We find that the dependence of on channel width is inconsistent with the time of contraction resulting from either viscoelastic or poroelastic timescales, which would predict constant and quadratic scalings respectively (Figure 3D). We next explored the influence of channel height ( = 75, 125, 150 m, all with width = 1.4 mm) and found that does not significantly vary in these channels (Figure 3E).
In all cases, the networks contracted to a similar final fraction, , of 0.77, irrespective of channel geometry (Figure 3F). Since the Taxol concentration was held constant, all experiments started with the same initial density of microtubules, regardless of the dimensions of the channel. Thus, all networks contracted to the same final density. By using fluorescence intensity as a proxy for tubulin concentration (see Materials and methods), we estimate the final concentration of tubulin in the network to be 30 M. Remarkably, this is comparable to the concentration of microtubules in reconstituted meiotic spindles in Xenopus extracts (Needleman et al., 2010), which is 60 M. As neither the simple viscoelastic nor poroelastic models are consistent with these results, we sought to construct an alternative model of the contraction process. Since Taxol stabilizes microtubules in these experiments, the density of microtubules is conserved throughout the contraction process, implying
where is the local velocity of the microtubule network. The velocity is set by force balance. If the relevant timescales are long enough that the microtubule network can be considered to be purely viscous, and if the network’s motion results in drag, then the equation for force balance is
where and are the viscosity and drag coefficients, respectively, and is an active stress caused by motor proteins which drive the contraction of the microtubule network. The observation that the timescale of contraction, , is independent of channel height (Figure 3E) shows that the drag does not significantly vary with channel height, and thus could arise from weak interactions between the microtubule network and the device wall.
We obtain an expression for the active stress, , by considering the microscopic behaviors of microtubules and motor proteins. As the contracting networks consist of microtubule asters (Figure 1D, E), and microtubule asters in meiotic extracts are thought to assemble by the dynein-induced clustering of microtubule minus ends (Verde et al., 1991), we hypothesize that the contraction process is also driven by dynein pulling microtubule minus ends towards each other (Figure 4A).
In an orientationally disordered suspension of microtubules, we expect dynein mediated collection of microtubule minus ends to drive a contractile stress which is proportional to the number of motor molecules and the local density of microtubules , (see Appendix).
As only a finite number of microtubules can fit near the core of an aster, steric collisions will counteract the contractile stress at high densities (Figure 4B).
Since most motion in the suspension is motor driven, thermal collisions can be ignored, and the extensile stress driven by steric interactions will be be proportional to the number of motor molecules and quadratic in the local density of microtubules (see Appendix).
Taken together, these two effects lead to the active stress
where is the strength of the active stress, is the final density at which the effects of dynein mediated clustering and steric repulsion between microtubules balance, and is a unit tensor (see Appendix).
Importantly, since the contractile and extensile parts of the active stress both depend linearly on the number of motor molecules, the prefered density that the suspension will reach after contraction depends only on the interaction geometry between microtubules and motors and not on the actual number of active motors. Only the strength of the active stress will be affected if the number of active motors could be changed.
Taken together, Equations (3,4,5) constitute an active fluid theory of microtubule network contraction by minus end clustering. We note that this theory could be reformulated, essentially without change, as the clustering of aster cores, again driven by dynein mediated clustering of minus ends. Isotropy of interactions remains a fundamental assumption.
We first investigated if this active fluid theory can explain the dependence of the timescale of contraction on sample geometry. An analysis of the equations of motion, Equations (3,4,5), near equilibrium predicts that the timescale of contractions obeys
where = 2.2 0.05 and = 0.085 0.006 are dimensionless constants, which we determined numerically (see Appendix). This predicted scaling is both consistent with the experimental data and simulations of the full theory (Figure 3D). Fitting the scaling relationship to the data allows combinations of the parameters to be determined, giving ) = 0.82 0.20 min and ) = 1.0 (mean standard error). Combining this measurement with an estimate for the network viscosity taken from measurements in spindles of (Shimamoto et al., 2011), we can estimate the dynein-generated active stress to be which is consistant with having 0.4 dynein per microtubule minus end each exerting an average force of 1 pN (Nicholas et al., 2015).
To further explore the validity of the active fluid theory of contraction by microtubule minus end clustering, we explored other testable predictions of the theory. This theory predicts that: (i) the preferred density of the network is constant and does not depend on the initial conditions. This is consistent with the constant measured experimentally (Figure 3F); (ii) since contractions are driven by stress gradients (Equation 4) and stress depends on microtubule density (Equation 5) the density discontinuity at the edge of the network should produce large-stress gradients, leading to an inhomogeneous density profile in the network during contraction; (iii) the magnitude of the active stress, s, is proportional to the number of active motors, but the final density of the network, , is independent of the number of molecular motors (see Appendix). Thus, reducing the number of motors should lead to slower contractions, but still yield the same final density.
We first examined prediction (ii), that the stress discontinuity at the edge of the network should lead to a material buildup in the film. To test this, we averaged the fluorescence intensity along the length of the channel (see Materials and Methods) and found that the microtubule density does indeed increase at the network’s edge during contraction (Figure 5A). We next explored if the inhomogeneous density profile could be quantitatively explained by our active fluid theory. We numerically solved Equations (3,4,5) and used least squares fitting to determine the simulation parameters which most closely matched the experimentally measured profiles (Figure 5B), yielding = 0.820.03 min, = 6.10.1 min/(), and = 0.32 0.01 (mean s.e.m., n=4 experiments). Within error, these values are the same as those determined from the dependence of the timescale of contraction on channel width (Figure 3D). The simulated profiles closely match the experimental ones for most of the contraction (Figure 5B), but at late times the simulated inhomogeneities dissipate in contrast to the experiments (Figure 5—figure supplement 1). This might be caused by a long-term aging of the network that is not incorporated into our simple model. To confirm that the density buildup was due to an increased velocity near the network’s edge, we measured the velocity throughout the network using Particle Image Velocimetry (PIV, see Materials and Methods) (Figure 5C) and found that the velocities increase superlinearly with distance from the network’s center, as predicted (Figure 5D).
