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The Mean-Field Survival Model for Stripe Formation in Zebrafish Exhibits Turing Instability

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A ring of coupled cell equations provably reproduces zebrafish stripe formation through a Turing instability, with exact parameter bounds.

desk verdict Rigorous, self-contained Turing analysis of a discrete cell model; the main theorems check out and the biological claims are honest, though the mean-field closure deserves scrutiny. read the letter →

arxiv 2411.15293 v3 pith:RKAEU7XA submitted 2024-11-22 q-bio.TO math.DSq-bio.PE

Robert Jencks This is my paper · ORCID
classification q-bio.TOmath.DSq-bio.PE MSC 34C2392C15
keywords Turinginstabilitymean-fieldsurvivalmodelcoupledODEsystemzebrafishstripeformationlinearstabilityanalysisdiscreteFouriertransformChebyshevpolynomialspattern
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the black and yellow stripes of zebrafish can be generated by a Turing instability—the classic mechanism in which a uniform state is destabilized by spatially periodic perturbations—in a ring of coupled ordinary differential equations. The model tracks the expected presence of melanophores and xanthophores at each cell location, with short-range inhibition and long-range promotion at distance h. The main theorems give a complete parameter classification: if the melanophore death rate d exceeds the melanophore birth parameter b (the set $P=\{0

What carries the argument

The load-bearing object is the linearization of the ring ODE system in discrete Fourier space. Because the ring is translation invariant, the $2N$-dimensional Jacobian splits into $N$ independent $2\times 2$ blocks $\hat{L}_k$ labeled by Fourier mode $k$, whose trace is always negative and whose determinant is $F(x_k, T_h(x_k), b, d)/((b+d)(b+2d))$. The sign of $F$ therefore decides stability: $F<0$ means an eigenvalue with positive real part and a Turing bifurcation. Lemma 1 establishes the geometry of the set where $F<0$ in the $(x,y)$-plane, including that it is nonempty exactly for $(b,d)\in P$ and that its left edge has infimum $1/3$, which is what forces the threshold $h\ge 3$. The proof then uses properties of Chebyshev polynomials—their rightmost local minimum at $x=\cos(\pi/h)$ approaching $x=1$ for large $h$, and the density of the points $x_k$ for large $N$—to turn the geometry into the parameter conditions of Theorems 1 and 2.

What would settle it

Run the stochastic survival model (system 2) at parameters inside $P$ with $h\ge 3$ and $N$ large enough for Theorem 1, starting from a uniform random state; if the homogeneous state persists and no periodic stripe mode grows, the mean-field closure—and with it the paper's central prediction—would be falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the mean-field survival model has a Turing bifurcation whose onset is exactly encoded by the sign of a single polynomial, $F(x_k, T_h(x_k), b, d)$, where $x_k=\cos(2\pi k/N)$ and $T_h$ is the $h$-order Chebyshev polynomial. The coexistence equilibrium $(d/(b+d), b/(b+2d))$ is linearly unstable exactly when some Fourier mode $k\neq 0$ makes $F<0$. Theorem 1 proves that for $(b,d)\in P$, sufficiently large $h$ and $N$ guarantee such a mode exists, and the fastest-growing mode satisfies $\lceil N/4h\rceil \le k \le \lceil N/2h\rceil$; for $(b,d)\notin P$ the equilibrium is linearly stable. Theorem 2 proves that $h\ge 3$ is necessary and sufficient for instability to be possible for large $N$, and Theorem 3 gives explicit vertical and slant asymptotes for the neutral stability curves. The smallest ring that can produce stripes is $N=6$ with $h=3$. These results imply that melanophore and xanthophore stripes always form out of phase with each other, and that stripe width should be between two and four projection lengths, matching the observed values in zebrafish.

Load-bearing premise

The load-bearing premise is that the expected values at neighboring cell locations are uncorrelated, so the stochastic survival model closes exactly to the ODE system; if spatial correlations are significant, the predicted Turing stripes may not occur in the stochastic process.

