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Control of osteoblast regeneration by a train of Erk activity waves

Nature volume 590, pages 129–133 (2021)Cite this article

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Abstract

Regeneration is a complex chain of events that restores a tissue to its original size and shape. The tissue-wide coordination of cellular dynamics that is needed for proper morphogenesis is challenged by the large dimensions of regenerating body parts. Feedback mechanisms in biochemical pathways can provide effective communication across great distances1,2,3,4,5, but how they might regulate growth during tissue regeneration is unresolved6,7. Here we report that rhythmic travelling waves of Erk activity control the growth of bone in time and space in regenerating zebrafish scales, millimetre-sized discs of protective body armour. We find that waves of Erk activity travel across the osteoblast population as expanding concentric rings that are broadcast from a central source, inducing ring-like patterns of tissue growth. Using a combination of theoretical and experimental analyses, we show that Erk activity propagates as excitable trigger waves that are able to traverse the entire scale in approximately two days and that the frequency of wave generation controls the rate of scale regeneration. Furthermore, the periodic induction of synchronous, tissue-wide activation of Erk in place of travelling waves impairs tissue growth, which indicates that wave-distributed Erk activation is key to regeneration. Our findings reveal trigger waves as a regulatory strategy to coordinate cell behaviour and instruct tissue form during regeneration.

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Fig. 1: Scale regeneration proceeds by patterned hypertrophy of osteoblasts.
Fig. 2: Waves of Erk activity travel across regenerating scales.
Fig. 3: Erk activity propagates as a reaction–diffusion trigger wave.
Fig. 4: Waves of Erk activity direct tissue growth in regenerating scales.

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Data availability

Reagents are available upon request. Transcriptomics data are available from Gene Expression Omnibus (GEO) under accession number GSE147551. The microscopy dataset consists of large files (>1 Tb); therefore, microscopy data are available from the corresponding authors, without limitation. Source data are provided with this paper.

Code availability

Zebrafish scale image processing, Erk activity and tissue flow quantification sample MATLAB code is available at https://github.com/desimonea/DeSimoneErkwaves2020.

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Acknowledgements

We thank J. Burris, S. Miller, K. Oliveri, C. Dolan, L. Frauen and D. Stutts for zebrafish care; J. M. Cook and the Duke Cancer Institute Flow Cytometry Facility for help with flow cytometry; A. Kawakami and B. Riley for sharing transgenic fish; I. Rask, H. Kim, S. Li and Z. Weishampel for help with data curation, imaging, fish husbandry and genotyping; V. Cigliola for advice regarding gene expression experiments; M. Bagnat, A. Puliafito, B. Shraiman, S. Streichan and M. Vergassola for scientific discussions and advice; and P. Gönczy, B. Hogan and B. Mathey-Prevot for critical reading of the manuscript. A.D. was supported by Early (P2ELP3_172293) and Advanced (P300PA_177838) Postdoc.Mobility fellowships from the Swiss National Science Foundation. B.D.C. and V.A.T. were supported by NSF Graduate Research Fellowships (1106401). This work was supported by an Innovation in Stem Cell Science Award from the Shipley Foundation, Inc. to S.D. and N.I.H. grant (R01-AR076342) to K.D.P. and S.D.

Author information

Author notes

  1. Ben D. Cox

    Present address: Department of Molecular and Cellular Biology, University of California, Davis, Davis, CA, USA

  2. Valerie A. Tornini

    Present address: Department of Genetics, Yale University School of Medicine, New Haven, CT, USA

Authors and Affiliations

  1. Regeneration Next, Duke University, Durham, NC, USA

    Alessandro De Simone, Maya N. Evanitsky, Luke Hayden, Ben D. Cox, Julia Wang, Valerie A. Tornini, Jianhong Ou, Anna Chao, Kenneth D. Poss & Stefano Di Talia

  2. Department of Cell Biology, Duke University Medical Center, Durham, NC, USA

    Alessandro De Simone, Maya N. Evanitsky, Luke Hayden, Ben D. Cox, Julia Wang, Valerie A. Tornini, Anna Chao, Kenneth D. Poss & Stefano Di Talia

