Modeling Stripe Formation on Growing Zebrafish Tailfins

Abstract

As zebrafish develop, black and gold stripes form across their skin due to the interactions of brightly colored pigment cells. These characteristic patterns emerge on the growing fish body, as well as on the anal and caudal fins. While wild-type stripes form parallel to a horizontal marker on the body, patterns on the tailfin gradually extend distally outward. Interestingly, several mutations lead to altered body patterns without affecting fin stripes. Through an exploratory modeling approach, our goal is to help better understand these differences between body and fin patterns. By adapting a prior agent-based model of cell interactions on the fish body, we present an in silico study of stripe development on tailfins. Our main result is a demonstration that two cell types can produce stripes on the caudal fin. We highlight several ways that bone rays, growth, and the body–fin interface may be involved in patterning, and we raise questions for future work related to pattern robustness.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. Bloomfield JM, Painter KJ, Sherratt JA (2011) How does cellular contact affect differentiation mediated pattern formation? Bull Math Biol 73(7):1529–1558

    MathSciNet  MATH  Article  Google Scholar 

  2. Bullara D, De Decker Y (2015) Pigment cell movement is not required for generation of Turing patterns in zebrafish skin. Nat Commun 6:6971

    Article  Google Scholar 

  3. Caicedo-Carvajal CE, Shinbrot T (2008) In silico zebrafish pattern formation. Dev Biol 315(2):397–403

    Article  Google Scholar 

  4. Eom DS, Inoue S, Patterson LB, Gordon TN, Slingwine R, Kondo S, Watanabe M, Parichy DM (2012) Melanophore migration and survival during zebrafish adult pigment stripe development require the immunoglobulin superfamily adhesion molecule Igsf11. PLoS Genet 8(8):e1002899

    Article  Google Scholar 

  5. Eom DS, Bain EJ, Patterson LB, Grout ME, Parichy DM (2015) Long-distance communication by specialized cellular projections during pigment pattern development and evolution. eLife 4:e12401

    Article  Google Scholar 

  6. Fadeev A, Krauss J, Frohnhöfer HG, Irion U, Nüsslein-Volhard C (2015) Tight junction protein 1a regulates pigment cell organisation during zebrafish colour patterning. eLife 4:e06545

    Article  Google Scholar 

  7. Frohnhöfer HG, Krauss J, Maischein H-M, Nüsslein-Volhard C (2013) Iridophores and their interactions with other chromatophores are required for stripe formation in zebrafish. Development 140(14):2997–3007

    Article  Google Scholar 

  8. Gaffney EA, Lee SS (2015) The sensitivity of Turing self-organization to biological feedback delays: 2d models of fish pigmentation. Math Med Biol 32(1):57–79

    MathSciNet  MATH  Article  Google Scholar 

  9. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39

    MATH  Article  Google Scholar 

  10. Goldsmith MI, Fisher S, Waterman R, Johnson SL (2003) Saltatory control of isometric growth in the zebrafish caudal fin is disrupted in long fin and rapunzel mutants. Dev Biol 259(2):303–317

    Article  Google Scholar 

  11. Goldsmith MI, Iovine MK, O’Reilly-Pol T, Johnson SL (2006) A developmental transition in growth control during zebrafish caudal fin development. Dev Biol 296(2):450–457

    Article  Google Scholar 

  12. Goodrich HB, Nichols R (1931) The development and the regeneration of the color pattern in brachydanio rerio. J Morphol 52(2):513–523

    Article  Google Scholar 

  13. Goodrich HB, Marzullo CM, Bronson WR (1954) An analysis of the formation of color patterns in two fresh-water fish. J Exp Zool 125(3):487–505

    Article  Google Scholar 

  14. Hamada H, Watanabe M, Lau HE, Nishida T, Hasegawa T, Parichy DM, Kondo S (2014) Involvement of Delta/Notch signaling in zebrafish adult pigment stripe patterning. Development 141(2):318–324

    Article  Google Scholar 

  15. Hirata M, Nakamura K-I, Kondo S (2005) Pigment cell distributions in different tissues of the zebrafish, with special reference to the striped pigment pattern. Dev Dyn 234(2):293–300

