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November  2012, 17(8): 2789-2814. doi: 10.3934/dcdsb.2012.17.2789

A Lattice model on somitogenesis of zebrafish

Kang-Ling Liao 1, and  Chih-Wen Shih 1,

1. 

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan, Taiwan

Received  April 2011 Revised  April 2012 Published  July 2012

Somitogenesis is the process of the development of somites which are segmental structure in vertebrate embryos. This process depends on the expression of segmentation clock genes. In this investigation, we consider lattice systems which describe the kinetics of the chief segmentation clock genes in zebrafish under negative feedback regulation with delay through interaction with the Delta-Notch signaling among neighboring cells. We first derive the analytical theories for the oscillation-arrested and synchronous oscillation in an autonomous lattice model. Based on the parameter regimes in the theories, we design suitable gradients of degradation rates and delays in a non-autonomous lattice model. Such a lattice system can generate synchronous oscillations, oscillatory traveling waves, oscillation slowing down, oscillation-arrested, and high-low expression levels. We further distinguish between different gradient structures which lead to normal and abnormal segmentations respectively and connect these structures to the dynamical regimes in the cell-cell model.
Keywords: traveling wave., oscillation-arrested, Somitogenesis, synchronous oscillation, lattice model.
Mathematics Subject Classification: Primary: 92B25, 37N25, 34K18, 34K1.
Citation: Kang-Ling Liao, Chih-Wen Shih. A Lattice model on somitogenesis of zebrafish. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2789-2814. doi: 10.3934/dcdsb.2012.17.2789
References:
[1]

N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030.  Google Scholar

[2]

N. J. Armstrong, K. J. Painter and J. A. Sherratt, Adding adhesion to a chemical signaling model for somite formation, Bulletin Math. Biol., 71 (2009), 1-24.  Google Scholar

[3]

R. E. Baker and S. Schnell, How can mathematics help us explore vertebrate segmentation?, HFSP J., 3 (2009), 1-5. Google Scholar

[4]

R. E. Baker, S. Schnell and P. K. Maini, A mathematical investigation of a clock and wavefornt model for somitogenesis, J. Math. Biol., 52 (2006), 458-482. doi: 10.1007/s00285-005-0362-2.  Google Scholar

[5]

M. Campanelli and T. Gedeon, Somitogenesis clock-wave initiation requires differential decay and multiple binding sites for clock protein, PLoS Comput. Biol., 6 (2010), e1000728.  Google Scholar

[6]

O. Cinquin, Repressor dimerization in the zebrafish somitogenesis clock, PLoS Comput. Biol., 3 (2007), 293-303.  Google Scholar

[7]

J. Cooke and E. C. Zeeman, A clock and wavefront model for control of the number of repeated structures during animal morphogenesis, J. Theoret. Biol., 58 (1976), 455-476. doi: 10.1016/S0022-5193(76)80131-2.  Google Scholar

[8]

J. Dubrulle, M. J. McGrew and O. Pourquié, FGF signaling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation, Cell, 106 (2001), 219-232. doi: 10.1016/S0092-8674(01)00437-8.  Google Scholar

[9]

J. Dubrulle and O. Pourquié, From head to tail: links between the segmentation clock and antero-posterior patterning of the embryo, Curr. Opin. Genet. Dev., 12 (2002), 519-523. doi: 10.1016/S0959-437X(02)00335-0.  Google Scholar

[10]

J. Dubrulle and O. Pourquié, fgf8 mRNA decay establishes a gradient that couples axial elongation to patterning in the vertebrate embryo, Nature, 427 (2004), 419-422. doi: 10.1038/nature02216.  Google Scholar

[11]

F. Giudicelli, E. M. Özbudak, G. J. Wright and J. Lewis, Setting the tempo in development: an investigation of the zebrafish somite clock mechanism, PLOS Biol., 5 (2007), e150, 1309-1323. Google Scholar

[12]

A. Goldbeter and O. Pourquié, Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways, J. Theoret. Biol., 252 (2008), 574-585. doi: 10.1016/j.jtbi.2008.01.006.  Google Scholar

[13]

E. Hanneman and M. Westerfield, Early expression of acetyl-choline-sterase activity in functionally distinct neurons of the zebrafish, J. Comp. Neurol., 284 (1989), 350-361. doi: 10.1002/cne.902840303.  Google Scholar

