  Interfaces in tissues are ubiquitous, both between tissue and environment as
well as between populations of different cell types. The propagation of an
interface can be driven mechanically. % e.g. by a difference in the respective
homeostatic stress of the different cell types. Computer simulations of growing
tissues are employed to study the stability of the interface between two
tissues on a substrate. From a mechanical perspective, the dynamics and
stability of this system is controlled mainly by four parameters of the
respective tissues: (i) the homeostatic stress (ii) cell motility (iii) tissue
viscosity and (iv) substrate friction. For propagation driven by a difference
in homeostatic stress, the interface is stable for tissue-specific substrate
friction even for very large differences of homeostatic stress; however, it
becomes unstable above a critical stress difference when the tissue with the
larger homeostatic stress has a higher viscosity. A small difference in
directed bulk motility between the two tissues suffices to result in
propagation with a stable interface, even for otherwise identical tissues.
Larger differences in motility force, however, result in a finite-wavelength
instability of the interface. Interestingly, the instability is apparently
bound by nonlinear effects and the amplitude of the interface undulations only
grows to a finite value in time.