  3. Department of Biology, Duke University, Durham, NC, USA

    Kenneth D. Poss

  4. Department of Medicine, Duke University Medical Center, Durham, NC, USA

    Kenneth D. Poss

  5. Department of Orthopaedic Surgery, Duke University Medical Center, Durham, NC, USA

    Stefano Di Talia

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Contributions

A.D., K.D.P. and S.D. conceived the project and designed the experiments; A.D., M.N.E., B.D.C. and J.W. conducted experiments; A.D., B.D.C., V.A.T. and A.C. generated transgenic fish; A.D. and B.D.C. developed the imaging platform; A.D. developed computational tools and performed data analysis with help from L.H. and J.O.; A.D., L.H. and S.D. developed the theory and wrote simulation codes; A.D. and L.H. performed and analysed the simulations; A.D., K.D.P. and S.D. wrote the paper with comments from all authors.

Corresponding authors

Correspondence to Kenneth D. Poss or Stefano Di Talia.

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Competing interests

The authors declare no competing interests.

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Peer review information Nature thanks Michiyuki Matsuda, Patrick Mueller, David Parichy and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Scale regeneration in zebrafish.

This figure contains data indicating that osteoblasts display minimal proliferation after 4 dpp and that their hypertrophic growth is patterned. a, Array of regenerating scales on the trunk of a fish. n > 50 fish from >5 independent experiments. b, Osteoblasts form a continuous monolayer in zebrafish scales (Supplementary Video 2). n > 50 fish from >5 independent experiments. Scale bar, 250 μm. c, Average cell area (error bars, mean with s.e.m.; n = 4 scales in 4 fish in a single trial) in scale regeneration. d, Fraction of EdU-positive nuclei during the proliferative and hypertrophic phases. Error bars, mean with s.d.; each circle represents the fraction of positive nuclei among 500–2,000 nuclei from an individual scale; 1–4 dpp, n = 4 fish; 1–7 dpp, n = 2 fish; 4–7 dpp, n = 4 fish; single trial; two-sided Wilcoxon’s rank-sum test P is indicated. Proliferative phase, fish are injected at 1 dpp and scales are collected at 4 dpp or 7 dpp, as indicated. Hypertrophic phase, fish are injected at 4.5 dpp and scales are collected at 7 dpp. e, Osteoblast nuclei tagged with the photoconvertible protein mEos2 are photoconverted during the hypertrophic phase (4.5 dpp), imaged daily and tracked thereafter. No nuclei were observed to divide, and almost all could still be detected after 4 days. n = 55/58 cells from 5 fish tracked from 4.5 to 8 dpp pooled from 2 independent experiments. Probability of cell division is less than 2% per day at 95% confidence. Scale bar, 50 μm. f, High magnification of e. Scale bar, 25 μm. g, Osteoblast nuclei tagged with the photoconvertible protein mEos2 are photoconverted during the proliferative phase (3 dpp) and imaged the day after. Cell division can be detected: 9 divisions, scored by the increase of converted nuclei, from 3 to 4 dpp in 55 converted cells from 5 scales from 2 fish (single trial); compatible with a total proliferation rate of 0.156 ± 0.003 per cell per day for the entire scale; white arrows indicates likely division events. Scale bar, 50 μm. h, High magnification of g. Scale bar, 25 μm. i, Examples of tissue velocity field \(\bar{v}\) (tissue flow, blue arrows) and its divergence \(\nabla \cdot \bar{v}\) (heat map), indicating the pattern of tissue expansion and contraction. n > 10 fish from 5 independent experiments. Tissue flows are calculated tracking individual cell movements for about 9 h (one frame every 3 h). Scale bar, 250 μm.