Finally, we sought to determine the molecular basis of the contraction process, and check prediction (iii), that the number of motors driving the contraction affects the rate of contraction, but not the final density the network contracts to. Aster assembly is dynein-dependent in Xenopus egg extracts (Gaglio et al.,1995; Verde et al., 1991), and dynein (Heald et al., 1996) and Kinesin-5 (Sawin et al., 1992) are two of the most dominant motors in spindle assembly in this system. We inhibited these motors to test their involvement in the contraction process. Extracts supplemented with STLC for Kinesin-5 inhibition or p150-CC1 for dynein inhibition were loaded into channels with a width, , of 0.9 mm and imaged at low magnification. Inhibiting Kinesin-5 had little effect on the contraction process (Figure 6—figure supplement 1). In contrast, inhibiting dynein caused a dose-dependent slowdown of the contraction (Figure 6A). In spindle assembly, inhibiting Kinesin-5 suppresses the morphological changes caused by dynein inhibition (Mitchison et al., 2005). We, therefore, tested how simultaneously inhibiting both motors influences the contraction process, but found that the effects of dynein inhibition were not rescued by the simultaneous inhibition of Kinesin-5 (Figure 6—figure supplement 1), suggesting that in this context, Kinesin-5 is not generating a counteracting extensile stress. This further suggests the possibility that in the spindle, the role of Kinesin-5 may be in orienting, polarity sorting, and sliding microtubules as opposed to active stress generation. Curves of (t) were fit using Equation (2) to extract the final fraction contracted, , and the characteristic time of contraction, . By varying the concentration of p150-CC1, the characteristic time, , could be tuned over a wide range from 3 min to 75 min (Figure 6B). Fitting a sigmoid function to the vs. p150-CC1 concentration curve yields an EC50 value of 0.22 . 02 M (mean standard error), similar to the value of 0.3 M reported for the effect of p150-CC1 on spindle length in Xenopus extracts (Gaetz and Kapoor, 2004), which is consistent with active stress generated by dynein being required for pole focusing. Despite this large change in the contraction timescale, we found no apparent differences in (Figure 6C). Thus, the microtubule networks contract to approximately the same final density irrespective of the concentration of p150-CC1. The observation that inhibiting dynein affects the timescale of contraction but not the final density to which the network contracts is consistent with the predictions of our model. We note that even at the highest p150-CC1 concentrations used, the network still undergoes a bulk contraction. This could possibly be due to incomplete inhibition of dynein by p150-CC1, or by another motor protein present in the extract that also contributes to the contraction process. As the characteristic time, , by comparing the characteristic times in the uninhibited and 2 M p150-CC1 cases, we can estimate that the strength of the active stress, , in the 2 M p150-CC1 condition is only 4% of the strength of the active stress in the uninhibited case, arguing that even if another motor is involved in the contraction, dynein contributes 96% of the active stress.
Here, we have shown that networks of stabilized microtubules in Xenopus egg extracts undergo a bulk contraction. By systematically varying the width of the microfluidic channel in which the network forms, we demonstrated that the timescale of contraction is not a poroelastic or viscoelastic timescale. A simple active fluid model of network contraction by dynein-driven clustering of microtubule minus ends correctly predicts the dependence of the contraction timescale on channel width, the nonuniform density profile in the network during contraction, and that inhibiting dynein affects the timescale of contraction but not the final density that the network contracts to. Parameters of this model can be measured by the scaling of the contraction timescale with channel width and by a detailed analysis of the inhomogeneities in the network that develop during contraction. Both methods give similar values.
Our results demonstrate that the behaviors of a complex biological system can be quantitatively described by a simple active matter continuum theory. These active matter theories aim to describe the behavior of cytoskeletal systems at large-length scales and long-timescales by effectively averaging all of the molecular complexity into a small set of coarse-grained parameters. Previously, these theories have been predominately applied to describe biological systems near non-equilibrium steady states (Prost et al., 2015; Brugués et al., 2014). In the present work, we augment previous theories with a nonlinear active stress term derived from microscopic considerations to capture the far from steady state dynamics of the contraction process. This approach allows us to quantitatively explain our experimental results using a theory with only four parameters, while a complete microscopic model would require understanding the behavior of the thousands of different proteins present in Xenopus egg extracts. Furthermore, the considerations of the model are general, and it will be interesting to consider whether the end clustering mechanism proposed here could contribute to contraction in actin networks as well.
In our model, the active stress which drives network contraction results from the motor-induced clustering of microtubule minus ends, the same process thought to be responsible for aster formation and spindle pole focusing (Gaglio et al., 1995; Mountain et al., 1999; Verde et al., 1991, Elting et al., 2014; Heald et al., 1996; Burbank et al., 2007; Khodjakov et al., 2003; Goshima et al., 2005). Our results, and previous data (Verde et al., 1991; Heald et al., 1996; Burbank et al., 2007), are consistent with minus end clustering in Xenopus egg extracts primarily arising from the activity of dynein. The ability of dynein to cluster microtubule minus ends could result from dynein being able to accumulate on the minus end of one microtubule, while simultaneously walking towards the minus end of another (Hyman and Karsenti, 1996; McKenney et al. 2014; Figure 4A). There is indication that such behaviors may indeed occur in spindles (Elting et al., 2014), and pursuing a better understanding of those processes is an exciting future direction that will help to clarify the function of dynein in spindles.
The observation that microtubule networks contract in Xenopus egg extracts suggests that motor-induced stresses in spindles are net contractile and not extensile as previously assumed (Brugués and Needleman, 2014). The contribution of dynein to spindle pole focusing may ultimately be due to these contractile stresses. The presence of contractile stresses from dynein might also explain both the observation that the fusion of spindles is dynein-dependent (Gatlin et al., 2009), and the apparently greater cohesion between microtubules at spindle poles, (where dynein is localized [Gatlin et al., 2010]). It is unclear what processes set the density of microtubules in the spindle, and the finding that the active stress generated from minus end clustering saturates at a preferred microtubule density could play an important role.
CSF-arrested extracts were prepared from Xenopus llaevis oocytes as previously described (Hannak and Heald, 2006). Crude extracts were sequentially filtered through 2.0, 1.2, and 0.2 micron filters, frozen in liquid nitrogen, and stored at −80°C until use.
