Decaying and expanding Erk gradients process memory of skeletal size during zebrafish fin regeneration

Amputated salamander limbs or fish fins precisely regenerate to their pre-injury size, providing a paradigm for positional memory. Although this phenomenon has been appreciated for centuries, how position-dependent cues control tissue growth remains unresolved. Here we quantify extracellular signal-regulated kinase (Erk) activity in whole populations of osteoblasts during zebrafish fin regeneration. We show that Erk activity scales with the amount of amputated tissue, predicts the likelihood of osteoblast cycling, and predicts the size of regenerated skeletal structures. We find that osteoblast Erk activity depends on fibroblast growth factors receptor signalling and organizes into millimetre-long gradients spanning from the distal tip to the amputation site. Mathematical modelling suggests gradients are established by acute, distally restricted deposition of ligand, whose activity is long-lived and transported by tissue growth. This mechanism is supported by the observed scaling of expression of the essential epidermal ligand Fgf20a with extents of amputation. Our work provides evidence that localized, scaled expression of pro-regenerative ligands instructs long-range signalling and cycling to control size in regenerating appendages.