Source data

Extended Data Fig. 2 Manipulation of Erk signalling during scale regeneration; Erk activity at 2–3 dpp.

a, Scale area increase (left) and average cell area increase (right) in fish treated with the Mek inhibitor PD0325901 and DMSO control (with s.e.m.; n = 6 scales from 6 fish in each condition pooled from 2 independent experiments; chi-squared test P is indicated). b, Scale area increase (left) and average cell area increase (right) as function of time in fish expressing a gene encoding a dominant negative version of the fibroblast growth factor receptor 1 (Fgfr1) downstream of the heat-shock promoter hsp70l (hsp70l:dnfgfr1-eGFP) and control siblings not carrying the transgene. Fish are heat-shocked every day, starting before the first time point at 4 dpp (with s.e.m.; n = 12 scales from 3 fish per condition in a single trial; chi-squared test P is indicated). c, d, Example (c) and quantification (d) of Erk activity in fish treated with the Mek inhibitor PD0325901 and DMSO control. Error bars, mean with s.d.; each circle represents a scale from an individual fish, pooled from 2 independent experiments; unpaired two-sided log-normal test P is indicated. e, Example of Erk activity in a regenerating scale at 2 and 3 dpp. Erk activity is activated in a uniform pattern at 2 dpp. n = 5 scales from 5 fish in a single trial. At around 3 dpp, Erk switches off starting from the scale centre. n = 6 scales from 5 fish in a single trial. Scale bars, 250 μm.

Source data

Extended Data Fig. 3 Pharmacological inhibition of Fgfr and Egfr signalling during scale regeneration.

a, b, Example (a) and quantification (b) of Erk activity in fish treated with the pan-Fgfr inhibitor BGJ398 and DMSO control. Error bars, mean with s.d.; n = 4 scales from 4 fish per condition; data from a single trial, replicated in 2 additional independent experiments; unpaired two-sided log-normal test P is indicated. c, Quantification of Erk activity in fish treated with the pan-Fgfr inhibitor BGJ398 for 1–3 h and DMSO control. Error bars, mean with s.d.; DMSO, n = 4 scales from 4 fish pooled from two independent experiments, BGJ398, n = 7 scales from 7 fish pooled from 3 independent experiments; unpaired two-sided log-normal test P is indicated. d, e, Example (d) and quantification (e) of Erk activity in fish treated with the pan-Fgfr inhibitor JNJ-42756493 and DMSO control. Error bars, mean with s.d.; DMSO, n = 6 scales from 6 fish in a single trial; JNJ-42756493, n = 4 scales from 4 fish in a single trial; unpaired two-sided log-normal test P is indicated. f, g, Example (f) and quantification (g) of Erk activity in fish treated with the Fgfr inhibitor SU5402 and DMSO control. Error bars, mean with s.d.; DMSO, n = 6 scales from 6 fish in a single trial; SU5402, n = 4 scales from 4 fish in a single trial; unpaired two-sided log-normal test P is indicated. Control fish are the same as in d, e. h, i, Example (h) and quantification (i) of Erk activity in fish treated with the Egfr inhibitor PD153035 and DMSO control. Error bars, mean with s.d.; DMSO, n = 3 scales from 3 fish in a single trial; PD153035, n = 4 scales from 4 fish per condition in a single trial; unpaired two-sided log-normal test P is indicated. Scale bars, 250 μm.

Source data

Extended Data Fig. 4 Expression of dnfgfr1 and overexpression of fgf3 and fgf20a during scale regeneration.

