Static morphogen scaling enables proportional growth in a tissue growth model inspired by axolotl limb regeneration

Axolotls can regenerate lost limbs throughout life, while they continue to grow. This poses the question of how the size and pattern of a regenerating limb is matched to a widely varying animal size. Two interacting signaling molecules, Sonic Hedgehog (SHH) and Fibroblast Growth Factor (FGF8), are produced at opposite sides of the regenerating limb and sustain tissue growth through a pair of oppositely oriented signaling gradients. As the size of the regrowing tissue can vary more than three-fold depending on the size of the animal, it is unclear how the activities of these mutually dependent morphogens are maintained and subsequently terminated to determine appropriate growth. Scaling of limb regeneration suggests a size-dependent adaptation of morphogen gradient parameters. Inspired by this biological example, we theoretically investigate general mechanisms of morphogen-controlled growth arrest and proportional growth. In the proposed mechanism, tissue growth increases the spatial distance between the two morphogen gradients, which eventually arrests morphogen activity and growth. We put forward two distinct scaling scenarios of morphogen gradients: either dynamic scaling with blastema size, where morphogen gradient parameters change dynamically with the growing tissue, or static scaling with animal size, where morphogen gradient parameters stay constant during blastema growth and only depend on animal size. We show that static scaling ensures proportional growth, but dynamic scaling does not. We compare theory predictions to experimental quantification of SHH and FGF8 morphogen gradient parameters at different time-points of regeneration in different-sized animals, indicating static scaling for some morphogen parameters, which is sufficient to ensure proportional growth in our model.