Figures and data in Theory of active self-organization of dense nematic structures in the actin cytoskeleton

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5 figures, 6 tables and 1 additional file

Figures

Figure 1

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Key model ingredients.

(a) The local state is defined by areal density ρ and by orientational order quantified by the nematic parameter S and by the nematic direction $n$ . (b) Isotropic active tension λ when the network is isotropic ( $S = 0$ ) and (c) anisotropic tension when $S \neq 0$ , controlled by $κ$ . Active tension is positive (contractile) in all directions whenever $| κ | < 1$ , but its deviatoric part is contractile when $κ > 0$ and extensile when $κ < 0$ . Orientational order is driven by (d) active forces conjugate to nematic order and characterized by parameter $λ_{⊙}$ and by (e) passive flow-induced alignment in the presence of deviatoric rate-of-deformation with coupling parameter β.

Figure 2 with 3 supplements

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Active patterns coupling nematic order and density driven by self-reinforcing flows.

(a) Illustration of the dimensionless parameters characterizing active tension anisotropy (κ) and pattern architecture, quantified by the relative orientation of nematic order and high-density structures ( $ω = ℓ_{s}^{2} / (ρ_{0}^{2} | A |) \int_{A} \nabla ρ \cdot q \cdot \nabla ρ d S$ ). (b) Order parameter of pattern architecture ω as a function of active tension anisotropy $κ$ obtained from nonlinear simulations, showing transition from states with nematic direction parallel to high-density structures (ω < 0, fibrillar patterns) for $κ < 0$ to states with nematic direction perpendicular to high-density structures (ω >0, banded patterns with perpendicular nematic organization) for $κ > 0$ . Because $| κ | < 1$ , the active tension is always positive in all directions. (c) Map of density, nematic order S, nematic direction (red segments), and flow field (green arrows) for quasi-steady fibrillar (I) and banded (III) patterns, and for a transition pattern of high-density droplets with high nematic order (II) corresponding to nearly isotropic active tension. (d) Illustration of the out-of-equilibrium quasi-steady states, maintained by self-reinforcing flows, diffusion, and turnover.

Figure 2—figure supplement 1

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Lengthscale of the pattern.

Illustration of the lengthscale of the pattern close to the onset of the instability (left) and deeper into the nonlinear regime (right). The linear stability estimate of the lengthscale of the pattern is $ℓ_{p a t t e r n} = 2 π / ν_{c r i t}$ , where $ν_{crit}$ is given by Equation 10 in terms of material parameters. The side length of the square periodic computational domain is $8 ℓ_{p a t t e r n}$ . Close to the onset (left), the pattern along the black arrow, with exactly 8 repeats, shows that the lengthscale in the nonlinear simulations closely follows the theoretical estimate. At activities 30% above the threshold (right), we sampled the typical separation between dense structures (green arrows) to approximate $ℓ_{p a t t e r n}^{s i m}$ from simulations, finding that $ℓ_{p a t t e r n}^{s i m} / ℓ_{p a t t e r n} = 0.87 \pm 0.02$ (mean and standard deviation). We robustly found across the parameter space that the pattern lengthscale from nonlinear simulations 30% above the threshold was a little smaller than the theoretical estimate $ℓ_{p a t t e r n}$ (see density maps in Figures 2 and 4).

Figure 2—video 1

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posterframe for video — Pattern formation in an active gel model not accounting for nematic order.

Figure 2—video 2

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Figure 3 with 1 supplement

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Pattern formation for a range of values of anisotropic active parameter $κ$ in the limit $λ_{⊙} \to 0$ .

(a) Pattern formation for material parameters used in Figure 2 except for $λ_{⊙} = 0$ while leaving $c_{0}$ unchanged. (b) Here, in addition to $λ_{⊙} = 0$ , we set $β^{2} = 2 η η_{r o t}$ to the largest value allowed by the entropy production inequality. (c) Same parameters as in (b), except for an increase in friction as detailed in Appendix 4—table 1.

Figure 3—video 1

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Figure 4 with 4 supplements

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Control of nematic bundle pattern orientation, connectivity, and dynamics.