Channel negatives were designed using AutoCAD 360 (Autodesk) and Silhouette Studio (Silhouette America) software, cut from 125-micron-thick tape (3M Scotchcal, St. Paul, MN) using a Silhouette Cameo die cutter, and a master was made by adhering channel negatives to the bottom of a petri dish. PDMS (Sylgard 184, Dow Corning, Midland, MI; 10:1 mixing ratio) was cast onto the masters and cured overnight at 60°C. Devices and coverslips were each corona treated with air plasma for 1 min before bonding. Channels containing a degassed solution of 5 mg/mL BSA (J.T. Baker, Center Valley, PA) supplemented with 2.5% w/w Pluronic F127 (Sigma, St. Louis, MO) were incubated overnight at 12°C. Unless stated otherwise, the microfluidic devices had a length of 18 mm, a height of 0.125 mm, and a width of 1.4 mm.
GST-tagged p150-CC1 plasmid was a gift from Thomas Surrey (Uteng et al., 2008). GST-p150-CC1 was expressed in E. coli BL21 (DE3)-T1R(Sigma) for 4 hr at 37°C. The culture was shifted to 18°C for 1 hr before adding 0.2 mM IPTG and the culture was grown overnight at 18°C. Cells were centrifuged, resuspended in PBS supplemented with Halt Protease Inhibitor Cocktail (Thermo Scientific, Rockford, IL) and benzonase (Novagen, San Diego, CA) before lysis by sonication. GST-p150-CC1 was purified from clarified lysate using a GSTrap column FF (G.E. Healthcare, Sweden) as per the manufacturer’s instructions. GST-p150-CC1 was dialyzed overnight into 20 mM Tris-HCl, 150 mM KCl, and 1 mM DTT. The GST tag was cleaved using Prescission Protease (overnight incubation at 4°C). After removing free GST and Prescission Protease using a GSTrap FF column, p150-CC1 was concentrated, frozen in liquid nitrogen, and stored at -80°C until use.
20 L aliquots of filtered extract were supplemented with ∼1 M Alexa-647 labeled tubulin and 2.5 M Taxol before being loaded into channels. For dynein inhibition experiments, 1 L of either p150-CC1 or buffer alone was added to the extract immediately before Taxol addition. For Kinesin-5 inhibition experiments, 100 M STLC (Sigma Aldrich) was added concurrently with Taxol. Channels were sealed with vacuum grease and imaged using a spinning disk confocal microscope (Nikon Ti2000, Yokugawa CSU-X1), an EMCCD camera (Hamamatsu), and a 2x objective using Metamorph acquisition software (Molecular Devices). t=0 is defined when the imaging begins, 1 min after Taxol addition to the extract. After a brief lag time, the microtubule networks spontaneously begin contraction. Images were analyzed using ImageJ and custom build MATLAB and Python software (available at https://github.com/peterjfoster/eLife). Parameters were fit to contraction data using timepoints where 0.1.
The final density was estimated using contraction experiments with 2.5 M Taxol in 0.9 mm channels. For each experiment, the frame closest to t = + was isolated, where and are the timescale of contraction and the offset time respectively, obtained from fits of the time course of contraction to Equation 2 of the main text. After correcting for the camera offset and inhomogeneous laser illumination, the average fluorescence intensity of the network, and the average fluorescence intensity in the channel outside the network, were calculated. The fluorescence intensity in the channel but outside the network comes from monomeric fluorescently labeled tubulin and was assumed to be constant throughout the channel. The fractional concentration was then estimated as . Using this measurement along with the fit curves for (t) and under the assumption that the network contracts in the z direction such that (t) in the z direction is the same as along the width, the inferred fractional concentration at t = was calculated as
Assuming the fluorescently labeled tubulin incorporates into microtubules at the same rate as endogenous tubulin, we can multiply the derived fractional density (t = ) by the tubulin concentration in extract, ≈18 M (Parsons and Salmon, 1997) to arrive a final network tubulin concentration of ≈30 M.
Images from contraction experiments were corrected for the camera offset and inhomogeneous laser illumination before being thresholded in order to segment the microtubule network from background fluorescence. Rotations of the channel relative to the CCD were detected by fitting linear equations to edges of the microtubule network. If the average of the slopes from the top and bottom of the network was greater than 1/(the number of pixels in the length of the image), a rotated, interpolated frame was constructed where pixels were assigned based on the intensity of the pixel in the original frame weighted by their area fraction in the interpolated pixel. Frames were averaged along the length of the channel before background signal subtraction. For density profiles compared with simulations, the edge peaks of the density profile were identified and pixels between the two peaks were retained. Profiles were normalized such that the integral of the profile was set to 1.
Particle Imaging Velocimetry was performed using PIVLab software (Thielicke and Stamhuis, 2014) using the FFT window deformation algorithm with a 16-pixel interrogation area and 8 pixel step for the first pass and an 8 pixel interrogation area with a 4-pixel step for the second pass. After PIV was performed, intensity images were thresholded to segment the microtubule network from the background, and only velocity vectors within the microtubule network that were 8 pixels from the network’s edges were retained.
We introduce a theoretical description of a confined active microtubule-motor gel immersed in a Newtonian fluid. We obtain generic equations of motion for this system closely following the logic outlined in (Doi and Onuki, 1992; Joanny et al., 2007). This generic description is augmented by including a density-dependent active stress, which is derived from a minimal microscopic description of microtubule-dynein interactions. Here, we present the equations of motion for the one dimensional system.
