Stability and Roughness of Interfaces in Mechanically Regulated Tissues

John. J. Williamson and Guillaume Salbreux

Phys. Rev. Lett. 121, 238102 – Published 7 December 2018

Abstract

Cell division and death can be regulated by the mechanical forces within a tissue. We study the consequences for the stability and roughness of a propagating interface by analyzing a model of mechanically regulated tissue growth in the regime of small driving forces. For an interface driven by homeostatic pressure imbalance or leader-cell motility, long and intermediate-wavelength instabilities arise, depending, respectively, on an effective viscosity of cell number change, and on substrate friction. A further mechanism depends on the strength of directed motility forces acting in the bulk. We analyze the fluctuations of a stable interface subjected to cell-level stochasticity, and find that mechanical feedback can help preserve reproducibility at the tissue scale. Our results elucidate mechanisms that could be important for orderly interface motion in developing tissues.

Received 9 August 2017
Revised 3 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.238102

Physics Subject Headings (PhySH)

Cell mechanics Cell migration Developmental biology

Cancer Tissues

Physics of Living Systems

Authors & Affiliations

John. J. Williamson and Guillaume Salbreux^*

The Francis Crick Institute, 1 Midland Road, London NW1 1AT, United Kingdom

^*Corresponding author. guillaume.salbreux@crick.ac.uk

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Vol. 121, Iss. 23 — 7 December 2018

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Images

Figure 1
(a) Side-on schematic of competing epithelial tissues. Each tissue is described with coarse-grained fields of cell density $ρ$ , stress $σ_{i j}$ , and velocity $v_{i}$ . Each has an elastic modulus $χ$ [Eq. (3)], substrate friction $ξ$ [Eq. (4)], and net division rate responsive to strain on a timescale $τ$ [Eq. (2)]. (b) Top-down view of the tissues illustrating an interface contour fluctuation. We use a 2D description, with $z$ a coordinate parallel to $x$ in the comoving frame of the interface.
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Figure 2
(a) Steady state density perturbation profile $Δ ρ$ (solid line) and velocity $v_{x}$ (dashed) for tissues $A$ ( $z < 0$ ) and $B$ ( $z > 0$ ). Parameters: homeostatic pressure difference $- Δ σ_{h} = 0.5 χ_{A}$ , frictions $ξ_{B} = ξ_{A}$ , elastic moduli $χ_{B} = χ_{A}$ , net division timescales $τ_{B} = τ_{A}$ , bulk motility forces $f_{A} = f_{B} = 0$ . (b) As (a), but with $Δ σ_{h} = 0$ , $f_{A} = 0.1 χ_{A} / ℓ_{A}$ . (c) As (a), but with $Δ σ_{h} = 0$ , $f_{B} = 0.1 χ_{A} / ℓ_{A}$ . (d) As (a), but with $Δ σ_{h} = 0$ , $f_{A} = f_{B} = 0.1 χ_{A} / ℓ_{A}$ . In this particular case the tissue moves uniformly and the density perturbation cancels to zero.
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Figure 3
(a) Example of numerically determined dispersion relations. Parameters: frictions $ξ_{B} = 1.5 ξ_{A}$ , moduli $χ_{B} = 0.5 χ_{A}$ , net division timescales $τ_{B} = τ_{A}$ , bulk motility forces $f_{A} = f_{B} = 0$ , and line tension $γ = 0.001 ℓ_{A} χ_{A}$ . The homeostatic pressure imbalance or leader-cell motility parameter varies, $Δ σ_{h} = - 0.1 χ_{A}, - 0.2 χ_{A}, \dots, - 0.5 χ_{A}$ , in the direction of the arrow, crossing a type $I_{s}$ instability transition [66]. (b) Phase diagram in $(Δ σ_{h}, ξ_{B})$ of the most unstable wave number $q^{*}$ (white if no instability), using $χ_{B} = 0.5 χ_{A}$ , $τ_{B} = τ_{A}$ , $f_{A} = f_{B} = 0$ , $γ = 0.001 ℓ_{A} χ_{A}$ . The dashed line indicates the approximation of the type $I_{s}$ transition line by Eq. (8). (c) Phase diagram in $(χ_{B}, ξ_{B})$ using $Δ σ_{h} = - 0.5 χ_{A}$ , $τ_{B} = τ_{A}$ , $f_{A} = f_{B} = 0$ , $γ = 0.001 ℓ_{A} χ_{A}$ . The meaning of the dashed line is as in (c), whereas the solid line indicates the type ${II}_{s}$ transition [Eq. (9)]. (d) Phase diagram in bulk directed motilities $(f_{A}, f_{B})$ , with other parameters $ξ_{B} = ξ_{A}$ , $χ_{B} = χ_{A}$ , $τ_{B} = τ_{A}$ , $γ = 0.001 ℓ_{A} χ_{A}$ . The black line is the transition given by Eq. (10). The dotted line is $V_{0} = 0$ , with the upper half-space $V_{0} > 0$ (tissue $A$ growing) and the lower $V_{0} < 0$ . In each quadrant, cartoons illustrate the direction of the bulk motilities.
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Figure 4
(a) Root-mean-squared interface deviation for a single tissue using example physical parameters [Eq. (12)]. The homeostatic pressure or leader-cell motility parameter $σ_{h}$ is varied, and the arrow shows increasing friction $ξ = 10^{3}, 10^{4}, 10^{5} Pa s / μ m$ . Other parameters: $χ = 10^{4} Pa μ m$ , $τ = 3.6 \times 10^{4} s$ , $γ = 10^{3} pN$ , $L_{⊥} = 1000 μ m$ , $D = 0.4 μ m^{2} / h$ . (b) Interface center-of-mass diffusion coefficient for a single tissue [Eq. (14)] using example physical parameters. The friction $ξ$ is varied, and the arrow shows increasing elastic modulus $χ = 10^{3}, 10^{4}$ , and $1 0^{5} Pa μ m$ . Other parameters: $τ = 3.6 \times 10^{4} s$ , $γ = 10^{3} pN$ , $L_{⊥} = 1000 μ m$ , and $D = 0.4 μ m^{2} / h$ .
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