Human mitotic spindles as active liquid crystals: From collective behaviors to discrete filaments

Significance

Human mitotic spindles are composed of microtubules (MTs), which interact with each other via molecular motors and other proteins, and contact chromosomes by attaching to kinetochores. It remains unclear how these constituents self-organize into the spindle. By integrating electron tomography and polarized light microscopy, we show that an active liquid-crystal theory can quantitatively explain the collective behaviors of MTs in human mitotic spindles, enabling inference of key organelle-scale material properties. We identify fundamental physical limits to this continuum description imposed by the intrinsic discreteness of the microtubule network. Our analysis also indicates that additional mechanisms, beyond those captured by the liquid-crystal framework, are required to explain the organization of kinetochore-attached MTs.

Abstract

How thousands of microtubules (MTs) and molecular motors self-organize into spindles remains poorly understood. By combining static, nanometer-resolution, large-scale electron tomography reconstructions and dynamic, optical-resolution, polarized light microscopy, we test an active liquid crystal continuum theory of mitotic spindles in human tissue culture cells. At micron length scales, probed by optical microscopy, the continuum theory accurately captures spindle morphology and fluctuation spectra, indicating that local interactions-polymerization, alignment, diffusion, and polar transport-govern the collective behaviors of MTs in human mitotic spindles. Electron tomography data enables tests of the continuum theory at submicron scales, revealing that chromosome-attached kinetochore microtubules (KMTs) show distinctive lateral organization not explained by the coarse-grained theory, while the non-KMTs that make up the bulk of the spindle follow the theory down to ∼300 nm length scales. At length scales below ∼300 nm, fluctuations arising from the intrinsic discreteness of the microtubule ensemble dominate over the collective correlations predicted from the continuum theory. Taken together, these findings show that an active liquid-crystal theory can quantitatively capture the self-organization of human mitotic spindles on long length scales and provides a means to measure the spindle’s material properties, while also pointing to the existence of additional processes contributing to the behaviors of KMTs.

The spindle is a highly dynamic cellular organelle that segregates chromosomes during cell division. It consists of rapidly polymerizing and depolymerizing microtubules (MTs) and a diverse array of motor and non-motor microtubule-associated proteins (MAPs) (1). This complex assembly provides a scaffold for force generation, ensuring high-fidelity chromosome segregation (2). While the molecular composition of spindles is now relatively well-characterized (3–5), the mechanisms driving its large-scale self-organization remain poorly understood.

One approach to studying spindle self-organization involves developing coarse-grained models, similar to those traditionally used in materials physics (6). In particular, continuum active hydrodynamic theories have emerged as powerful tools for studying cytoskeletal systems in which filaments are transiently coupled by a large number of motors and passive cross-linkers, which bind and unbind on timescales that are fast compared to the long-time, large-scale dynamics of the system. These theories have been instrumental in explaining the results of a series of experiments on a variety of microtubule-based systems, including purified in vitro mixtures in which cytoskeletal filaments and motors spontaneously self-organize into dynamic ordered structures (7–9), Xenopus laevis egg extract spindles (10, 11), and spindles in intact mammalian cells (12, 13). Of these systems, extract spindles are the best-characterized, in part because their large sizes (∼40 $\mathrm{\mu }\text{m}$) and long lifetimes (∼hours) facilitate detailed measurements of material properties, using either force probes or light microscopy (10, 14). As a result, much of our understanding of spindle material physics comes from studies of Xenopus extract spindles.

There are significant differences between Xenopus extract spindles and other spindles, such as those in human mitotic cells. While the self-organization of extract spindles is believed to result from the mutual interactions of their constituent MTs, it has been hypothesized that in human mitotic spindles there are equally important interactions of MTs with centrosomes and with chromosomes (15, 16). Furthermore, it has been proposed that both the chromosome associated kinetochore microtubules (KMTs) and the non-KMTs that make up the bulk of human mitotic spindles are organized into bundles, while the simplest active liquid crystal theories lack a description of a bundling mechanism (17–21). Finally, human mitotic spindles contain approximately one tenth the number of MTs as Xenopus extract spindles, ∼10,000 in total, and the average length of these MTs is 2 to 3 $\mathrm{\mu }\text{m}$, comparable to the overall spindle dimensions (∼10 $\mathrm{\mu }\text{m}$) (22), suggesting that a continuum description of these spindles may be inappropriate. These differences raise the possibility that the coarse-grained theory that explains Xenopus extract spindles self-organization might not capture the behaviors of more complex human mitotic spindles.

Recent advances in serial-section electron tomography have enabled full three-dimensional reconstructions of the positions and configurations of every microtubule in metaphase spindles in human tissue culture cells (HeLa cells) at a single instant in time (22, 23). Separately, noninvasive polarization microscopy provides access to the time dynamics of the microtubule network in living spindles, but at a spatial resolution limited by the wavelength of visible light, ∼500 nm. By combining data obtained using these complementary techniques, we probe the physical basis of microtubule organization in human metaphase spindles, and demonstrate that large-scale spatiotemporal patterns in microtubule orientations and densities are predicted by a minimal coarse-grained model based on the physics of active liquid crystals (24). These results indicate that spindle-scale microtubule organization is driven by their turnover dynamics and collective coalignment, diffusion, and polar transport. Thus, the large-scale behaviors of the MTs which form the bulk of human spindles can be quantitatively understood without the need to invoke specific bundling mechanisms or interactions with other structures. Furthermore, the model enables inference of key organelle-scale material properties, including parameters describing nematic elasticity and microtubule diffusivity.

The nanoscale resolution of electron tomography allowed us to test the active liquid crystal theory at spatial scales an order of magnitude smaller than can be accessed by light microscopy. These analyses revealed that MTs attached to kinetochores exhibit spatial organization not captured by the theory, indicating that additional mechanisms govern their behaviors. At still smaller scales, we find that measured correlations become dominated by fluctuations arising from the intrinsic discreteness of the microtubule ensemble, which overwhelm the collective fluctuations predicted by the continuum description. Taken together, this work shows that active liquid-crystal theory provide a quantitative framework for understanding microtubule self-organization and inferring organelle-scale material properties in human mitotic spindles, while also pointing toward additional mechanisms contributing to the behaviors of KMTs.

Results

Microtubule Density and Orientation Fields Estimated from Electron Tomography Data Are Consistent with Active Liquid Crystal Theory.

Electron tomography provides detailed ultrastructural information on MTs and chromosomes within spindles, allowing for precise measurements of each microtubule’s length, position, and morphology within the spindle apparatus (23, 25), Fig. 1A]. We used recent electron tomography reconstructions of HeLa spindles (22) to estimate continuous orientation and density fields in the spindle bulk. To do this, we divided microtubule p into $M_p$ subsegments, with each subsegment m characterized by its three-dimensional (3D) position ${\mathbf{R}}_m^p$, and 3D orientation vector ${\widehat{\mathbf{N}}}_m^p$, both determined from the endpoint locations of the segment (SI Appendix, section 1). In the three complete electron tomography reconstructions we analyzed, the total number of MTs ($N_{\text{MT}}$) varies between 4,000 and 7,300, with an average of approximately 110 subsegments per microtubule. The coarse-grained fields describing the microtubule orientation $\widehat{\mathbf{N}}(\mathbf{R})$ and density $\rho (\mathbf{R})$ at position $\mathbf{R}$ are estimated from the positional and orientational information of the subsegments (6),

$$\begin{array}{c}\hfill \rho (\mathbf{R})={\displaystyle \sum _{p=1}^{N_{\text{MT}}}}{\displaystyle \sum _{m=1}^{M_p}}\delta (\mathbf{R}-{\mathbf{R}}_m^p)\end{array}$$ [1]

$$\begin{array}{c}\hfill \widehat{\mathbf{N}}(\mathbf{R})=\frac{V}{N_{\text{seg}}}{\displaystyle \sum _{p=1}^{N_{\text{MT}}}}{\displaystyle \sum _{m=1}^{M_p}}{\widehat{\mathbf{N}}}_m^p\delta (\mathbf{R}-{\mathbf{R}}_m^p)\end{array}$$ [2]

Fig. 1.

