Mechanical Heterogeneity in Tissues Promotes Rigidity and Controls Cellular Invasion

Xinzhi Li, Amit Das, and Dapeng Bi

Phys. Rev. Lett. 123, 058101 – Published 31 July 2019

Abstract

We study the influence of cell-level mechanical heterogeneity in epithelial tissues using a vertex-based model. Heterogeneity is introduced into the cell shape index ( $p_{0}$ ) that tunes the stiffness at a single-cell level. The addition of heterogeneity can always enhance the mechanical rigidity of the epithelial layer by increasing its shear modulus, hence making it more rigid. There is an excellent scaling collapse of our data as a function of a single scaling variable $f_{r}$ , which accounts for the overall fraction of rigid cells. We identify a universal threshold $f_{r}^{*}$ that demarcates fluid versus solid tissues. Furthermore, this rigidity onset is far below the contact percolation threshold of rigid cells. These results give rise to a separation of rigidity and contact percolation processes that leads to distinct types of solid states. We also investigate the influence of heterogeneity on tumor invasion dynamics. There is an overall impedance of invasion as the tissue becomes more rigid. Invasion can also occur in an intermediate heterogeneous solid state that is characterized by significant spatial-temporal intermittency.

Received 18 March 2019
Revised 5 June 2019

DOI:https://doi.org/10.1103/PhysRevLett.123.058101

Physics Subject Headings (PhySH)

Cancer Cell mechanics Cell migration Jamming Percolation

Living matter & active matter Packing & jamming problems Tumors

Theories of collective dynamics & active matter

Polymers & Soft MatterPhysics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Xinzhi Li ^*, Amit Das, and Dapeng Bi

Department of Physics, Northeastern University, Boston, Massachusetts 02115, USA

^*li.xinz@husky.neu.edu

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Vol. 123, Iss. 5 — 2 August 2019

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Images

Figure 1
Tissue mechanical property. (a) Shear modulus $G$ (in units of $K_{P}$ ) vs $μ$ at $σ = 0, 0.07, 0.09, 0.11, 0.13, 0.15, 0.17, 0.19, 0.21, 0.23, 0.25, 0.27, 0.29$ for $N = 400$ , $κ_{A} = 1$ . (Inset) $G / σ$ vs $μ$ . (b) $G / σ$ vs $σ / | μ - μ^{*} |$ in log-log scale. (Inset) $G / σ$ vs the fraction of rigid cells $f_{r}$ [defined in Eq. (4)].
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Figure 2
Rigidity and contact percolations do not coincide. (a) Connected clusters of rigid cells shown in red. White cells correspond to $p_{0}^{i} \geq μ^{*}$ . (b) Typical snapshots for the tension network. Edges with finite tensions are indicated by thick black lines, while other edges have $τ = 0$ . (c) Probability of contact percolation for rigid cells, $P_{contact}$ vs $f_{r}$ . Colors indicate different tissue sizes in the range $N = 36 - 900$ (see the SM [53] for the method of calculating $P_{contact}$ ). (Inset) Finite-size scaling yields a contact percolation threshold of $f_{r}^{c} = 0.475 \pm 0.005$ and $ν = 1.60 \pm 0.03$ . (d) Probability to obtain a system-spanning tension cluster vs $f_{r}$ . Finite-size analysis yields a transition at $f_{r}^{*} = 0.21 \pm 0.01$ .
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Figure 3
Spatial distribution of cellular tensions dictates tissue rigidity. (a) Distribution of tensions $p (τ)$ in semilog scale at $κ_{A} = 1$ , $N = 400$ for various $f_{r}$ . (b) $G / σ$ and $τ / σ$ vs $f_{r}$ in semilog scale at $κ_{A} = 1$ , $σ = 0.1$ , $N = 400$ . $G / σ$ vanishes at $f_{r}^{*} = 0.21$ , while the tension remains finite even in fluid states. $τ$ has units of $K_{P} \sqrt{\bar{A}}$ , and $G$ has units of $K_{P}$ . (c) A phase diagram in ( $μ$ , $σ$ ). The solid black line indicates rigidity percolation and corresponds to values of $μ$ and $σ$ where $f_{r} = f_{r}^{*}$ . The red line is the mean field estimate for the bond percolation on a random Voronoi tessellation. The blue dashed line indicates the contact percolation transition of rigid cells at $f_{r}^{c}$ . The mismatch between rigidity and contact percolations results in an intermediate “heterogeneous solid state” which exists only for $σ > 0$ .
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Figure 4
The effect of heterogeneity depends on cell area elasticity. (a) $G / σ$ vs $f_{r}$ in semilog scale at various $κ_{A}$ values. $G$ has unit of $K_{P}$ . (b) Phase diagram of mechanical property as a function of $f_{r}$ and $κ_{A}$ . The rigidity and contact percolations are distinct when $κ_{A} > 0$ but coincide at $κ_{A} = 0$ . The region of ( $κ_{A} > 0$ , $f_{r}^{*} < f_{r} < f_{r}^{c}$ ) corresponds to a solid that follows the lower branch in Fig. 1, where the shear modulus is sharply enhanced by increasing heterogeneity.
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Figure 5
Heterogeneity and cellular invasion. (a) Analysis of the time-averaged displacement (in units of $\sqrt{\bar{A}}$ ) of a single cell attempting to move through the tissue with constant $v_{0} = 0.4$ at various $f_{r}$ . Blue points represent the mean displacement of the cell taken over twice the persistence time. Red points represent the kurtosis of short time displacements indicating a growing intermittency of the invasion dynamics. (b) Representative traces of the invading cell velocities in units of $K_{P} \sqrt{\bar{A}} / Γ$ . (Top panel) The fluid phase at $f_{r} = 0.14$ . (Middle panel) The intermediate solid phase at $f_{r} = 0.47$ . (Bottom panel) The rigid phase when $f_{r} = 0.98$ .
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