Contractility-Induced Phase Separation in Active Solids

Sifan Yin and L. Mahadevan

Phys. Rev. Lett. 131, 148401 – Published 2 October 2023

Abstract

Experiments over many decades are suggestive that the combination of cellular contractility and active phase separation in cell-matrix composites can enable spatiotemporal patterning in multicellular tissues across scales. To characterize these phenomena, we provide a general theory that incorporates active cellular contractility into the classical Cahn-Hilliard-Larché model for phase separation in passive viscoelastic solids. Within this framework, we show how a homogeneous cell-matrix mixture can be destabilized by activity via either a pitchfork or Hopf bifurcation, resulting in stable phase separation and/or traveling waves. Numerical simulations of the full equations allow us to track the evolution of the resulting self-organized patterns in periodic and mechanically constrained domains, and in different geometries. Altogether, our study underscores the importance of integrating both cellular activity and mechanical phase separation in understanding patterning in soft, active biosolids in both in vivo and in vitro settings.

Received 31 October 2022
Accepted 3 August 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.148401

Physics Subject Headings (PhySH)

Biological self-organization Cell mechanics Developmental pattern formation Morphogenesis

Living matter & active matter

Physics of Living Systems

Authors & Affiliations

Sifan Yin ¹ and L. Mahadevan ^1,2,3,*

¹School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138, USA
³Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*lmahadev@g.harvard.edu

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Issue

Vol. 131, Iss. 14 — 6 October 2023

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Images

Figure 1
Illustration of biological phase separation process. (a),(b) In vivo condensation of tracheal cartilage rings of mouse embryo at (a) early and (b) late stages [7]. (Scale bar: $300 μ m$ is estimated from [11].) The blue and pink regions are mesenchymal cells and ECM, respectively. (c) Schematic of the cell-ECM mixture. Initially homogeneous soft cells (blue) scattered in ECM (gray) can contract and deform the viscoelastic biphasic mixture. Cellular contractility and movement are indicated by red arrows, and the gray color map denotes the local ECM fraction. (d) Profiles of the cell fraction $ρ$ and active contractility $C$ at (left) homogeneous and (right) patterned states.
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Figure 2
Phase diagrams in the normalized $ρ_{0} - \tilde{C}$ plane obtained from linear stability analysis on a 1D model [Eq. (7)]. Colored regions depict $Re (ω) > 0$ . (a) For diffusivity $\tilde{D} = 0$ , pink indicates $Im (ω) = 0$ for all wave numbers. The inset shows a typical oscillatory pattern with nearly zero diffusivity $\tilde{D} = 0.0001$ [Video S1(a)], with color representing cell fraction $ρ$ from 0 (gray) to 1 (blue), and the horizontal axis covering time $t \in [0, 20] τ$ and the vertical axis $x \in [- 5, 5] ℓ$ . Other parameters are $\tilde{C} = 5$ and $ρ_{0} = 0.5$ . (b) For nonzero diffusivity, $\tilde{D} = 0.01$ , three regions are distinguished from linear stability analysis [Eq. (7)]: (I) stable (white), (II) stationary phase separation (green), and (III) oscillatory and traveling waves (blue). The white region (I) represents all the modes being stable with $Re (ω) < 0$ ; the green region (II) represents a finite band of modes with $Re (ω) > 0$ whereas $Im (ω) = 0$ ; and the blue region (III) represents a finite band of complex eigenvalues with positive real parts. Colored symbols show the systematic numerical results of the full 1D model [Eq. (6)], with periodic boundary conditions. Black dots indicate no pattern formation, red up-triangles are used for phase separation [Video S1(b)], and blue down-triangles are used for traveling waves [Video S1(c)]. Additional complex and irregular patterns (gray squares) near the interface of regions (II) and (III) are observed, which exhibit two-direction traveling and continuous fission and merging [Video S1(d)]. Three typical spatiotemporal plots of the cell fraction $ρ$ —including traveling waves ( $\tilde{C} = 3$ , $ρ_{0} = 0.7$ ), stationary phase separation ( $\tilde{C} = 5$ , $ρ_{0} = 0.2$ ), and complex transition patterns ( $\tilde{C} = 4$ , $ρ_{0} = 0.5$ )—are shown in the insets, with the horizontal axis covering time $t \in [0, 100] τ$ , and the vertical axis $x \in [- 5, 5] ℓ$ .
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Figure 3
Experimental and numerical simulations of representative two-dimensional dynamic patterns. (a),(b) The left column shows the images of typical in vitro (a) spots [12] and (b) stripes [30], patterns of precartilage condensations in micromass cultures of mesenchymal cells from mouse embryos. The right column exhibits 2D simulations of spots and labyrinth. (c),(d) Experimental (c) normal and (d) abnormal tracheal cartilage rings [8] and numerical simulations. The parameters used are (a) $\tilde{C} = 10$ , $\tilde{D} = 0.001$ , $ρ_{0} = 0.3$ , $\tilde{γ} = 0.01$ at $\tilde{t} = 50$ , periodic BCs; (b) $\tilde{C} = 10$ , $\tilde{D} = 0.01$ , $ρ_{0} = 0.3$ , $\tilde{γ} = 0.01$ at $\tilde{t} = 25$ , periodic BCs. Panels (c) and (d) show constant velocity $\tilde{v} = 0.001 ℓ / τ$ on the top boundary with $\tilde{C} = 5$ , $\tilde{D} = 0.01$ , $ρ_{0} = 0.5$ , $\tilde{γ} = 0.1$ , ${\tilde{L}}_{y} = 10 ℓ$ , with ${\tilde{L}}_{x} = 2 ℓ$ in (c) and $5 ℓ$ in (d). Snapshots are captured at $\tilde{t} = 100$ .
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