a, b, Example (a) and quantification (b) of Erk activity in fish expressing a gene encoding a dominant negative version of the fibroblast growth factor receptor 1 (Fgfr1) downstream of the heat-shock promoter hsp70l (hsp70l:dnfgfr1-eGFP) and control siblings not carrying the transgene. Error bars, mean with s.d.; control, n = 3 scales from 2 fish in a single trial; dnfgfr1, 5 scales from 3 fish in a single trial; unpaired two-sided log-normal test P is indicated. In 1/5 scales in hsp70l:dnfgfr1-eGFP fish, we observed that a new wave originated at 108 hpp. Erk peak activity after 24 h treatment could not be measured as waves reached the scale border. c, d, Example (c) and quantification (d) of Erk activity in fish overexpressing fgf20a downstream of the heat-shock promoter hsp70l (hsp70l:mCherry-2a-fgf20a) and in control siblings. Error bars, mean with s.d.; 4 and 7 dpp, n = 4 scales from 4 fish per condition in a single trial; 5 dpp control, n = 5 scales from 4 fish pooled from 2 independent experiments, hsp70l:mCherry-2a-fgf20a, n = 4 scales from 4 fish pooled from 2 independent experiments; two-sided Wilcoxon’s rank-sum test P is indicated. Heat-shock was performed everyday starting from 4 dpp, and Erk activity was measured thereafter (approximately 6 h after the start of the heat-shock) (Extended Data Fig. 8a, b). e, f, Example (e) and quantification (f) of Erk activity in fish overexpressing fgf3 downstream of the heat-shock promoter hsp70l (hsp70l:fgf3) and in control siblings. Error bars, mean with s.d.; n = 5 scales from 5 fish per condition in a single trial; two-sided Wilcoxon’s rank-sum test P is indicated. Heat-shock was performed everyday starting from 4 dpp, and Erk activity was measured thereafter (approximately 6 h after the start of the heat-shock) (Extended Data Fig. 9a, b). Scale bars, 250 μm.

Source data

Extended Data Fig. 5 Sequencing strategy for osteoblasts indicates increased transcript abundance of Erk inhibitors in Erk active cells.

a, Erk activity and osx:Venus–hGeminin signal in regenerating scales at different time-points are shown. n > 10 fish from >5 independent experiments. b,Venus–hGeminin signal as a function of time from Erk peak. Error bars, mean with s.e.m.; n = 89 cells from 3 scales from 3 fish in a single trial; for each cell track, t = 0 is the time of the Erk peak; Erk data are the same as presented in Fig. 2f. hGeminin nuclear signal is normalized to cytoplasmic signal. c, d, osx:hGeminin signal and osx:mCherry–zCdt1 signal (normalized for the respective cytoplasmic signals) in individual cells of a representative scale (n = 4 scales from 4 fish from a single experiment, replicated in 2 additional independent experiments; quantified in e) during the proliferative (c) and hypertrophy phases (d). e, Fraction of osteoblasts in proliferative state (normalized Venus–hGeminin > normalized mCherry–zCdt1) during the proliferative and hypertrophic phases of scale regeneration. Error bars, mean with s.d.; each circle is a scale from an individual fish in a single trial; two-sided Wilcoxon’s rank-sum test is indicated. f, Flow-cytometry strategy to sort two populations of osteoblasts (H2A–mCherry+): one enriched for Venus–hGeminin− Erk active cells (Erk−) (D, 9 × 104, 8 × 104 and 5 × 104 cells in the three samples) and one enriched for Venus–hGeminin+ Erk− inactive cells (Erk+) (E, 4 × 104, 5 × 104, 2 × 104 cells in the three samples). g, Gene set enrichment analysis for the Kyoto Encyclopedia of Genes and Genomes MAPK signalling pathway, Gene Ontology bone mineralization, morphogenesis and growth. FDR, false discovery rate. Data from 3 samples pooled from 2 independent experiments. h, Normalized counts for expressed Erk-related dusp genes (dusp1, dusp2, dusp4, dusp5, dusp6, dusp7, dusp22a and dusp22b), sprouty (spry1, spry2 and spry4) genes and the transmembrane proteoglycan syndecan 4 (sdc4). DeSeq2 P-adjusted is indicated. i, Fold change of Erk inhibitory gene transcripts in regenerating scales of fish treated with the Mek inhibitor PD0325901 with respect to DMSO controls. Error bars, mean with s.d.; unpaired two-sided log-normal test P is indicated; 4 dpp scales are used; DMSO, n = 2 samples in a single trial; PD0325901, n = 3 samples in a single trial. j, Enrichment of Erk target spry4 in scales of fish expressing a gene encoding a dominant negative version of the fibroblast growth factor receptor 1 (Fgfr1) downstream of the heat-shock promoter hsp70l (hsp70l:dnfgfr1-eGFP) with respect to control siblings. Error bars, mean with s.d.; unpaired two-sided log-normal test P is indicated; 4 dpp scales are used; n = 3 samples per condition in a single trial. Heat-shocked (hs) hsp70l:dnfgfr1-eGFP fish are compared with heat-shocked siblings not carrying the transgene. As an additional control, not heat-shocked (not hs) hsp70l:dnfgfr1-eGFP fish are compared with not heat-shocked siblings not carrying the transgene. Scale bar, 250 μm.

