SUMMARY
Regenerating fish fins return to their original size and shape regardless of the nature or extent of injury. Prevailing models for this longstanding mystery of appendage regeneration speculate fin cells maintain uncharacterized positional identities that instruct outgrowth after injury. Using zebrafish, we show differential Wnt production by pools of progenitor-maintaining niche cells promotes the correct extent of regeneration across the fin. We identify Dachshund transcription factors as niche markers and show the niche derives from mesenchyme populating cylindrical and progressively tapered fin rays. The niche, and consequently Wnt, steadily dissipates as regeneration proceeds; once exhausted, ray and fin growth stops. Supported by mathematical modeling, we show longfint2 zebrafish regenerate exceptionally long fins due to a broken niche “countdown timer”. We conclude regenerated fin size is dictated by the amount of niche formed upon damage – which is simply dependent on the availability of intra-ray mesenchyme defined by skeletal girth at the injury site. This “transpositional scaling” model contends mesenchyme-niche state transitions and self-restoring skeletal geometry rather than cell memories determine a regenerated fin’s size and shape.
MAIN TEXT
Regenerating organs restore their original size and shape after injury. Vertebrate appendage regeneration, including that of teleost fish fins, provides a striking example of this phenomenon. Major fin amputations, tiny resections, and cuts of diverse geometry all produce the same outcome – a restored fin matching the original’s form and in scale with the animal’s body. Spallanzani, Broussonet, and T. H. Morgan pioneered studies of this longstanding mystery of regeneration in the 18th and 19th centuries1, 2. For example, Morgan used oblique caudal fin resections to show that regeneration rates initially correlate with the amount of tissue lost and then progressively slow, ultimately stopping growth when the original size is regained1.
Prevailing models, now largely from zebrafish studies, posit that fin tissue adjacent to damage sites interpret Cartesian coordinate-like positional information to trigger the appropriate rate and extent of re-growth. This concept supposes an extensive array of positional identities, some epigenetic mechanism for cells to store such identities, and, perhaps most puzzling, for the positional information to be restored during the regeneration process. Ultimately, however, the rate and extent of outgrowth is dictated by production of growth factors, including FGF3–6 and Wnt7–10, with the net effect of promoting cellular proliferation. Therefore, understanding growth factor production dynamics provides a logical entry to uncover fin size and shape restoration mechanisms.
Wnt signals indirectly promote cell proliferation9, including by maintaining progenitor cells of the bone-producing osteoblast lineage10, to drive outgrowth until the fin is fully restored. Several Wnts including wnt5a, wnt5b, and wnt10a are produced in the distal regenerating fin7–11, including by distal blastema cells that we term the niche. We hypothesized differential Wnt production determines the rate and ultimately extent of regeneration. Revisiting Morgan’s experiments1, we obliquely amputated adult zebrafish fins from the distal tip of the dorsal side to a proximal location ventrally (see Extended Data Fig. 1 for fin amputation planes and anatomical definitions). We then variably inhibited Wnt secretion by graded dosing of low concentrations of Wnt-C5910, 12. Partial perturbation of Wnt signaling prevented dorsal but not ventral tissue regeneration (Fig. 1A-E). Similarly, medial tissue was more sensitive than lateral tissue to Wnt inhibition following perpendicular resections (Extended Data Fig. 2). We conclude differential Wnt production underlies reacquisition of the stereotypical shape of a zebrafish fin. Concordantly, levels of wnt5a, a representative distal blastema-expressed Wnt7–11, correlated with the demand for regenerative growth along the proximal-distal axis four days after oblique fin amputations (Fig. 1F, G).
