Active condensation of filaments under spatial confinement

doi:10.3389/fphy.2022.897255

. 2022:10:897255.

doi: 10.3389/fphy.2022.897255. Epub 2022 Jun 24.

Active condensation of filaments under spatial confinement

Saad Ansari¹, Wen Yan², Adam Lamson², Michael J Shelley^{2

3}, Matthew A Glaser^{1

2}, Meredith D Betterton^{1

2

4}

Affiliations

¹ Department of Physics, University of Colorado Boulder, Colorado, USA.
² Center for Computational Biology, Flatiron Institute, New York, USA.
³ Courant Institute, New York University, New York, USA.
⁴ Department of Molecular, Cellular, and Developmental Biology, University of Colorado Boulder, Colorado, USA.

PMID: 38116396
PMCID: PMC10730113
DOI: 10.3389/fphy.2022.897255

Active condensation of filaments under spatial confinement

Saad Ansari et al. Front Phys. 2022.

. 2022:10:897255.

doi: 10.3389/fphy.2022.897255. Epub 2022 Jun 24.

Authors

Saad Ansari¹, Wen Yan², Adam Lamson², Michael J Shelley^{2

3}, Matthew A Glaser^{1

2}, Meredith D Betterton^{1

2

4}

Affiliations

¹ Department of Physics, University of Colorado Boulder, Colorado, USA.
² Center for Computational Biology, Flatiron Institute, New York, USA.
³ Courant Institute, New York University, New York, USA.
⁴ Department of Molecular, Cellular, and Developmental Biology, University of Colorado Boulder, Colorado, USA.

PMID: 38116396
PMCID: PMC10730113
DOI: 10.3389/fphy.2022.897255

Abstract

Living systems exhibit self-organization, a phenomenon that enables organisms to perform functions essential for life. The interior of living cells is a crowded environment in which the self-assembly of cytoskeletal networks is spatially constrained by membranes and organelles. Cytoskeletal filaments undergo active condensation in the presence of crosslinking motor proteins. In past studies, confinement has been shown to alter the morphology of active condensates. Here, we perform simulations to explore systems of filaments and crosslinking motors in a variety of confining geometries. We simulate spatial confinement imposed by hard spherical, cylindrical, and planar boundaries. These systems exhibit non-equilibrium condensation behavior where crosslinking motors condense a fraction of the overall filament population, leading to coexistence of vapor and condensed states. We find that the confinement lengthscale modifies the dynamics and condensate morphology. With end-pausing crosslinking motors, filaments self-organize into half asters and fully-symmetric asters under spherical confinement, polarity-sorted bilayers and bottle-brush-like states under cylindrical confinement, and flattened asters under planar confinement. The number of crosslinking motors controls the size and shape of condensates, with flattened asters becoming hollow and ring-like for larger motor number. End pausing plays a key role affecting condensate morphology: systems with end-pausing motors evolve into aster-like condensates while those with non-end-pausing crosslinking motor proteins evolve into disordered clusters and polarity-sorted bundles.

Keywords: active matter; condensation; confinement; crosslinking motors; filaments; microtubules; self-organization.

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Conflict of interest statement

CONFLICT OF INTEREST STATEMENT The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Figures

**Figure 1.**
Self-organization of filaments and crosslinking motor proteins is affected by confinement geometry. (A) End-pausing crosslinking motors (pink) bind to filaments (minus end green, plus end black) and each motor head walks to the plus end of the filament to which it is bound. Upon reaching the plus end, the motor heads stop walking and pause there. (B-E) Simulation snapshots at a filament packing fraction of $ϕ = 0.08$ . This corresponds to 1600 filaments in B, D, E, and 181 filaments in C. The number of crosslinking motors in each geometry is three times the number of filaments. (B) Simulation snapshot of an unconfined system of filaments and crosslinking motors in a 3D periodic box. End pausing of crosslinking motor heads leads to tip accumulation of crosslinking motors on filament plus ends. (C) Spherical confinement. Filaments (left) self-organize into a symmetric 3D aster with the motors (right) forming a core near the center of the aster. (D) Cylindrical confinement. Filaments (top) form a polarity sorted bilayer oriented along the cylindrical axis. Motors (bottom) polarity sort the filaments. (E) Planar confinement. Filaments (left) self-organize into flattened asters due to activity of motors (right). Top (3D view) and bottom (top view).