Spatial correlations in fluctuations calculated from electron tomography reconstructions of HeLa spindles. (A) Electron tomography enables complete reconstruction of HeLa spindle MTs. Only MTs with contour length >1 $\mathrm{\mu }\text{m}$ are displayed here, but all MTs are used in subsequent calculations. The light blue plane indicates $z=0$. (B) Active liquid crystal model of microtubule orientation. Gray curves show streamlines of the numerically predicted director field ${\widehat{\mathbf{N}}}_0(\mathbf{R})$; black curve in $z=0$ plane indicates the outline of the ellipsoid that best fits the boundary of the spindle shown in A. (C) At each microtubule segment, the director fluctuation $\delta \mathbf{N}(\mathbf{R})$ (Right) is calculated by subtracting the predicted director ${\widehat{\mathbf{N}}}_0(\mathbf{R})$ (Middle) from the observed director $\widehat{\mathbf{N}}(\mathbf{R})$ (Left). In the $\widehat{\mathbf{N}}(\mathbf{R})$ plot, colored curves show MTs that lie within $0.5 \mathrm{\mu }\text{m}$ of the plane $z=0$. In all plots, black circles indicate centriole positions; black dashed box shows the region (dimensions $L_x\times L_{\perp }\times L_{\perp }$) where correlations are calculated. (D) Director–director, density–density, and director–density correlation functions $s_{\text{nn}}(\mathbf{q})$, $s_{\mathit{cc}}(\mathbf{q})$, and $s_{\text{n}c}(\mathbf{q})$ along wave-vector components $q_x$ and $q_y$, parallel and perpendicular to the spindle axis respectively, for $n_{\text{ET}}=3$ different spindles. The cross-correlation $s_{\text{n}c}(\mathbf{q})$ is shown as a function of $q_y$ only (rightmost plot). For all curves plotted as a function of $q_y$, correlation functions were averaged over 10 rotations of the spindle about its central axis $\widehat{\mathbf{x}}$. Dashed black triangles in all plots indicate slope −2, corresponding to a functional form $\propto q^{-2}$.

where V is the analysis volume, $N_{\text{seg}}=\sum _p{M}_p$ is the total number of microtubule segments in the volume, and the nematic director $\widehat{\mathbf{N}}(\mathbf{R})$ is normalized to unit length at each point $\mathbf{R}$.

To interpret these coarse-grained estimates, we compared them to predictions from an active liquid crystal theory of dynamic MTs collectively interacting via cross-linkers and molecular motors. We use a theory in which the orientation of MTs is determined by nematic interactions, with polar transport of MTs along the nematic director. Neglecting random noise terms and considering a state without hydrodynamic flows, as has previously been argued is appropriate for metaphase spindles, this theory predicts that the fields $\rho (\mathbf{R},t)$ and $\widehat{\mathbf{N}}(\mathbf{R},t)$ obey the equations of motion (10), SI Appendix, section 3]

$$\frac{\partial \rho }{\partial t}={\text{Γ}}_0-\text{Θ}\rho +\nabla ·\left[D\nabla \rho -v_1\rho \widehat{N}\right]$$ [3]

$$\frac{\partial \widehat{N}}{\partial t}=K(I-\widehat{N}\widehat{N}){\nabla }^2\widehat{N}.$$ [4]

In the equation for microtubule density ρ, the first term represents microtubule nucleation with rate ${\mathrm{\Gamma }}_0$. The second term describes microtubule disassembly events (catastrophes) occurring at rate Θ. The term inside the divergence operator captures spatial transport of MTs by both diffusive-like motion and directed (polar) transport. Microtubule diffusive-like motion is characterized by a diffusivity D, while directed (polar) transport follows the vector field $v_1\widehat{\mathbf{N}}$. The latter term explicitly violates nematic symmetry and is uniquely active in origin, since it breaks detailed balance (24). The equation for the director results from using the one-Frank constant approximation with parameter K, the nematic diffusivity, given by the ratio of the nematic elastic constant to the rotational drag coefficient (10, 12). The projection operator $\mathbf{I}-\widehat{\mathbf{N}}\widehat{\mathbf{N}}$ preserves the unit magnitude of the orientation field. To the lowest order in spatial gradients, the theory predicts that steady-state microtubule density ${\rho }_0$ is constant in space and time, while the steady-state director ${\widehat{\mathbf{N}}}_0$ obeys the projected vectorial Laplace equation,

$$\begin{array}{c}\hfill {\rho }_0=\frac{{\mathrm{\Gamma }}_0}{\mathrm{\Theta }};\hspace{1em}\left(\mathbf{I}-{\widehat{\mathbf{N}}}_0{\widehat{\mathbf{N}}}_0\right){\mathrm{\nabla }}^2{\widehat{\mathbf{N}}}_0=0.\end{array}$$ [5]

It was previously shown that Eq. 5 for the director $\widehat{\mathbf{N}}$, when combined with anchoring conditions derived from the geometry of the best-fit spindle boundary, accurately describes the average microtubule orientation field in HeLa cells (13). This result agrees quantitatively with our independent analysis of microtubule orientation patterns across the spindle (Fig. 1B and SI Appendix, Fig. S1). Additionally, within the spindle interior, the coarse-grained microtubule density is approximately uniform, varying by at most ∼30% near the metaphase plate, where condensed chromosomes occupy a significant fraction of the available volume (SI Appendix, Fig. S2). Taken together, the spindle-scale patterns of microtubule concentration and orientation quantitatively agree with the theory.

Properties of Density and Orientation Correlation Functions from Electron Tomography Data.

We next investigated spatial patterns in fluctuations of microtubule density and orientation. These patterns, as quantified by correlation functions, provide a validation of the equations of motion Eqs. 3 and 4, and allow estimation of the physically meaningful parameters in those equations. To remove the dependence of ρ on the density of microtubule subsegments, which depends on experimental details such as sampling density, we instead analyze the normalized density, $C=\rho /{\rho }_0=\rho V/N_{\text{seg}}$. Here, V is the volume of the analysis box, defined as the largest-volume cuboid contained within the spindle boundary; the dimensions of V are $L_x\times L_{\perp }\times L_{\perp }$ (Fig. 1C, black dashed box). This normalization procedure also facilitates direct comparison with polarization microscopy measurements (next section). Fluctuations in normalized density are defined as

$$\begin{array}{c}\hfill \delta C(\mathbf{R})=\frac{V}{N_{\text{seg}}}{\displaystyle \sum _{p=1}^{N_{\text{MT}}}}{\displaystyle \sum _{m=1}^{M_p}}\delta (\mathbf{R}-{\mathbf{R}}_m^p)-1.\end{array}$$ [6]

Fluctuations in orientation are calculated by subtracting the director field predicted by Eq. 5 from the observed director field (Fig. 1C),

$$\begin{array}{c}\hfill \delta \mathbf{N}(\mathbf{R})=\widehat{\mathbf{N}}(\mathbf{R})-{\widehat{\mathbf{N}}}_0(\mathbf{R}).\end{array}$$ [7]

To quantify spatial patterns in fluctuations of microtubule concentration and orientation fields, we plot equal time correlation functions, measuring how fluctuations at different spatial locations are related at the same instant in time (Materials and Methods). We calculate density–density, director–director, and director–density correlation functions in Fourier space, where they are predicted to take relatively simple functional forms (6), and plot them as a function of the wavevector components along the long and short axes of the spindle (Fig. 1D). To facilitate comparison with polarization microscopy measurements (following section), we project the fluctuating fields over the $\widehat{\mathbf{z}}$-axis and compute the correlation functions in the xy-plane (Materials and Methods, SI Appendix, section 3). For each of the $n_{\text{ET}}=3$ electron tomography reconstructions, we first plot the director–director $s_{\text{nn}}(q_x,q_{0y})$ and density–density $s_{\mathit{cc}}(q_x,q_{0y})$ correlation function along the wave vector component $q_x$ parallel to the spindle axis. We plot these correlation functions in the long-wavelength limit, i.e., with the perpendicular component $q_y$ fixed at the lowest available mode $q_{0y}=2\pi /L_{\perp }$ (Materials and Methods). We also plot $s_{\text{nn}}$ and $s_{\mathit{cc}}$ as a function of the wavevector component $q_y$ perpendicular to the spindle’s axis, with the parallel component fixed at the lowest available mode $q_{0x}=2\pi /L_x$. All these correlation functions approximately follow inverse-square power-law scaling, $1/q^2$, consistent with a minimal model in which orientation and density dynamics are governed by independent local processes, for example, nematic elasticity and microtubule diffusive-like motion respectively (6), SI Appendix, section 3). However, we also observe that the director–density cross-correlation $s_{\text{n}c}(q_{0x},q_y)$ is finite (Fig. 1D, Far Right), with magnitude and functional form comparable to the self-correlations. This observation indicates that density and orientation dynamics are not independent and suggests that coupling between the fields significantly influences their behavior.