(a) Effect of orientational bias. (I) A uniform isotropic gel self-organizes into a labyrinth pattern with defects. (II) A small background anisotropic strain rate efficiently orients nematic bundles. (III) A slight initial network alignment ( $S_{0} = 0.05$ ) orients bundles, which later lose stability, bend, and generate/anneal defects. See Figure 4—video 1. We recall that the nematic order parameter in the quiescent and uniform equilibrium state is $S_{0} = 0$ if $c_{0} \geq 0$ and $S_{0} = \sqrt{- c_{0} / b}$ otherwise, with $c_{0} = 2 a - ρ_{0} λ_{⊙}$ . (b) Depending on active tension anisotropy, nematic bundles are contractile and straighten (I, $κ = - 0.2$ ), leading to quasi-steady networks, or extensile and wrinkle (II, $κ = - 0.8$ ), leading to bundle breaking and recombination, and persistently dynamic networks (III). See Figure 4—video 2. The contractility or extensibility of the nematic bundles can be appreciated in the maps of the difference between the stress along the nematic direction ( $σ_{∥}$ ) and the stress perpendicular to it ( $σ_{⟂}$ ), normalized by the constant $σ_{crit} = λ_{crit} ρ_{0}$ . (c) Promoting mechanical interaction between bundles. (I) Dynamical pattern obtained by reducing friction, and thereby increasing $\overline{a}$ , $\overline{b}$ , ${\overline{λ}}_{⊙}$ , and ${\overline{k}}_{d}$ . Time sequence in the bottom indicates a typical reconfiguration event during which weak bundles (dashed) become strong ones (solid) and vice versa. (II) Nearly static pattern obtained increasing ${\overline{k}}_{d}$ (III) which becomes highly dynamic by further increasing ${\overline{k}}_{d}$ . Time sequence in the bottom indicates the collapse (black), expansion (purple), and splitting (green) of cells in the network. See Figure 4—video 3. We indicate by ∗ dynamical patterns exhibiting spatiotemporal chaos.

Figure 4—figure supplement 1

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Tension distribution along ( $σ_{| |}$ ) and perpendicular to ( $σ_{⟂}$ ) the dense nematic bundles.

The left column shows the density distribution, the middle column the total tension along (solid) and perpendicular (dashed) to nematic bundles, and the right column the different contributions to the total tension dominated by the active (black) and the viscous (blue) components. In all cases, we consider for convenience fully nonlinear simulations in 1D to easily define the orthogonal directions relative to the self-organized pattern. (a) to (c) show patterns obtained for negative tension anisotropy $κ$ of increasing magnitude and correspond to Figure 4b in the main text, whereas (d) shows a chaotic pattern resulting from a large hydrodynamic length and corresponds to Figure 4cI in the main text. The model parameters used in the plots are the same as in Figures 2 and 4 and are described in Appendix 4—table 1 and Appendix 4—table 2.

Figure 4—video 1

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Figure 4—video 2

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Figure 4—video 3

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Figure 5 with 3 supplements

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Assessment of activity parameters $κ$ and $λ_{⊙}$ through discrete network simulations.

(a) Illustration of the computational domain of the discrete network as a uniform representative volume element of the gel. (b) Sketch of model ingredients and setup to compute tension along and perpendicular to the nematic direction. (c) Typical time signal for parallel and perpendicular tensions following addition of crosslinkers and motors (translucent lines) along with time average (solid lines) for isotropic and anisotropic networks. Tension is normalized by mean tension $\overline{σ} = (σ_{| |} + σ_{⊥}) / 2$ , computed from time averages and time by actin turnover time. (d) Mean tension as a function of network density for several nematic parameters $S_{0}$ , where both quantities are normalized by their values for the lowest density. With this normalization, Equation 11 predicts a linear dependence with slope $λ = 1$ (dashed line). Error bars span two standard deviations. (e) Deviatoric tension as a function of nematic order for different densities. The dashed line is a linear regression to simulation data. (f) Dynamics of nematic order in a periodic network following addition of crosslinkers and motors for three initial values of nematic order. (g) Rate of change of nematic order normalized by turnover rate as a function of initial nematic order at zero and finite temperature.

Figure 5—figure supplement 1

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Detailed description of the discrete network simulations.

