A coarse-grained active liquid crystal theory has been shown to quantitatively reproduce the morphology, fluctuation spectra, and cross-sectional organization of microtubules in human metaphase spindles, yielding measured organelle-scale parameters: microtubule diffusivity, nematic diffusivity, polar transport speed, and fluctuation noise amplitudes. The open problem is to establish the molecular-scale basis and validity domain of this continuum description: (i) derive the measured coarse-grained parameters from specified molecular-scale interactions (motor types and kinetics, cross-linker mechanics, microtubule length distributions, nucleation pathways); (ii) directly measure the spindle-scale microtubule polarity field and its fluctuations at high resolution, to test the polar-transport interpretation of the observed density-director coupling and to resolve unexplained nematic symmetry-breaking at the spindle mid-plane; and (iii) construct a statistical or active-liquid theory incorporating molecular-scale cross-linker geometry and short-range interactions that explains microtubule organization below ~300 nm, where the continuum description fails and discreteness of the finite microtubule ensemble dominates.

Background

The mitotic spindle is the cellular machine that segregates chromosomes during cell division. It is built from thousands of microtubules - stiff protein filaments with an intrinsic structural polarity (a chemically distinct plus end and minus end) - together with motor proteins that step directionally along microtubules and passive cross-linkers that transiently bind pairs of microtubules. How this molecular parts list produces an organelle with coherent, reproducible organization and measurable material properties is a central question of cytoskeletal physics.

Active liquid crystal (active gel) theory offers a coarse-grained route to this question. In these continuum theories the filament network is treated as a nematic liquid crystal: filaments are collectively aligned along a director field N^\widehat{\mathbf{N}}, and deviations from uniform alignment cost elastic energy parameterized by a nematic diffusivity KK (the coefficient governing relaxation of orientational fluctuations). 'Active' terms, generated by motor proteins that consume chemical energy, supplement the passive liquid-crystal dynamics and can drive contraction, transport, and instabilities (Marchetti et al. 2013). Such theories have explained purified in vitro mixtures of microtubules and motors, including the self-straining of actively crosslinked microtubule networks (Fürthauer et al. 2019), and spindle organization in Xenopus egg extracts, where nucleation and transport organize microtubules (Brugués et al. 2012). Mammalian somatic-cell spindles are much smaller (~10 microns) and shorter-lived (metaphase steady state lasting ~10 minutes) than Xenopus spindles, so their material physics was long inaccessible.

Kelleher et al. (arXiv:2507.22273; published in revised form as Maddu et al., PNAS 2026) changed this by combining two complementary data sources on HeLa human tissue-culture spindles in metaphase: serial-section electron tomography reconstructions that record the position, length, and orientation of every microtubule at a single instant (Kiewisz et al. 2022), and LC-PolScope polarized light microscopy, a polarization-imaging technique that measures birefringence (the optical retardance produced by aligned microtubules) and thereby the local alignment direction in living spindles, supplying the time dynamics. A minimal active liquid crystal theory for the microtubule density and director, including microtubule turnover (nucleation and catastrophe), quantitatively reproduced the measured static and dynamic correlation functions. Fitting the combined data yielded the spindle's material parameters: microtubule diffusive-like diffusivity D=0.0043±0.0023 μm2s−1D=0.0043\pm0.0023\ \mu\mathrm{m}^2\mathrm{s}^{-1}, nematic diffusivity K=0.0021±0.0002 μm2s−1K=0.0021\pm0.0002\ \mu\mathrm{m}^2\mathrm{s}^{-1}, polar transport speed v1=8.4±6.2 μm min−1v_1=8.4\pm6.2\ \mu\mathrm{m\,min}^{-1}, concentration and director noise amplitudes s0c=0.0265±0.0039s_0^{c}=0.0265\pm0.0039 and s0n=0.0110±0.0005 μm s−1/2s_0^{\mathrm{n}}=0.0110\pm0.0005\ \mu\mathrm{m\,s}^{-1/2}, and turnover rate Θ0=0.047 s−1\Theta_0=0.047\ \mathrm{s}^{-1}. The theory also reproduced the statistics of microtubule arrangements in spindle cross-sections near the metaphase plate without additional fitting parameters - but only down to length scales λ∗≈0.3 μm\lambda^{*}\approx0.3\ \mu\mathrm{m} (wavenumbers qs≲20 rad μm−1q_s\lesssim20\ \mathrm{rad}\ \mu\mathrm{m}^{-1}); the typical distance between neighboring microtubules is ~50 nm, and sub-300-nm arrangements are presumably dictated by the molecular properties of cross-linkers.