[14]

S. A. Holley, The genetics and embryology of zebrafish metamerism, Dev. Dyn., 236 (2007), 1422-1449. doi: 10.1002/dvdy.21162.  Google Scholar

[15]

K. Horikawa, K. Ishimatsu, E. Yoshimoto, S. Kondo and H. Takeda, Noise-resistant and synchronized oscillation of the segmentation clock, Nature, 441 (2006), 719-723. doi: 10.1038/nature04861.  Google Scholar

[16]

Y.-J. Jiang, B. L. Aerne, L. Smithers, C. Haddon, D. Ish-Horowicz and J. Lewis, Notch signaling and the synchronization of the somite segmentation clock, Nature, 408 (2000), 475-479. doi: 10.1038/35044091.  Google Scholar

[17]

A. Kawamura, S. Koshida, H. Hijikata, T. Sakaguchi, H. Kondoh and S. Takada, Zebrafish hairy/enhancer of split protein links FGF signaling to cyclic gene expression in the periodic segmentation of somites, Genes Dev., 19 (2005), 1156-1161. doi: 10.1101/gad.1291205.  Google Scholar

[18]

J. Lewis, Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator, Curr Biol., 13 (2003), 1398-1408. doi: 10.1016/S0960-9822(03)00534-7.  Google Scholar

[19]

K.-L. Liao, C.-W. Shih and J.-P. Tseng, Synchronized oscillations in a mathematical model of segmentation in zebrafish, Nonlinearity, 25 (2012), 869-904. doi: 10.1088/0951-7715/25/4/869.  Google Scholar

[20]

K.-L. Liao, "Analysis on Mathematical Models of Somitogenesis in Zebrafish," Ph.D thesis, National Chiao Tung University, Hsinchu, 2012. Google Scholar

[21]

A. Mara, J. Schroeder, C. Chalouni and S. A. Holley, Priming, initiation and synchronization of the segmentation clock by deltaD and deltaC, Nat. Cell. Biol., 9 (2007), 523-530. Google Scholar

[22]

L. G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher and A. C. Oates, Delayed coupling theory of vertebrate segmentation, HFSP J., 3 (2009), 55-66. Google Scholar

[23]

E. M. Özbudak and J. Lewis, Notch signalling synchronizes the zebrafish segmentation clock but is not needed to create somite boundaries, PLoS Genet., 4 (2008), e15. Google Scholar

[24]

O. Pourquié, The chick embryo: a leading model in somitogenesis studies, Mech. Dev., 121 (2004), 1069-1079. doi: 10.1016/j.mod.2004.05.002.  Google Scholar

[25]

I. H. Riedel-Kruse, C. Müller and A. C. Oates, Synchrony dynamics during initiation, failure, and rescue of the segmentation clock, Science, 317 (2007), 1911-1915. doi: 10.1126/science.1142538.  Google Scholar

[26]

J. A. Sherratt and M. J. Smith, Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models, J. R. Soc. Interface, 5 (2008), 483-505. Google Scholar

[27]

C.-W. Shih and J.-P. Tseng, Convergent dynamics for multistable delayed neural networks, Nonlinearity, 21 (2008), 2361-2389. doi: 10.1088/0951-7715/21/10/009.  Google Scholar

[28]

C.-W. Shih and J.-P. Tseng, Global synchronization and asymptotic phases for a ring of identical cells with delayed coupling, SIAM J. Math. Analy., 43 (2011), 1667-1697. doi: 10.1137/10080885X.  Google Scholar

[29]

D. Sieger, B. Ackermann, C. Winkler, D. Tautz and M. Gajewski, her1 and her13.2 are jointly required for somitic border specification along the entire axis of the fish embryo, Dev. Biol., 293 (2006), 242-251. doi: 10.1016/j.ydbio.2006.02.003.  Google Scholar

[30]

K. Uriu, Y. Morishita and Y. Iwasa, Traveling wave formation in vertebrate segmentation, J. Theoret. Biol., 257 (2009), 385-396. doi: 10.1016/j.jtbi.2009.01.003.  Google Scholar

[31]