Wnt signaling “strength” could be encoded either by the amount of Wnt produced per niche cell or the number of Wnt-producing niche cells. Regardless, we reasoned that understanding the properties of niche cells was key to deciphering size control mechanisms. We used RNA-Seq of 4 days post amputation (dpa) distal vs. proximal fin regenerates, as demarcated by the distal epidermal marker Tg(shha:EGFP)13, to identify niche-characterizing factors (Extended Data Fig. 3). Among transcription factors, the dachshund (dach) family members14, 15 dacha (5.9-fold) and dachc (6.0-fold) were notably distally enriched (Extended Data Fig. 3). We also used the RNA-Seq dataset to confirm the distal enrichment of several Wnts, with particularly high levels of wnt5a and wnt5b (Extended Data Fig. 3). dachc mimicked wnt5a in having notably higher expression levels in 4 dpa proximal tissue of diagonally amputated regenerating fins (Fig. 1H, I).
We used antibody staining to conclusively identify caudal fin cells that express Dacha/c (hereafter referred to as Dach) over the course of fin regeneration (Fig. 1J-O). Dach-expressing cells were rarely observed before 3 days post amputation (dpa) (Fig. 1J). At 3 dpa, Dach became robustly expressed in distal cells adjacent to, but distinct from, distal Runx2+ pre-osteoblasts (Fig. 1K). Dach+ niche cells then steadily reduced over the course of regeneration to a small residual population also found in distal tissue of unamputated fins (Fig. 1L-O). We conclude Dach transcription factors mark Wnt-producing niche cells throughout the course of regeneration. The three-day delay prior to the appearance of Dach+ niche corresponds with the view that fin regeneration proceeds by an acute injury repair phase followed by a prolonged period of progressive outgrowth16. Finally, differential numbers of niche cells produced upon injury and then maintained through regeneration likely accounts for our observation that variable Wnt strength underlies fin outgrowth extents.
Canonical Wnt/β-catenin signaling during fin regeneration is confined to distal blastema / niche cells and pre-osteoblasts in which it indirectly promotes further proximal mesenchyme proliferation9 and maintains a progenitor state10, respectively. However, a potential role for Wnt in maintaining the niche pool itself is unexplored. Using the new Dach marker, we found that pan-Wnt inhibition initiated at 4 dpa using Wnt-C59 rapidly depleted niche cells (Extended Data Fig. 4). Wnt inhibition prevented outgrowth but did not disrupt osteoblast differentiation, joint formation, or skeletal maturation (Extended Data Fig. 4), reinforcing our previous insight that Wnt/β-catenin promotes bone progenitor maintenance upstream of and in opposition to differentiation10. Wnt, whether canonical or non-canonical, appears to have an analogous autocrine role – maintaining the niche population in a state whereby it can orchestrate continued regeneration.
The loss of niche cells upon Wnt inhibition allowed us to test if artificial depletion of the niche pool mimics normal termination of regeneration – as supported by our observation that Dach+ niche cells progressively decrease over regeneration. Wnt inhibition irreversibly blocked fin regeneration, with only rare outgrowth seen long after drug removal (Extended Data Fig. 4). In contrast, re-amputating Wnt-inhibited fins re-initiated full regeneration. Small molecule inhibition of FGF receptor signaling prevented outgrowth to the same degree as Wnt inhibition (Extended Data Fig. 4). However, as FGF inhibition did not deplete Dach+ niche cells (Extended Data Fig. 4), regeneration resumed following drug washout, matching previous observations5. Therefore, mitogenic FGF, which appears niche-expressed3 and is Wnt-dependent9, acts downstream of Wnt’s niche maintenance role. Niche cell numbers then set dynamic levels of both signaling proteins and therefore outgrowth rate. Regeneration ceases when the niche pool, and correspondingly Wnt and FGF production, depletes below an effective level. These observations underscore that understanding instructive growth control mechanisms requires revealing niche cell origins and fates.
Regenerated fin tissues are derived from progenitor cells formed by de-differentiation of mature cells extant at the injury site16. We used a mosaic lineage tracing system to permanently label different cell populations in uninjured fins17 and then track which lineage generates Dach+ niche cells during regeneration. Intra-ray mesenchyme, cells within the cylindrical bony rays, generated Dach+ niche cells (Fig. 2A-D). Many blastema cells derive from intra-ray mesenchymal cells that upregulate tryptophan hydroxylase 1b (tph1b) upon injury18. These cells, including those that populate the distal blastema, are a source of replacement ray mesenchyme18. We confirmed Dach+ niche cells share this lineage using the Tg(tph1b-mCherry) reporter line (Fig. 2E-F). Similarly, the distal-most Msx-expressing blastema cells were Dach-positive (Fig. 2G).