We begin by stating the conservation laws an active gel permeated by a Newtonian fluid obeys. The system shall be incompressible such that the total density is a constant. Here and are the densities of gel and fluid, respectively. The gel density obeys the continuity equation,
where is the velocity of the gel. Similarly the fluid permeating the gel obeys,
where and are the fluid density and velocity fields, respectively. Since the overall system is incompressible Force balance in the gel requires
where the gel stress consists of a viscous stress , an active stress , and a hydrostatic pressure . The friction coefficient quantifies the momentum transfer between the gel and its confinement and quantifies the momentum transfer between and the gel and the fluid. The momentum continuity equation of the permeating fluid is
where is the fluid viscosity. In our experiments, changing the height of the chamber does not appreciably change the timescale of the observed contractions, see Figure 3F. Furthermore, there is little observed motion of the extract surrounding the film. Presumably since the system is relatively dilute, i.e. , and the length-scale is large compared to the chamber height. We thus simplify Equation (9) to
where . Equation (11) is the force balance equation we henceforth use for the gel to quantitatively capture the experimental dynamics. Note that also allowed us to neglect the hydrostatic pressure in the gel. We complement Equation (11) by the stress boundary condition at the edges of the film at
The width of the film obeys
We next seek to obtain an expression for the active stress by coarse-graining a microscopic model of interactions between dynein molecular motors and microtubules. Here, we assume that dynein builds up near microtubule minus ends as previously suggested (Elting et al., 2014; Surrey et al., 2001), and hence forces are exchanged between microtubules through the microtubules’ minus ends. We introduce the positions of microtubule minus ends , such that the film density can be written as
The force exerted by the -th on the -th filament is , with as required by momentum conservation. The active stress generated in this context is defined by the force balance equation
up to an arbitrary constant of integration. Note that averaging Equation (15) over an appropriate mesoscopic volume yields the well-known Kirkwood formula. To model microtubule-dynein interactions, we propose that , where is a dynein-mediated attractive force between minus ends, and is a repulsive force from steric interactions between nearby filaments (Figure 4). Generically, and depend on the relative positions and orientations of microtubules and . Since we are concerned with a disordered assembly of microtubules in which all orientations occur with the same likelihood it is sufficient for our purposes to only think of microtubule positions, and orientation effects average out. The average attractive force that motors bound to the minus end of filament exert on filament can be expressed locally as the product
where is the probability that a motor connects the minus end of filament to filament and is the force which the motor exerts if a connection is made. Since each dynein can link at most two filaments,
where is the fraction of filaments that carry an active motor at their minus end and is a typical interaction distance. Here, denotes the Heaviside function which is equal to one for positive and zero otherwise. If is an odd function of the separation vector , it can be expressed by the series . Using Equation (14), the force density field generated by motor contractions becomes to lowest order in ,
which corresponds to an active stress contribution .
We next discuss the average steric force that filament exerts on filament . Given the force of a collision event we find
Equation (19) is linear in the motor density , since only filaments that are being actively moved will sterically displace their neighbors. Note that here we chose the typical interaction distance to be the same in Equation (19) and Equation (16) for simplicity. If is an odd function of the displacement between the microtubule ends and the force density field generated by steric interactions is
which corresponds to an active stress contribution . The total active stress is thus given by,
with and . Together with Equations (7,11,21) are the equations of motions of our system.
We asked how the characteristic time of contractions scales as a function of the width of the confining chamber, according to our theory. For this, we rewrite the equations of motion, Equations (7,11), in dimensionless form
where , , , and and . The boundary condition then becomes
with . To further simplify our analysis, we move to the 'Lagrangian' frame defined by , where the equations of motion become
with the boundary conditions
This system of equations has steady-states for , , , where is the final width of the film. We next linearize around this steady state, i.e., choose , , , where is a small quantity. To linear order the equations of motion then become
Using Equations (28,29), we find
and the boundary condition
We solve this equation by making the Ansatz and find
with , and . Thus,
where is an integration constant determined from the initial condition. In the following we shall consider the case , i.e. we start with a uniformly stretched film, for which .
To determine the timescale of the width contractions we need to remember the conservation of mass
We determine the timescale of contractions from and find
Thus, the dynamics is governed by multiple relaxation processes with varying timescales. In particular
In the experimental parameter regime, the timescale we measure is presumably intermediate,
where and are dimensionless quantities which we determine numerically. To obtain and for a given set of input parameters, we numerically solve Equations (7,11,13) and extract the timescale for several initial widths. We then fit the results to the functional form of Equation (40).
In the experimental regime, using the parameters for which the theoretical contraction profiles best agree with the numerical one (see Figure 5B), we estimate and . The error estimates were obtained by sampling and over a range of input parameters between half and twice the best fit values, and evaluating a standard error on the computed values.
To solve Equations (7,11,13) numerically, we discretize the system by representing on an equispaced grid between and , where is the instantaneous width of the contracting film. The instantaneous film velocity is determined from Equation (11) using a second order finite difference scheme. The boundary conditions at , are implemented using an asymmetric second order finite difference stencil, see (Tornberg and Shelley, 2004). We determine the time derivative of density using Equation (26) with the boundary condition specified in Equation 27, which account for the grid contracting with the width of the film. We time-evolve Equations (26,13) using a adaptive second order time stepping provided by Scientific Python project (Jones and Oliphant, 2001).
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Anna AkhmanovaReviewing Editor; Utrecht University, Netherlands
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Thank you for submitting your work entitled "Active Contraction of Microtubule Networks" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Naama Barkai as the Senior Editor. One of the three reviewers, Gijsje Koenderink, has agreed to reveal her identity.
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This study describes the macroscopic contraction of the microtubule network induced in Xenopus egg extracts by the addition of Taxol. It was shown before that with these conditions, the motor dynein induce the formation of asters (radial structures) that have a size that is commensurate with the length of the microtubules induced by Taxol. In this study, the larger scale is considered for the first time, and it is shown that the asters will coalesce, inducing the whole network to contract. This is reminiscent of the actin-gel contraction that is observed in the same system, but with a network of microtubules. The observations are interesting and fairly systematic. The quantitative nature of the experimental data allows for the development of phenomenological theoretical models that can describe the data. The message is clear and the article will be a worthwhile addition to the literature, in particular because it shows that contraction is not a unique feature of the actin world, as it occurs also in microtubule networks.
1) The claim "Our results suggest that the dynein-driven clustering of microtubule minus ends causes both aster formation and network contraction" (Introduction) is not sufficiently well supported by the data. Firstly, the aster nature of the structures is only weakly supported by the data (the data being shown in Figure 1B and C; especially panel 1C is unconvincing); it would be more convincing to visualize the MT minus ends or the motors. Secondly, the claim that dynein drives minus-end clustering is not well-supported since dynein inhibition only slows contraction, but does not stop it. Is the inhibition not 100% effective, or are other processes involved in contraction?