Source data

Extended Data Fig. 6 Tests and consequences of trigger wave model.

This figure contains extended details on the mathematical model of Erk waves, on the predictions of the models and their experimental tests. a, Mathematical model of Erk dynamics including a diffusible activator, such as Fgf, in turn activated by Erk, and a delayed inhibitor. Activator and inhibitor concentrations (heat map) as a function of time are shown. Red dashed region, activator source region. b, c, Examples of Erk activity and quantifications of wave speed (corrected for tissue growth) in regenerating scales in fish treated with cycloheximide at 4 dpp and controls. Error bars, mean with s.d.; each circle represents a scale from an individual fish; single trial. d, Example of Erk activity in fish treated with a concentration of the Mek inhibitor PD0325901 that slows wave propagation but does not completely impair it, and DMSO control (see Fig. 3f for Erk wave speed quantification, n = 7 scales from 7 fish pooled from 3 independent experiments). e, Example of Erk activity, organized in an expanding ring (arrowheads), in a developing scale in a juvenile fish throughout time. n > 15 scales in 4 fish in a single trial. f, Wave width for different fold-increases of the activator diffusion constant (simulation; with respect to the standard simulation of Fig. 3, Methods). g, Simulation of growth factor concentration as a function of time in a simple diffusion model and in the Erk trigger wave model. In both models, D is approximately 0.1 μm2 s−1 (Methods). Scale bars, 250 μm.

Source data

Extended Data Fig. 7 Erk activity is required for tissue expansion during zebrafish scale hypertrophy.

a, Time delay of expansion peak position versus Erk peak position. Time delay is measured fitting the relationship between expansion peak position and Erk peak position for different lag times with a linear fit (unitary slope; error bars, best fit with 95% confidence interval; n = 18 expansion peaks from 16 scales from 12 fish pooled from >5 independent experiments). The expansion peak is taken at the start of the 9-h-long time-window considered to calculate flows. For clarity, a positive lag time means that the position of the Erk peak is taken at a time subsequent to the initial point of the time window used to calculate the expansion peak. b, Average single-cell area increase as a function of time from Erk peak. Error bars, mean with s.e.m.; n = 89 cells from 3 scales from 3 fish in a single trial; for each cell track, t = 0 is the time of the Erk peak; Erk data are the same presented in Fig. 2f. Each cell area is normalized to cell area at the time of Erk peak (cell area was measured manually using the Erk KTR–mCerulean signal). Dashed lines, linear fit of normalized cell areas before and after the Erk peak. Slope before peak: (−0.002 ± 0.002) h−1; slope after peak: (0.006 ± 0.002) h−1; with 68% confidence interval. c, Scale area growth as a function of Erk wave speed, corrected for tissue growth, in ontogenetic and regenerating scales. Circles represent individual scales from 12 (regeneration, pooled from >5 independent experiments) and 3 (juvenile development, single trial) fish; Spearman’s correlation coefficient 0.75, P = 2 × 10−4. d, Erk activity, tissue velocity field \(\bar{v}\) (tissue flow, blue arrows) and its divergence \(\nabla \cdot \bar{v}\) (heat map) in scales in fish treated with the Mek inhibitor PD0325901 and DMSO control. n = 4 scales from 4 fish per condition pooled from 2 independent experiments. In d–f, fish are treated at 4 dpp and imaged about 24 h later for about 12 h at 3-h frame rate. e, f, Total expansion rate of expanding regions (e), normalized for scale area, and anterior–posterior (AP)-velocity component (f) in fish treated with PD0325901 (10 μM) and DMSO control. Error bars, mean with s.d.; n = 4 scales from 4 fish per condition pooled from 2 independent experiments; unpaired two-sided Student’s t-test is shown. Fish are treated at 4 dpp and imaged about 24 h later for around 12 h (1 frame every 3 h). g, Cumulative number of waves and scale area as a function of time throughout entire scale regeneration (single trial). h, Cumulative number of waves and scale area as a function of time in scales treated with the pan-Fgfr inhibitor BGJ398 (10 μM) for around 3 h at 4 dpp (orange area) and transferred to fresh water thereafter. Two pooled independent experiments. Scale bar, 250 μm.