(I) Microstructural modeling approach using cytosim. (a) The nematic ordering of the network, S, measured with respect to a director $\hat{n}$ , is controlled by a system-wide restraining energy. (b) Actin filaments are represented by connected points kept at a fixed distance and have finite bending rigidity $κ_{A}$ . Crosslinkers are Hookean springs with stiffness $K_{X}$ and resting length $ℓ_{X}$ . Myosin motors are Hookean springs with stiffness $K_{M}$ and resting length $ℓ_{M}$ ; their ends can walk on filaments at a speed $v_{M}$ , which is affected by force application. (II) Simulation protocol to quantify active tension anisotropy. (a) Initial fiber seeding without orientational bias. (b) Athermal imposition of desired order parameter S₀ using a restraining potential. (c) Introduction of boundary anchors. (d) Deactivation of orientational restraining potential and addition of crosslinkers and myosin motors, which drive the system out-of-equilibrium. Reactions at anchors allow us to track tensions at the boundary (**e, f**). (III) Protocol to quantify the orientational activity parameter $ρ λ_{⊙}$ . (a) Initial fiber seeding without orientational bias in a periodic box. (b) Athermal imposition of desired order parameter S₀ by restraining potential. (**c, d**) Activation of temperature, deactivation of orientational restraining potential, and addition of crosslinkers and myosin motors. The dynamics of orientational order of the system driven out-of-equilibrium are then tracked during a time period (**c, d**). (e) Estimation of the model parameter $η_{rot}$ , which characterizes the athermal dynamics of network reorientation in the presence of a restraining potential of stiffness $K_{S}$ ; note that a single value of $η_{rot}$ fits the dynamics for all considered S₀. (f) The rate of change of nematic order, $\dot{S}$ , is estimated from a 10 time interval.

Figure 5—video 1

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Figure 5—video 2

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Tables

Appendix 4—table 1

Model parameters used in figures.

Parameters	Figure 2	Figure 4a	Figure 4b	Figure 4cI	Figure 4cII,III	Figure 3a	Figure 3b	Figure 3c
${\bar{k}}_{d}$	0.1	0.1	0.1	20	10, 20	0.1	0.1	0.01
L	1	1	1	1	1	1	1	1
${\bar{c}}_{0}$	4	4, 4, –0.05	4	200	4	4	4	0.4
b	20	20	20	4000	20	20	20	2
${\overline{η}}_{r o t}$	1	1	1	1	1	1	1	1
$\overline{β}$	–0.2	–0.2	–0.2	–0.2	–0.2	–0.2	– √2	– √2
${\bar{λ}}_{⊙}$	6	6, 6, 10.05	6	1800	6	0	0	0
$\overline{λ}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$	$1.3 {\bar{λ}}_{c r i t}$
κ	[–0.8, 0.8]	–0.2	–0.2, –0.8	–0.2	–0.2	–1.5, –0.75, 0, 0.75	–1.5, –0.75, 0, 0.75	–0.75, –0.375, 0, 0.75
$D o m a i n S i z e \times ν_{c r i t} / (2 π)$	8	8	8	8	8	8	8	8

Appendix 4—table 2

Model parameters used in movies that do not directly reproduce figures.

Parameters	Figure 2—video 1	Figure 3—video 1 (top)	Figure 3—video 1 (bottom)	Figure 4—video 2
${\overline{k}}_{d}$	0.01	0.1	0.1	0.1
L	1	1	1	1
${\overline{c}}_{0}$	0.4	4	4	4
b	2	20	20	20
${\overline{η}}_{r o t}$	0	1	1	1
$\overline{β}$	0	–0.2, 0	– √2 , –0.2	–0.2
${\overline{λ}}_{⊙}$	0	6	6	6
$\overline{λ}$	1.3 ${\overline{λ}}_{c r i t}$	1.3 ${\overline{λ}}_{c r i t}$	1.3 ${\overline{λ}}_{c r i t}$	1.3 ${\overline{λ}}_{c r i t}$
$κ$	0	–0.2	0.2	–0.2, –0.5, –0.8
$Domain Size \times ν_{c r i t} / (2 π)$	8	8	8	8

Appendix 5—table 1

Global parameters adopted in this study and in previous microstructural models that used cytosim.

Reference		Geometry and size (μm)	$k T$ (pN μm)	Time step (S)	$ν$ ( $P a s$ )
Present work		Square: $ℓ = 5$	0 or 0.0042	0.005	1
Cortes et al., 2020	MM1	Rectangle: 2 × 0.2	0.0042	0.001	1
	MM4	Rectangle: 9.424 × 1	0.0042	0.001	1
	MM3 and MM5	Circle: $R = 1.5$	0.0042	0.001	1
Cortex simulations in Wollrab et al., 2019		Square: $L = 8$	0.0042	0.002	0.18
Bun et al., 2018		Circle: $R = 10$	0.0042	0.01 or 0.1	0.3
Model with turnover in Belmonte et al., 2017		Square: $L = 16$	0.0042	0.001	0.1
Descovich et al., 2018		Circle: $R = 15$	0.0042	0.002	1
Ding et al., 2017		Circle: $R = 10$	0.0042	0.001	0.1
Ennomani et al., 2016		Ring: $R = 4.5$ , $t \approx 0.5$	0.0042	0.01	0.18

Appendix 5—table 2

Actin filament parameters adopted in this study and in previous microstructural models that used cytosim.