2) The biological relevance of the work must be made more explicit. The authors claim that the observation of dynein-driven clustering of microtubule minus ends has "strong implications for understanding the role of dynein in spindle assembly and pole formation." It is, however, not clear specifically how the insights obtained here can be translated to those situations. The statement in the Results section "suggesting that dynein is performing the same activity in spindles as in the contracting networks" is particularly unclear.
3) To develop the model (Appendix, “Active stresses from dynein-mediated microtubule interactions), the authors consider point-like objects in 1D, which are said to represent the "minus ends of microtubules". However the attractive interaction between these points extends symmetrically left and right, as if the microtubules were not orientated. The effective steric interaction between microtubules is also only dependent on the position of the minus-end, ignoring the fact that it is really the polymer mass that would induce steric forces, and not just the minus-ends. The plus-ends would be equally relevant. In other words, it seems in this theory that the minus-end located in the middle of the filament. To avoid this inconsistency with the accepted molecular picture, the authors should consider reformulating the model as a theory of many asters coming together. The basic unit would not be a microtubule, but a bunch of them already connected at their minus-ends. The variables 'Xi' could represent the positions of the centres of these asters, and assuming that they are symmetric, it would be justified to use the functional form of the interactions that are surmised. Doing so will not require changing the algebra in the theory, and rewriting its description would be sufficient.
4) The authors briefly mention that networks confined within 10-micron thick chambers do not exhibit any macroscopic contraction. Is there a way to understand this experimental observation and can it be accounted for by the theoretical model?
5) It seems that the contraction is a two-step process in which microtubules first form localized aster. Subsequently, these asters interact with each other in order to form a bulk contractile gel. Their proposed mechanism of end motor clustering explains local contractions that lead to formation of asters. However, presumably once the asters are formed all the microtubule ends are localized at the center of asters. In view of this information, please explain better why interacting asters would form a bulk contractile structure.
6) Previous work by Leibler, Surrey and Nedelec has investigated the behavior of purified systems of molecular motors and microtubules. They have not described bulk contraction but only formation of aster-like structures, which perhaps could be considered to be localized contractions. It would be good to explain the differences between the two analyzed systems and discuss whether the comparison between the two studies might suggest a way to assemble bulk microtubule contractile networks from purified components.
7) Model and data description:
A) Regarding the kinetics of contraction, several important aspects are unclear: what defines t=0? What is the origin of the lag time? And what process actually triggers constriction? A more detailed description of the kinetics of microtubule assembly versus that of motor-driven remodeling would be helpful.
B) It is not evident why the width W(t) is used as a read-out for the densification. It seems that fluorescence intensity would be a much more direct measure of concentration. In particular, since the ε does not take into account the thickness of the gel. xz/yz-projections could help to justify this choice.
D) Viscoelastic and poroelastic models can be nicely ruled out by showing the dependence of (W0). The new model clearly is more consistent with the data, but actually does not reproduce the trend very well. The authors should critically comment on this, as well as on other aspects of their model.
E) The model derivation (assumptions, s, 0) should be better explained in the main text to allow the reader to follow the reasoning.
F) The authors should provide a more detailed explanation of what s and 0 depend on, and give a simple estimate of physically reasonable values. In general, the authors should comment on the connection between the parameters in the continuum theory with microscopic parameters. As it stands, the best-fit parameter values (Results) are meaningless.
G) It is unclear how alpha and beta are derived. What are the input parameters? (Please see the subsection “Scaling analysis of the equations of motion”).
I) The cytochalasin D concentration should be specified.
J) For the PIV analysis, the settings should be specified.
K) In Figure 5C: is the initial systematic decrease of ε∞a real effect?
8) What are the implications of this new mechanism for other, more well studied, contractile systems, in particular actin-myosin?
9) The statements on alternative methods of study appear to be a bit superficial. The authors are encouraged to discuss better the shortcomings of their approach.https://doi.org/10.7554/eLife.10837.018
1. The claim "Our results suggest that the dynein-driven clustering of microtubule minus ends causes both aster formation and network contraction." (Line 94 p4) is not sufficiently well supported by the data. Firstly, the aster nature of the structures is only weakly supported by the data (the data being shown in Figures 1B and C; especially panel 1C is unconvincing); it would be more convincing to visualize the MT minus ends or the motors. Secondly, the claim that dynein drives minus-end clustering is not well-supported since dynein inhibition only slows contraction, but does not stop it. Is the inhibition not 100% effective, or are other processes involved in contraction?
We thank the reviewers for their comments. In order to address the aster nature of the structures, we repeated the contraction experiment, but with the addition of Alexa Fluor 488 anti-NUMA antibodies. It has previously been shown that NUMA forms a complex with dynein  localizes to the cores of Taxol asters in mitotic extracts  and has been used as a minus end marker to infer microtubule polarity . Figure 1 was updated to include results showing NUMA localization. Figure 1B illustrates that isolated asters in our system have NUMA at their core as well. As evidence that microtubules in this system can form asters that can fuse together, a basic assumption of our model, Figure 1C and Video 1 were added, directly showing this process for isolated asters. An example field of view for the bulk network is displayed in Figure 1F, where tubulin is shown in magenta and NUMA is shown in green. From this image, one can see that NUMA tends to lie in the interior of the structures, arguing that the microtubule minus ends are being focused in the interior, as is the case in asters, providing further evidence of their aster-like nature.
The data presented argue that dynein contributes strongly to the contraction process due to the fact that the contraction timescale can be tuned from a few minutes to well over an hour by inhibiting dynein with varying amounts of p150-CC1. p150-CC1 is a subunit of the dynactin complex that inhibits dynein activity by competing off full length dynactin. Due to this mechanism of action, it remains possible that the entire pool of dynein in the extracts is not being fully inactivated, even with the addition of 2 μM p150-CC1. Thus, it remains possible that the inhibition is not 100% effective and that some small residual pool of active dynein is sufficient to drive contraction, albeit at much slower timescales. A second possibility is that there exists another motor protein that is sufficient to drive contraction in the absence of dynein activity. One possible candidate would be Kinesin-14, a minus end directed motor that can crosslink and slide microtubules, organizes microtubules into asters in purified systems, and is known to be present in Xenopus extracts where it has been shown to have a small effect on spindle morphology . However, as the timescale is proportional to 1/s, where s is the strength of the active stress, if we compare the measured timescales between the case with 2 μM p150-CC1 and the case with no added p150-CC1, then it suggests that in the case where dynein is inhibited, the active stress is only ~ 4% of the active stress in the unperturbed case. Thus, even if there is another motor partially involved in the contraction process, ~96% of the stress driving the contraction is being generated by dynein.
