Geometric Frustration and Solid-Solid Transitions in Model 2D Tissue

We study the mechanical behavior of two-dimensional cellular tissues by formulating the continuum limit of discrete vertex models based on an energy that penalizes departures from a target area $A_0$ and a target perimeter $P_0$ for the component cells of the tissue. As the dimensionless target shape index $s_0=(P_0/\sqrt{A_0})$ is varied, we find a transition from a soft elastic regime for a compatible target perimeter and area to a stiffer nonlinear elastic regime frustrated by geometric incompatibility. We show that the ground state in the soft regime has a family of degenerate solutions associated with zero modes for the target area and perimeter. The onset of geometric incompatibility at a critical $s_0^c$ lifts this degeneracy. The resultant energy gap leads to a nonlinear elastic response distinct from that obtained in classical elasticity models. We draw an analogy between cellular tissues and anelastic deformations in solids.

Figure 1

Strain measurement with respect to a family of reference configurations in a tissue that penalizes only perimeter discrepancies. (a) and (b) are two configurations with the same perimeter $P_0$; (c) and (d) are two deformed states. Strain should be measured by comparing each deformed configuration to the closest reference configuration; thus, (c) should be compared to (a), and (d) compared to (b).

Figure 2

(a) Dimensionless energy $E(\mathrm{\Delta })$ versus strain on a linear scale for $s_0=3.874$ (compatible, green), $s_0=3.722$ (at threshold, red), and $s_0=3.577$ (incompatible, blue). (Inset) Two tissue configurations of equal minimal energy for $s_0=3.874$. The incompatible tissue has an energy gap at zero strain, corresponding to the residual stresses associated with the onset of strain stiffening. This is shown in (b), which displays the magnitude of the force $F=\partial E/\partial \mathrm{\Delta }$ as a function of strain $\mathrm{\Delta }$ on a log-log scale for $\zeta =1$ and $s_0=3.577$. The dashed line has the slope 1.

Figure 3

(a) Effective Young’s modulus, as defined by the response to stretching, versus target shape index $s_0$ for $\zeta =0.125$, 0.25, 0.5, 1, 2, 4, 8 of a uniform tissue of hexagonal cells. A transition occurs at the critical value $s_0^c=\sqrt{8\sqrt{3}}$ (vertical dashed line), below which the perimeter and area are incompatible. (b) Energy as a function of strain ${\mathrm{\Delta }}_{xx}$ for various values of fixed transverse strain ${\mathrm{\Delta }}_{yy}=0.15$, 0.6, 0.75, 0.9, 1.05, 1.35, for an incompatible tissue with $s_0=3.438$ and $\zeta =1$.