The mesenchymal cell state marker Snail19 was expressed in non-injured intra-ray mesenchyme and proximal blastema cells (Fig. 2H, I). In contrast, Snail was absent from Dach+ niche cells except for a small population of transitioning double Dach+/Snail+ cells (Fig. 2I). Concordantly, snai2 transcripts were abundant in blastema mesenchyme but undetectable in distal-most tissue (Extended Data Fig. 5). Dach-positive cells incorporated EdU even with a brief 4-hour pulse prior to tissue collection, showing the niche population includes actively proliferating cells (Extended Data Fig. 5). These observations support a cyclical model whereby fin amputation induces a delayed transition of Snail+ intra-ray mesenchyme to a Dach+ niche state (Fig. 2J, K; Extended Data Fig. 6). The Dach-expressing niche cells both proliferate and progressively convert back to Snail+ mesenchyme that re-populates regenerated bony rays.
The unique origin of niche cells from intra-ray mesenchyme led us to consider that skeletal geometry might instruct how much niche is formed upon amputation. By this model, resecting a larger bony ray, with a relatively high volume capacity for mesenchyme, would produce a larger initial niche population and therefore more regeneration. In support, the size of fin ray bone segments noticeably tapers along the proximal-to-distal axis (Fig. 3A-E). Further, longer lateral rays are wider at their base than shorter medial rays. 3-D micro-CT analysis confirmed this assessment and underscored that the internal geometry of the rays approximates cylinders (Extended Data Fig. 7, Supplementary Video 1). We measured the radius of the medial sixteen rays at a fixed proximal-distal position and then plotted the squared values (as a proportional surrogate for bone segment cylindrical volume) as adjacent bars. The V-shape formed by the bars’ distribution neatly matched that of a zebrafish fin by closely approximating distributed ray lengths (Fig. 3A).
This model predicts niche amount should vary by proximal-distal amputation position, correlating with the tapering bony rays. Whole-mount in situ hybridization for dachc transcripts at 3 dpa, the time of niche formation, revealed that proximal amputations produced large dachc-expressing niche pools while distal cuts generated progressively smaller niches (Fig. 3F-H). Further, dachc expression across the dorsal-to-ventral fin axis correlated with regenerative demand upon perpendicular resections (Extended Data Fig. 8) – as with diagonal amputations (Fig. 1H, I). Lateral regions that re-form the longest rays expressed high levels of dachc. In contrast, medial tissue that generates short rays to restore the original V-shaped fin produced small dachc-marked niches.
We used mathematical modeling to further explore if skeletal geometry coupled with progressive niche depletion could account for robust size restoration during fin regeneration. We started by assuming outgrowth rate is proportional to the current number of distal niche cells. The bony ray area exposed upon resection, as a linear proxy for cylinder volume, determines the number of initial niche cells. A niche that collectively re-differentiates faster than it proliferates establishes a “countdown timer”. The number of initial niche cells sets the timer, which ends outgrowth when the niche population depletes below some effective level. As such, bony ray size at the amputation site ultimately instructs regenerated ray length.
We solved differential equations to derive a final equation describing regenerated ray length as a function of ray radius at the amputation position: where kl is the rate of length growth per niche cell, α is the difference in niche cell growth rate versus rate of differentiation back to intra-ray mesenchymal cells, NΦ is the number of niche cells below which growth stops, and β is the number of niche cells released per ray cross-sectional area. Mathematically exploring the formula reinforced the conclusion that α must be negative to provide net niche depletion. Solving expected regenerated ray lengths using measured ray radii and optimized constants produced a distribution largely matching actual measured lengths (Fig. 3I) – naturally matching normalized radius2 distributions. The linear relationship between ray radius squared and remaining ray length from the measurement position held across the proximal-distal fin axis (Fig. 3J, R-squared = 0.7). Likewise, existent ray lengths from a hypothetical injury point largely matched theoretical regenerated lengths derived from radius measurements and our growth equation (Fig. 3K).