In order to clarify this point, the following has been added to the main text.
“We note that even at the highest p150-CC1 concentrations used, the network still undergoes a bulk contraction. This could possibly be due to incomplete inhibition of dynein by p150-CC1, or by another motor protein present in the extract that also contributes to the contraction process. As the characteristic time, τ ∝ 1/s, by comparing the characteristic times in the uninhibited and 2 μM p150-CC1 cases, we can estimate that the strength of the active stress, s, in the 2 μM p150-CC1 condition is only ≈ 4% of the strength of the active stress in the uninhibited case, arguing that even if another motor is involved in the contraction, dynein contributes ≈ 96% of the active stress.”
2.The biological relevance of the work must be made more explicit. The authors claim that the observation of dynein-driven clustering of microtubule minus ends has "strong implications for understanding the role of dynein in spindle assembly and pole formation." It is, however, not clear specifically how the insights obtained here can be translated to those situations. The statement on p10 lines 259 "suggesting that dynein is performing the same activity in spindles as in the contracting networks" is particularly unclear.
A substantial amount is known about the behavior of dynein, both from its behavior in purified systems  as well as in spindles . Much of the previous work has focused on dynein’s ability to slide microtubules and cluster minus ends together, leading to either aster or spindle pole formation. Models based on this view, e.g. so called “slide-and-cluster models”  have had some success describing spindle pole formation, but are essentially one dimensional models and consequently cannot address aspects of the spindle pole formation process such as the “pinching down” of the poles. In this work, we showed how the behavior of individual filaments leads to emergent contractile stresses, and based on our data we think that dynein could be contributing contractile stresses in spindles. By thinking about motor activity in terms of stresses, one has a framework from which to consider more macroscopic aspects of spindle pole formation. For example, in our system, inhibiting dynein leads to a lower contractile stress. Applying this idea to spindles naturally explains why spindle poles unfocus in the absence of dynein activity. Furthermore, the form of the active stress presented here naturally gives insight into how dynein activity regulates the density of microtubules.
To clarify the statement on “dynein performing the same activity” on p10 line 259, the relevant sentence has been changed as follows:
“Fitting a sigmoid function to the τ vs. p150-CC1 concentration curve yields an EC50 value of 0.22 ±. 02 μM (mean ± standard error), similar to the value of ≈ 0.3 μM reported for the effect of p150-CC1 on spindle length in Xenopus extracts (Gaetz and Kapoor, 2004), which is consistent with active stress generated by dynein being required for pole focusing.”
3. To develop the model (5.1.2), the authors consider point-like objects in 1D, which are said to represent the "minus ends of microtubules". However the attractive interaction between these points extend symmetrically left and right, as if the microtubules were not orientated. The effective steric interaction between microtubules is also only dependent on the position of the minus-end, ignoring the fact that it is really the polymer mass that would induce steric forces, and not just the minus-ends. The plus-ends would be equally relevant. In other words, it seems in this theory that the minus-end located in the middle of the filament. To avoid this inconsistency with the accepted molecular picture, the authors should consider reformulating the model as a theory of many asters coming together. The basic unit would not be a microtubule, but an bunch of them already connected at their minus-ends. The variables 'Xi' could represent the positions of the centres of these asters, and assuming that they are symmetric, it would be justified to use the functional form of the interactions that are surmised. Doing so will not require changing the algebra in the theory, and rewriting its description would be sufficient.
The referee is correct. Our microscopic model considers all orientations of microtubules to occur with equal likelihood. This choice is consistent with experimental observations, which do not show strong alignment occurring during contractions. We assume that interactions are statistically isotropic, and make the further simplifying abstraction that steric interactions can be usefully captured by only accounting for MT minus-ends. These assumptions yield a simple model that accounts for the experimental observations.
As the referee correctly points out, our model could be rephrased in terms of higher order fundamental units such as MT-asters, without changing the algebra. We could then (consistently with experiments) postulate that asters stay circular over the contraction process, such that aster-aster interactions occur with equal likelihood in all directions. The aster based formulation, is equivalent to the MT-minus end formulation in all aspects, including the isotropy assumptions required.
In the revised version of the manuscript we chose to keep our explanation of the origin of contractions formulated in terms of MT minus ends as the fundamental units, since this framework allows to explicitly name the microscopic processes that we think are central to this system.
To address the referee's point, we now explicitly state that under the assumption of orientational isotropy, the effects of relative orientations of microtubules average out. Furthermore, we added a sentence in the main text stating the equivalence between our MT-minus end based description and a description constructed by tracking aster cores.
4. It seems that the contraction is a two-step process in which microtubules first form localized aster. Subsequently, these asters interact with each other in order to form a bulk contractile gel. Their proposed mechanism of end motor clustering explains local contractions that lead to formation of asters. However, presumably once the asters are formed all the microtubule ends are localized at the center of asters. In view of this information, please explain better why interacting asters would form a bulk contractile structure.
Our interpretation is that microtubules can extend from one aster into the the core of a neighboring aster where forces can be exerted that drive the aster cores together (see Video 1). If there are many asters all trying to coalesce with their neighbors, then the stress on the entire aster network will necessarily be net contractile. However, we don’t think that this process occurs in two discreet steps, where first asters are made and then the asters interact. Rather, what we think is that both of these processes are happening simultaneously and the asters are beginning to interact and coalesce before they are completely formed.
In order to clarify that neighboring asters can fuse, even if the asters are isolated from the network, Video 1 and Figure 1C have been added showing this process occurring in asters that form outside of the main contracting network.
5. Model and data description:
a. Regarding the kinetics of contraction, several important aspects are unclear: what defines t=0? What is the origin of the lag time? And what process actually triggers constriction? A more detailed description of the kinetics of microtubule assembly versus that of motor-driven remodeling would be helpful.
