Nonlinear and nonlocal elasticity in coarse-grained differential-tension models of epithelia

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Nonlinear and nonlocal elasticity in coarse-grained differential-tension models of epithelia

Pierre A. Haas and Raymond E. Goldstein

Phys. Rev. E 99, 022411 – Published 19 February 2019

Abstract

The shapes of epithelial tissues result from a complex interplay of contractile forces in the cytoskeleta of the cells in the tissue and adhesion forces between them. A host of discrete, cell-based models describe these forces by assigning different surface tensions to the apical, basal, and lateral sides of the cells. These differential-tension models have been used to describe the deformations of epithelia in different living systems, but the underlying continuum mechanics at the scale of the epithelium are still unclear. Here, we derive a continuum theory for a simple differential-tension model of a two-dimensional epithelial monolayer and study the buckling of this epithelium under imposed compression. The analysis reveals how the cell-level properties encoded in the differential-tension model lead to linear and nonlinear elastic as well as nonlocal, nonelastic behavior at the continuum level.

Received 22 October 2018

DOI:https://doi.org/10.1103/PhysRevE.99.022411

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Developmental biology Morphogenesis

Tissues

Physics of Living Systems

Authors & Affiliations

Pierre A. Haas^* and Raymond E. Goldstein^†

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

^*P.A.Haas@damtp.cam.ac.uk
^†R.E.Goldstein@damtp.cam.ac.uk

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Vol. 99, Iss. 2 — February 2019

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Images

Figure 1
Model epithelium. Isosceles trapezoidal cells of apical and basal bases $L_{a}$ and $L_{b}$ are connected along their lateral sides, which have length $L_{ℓ}$ . The continuous midline and shaded area provide a cartoon of the continuum limit, in which the epithelium is characterized by the deformed arclength $S$ of the midline and the tangent angle $ψ$ of the midline below the horizontal.
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Figure 2
Cell Geometry. (a) Geometry of a single isosceles trapezoidal cell of mean base $K$ and height $L$ , and sidelengths $L_{a}, L_{b}, L_{ℓ}$ , the lateral sides being at an angle $2 ϕ$ to each other. (b) Definition of the tangent angle $ψ$ of the midline below the horizontal. The geometry of contiguous cells defines the relation between $ϕ$ and $ψ$ , as expressed in Eq. (11).
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Figure 3
Numerical buckling results. (a) Plot of $Λ$ against relative compression $D$ . Above a critical compression $D_{▵}$ (buckled shape for $D = D_{▵}$ shown), solution shapes at the energy minimum begin to self-intersect (dotted line for $D > D_{▵}$ ). Inset: zoomed plot of $Λ$ (filled marks) against $D$ close to the buckling threshold $D_{*}$ . (b) Plot of compressive force $μ$ against relative compression $D$ , showing numerical results (solid line and dotted line for $D > D_{▵}$ ) in agreement with asymptotic results (dashed line). Inset: zoomed plot of $μ$ against $D$ close to $D_{*}$ . Parameter values for numerical calculations: $Ξ = 20$ , $δ = 1$ , $ℓ_{0} = \sqrt{10}$ . (c) Critical compression $D_{▵}$ against $r = Ξ / ℓ_{0}^{2}$ , for different values of $δ$ , at fixed $ℓ_{0} = \sqrt{10}$ , and approximation (45) thereof. Insets show buckled shapes at $D = D_{▵}$ and $δ = 1$ , for different values of $Ξ$ .
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Figure 4
Buckling for $D > D_{▵}$ . Two scenarios are possible: (a) If $\partial \dot{x} (0) / \partial Λ > 0$ or $Ξ < Ξ_{*} (δ)$ , buckled shapes with increased $Λ$ do not self-intersect. (b) If $\partial \dot{x} (0) / \partial Λ < 0$ or $Ξ > Ξ_{*} (δ)$ , buckled shapes with decreased $Λ$ do not self-intersect. Numerical results: (c) Plot of $Λ$ (for buckled shapes of minimal energy without self-intersections) against relative compression $D$ , for $Ξ_{1} < Ξ_{*}$ (thick lines) and $Ξ_{2} > Ξ_{*}$ (thin lines). For $Ξ < Ξ_{*}$ , $Λ \to {(1 - D)}^{- 1}$ for $D > D_{▵}$ . Insets show buckled shapes at $D = D_{▵}$ . (d) Corresponding plots of scaled compressive force $μ / ξ^{2}$ against $D$ . Dotted lines for $D > D_{▵}$ correspond to self-intersecting shapes at the energy minimum. Parameter values for numerical calculations: $δ = 1$ , $Ξ_{1} = 20$ , $Ξ_{2} = 34$ , $ℓ_{0} = \sqrt{10}$ .
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Figure 5
Eigenmodes of a buckled epithelium: plot of relative end-to-end shortening $D$ against $Δ$ . Parameter value: $Ξ = 20$ . No eigenmodes were found for $Δ < Δ_{*} (Ξ)$ . On the dashed part of the branch, $E < 2$ . Continuation failed at the point marked $\times$ . Inset: plot of $Δ_{*}$ against $Ξ$ (solid line) and power-law fit (dashed line).
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