Several zebrafish mutants develop and regenerate long fins, including dominant longfint2 20, well known as one of two mutations characterizing the Tüpfel longfin (TL) strain21 (Fig. 4A, B). longfint2 ray radius measurements predicted overall fin shape but greatly under-anticipated ray lengths (Fig. 4C). Therefore, the skeleton was not proportionally re-scaled during development and instead suggesting longfint2 disrupts a fin outgrowth regulatory mechanism. We used our mathematical model to predict the growth-determining parameter disrupted in longfint2 fish. The cell proliferation rate (kl) could be larger (e.g. increased mitogen production or mitogen sensitivity), β could be larger, establishing a larger niche pool upon injury, or α could be “less negative”, leading to niche perdurance. Changing kl or β vs. α had distinct theoretical effects when plotting expected regenerative growth over time and growth rate over time (Fig. 4D, E). Following amputation, longfint2 fish showed prolonged regenerative growth rather than increased initial growth or peak growth rate (Fig. 4F, G; Extended Data Fig. 9), suggesting they have a persistent niche (less negative α) rather than enhanced niche generation (β) or increased cellular growth rate (kl).
We examined dachc expression in longfint2 heterozygotes following fin resection to see if niche dynamics were disrupted as anticipated by our modeling (Fig. 4H-Q). longfint2 fish initially formed a normal-sized niche. However, unlike wildtype clutchmates, their dachc-expressing niche only slowly depleted, implicating a less negative α constant. Increasing α while inputting longfint2 ray radii measurements largely predicted actual longfint2 regenerated ray lengths (Extended Data Fig. 10). These results support the robustness of our model and show that longfint2 disrupts the niche depletion system.
We conclude ray widths are transposed into regenerated ray lengths, and hence fin form, via the amount of intra-ray mesenchyme exposed upon ray resection. The mesenchyme transitions into a concordantly sized outgrowth-promoting niche pool that then steadily depletes, ultimately returning the regenerated fin to its original size and shape. In 1929, Samuel Nabrit similarly concluded: “… in [fin] regeneration the rate of growth and consequently the form is controlled locally by the cross-sectional area of the fin rays exposed”22. The mesenchyme itself is therefore Nabrit’s predicted growth promoting “fin ray end substance”. Positional information stored in skeletal geometry is self-restoring since a depleting niche progressively directs slower outgrowth and narrower bones due to its diminishing progenitor-maintenance capacity. We use the term “transpositional scaling” to describe this model for organ size determination – a new answer to an age-old question of appendage regeneration.
Vertebrate appendage regeneration, including that in zebrafish fins, is used as compelling evidence for pattern formation by Cartesian coordinate-defined cellular positional identities23. However, our transpositional scaling model does not require fin cells retain any such molecular information. Similarly, the hypothesized “pre-pattern” established early in a regenerating blastema that directs regional outgrowth18, 24–26 can be explained by skeletal geometry-defined niche cell numbers alone. The state transition mechanisms that generate the niche and drive its progressive depletion are then identical throughout the fin but produce differential, predictable output depending on the starting condition defined by ray widths.
Could similar models apply in other circumstances where an organ is restored to form and scale? The hemi-ray bones in zebrafish fins appose to approximate cylinders, conveniently enabling consideration of the geometric storage of a growth-promoting cell lineage. Urodele amphibian limb regeneration is another striking model of size and pattern restoration. Matching our idea, an older model proposes that initial blastema mass rather than origin determines which proximal-distal limb skeletal elements regenerate27. However, urodele limb cartilage/bones are solid and therefore position-defining geometric reservoirs of a hypothetical niche lineage would have to lie around rather than inside skeletal structures. Intriguingly, urodele limb positional information is stored in connective tissue mesenchyme surrounding bones, albeit reportedly by molecular memories28–30. Regardless, progenitor cell supporting-niche populations are a common component of organ growth, homeostasis, and regeneration31, 32. Our simple model, inherently dependent on regulatory networks with the positional information stored in tissue geometry rather than molecular information, establishes a compelling paradigm to explore in many contexts.
