Physical basis of spindle self-organization

Significance

The spindle segregates chromosomes during cell division and is composed of microtubules and hundreds of other proteins, but the manner in which these molecular constituents self-organize to form the spindle remains unclear. Here we use a holistic approach, based on quantitative measurements in spindles of the spatiotemporal correlation functions of microtubule density, orientation, and stresses, to identify the key processes responsible for spindle self-organization. We show that microtubule turnover and the collective effects of local microtubule interactions, mediated via motor proteins and cross-linkers, can quantitatively account for the dynamics and the structure of the spindle. We thus reveal the physical basis of spindle self-organization and provide a framework that may be useful for understanding cytoskeletal function in vivo.

Abstract

The cytoskeleton forms a variety of steady-state, subcellular structures that are maintained by continuous fluxes of molecules and energy. Understanding such self-organizing structures is not only crucial for cell biology but also poses a fundamental challenge for physics, since these systems are active materials that behave drastically differently from matter at or near equilibrium. Active liquid crystal theories have been developed to study the self-organization of cytoskeletal filaments in in vitro systems of purified components. However, it has been unclear how relevant these simplified approaches are for understanding biological structures, which can be composed of hundreds of distinct proteins. Here we show that a suitably constructed active liquid crystal theory produces remarkably accurate predictions of the behaviors of metaphase spindles—the cytoskeletal structure, composed largely of microtubules and associated proteins, that segregates chromosomes during cell division.

Continuum theories form the basis of our understanding of much of the material world, but it has been unclear if such theories can be used to study self-organizing biological structures, due to the complexity and fundamentally nonequilibrium nature of these systems. One example of these structures that has been extensively studied for over a century is the metaphase spindle, an ensemble of microtubules, molecular motors, and other associated proteins that segregates chromosomes during cell division. Hundreds of proteins have been found to contribute to spindle assembly (1), but while various principles have been proposed to explain how these constituents self-organize to form the spindle—including gradients of signaling molecules (2), a mechanical matrix (3), and the regulated “feeding” of microtubule depolymerizers by microtubule sliding (4)—the physical basis of spindle assembly is currently unknown (5). Complementary to this in vivo work, in vitro experiments have shown that mixtures consisting solely of purified cytoskeletal filaments and motors can spontaneously self-organize into structures (6, 7) and display complex dynamics (8–10). Sophisticated theories of these simplified systems have been developed to explain how the collective effects of the local interactions of microtubules, mediated by motors, give rise to these large-scale behaviors (11, 12). It is unclear if the principles learned from these reconstituted systems apply to the self-organization of the spindle and other cytoskeletal structures in vivo.

Here we apply a holistic approach to study the physical principles that give rise to spindle self-organization by quantitatively studying the collective behaviors of microtubules in spindles. Our approach uses the intimate connection between spatiotemporal correlation functions of the spontaneous fluctuations of microtubule density, orientation and stress, and the underlying physical processes that drive them. Comparing measurements of the correlation function with predictions from theory provides both a rigorous test of the validity of that theory and a sensitive means of determining its parameters. Similar approaches are commonly used in condensed matter physics to quantitatively test explanations of phenomena (13) and provide the basis of several widely used experimental techniques, such as Fluorescence Correlation Spectroscopy and Dynamic Light Scattering. We therefore sought to experimentally measure correlation functions associated with density, orientation, and stress in metaphase arrested spindles assembled in Xenopus laevis egg extracts. All of our measurements on correlation functions and spindle morphology can be quantitatively accounted for using an active liquid crystal theory that we construct. Our combined theoretical and experimental approach provides a general framework for understanding the structure and dynamics of the spindle and its responses to physical and molecular perturbations.