Microtubule Density and Orientation Fields Estimated from LC-PolScope Data Are Consistent with Active Liquid Crystal Theory.

Next, we sought to measure the time-resolved dynamics of microtubule density and orientation fluctuations to complement the spatial correlation measurements obtained from static electron tomography reconstructions. We used the LC-PolScope (26, 27), a form of noninvasive, label-free polarized light microscopy, to characterize the structure and dynamics of the microtubule network in HeLa spindles. The LC-PolScope simultaneously measures the optical retardance $r(\mathbf{r},t)$ and optical slow axis $\theta (\mathbf{r},t)$ at a given time t and each position $\mathbf{r}$ in a two-dimensional (2D) image (27). Although the LC-PolScope has lower spatial resolution than electron tomography (SI Appendix, section 2), it allows us to extract time-resolved information regarding the microtubule cross-sectional density ${\rho }_{2D}(\mathbf{R},t)$ and the director $\widehat{\mathbf{N}}(\mathbf{r},t)$ ((12, 13). In general, the retardance value depends on the angle between the director and the optical axis $\widehat{\mathbf{z}}$ (SI Appendix, section 2). However, if MTs are perpendicular to the optical axis $\widehat{\mathbf{z}}$, $\widehat{\mathbf{N}}·\widehat{\mathbf{z}}\approx 0$, as is the case for MTs in the central spindle region (Fig. 2 A and B),

$$r(r,t)\approx A_0{\displaystyle \underset{T}{\int }}{\rho }_{2D}(R,t)dz;$$ [8]

$$\widehat{n}(r,t)\equiv (\cos \theta (r,t),\sin \theta (r,t))\approx \frac{{{\displaystyle \int }}_T\widehat{N}(R,t)dz}{{{\displaystyle \int }}_T\widehat{N}(R,t)dz},$$ [9]

Fig. 2.

LC-PolScope characterization of spindles in living HeLa cells. (A) Left: LC-PolScope images take advantage of the intrinsic birefringence of spindle MTs to provide information about their density and orientation, projected over the optical axis $\widehat{\mathbf{z}}$. Right: LC-PolScope images are compared to the predictions of a model where the 3D streamlines predicted by the nematic model are projected over $\widehat{\mathbf{z}}$ using Eq. 9. (B) Top and Middle Rows: Instantaneous and time-averaged images of LC-PolScope retardance (related to projected microtubule density) and optical slow axis (related to projected orientation). Time averages taken over 5 min (34 frames). White dashed curve in the retardance images shows the best-fit ellipse. Bottom Row: Predictions of retardance and slow axis from numerical solutions of Eq. 5 supplemented with boundary conditions based on the geometry of this spindle (SI Appendix, section 3). Defect positions (black dots) are chosen so that the predicted orientation field fits the observed time-averaged orientation field. (Scale bar, 2 $\mathrm{\mu }\text{m}$.) (C) 2D-projected director fluctuations $\delta \mathbf{n}(\mathbf{r},t)$ are calculated by subtracting the numerically predicted 2D director ${\widehat{\mathbf{n}}}_0(\mathbf{r})$ from the measured 2D director $\widehat{\mathbf{n}}(\mathbf{r},t)$. (D) Instantaneous plot of $\delta n_y(\mathbf{r},t)$, the y-component of $\delta \mathbf{n}(\mathbf{r},t)$. Near the middle of the spindle, $\delta \mathbf{n}\approx \delta n_y\widehat{\mathbf{y}}$. Correlations are calculated in the white dashed box, dimensions $L_0\times L_0=6\mathrm{\mu }\text{m}\times 6\mathrm{\mu }\text{m}$. (E) Left to Right: Spatiotemporal director–director and density–density correlation functions $c_{\text{nn}}({\mathbf{q}}_0,\omega )$ and $c_{\mathit{cc}}({\mathbf{q}}_0,\omega )$, evaluated at the longest available wave-vector ${\mathbf{q}}_0=(2\pi /L_0,2\pi /L_0)$, as a function of frequency ω. The spatial correlation functions $s_{\text{nn}}(\mathbf{q})$ and $s_{\mathit{cc}}(\mathbf{q})$, plotted as functions of wave-vector components $q_x$ and $q_y$, parallel and perpendicular to the spindle axis respectively. Dashed black triangles in the first five plots indicate a slope of $-2$ ($\propto q^{-2}$ scaling); the bold red dashed triangle in the final plot indicates a slope of $-4$ ($\propto q^{-4}$ scaling). Each colored curve represents data from one spindle; data from $n_{\text{Pol}}=11$ different spindles is shown.

where T is the sample thickness, $A_0\approx 7.5$ nm² is a previously measured constant that characterizes the retardance contribution of a single microtubule (12, 28), and ${\rho }_{2D}(\mathbf{R})$ is the 2D microtubule cross-sectional density in the plane perpendicular to $\widehat{\mathbf{N}}(\mathbf{R})$. In the central spindle, MTs are approximately parallel to the spindle axis $\widehat{\mathbf{x}}$, and ${\rho }_{2D}(\mathbf{R})=\lambda \rho (\mathbf{R})$, where λ is the electron tomography sampling length along the MTs, $\lambda \approx 20$ nm (22). It was previously shown that retardance and slow axis measurements of HeLa spindles are quantitatively consistent both with electron tomography reconstructions and with the predictions of Eq. 5 (13). This is also consistent with our independent measurements and analysis (SI Appendix, Fig. S5).

Properties of Density and Orientation Correlation Functions from LC-PolScope Data.

We next sought to compare in more detail the structure and dynamics of the microtubule network, as measured by LC-PolScope, with the results of our analysis of electron tomography data. To perform this comparison, we computed the spatiotemporal correlation functions of director and density fluctuations from $n_{\text{Pol}}=11$ LC-PolScope datasets, focusing on several long-wavelength limits (Fig. 2C). To facilitate comparison with electron tomography data, we normalize the retardance measurements in each spindle by its time-averaged value inside the analysis box, $c(\mathbf{r},t)=r(\mathbf{r},t)/{⟨r(\mathbf{r})⟩}_t$ (Materials and Methods). To minimize interference from metaphase chromosomes, we analyzed fluctuations in an analysis box of dimensions $L_0\times L_0=6\mathrm{\mu }\text{m}\times 6\mathrm{\mu }\text{m}$, centered at $(-3\mathrm{\mu }\text{m},0)$ (Fig. 2D). Fluctuations are defined via continuous two-dimensional analogs of Eqs. 6 and 7,

$$\begin{array}{cc}\hfill \delta c(x,y)& =c(x,y)-1;\hfill \\ \hfill \delta \mathbf{n}(x,y)& =\widehat{\mathbf{n}}(x,y)-{\widehat{\mathbf{n}}}_0(x,y),\hfill \end{array}$$

where ${\widehat{\mathbf{n}}}_0(x,y)$ is the orientation field predicted by Eq. 5, projected into 2D along a direction perpendicular to the spindle axis using Eq. 9. We observed that the autocorrelation functions, $c_{\text{nn}}({\mathbf{q}}_0,\omega )$ and $c_{\mathit{cc}}({\mathbf{q}}_0,\omega )$, with wavenumbers fixed at the lowest nonzero modes ${\mathbf{q}}_0=(q_0,q_0)=(2\pi /L_0,2\pi /L_0)$ (Fig. 2E), decay roughly proportional to $1/{\omega }^2$. At large q, the equal-time correlation functions $s_{\text{nn}}(q_x,q_0)$ and $s_{\mathit{cc}}(q_x,q_0)$ appear consistent with inverse-square power-law scaling, similar to the electron tomography measurements. At large $q_y$, however, the density correlation $s_{\mathit{cc}}(q_0,q_y)$ displays scaling between $1/q_y^2$ and $1/q_y^4$, inconsistent with collective dynamics controlled by microtubule diffusion alone. At low wavenumbers (long wavelengths), all equal-time correlation functions calculated from LC-PolScope data deviate from the high-q scaling; this trend is most clearly evident in $s_{\mathit{cc}}(q_0,q_y)$. This could be a consequence of the finite size of our analysis box, since power-law behavior is only expected in the infinite system-size limit. Alternatively, it could indicate that other processes, for instance microtubule turnover, become dominant at the longest wavelengths.