ENDNOTES
Acknowledgements
We thank C. Kimmel and V. Devasthali for feedback on the manuscript; the Stankunas lab for discussions; E. Shaw for technical assistance; the University of Oregon X-ray Imaging Core and A. Lin for assisting with micro-CT; K. Poss and ZIRC for providing zebrafish lines; and the DSHB for antibodies. The National Institutes of Health (NIH) provided research funding (1R01GM127761 (K. S. and S. S.) and 5R03AR067522 (S. S.)). G. A. Y. was supported by a NIH NRSA graduate fellowship (5F31AR071283). H. K. L. received support from the University of Oregon Developmental Biology Training Program (5T32HD007348).
Author contributions
S. S., G. A. Y., H. K. L. and K. S. designed experiments. S.S., G. A. Y., H. K. L., A. L. H., J. A. B., and J. C. performed experiments. K. S. and M. J. H. led the mathematical modeling. K. S. and S. S. prepared and wrote the manuscript with input from M. J. H., G. A. Y., H. K. L., and A. L. H.
Author Information
The authors declare no competing interests.
Correspondence and requests for materials should be addressed to kryn{at}uoregon.edu
METHODS
Zebrafish
Wildtype AB (University of Oregon Zebrafish Facility), longfint2 1, 2, Tg(-2.4shha:gfp:ABC)3, Tg(tph1:mCherry)4, Tg(dusp6:CreERT)5, and Tg(eab:FlEx)6 zebrafish lines were used. Zebrafish were housed in the University of Oregon Aquatic Animal Care Services facility at 28-29°C. The University of Oregon Institutional Animal Care and Use Committee oversaw animal use.
Transgenic mosaic fin lineage tracing
Zebrafish embryos carrying the Tg(dusp6:CreERT) and Tg(eab:FlEx) lines were treated with low doses of tamoxifen, reared to adulthood, screened for mosaic labeling, caudal fin-amputated, and analyzed by whole mount imaging and antibody staining of sectioned tissue as described previously5.
Small molecule studies of fin regeneration
The Porcupine inhibitor Wnt-C597, which blocks Wnt secretion, and the FGF receptor tyrosine kinase inhibitor PD1730748 were dissolved and diluted in DMSO. Caudal fins were amputated at a fixed proximal-distal position or common oblique plane (illustrated in Fig. S1). Fin-resected fish were then immediately placed in drug-containing aquarium water. Animals were transferred to fresh drug-containing water every 48 hours. The final concentration of DMSO in fish water was 0.01% for all conditions.
For niche depletion experiments with delayed small molecule addition (Fig. S4), fin-resected AB wildtype fish first were allowed to regenerate for 4 days. The animals then were treated with DMSO (0.01%, n = 10), Wnt-C59 (500 nM, n = 10), or PD173074 (5 μM, n = 10) for another 4 days. Animals were fed and changed to fresh drug-containing water after 48 hours. At 8 days post-amputation (dpa), animals were returned to plain fish water. In some cases, fins were subjected to a second amputation proximal to the original amputation site at 12 dpa. Animals were followed for up to 60 dpa, when each fish was imaged using a Leica M165 FC stereomicroscope. Representative fish for each condition were visualized using high resolution stitched imaging (Nikon Eclipse Ti widefield microscope and NIS-Elements software) before drug treatment, immediately after drug treatment, and at the experiment’s conclusion. Regeneration was quantified by scoring the number of fully regenerated individual rays. This complete experiment was repeated three times with the same outcomes.
RNA-Seq
Caudal fins of adult Tg(-2.4shha:gfp:ABC) zebrafish were resected. At 4 dpa, dissected regenerated tissue at and beyond the distal GFP-marked epidermal domains was collected as “distal regenerate” samples. The remaining proximal regenerated tissue also was harvested. Tissue pooled from four animals constituted matched replicate samples. Tissue was immediately placed in TRIzol reagent (Thermo Fisher) and homogenized. RNA was isolated following the manufacturer’s instructions with minor alterations. The RNA was precipitated overnight at −80°C and then pelleted at 15,000 rpm for 30 minutes at 4°C. Pellets were washed twice with 70% ethanol, dried for 10 minutes at room temperature, and resuspended in RNase-free water (Thermo Fisher).