Editorial extensions

If this is right

  • If the theorem is right, any zebrafish large enough to have at least six cells around its circumference and projections at least three cells long lies in a parameter regime where the uniform pigment state is unstable, so stripes can form without cell movement or iridophores.
  • The bound $\lceil N/4h\rceil \le k \le \lceil N/2h\rceil$ fixes stripe width between two and four projection lengths; combined with observed stripe widths of 7–12 cell diameters, it selects the realistic range $4\le h\le 13$, $112\le N\le 192$.
  • Because $h=1,2$ never yield instability for any birth and death rates, the model makes a sharp biological prediction: melanophore projections shorter than three cell diameters cannot produce stripes by this mechanism.
  • The finite-$N$ ODE analysis and its $N\to\infty$ limit differ qualitatively from the earlier PDE continuum model, which predicts instability for arbitrarily large birth parameter $b$; the ODE analysis says the unstable region is confined to $0<b<d$.
  • For a growing fish, the bounds imply stripe width stays near a fixed chemical wavelength while the number of stripes increases roughly in proportion to circumference.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the same discrete-Fourier-plus-Chebyshev method should transfer to any ring or network model whose coupling is a translation-invariant average, with the curve $y=T_h(x)$ replaced by the ratio of the Fourier symbols of the short-range and long-range coupling operators.
  • Editorial inference: the discrepancy with the PDE limit suggests a testable mathematical question—whether a higher-order Taylor expansion or a nonlocal integral kernel converges to the ODE instability region as $N\to\infty$; the paper states no such general result exists.
  • Editorial inference: the mean-field assumption of no spatial correlation could be checked by simulating the stochastic survival model directly; if correlations suppress the predicted modes, the deterministic predictions would fail precisely at biologically relevant cell densities.
  • Editorial inference: the model's stripe-count growth prediction could be compared with time-lapse observations of regenerating or growing fin stripes, where the wavenumber should increase in stepwise fashion as circumference crosses multiples of the preferred wavelength.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This paper performs a linear stability analysis of the mean-field survival model (1), a 2N-dimensional coupled ODE system on a ring describing xanthophore and melanophore probabilities, with nearest-neighbor inhibitory coupling and distance-h excitatory coupling. The main results characterize when the nontrivial homogeneous equilibrium is unstable to spatially periodic perturbations: Theorem 1 states that for (b,d) in P={0<b<d}, instability occurs for sufficiently large h and N, with the most unstable wavenumbers lying between ceil(N/4h) and ceil(N/2h), and that the equilibrium is linearly stable outside P; Theorem 2 shows that h>=3 is necessary and sufficient for instability to be possible for large N; Theorem 3 and the corollaries give vertical and slant asymptotes of the neutral stability curves, including the N->infinity limit. The paper also compares the ODE predictions with the earlier PDE limit, reports numerical simulations consistent with the linear analysis, and draws qualitative comparisons with zebrafish stripe widths and melanophore projection lengths.

Significance. If the main theorems are correct, this is a valuable contribution: it provides a complete, essentially parameter-free linear-stability characterization of a coupled ODE model that is not obtained by discretizing a PDE, and it shows that the N->infinity ODE limit differs qualitatively from the previously studied PDE limit. The derivations are self-contained and the linear algebra is checked in detail: the sign of the determinant is controlled by the explicit function F in Eq. (5), Lemma 1 characterizes its negativity region, and the density arguments in Theorems 1 and 2 are valid. The numerical simulations corroborate the predicted unstable modes. The biological discussion is appropriately cautious, although the mean-field closure assumption, disclosed in Section 1.1, is not tested against stochastic simulations of the underlying survival model; this limits the strength of the biological extrapolation but does not affect the ODE theorem.

major comments (2)
  1. [Theorem 3, Eq. (10c); Corollary 1, Eq. (12c); Corollary 2, Eq. (16d)]
  2. [Corollary 2, Eq. (16b)]
minor comments (4)
  1. [Proof of Theorem 2]
  2. [Proof of Theorem 3]
  3. [Figure 4 and Figure 5 captions]
  4. [Section 1, affiliation]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the linear-stability theorems are derived from the model equations with no fitted parameters and no load-bearing self-citation.