**Figure 2.**
Confinement lengthscale alters filament self-organization and steady-state morphology. Simulation snapshots showing filaments (minus end green, plus end black) of length $L$ and end-pausing crosslinking motors (pink). Filaments in the vapor have been grayed out. (A) Spherical confinement in a sphere of varying diameter $D$ . (A1) $D = L$ . Small spherical boundary inhibits extensile force generation of the motor proteins and no tip accumulation is observed. (A2) $D = 2 L$ . The confining sphere is just large enough to allow for extensile stress generation, and crosslinking motor proteins accumulate at filament plus ends to form a half aster whose core is visible at the edge of the sphere. (A3) $D = 3 L$ . Filaments self-assemble into a symmetric aster centered in the confining sphere. (B) Cylindrical confinement of 1600 filaments and 4800 crosslinking motors in cylinders of varying diameter $D$ . Snapshots show a 2μm long section of the simulation cylinder. (B1) $D = L$ . Filaments form polarity-sorted bilayers, each one held together by a core of crosslinking motors. (B2) $D = 2 L$ . An asymmetric aster showing a degree of orientational alignment of microtubules along the axis of the cylinder. (B3) $D = 3 L$ . A bottle-brush-like aster with an elongated core of crosslinking motors. Filaments point radially in towards the surface of the crosslinker core. (C) Planar confinement of 1600 filaments and 4800 crosslinking motor proteins between two surfaces with separation $H .$ (C1–2) $H = L$ and $H = 2 L$ . Flattened asters with a hollow filament core. (C3) $H = 3 L$ . The surface separation is large enough to accommodate a full, albeit slightly elongated, aster.

**Figure 3.**
Filaments and end-pausing crosslinking motor proteins self-assemble into flower-like and ring-like asters when confined between two planar surfaces with separation $H = L$ . (A) Simulation snapshots for varying crosslinker number ratio ${\tilde{N}}_{c} = N_{c} / N$ . End-pausing crosslinking motors (black) bind and walk on filaments (colored according to their $X Y$ orientation), before pausing at the filament plus end. Condensate morphology changes from flower-like ( ${\tilde{N}}_{c} = 1, 2, 3$ ) to ring-like ( ${\tilde{N}}_{c} = 4$ ). (B) Radial distribution function $g (r)$ computed for filament plus ends. The first peak at 25nm, equal to the filament diameter, is due to steric exclusion between filament pairs. A second peak at 78nm, the sum of filament diameter and crosslinking motor rest length, arises from crosslinked filament pairs. As ${\tilde{N}}_{c}$ increases, the size of the condensate increases, as evidenced by a widening of $g (r)$ . (C) Fraction of condensed filaments increases to 0.45, before decreasing to 0.3 for ${\tilde{N}}_{c} = 4$ . This dip in condensed fraction is due to the presence of a single aster, as opposed to multiple. (D) As ${\tilde{N}}_{c}$ increases, the number of filaments per condensate increases non-linearly. This is determined by the number of crosslinking motors per condensate. (E) The astral order parameter $α_{x y}$ is negative for all condensates, indicating that the filament orientations in the $X Y$ plane tend to point toward the center of the condensate. The increase in the absolute value of $α_{x y}$ with increasing ${\tilde{N}}_{c}$ is evidence for an increase in radial symmetry. (F) Scalar order parameter $S_{z}$ of the condensed filaments decreases with increasing ${\tilde{N}}_{c}$ , indicating an increase in alignment in the $X Y$ plane. (G) Radial distribution of crosslinking motors, measured relative to the center of the condensate. With increasing ${\tilde{N}}_{c}$ , the crosslinker core widens and becomes hollow for ${\tilde{N}}_{c} = 4$ . Note: C-F show mean values averaged over 10s of a single simulation and error bars show standard deviation. Where not visible, errors are to small.