    Article  Google Scholar 

  16. Inaba M, Yamanaka H, Kondo S (2012) Pigment pattern formation by contact-dependent depolarization. Science 335(6069):677–677

    Article  Google Scholar 

  17. Inoue S, Kondo S, Parichy DM, Watanabe M (2014) Tetraspanin 3c requirement for pigment cell interactions and boundary formation in zebrafish adult pigment stripes. Pigment Cell Melanoma Res 27(2):190–200

    Article  Google Scholar 

  18. Iovine MK (2007) Conserved mechanisms regulate outgrowth in zebrafish fins. Nat Chem Biol 3(10):613–618

    Article  Google Scholar 

  19. Iovine MK, Johnson SL (2000) Genetic analysis of isometric growth control mechanisms in the zebrafish caudal fin. Genetics 155(3):1321–1329

    Google Scholar 

  20. Iwashita M, Watanabe M, Ishii M, Chen T, Johnson SL, Kurachi Y, Okada N, Kondo S (2006) Pigment pattern in jaguar \(/\)obelix zebrafish is caused by a kir7.1 mutation: implications for the regulation of melanosome movement. PLoS Genet 2(11):e197

    Article  Google Scholar 

  21. Kizil C, Otto GW, Geisler R, Nüsslein-Volhard C, Antos CL (2009) Simplet controls cell proliferation and gene transcription during zebrafish caudal fin regeneration. Dev Biol 325(2):329–340

    Article  Google Scholar 

  22. Kondo S, Watanabe M (2015) Black, yellow, or silver: which one leads skin pattern formation? Pigment Cell Melanoma Res 28(1):2–4

    Article  Google Scholar 

  23. Lopes SS, Yang X, Müller J, Carney TJ, McAdow AR, Rauch G, Jacoby AS, Hurst LD, Delfino-Machín M, Haffter P, Geisler R, Johnson SL, Ward A, Kelsh RN (2008) Leukocyte tyrosine kinase functions in pigment cell development. PLoS Genet 4(3):1–13

    Article  Google Scholar 

  24. Mahalwar P, Walderich B, Singh AP, Nüsslein-Volhard C (2014) Local reorganization of xanthophores fine-tunes and colors the striped pattern of zebrafish. Science 345(6202):1362–1364

    Article  Google Scholar 

  25. Mahalwar P, Singh AP, Fadeev A, Nüsslein-Volhard C, Irion U (2016) Heterotypic interactions regulate cell shape and density during color pattern formation in zebrafish. Biol Open 5(11):1680–1690

    Article  Google Scholar 

  26. McMenamin SK, Chandless MN, Parichy DM (2016) Working with zebrafish at postembryonic stages. Methods Cell Biol 134:587–607

    Article  Google Scholar 

  27. Mellgren EM, Johnson SL (2006) Pyewacket, a new zebrafish fin pigment pattern mutant. Pigment Cell Res 19(3):232–238

    Article  Google Scholar 

  28. Mills MG, Nuckels RJ, Parichy DM (2007) Deconstructing evolution of adult phenotypes: genetic analyses of kit reveal homology and evolutionary novelty during adult pigment pattern development of Danio fishes. Development 134(6):1081–1090

    Article  Google Scholar 

  29. Moreira J, Deutsch A (2005) Pigment pattern formation in zebrafish during late larval stages: a model based on local interactions. Dev Dyn 232(1):33–42

    Article  Google Scholar 

  30. Nakamasu A, Takahashi G, Kanbe A, Kondo S (2009) Interactions between zebrafish pigment cells responsible for the generation of Turing patterns. Proc Natl Acad Sci USA 106(21):8429–8434

    Article  Google Scholar 

  31. Nüsslein-Volhard C, Singh AP (2017) How fish color their skin: a paradigm for development and evolution of adult patterns. BioEssays 39(3):1600231

    Article  Google Scholar 

  32. Painter KJ, Bloomfield JM, Sherratt JA, Gerisch A (2015) A nonlocal model for contact attraction and repulsion in heterogeneous cell populations. Bull Math Biol 77(6):1132–1165