K. Uriu, Y. Morishita and Y. Iwasa, Synchronized oscillation of the segmentation clock gene in vertebrate development, J. Math. Biol., 61 (2010), 207-229. doi: 10.1007/s00285-009-0296-1.  Google Scholar

[32]

M. B. Wahl, C. Deng, M. Lewandoski and O. Pourquié, FGF signaling acts upstream of the NOTCH and WNT signaling pathways to control segmentation clock oscillations in mouse somitogenesis, Dev., 134 (2007), 4033-4041. doi: 10.1242/dev.009167.  Google Scholar

[33]

D. M. Young, "Iteration Solution of Large Linear Systems," Academic Press, New York-London, 1971.  Google Scholar

show all references

References:
[1]

N. J. Armstrong, K. J. Painter and J. A. Sherratt, A continuum approach to modelling cell-cell adhesion, J. Theoret. Biol., 243 (2006), 98-113. doi: 10.1016/j.jtbi.2006.05.030.  Google Scholar

[2]

N. J. Armstrong, K. J. Painter and J. A. Sherratt, Adding adhesion to a chemical signaling model for somite formation, Bulletin Math. Biol., 71 (2009), 1-24.  Google Scholar

[3]

R. E. Baker and S. Schnell, How can mathematics help us explore vertebrate segmentation?, HFSP J., 3 (2009), 1-5. Google Scholar

[4]

R. E. Baker, S. Schnell and P. K. Maini, A mathematical investigation of a clock and wavefornt model for somitogenesis, J. Math. Biol., 52 (2006), 458-482. doi: 10.1007/s00285-005-0362-2.  Google Scholar

[5]

M. Campanelli and T. Gedeon, Somitogenesis clock-wave initiation requires differential decay and multiple binding sites for clock protein, PLoS Comput. Biol., 6 (2010), e1000728.  Google Scholar

[6]

O. Cinquin, Repressor dimerization in the zebrafish somitogenesis clock, PLoS Comput. Biol., 3 (2007), 293-303.  Google Scholar

[7]

J. Cooke and E. C. Zeeman, A clock and wavefront model for control of the number of repeated structures during animal morphogenesis, J. Theoret. Biol., 58 (1976), 455-476. doi: 10.1016/S0022-5193(76)80131-2.  Google Scholar

[8]

J. Dubrulle, M. J. McGrew and O. Pourquié, FGF signaling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation, Cell, 106 (2001), 219-232. doi: 10.1016/S0092-8674(01)00437-8.  Google Scholar

[9]

J. Dubrulle and O. Pourquié, From head to tail: links between the segmentation clock and antero-posterior patterning of the embryo, Curr. Opin. Genet. Dev., 12 (2002), 519-523. doi: 10.1016/S0959-437X(02)00335-0.  Google Scholar

[10]

J. Dubrulle and O. Pourquié, fgf8 mRNA decay establishes a gradient that couples axial elongation to patterning in the vertebrate embryo, Nature, 427 (2004), 419-422. doi: 10.1038/nature02216.  Google Scholar

[11]

F. Giudicelli, E. M. Özbudak, G. J. Wright and J. Lewis, Setting the tempo in development: an investigation of the zebrafish somite clock mechanism, PLOS Biol., 5 (2007), e150, 1309-1323. Google Scholar

[12]

A. Goldbeter and O. Pourquié, Modeling the segmentation clock as a network of coupled oscillations in the Notch, Wnt and FGF signaling pathways, J. Theoret. Biol., 252 (2008), 574-585. doi: 10.1016/j.jtbi.2008.01.006.  Google Scholar

[13]

E. Hanneman and M. Westerfield, Early expression of acetyl-choline-sterase activity in functionally distinct neurons of the zebrafish, J. Comp. Neurol., 284 (1989), 350-361. doi: 10.1002/cne.902840303.  Google Scholar

[14]

S. A. Holley, The genetics and embryology of zebrafish metamerism, Dev. Dyn., 236 (2007), 1422-1449. doi: 10.1002/dvdy.21162.  Google Scholar

[15]

K. Horikawa, K. Ishimatsu, E. Yoshimoto, S. Kondo and H. Takeda, Noise-resistant and synchronized oscillation of the segmentation clock, Nature, 441 (2006), 719-723. doi: 10.1038/nature04861.  Google Scholar

[16]