For each experiment, Taxol is added to the extract before the extract is loaded into the microfluidic channels. The channels are then sealed to prevent evaporation, and then placed on the microscope before imaging begins. t=0 is defined as the time that imaging begins, and it occurs approximately one minute after Taxol is added to the extract. The lag time includes contributions from many components, including the time it takes between Taxol addition and the beginning of imaging, the time it takes for microtubules to assemble, and the time for the microtubule network to percolate and remodel. Other than the addition of Taxol to the extract, there is no process that we do that actually triggers the constriction; it happens spontaneously.
To better explain this, to section 4.4 Bulk Contraction Assay the following text has been added.
“t=0 is defined when the imaging begins, ≈ 1 minute after Taxol addition to the extract. After a brief lag time, the microtubule networks spontaneously begin contraction”
b. It is not evident why the width W(t) is used as a read-out for the densification. It seems that fluorescence intensity would be a much more direct measure of concentration. In particular, since the epsilon does not take into account the thickness of the gel. xz/yz-projections could help to justify this choice.
We consider both the width W(t) (Figures 2,3,6) as well as the fluorescence intensity (Figure 5) as read-outs for densification, and model parameters extracted from fitting the characteristic time scaling with channel width found from fitting W(t) are in agreement with the parameters extracted from the density profile fitting in Figure 5, arguing that there is no systematic bias in choosing one readout over the other. In addition, our value for the final network density, ρ0, comes from measurements based on using fluorescence in the network as a proxy for tubulin concentration.
One example fit of epsilon(t) was shown in the inset to Figure 1F, though it was difficult to see both the data and the fit due to the size of the lines used. In order to make it more clear, the thickness of the data line was increased. Fits of epsilon(t) for all of the data previously in Figure 1 are now included as Figure 2 - figure supplement 1 and as a representative of the data previously shown in Figure 5, fits of epsilon(t) for the 2 μM p150-CC1 data are included as Figure 6 - figure supplement 2. In all cases, the data are adequately described by the exponential fits.
d.Viscoelastic and poroelastic models can be nicely ruled out by showing the dependence of tau(W_o). The new model clearly is more consistent with the data, but actually does not reproduce the trend very well. The authors should critically comment on this, as well as on other aspects of their model.
We agree that the fit to the tau vs W_0 data does not perfectly reproduce the data, but only the general trend. This could be due to several reasons, including the relative simplicity of our model. It would be possible to extend the model so that the fit is improved, but at the cost of additional model complexity. With the simplicity of the model taken into account, we find the close agreement between the parameters found by fitting the tau vs W_0 data and the parameters extracted from the density profile fits to be remarkable.
e. The model derivation (assumptions, s, Rho0) should be better explained in the main text to allow the reader to follow the reasoning.
We agree, and followed the referees advice. The discussion of the model in the main text has been expanded and enhanced.
f. The authors should provide a more detailed explanation of what s and Rho0 depend on, and give a simple estimate of physically reasonable values. In general, the authors should comment on the connection between the parameters in the continuum theory with microscopic parameters. As it stands, the best-fit parameter values (e.g. bottom p8 and p9) are meaningless.
ρ0 represents the microtubule density at which the dynein driven attraction between minus ends balances the steric repulsion between filaments, and is presumably set by the packing of microtubules. In Figure 6C, we show ε∞ and hence ρ0 to be independent of the total concentration of motor proteins. We also provided an estimate for ρ0, but were not clear on this point. To clarify that 30 μM is an estimate for ρ0, the text was modified as follows.
“By using fluorescence intensity as a proxy for tubulin concentration (see Materials and Methods), we estimate the final concentration of tubulin in the network to be ρ0 ≈ 30 μM.”
To estimate the magnitude of the active stress, we begin with our estimate of ρ0 = 30 μM = 3 x 10-20 mol/μm3. We then estimate the average length of a microtubule to be ≈ 6.5 μm. Since microtubules have 1625 heterodimers per μm  we can estimate that there are ≈ 1.8 x 10-20 mol/MT. Combining these two estimates leads to a microtubule density of ≈ 1.7 MT/μm3. We note that this is also the density of microtubule minus ends. If there are γ dynein per microtubule minus end, each dynein exerts an average force of ≈ 1 pN, and the characteristic interaction length is the average length of a microtubule ≈ 6.5 μm, we can estimate the dipole moment per microtubule to be 6.5 γ pN⋅μm/MT. Combining this with our microtubule density measurement leads to an estimate of the active stress, sρ02 ≈ 11 γ Pa. As an estimate of the viscosity of the microtubule network, we take a value measured in spindles in Xenopus extracts, η ≈ 2 x 102 Pa⋅s . Combining this with the value of η / sρ02 = 0.82 min we measure, gives an estimate of sρ02 ≈ 4 Pa. Combining this with our stress estimate derived above from microscopic considerations gives an estimate of γ ≈ 0.4 motors per minus end. Thus, our measured timescale is consistent with our measured microtubule density given an average force per motor of ≈ 1pN and 0.4 dynein per minus end. To provide an estimate of the total active stress as well as things it will depend on, the following has been added to the main text.
“Combining this measurement with an estimate for the network viscosity taken from measurements in spindles of η ≈ 2 x 102 Pa⋅s (Shimamoto et al., 2011), we can estimate the dynein generated active stress to be sρ02 ≈ 4 Pa which is consistent with having ≈ 0.4 dynein per microtubule minus end each exerting an average force of 1 pN (Nicholas et al., 2015).”
g. It is unclear how alpha and beta are derived, what are the input parameters? (see p22)
In principle alpha and beta can depend on dimensionless combinations of all model parameters and thus need to be determined numerically. To obtain alpha and beta for a given set of input parameters we numerically evaluate the time scale τ(W0) for several initial widths and fit the results to Appendix Eqn 40.
To estimate for alpha and beta in the experimental regime, we sampled a large number of sets of the input parameters around the best fit values for which best agreement between experimental and theoretical contraction dynamics was found (Figure 5B), varying all parameters by a factor of up to 4. We find that throughout this large region of parameter space alpha and beta vary only weakly and report their mean value and standard deviation.
We thank the referee for pointing out our lack of clarity and we now better explain this procedure in the Appendix.
In this work we approximate the microtubule gel as a smooth continuum, and do not account for the local structure which exists on length scales smaller then typically 100 micron. Such a description is appropriate since, both the size of the system (cms) and the length over which local contractions propagate in the film (, mm) are much larger then the local structures (10-100 microns).
i. The cytochalasin D concentration should be specified.