    MathSciNet  MATH  Article  Google Scholar 

  33. Parichy DM, Spiewak JE (2015) Origins of adult pigmentation: diversity in pigment stem cell lineages and implications for pattern evolution. Pigment Cell Melanoma Res 28(1):31–50

    Article  Google Scholar 

  34. Parichy DM, Turner JM (2003) Temporal and cellular requirements for Fms signaling during zebrafish adult pigment pattern development. Development 130(5):817–833

    Article  Google Scholar 

  35. Parichy DM, Elizondo MR, Mills MG, Gordon TN, Engeszer RE (2009) Normal table of postembryonic zebrafish development: staging by externally visible anatomy of the living fish. Dev Dyn 238(12):2975–3015

    Article  Google Scholar 

  36. Patterson LB, Parichy DM (2013) Interactions with iridophores and the tissue environment required for patterning melanophores and xanthophores during zebrafish adult pigment stripe formation. PLoS Genet 9(5):e1003561

    Article  Google Scholar 

  37. Patterson LB, Bain EJ, Parichy DM (2014) Pigment cell interactions and differential xanthophore recruitment underlying zebrafish stripe reiteration and Danio pattern evolution. Nat Commun 5:5299

    Article  Google Scholar 

  38. Pfefferli C, Jaźwińska A (2015) The art of fin regeneration in zebrafish. Regeneration 2(2):72–83

    Article  Google Scholar 

  39. Quigley IK, Manuel JL, Roberts RA, Nuckels RJ, Herrington ER, MacDonald EL, Parichy DM (2005) Evolutionary diversification of pigment pattern in Danio fishes: differential fms dependence and stripe loss in D. albolineatus. Development 132(1):89–104

    Article  Google Scholar 

  40. Rawls JF, Johnson SL (2000) Zebrafish kit mutation reveals primary and secondary regulation of melanocyte development during fin stripe regeneration. Development 127(17):3715–3724

    Google Scholar 

  41. Rolland-Lagan A-G, Paquette M, Tweedle V, Akimenko M-A (2012) Morphogen-based simulation model of ray growth and joint patterning during fin development and regeneration. Development 139(6):1188–1197

    Article  Google Scholar 

  42. Singh AP, Nüsslein-Volhard C (2015) Zebrafish stripes as a model for vertebrate colour pattern formation. Curr Biol 25(2):R81–R92

    Article  Google Scholar 

  43. Singh AP, Schach U, Nüsslein-Volhard C (2014) Proliferation, dispersal and patterned aggregation of iridophores in the skin prefigure striped colouration of zebrafish. Nat Cell Biol 16(6):604–611

    Article  Google Scholar 

  44. Singh AP, Frohnhöfer H-G, Irion U, Nüsslein-Volhard C (2015) Response to comment on “Local reorganization of xanthophores fine-tunes and colors the striped pattern of zebrafish”. Science 348(6232):297–297

    Article  Google Scholar 

  45. Takahashi G, Kondo S (2008) Melanophores in the stripes of adult zebrafish do not have the nature to gather, but disperse when they have the space to move. Pigment Cell Melanoma Res 21(6):677–686

    Article  Google Scholar 

  46. Tu S, Johnson SL (2010) Clonal analyses reveal roles of organ founding stem cells, melanocyte stem cells and melanoblasts in establishment, growth and regeneration of the adult zebrafish fin. Development 137(23):3931–3939

    Article  Google Scholar 

  47. Tu S, Johnson SL (2011) Fate restriction in the growing and regenerating zebrafish fin. Dev Cell 20(5):725–732

    Article  Google Scholar 

  48. Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B Biol Sci 237(641):37–72

    MathSciNet  MATH  Article  Google Scholar 

  49. Volkening A, Sandstede B (2015) Modelling stripe formation in zebrafish: an agent-based approach. J R Soc Interface 12(112):20150812

    Article  Google Scholar 

  50. Volkening A, Sandstede B (2018) Iridophores as a source of robustness in zebrafish stripes and variability in Danio patterns. Nat Commun 9:3231

