Tissue morphogenesis: a surface buckling mechanism.

Surface patterns can emerge during growth of anisotropic tissues because of surface buckling. This morphogenetic scenario is examined in the present paper based on a simple phenomenological theory of tissue growth. In particular, we show that constrained growth can lead to tissue compression, which in turn may result in surface buckling of the tissue. The latter means the appearance of wavy patterns on the surface. These patterns decay away from the surface. It is interesting that the critical magnitude of the parameter of mass supply, which corresponds to surface buckling, is independent of the pattern wavelength and various patterns can generally be generated in growth. Results of theoretical analysis show that the surface buckling scenario is realistic if the growing tissue matches the following two conditions. First, compression should appear during tissue growth. Second, the tissue should exhibit strong anisotropy. The former condition does not necessarily mean geometric constraints: inhomogeneous growth or material inhomogeneity and anisotropy can lead to the appearance of compressive stresses. The latter condition is typical of some tissues with fiber reinforcement in planes parallel to the surface. In the latter case, the tissue material is much softer in the out-of-plane direction than in plane. The creation of patterns by restraining tissue growth and forcing the surface to buckle represents a challenging experimental problem.