Cytochalasin D was added to the extracts at a final concentration of 10 μg/mL, as is standard . The concentration is now given in the text.
j. For the PIV analysis, the settings should be specified.
Section 4.7 Particle Imaging Velocimetry has been updated to include the settings and algorithm used. The relevant sentence now reads:
“Particle Imaging Velocimetry was performed using PIVLab software (Thielicke and Stamhuis, 2014) using the FFT window deformation algorithm with a 16 pixel interrogation area and 8 pixel step for the first pass and an 8 pixel interrogation area with a 4 pixel step for the second pass.”
k. In Figure 5c: is the initial systematic decrease of epsilon_infinity a real effect?
While we cannot conclusively demonstrate whether or not the effect is real, we suspect that it is not. Still, the effect is interesting and is an aspect worth following up on. In either case, the main trend that epsilon_infinity does not appreciably vary with dynein inhibition is clearly shown, and is consistent with the proposed theoretical model.
6. What are the implications of this new mechanism for other, more well-studied, contractile systems, in particular actin-myosin?
We find the comparisons between the similarities and differences between the contracting microtubule networks shown here and the more well studied contractions in the actin-myosin system to be a fascinating area of consideration. On the microscopic scale, clear differences exist between the two systems, including the orders of magnitude greater persistence length of microtubules relative to actin, and the fundamental differences between myosin-filaments, which can have many myosin heads functioning in parallel, and dynein, which has only two motor domains. Here we present data that argues motor induced minus end clustering leads to active stresses that drive the contraction process. As far as we’re aware, it is not known how myosin interacts with the ends of actin filaments. Thus, the mechanism of end clustering contributing to contractile stresses could be a plausible mechanism for the actin system as well, in addition to mechanisms previously proposed.
To further elaborate on the connection between our system and contractile actin networks the following has been added to the main text.
“Furthermore, the considerations of the model are general, and it will be interesting to consider whether the end clustering mechanism proposed here could contribute to contraction in actin networks as well.”
7. The statements on alternative methods of study appear quite shallow. The authors are encouraged to discuss better the shortcomings of their approach.
We view our work as a complementary approach to other methods of study, and there are several other approaches that could be used to study microtubule organization, each with their own advantages and disadvantages. One alternate approach would be to measure stresses in the spindle directly to test whether the net motor generated stresses are contractile or extensile. This approach has the clear advantage of being the most biologically relevant measurement, however it would be extremely challenging technically. Furthermore, stresses in the spindle might reflect the coupled effects of motor activity with contributions from other factors, e.g. microtubule polymerization. Still, it will be interesting to see, at least from a theory perspective, to what extent the dynamics of spindle assembly can be quantitatively understood using a framework based on the dynein induced contractile stress presented here. Another possible approach would be to purify fluorecently labeled dynein and study how it slides pairs of filaments and to measure the behavior of dynein at the filament minus ends. This has the advantage of giving direct insights into how dynein dictates the motion of individual filaments, yet has the disadvantage of the difficulty in relating this microscopic behavior with mesoscale network organization. Our approach has the advantage of directly considering large scale network behaviors, but the disadvantage of top down approaches like the one considered here is that microscopic behaviors of individual components are inferred rather than directly measured. To fully understand the behaviors of a complex system like the one considered here, an integrated approach is needed that combines results considering the system from different perspectives.
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- Daniel J Needleman
- Daniel J Needleman
- Daniel J Needleman
- Michael J Shelley
- Michael J Shelley
- Sebastian Fürthauer
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
The authors would like to thank Bryan Hassell for assistance fabricating the microfluidic devices, Thomas Surrey for the generous gift of the GST-p150-CC1 plasmid, and Tim Mitchison for the gift of labeled NUMA antibody. SF acknowledges support by Human Frontiers Science Program. This work was supported by National Science Foundation Grants PHY-0847188, PHY-1305254, and DMR-0820484 to DJN and Grant DMR-1420073 to MJS, and National Institutes of Health Grant 1R01GM104976-01 to MJS.
Animal experimentation: All animals were handled according to approved institutional animal care and use committee (IACUC) protocols (#28-18) of Harvard University.
- Anna Akhmanova, Utrecht University, Netherlands
© 2015, Foster 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.
Genotype imputation is a foundational tool for population genetics. Standard statistical imputation approaches rely on the co-location of large whole-genome sequencing-based reference panels, powerful computing environments, and potentially sensitive genetic study data. This results in computational resource and privacy-risk barriers to access to cutting-edge imputation techniques. Moreover, the accuracy of current statistical approaches is known to degrade in regions of low and complex linkage disequilibrium. Artificial neural network-based imputation approaches may overcome these limitations by encoding complex genotype relationships in easily portable inference models. Here we demonstrate an autoencoder-based approach for genotype imputation, using a large, commonly used reference panel, and spanning the entirety of human chromosome 22. Our autoencoder-based genotype imputation strategy achieved superior imputation accuracy across the allele-frequency spectrum and across genomes of diverse ancestry, while delivering at least 4-fold faster inference run time relative to standard imputation tools.
How cells control gene expression is a fundamental question. The relative contribution of protein-level and RNA-level regulation to this process remains unclear. Here, we perform a proteogenomic analysis of tumors and untransformed cells containing somatic copy number alterations (SCNAs). By revealing how cells regulate RNA and protein abundances of genes with SCNAs, we provide insights into the rules of gene regulation. Protein complex genes have a strong protein-level regulation while non-complex genes have a strong RNA-level regulation. Notable exceptions are plasma membrane protein complex genes, which show a weak protein-level regulation and a stronger RNA-level regulation. Strikingly, we find a strong negative association between the degree of RNA-level and protein-level regulation across genes and cellular pathways. Moreover, genes participating in the same pathway show a similar degree of RNA- and protein-level regulation. Pathways including translation, splicing, RNA processing, and mitochondrial function show a stronger protein-level regulation while cell adhesion and migration pathways show a stronger RNA-level regulation. These results suggest that the evolution of gene regulation is shaped by functional constraints and that many cellular pathways tend to evolve one predominant mechanism of gene regulation at the protein level or at the RNA level.