Axolotls regenerate limbs throughout life while their body size changes more than tenfold, so the regrowing limb must arrest growth at a size matched to the animal. Theory and mRNA-level measurements (Lyubaykina et al., PNAS 2025) show that proportional regeneration requires 'static scaling' — morphogen gradient parameters that are set by animal size and held fixed during blastema growth; in their model, scaling of the signaling range can be achieved by an effective diffusion coefficient scaling as D∼La2D\sim L_a^2 (D-scaling) or by a degradation rate scaling as β∼La−2\beta\sim L_a^{-2} (beta-scaling), but exactly proportional final size is obtained only for D-scaling, whereas for beta-scaling the final size increases superlinearly with animal size — yet the molecular machinery that implements this scaling is unknown, the growth-arrest regime operating in vivo is undetermined, and morphogen gradients have never been quantified at the protein level across animal sizes. A 2026 preprint (Tardivo et al.) shows that SHH signaling scaling saturates in the largest adults, explaining allometric regenerative defects, but the molecular mechanism, the in vivo regime, and protein-level gradient measurements all remain open. The open problem is mechanistic size control during the SHH–FGF-driven blastema outgrowth phase of regenerating axolotl limbs.

Background

The axolotl (Ambystoma mexicanum) can regenerate an amputated limb at any post-embryonic stage, over an order-of-magnitude range of body sizes. Regeneration proceeds through a blastema, a proliferating mass of progenitor-like cells that first regrows the limb's pattern and then brings it to the correct final size. Growth of the early blastema is sustained by two morphogens — signaling molecules whose concentration gradients encode positional information — namely SHH (Sonic hedgehog), produced in the posterior part of the blastema, and FGF8 (fibroblast growth factor 8), produced anteriorly; the two form a mutual positive-feedback loop, with SHH maintaining FGF8 expression and vice versa. After patterning and tissue differentiation, the young regenerate is proportionally small (the 'tiny-limb' stage) and enters a second growth phase that depends on limb nerve supply rather than SHH–FGF signaling (Wells et al., eLife 2021). Empirically, blastema size and the Shh and Fgf8 expression domains grow sub-proportionally (allometrically) with limb size, while the ratio of the Shh- to Fgf8-dominated signaling regions is nearly scale-invariant (Furukawa et al., Development 2024).

Lyubaykina et al. (PNAS 2025) asked how such a system knows when to stop growing at the size appropriate to the animal. In their minimal model, tissue grows where the two oppositely oriented morphogen gradients overlap above a threshold; growth pushes the two sources apart, shrinking the overlap and thus terminating growth — a robust negative feedback. They distinguish two scaling scenarios: dynamic scaling, where gradient parameters change with the growing blastema, and static scaling, where the parameters depend only on animal size LaL_a and stay constant during growth. Analytically and numerically, static scaling of at least a subset of gradient parameters is necessary and sufficient for proportional growth (final size proportional to animal size), while dynamic scaling is not. Static scaling of the signaling range requires either D-scaling (effective diffusion coefficient D∼La2D\sim L_a^2) or beta-scaling (degradation rate β∼La−2\beta\sim L_a^{-2}), in addition to or instead of scaling of the source-region width; the two subscenarios are not equivalent for final size — proportional growth (L∗∼LaL^*\sim L_a) is obtained exactly for D-scaling, whereas for beta-scaling L∗L^* increases superlinearly with LaL_a. Their whole-mount HCR (hybridization chain reaction, an amplification-based method for imaging mRNA expression domains in intact tissue) RNA in situ hybridization data across animal sizes and time points indicate static scaling of the FGF8 source region, while the SHH source region may scale dynamically. Candidate molecular implementations are known only from other systems: 'expander' molecules that regulate morphogen diffusion or degradation, either diluted by tissue growth (expander-dilution) or repressed by the morphogen itself (expander-repression, established for the DPP (Decapentaplegic, the Drosophila BMP-family morphogen) gradient in Drosophila), and SCUBE-family molecules (secreted proteins that bind Hedgehog ligands and modulate their extracellular spreading), shown to affect SHH transport in the zebrafish neural tube; none have been tested in the axolotl limb. The paper's discussion leaves a cluster of unresolved questions: the molecular implementation of static scaling; which of the two possible growth-arrest regimes of the model operates in vivo (arrest when the growth threshold is crossed first, with morphogen production still on, versus the feedback threshold being crossed first, which switches off both morphogen sources and growth as a tissue-level ON/OFF switch); whether the in vivo growth response to SHH and FGF8 is switch-like or graded (theory shows graded responses preserve the same scaling requirements); direct protein-level quantification of the SHH and FGF8 gradients (current measurements are mRNA-based and may be reshaped by post-transcriptional regulation); and the origin of the allometric breakdown of proportional growth in the largest animals (limbs ≳10 cm), where regeneration is sub-proportional. The paper explicitly limits its analysis to the first growth phase of blastema outgrowth; the subsequent nerve-dependent growth phase (Wells et al., eLife 2021) is noted there as outside its scope rather than posed as an open question.