Average Orientational Data from Electron Tomography and LC-PolScope is Consistent with Active Liquid Crystal Theory.

We next sought to compare orientation data, simultaneously averaged over all electron tomography and all LC-PolScope datasets, with the predictions of Eq. 5. Averaging orientation data across multiple spindles is nontrivial because predictions from Eq. 5 depend explicitly on the geometry of each individual spindle. In our model, spindle geometry is characterized by three independent parameters (SI Appendix, section 1): the best-fit spindle half-length a, best-fit spindle half-width b, and half-distance between spindle poles d. (The quantity d is measured directly from electron tomography data, but is treated as fit parameter when determining the predicted orientation fields for LC-PolScope slow axis data). To average over spindles, we first rescaled the spatial coordinates $(x,y,z)$ of each dataset by its interpolar half-distance d, and then computed the average over all available datasets:

$$\begin{array}{c}\hfill ⟨\theta ⟩(x/d,s/d)={⟨\theta (x/d,s/d)⟩}_{\text{all spindles}}.\end{array}$$

where $s=\sqrt{y^2+z^2}$ denotes the radial coordinate. When taking the average over spindles, we give equal weight to the $n_{\text{ET}}=3$ $\widehat{\mathbf{z}}$-projected electron tomography datasets and the $n_{\text{Pol}}=11$ time-averaged LC-PolScope datasets. We compare this combined dataset to numerical simulations in which the spindle poles are positioned at $(\pm 1,0,0)$. In these simulations, we enforce tangential boundary conditions on an ellipsoid whose long and short semi-axes were set to the averages of the rescaled spindle dimensions $a/d$ and $b/d$, with averages again taken over all datasets (equal weights given to electron tomography and LC-PolScope). Without using additional fit parameters, the combined dataset shows excellent agreement with the numerical predictions (Fig. 3 A and B).

Fig. 3.

Active liquid crystal theory accurately describes microtubule orientation in HeLa spindles, as well as spatiotemporal correlations in microtubule density and orientation. (A) Left: The $\widehat{\mathbf{z}}$-projected orientation field and best-fit spindle boundary, averaged over both electron tomography (ET) and LC-PolScope datasets. Prior to averaging, x- and y-coordinates of all datasets were normalized by the interpolar half-spacing d. Right: Numerical prediction of $\widehat{\mathbf{z}}$-projected orientation field, found by solving Eq. 5, supplemented with boundary conditions based on the average spindle geometry (SI Appendix, section 3). (B) Vertical (Left) and horizontal (Right) slices through the data shown in A. Error bars represent the discrepancy between LC-PolScope and electron tomography estimates; solid black lines show theoretical predictions. (C) Selected director–director, density–density, and director–density correlation functions, for LC-PolScope data (green circles), electron tomography data (ET, yellow squares), and averaged (black diamonds). Error bars for LC-PolScope and electron tomography data are given by the SE over individual-spindle datasets; error bars in the averaged correlation functions indicate the discrepancy between the two measurements. For the spatiotemporal correlations $c_{\text{nn}}({\mathbf{q}}_0,\omega )$ and $c_{\mathit{cc}}({\mathbf{q}}_0,\omega )$ (left column), only LC-PolScope data is available. The thin black line shows fits to Eq. 11, with parameters reported in Table. 1.

Active Liquid Crystal Theory Quantitatively Explains Combined Electron Tomography and LC-PolScope Fluctuation Spectra.

We then investigated whether estimates of spatiotemporal correlation functions, obtained from electron tomography and LC-PolScope data, are consistent with the predictions of Eqs. 3 and 4. To do this, we first binned each dataset and plotted them on a common set of axes (Fig. 3C). This reveals that the two estimates of $s_{\text{nn}}(q_x,q_0)$ and $s_{\text{nn}}(q_x,q_0)$ agree, within experimental uncertainty, for all values of $q_x$. Along $q_y$, the two estimates of the director–director correlation function agree, except at the very lowest values of $q_y$. The density–density correlation functions and the density–director correlation function have a similar shape between both datasets, but the LC-PolScope estimates were systematically lower. The close agreement between these independent measurements–achieved without using any adjustable parameters–is remarkable, and indicates that experimental errors are relatively minor, such as potential fixation and reconstruction artifacts in the electron tomography data and potential optical projection artifacts and camera noise in the LC-PolScope data.

Having established that LC-PolScope and electron tomography measurements yield similar correlation spectra, we next focused on using all available fluctuation data to test the active liquid crystal model and to estimate the phenomenological parameters it contains. To do this, we combined the spatial correlation function sets by directly averaging the LC-PolScope and electron tomography estimates, giving equal weight to each dataset (Fig. 3C, black points with error bars). For the temporal correlation functions, only LC-PolScope data were available (Fig. 3C, green circles with error bars). To compare the combined correlation functions with the predictions of Eqs. 3 and 4, we introduced noise terms into the equations of motion and calculated the resulting spatiotemporal density–density, $c_{\mathit{cc}}(\mathbf{q},\omega )$, and director–director, $c_{\text{nn}}(\mathbf{q},\omega )$, correlations in Fourier space (SI Appendix, section 3),

$$\begin{array}{cc}\hfill c_{\mathit{cc}}(\mathbf{q},\omega )& =\frac{q_y^2{v}_1^2{(s_0^{\text{n}})}^2+(K^2{q}^4+{\omega }^2){(s_0^c)}^2}{(K^2{q}^4+{\omega }^2)\left({(q_x{v}_1+\omega )}^2+{(D{q}^2+\mathrm{\Theta })}^2\right)};\hfill \\ \hfill c_{\text{nn}}(\mathbf{q},\omega )& =\frac{{(s_0^{\text{n}})}^2}{{\omega }^2+K^2{q}^4},\hfill \end{array}$$ [10]

and the equal-time density–density, $s_{\mathit{cc}}(\mathbf{q})$, director–director, $s_{\text{nn}}(\mathbf{q})$, and density–director, $s_{\text{n}c}(\mathbf{q})$, correlations in Fourier space,

$$\begin{array}{c}\hfill s_{\mathit{cc}}(\mathbf{q})=\frac{{(s_0^c)}^2}{2(D{q}^2+\mathrm{\Theta })}+\hspace{1em}\\ \hfill \frac{q_y^2{(s_0^{\text{n}})}^2{v}_1^2((D+K)q^2+\mathrm{\Theta })}{2K{q}^2(D{q}^2+\mathrm{\Theta })(q_x^2{v}_1^2+{((D+K)q^2+\mathrm{\Theta })}^2)};\\ \hfill s_{\text{nn}}(\mathbf{q})=\frac{{(s_0^{\text{n}})}^2}{2K{q}^2};\hspace{1em}\\ \hfill s_{\text{n}c}(\mathbf{q})=-\frac{i{q}_y{v}_1{(s_0^{\text{n}})}^2}{2K{q}^2((D+K)q^2+i{q}_x{v}_1+\mathrm{\Theta })}.\end{array}$$ [11]

In these equations, $s_0^c$ and $s_0^{\text{n}}$ are the magnitudes of the Gaussian-distributed random noise in the density and orientation equations, respectively. All other parameters are defined in Eqs. 3 and 4 and described in the text immediately following those equations. This model for fluctuations is similar to that presented in ref. 10, with one important and nontrivial modification: whereas the model in that work only contained random noise in the orientation field, the model presented here includes independent sources of noise in both fields. This modification is needed to explain the magnitude and form of the density correlations $s_{\mathit{cc}}(q_x,q_{0y})$ in the direction parallel to the spindle axis (Fig. 3C, second column), and the cross-correlations between the director and density fields. Nonconservative noise in the density equation captures fluctuations in the total microtubule number, which could result from spatiotemporal fluctuations in the nucleation rate.