    Article  Google Scholar 

  51. Walderich B, Singh AP, Mahalwar P, Nüsslein-Volhard C (2016) Homotypic cell competition regulates proliferation and tiling of zebrafish pigment cells during colour pattern formation. Nat Commun 7:11462

    Article  Google Scholar 

  52. Watanabe M, Kondo S (2015a) Comment on “Local reorganization of xanthophores fine-tunes and colors the striped pattern of zebrafish”. Science 348(6232):297–297

    Article  Google Scholar 

  53. Watanabe M, Kondo S (2015b) Is pigment patterning in fish skin determined by the Turing mechanism? Trends Genet 31(2):88–96

    Article  Google Scholar 

  54. Watanabe M, Sawada R, Aramaki T, Skerrett IM, Kondo S (2016) The physiological characterization of connexin41.8 and connexin39.4, which are involved in the striped pattern formation of zebrafish. J Biol Chem 291(3):1053–1063

    Article  Google Scholar 

  55. Woolley TE, Maini PK, Gaffney EA (2014) Is pigment cell pattern formation in zebrafish a game of cops and robbers? Pigment Cell Melanoma Res 27(5):686–687

    Article  Google Scholar 

  56. Yamaguchi M, Yoshimoto E, Kondo S (2007) Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism. Proc Natl Acad Sci USA 104(12):4790–4793

    Article  Google Scholar 

  57. Yamanaka H, Kondo S (2014) In vitro analysis suggests that difference in cell movement during direct interaction can generate various pigment patterns in vivo. Proc Natl Acad Sci USA 111(5):1867–1872

    Article  Google Scholar 

Download references

Acknowledgements

We thank the Nüsslein-Volhard lab for feedback during early model development and are particularly grateful to April Dinwiddie for sharing her expertise on zebrafish fin patterns. We also recognize Emily Briggs, who contributed to earlier discussions on fin growth during an independent study with B.S. and A.V. We thank ICERM for hosting the undergraduate research component of this project.

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Volkening.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The work of A.V. has been supported in part by the National Science Foundation (NSF) through DMS-1148284, DMS-1764421, and DMS-1440386; by the Mathematical Biosciences Institute; and by the Simons Foundation/SFARI under 597491-RWC. M.R.A., N.C., B.D., F.L., and D.S. were supported by Brown University and ICERM through DMS-1439786. The work of B.S. was partially supported by the NSF through DMS-1408742 and DMS-1714429.

Appendices

A Appendix: Parameters

With the exception of the two switch parameters [namely \(\zeta \) in Eq. (12) and d in Eqs. (13)–(14)] that we use to implement Mechanism II and IV, we summarize all of the parameters involved in our rules for cell migration, birth, and death in Tables 2, 3, and 4, respectively. We give the simulation-specific values of the switch parameters by figure in “Appendix B.5.” We set \(d = 150\,\upmu \hbox {m}\) when we include Mechanism II, and we set \(\zeta = 1000\) cells when we test Mechanism IV.

Table 2 Summary of parameters for cell migration in Eqs. (9)–(12) and (18)
Table 3 Summary of our parameters for cell birth [see Eqs. (13)–(14)] and model length scales
Table 4 Summary of our parameters for cell death in Eqs. (15)–(16)

B Appendix: Simulation Conditions

We used MATLAB 9.3, The MathWorks, Inc., Natick, MA, USA, to simulate our model of cell interactions on growing fins. Our code is available from the corresponding author on request. We now describe our model implementation (“Appendix B.1”), our boundary conditions (“Appendix B.2”), our initial conditions (“Appendix B.3”), and our methods for selecting cell-birth locations in tailfin domains and implementing Mechanism V (“Appendix B.4”). In “Appendix B.5,” we summarize the parameters and simulation conditions associated with our simulations in Figs. 5, 6, and 7.

B.1 Model Implementation

Our simulation begins with an initial condition (see “Appendix B.3”) at \(t=18\) dpf. We update cell positions from day t to day \(t+1\) in seven steps:

  1. 1.