We used an LC-Polscope (14), a form of polarized light microscopy, to quantitatively measure the retardance (the birefringence integrated over the optical volume) and the optical slow axis, also called the nematic director (a measure of microtubule orientation), at every pixel in time-lapse movies of spindles (Fig. 1A and Movies S1 and S2). LC-Polscope movies reveal that at steady state, microtubules are highly oriented along the spindle long axes, and display large orientational fluctuations (see Movie S1). We used these movies to compute the spatiotemporal correlations of director fluctuations (see Methods Summary and Fig. 1 A and B) and found that the equal time director autocorrelation function decays with wavelength as $1/q^2$ and decays with frequency as $1/{\omega }^2$ (for $q\to 0$). Fluctuations of this form indicate that nearby microtubules tend to orient in the same direction while microtubules that are farther apart are less well aligned, and are quantitatively consistent with microtubules being oriented by their local interactions from cross-linkers, motors, and steric effects (see SI Text) (12, 15). This form of the director autocorrelation function is inconsistent with microtubules being embedded in an elastic matrix (which would cause a plateau in the orientational correlation function at long wavelength due to elastic deformation of the matrix), or microtubules being aligned independently of each other by an external field or global signal (which would result in a flat equal time correlation function because local fluctuations would not collectively accumulate).

Fig. 1.

Measurement of director orientation, density, and two-point correlation functions. (A) LC-Polscope images provide a quantitative measurement of the director orientation (Left) and retardance (Right). The red square illustrates a typical region used to calculate the spatiotemporal correlations of the orientation fluctuations (dashed white arrow) with respect to the mean orientation at that location (solid white arrow). See also Movies S1 and S2. (B) (Left) Equal time director autocorrelation functions from 14 spindles as a function of the wave number, $q_{\perp }$, for $q_z=0.39$ rad/μm, the lowest wave number measured along this direction (<<$q_{\perp }$). Dashed line represents a $1/q_{\perp }^2$ decay. (Right) Long-wavelength limit of the spatiotemporal correlation as a function of the frequency. Dashed line represents a 1/ω² scaling. (C) We used 3D confocal fluorescence time-lapse movies to measure spatiotemporal correlations of the microtubule density fluctuations. The red cube illustrates a typical 3D region used to measure these correlations. See also Movie S3. (D) (Left) The amplitude of the equal time density autocorrelation function (n = 14 spindles) plateaus for long wavelengths and decays as$1/q_{\perp }^4$ for short wavelengths (dashed black lines) and in marked contrast to the $1/q_{\perp }^2$ scaling predicted without turnover (dashed red line). (Right) Long-wavelength limit of the spatiotemporal correlation. Dashed line represents a 1/ω² scaling. (E) (Left) A frame of single molecule time-lapse movie of fluorescently labeled microtubules. The dashed white contour represents the boundary of the spindle. Red circles highlight two single fluorescence molecules. Solid orange represents particle displacements (ε⁽¹⁾, ε⁽²⁾). Orange dashed line represents distance (R) between particles. (Right) Detail of two speckles and their spatial displacements over time (blue line). See also Movie S4. (F) Cross-correlations of pairs of tracer particles ($C_{\epsilon \epsilon }$; see Methods Summary) normalized by their initial distance as a function of time lag (τ) show superdiffusive behavior. Dashed line corresponds to an exponent of 1.8; each trajectory (13 in the figure) corresponds to the average two point cross-correlations of a spindle. (G) Spatial dependence of the two-point cross-correlations $C_{\epsilon \epsilon }$ for the smallest measured time lag, τ = 2.1 s, decays as 1/R, as expected in a continuum theory.

We measured microtubule density in spindles by obtaining 3D time-lapse spinning disk confocal microscopy movies of spindles labeled with high concentration of fluorescent tubulin (Fig. 1C and Movie S3). The equal time autocorrelation function of density fluctuations along the direction perpendicular to the spindle axis (⊥) plateaus for small $q_{\perp }$, and decays as $1/q_{\perp }^4$ for large $q_{\perp }$ (Fig. 1D). The relative sliding of microtubules, by the activity of cross-linking molecular motors (16), will produce a coupling between the orientation and density of microtubules, so fluctuations in orientation will lead to fluctuations in density. If the dominant processes that controlled microtubule arrangement in the spindle were this relative sliding and the tendency of microtubules to locally orient each other, revealed from our LC-Polscope measurements, then the density fluctuations would diverge as 1/q² (see SI Text and ref. 17), in marked contrast with their observed behavior (Fig. 1D). Microtubules in the spindle are also continuously being nucleated, growing, and shrinking (2). The combination of this rapid turnover with the mutual orientation and sliding of microtubules predicts density correlation functions of the form that is experimentally observed (see SI Text). The plateau in the density correlation function at small q arises because microtubules turn over too rapidly to be transported significant distances by sliding, so large-scale motor driven fluctuations are suppressed. The 1/q⁴ decay at short wavelengths results from the orientation and sliding interactions cooperating with stochastic, diffusive-like motions of microtubules (an intermediate 1/q² decay would be observed if either diffusion or orientation interactions dominate for intermediate wavelengths, but an appreciable regime of this nature is not seen, indicating that these two process have similar magnitudes). Thus, the scaling of the orientational and density fluctuations can be explained by the local, mutual interactions of microtubules sliding and orienting each other, while polymerizing and depolymerizing.