full rationale

The paper's central claim is a mathematical characterization of Turing instability for the fixed ODE system (1), derived from the linearization of that system. The instability criterion is the sign of F in Eq. (5), computed directly from the Jacobian determinant; Theorems 1 and 2 are then proved from Lemma 1 and density arguments on the Chebyshev curve, with no parameter fitted to the target conclusion. The model itself is taken from Konow et al. [26] with explicitly stated simplifying assumptions, and the paper is an analysis of that model rather than a derivation of the model from the observed stripe widths, so carrying the model forward is not circular. The biological comparisons (h >= 3 matching observed projection lengths; stripe-width bounds 2h to 4h) are derived bounds compared with independent observations, not constants selected to make the predictions match. The mean-field closure assumption, that stochastic transitions are uncorrelated across space, is disclosed in Section 1.1 and is a limitation on biological interpretation rather than a circular step in the mathematical derivation. No uniqueness theorem or central premise is imported from the author's own prior work, and the only self-referential note about circularity is the paper's own assurance that the proof of Theorem 3 does not depend on Theorem 2, which is consistent with the appendix's independent polar-coordinate argument. Therefore the derivation chain is self-contained and the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the mean-field closure and the simplified parameter regime inherited from Konow et al., plus standard linear stability and Chebyshev polynomial facts. The paper introduces no new free parameters or invented entities; b, d, h, N are model inputs explored across their ranges, and the biological comparisons use independent observations.

assumptions (6)
  • domain assumption Mean-field closure: the expected values ⟨Xj⟩ and ⟨Mj⟩ evolve according to the closed ODE system (1), i.e., spatial correlations are zero.
    Stated in Section 1.1 as the assumption that there is no correlation across space in the stochastic transitions; this is the bridge from the survival model to the ODE system analyzed.
  • domain assumption Simplified parameter regime: s_x = s_M = 1, b_X = 1, d_X = 0, d_MX = 0, inherited from Konow et al.
    These simplifications are stated in Section 1.1 and define system (1); the analysis does not cover the full parameter space.
  • domain assumption Ring geometry with periodic boundary conditions and h ≤ N/2.
    Model (1) is defined on a ring with indices modulo N, stated in Section 1; the restriction 1 ≤ h ≤ N/2 is stated to avoid wrapping duplication.
  • domain assumption Restriction to b > 0, d > 0 so that the coexistence equilibrium is physically relevant.
    Section 1 states probabilities require b > 0 and d > 0; the focus is on the coexistence equilibrium for b ≥ 0.
  • standard math Standard linear stability theory: instability is equivalent to the Jacobian having an eigenvalue with positive real part; for the 2x2 L_k with Tr < 0, this is equivalent to Det < 0.
    Used throughout; the trace is shown negative for admissible parameters, reducing stability to the sign of F.
  • standard math Properties of Chebyshev polynomials (roots, extrema, values) used to locate the curve y = T_h(x) relative to the instability region.
    Stated in the proof of Theorem 1 and used for the bounds on k.

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Cite this review

Pith. "Pith review of The Mean-Field Survival Model for Stripe Formation in Zebrafish Exhibits Turing Instability." pith.science (2026). https://pith.science/paper/RKAEU7XA

@misc{pith2026241115293,
  author       = {Pith},
  title        = {Pith review of: The Mean-Field Survival Model for Stripe Formation in Zebrafish Exhibits Turing Instability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKAEU7XA}},
  note         = {Machine review of arXiv:2411.15293}
}
read the original abstract

Zebrafish have been used as a model organism in many areas of biology, including the study of pattern formation. The mean-field survival model is a coupled ODE system describing the expected evolution of chromatophores coordinating to form stripes in zebrafish. This paper presents analysis of the model focusing on parameters for the number of cells, length of distant-neighbor interactions, and rates related to birth and death of chromatophores. We derive the conditions on these parameters for a Turing bifurcation to occur and show that the model predicts patterns qualitatively similar to those in nature. In addition to answering questions about this particular model, this paper also serves as a case study for Turing analysis on coupled ODE systems. The qualitative behavior of such coupled ODE models may deviate significantly from continuum limit models. The ability to analyze such systems directly avoids this concern and allows for a more accurate description of the behavior at physically relevant scales.