Y.-J. Jiang, B. L. Aerne, L. Smithers, C. Haddon, D. Ish-Horowicz and J. Lewis, Notch signaling and the synchronization of the somite segmentation clock, Nature, 408 (2000), 475-479. doi: 10.1038/35044091.  Google Scholar

[17]

A. Kawamura, S. Koshida, H. Hijikata, T. Sakaguchi, H. Kondoh and S. Takada, Zebrafish hairy/enhancer of split protein links FGF signaling to cyclic gene expression in the periodic segmentation of somites, Genes Dev., 19 (2005), 1156-1161. doi: 10.1101/gad.1291205.  Google Scholar

[18]

J. Lewis, Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator, Curr Biol., 13 (2003), 1398-1408. doi: 10.1016/S0960-9822(03)00534-7.  Google Scholar

[19]

K.-L. Liao, C.-W. Shih and J.-P. Tseng, Synchronized oscillations in a mathematical model of segmentation in zebrafish, Nonlinearity, 25 (2012), 869-904. doi: 10.1088/0951-7715/25/4/869.  Google Scholar

[20]

K.-L. Liao, "Analysis on Mathematical Models of Somitogenesis in Zebrafish," Ph.D thesis, National Chiao Tung University, Hsinchu, 2012. Google Scholar

[21]

A. Mara, J. Schroeder, C. Chalouni and S. A. Holley, Priming, initiation and synchronization of the segmentation clock by deltaD and deltaC, Nat. Cell. Biol., 9 (2007), 523-530. Google Scholar

[22]

L. G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher and A. C. Oates, Delayed coupling theory of vertebrate segmentation, HFSP J., 3 (2009), 55-66. Google Scholar

[23]

E. M. Özbudak and J. Lewis, Notch signalling synchronizes the zebrafish segmentation clock but is not needed to create somite boundaries, PLoS Genet., 4 (2008), e15. Google Scholar

[24]

O. Pourquié, The chick embryo: a leading model in somitogenesis studies, Mech. Dev., 121 (2004), 1069-1079. doi: 10.1016/j.mod.2004.05.002.  Google Scholar

[25]

I. H. Riedel-Kruse, C. Müller and A. C. Oates, Synchrony dynamics during initiation, failure, and rescue of the segmentation clock, Science, 317 (2007), 1911-1915. doi: 10.1126/science.1142538.  Google Scholar

[26]

J. A. Sherratt and M. J. Smith, Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models, J. R. Soc. Interface, 5 (2008), 483-505. Google Scholar

[27]

C.-W. Shih and J.-P. Tseng, Convergent dynamics for multistable delayed neural networks, Nonlinearity, 21 (2008), 2361-2389. doi: 10.1088/0951-7715/21/10/009.  Google Scholar

[28]

C.-W. Shih and J.-P. Tseng, Global synchronization and asymptotic phases for a ring of identical cells with delayed coupling, SIAM J. Math. Analy., 43 (2011), 1667-1697. doi: 10.1137/10080885X.  Google Scholar

[29]

D. Sieger, B. Ackermann, C. Winkler, D. Tautz and M. Gajewski, her1 and her13.2 are jointly required for somitic border specification along the entire axis of the fish embryo, Dev. Biol., 293 (2006), 242-251. doi: 10.1016/j.ydbio.2006.02.003.  Google Scholar

[30]

K. Uriu, Y. Morishita and Y. Iwasa, Traveling wave formation in vertebrate segmentation, J. Theoret. Biol., 257 (2009), 385-396. doi: 10.1016/j.jtbi.2009.01.003.  Google Scholar

[31]

K. Uriu, Y. Morishita and Y. Iwasa, Synchronized oscillation of the segmentation clock gene in vertebrate development, J. Math. Biol., 61 (2010), 207-229. doi: 10.1007/s00285-009-0296-1.  Google Scholar

[32]

M. B. Wahl, C. Deng, M. Lewandoski and O. Pourquié, FGF signaling acts upstream of the NOTCH and WNT signaling pathways to control segmentation clock oscillations in mouse somitogenesis, Dev., 134 (2007), 4033-4041. doi: 10.1242/dev.009167.  Google Scholar

[33]

D. M. Young, "Iteration Solution of Large Linear Systems," Academic Press, New York-London, 1971.  Google Scholar

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