To test whether these predictions are consistent with the combined correlation functions, we estimated $\mathrm{\Theta }=0.047s^{-1}$ from previous measurements in HeLa cells (13), and used all remaining parameters ($s_0^c$, $s_0^{\text{n}}$, D, K, and $v_1$) as fit parameters. (When estimating Θ, we used the average of the turnover frequencies of kinetochore and non-KMTs, weighted to account for the relative abundance of these populations.) With the best-fit parameters given in Table. 1, the model produces excellent fits to the observed data (Fig. 3C, black curves). As in the previous analysis of fluctuations in Xenopus egg extract spindles, we found that the best-fit microtubule diffusion constant, $D=(0.0043\pm 0.0023)\mathrm{\mu }{\text{m}}^2$ s⁻¹, is similar to the best-fit nematic diffusivity, $K=(0.0021\pm 0.0002)\mathrm{\mu }{\text{m}}^2$ s⁻¹ [uncertainties estimated via nonreplacement subsampling, (Materials and Methods)]. The values we measured are lower than the Xenopus values by a factor of approximately five (10), which might be caused by MTs being shorter in HeLa spindles than in Xenopus extract spindles, ∼2 $\mathrm{\mu }\text{m}$ and ∼6 $\mathrm{\mu }\text{m}$, respectively, or MTs being less dense in HeLa spindles than in Xenopus extract spindles, ∼13 MTs$/\mathrm{\mu }{\text{m}}^2$ and ∼50 MTs$/\mathrm{\mu }{\text{m}}^2$ in cross-section, respectively [SI Appendix, sections 1 and 2, (29)]. The value we measured for $v_1$, $(8.4\pm 6.2) \mathrm{\mu }\text{m}$/min, is larger than the speed of microtubule sliding in these spindles, $\sim 1 \mathrm{\mu }\text{m}$/min (13), but smaller than the speed of microtubule polymerization in these spindles, ∼15 $\mathrm{\mu }\text{m}$/min. Such an intermediate polar transport speed could result from microtubule-mediated microtubule nucleation (30–32).

Table 1.

Table of best-fit parameters for spindle material properties

Parameter Name	Symbol	Value	Units
Concentration noise amplitude	$s_0^c$	$0.0265\pm 0.0039$	$\mathrm{\mu }\text{m} {\text{s}}^{-1/2}$
Director noise amplitude	$s_0^{\text{n}}$	$0.0110\pm 0.0005$	$\mathrm{\mu }\text{m} {\text{s}}^{-1/2}$
Microtubule diffusion constant	D	$0.0043\pm 0.0023$	$\mathrm{\mu }{\text{m}}^2{\text{s}}^{-1}$
Nematic diffusivity	K	$0.0021\pm 0.0002$	$\mathrm{\mu }{\text{m}}^2{\text{s}}^{-1}$
Polar microtubule transport speed	$v_1$	$8.4\pm 6.2$	$\mathrm{\mu }\text{m}$ min−1
Microtubule turnover rate	${\mathrm{\Theta }}_0$	0.047	${\text{s}}^{-1}$

The value of Θ₀ was calculated from a weighted average of previously measured kinetochore and nonkinetochore microtubule catastrophe rates (13). Estimates are found by fitting combined electron tomography and LC-PolScope correlation functions (Fig. 3); uncertainties are found via nonreplacement subsampling (Materials and Methods).

The Statistics of Microtubule Arrangements in Spindle Cross-Sections Is Explained by Active Liquid Crystal Theory and the Discreteness of Filaments (Shot Noise).

MTs in spindles, both KMTs and non-KMTs, are often described as being organized into bundles (17–21). The active liquid crystal theory presented here successfully describes aspects of microtubule organization in the spindle without explicitly invoking bundling. In this theory, a “bundle,” if present, would be a transient density fluctuation, not requiring any additional mechanism beyond the local interactions of MTs (and microtubule turnover), which underlie the basis of the active liquid crystal theory. However, the correlation functions used to compare theory and experiments in Fig. 4 are all evaluated in the long wavelength limit: i.e., in the low-$q_x$ limit to investigate the $q_y$ dependence, in the low-$q_y$ limit to investigate the $q_x$ dependence. These limits effectivity average over all MTs aligned parallel or perpendicular to the spindle axis. In contrast, some studies investigating bundling in spindles have used optical microscopy to analyze the arrangements of MTs in spindle cross-sections (33–35). A cross-section is a thin slice, and thus contains short-wavelength information not present in the limits above. We therefore asked whether the active liquid crystal theory–which accurately predicts large-scale fluctuation spectra–also describes the lateral organization of MTs in cross-sections near the metaphase plate.

Fig. 4.

At distances less than 300 nm, cross-sectional correlations are dominated by the statistical effects of finite microtubule numbers (shot noise); at longer distances, correlations among non-KMTs are predicted by the active liquid-crystal model, but correlations among KMTs are not. (A) Cross-section of the spindle at the metaphase plate ($x=0$). (B) Director configurations in the $x=0$ plane. From Left to Right: All MTs; KMTs; randomly chosen sample of non-KMTs equal in number to the KMTs. The black dashed square indicates the region used for correlation-function analysis. (C) Density–density ($R_{\mathit{CC}}$) and director–director ($R_{\mathit{NN}}$) correlation functions in the $x=0$ cross-section. Dotted lines show the active liquid-crystal predictions (Eqs. 12 and 13), using parameters estimated from long-wavelength fits (Fig. 3); dashed lines show the predicted shot-noise contributions (SI Appendix, section 3G), given by the mean values of $A/N_s$ (for $R_{\mathit{CC}}$) and $\mathrm{Var}(\delta N_y)A/N_s$ (for $R_{\mathit{NN}}$), where the mean is taken over the $n_{\text{ET}}=3$ ET datasets. Solid curves show the sum of active liquid crystal and shot noise predictions. At high wavenumbers $q_s^{\ast }\gtrsim 20 \mathrm{rad}/\mu \mathrm{m}$ (black arrows), corresponding to a real-space scale ${\lambda }^{\ast }=2\pi /q_s^{\ast }\approx 300\mathrm{nm}$, correlations are shot-noise dominated. (D) $R_{\mathit{CC}}$ and $R_{\mathit{NN}}$ computed from random subsamples of $N_s=50,100,250$ MTs, together with the values for the full set of MTs ($⟨N_s⟩=564$, where the average is taken over $n_{\text{ET}}$ spindles). For each dataset, black dashed lines indicate the average high-$q_s$ plateau values, calculated using data points with $q_s>q_s^{\ast }$. (E) Measured plateau values of $R_{\mathit{CC}}$ and $R_{\mathit{NN}}$ as a function of $N_s$ (points), compared with the predicted shot-noise contributions (solid lines): $A/N_s$ for density fluctuations and $\mathrm{Var}(\delta N_y)A/N_s$ for director fluctuations. (F) $R_{\mathit{CC}}$ and $R_{\mathit{NN}}$ for KMTs only (Left) and for an equal-sized random sample of non-KMTs (Right). For each dataset, predictions of active liquid-crystal theory, shot noise, and their sum are shown (see legend in C). In C, D, and F, log-scaled data from $nET=3$ spindles are binned and averaged; error bars represent SE.

