  The vertex model of epithelia describes the apical surface of a tissue as a
tiling of polygonal cells, with a mechanical energy governed by deviations in
cell shape from preferred, or target, area, $A_0$, and perimeter, $P_0$. The
model exhibits a rigidity transition driven by geometric incompatibility as
tuned by the target shape index, $p_0 = P_0 / \sqrt{A_0}$. For $p_0 > p_*(6) =
\sqrt{8 \sqrt{3}} \approx 3.72$, with $p_*(6)$ the perimeter of a regular
hexagon of unit area, a cell can simultaneously attain both the preferred area
and preferred perimeter. As a result, the tissue is in a mechanically soft
compatible state, with zero shear and Young's moduli. For $p_0 < p_*(6)$, it is
geometrically impossible for any cell to realize the preferred area and
perimeter simultaneously, and the tissue is in an incompatible rigid solid
state. Using a mean-field approach, we present a complete analytical
calculation of the linear elastic moduli of an ordered vertex model. We analyze
a relaxation step that includes non-affine deformations, leading to a softer
response than previously reported. The origin of the vanishing shear and
Young's moduli in the compatible state is the presence of zero-energy
deformations of cell shape. The bulk modulus exhibits a jump discontinuity at
the transition and can be lower in the rigid state than in the fluid-like
state. The Poisson's ratio can become negative which lowers the bulk and
Young's moduli. Our work provides a unified treatment of linear elasticity for
the vertex model and demonstrates that this linear response is
protocol-dependent.