The published version sharpened the picture in three ways. First, the non-kinetochore microtubules that make up the bulk of the spindle (those not attached to chromosomes) obey the continuum theory down to ~300 nm, below which fluctuations from the intrinsic discreteness of the finite microtubule ensemble (a Poisson-like shot-noise regime) dominate over the predicted collective correlations, setting a fundamental limit on validating continuum theories in this regime. Second, kinetochore microtubules (KMTs, the bundles connecting chromosomes to spindle poles) show lateral organization inconsistent with the coarse-grained theory, suggesting additional mechanisms - possibly direct shaping by chromosome arrangement, by analogy with the long-range chromosome repulsion measured in mammalian oocyte spindles (Kelleher, Rana and Needleman 2024). Third, the data require a density-director coupling term v1cN^v_1 c\widehat{\mathbf{N}} describing polar (directional) transport of microtubules that explicitly breaks the nematic (head-tail) symmetry of the equations; but the observed spindle-scale symmetry-breaking in electron-tomography regions centered at the spindle mid-plane, where polarity symmetry along the spindle axis is expected, remains unexplained. Existing measurements of microtubule polarity in spindles come from second-harmonic generation microscopy (Yu et al. 2014; Bancelin et al. 2017), a nonlinear-optical technique sensitive to the average polarity of aligned microtubule lattices; these measurements are static and of limited resolution and throughput for human spindles, and no direct dynamic measurement of the polarity field exists.

Problem Statement

Establish the molecular-scale basis and validity domain of the coarse-grained active liquid crystal description of the human mitotic spindle. The source work poses three linked questions, which together bridge the validated continuum theory and the molecular mechanisms it coarse-grains:

  1. Can the inferred organelle-scale parameters - microtubule diffusivity D=0.0043±0.0023 μm2s−1D=0.0043\pm0.0023\ \mu\mathrm{m}^2\mathrm{s}^{-1}, nematic diffusivity K=0.0021±0.0002 μm2s−1K=0.0021\pm0.0002\ \mu\mathrm{m}^2\mathrm{s}^{-1}, polar transport speed v1=8.4±6.2 μm min−1v_1=8.4\pm6.2\ \mu\mathrm{m\,min}^{-1}, and the concentration and director noise amplitudes s0c=0.0265±0.0039s_0^{c}=0.0265\pm0.0039 and s0n=0.0110±0.0005 μm s−1/2s_0^{\mathrm{n}}=0.0110\pm0.0005\ \mu\mathrm{m\,s}^{-1/2} - be derived quantitatively from specified molecular-scale interactions: motor types and their kinetics, cross-linker mechanics, microtubule length distributions, and nucleation mechanisms? A satisfactory answer maps independently measurable molecular inputs onto the coarse-grained coefficients and their uncertainties, so that specific molecular perturbations (for example, motor inhibition) predict specific parameter changes.

  2. What is the source and spatial patterning of spindle-scale microtubule polarity? Direct, high-resolution measurements of the polarity field (microtubule plus-end orientation) and its fluctuations in living spindles are needed to test the polar-transport interpretation of the v1cN^v_1 c\widehat{\mathbf{N}} density-director coupling and to resolve the unexplained nematic symmetry-breaking observed in electron-tomography regions centered at the spindle mid-plane, where polarity symmetry along the spindle axis is expected. A satisfactory answer to this component fixes the spindle-scale polarity field and its fluctuations quantitatively and uses them to test the polar-transport interpretation and to account for the mid-plane symmetry-breaking; a measurement scheme specified quantitatively and defended together with its feasibility and limitations resolves this component, whereas restating that such measurements are lacking does not.

  3. Can a statistical or active-liquid theory incorporating molecular-scale cross-linker geometry and short-range interactions explain microtubule organization at length scales below approximately 0.3 μm0.3\ \mu\mathrm{m} (wavenumbers qs≳20 rad μm−1q_s\gtrsim20\ \mathrm{rad}\ \mu\mathrm{m}^{-1}) - including bundle formation, nearest-neighbor spacing on the ~50 nm scale, the discreteness-dominated (shot-noise) regime, and the anomalous lateral organization of kinetochore microtubules - where the continuum description fails?

These three components jointly support the single objective of closing the gap between molecular-scale interactions and the organelle-scale active liquid crystal description: the derivation (component 1) supplies the mapping from molecules to continuum coefficients, the polarity measurement (component 2) tests the interpretation of the symmetry-breaking transport term in those coefficients, and the sub-continuum theory (component 3) fixes where and why the coarse-grained description ceases to apply. The source's discussion also lists methodological improvements to the imaging/inference pipeline and extension to other cell types and perturbations; these two further programs are distinct from the three components above and fall outside its completion boundary. All three components are required, so resolving only a subset (for example, the derivation and the sub-continuum theory without the polarity component) leaves the objective unresolved.

References

  1. Colm P. Kelleher, Suryanarayana Maddu, Mustafa Basaran, Thomas Müller-Reichert, Michael J. Shelley, Daniel J. Needleman (2025). Active Liquid Crystal Theory Explains the Collective Organization of Microtubules in Human Mitotic Spindles. arXiv:2507.22273, https://arxiv.org/abs/2507.22273
  2. Suryanarayana Maddu, Colm P. Kelleher, Mustafa Basaran, Thomas Müller-Reichert, Michael J. Shelley, Daniel J. Needleman (2026). Human mitotic spindles as active liquid crystals: From collective behaviors to discrete filaments. Proceedings of the National Academy of Sciences 123(7):e2520490123, DOI 10.1073/pnas.2520490123, https://doi.org/10.1073/pnas.2520490123
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