About the equilibrium shape of fibred structures, and biological shapes

Biological morphogenesis has often been modeled with reaction-diffusion models [A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237 (1952) 37-72]. The interplay of bio-chemical fields is supposed to generate shapes by positional information carried by the values in the field. However, the structure of the biological tissue at the microscopic scale is absent from these models. We show that the fibred nature of biological tissue induces specific morphogenic properties. Fibred shapes can be calculated from physical principles borrowed from the theory of crystallogenesis. These give an intuitive insight into the shape of fruits or vegetables, buds and pins in botany, fingers, muscles, insects abdomen and heart in the animal realm, and also into other fibred structures such as the mitotic spindle. We predict the existence of bumps, apices or cusps at poles of fibred structures. An extrapolation to out-of-equilibrium growth predicts that these structures grow forward in the direction of the cusp, and that fibred organs should have a regular branching ordering. However, our model does not take into account the elasto-plastic properties, or the composite nature of the living material.