Figures

Figures reproduced from arXiv: 2411.15293 by the authors.

Figure 1
Figure 1. Schematic of ring geometry of model (1). Red arrows rep [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. The curves y = Th(x) (solid black) and F(x, y, b, d) = 0 (dashed black) in the xy￾plane. The set with F < 0 is the shaded region below and to the right of the F = 0 curve. Turing instability occurs when one or more of the points (xk, Th(xk)) (black markers) lie in the region with F < 0 for some (b, d). All markers are labeled with the mode(s) that they correspond to. For both plots, h = 3 and N = 6. (a) For (b, d) =… view at source ↗
Figure 3
Figure 3. The curves y = Th(x) (solid black) and F(x, y, b, d) = 0 (dashed black) in the xy-plane. The set with F < 0 is the shaded region below and to the right of the F = 0 curve. Turing instability occurs when one or more of the points (xk, Th(xk)) (black markers) lie in the region with F < 0 for some (b, d). The rightmost marker corresponds to the k = 0 mode, the next marker corresponds to the k = ±1 modes, etc. The modes… view at source ↗
Figures from the paper (7 more)
Figure 3
Figure 3. Figure 3: Figure 3c shows the F = 0 level set in the xy-plane for the same h and N values, with (b, d) = (3, 10). The point corresponding to the k = ±1 modes lies between the rightmost minimum and rightmost root of y = Th(x), whereas the k = ±3 modes do not. This is the geometri…
Figure 4
Figure 4. Figure 4: The curves, F(xk, Th(xk), b, d) = 0, in the bd-plane. The bifurcation curve (solid blue) represents the boundary where the equilibrium is neutrally stable. The neutral sta￾bility curves (dashed blue) for specific modes represent where the equilibrium is neutrally stabl…
Figure 5
Figure 5. Figure 5: Scatter plots of linear instability behavior in the [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: The numerical steady state for a simulation with paramete [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: The time series data of the same simulation as Figure 6 for se [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: Time series data for simulations with differing parameter valu [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: The bifurcation curve (solid blue) for the ODE system with [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]

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Works this paper leans on

33 extracted references · 19 canonical work pages

  1. [1]

    A. M. Turing, The chemical basis of morphogenesis, Philos. Trans . R. Soc. Lond. B 237 (641) (1952) 37–72. doi:10.1098/rstb.1952.0012

    arXiv 1952

  2. [2]

    Eckhaus, Studies in non-linear stability theory, Springer Tracts in Natural Philosophy, Springer, Berlin, Germany, 1965

    W. Eckhaus, Studies in non-linear stability theory, Springer Tracts in Natural Philosophy, Springer, Berlin, Germany, 1965. doi:10.1007/978-3-642-88317-0

    doi:10.1007/978-3-642-88317-0 1965

  3. [3]

    Edelstein-Keshet, Mathematical models in biology, Classics in Ap - plied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005

    L. Edelstein-Keshet, Mathematical models in biology, Classics in Ap - plied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2005. doi:10.1137/1.9780898719147.fm

    work page doi:10.1137/1.9780898719147.fm 2005

  4. [4]

    J. D. Murray, Mathematical Biology, Interdisciplinary Applied Mat he- matics, Springer, Berlin, Germany, 1993. doi:10.1007/b98868

    doi:10.1007/b98868 1993

  5. [5]

    Walgraef, Spatio-temporal pattern formation, Partially Ordered Systems, Springer, New York, NY, USA, 1996

    D. Walgraef, Spatio-temporal pattern formation, Partially Ordered Systems, Springer, New York, NY, USA, 1996. doi:10.1007/978-1-4612-1850-0 . 48

    work page doi:10.1007/978-1-4612-1850-0 1996

  6. [6]

