# Physical basis of spindle self-organization

^{a}School of Engineering and Applied Sciences, Department of Molecular and Cellular Biology, and FAS Center for Systems Biology, Harvard University, Cambridge, MA 021382;^{b}Max Planck Institute of Molecular Cell Biology and Genetics, 01307 Dresden, Germany; and^{c}Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany

See allHide authors and affiliations

Edited by Boris I. Shraiman, University of California, Santa Barbara, CA, and approved November 7, 2014 (received for review May 29, 2014)

## 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. 1*A* 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 *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).

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. 1*C* and Movie S3). The equal time autocorrelation function of density fluctuations along the direction perpendicular to the spindle axis (⊥) plateaus for small *D*). 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/q*^{2} (see *SI Text* and ref. 17), in marked contrast with their observed behavior (Fig. 1*D*). 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/q*^{4} decay at short wavelengths results from the orientation and sliding interactions cooperating with stochastic, diffusive-like motions of microtubules (an intermediate *1/q*^{2} 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. 1*G*), 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. 1*F*), 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 *A* and *B*) leading to *S*_{⊥} = 0.021 ± 0.002 rad s^{3/2}⋅μm^{3/2}, *K* = 0.022 ± 0.009 μm^{2}⋅s^{–1}. The measured equal time density autocorrelation function is well fit by the predicted form, *C*; see *SI Text*), which has a plateau for small *q* of ^{–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 μm^{2}⋅s^{–1} (Fig. 2*C*). This value of Θ corresponds to a microtubule turnover time of *H* and *SI Text*), giving ^{2}, 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. 2*H*, 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

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. 2*C*). The measured equal time and frequency-dependent (for *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. 3*C*; 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. 3*D* 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.

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 *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 ^{(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. 1*E*). See *Stress Correlation Functions and Two-Point Microrheology*.

## Acknowledgments

We thank Andy Lau, Howard Stone, Frank Jülicher, Pierre Sens, Philippe Cluzel, and Reza Farhadifar for useful discussions, and Rudolf Oldenbourg and Grant Harris for assistance with the LC-Polscope. This work was supported by Nation Science Foundation Grants PHY-0847188 and DMR-0820484 and by the Human Frontiers Science Program.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: brugues{at}mpi-cbg.de.

Author contributions: J.B. and D.N. designed research; J.B. and D.N. performed research; J.B. contributed new reagents/analytic tools; J.B. and D.N. analyzed data; and J.B. and D.N. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1409404111/-/DCSupplemental.

## References

- ↵
- ↵.
- Karsenti E,
- Vernos I

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Sanchez T,
- Welch D,
- Nicastro D,
- Dogic Z

- ↵
- ↵
- ↵.
- Chaikin PM,
- Lubensky TC

- ↵
- ↵.
- Juelicher F,
- Kruse K,
- Prost J,
- Joanny J-F

- ↵.
- Miyamoto DT,
- Perlman ZE,
- Burbank KS,
- Groen AC,
- Mitchison TJ

*Xenopus laevis*egg extract spindles. J Cell Biol 167(5):813–818 - ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Heald R,
- Tournebize R,
- Habermann A,
- Karsenti E,
- Hyman A

*Xenopus*egg extracts: Respective roles of centrosomes and microtubule self-organization. J Cell Biol 138(3):615–628 - ↵
- ↵.
- Needleman DJ, et al.

- ↵.
- Mizuno D,
- Tardin C,
- Schmidt CF,
- Mackintosh FC

- ↵
- ↵.
- Taylor G

- ↵
- ↵
- ↵.
- Tirnauer JS,
- Salmon ED,
- Mitchison TJ

*Xenopus*egg extract spindles. Mol Biol Cell 15(4):1776–1784 - ↵
- ↵
- ↵
- ↵.
- Brugués J,
- Needleman D

*Proc SPIE*7618:76180L - ↵.
- Crocker J

- ↵.
- Austin RT,
- England AW,
- Wakefield GH

*IEEE Trans Geosci Remote Sens*, 32(4):928–939

## Citation Manager Formats

## Article Classifications

- Physical Sciences
- Physics

- Biological Sciences
- Biophysics and Computational Biology