To test this, we take $x=0$ cross-sections of each spindle and extract from the electron tomography data the y-z coordinates ${\mathbf{r}}_i$ of each microtubule intersection (Fig. 4B). We then evaluate the slice correlation functions $R_{\mathit{CC}}$ and $R_{\mathit{NN}}$ (Materials and Methods). The theoretical prediction for these quantities is obtained by marginalizing the 3D spatial correlation function over the wavevector component $q_x$ parallel to the spindle-axis (SI Appendix, section 3),

$$\begin{array}{cc}& R_{\mathit{CC}}(q_s)=\frac{L_z{(s_0^c)}^2}{4\sqrt{D}\sqrt{D{q}_s^2+\mathrm{\Theta }}}+\hfill \\ \hfill & \frac{L_z{(s_0^{\text{n}})}^2{q}_{s^{″}}^2}{8K^{\prime 2}}[\frac{2\sqrt{2}{e}_1(e_1^2-w^2)+2q_{s^{\prime }}^2\left(\alpha -3\sqrt{2}{e}_1\right)+{\alpha }^{\prime }(q_s)}{D{q}_{s^{\prime }}^2{e}_1\left({(q_{s^{\prime }}^2-e_1^2)}^2-e_1^2{w}^2\right)\alpha (q_s)}\hfill \\ \hfill & +\frac{2\sqrt{2}{e}_2(e_2^2-w^2)+2q_{s^{\prime }}^2\left(\alpha -3\sqrt{2}{e}_2\right)+{\alpha }^{\prime }(q_s)}{K{q}_{s^{\prime }}^2{e}_2\left({(q_{s^{\prime }}^2-e_2^2)}^2-e_2^2{w}^2\right)\alpha (q_s)}]\hfill \end{array}$$ [12]

$$\begin{array}{c}\hfill R_{\mathit{NN}}(q_s)=\frac{L_z{(s_0^{\text{n}})}^2}{4K{q}_s}\\ \hfill \end{array}$$ [13]

where $q_s=|{\mathbf{q}}_{\perp }|;K^{\prime }=D+K;R=\sqrt{4q_{s^{\prime }}^2+w^2};{\alpha }_{\pm }=\sqrt{2q_{s^{\prime }}^2+w(w\pm R)};\alpha ={\alpha }_{+}+{\alpha }_{-};{\alpha }^{\prime }(q_s)=w\left(R({\alpha }_{-}-{\alpha }_{+})+w\alpha \right);q_{s^{\prime }}^2=q_s^2+\frac{\mathrm{\Theta }}{D+K};w^2=\frac{v_1^2}{{(D+K)}^2};e_1^2=q_s^2+\frac{\mathrm{\Theta }}{D};e_2^2=q_s^2;q_{s^{″}}^2=q_s^2{v}_1^2.$

Using the parameters found previously by fitting the long-wavelength correlation functions, the active liquid crystal theory correctly predicts the shapes of the measured slice correlation functions for $q_s\le 20$ rad$\mathrm{\mu }{\text{m}}^{-1}$ (corresponding to real-space distances $\ge 300$ nm), but not their amplitudes (Fig. 4C, points with error bars vs. dotted curves). At higher $q_s$, both $R_{\mathit{CC}}(q_s)$ and $R_{\mathit{NN}}(q_s)$ exhibit plateaus that are in strong disagreement with the predictions of the active liquid crystal theory.

One clear difference between the active liquid crystal theory and the electron tomography data is that the data consists of the locations and orientations of discrete filaments, not continuous fields. We next considered the possibility that the differences between the measured slice correlation functions and the predictions of the active liquid crystal theory result from the discreteness of the microtubule ensemble. The discrete structure factor can be decomposed into i) self-correlations, which reflect the discreteness of the sampled field, and ii) connected correlations, which encode the true spatial organization of the coarse-grained density (SI Appendix, section 3G). The self-correlation is a constant “shot-noise” background arising from finite sampling, which, for $N_s$ MTs in an analysis box of area A, is predicted to be $A/N_s$ for density correlations and $(A/N_s)\text{Var}(\delta N_y)$ for nematic correlations. Combining this shot noise with the active liquid crystal theory results in predictions which closely match the measured slice correlation functions with no additional fitting (Fig. 4C, solid curve). The length-scale at which shot noise becomes dominant is comparable to the mean microtubule spacing in the cross-section, $\sqrt{A/N_s}\approx$270 nm.

To further test if the measured plateaus in the slice correlation functions result from shot noise, we recalculated the density and director correlation functions for $N_s$ randomly selected microtubule intersections (where $N_s=50,100,$ and 250), and compared them with the full-sample slice correlation functions ($⟨N_s⟩=564$, where the average is taken over $n_{\text{ET}}=3$ spindles, Fig. 4D). These plateau values scale as $A/N_s$ for the density correlations and as $(A/N_s)\text{Var}(\delta N_y)$ for the nematic correlations, precisely as shot noise is predicted to do Fig. 4E). This provides strong evidence that the statistics of microtubule arrangements in cross-sections of the spindle are dominated i) at small length scales by shot noise due to the discrete nature of filaments and ii) at large length scales by collective fluctuations, as described by the active liquid crystal theory, without the need to invoke other processes such as specific microtubule bundling mechanisms.

The Arrangements of KMTs Are Inconsistent with Active Liquid Crystal Theory.

Since KMTs make up only ∼15% of the MTs in human spindles, the statistics calculated above are largely determined by the behaviors of the non-KMTS which make up the bulk of the spindle (22). We next investigated if the organization of KMTs is different from those of non-KMTs. The nanometer resolution of the electron tomography data enables each microtubule to be identified as a KMT or a non-KMT (22). This allows us to compare, in the same spindle cross-section, the spatial arrangement of KMTs and non-KMTs.

We compared the slice correlation functions of KMTs with those of a randomly chosen subset of non-KMTs equal in number to the KMTs intersecting the spindle cross-section. Whereas the non-KMT correlations agree with the predictions from the active liquid crystal plus shot noise, the KMT slice correlations deviate markedly from theory at low $q_s$, in both the density and nematic correlations (Fig. 4F). These observations indicate that the arrangements of KMTs are shaped by processes not captured by the active liquid crystal theory, perhaps due to geometric or mechanical constraints imposed by kinetochore attachment, or bundling proteins that specifically interact with KMTs.

Discussion

This study integrates static micromechanical information from high-resolution electron tomography reconstructions with dynamic light microscopy measurements to investigate the physical mechanisms underlying microtubule self-organization in metaphase human tissue culture spindles. Using a coarse-grained active liquid crystal theory, we quantitatively describe the emergent fluctuation spectra of microtubule density and orientation fields. Our findings highlight distinct physical mechanisms driving spindle organization along its long and short axes, providing insights into the anisotropic material properties of the spindle.

The LC-PolScope has limitations such as projection artifacts, limited spatial resolution, correlated background noise, and low sensitivity to weak birefringence. Similarly, serial-section electron tomography faces challenges like sample preparation difficulties, sectioning artifacts, and alignment issues. Despite this, the fluctuation spectra measured by these two methods show remarkable agreement, highlighting their complementary strengths in studying microtubule organization.

Our data suggests that nematic elasticity alone governs the relaxation of orientational fluctuations in the spindle along both axes, but that relaxation of microtubule density is more complex. Along the spindle’s long axis, microtubule density dynamics are decoupled from orientation and are primarily governed by diffusive-like motion. In contrast, along the short axis, the relaxation of density fluctuations is driven by a combination of alignment interactions (i.e., nematic elasticity), diffusive-like motion, and microtubule turnover. In particular, the observed deviation of $s_{\mathit{cc}}(q_0,q_y)$ from $\propto q_y^{-2}$ behavior at high $q_y$ suggests that density dynamics are coupled to the director configuration (Fig. 3C, Bottom Right plot). Additionally, the nonzero cross-correlation function directly confirms coupling between the fields (Fig. 3C, Top Right plot).

In our active liquid crystal theory, density–director coupling is captured by the $v_{1}c\widehat{\mathbf{N}}$ term in Eq. 3, which describes active transport of MTs along a specific (vector) direction in space, and explicitly breaks the nematic symmetry of the equations of motion. If this term was absent, there would be no coupling between the fields (SI Appendix, section 3I), which is qualitatively inconsistent with both the LC-PolScope and electron tomography data. Thus, activity, density–director coupling, and nematic symmetry-breaking are intimately related in this theory. Nematic symmetry-breaking is anticipated in the LC-PolScope data, as analysis box spans are centered on one half of the spindle, where coherent polar ordering of MTs is known to influence transport (36, 37). In contrast, the electron tomography data analysis regions are centered at the spindle mid-plane, in which case symmetry in microtubule polarity along the spindle axis is expected. Thus, the observed spindle-scale symmetry-breaking in this region remains unclear. More detailed measurements of the microtubule polarity field (and its fluctuations) may help to answer this question (29, 38, 39).