    Set the cycle counter c for migration and birth to \(c=1\).

  2. 2.

    Increment \(N^\text {M}_\text {diff}(t)\) and \(N^\text {X}_\text {diff}(t)\) by 20 locations each, so that \(N^\text {M}_\text {diff}(t) = n^\text {M}_\text {diff} + 20(t-18)\) locations and \(N^\text {X}_\text {diff}(t) = n^\text {X}_\text {diff} + 20(t-18)\) locations.

  3. 3.

    Perform one step \(\varDelta t_\text {mig,birth}\) of migration [e.g., solve Eqs. (9) and (11) using the forward Euler scheme with time step \(\varDelta t_\text {mig,birth}\)]. All of our cells migrate simultaneously. We also specify repulsive forces from the discretized fin boundaries at each step \(\varDelta t_\text {mig,birth}\) of migration (see “Appendix B.2”).

  4. 4.

    Select \(N^\text {M}_\text {diff}(t+1)\) and \(N^\text {X}_\text {diff}(t+1)\) potential locations for M and X birth on the fin domain (outlined by our boundary curve at time \(t+1\)), respectively. Evaluate these locations (simultaneously) for cell birth based on Eqs. (13)–(14) and random birth if included (e.g., if \(p_\text {M} >0\) and \(p_\text {X} >0\)). Add the newly born cells to the domain at time \(t+c\varDelta t_\text {mig,birth}\).

  5. 5.

    If \(c \varDelta t_\text {mig,birth} = 1\) day, one day of migration and birth has been completed: go to Step 6. Otherwise, increment the cycle counter c for migration and birth (\(c= c+1\)) and return to Step 3 with the cell positions at time \(t+c\varDelta t_\text {mig,birth}\).

  6. 6.

    Evaluate all of the cells for possible death (simultaneously) through Eqs. (15)–(16). Remove any cells that have died from the domain at day \(t+1\). We note that the time step for our cell death rules is always \(\varDelta t_\text {death} =1\) day.

  7. 7.

    If uniform domain growth is included, scale the cell positions at time \(t+1\) using the fin domains and bone rays at times \(t+1\) and \(t+2\) days as we describe in Sect. 3.1. If distal epithelial growth is included, do not scale the cell positions. The result of this process is the updated cell positions at time \(t+1\) days.

B.2 Boundary Conditions

To help keep cells in the fin domain, we include wall-like boundary conditions. For each simulated day t, we discretize the associated fin boundary curve at time t into 500 points: \(\{\mathbf{F}_i(t)\}_{i=1,\ldots ,500}\). We then specify repulsive forces from these boundary points to our cell agents at each time step of migration \(\varDelta t_\text {mig,birth}\):

$$\begin{aligned} \text {boundary force on } i\text {th cell at position } \mathbf{C} _i = -\sum _{j=1}^{500} \triangledown Q^\text {bnd}(\mathbf{F} _j - \mathbf{C} _i) \end{aligned}$$
(17)

where \(\mathbf{C} \in \{\mathbf{M}, \mathbf{X}\}\) and

$$\begin{aligned} Q^\text {bnd}(\mathbf{d} ) = R^\text {bnd} e^{-|\mathbf{d} |/r_\text {bnd}}. \end{aligned}$$
(18)

These rules are an approximation of Neumann boundary conditions, and we note that a small number of cells escape from our domain in some simulations. Cells are more likely to escape with increasing time, suggesting that it may be useful for future work to increase the number of boundary agents as the fin grows. When cells escape, we remove them from our final simulated images in post-processing.

B.3 Initial Condition

The initial condition for our simulations is motivated by images in Parichy et al. (2009). First, we specify a single horizontal strip of M cells (separated \(30\,\upmu \hbox {m}\) apart) at the center of our fin domain (e.g., with y-coordinate 0). Second, for X cells, we consider a random distribution of cells that are concentrated more highly toward the proximal edge of the fin. We choose the y-coordinates of these positions by selecting 500 points uniformly at random between the maximum and minimum y-coordinates for the discretized boundary curve that represents our initial domain at 18 dpf (see “Appendix B.2”). We choose the x-coordinates for these points by taking the absolute value of 500 points sampled from a normal distribution with mean 0 and standard deviation \(\sigma = 0.25 x_\text {max}(t)\), where \(x_\text {max}(t)\) is the maximum x-coordinate for our discretized boundary curve at 18 dpf.