RNA-Seq libraries were prepared from 1 μg of isolated RNA using a Kapa Biosystems Stranded mRNA-Seq kit. Bar-coded libraries were pooled and sequenced using a NextSeq 500 (Illumina). Illumina reads were aligned to the zebrafish genome (GRCz11) using TopHat29. Gene-assigned aligned reads were counted using HTseq10. DEseq11 was used to determine differential expression of 1147 genes annotated to have “transcription regulator activity” (GO:0140110) by AmiGO 212–14. Genes below the 40% quantile for total counts were filtered out to improve the statistical analysis of relatively highly expressed genes.
Immunostaining and in situ hybridization
Immunostaining was performed on paraffin sections as described15. Antibodies used were: Runx2 (Santa Cruz Biotechnology, 27-K, 1:500-1000 dilution); Dach (Proteintech Group, 1:2000); Msx (Developmental Studies Hybridoma Bank, tissue culture supernatant, 1:15-20); Snail (Developmental Studies Hybridoma Bank, ascites, 1:100); mCherry (Takara Bio or Novus Biologicals, 1:100). For Snail/Dach double immunostaining, Dach antibodies were used at 1:1000 and sodium dodecyl sulfate (SDS) was added to 0.005% during the primary antibody incubation step. Immunostained sections were imaged using either Olympus FV1000 or Zeiss LSM 880 laser scanning confocal microscopes.
Whole mount in situ hybridization using DIG-labeled probes followed established procedures15. The wnt5a probe was described previously15. Templates for dacha and dachc probes were amplified by PCR from regenerating fin cDNA using the indicated primers, where the reverse primer contains a T7 promoter sequence (in caps) for in vitro transcription using DIG labeling mixtures (Roche) and T7 RNA polymerase:
dacha forward: 5’-atggccgtatctgcaactcctccggtgc-3’
dacha reverse: 5’-TAATACGACTCACTATAGGGtcagtacatgatggggggtttggagtagg-3’
dachc forward: 5’-atggcccacgcgcctccga-3’
dachc reverse: 5’-TAATACGACTCACTATAGGGtcagtacatcattgtggactttagaaagagcctt-3’
The snai2 probe template was generated by PCR amplifying its coding sequence from regenerating fin cDNA using the following primers followed by ligation to the pCRII vector (Thermo Fisher).
snai2 forward: 5’-atgcctcgttcattcctagtaaagaagc-3’
snai2 reverse: 5’-tcagtgtgcgatgcaacagccag-3’
The resulting plasmid was linearized and a DIG-labeled probe synthesized by in vitro transcription using SP6 RNA polymerase.
5-ethynyl-2’-deoxyuridine (EdU) incorporation and staining
10 μg of EdU in saline was intraperitoneally injected into wildtype fish 92 hours post caudal fin amputation. Fin tissue was collected 4 hours later, fixed, and processed for paraffin sectioning. EdU detection used a Click-iT kit (Thermo Fisher) combined with immunostaining as described above using Dach antibodies.
X-ray micro computed-tomography imaging
An adult wildtype zebrafish was sacrificed and fixed in 4% PFA for 24 hours. The specimen was stabilized within a 15 ml conical tube and its caudal fin scanned using a VivaCT 80 (SCANCO Medical) with 55 kVp X-ray source energy, 145 µA current, 6.5 µm pixel size and 1000 ms per projection integration time. Slice images were reconstructed to a 1590×1590 pixel matrix using an automated cone beam convolution backprojection algorithm. The resulting 2-D images were output for 3-D reconstruction using Imaris 9.3 software (Bitplane).
