  Living tissues undergo wetting transitions: On a surface, they can either
form a droplet-like cell aggregate or spread as a monolayer of migrating cells.
Tissue wetting depends not only on the chemical but also on the mechanical
properties of the substrate. Here, we study the role of substrate stiffness in
tissue spreading, which we describe by means of an active polar fluid model.
Taking into account that cells exert larger active traction forces on stiffer
substrates, we predict a tissue wetting transition at a critical substrate
stiffness that decreases with tissue size. On substrates with a stiffness
gradient, we find that the tissue spreads faster on the stiffer side. Further,
we show that the tissue can wet the substrate on the stiffer side while
dewetting from the softer side. We also show that, by means of viscous forces
transmitted across the tissue, the stiffer-side interface can transiently drag
the softer-side interface towards increasing stiffness, against its spreading
tendency. These two effects result in directed tissue migration up the
stiffness gradient. This phenomenon -- tissue durotaxis -- can thus emerge both
from dewetting at the soft side and from hydrodynamic interactions between the
tissue interfaces. Overall, our work unveils mechanisms whereby substrate
stiffness impacts the collective migration and the active wetting properties of
living tissues, which are relevant in development, regeneration, and cancer.