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Extended Data Fig. 8 Effects of ectopic and tissue-wide Fgf20a pulses on scale regeneration.

This figure indicates that tissue-wide and synchronous Erk oscillations induced by Fgf20a ectopic expression display similar temporal dynamics to baseline, wave-dependent Erk activation, and that they impair tissue growth. a, Quantification of spatial pattern of Erk activity in 4-dpp regenerating scales in fish expressing fgf20a downstream of the heat-shock promoter hsp70l (hsp70l:mCherry-2a-fgf20a) and control siblings not carrying the transgene. Erk activity is averaged along a 240 μm-wide stripe passing through the scale centre and the wave origin. n = 4 scales from 4 fish per condition, data from a single trial, replicated in 2 additional independent experiments. b, Erk activity in initially active cells (cytoplasmic Erk KTR >1.1 nuclear Erk KTR) in hsp70l:mCherry-2a-fgf20a fish and control siblings not carrying the transgene. Error bars, mean with s.e.m.; n = 3 scales from 3 fish per condition pooled from 2 independent experiments. Transgenic fgf20a expression was induced by heat-shock at 3.5 dpp. Scales were imaged starting 4 h after the start of heat-shock for around 12 h (1 frame every 3 h). c, Erk activity, tissue velocity field \(\bar{v}\) (tissue flow, blue arrows) and its divergence \(\nabla \cdot \bar{v}\) (heat map) indicating tissue expansion, in regenerating scales in hsp70l:mCherry-2a-fgf20a fish and control siblings not carrying the transgene. Transgenic fgf20a expression was induced by heat-shock at 3.5 and at 4.5 dpp. Scales were imaged starting 4 h after heat-shock for around 12 h (1 frame every 3 h). Quantifications in d. d, Tissue expansion in regenerating scales in hsp70l:mCherry-2a-fgf20a fish and control siblings not carrying the transgene. Error bars, mean with s.d.; each circle represents a scale from an individual fish, except for ‘Control sibling’ in ‘2nd heat-shock’ in which n = 4 scales from 3 fish are shown; pooled from 2 independent experiments; unpaired two-sided Student’s t-test P is indicated. Transgenic fgf20a expression was induced by heat-shock at 3.5 and at 4.5 dpp. Scales were imaged starting 4 h after heat-shock for about 9 h (first heat-shock) or about 12 h (second heat-shock) (1 frame every 3 h). e, Average cell area increase as function of time in regenerating scales in hsp70l:mCherry-2a-fgf20a fish and control siblings not carrying the transgene. Error bars, mean with s.e.m.; n = 4 scales from 4 fish per condition in a single trial; chi-squared test P is indicated. Transgenic fgf20a expression is induced by heat-shocking fish every day at the same time (Methods), starting after the first time point. f, g, Example of scale morphology as a function of time in fish expressing hsp70l:mCherry-2a-fgf20a and control siblings not carrying the transgene. n = 4 scales from 4 fish per condition in a single trial (f). Average deviation of scale morphology (g) (error bars, mean with s.e.m., n = 4 scales from 4 fish per condition in a single trial) is measured with respect to before heat-shock by calculating the total discrepancy of the rescaled scale borders in polar coordinates (Methods). Chi-squared test P is indicated. Transgenic fgf20a expression is induced by heat-shocking fish every day at the same time (Methods), starting after the first time point. Scale bar, 250 μm.