    I. R. Epstein, J. A. Pojman, An introduction to nonlin- ear chemical dynamics: Oscillations, waves, patterns, and chaos, Oxford University Press, New York, NY, USA, 1998. doi:10.1093/oso/9780195096705.001.0001

    arXiv 1998

  7. [7]

    Kondo, T

    S. Kondo, T. Miura, Reaction-diffusion model as a framework for under- standing biological pattern formation, Science 329 (5999) (2010) 1616–

    work page 2010

  8. [8]

    Van Saarloos, Front propagation into unstable states, Phys

    W. Van Saarloos, Front propagation into unstable states, Phys . Rep. 386 (2-6) (2003) 29–222. doi:10.1016/j.physrep.2003.08.001

    doi:10.1016/j.physrep.2003.08.001 2003

Show all 33 references
  1. [9]

    Prigogine, G

    I. Prigogine, G. Nicolis, Self-organisation in nonequilibrium systems : Towards a dynamics of complexity, in: M. Hazewinkel, R. Jurkovich, J. H. P. Paelinck (Eds.), Bifurcation Analysis, Springer, Dordrecht , Netherlands, 1985, pp. 3–12. doi:10.1007/978-94-009-6239-2\_1

    1985 doi

  2. [10]

    A. L. Krause, E. A. Gaffney, P. K. Maini, V. Klika, Modern perspe ctives on near-equilibrium analysis of Turing systems, Phil. Trans. R. Soc. A 379 (2213) (2021) 20200268. doi:10.1098/rsta.2020.0268

    2021

  3. [11]

    Kondo, An updated kernel-based Turing model for studying the mech- anisms of biological pattern formation, J

    S. Kondo, An updated kernel-based Turing model for studying the mech- anisms of biological pattern formation, J. Theor. Biol. 414 (2017) 1 20–

    2017

  4. [12]

    T. J. Jewell, A. L. Krause, P. K. Maini, E. A. Gaffney, Patterning of nonlocal transport models in biology: The impact of sp atial dimension, Mathematical Biosciences 366 (2023) 109093. doi:10.1016/j.mbs.2023.109093. URL https://www.sciencedirect.com/science/article/pii/S002555...

    2023

  5. [13]

    Y. Ide, H. Izuhara, T. Machida, Turing instability in reaction-diff usion models on networks, Physica A: Statistical Mechanics and its Applica - tions 457 (2016) 331–347. doi:10.1016/j.physa.2016.03.055

    2016 doi

  6. [14]

    Wolfram, The Turing bifurcation in network systems: Collectiv e pat- terns and single differentiated nodes, Physica D: Nonlinear Phenome na 241 (16) (2012) 1351–1357

    M. Wolfram, The Turing bifurcation in network systems: Collectiv e pat- terns and single differentiated nodes, Physica D: Nonlinear Phenome na 241 (16) (2012) 1351–1357. doi:10.1016/j.physd.2012.05.002. 49

    2012 doi

  7. [15]

    Nakao, A

    H. Nakao, A. S. Mikhailov, Turing patterns in network-organize d activator-inhibitor systems, Nature Phys 6 (2012) 544–550. doi:10.1038/nphys1651

    2012 doi

  8. [16]

    McCullen, T

    N. McCullen, T. Wagenknecht, Pattern formation on networks : from localised activity to Turing patterns, Sci. Rep. 6 (2016) 27397. doi:10.1038/srep27397

    2016 doi

  9. [17]

    A. P. Singh, U. Schach, C. N¨ usslein-Volhard, Proliferation, dis per- sal and patterned aggregation of iridophores in the skin prefigure striped colouration of zebrafish, Nat. Cell Biol. 16 (2014) 604–611 . doi:10.1038/ncb2955

    2014 doi

  10. [18]

    Mahalwar, B

    P. Mahalwar, B. Walderich, A. P. Singh, C. N¨ usslein-Volhard, Local reorganization of xanthophores fine-tunes and colors the striped pattern of zebrafish, Science 345 (6202) (2014) 1362–1 364. doi:10.1126/science.1254837