Our work also demonstrates the utility of a phenomenological coarse-grained active liquid crystal framework for inferring collective properties of the microtubule network, such as nematic elasticity, diffusion, and polar transport. The measured parameters provide a quantitative link between microscopic interactions and emergent mesoscale spindle behaviors. Future work should focus on connecting these coarse-grained parameters to molecular-scale interactions to refine the theoretical framework further. Additionally, extending this methodology to study spindle dynamics in other cell types and experimental conditions will provide a broader understanding of the principles governing microtubule organization in diverse biological systems.

The collective behaviors of the non-KMTs that make up the bulk of human spindles are well described by active liquid crystal theory down to length scales of ∼300 nm, where fluctuations due to the discrete nature of the filaments become dominant. This argues that the organization of these non-KMTs are determined by their local, mutual interactions, with no need to invoke specific bundling mechanisms or other processes. In contrast, the statistics of KMTs arrangements are inconsistent with the predictions of this coarse-grained theory, suggesting that additional mechanisms govern their organization. Recent work in mammalian oocyte spindles reports that metaphase chromosomes exhibit spatial anticorrelations driven by long-range repulsive interactions (12). Because KMTs are anchored to chromosomes via the kinetochore, their organization may be shaped directly by chromosome arrangement, a hypothesis worth exploring in future studies.

Our results also reveal a fundamental physical limit on testing active liquid-crystal theories in the spindle. At sufficiently high wavenumbers, the measured correlations enter a shot-noise regime in which all spatial structure is obscured by Poisson statistics arising from the finite number of MTs in the slice. Beyond this scale set roughly by the mean microtubule spacing, collective fluctuations cannot be distinguished from discreteness-induced noise, and continuum predictions can no longer be meaningfully validated in this regime.

Materials and Methods

Cell Lines.

Electron tomography data were obtained from HeLa Kyoto cells, cultured, and prepared as described previously (22). For live-cell imaging experiments, we used a stable HeLa cell line expressing EGFP-tagged centrin to fluorescently label centrioles and sfGFP-tagged CENPA to label since centromeres; the establishment and characterization of this cell line was previously reported in ref. 13. All cell culture and imaging procedures adhered strictly to protocols detailed in these prior studies.

LC-PolScope Imaging, Metaphase Spindle Selection, Definition of Spindle-Referenced Coordinate System.

To acquire live-cell images, we recorded movies consisting of 34 frames captured over approximately 5 min. Each movie frame comprised a multiplexed z-stack combining LC-PolScope and epifluorescence imaging, which allowed simultaneous visualization of MTs and centrioles, respectively. To acquire polarization images, LC-PolScope hardware (Cambridge Research Instruments) was mounted on a Nikon TE2000-E microscope equipped with a numerical aperture (NA) 0.52 air condenser lens, a 100×, NA 1.45, oil-immersion objective lens, and a monochromatic bandpass filter centered at ${\lambda }_{\text{im}}=546$ nm. To image centrioles, the microscope was equipped with an LED epifluorescence light source (Thorlabs) and a standard EGFP filter cube. We controlled the microscope hardware using the OpenPolScope MicroManager software package (https://www.openpolscope.org).

Metaphase cells were identified for analysis based on spindle geometry and stability criteria. Specifically, we selected datasets where the interpolar distance remained approximately constant (SD of interpolar distance less than 10%) throughout imaging. Additionally, to ensure consistent orientation, we included only those datasets in which the vertical positions ($\widehat{\mathbf{z}}$-direction) of the centrioles remained within ±1μm of each other for the entire duration of the movie. Datasets not meeting these criteria were excluded from further analysis. Live-cell imaging experiments were conducted in four independent experimental replicates, each involving separately plated cells. To correct for spindle translation and rotation in the image plane (xy-plane), we performed all analysis in a coordinate system in which the origin is fixed to the center of the spindle and the spindle long axis defines the $\widehat{\mathbf{x}}$-direction (SI Appendix, Fig. S3).

Calculation of Experimental Correlation Functions.

From Electron Tomography.

To estimate correlation functions from electron tomography reconstructions, we first evaluate the discrete Fourier transforms of the fluctuating director $\widehat{\mathbf{N}}(\mathbf{R})$ and the normalized density field $C(\mathbf{R})$ as follows,

$$\begin{array}{cc}\hfill \delta C(\mathbf{Q})& =\frac{V}{N_{\text{seg}}}{\displaystyle \sum _{p=1}^{N_{\text{MT}}}}{\displaystyle \sum _{m=1}^{M_p}}exp(-i\mathbf{Q}·{\mathbf{R}}_m^p)-\delta (\mathbf{Q})\hfill \\ \hfill \delta \mathbf{N}(Q)& =\frac{V}{N_{\text{seg}}}{\displaystyle \sum _{p=1}^{N_{\text{MT}}}}{\displaystyle \sum _{m=1}^{M_p}}{\widehat{\mathbf{N}}}_m^{p}exp(-i\mathbf{Q}·{\mathbf{R}}_m^p)-{\widehat{\mathbf{N}}}_0\delta (\mathbf{Q}),\hfill \end{array}$$

where $\mathbf{Q}=(q_x,q_y,q_z)$, and ${\widehat{\mathbf{N}}}_0(\mathbf{R})$ is the solution to Eq. 5 with physically realistic boundary conditions, obtained numerically (SI Appendix, section 3). We estimate the spatial correlation functions in three dimensions using the direct summation method, i.e.,

$$\begin{array}{cc}\hfill S_{\mathit{CC}}(\mathbf{Q})& =\frac{1}{V}\delta c(Q)\delta c(-Q);\hfill \\ \hfill S_{\text{NN}}(\mathbf{Q})& =\frac{1}{V}\delta \mathbf{N}(Q)\delta \mathbf{N}(-Q);\hfill \\ \hfill S_{N_{y}C}(\mathbf{Q})& =\frac{1}{V}\delta \mathbf{N}(Q)\delta \mathbf{c}(-Q),\hfill \end{array}$$

where V is the volume of the analysis box. In three dimensions, all spatial correlation functions have units of $\sim \mathrm{\mu }{\text{m}}^3$. Two-dimensional, $\widehat{\mathbf{z}}$-projected correlation functions are calculated from the 3D ones (Materials and Methods)

$$\begin{array}{cc}\hfill s_{\mathit{cc}}(\mathbf{q})& =\frac{1}{L_z}{S}_{\mathit{CC}}(q_x,q_y,0);s_{\text{nn}}(\mathbf{q})=\frac{1}{L_z}{S}_{\text{NN}}(q_x,q_y,0);\hfill \\ \hfill & s_{\text{n}c}(\mathbf{q})=\frac{1}{L_z}{S}_{N_{y}C}(q_x,q_y,0),\hfill \end{array}$$

where $L_z$ is the average spindle thickness at the metaphase plate (the $x=0$ section of the best-fit ellipsoid), $L_z=b^{-1}{\int }_{-b}^b(b^2-y^2)\text{d}y=\pi b/2$. The two-dimensional, $\widehat{\mathbf{z}}$-projected correlation functions have units of $\sim {\mu }^2$.

Slice correlations from electron tomography.

