Nestor-Bergmann A et al. (2018), Mechanical characterization of disordered and a...

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Phys Rev E 2018 May 01;975-1:052409. doi: 10.1103/PhysRevE.97.052409.

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Mechanical characterization of disordered and anisotropic cellular monolayers.

Nestor-Bergmann A , Johns E , Woolner S , Jensen OE .

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We consider a cellular monolayer, described using a vertex-based model, for which cells form a spatially disordered array of convex polygons that tile the plane. Equilibrium cell configurations are assumed to minimize a global energy defined in terms of cell areas and perimeters; energy is dissipated via dynamic area and length changes, as well as cell neighbor exchanges. The model captures our observations of an epithelium from a Xenopus embryo showing that uniaxial stretching induces spatial ordering, with cells under net tension (compression) tending to align with (against) the direction of stretch, but with the stress remaining heterogeneous at the single-cell level. We use the vertex model to derive the linearized relation between tissue-level stress, strain, and strain rate about a deformed base state, which can be used to characterize the tissue's anisotropic mechanical properties; expressions for viscoelastic tissue moduli are given as direct sums over cells. When the base state is isotropic, the model predicts that tissue properties can be tuned to a regime with high elastic shear resistance but low resistance to area changes, or vice versa.

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	Fig. 2. (a) Epithelial apical layer of a Xenopus laevis animal cap, showing 801 cells rendered as polygons superimposed on the original microscopy image. Pαeff for each cell was calculated assuming Peff¯ = 0 and (Λ,Γ) = (−0.259,0.172). Line segments indicate the principal axis of shape and stress for each cell. Darker (lighter) cells have Pαeff>0 (<0) and exert a net inward (outward) force along each line segment. (b) The apical layer in (a) following a 35% instantaneous uniaxial stretch (horizontal) of the membrane beneath the basal cells, resulting in a 19.67 ± 1.91% (95% confidence interval) uniaxial stretch of the apical cells. (c), (d) Histograms showing the frequency density of orientation of the principal axis of stress for cells under tension (darker) and compression (lighter), for apical layers given in (a) [corresponding to (c)] and (b) [corresponding to (d)]; bin areas integrate to unity. Bin size was selected using the Freedman-Diaconis rule.
	Fig. 3. (a) A simulation of a representative monolayer realization satisfying Peff¯ = 0 with 800 cells, for (Λ,Γ) = (−0.259,0.172) (see Fig. 1 for the location in parameter space). Cell shadings and line segments follow the scheme in Fig. 2. (b) The monolayer in (a) following a 20% area-preserving uniaxial stretch and subsequent relaxation. (c), (d) Histograms showing the orientation of the principal axis of stress for cells under tension (darker) and compression (lighter), for monolayers given in (a) [corresponding to (c)] and (b) [corresponding to (d)]. Bin size was selected using the Freedman-Diaconis rule.
	Fig. 4. (a) The effect of incremental stretch on effective tissue pressure. The tissue shown in Fig. 3(a) was subjected to a 30% area-preserving uniaxial stretch in a varying number of steps (straight lines are drawn between data points). The total stretch was divided into equally spaced increments and the tissue was relaxed after every stretch. The tissue starts at Peff¯ = 0 and ends at approximately Peff¯ = 0.57, regardless of how many steps were used. Translucent shading indicates 95% confidence intervals over the five simulations. (b) Shear stress ξ vs magnitude of stretch with (Λ,Γ) = (−0.259,0.172) (solid; the monolayer given in Fig. 3) and (Λ,Γ) = (−0.569,0.145) (dashed). See Fig. 1(b) for locations in parameter space. Each data point represents an instantaneous stretch performed on the same initial isotropic monolayer satisfying Pext = 0. The monolayers were relaxed to equilibrium following stretch.
	Fig. 5. Perturbation stress response to small-amplitude deformations in a prestretched monolayer. The stretched monolayers used were the same as those for the solid line in Fig. 4(b), with (Λ,Γ) = (−0.259,0.172) and the magnitude of prestretch is indicated on the x axis. The equilibrium prestretched monolayers were subjected to small deformations in the x [EX = diag(λ,0)] and y [Ey = diag(0,λ)] directions, with λ = 0.01. The component of the perturbation stress tensor in the direction of stretch is indicated on the y axis, with Δσxxℳ(EY) giving the x-directed stress following EX (lower line) and Δσyyℳ(EX) giving the y-directed stress following Ey (upper line). Solid lines indicate values directly evaluated using (12) and dashed lines are predicted values using (28).
	Fig. 6. (a) Heat map across discrete intervals of (Λ,Γ) parameter space showing the value of the analytic bulk elastic modulus Ke [calculated using (33)] of a disordered isotropic monolayer with 800 cells. (b) Equivalent plot of the analytic shear modulus [calculated using (37)]. (c) Computationally estimated shear modulus across parameter space. The elastic shear modulus Ge was estimated from the global perturbation stress as Δσxyℳ=−κGe, following a 1% simple-shear deformation (k = 0.01) on the simulated monolayers used in (b). Each data point is taken as an average from five realizations of a monolayer with 800 cells, with Pext = 0. (d) Percentage difference (\|Ge−Geiso\|/[0.5(Ge+Geiso)]) between the exact modulus Ge (37) and the approximation Geiso (38), across parameter space; the difference is below 2% in the hatched region.
	Fig. 7. Heat map across discrete intervals of (Λ,Γ) parameter space showing the value of log(ΔPeff¯/ξ^). The dashed line represents the contour where ΔPeff¯=ξ^: Tissues show dominant resistance to area change in the shaded region to the left of the dashed line [where log(ΔPeff¯/Δξ)>0, region A], and to shear in the region to the right [where log(ΔPeff¯/Δξ)<0, region B]. A log scale is used to help display the differences across a large range of values. The monolayers used for all heat maps were the same as those used in Fig. 6, where each data point is taken as an average from five realizations of a monolayer with 800 cells, with Peff¯=0.

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Alt, Vertex models: from cell mechanics to tissue morphogenesis. 2017, Pubmed