**Figure 4.**
Condensate formation in the presence of non-end-pausing crosslinking motor proteins under planar confinement with surface separation $H = L$ . (A) Simulation snapshots at packing fraction $ϕ = 0.04$ (top row) and 0.16 (bottom row) and varying crosslinker number ratio ${\tilde{N}}_{c}$ . Crosslinking motors (black) bind and walk on filaments (colored according to their $X Y$ orientation) leading to extensile stress generation due to antiparallel sliding of anti-aligned filament pairs. At $ϕ = 0.04$ , motor activity leads to formation of condensates with low orientational order. At $ϕ = 0.16$ , no condensation is observed for ${\tilde{N}}_{c} = 1$ , but increasing ${\tilde{N}}_{c}$ leads to condensation into bundles of increasing length and with a high degree of orientational order. (B) Center-of-mass radial distribution functions $g (r)$ for aligned and anti-aligned filaments. For both $ϕ = 0.04$ (orange) and 0.16 (blue), the density of aligned filaments (solid lines) is greater than the density of anti-aligned filaments (dotted lines) for small pair separations ( $r < 0.3 μm$ ), evidence for polarity sorting. (C) The global nematic order parameter $S$ for the system is independent of ${\tilde{N}}_{c}$ for $ϕ = 0.04$ but increases linearly with ${\tilde{N}}_{c}$ for $ϕ = 0.16$ . (D) The fraction of condensed filaments increases linearly with ${\tilde{N}}_{c}$ for both values of $ϕ$ . However, for $ϕ = 0.04$ , the condensed fraction plateaus at ~ 0.3. (E) Probability density of the local nematic order parameter $S_{i}$ within the condensate. For $ϕ = 0.16$ , condensed filaments have a high degree of local orientational order, evidenced by peaks at values greater than 0.8. (F) Probability density of the local polar order parameter $P_{i}$ for all condensed filaments $i$ . The presence of peaks in the density at positive values of $P_{i}$ is evidence of polarity sorting. (G) Residence time $τ_{R}$ for filaments within condensates. Filaments experience random directed motion in condensates with low $ϕ$ and low orientational order, which leads to an increased residence time. For $ϕ = 0.16$ , filaments experience coherent directed motion which is more efficient at ejecting them from the condensate. Note: C-D show mean values averaged over 10s of a single simulation and error bars show standard deviation. Where not visible, errors are too small.

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References

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[1] Foster PJ, Furthauer S, Shelley MJ, Needleman DJ. Active contraction of microtubule networks. Elife 4 (2015) e10837. - PMC - PubMed

[2] Foster PJ, Furthauer S, Shelley MJ, Needleman DJ. Active contraction of microtubule networks. Elife 4 (2015) e10837. - PMC - PubMed

[3] Gompper G, Winkler RG, Speck T, Solon A, Nardini C, Peruani F, et al. The 2020 motile active matter roadmap. Journal of Physics: Condensed Matter 32 (2020) 193001. - PubMed

[4] Gompper G, Winkler RG, Speck T, Solon A, Nardini C, Peruani F, et al. The 2020 motile active matter roadmap. Journal of Physics: Condensed Matter 32 (2020) 193001. - PubMed

[5] Moore JM, Thompson TN, Glaser MA, Betterton MD. Collective motion of driven semiflexible filaments tuned by soft repulsion and stiffness. Soft Matter 16 (2020) 9436–9442. - PubMed

[6] Moore JM, Thompson TN, Glaser MA, Betterton MD. Collective motion of driven semiflexible filaments tuned by soft repulsion and stiffness. Soft Matter 16 (2020) 9436–9442. - PubMed

[7] Bricard A, Caussin JB, Desreumaux N, Dauchot O, Bartolo D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503 (2013) 95–98. - PubMed

[8] Bricard A, Caussin JB, Desreumaux N, Dauchot O, Bartolo D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503 (2013) 95–98. - PubMed

[9] Roostalu J, Rickman J, Thomas C, Ned´ elec F, Surrey T. Determinants of polar versus nematić organization in networks of dynamic microtubules and mitotic motors. Cell 175 (2018) 796–808. - PMC - PubMed

[10] Roostalu J, Rickman J, Thomas C, Ned´ elec F, Surrey T. Determinants of polar versus nematić organization in networks of dynamic microtubules and mitotic motors. Cell 175 (2018) 796–808. - PMC - PubMed

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Active condensation of filaments under spatial confinement

Affiliations

Active condensation of filaments under spatial confinement

Authors

Affiliations

Abstract

Conflict of interest statement

Figures

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