Reference		$ℓ_{A}$ (μm)	$ρ_{A}$ (μm/μm²)	$s_{A}$ (μm)	$κ_{A}$ (pN μm²)	$r_{A}$ (%/s)
Present work		1.3	78, 156, or 234	0.2	0.1	20
Cortes et al., 2020	MM1	2	1800	0.05−0.1	0.06	0
	MM3	1.3±0.3	5.09−81.5	0.05−0.1	0.06	0
	MM4	1.3±0.3	38.2−61.1	0.05−0.1	0.06	0
	MM5	1.3±0.3	50.9−81.5	0.05−0.1	0.06	0
Cortex simulations in Wollrab et al., 2019		0.1−4.0	5.5−218.8	0.1−0.4	0.075	0
Bun et al., 2018		1.5	23.9	0.15	0.07	0
Model with turnover in Belmonte et al., 2017		5	27.3	0.1−0.2	0.075	1.1–18.3
Descovich et al., 2018		1.3±0.3	3.4−5.4	0.1	0.06	0
Ding et al., 2017		2.2	35	–	0.05	0
Ennomani et al., 2016		0.95–1.75	117.2−154.2	–	0.042−0.063	0

Appendix 5—table 3

Myosin motor parameters adopted in this study and in previous microstructural models that used cytosim.

Reference	$ρ_{M}$ (1/μm actin)	$K_{M}$ (pN/μm)	$ℓ_{M}$ (μm)	$r_{M}^{b}$ (1/s)	$d_{M}^{b}$ (μm)	$r_{M}^{u, 0}$ (1/s)	$f_{M}^{u}$ (pN)	$f_{M}^{s t a l l}$ (pN)	$v_{M}^{m a x}$ (μm/s)
Present work	0.8	250	0	50	0.02	50	∞	6	0.3
Weißenbruch et al., 2021	0.5	100	0.3	0.2−3.6	0.06−0.12	0.8−1.71	5	3.85−15	0.137−0.6
Weißenbruch et al., 2021	0.6−1.0	100	0.3	0.2−3.6	0.06−0.12	0.8−1.71	5	3.85−15	0.137−0.6
Cortex simulations in Wollrab et al., 2019	2−64	100	0	10	0.01	0.5	∞	4	2
Bun et al., 2018	2.7	250	0.01	10	0.01	0.1	3	6	0.02
Model with turnover in Belmonte et al., 2017	3.2	500	0	10	0.01	0.3	∞	6	0.2
Descovich et al., 2018	0.5−0.8	1400	0.32	0.2	0.33	0.3	3.85	24.5	0.1
Ding et al., 2017	0 or 5.8	250	0	10	0.01	0.5	∞	6	0.5
Ennomani et al., 2016	0.3−0.4	100	0.03	5	0.05	0	3.65	2	0.3

Appendix 5—table 4

Crosslinker parameters adopted in this study and in previous microstructural models that used cytosim.

Reference		$ρ_{X}$ [1/μm actin]	$K_{X}$ [pN/μm]	$ℓ_{X}$ [μm]	$r_{X}^{b}$ [1/s]	$d_{X}^{b}$ [μm]	$r_{X}^{u, 0}$ [1/s]	$f_{X}^{u}$ [pN]
Present work		15.4	250	0	50	0.02	50	∞
Cortes et al., 2020	MM1	1.4–16.7	100	0.04	10	0.05	0.1–0.3	5
Cortes et al., 2020	MM3 - MM5	1.7–2.8	100	0.04	10	0.05	0.1–0.3	5
Cortex simulations in Wollrab et al., 2019		2–64	50	0	15	0.02	0.3	1
Bun et al., 2018		2.7	250	0.01	10	0.01	0.1	3
Model with turnover in Belmonte et al., 2017		0.8	500	0	10	0.01	0.3	∞
Descovich et al., 2018		11.5–18.3	100	0.04	10	0.1	0	5
Ding et al., 2017		0.2–11.6	250	0	10	0.01	0.5	∞
Ennomani et al., 2016		0.6–1.2	2	0	5	0.03	0.05	0.05

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Waleed Mirza
Marco De Corato
Marco Pensalfini
Guillermo Vilanova
Alejandro Torres-Sánchez
Marino Arroyo

(2025)

Theory of active self-organization of dense nematic structures in the actin cytoskeleton

eLife 13:RP93097.

https://doi.org/10.7554/eLife.93097.3

Figures

Key model ingredients.

Active patterns coupling nematic order and density driven by self-reinforcing flows.