As the penultimate step in setting our initial condition, we remove any M or X cells that fall within \(25\,\upmu \hbox {m}\) of our discretized boundary curve in Eq. (18). Finally, if more than 300 of our selected X locations fall in the domain, we use only the first 300 such locations in our initial condition.

For the special case of Fig. 5c, after specifying our initial condition as above, we scale the cell positions to account for one day of uniform epithelial growth (for details on how we implement domain growth, see Sect. 3.1). We use these scaled cell positions as our initial condition for the simulations in Fig. 5c.

B.4 Selecting Cell Birth Locations

We consider two methods for selecting \(N^\text {M}_\text {diff}(t)\) possible locations for M birth:

  • Control case (similar to body-model birth) We choose \(2\times N^\text {M}_\text {diff}(t)\) locations uniformly at random in a rectangular region surrounding the fin, and we evaluate the first \(N^\text {M}_\text {diff}(t)\) of these positions that are inside the fin domain for possible birth simultaneously.

  • Mechanism V Under Mechanism V, we first choose \(N^\text {M}_\text {diff}(t)\) points uniformly at random from our discretized bone rays, namely \(\{\mathbf{B}_i^j\}\) in Eq. (1). For each such point \(\mathbf{B} _k\), we then choose a corresponding location to evaluate for M birth by selecting a point uniformly at random in a ball of radius \(r \sim \mathcal {N}(0, 2)\,\upmu \hbox {m}\) around \(\mathbf{B} _k\).

We always select potential X birth locations in the same way as in the M control case.

Lastly, prior to applying our rules for cell birth to the locations that we randomly selected as we outlined above, we require that these positions are strictly greater than \(25\,\upmu \hbox {m}\) away from our discretized boundary curve [see Eq. (17) in “Appendix B.2”]. If a randomly selected location is within \(25\,\upmu \hbox {m}\) of a discretized boundary point in \(\{\mathbf{F}_i(t)\}_{i=1,\ldots ,500}\), we do not allow cell birth to occur at that location.

B.5 Instructions for Reproducing Our Figures

We summarize the parameters for our simulated patterns in Figs. 5, 6, and 7 below:

  • Figure 5a, Mechanism I (distal epithelial growth)

    • Cell migration: We use the body-model values in Table 2.

    • Cell birth: We use the body-model values in Table 3 with \(n^\text {M}_\text {diff}= n^\text {X}_\text {diff} = 600\) locations.

    • Cell death: We use the body-model values in Table 4.

    • Skin growth: We do not scale cell positions with domain growth.

    • Switch parameters: In Eq. (12), we use \(\zeta = -1\) cells. In Eqs. (13)–(14), we use \(d = -1\,\upmu \hbox {m}\).

  • Figure 5b, Mechanisms I and III (distal epithelial growth with alignment cues from the body)

    • Cell migration: We use the body-model values in Table 2 with one exception: \(\varDelta t_\text {mig,birth} = 0.25\) days (more frequent cell birth and migration than we use in Fig. 5a).

    • Cell birth: We use the body-model values in Table 3 with three exceptions: \(\varDelta t_\text {mig,birth} = 0.25\) days, \(p_\text {M} = 0\), and \(p_\text {X} = 0\) (no random birth). We use \(n^\text {M}_\text {diff}= n^\text {X}_\text {diff} = 600\) locations.

    • Cell death: We use the body-model values in Table 4.

    • Skin growth: We do not scale cell positions with domain growth.

    • Switch parameters: In Eq. (12), we use \(\zeta = -1\) cells. In Eqs. (13)–(14), we use \(d = 150\,\upmu \hbox {m}\) (special body–fin interface dynamics; see Fig. 4g).

  • Figure 5c, Mechanisms II and III (uniform epithelial growth with alignment cues from the body)

    • Cell migration: We use the body-model values in Table 2.