To study the production and propagation of forces, we measured stress fluctuations in spindles using a combination of passive two-point particle displacements and active microrheology measurements (18–20). This method relies on the generalized Stokes−Einstein equation to relate the correlated motions between pairs of particles to the mechanics of the intervening media and the stresses that drive their motion, and is valid for incrompressible, viscoelastic continuous media in the presence of both active and thermal stress fluctuations (19, 20) (see SI Text). We obtained two-point particle displacements by tracking single molecules of fluorescently labeled tubulin incorporated into microtubules in the spindle, and computed the two-point correlation between these single molecules along the direction perpendicular to the spindle axis (see Fig. 1 E and F). The two-point displacements decay as the inverse of the particle separation, R (Fig. 1G), consistent with stresses being propagated by the local interactions between microtubules [and as would be expected in any media that can be approximated as a continuum (19)]. These two-point displacements exhibit super-diffusive motion with an exponent α ∼1.8 (Fig. 1F), which, when combined with the active microrheology measurements of the frequency-dependent shear modulus of the spindle by Shimamoto et al. (21), reveals that stress fluctuations in the spindle increase linearly with time lag (see below). This spectrum of stress fluctuations is expected from motors exerting forces between microtubules they cross-link, giving rise to dipolar stresses (see SI Text and ref. 17).

Our investigation of the scaling of spatiotemporal correlation functions of microtubule orientation, microtubule density, and stresses in the spindle reveals the role of local interactions between microtubules in orienting, sliding, and generating and propagating stresses between microtubules, as well as the importance of microtubule turnover. To more thoroughly study these phenomena, we constructed a minimal model of these processes based on the shape of the correlation functions of density, orientation, and stress that is consistent with relevant conservation laws and the known symmetries of microtubules in the spindle, and explored predictions beyond scaling (see SI Text). The prediction for the equal time director autocorrelation function in the perpendicular direction to the spindle is $S_{\perp }^2/2K{q}_{\perp }^2$, and the frequency dependence of the director autocorrelation function (for $q\to 0$) is $S_{\perp }^2/{\omega }^2$, where $S_{\perp }$ is the magnitude of orientation noise (assumed Gaussian), and $K$ is the strength of the nematic elasticity, a measure of the tendency of local interactions to orient microtubules. Thus, these two parameters can be determined by fitting the director correlation functions (Fig. 2 A and B) leading to S_⊥ = 0.021 ± 0.002 rad s^3/2⋅μm^3/2, K = 0.022 ± 0.009 μm²⋅s^–1. The measured equal time density autocorrelation function is well fit by the predicted form, $S_{cc}=\{q_{\perp }^2{v}_0^2{c}_0^2{S}_{\perp }^2/\left[2\left(K^2{q}_{\perp }^4+{\left(\mathrm{\Theta }+\text{D}{q}_{\perp }^2\right)}^2\right)\right]\}·\left[1/\left(K{q}_{\perp }^2\right)+\left(1/\mathrm{\Theta }+\text{D}{q}_{\perp }^2\right)\right]$ (Fig. 2C; see SI Text), which has a plateau for small q of $S_{cc}\sim v_0^2{c}_0^2{S}_{\perp }^2/2K{\mathrm{\Theta }}^2$, where $c_0$ is the average density of microtubules, $v_0$ is the velocity of microtubule transport, and Θ is the turnover rate, and transitions to the $1/q_{\perp }^4$ regime at a wavelength determined by the square root of the ratio of $K$ and Θ, $S_{cc}\approx \left[\left(v_0^2{c}_0^2{S}_{\perp }^2\right)/\left(2q_{\perp }^4\right)\right]\{\left(D+K\right)/\left[KD\left(K^2+D^2\right)\right]\}$. Thus, this fit provides a means to measure the stationary flux, $c_0{v}_0$ = 3.3 ± 0.4 microtubules μm^–2⋅s^–1, the turnover rate, Θ = 0.06 ± 0.02 s^–1, and the coefficient characterizing diffusive-like motion of microtubules in the spindle, D = 0.022 ± 0.006 μm²⋅s^–1 (Fig. 2C). This value of Θ corresponds to a microtubule turnover time of $1/\mathrm{\Theta }$ ≈ 17 s, which, within error, is the same as the average lifetime of microtubules in the spindle measured from single molecule studies (24). The prediction for the spectrum of stress fluctuations, $\mathrm{\Delta }(\omega )=4W^2{c}_0^2{S}_{\perp }^2/{\omega }^2$, provides a good fit to our measurements (see Fig. 2H and SI Text), giving $W{c}_0$ = 69 ± 2 pN/μm², where W is the active stress per unit microtubule density. This magnitude of stress fluctuations is consistent with ∼1 molecular motor per micrometer of microtubule (assuming an average force per motor of ∼10 pN). The measured stress fluctuations are orders of magnitude greater, and have a different temporal dependence, than predicted from the equilibrium fluctuation dissipation relation (Fig. 2H, black line), based on thermal fluctuations and the measured rheology of the spindle (21). The strong violation of the equilibrium fluctuation dissipation relationship demonstrates the intrinsic out-of-equilibrium nature of the spindle. The linear temporal increase of the stress fluctuations corresponds to a $1/{\omega }^2$ frequency dependence, which has previously been reported for reconstituted cytoskeletal systems (25) and cells (19, 26).