Lengthscale of the pattern.

Pattern formation in an active gel model not accounting for nematic order.

Pattern formation for different values of the tension anisotropy coefficient $κ$ , leading to (I) fibrillar patterns for $κ < 0$ , (II) tactoids for $κ \approx 0$ , and (III) banded patterns for $κ > 0$ .

Pattern formation for a range of values of anisotropic active parameter $κ$ in the limit $λ_{⊙} \to 0$ .

Effect of flow-alignment coupling coefficient β on pattern formation for positive and negative tension anisotropy coefficient $κ$ .

Control of nematic bundle pattern orientation, connectivity, and dynamics.

Tension distribution along ( $σ_{| |}$ ) and perpendicular to ( $σ_{⟂}$ ) the dense nematic bundles.

Effect of orientational bias on fibrillar patterns.

Contractile vs extensile bundles as the magnitude of $κ$ increases, leading to active turbulence of fibrillar patterns.

Changes in network geometry and dynamics following enhanced inter-bundle mechanical communication.

Assessment of activity parameters $κ$ and $λ_{⊙}$ through discrete network simulations.

Detailed description of the discrete network simulations.

Illustration of simulation protocol to estimate tension anisotropy from discrete network simulations for an isotropic (left) and anisotropic (right) representative volume elements.

Illustration of simulation protocol to estimate the generalized active tension conjugate to nematic order for an isotropic (left) and anisotropic (right) representative volume elements with periodic boundary conditions.

Tables

Model parameters used in figures.

Model parameters used in movies that do not directly reproduce figures.

Global parameters adopted in this study and in previous microstructural models that used cytosim.

Actin filament parameters adopted in this study and in previous microstructural models that used cytosim.

Myosin motor parameters adopted in this study and in previous microstructural models that used cytosim.

Crosslinker parameters adopted in this study and in previous microstructural models that used cytosim.

Additional files

MDAR checklist

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Key model ingredients.

Active patterns coupling nematic order and density driven by self-reinforcing flows.

Lengthscale of the pattern.

Pattern formation in an active gel model not accounting for nematic order.

Pattern formation for different values of the tension anisotropy coefficient κ, leading to (I) fibrillar patterns for κ<0, (II) tactoids for κ≈0, and (III) banded patterns for κ>0.

Pattern formation for a range of values of anisotropic active parameter κ in the limit λ⊙→0.

Effect of flow-alignment coupling coefficient β on pattern formation for positive and negative tension anisotropy coefficient κ.

Control of nematic bundle pattern orientation, connectivity, and dynamics.

Tension distribution along (σ||) and perpendicular to (σ⟂) the dense nematic bundles.

Effect of orientational bias on fibrillar patterns.

Contractile vs extensile bundles as the magnitude of κ increases, leading to active turbulence of fibrillar patterns.

Changes in network geometry and dynamics following enhanced inter-bundle mechanical communication.

Assessment of activity parameters κ and λ⊙ through discrete network simulations.

Detailed description of the discrete network simulations.

Illustration of simulation protocol to estimate tension anisotropy from discrete network simulations for an isotropic (left) and anisotropic (right) representative volume elements.

Illustration of simulation protocol to estimate the generalized active tension conjugate to nematic order for an isotropic (left) and anisotropic (right) representative volume elements with periodic boundary conditions.

Model parameters used in figures.

Model parameters used in movies that do not directly reproduce figures.

Global parameters adopted in this study and in previous microstructural models that used cytosim.

Actin filament parameters adopted in this study and in previous microstructural models that used cytosim.

Myosin motor parameters adopted in this study and in previous microstructural models that used cytosim.

Crosslinker parameters adopted in this study and in previous microstructural models that used cytosim.

MDAR checklist

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

Pattern formation for different values of the tension anisotropy coefficient $κ$ , leading to (I) fibrillar patterns for $κ < 0$ , (II) tactoids for $κ \approx 0$ , and (III) banded patterns for $κ > 0$ .

Pattern formation for a range of values of anisotropic active parameter $κ$ in the limit $λ_{⊙} \to 0$ .

Effect of flow-alignment coupling coefficient β on pattern formation for positive and negative tension anisotropy coefficient $κ$ .

Tension distribution along ( $σ_{| |}$ ) and perpendicular to ( $σ_{⟂}$ ) the dense nematic bundles.

Contractile vs extensile bundles as the magnitude of $κ$ increases, leading to active turbulence of fibrillar patterns.

Assessment of activity parameters $κ$ and $λ_{⊙}$ through discrete network simulations.