    • Cell birth: We use the body-model values in Table 3 with \(n^\text {M}_\text {diff}= 1200\) and \(n^\text {X}_\text {diff} =600\) locations.

    • Cell death: We use the body-model values in Table 4.

    • Skin growth: We scale cell positions along the bone rays with fin growth as we describe in Sect. 3.1.

    • Switch parameters: In Eq. (12), we use \(\zeta = -1\) cells. In Eqs. (13)–(14), we use \(d = 150\,\upmu \hbox {m}\).

  • Figure 6, Mechanisms I, III, and IV (M migration along bone rays with alignment cues from the body and distal epithelial growth)

    • Cell migration: We use the fin-specific values in Table 2.

    • Cell birth: We use the fin-specific values in Table 3.

    • Cell death: We use the fin-specific values in Table 4.

    • Skin growth: We do not scale cell positions with domain growth.

    • Switch parameters: In Eq. (12), we use \(\zeta =1000\) cells (so that M movement is always projected along the bones). In Eqs. (13)–(14), we use \(d = 150\,\upmu \hbox {m}\).

    • Additional note in Fig. 6b: When we calculate the nearest-neighbor distances between cells, we only consider MM and XX distances that are less than \(200\,\upmu \hbox {m}\) (this ensures that cells that have escaped our fin domain or appear at low density do not affect our measurements of cell–cell distances in developing stripes). When we calculate MX distances at stripe–interstripe boundaries, we only consider measurements that are less than \(110\,\upmu \hbox {m}\). We made this choice so that our MX distances measure stripe–interstripe separation (in comparison, Takahashi and Kondo (2008) showed that M and X cells are roughly \(82\,\upmu \hbox {m}\) apart at stripe–interstripe boundaries).

  • Figure 7, Mechanisms I, III, and V (M birth in association with bone rays, with alignment cues from the body and distal epithelial growth)

    • Cell migration: We use the fin-specific values in Table 2.

    • Cell birth: We use the fin-specific values in Table 3. Additionally, we select locations for cell birth using the coordinates of the discretized bone rays in Eq. (1) (see “Appendix B.4” for details).

    • Cell death: We use the fin-specific values in Table 4.

    • Skin growth: We do not scale cell positions with domain growth.

    • Switch parameters: In Eq. (12), we use \(\zeta = -1\) cells. In Eqs. (13)–(14), we use \(d = 150\,\upmu \hbox {m}\).

    • Additional note in Fig. 7b: To calculate the distances that cells move per day, we consider the differences in their locations between consecutive days. In particular, the distance the ith M cell moves in one day is \(||\mathbf{M} _i(t) - \mathbf{M} _i (t + \varDelta t_\text {mig,birth})|| + ||\mathbf{M} _i(t+\varDelta t_\text {mig,birth}) - \mathbf{M} _i (t + 2\varDelta t_\text {mig,birth})|| +||\mathbf{M} _i(t +2\varDelta t_\text {mig,birth}) - \mathbf{M} _i (t + 1)||\), since \(\varDelta t_\text {mig,birth} = 1/3\) days in this simulation. If a new cell is born at position \(\mathbf{M} _j\) at, for example, time \(t+\varDelta t_\text {mig,birth}\), then we define the distance that cell agent moved between day t and day \(t+1\) as just \(||\mathbf{M} _j(t+\varDelta t_\text {mig,birth}) - \mathbf{M} _j (t + 2\varDelta t_\text {mig,birth})|| +||\mathbf{M} _j(t +2\varDelta t_\text {mig,birth}) - \mathbf{M} _j (t + 1)||\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Volkening, A., Abbott, M.R., Chandra, N. et al. Modeling Stripe Formation on Growing Zebrafish Tailfins. Bull Math Biol 82, 56 (2020). https://doi.org/10.1007/s11538-020-00731-0

Download citation

Keywords

  • Zebrafish
  • Pattern formation
  • Agent-based model
  • Tailfin
  • Self-organization
  • Growing domain

Mathematics Subject Classification

  • 92