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Extended Data Fig. 9 Effects of ectopic and tissue-wide Fgf3 pulses on scale regeneration.

This figure indicates that Fgf3-induced tissue-wide and synchronous Erk oscillations lead to impaired tissue growth. a, b, Quantification of spatial (a) and temporal (b) patterns of Erk activity in regenerating scales in fish expressing fgf3 downstream of the heat-shock promoter hsp70l (hsp70l:fgf3) and control siblings not carrying the transgene. n = 5 scales from 5 fish per condition from a single trial. Transgenic fgf3 expression was induced by heat-shock every day; fish were imaged before and after heat-shock (shaded regions in b). In a, Erk activity is averaged along a 240 μm-wide stripe passing through the scale centre and the wave origin. In b, the fraction of active Erk cells with respect to total is calculated in the entire scale (cytoplasmic Erk KTR >1.1 nuclear Erk KTR). c, Average cell area increase (with s.e.m.) as function of time in regenerating scales in hsp70l:fgf3 fish and control siblings not carrying the transgene. Error bars, mean with s.e.m.; n = 5 scales from 5 fish per condition in a single trial; chi-squared test P is indicated. Transgenic fgf3 expression was induced by heat-shock every day starting from 3.5 dpp. Fish were imaged before and after heat-shock (shaded regions in b). d, e, Example of scale morphology as a function of time in fish expressing hsp70l:fgf3 and control siblings not carrying the transgene. n = 5 scales from 5 fish per condition in a single trial (d). Average deviation of scale morphology (e) (error bars, mean with s.e.m., n = 5 scales from 5 fish per condition in a single trial) is measured with respect to before heat-shock by calculating the total discrepancy of the rescaled scale borders in polar coordinates (Methods). Chi-squared test P is indicated. Transgenic fgf3 expression was induced by heat-shock every day; fish were imaged before and after heat-shock (shaded regions in b). Scale bar, 250 μm.

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Extended Data Fig. 10 Minimal mechanical model of tissue growth and tissue flow properties.

a–d, Tissue density (a), tissue expansion rate (b, c) (calculated as \(\frac{\partial v}{\partial x}\)) and total tissue growth (d) in the case of wave-like and uniform basal tissue growth in 1D mathematical model of tissue growth (Supplementary Notes). e, Two-point correlator of tissue flow velocities in control scales. Error bars, mean with s.d.; n = 5 scales from 5 fish pooled from 2 independent experiments. λ is the flow velocity correlation length (with 95% confidence interval). f–h, Vorticity of tissue velocity field (f), its two-point correlator (g) (error bars, mean with s.d.; n = 5 scales from 5 fish pooled from 2 independent experiments) and adimensionalized vorticity (h) (\(v\,\)is the average flow absolute velocity and λvorticity is the vorticity correlation length, estimated to be about 30 μm). λvorticity is similar to the length used to calculate the vorticity itself, so it represents an upper limit; smaller values of λvorticity would further decrease adimensionalized vorticity. Average absolute tissue flow vorticity for 5 scales from 5 fish pooled from 2 independent experiments: 0.0007 h−1, 0.0009 h−1, 0.0009 h−1, 0.0008 h−1 and 0.0012 h−1. Average tissue flow speed for 5 scales from 5 fish pooled from 2 independent experiments: 0.4 μm h−1, 0.5 μm h−1, 0.5 μm h−1, 0.4 μm h−1 and 0.5 μm h−1. i, Position of the divergence trough, that is, compression peak (distance from scale centroid; calculated over 9 h), as a function of the position of the trough of \(-\frac{{d}^{2}{\rm{Erk}}}{d{x}^{2}}\) in which x is the distance from scale centroid. n = 16 scales from 12 fish from >5 independent experiments; Pearson’s correlation coefficient 0.87, P = 3 × 10−8; dashed line: bisector of the axis. The divergence trough has intensity (−0.002 ± 0.002) h−1 (s.d.; s.e.m. 0.0003 h−1). j, Quantification of average AP component of tissue flow velocity, normalized to the norm of the velocity vector, in regenerating scales in hsp70l:mCherry-2a-fgf20a fish and control siblings not carrying the transgene. Error bars, mean with s.d.; each circle represents a scale from an individual fish, except for ‘Control sibling’ in ‘2nd heat-shock’ in which n = 4 scales from 3 fish are shown and for ‘PD03 followed by heat-shock’ in which n = 5 scales from 3 fish are shown; first and second heat-shock: pooled from 2 independent experiments each, PD03 followed by heat-shock: single trial; unpaired two-sided Student’s t-test P is indicated. First heat-shock, transgenic fgf20a expression was induced by heat-shock at 3.5 dpp; second heat-shock, 3.5 and 4.5 dpp; PD03 followed by heat-shock, fish were treated with PD0325901 (10 μM) for 24 h at 4 dpp, then they were transferred to fresh water and heat-shocked. Finally, fish were returned to the chemical treatment and imaged. Scales were imaged starting 5 h after heat-shock for about 9 h (first heat-shock and PD03 followed by heat-shock) or 12 h (second heat-shock). k, Erk activity in scales treated with PD0325901 (10 μM) and then heat-shocked in fresh water, as described in j, but without returning them to chemical treatment, and imaged. Fraction of Erk active cells with respect to total from 4 scales from 2 fish in a single trial: 0.73, 0.75, 0.64, 0.45. l, Tissue velocity field \(\bar{v}\) (tissue flow, blue arrows) and its divergence \(\nabla \cdot \bar{v}\) (heat map) indicating tissue expansion, in regenerating scales in hsp70l:mCherry-2a-fgf20a fish treated with PD0325901 (10 μM) for 24 h, then heat-shocked, returned to chemical treatment and imaged, as described in j. Quantifications are in j (single trial). Scale bars, 250 μm.