Fig. 2.

Comparison between experimental data and expected periodogram of the theoretical predictions (see Methods Summary and SI Text) for density, director orientation, and stress correlation functions. Solid red lines show the fits used to determine the six parameters of the theory, and the dashed red lines are the resulting predictions. (A) Average equal time correlation functions for the director−director fluctuations obtained from the LC-Polscope, and corresponding fit to the theoretical prediction $S_{nn}=S_{\perp }^2/2K{q}_{\perp }^2$ (solid red line). (B) Average frequency dependence of the spatiotemporal correlation functions for the director−director fluctuations obtained from the LC-Polscope, and corresponding fit to the theoretical prediction, $C_{nn}=S_{\perp }^2/\left({\omega }^2+K^2{q}_0^4\right)$ (solid red line), where $q_0=0.39$ rad/μm is the lowest wave number measured. (C) Average equal time correlation functions for density−density fluctuations obtained from fluorescence (Fig. 1D, dark green) and LC-Polscope retardance measurements (light green). The fluorescence density correlations were rescaled to the retardance correlations, which have units of square nanometers. These correlations were fit to the theoretical prediction for the equal time density correlations, $S_{cc}=\{\left(q_{\perp }^2{v}_0^2{c}_0^2{S}_{\perp }^2\right)/\left[2\left(K^2{q}_{\perp }^4+{\left(\mathrm{\Theta }+\text{D}{q}_{\perp }^2\right)}^2\right)\right]\}\left[1/K{q}_{\perp }^2+1/\left(\mathrm{\Theta }+\text{D}{q}_{\perp }^2\right)\right]$ (solid red line). To convert density to number of microtubules per volume, we normalized $c_0{v}_0$ by the average retardance and rescaled it to the number of microtubule per unit volume using EM images of Xenopus spindles (22), 7.14 microtubules/μm³ [50 microtubules/μm² in a cross-section, divided by the average microtubule length in a spindle, 7 μm (23)]. (D) Average frequency dependence of the spatiotemporal correlation functions for density−density fluctuations obtained from the retardance measurements. These correlations agree quantitatively with the theoretical prediction for the frequency-dependent density correlations, $C_{cc}=\left(q_0^2{v}_0^2{c}_0^2{S}_{\perp }^2\right)/\{\left({\omega }^2+K^2{q}_0^4\right)\left[{\omega }^2+{\left(\mathrm{\Theta }+\text{D}{q}_0^2\right)}^2\right]\}$ (dashed red line). (E and F) Theoretical predictions (dashed red lines) for the equal time and frequency dependence spatiotemporal density−director cross-correlation functions, (E) $\left|S_{cn}\right|=\left(v_0{c}_0{S}_{\perp }^2\right)/\left\{2K{q}_{\perp }\left[\mathrm{\Theta }+\left(\text{K}+\text{D}\right)q_{\perp }^2\right]\right\}$ and (F) $\left|C_{cn}\right|=\left(v_0{c}_0{q}_{\perp }{S}_{\perp }^2\right)/\left\{\left({\omega }^2+K^2{q}_0^4\right){\left[{\omega }^2+{\left(\mathrm{\Theta }+\text{D}{q}_0^2\right)}^2\right]}^{1/2}\right\}$, respectively, and their corresponding experimental measurements. (G) Spatial dependence of the two-point cross-correlations for the smallest measured time lag (Fig. 1G) and its theoretical prediction $C_{\epsilon \epsilon }=A/R,$ where $A$ is a constant (see SI Text). (H) Temporal dependence of the stress fluctuations and its theoretical fit, $\mathrm{\Delta }\left(\tau \right)=2W^2{c}_0^2{S}_{\perp }^2\tau$. Black line corresponds to the prediction from the fluctuation dissipation theorem. Error bars are SDs.