    2014 doi

  11. [19]

    D. M. Parichy, J. M. Turner, Zebrafish puma mutant decouples pigment pattern and somatic metamorphosis, Dev. Biol. 256 (2) (2003) 242 –257. doi:10.1016/S0012-1606(03)00015-0

    2003 doi

  12. [20]

    H. G. Frohnh¨ ofer, J. Krauss, H.-M. Maischein, C. N¨ usslein-Volhard, Iri- dophores and their interactions with other chromatophores are r equired for stripe formation in zebrafish, Development 140 (14) (2013) 29 97–

    2013

  13. [21]

    Takahashi, S

    G. Takahashi, S. Kondo, Melanophores in the stripes of adult ze brafish do not have the nature to gather, but disperse when they have th e space to move, Pigment Cell Melanoma Res. 21 (6) (2008) 677–686. doi:10.1111/j.1755-148X.2008.00504.x

    2008 arXiv

  14. [22]

    Nakamasu, G

    A. Nakamasu, G. Takahashi, A. Kanbe, S. Kondo, Interaction s be- tween zebrafish pigment cells responsible for the generation of Tur ing patterns, Proc. Natl. Acad. Sci. U. S. A. 106 (21) (2009) 8429–8 434. doi:10.1073/pnas.0808622106

    2009 doi

  15. [23]

    Hamada, M

    H. Hamada, M. Watanabe, H. E. Lau, T. Nishida, T. Hasegawa, D . M. Parichy, S. Kondo, Involvement of Delta/Notch signaling in zebrafis h adult pigment stripe patterning, Development 141 (2) (2014) 318– 324. doi:10.1242/dev099804. 50

    2014 doi

  16. [24]

    Hirata, K.-I

    M. Hirata, K.-I. Nakamura, T. Kanemaru, Y. Shibata, S. Kondo , Pig- ment cell organization in the hypodermis of zebrafish, Dev. Dyn. 22 7 (4) (2003) 497–503. doi:10.1002/dvdy.10334

    2003 doi

  17. [25]

    Bullara, Y

    D. Bullara, Y. De Decker, Pigment cell movement is not required f or generation of Turing patterns in zebrafish skin, Nature Communica tions 6 (1) (2015) 6971. doi:10.1038/ncomms7971

    2015 doi

  18. [26]

    Konow, Z

    C. Konow, Z. Li, S. Shepherd, D. Bullara, I. R. Epstein, Influen ce of survival, promotion, and growth on pattern formation in zebrafish skin, Sci. Rep. 11 (1) (2021) 9864. doi:10.1038/s41598-021-89116-4

    2021 doi

  19. [27]

    Volkening, B

    A. Volkening, B. Sandstede, Modelling stripe formation in zebrafi sh: an agent-based approach, J. R. Soc. Interface 12 (112) (2015) 2 0150812. doi:10.1098/rsif.2015.0812

    2015

  20. [28]

    Volkening, B

    A. Volkening, B. Sandstede, Iridophores as a source of robus tness in zebrafish stripes and variability in Danio patterns, Nat. Commun. 9 (2018) 3231. doi:10.1038/s41467-018-05629-z

    2018 doi

  21. [29]

    Kondo, M

    S. Kondo, M. Watanabe, S. Miyazawa, Studies of Turing patter n forma- tion in zebrafish skin, Phil. Trans. R. Soc. A 379 (2213) (2021) 2020 0274. doi:10.1098/rsta.2020.0274

    2021

  22. [30]

    D. M. Parichy, M. R. Elizondo, M. G. Mills, T. N. Gordon, R. E. En- geszer, Normal table of postembryonic zebrafish development: S taging by externally visible anatomy of the living fish, Dev. Dyn. 238 (12) (2009) 2975–3015. doi:10.1002/dvdy.22113. 51

    2009 doi

  23. [127]

    doi:10.1016/j.jtbi.2016.11.003

    2016 doi

  24. [1620]

    doi:10.1126/science.1179047

    doi

  25. [3007]

    doi:10.1242/dev096719

    doi

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