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Supplementary information

Supplementary Information

Supplementary Methods: relative Erk activity peak amplitude, equations and parameters of Erk activity propagation model, corrected wave velocity. Supplementary Notes: effects of Fgf perturbations; sorting strategy for Erk-active osteoblasts and sequencing results; discussion on signal spreading; mathematical model and properties of tissue growth; initial scale/cell areas in perturbation experiments.

Reporting Summary

Supplementary Table

Supplementary Table 1: KEGG and Gene Ontology Gene Set Enrichment Analysis. Gene Set Enrichment Analysis (GSEA) in RNAseq dataset from Geminin-/Erk+ and Geminin+/Erk- osteoblast enriched populations. Size: number of genes in the expressed set. ES: enrichment score. NES: enrichment score normalized across analysed gene sets. NOM: nominal P-value. FDR: False Discovery Rate. FWER: family-wise error-rate.

Video 1

Erk activity propagates as a travelling wave in scale regeneration. Erk activity (heat map) during the hypertrophic phase of scale regeneration is shown. Hpp: hours post-plucking. Scale bar: 250 µm.

Video 2

A train of Erk activity waves propagates across the osteoblast tissue during scale regeneration. Long-term imaging (~2 weeks) of Erk waves during the hypertrophic phase of scale regeneration. Equalized osx:ErkKTR-mCerulean (left) and quantified Erk activity (right; heat map) are shown. Dpp: days post plucking. Scale bar: 250 µm.

Video 3

Mathematical model of Erk activity trigger waves in a reaction-diffusion system. A diffusible Erk activator, such as a growth factor, is produced in a source region (red circle) and activates Erk. Active Erk cells produce more diffusible activator, thus generating an expanding wave front. Erk activates its own inhibitor, generating a delayed negative feedback loop. Scale bar: 250 µm.

Video 4

Erk activity trigger waves orchestrate tissue growth. Schematic representation of Erk activity waves that propagate across zebrafish scales and instruct osteoblast hypertrophic growth.

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De Simone, A., Evanitsky, M.N., Hayden, L. et al. Control of osteoblast regeneration by a train of Erk activity waves. Nature 590, 129–133 (2021). https://doi.org/10.1038/s41586-020-03085-8

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