As described above, if microtubule movements are driven by their mutual, local interactions, then the orientation and density of microtubules will be coupled. This coupling should be fundamental for driving the dynamics of microtubules in the spindle because large orientational fluctuations, which result from the collective accumulation between microtubules, should dominate other sources of density fluctuations. To explore the validity of these ideas and to directly probe this coupling, we sought to measure the cross-correlation between microtubule density and orientation in the spindle. To experimentally determine density−director cross-correlations, it is necessary to simultaneously measure both these fields, which is possible using the pair of images simultaneously provided by the LC-Polscope: The orientation of the optical slow axis is the orientation of the director, while the retardance is proportional to microtubule density, as expected from of the high degree of ordering of microtubules in spindles and as is directly demonstrated by the quantitative agreement between the retardance and fluorescence autocorrelation functions (Fig. 2C). The measured equal time and frequency-dependent (for $q\to 0$) density−director cross-correlations are in quantitative agreement with predictions using the previously measured parameters (Fig. 2 E and F), thus strongly supporting the validity of the proposed mechanisms. Alternatively, simultaneously fitting all eight correlation functions (wavelength and frequency dependence of density−density, director−director, density−director, and stress) does not significantly change the results. Explaining these eight correlation functions with a theory with only six parameters is a strong validation of the theory because even if all of the correlation functions were simple power laws (and they actually display more complex structure), it would be necessary to use 16 free parameters to empirically characterize the measured curves. In conclusion, all of our quantitative measurements of microtubule orientation, density, and the generation and propagation of stress in the spindle are consistent with spindle self-organization arising from the local interactions of microtubules, mediated by cross-linkers and motors, and microtubule polymerization dynamics.

Having shown that the local interactions of microtubules and microtubule turnover are sufficient to account for the internal dynamics of the spindle, we next sought to investigate if these same processes could explain the morphology of the spindle. In this view, the spindle is similar to a droplet of liquid crystal, but it is composed of a nonequilibrium material with properties determined from our measurements of correlation functions. Building off of theories of the shape of liquid droplets (27, 28), appropriately modified to account for the active nature of the spindle, we approximated the spindle as an ellipse with constant density and fixed volume, and calculated the distribution of microtubule orientation and polarity inside the spindle and the aspect ratio of the spindle. The orientation of microtubules is governed by nematic elasticity—arising from the tendency of microtubules to mutually align each other—and is thus solely determined by the solution of Laplace’s equation with the appropriate geometry and boundary conditions, which we take to be tangential anchoring with two half defects at the poles (see SI Text). The calculated orientation of microtubules throughout the spindle quantitatively agrees with our LC-Polscope measurements (Fig. 3 A and B), which is a strong confirmation of the theory, as this prediction is not based on a fit and in fact involves no parameters at all. We reproduced the observed spatial variation of polarity (23) by imposing vanishing polarity at the center of the spindle and fitting a single parameter given by the ratio of polarity-dependent active transport and the preferred value of polarity due to motor activity (Fig. 3C; see SI Text). Finally, we calculated the aspect ratio of the spindle by balancing the active stress from motor activity with surface tension, which is a consequence of microtubules cross-linking each other. A surface tension of 143 ± 24 pN/μm reproduced an aspect ratio of 1.7 ± 0.5 and a shape that closely agrees with observation (see Fig. 3D and SI Text). This quantitative agreement between calculations and measurements demonstrate that local interactions between microtubules are sufficient to account for the morphology of the spindle.