To calculate the correlation functions in the $x=0$ plane (“slice correlations”), we computed, for each ET dataset:

$$\begin{array}{cc}\hfill R_{\mathit{CC}}({\mathbf{q}}_{\perp })& =\frac{A}{N_s^2}{\displaystyle \sum _{m,n=1}^{N_s}}exp(-i{\mathbf{q}}_{\perp }·({\mathit{r}}_{\perp m}-{\mathit{r}}_{\perp n}))\hfill \end{array}$$

$$\begin{array}{cc}\hfill R_{\mathit{NN}}({\mathbf{q}}_{\perp })& =\frac{A}{N_s^2}{\displaystyle \sum _{m,n=1}^{N_s}}\delta N_{\perp }^m\delta N_{\perp }^{n}exp(-i{\mathbf{q}}_{\perp }·({\mathit{r}}_{\perp m}-{\mathit{r}}_{\perp n}))\hfill \end{array}$$

where A is the area of the analysis box (defined as the square $x=0$ cross-section of the analysis cuboid defined in the paragraph preceding Eq. 6) and $N_s$ is the number of MTs intersecting $x=0$. The two-dimensional vectors ${\mathbf{r}}_{\perp }=(r_y,r_z)$ and ${\mathbf{q}}_{\perp }=(q_y,q_z)$ denote real- and Fourier-space coordinates in the $x=0$ plane. In the $n_{\text{ET}}=3$ spindles we analyzed, we identified $N_s=$ 496, 748, and 447 MTs intersecting with $x=0$; the corresponding analysis boxes have areas 33.4, 50.8, and 40.2 ${\mu }^2$. To generate the data shown in Fig. 4C, we binned and averaged both sets of correlation functions to obtain the plotted points with error bars. To obtain the data displayed in Fig. 4D, we randomly subsampled $N_s$ microtubule intersections from each spindle midplane, where $N_s=50,100,$ and 250, computed $R_{\mathit{CC}}$ and $R_{\mathit{NN}}$ using only those points, and finally averaged the results over the $n_{\text{ET}}=3$ datasets. To compute KMT slice correlation functions for each spindle, we calculated the correlation functions with only those KMTs that intersected the analysis box ($N_s^{\text{KMT}}=$ 67, 161, 50); we compared the resulting data with non-KMT correlation functions computed with $N_s^{\text{KMT}}$ microtubule intersections, randomly selected from the non-KMTs in each dataset. For both KMT and non-KMT data, we binned and averaged the results over the $n_{\text{ET}}=3$ spindles to obtain the data shown in Fig. 4F.

From LC-PolScope measurements.

To estimate correlation functions from LC-PolScope data, we first compute the normalized density field $c(\mathbf{r},t)=r(\mathbf{r},t)/{⟨r⟩}_t(\mathbf{r})$. The time-averaged retardance field ${⟨r⟩}_t(\mathbf{r})$ is calculated by fitting the computed time-averaged retardance image to (Fig. 2B, middle row) to a paraboloid. We use Mathematica’s inbuilt Fourier[] function to approximate the (2 + 1)D Fourier transforms of the fluctuations $\delta c=c(\mathbf{r},t)-1$, $\widetilde{\delta c}(\mathbf{q},\omega )$ and ${\widetilde{\delta n}}_y(\mathbf{q},\omega )$. To mitigate boundary effects, we apply mirror padding in space and time to all data arrays before computing all self-correlation functions (nn, cc correlations) (40). We omit the mirror padding step when computing the cross-correlations (nc correlations), since mirror padding forces the imaginary component of the cross-correlation to vanish artificially. After computing the Fourier transforms $\widetilde{\delta c}(\mathbf{q},\omega )$ and ${\widetilde{\delta n}}_y(\mathbf{q},\omega )$, we calculate the spatiotemporal and equal-time correlation functions:

$$\begin{array}{cc}\hfill c_{\mathit{cc}}(\mathbf{q},\omega )& =\frac{1}{{\tau }_0{\lambda }_0^2}\widetilde{\delta c}(\mathbf{q},\omega )\widetilde{\delta c}(-\mathbf{q},-\omega );\hfill \\ \hfill c_{\text{nn}}(\mathbf{q},\omega )& =\frac{1}{{\tau }_0{\lambda }_0^2}{\widetilde{\delta n}}_y(\mathbf{q},\omega ){\widetilde{\delta n}}_y(-\mathbf{q},-\omega );\hfill \\ \hfill c_{\text{n}c}(\mathbf{q},\omega )& =\frac{1}{{\tau }_0{\lambda }_0^2}{\widetilde{\delta n}}_y(\mathbf{q},\omega )\widetilde{\delta c}(-\mathbf{q},-\omega );\hfill \\ \hfill & s_{\mathit{cc}}(\mathbf{q})=\frac{1}{2\pi }\int c_{\mathit{cc}}(\mathbf{q},\omega )\text{d}\omega ;s_{\text{nn}}(\mathbf{q})=\frac{1}{2\pi }\int c_{\text{nn}}(\mathbf{q},\omega )\text{d}\omega ;\hfill \\ \hfill s_{\text{n}c}(\mathbf{q})& =\frac{1}{2\pi }\int c_{\text{n}c}(\mathbf{q},\omega )\text{d}\omega ,\hfill \end{array}$$

where ${\tau }_0$ is the total time of the movie being analyzed and the ω-integrals are taken over all available angular frequencies ω. To correct for Gaussian-distributed background noise in the retardance and slow-axis signals, we initially fit each temporal correlation function at fixed wavevector to functions of the form:

$$\begin{array}{c}\hfill c_{\text{nn}}({\mathbf{q}}_0,\omega )=\frac{A_{\text{nn}}}{{\omega }^2}+B_{\text{nn}};\\ \hfill c_{\mathit{cc}}({\mathbf{q}}_0,\omega )=\frac{A_{\mathit{cc}}}{{\omega }^2}+B_{\mathit{cc}}.\end{array}$$

We then subtract the fitted additive constants from the original correlation function,

$$\begin{array}{c}\hfill c_{\text{nn}}({\mathbf{q}}_0,\omega )\to c_{\text{nn}}({\mathbf{q}}_0,\omega )-B_{\text{nn}};\\ \hfill c_{\mathit{cc}}({\mathbf{q}}_0,\omega )\to c_{\mathit{cc}}({\mathbf{q}}_0,\omega )-B_{\mathit{cc}}.\end{array}$$

To correct for spurious high-q correlations induced by the finite optical resolution of the LC-PolScope, we deconvolve the correlation functions $s_{\mathit{cc}}(\mathbf{q})$, $s_{\text{nn}}(\mathbf{q})$, and $s_{\text{n}c}(\mathbf{q})$ by the experimentally measured optical impulse function of the LC-PolScope (SI Appendix, Fig. S6),

$$\begin{array}{c}\hfill s_{\mathit{cc}}(\mathbf{q})\to e^{{\sigma }_0^2{\mathbf{q}}^2}{s}_{\mathit{cc}}(\mathbf{q});\hspace{1em}{s}_{\text{nn}}(\mathbf{q})\to e^{{\sigma }_0^2{\mathbf{q}}^2}{s}_{\text{nn}}(\mathbf{q});\\ \hfill s_{\text{n}c}(\mathbf{q})\to e^{{\sigma }_0^2{\mathbf{q}}^2}{s}_{\text{n}c}(\mathbf{q}),\end{array}$$

where ${\sigma }_0=61$ nm characterizes the point spread function. Finally, we discard data for which $|\mathbf{q}|>2\pi /{\lambda }_{\text{im}}$, where ${\lambda }_{\text{im}}=273$ nm is half the microscope illumination wavelength.

Model Fitting and Parameter Estimation.

The predictions of the model (Eq. 11) were fitted to the log-transformed experimental data (black points in Fig. 3). Best-fit values were obtained by minimizing the sum of squared residuals across all log-transformed datasets simultaneously. Parameter uncertainties were estimated via nonreplacement subsampling (41): in each trial, we randomly selected 6 of the 11 LC-PolScope single-spindle datasets and 2 of the 3 single-spindle ET datasets. For the spatial correlation functions, we averaged the selected datasets, giving equal weight to LC-PolScope and ET data; the model was then fitted to the composite subsampled dataset. This procedure was repeated 50 times, and the means and SD of the distributions of best-fit parameters were estimated. For each parameter, the reported uncertainty (Main Text and Table. 1) is the larger of the following two quantities: i) the SD over subsamples; ii) the absolute difference between the mean of the subsample distribution, and the best-fit value obtained by fitting all datasets simultaneously.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All data and code (Excel and .dat) and Modified Active Nematic Model of HeLa Spindles data have been deposited in OSF https://doi.org/10.17605/OSF.IO/REQX4 (42).