Problem Statement

Explain, mechanistically and experimentally, how size-matched growth is achieved during the SHH–FGF-driven blastema outgrowth phase of the regenerating axolotl limb — the target fixed by Lyubaykina et al. (PNAS 2025), whose analysis addresses only this first growth phase. This comprises four linked sub-questions:

  1. Molecular implementation of static scaling: identify the molecular component(s) or process(es) by which the parameters of the SHH and FGF8 morphogen gradients in the regenerating limb depend on animal size — in particular how the effective diffusion coefficient DD could scale with animal size squared (D∼La2D\sim L_a^2), how the morphogen degradation rate β\beta could scale inversely with animal size squared (β∼La−2\beta\sim L_a^{-2}), or how source-region widths are set — including whether expander-type mechanisms (expander-dilution or expander-repression), transport modulators such as SCUBE-family molecules, or extracellular-matrix properties implement the scaling, and establish the molecular basis of the observed saturation of SHH signaling scaling in the largest adults (measured in Tardivo et al. 2026, a non-peer-reviewed preprint, whose model attributes the breakdown to a limited expander diffusion range without identifying the expander).

  2. In vivo growth-arrest regime and growth rule: determine which of the two regimes of the mutual SHH–FGF feedback identified in the model operates in the regenerating blastema — arrest when the growth threshold is crossed first (morphogen production still active) or when the feedback threshold is crossed first (source regions shrink and both morphogens switch off, halting growth tissue-wide) — and characterize the actual growth response to SHH and FGF8 in vivo, i.e. whether it is switch-like or graded.

  3. Protein-level gradient quantification: measure the SHH and FGF8 protein concentration gradients directly (rather than mRNA or downstream-target readouts) as a function of animal size and regeneration time, sufficient to test the static-scaling predictions of the theory, including any post-transcriptional reshaping of the gradients.

  4. Allometric breakdown: account for the observed breakdown of proportional blastema growth into allometric scaling in the largest animals (limbs ≳10 cm), for which the source lists alternative explanations (limits of scaling, a combination of dynamic and static scaling, or a separate timer mechanism), in light of the measured saturation of SHH signaling scaling (Tardivo et al. 2026, preprint).

Sub-question 3 calls for a concrete measurement-and-analysis design that would test the static-scaling predictions at the protein level. Sub-question 1 owns the molecular basis of the large-adult saturation of SHH signaling scaling; sub-question 4 takes that saturation as given and accounts for the allometric breakdown it produces.

References

  1. Lyubaykina N, Knapp D, Tardivo P, Kotz M, Sandoval-Guzmán T, Friedrich BM (2025) Static morphogen scaling enables proportional growth in a tissue growth model inspired by axolotl limb regeneration. Proceedings of the National Academy of Sciences 122(47):e2503086122. doi:10.1073/pnas.2503086122 https://doi.org/10.1073/pnas.2503086122 (bioRxiv preprint doi:10.1101/2025.02.14.638295)
  2. Wells KM, Kelley K, Baumel M, Vieira WA, McCusker CD (2021) Neural control of growth and size in the axolotl limb regenerate. eLife 10:e68584. doi:10.7554/eLife.68584 https://doi.org/10.7554/eLife.68584
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