Fig. 3.

Shape of the spindle: director orientation, polarity, and boundary. (A) Prediction of director orientation in the spindle (Left) and corresponding measurement (Right) from averaging ∼5 min of an LC-Polscope movie (100 frames) of a spindle. (B) Spatial dependence of director orientation for several sections in the spindle. Red dot in right plot corresponds to the same orientation value in the three sections. Error bars correspond to SDs. (C) Polarity measurements from ref. 23 and corresponding theoretical fit. (D) Representative retardance image of a spindle with the best fit to an ellipse (solid red).

The active liquid crystal theory of the spindle that our work has validated should be a powerful framework for understanding the spindle. The theory provides a basis for investigating the wide range of spindle phenomenology that has been observed, such as the fusion of two spindles (29) or the response of the spindle to physical perturbations (30). Molecular perturbations should act to change the parameters of the theory, such as K, the orientational elasticity, or W, the strength of the active stress, which could result in changes in spindle structure and dynamics. While additional work will be required to predict which parameters will be affected by specific molecular perturbations, this could be investigated empirically by using fluctuations to measure how molecular perturbations influence the theory’s parameters. More broadly, the success of such a simple description demonstrates that, despite the extreme molecular complexity of spindles (31), their structure and dynamics at cellular scales are quantitatively accounted for using just a few effective parameters and argues that active liquid crystal theories are a promising route for developing predictive theories of cell biology (32).

Methods Summary

Spindle Assembly.

We prepared CSF-arrested egg extracts from Xenopus laevis female oocytes as described previously (33). To measure microtubule density fluctuations, we added 0.5 μM Atto565-labeled tubulin to spindles and acquired 3D fluorescence movies. For two-point particle tracking, we added ∼100 pM Alexa647-labeled tubulin. We used a spinning disk confocal microscope (Nikon Ti2000), an EMCCD camera (Hamamatsu), and a 60× objective for acquisition of 3D fluorescence and two-point microrheology images. We used a LC-Polscope and a 100× objective for acquisition of orientation and retardance images.

Image Analysis.

Before computing the spatiotemporal correlations, we registered spindles from a time lapse using a custom Matlab (The MathWorks) routine (34). For the two-point particle tracking measurements, we used custom-written Matlab code and routines from people.umass.edu/kilfoil/downloads.html (35).

Computation of Correlation Functions.

We computed spatiotemporal correlations (3 space + 1 time for fluorescence and 2 space + 1 time for retardance movies) by first subtracting the temporal mean at each pixel of a time-lapse movie, and then using the periodogram formula $C\left(\overrightarrow{q},w\right)=F\left(\delta i\left(\overrightarrow{x},t\right)\right)conj(F\left(\delta i\left(\overrightarrow{x},t\right)\right)/V$, where δi is the fluctuation as a function of space and time, V is the product of time and space dimensions, and F is the discrete fast Fourier transform. We corrected for distortions in the empirically measured correlation functions due to the point spread function and finite exposure time by dividing the data by the Fourier transform and power spectrum of these response functions. When comparing theoretically predicted and measured correlation functions, we compensated for artifacts caused by calculating power spectra of finite data by analytically computing the expected periodogram of the theoretical prediction. Alternatively, applying different windowing functions to the data did not significantly change any of our results (36). We defined the cross-correlation of two pairs of tracer particles as $C_{\epsilon \epsilon }\left(R,t\right)=\frac{1}{2}<{\epsilon }^{\left(1\right)}\left(t\right){\epsilon }^{\left(2\right)}\left(t\right)>$, where ε⁽¹⁾ and ε⁽²⁾ are the displacements with respect to their initial position as a function of time of two tracer particles initially separated a distance R apart (see Fig. 1E). See Stress Correlation Functions and Two-Point Microrheology.