Morphometrics and mathematical modeling
Rays were considered as simple cylinders to facilitate modeling regenerative fin outgrowth as a function of skeletal geometry. A given bony ray segment’s volumetric capacity for injury-activated, niche-forming intra-ray mesenchyme4, then could be estimated by first measuring the ray’s width (= diameter) at a hypothetical amputation site. For modeling purposes, the square of the ray’s radius was used as a proxy as normalized areas and volumes (for cylinders of a fixed length) are mathematically identical.
Caudal fins of five-month-old longfint2/+ and wildtype clutchmate fish were imaged using differential interference contrast microscopy and multi-field stitching (Nikon Eclipse Ti and NIS-Elements) for morphometrics. Fish were euthanized by tricaine bath overdose and mounted in 0.75% low melt agarose / PBS on microscope slides. The fins were gently fanned out using a horse hair brush to fully extend the tissue and separate rays. The initial proximal-distal positions for ray measurements were determined by first identifying the end of the innermost truncated rays (lateral “mini-rays” that, unlike the standard 18 caudal rays, do not extend the full distance of the fin, shown in Extended Data Fig. 1). A line was drawn across the fin that intersected points two ray bone segments preceding the end of the mini-rays on the fin’s dorsal and ventral sides. The width of each ray was measured where it intersected this transverse line.
Similarly, polylines (using anchored points to allow for shifts in ray orientation) starting from the same proximal-distal position were drawn along each ray and then directly in between branches of a given “mother ray” until reaching the fin tip. The lengths of these lines defined how much each ray would have to regenerate to restore the fin’s original size. Lengths were normalized to the longest wildtype ray. Repeated measures taken every three bone segments along the proximal-distal axis for rays 3, 4, 15, 16 used the same procedure, with lengths representing the distance from the diameter-measurement position to the end of the ray.
We derived a mathematical model for “Transposition Scaling” that predicts the extent of regenerative outgrowth as a function of skeletal geometry – the bony ray’s capacity for niche-originating mesenchyme. First, we considered the number of niche cells released at a cut site as: where r is the radius of the ray at the cut site and β; is the average number of niche cells released per unit area. Niche cells can then proliferate (given by kg) or convert to a differentiated, mesenchymal state (given by kq). The change in the number of niche cells per unit time is:
The difference in these rates can be summarized by α ≡ (kg – kq) Using the boundary condition that N(0) = N0, the solution to this first-order differential equation is:
We model bony ray growth rate as directly proportional to the number of niche cells, controlled by kl:
Using the boundary condition that, the solution to this differential equation is:
To link the number of cells released to the final ray length, we assume that skeletal growth stops when the number of cells reaches Nϕ, the minimum number of cells required to support growth. This happens at tend:
We then solve for tend: and substitute into our model for l(t):
This gives the final relationship: where kl is the length growth rate per niche cell, is the difference between the niche cell growth rate and their conversion back to intra-ray mesenchymal cells, Nɸ is the minimum number of cells required to support growth, and β reflects how many niche cells are released per bony ray cross-sectional area.
Fin regeneration outgrowth measurements
Caudal fins from clutchmate longfin+/+ and longfint2/+ animals were amputated at the same position along the proximal-distal axis (Extended Data Fig. 1) and allowed to regenerate for 30 days. Outgrowth measurements were taken every 24 hours from 1-8 dpa and every 48 hours from 10-30 dpa. The length of the third ray (Extended Data Fig. 1) from the amputation site to the fin’s tip was measured from images acquired with a Leica M205 FA stereomicroscope. Stitched fin images of zebrafish euthanized with tricaine and mounted in 0.75% low melting agarose were captured using a Nikon Eclipse Ti widefield microscope and NIS-Elements software. Data points were fit to user-defined transpositional scaling equations (see Mathematical modeling) by nonlinear regression using GraphPad Prism. The α, β, and kl parameters were allowed to vary to generate best-fit curves. β and kl were nearly identical between the wildtype and longfint2/+ data sets. In contrast, α was 3.0x higher for longfint2/+, similar to the α difference (2.1x) derived from the geometrical (rather than growth rate) analysis.
Data availability
RNA-Seq data are deposited at the NCBI Gene Expression Omnibus (GSE125859).