Bifurcation Analysis About a Mathematical Model of Somitogenesis Based on the Runge–Kutta Method

Article

Abstract

In this paper, we investigated the modified two dimensional model which can explain somite patterning in embryos. It is suitable for exploring a design space of somitogenesis and can explain aspects of somitogenesis that previous models cannot. Here, we mainly studied the non-diffusing case. We have used the Hopf bifurcation theorem, the Center manifold theorem and Runge–Kutta method in our investigation. First, we investigate its dynamical behaviors and put forward a sufficient condition for the oscillation of the small network. Then, we give the mathematical simulation based on the Runge–Kutta method. In the process of solving ordinary differential equations, the four order Runge–Kutta method has the advantages of high accuracy, convergence and stability (under certain conditions), which can change the step size and do not need to calculate higher order derivatives. Therefore, it has become the most commonly used numerical solution. At the same time, we get the sufficient condition in which the bistable state of the system exists and give the numerical simulation. Because somitogenesis occupies an important position in the process of biological development, and as a pattern process can be used to study pattern formation and many aspects of embryogenesis. So our study have a great help for embryonic development, gene expression, cell differentiation. In addition, it is beneficial to study the clone animal variation problem of spinal bone number and is of great help to the treatment and prevention of defects of human spine disease.

Keywords

Hopf bifurcation Bistable Somitogenesis Runge–Kutta method 

Notes

Acknowledgements

This work is supported by National Natural Science Foundation of China (No. 11772291, 11572278), Innovation Scientists and Technicians Troop Construction Projects of Henan Province (No. 2017JR0013).

Compliance with ethical standards

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

References

  1. 1.
    Cooke, J. (1975). Control of somite number during morphogenesis of a vertebrate, Xenopus laevis[J]. Nature, 254(5479), 196–199.CrossRefGoogle Scholar
  2. 2.
    McGrew, M. J., & Pourquie, O. (1998). Somitogenesis: Segmenting a vertebrate. Current Opinion in Genetics & Development, 8(4), 487–493.CrossRefGoogle Scholar
  3. 3.
    Cotterell, J., Robert-Moreno, A., & Sharpe, J. (2015). A local, self-organizing reaction-diffusion model can explain somite patterning in embryos. Cell Systems, 1(4), 257–269.CrossRefGoogle Scholar
  4. 4.
    Kulesa, P. M., Schnell, S., Rudloff, S., Baker, R. E., & Maini, P. K. (2007). From segment to somite: Segmentation to epithelialization analyzed within quantitative frameworks. Developmental Dynamics, 236, 1392–1402.CrossRefGoogle Scholar
  5. 5.
    Cooke, J., & Zeeman, E. C. (1976). A clock and wavefront model for control of the number of repeated structures during animal morphogenesis. Journal of Theoretical Biology, 58, 455–476.CrossRefGoogle Scholar
  6. 6.
    Meinhardt, H. (1996). Model of biological pattern formation: Common mechanism in plant and animal development. International Journal of Developmental Biology, 40, 123–134.Google Scholar
  7. 7.
    Polezhaev, A. A. (1992). A mathematical model of the mechanism of vertebrate somitic segmentation. Journal of Theoretical Biology, 156(2), 169–181.CrossRefGoogle Scholar
  8. 8.
    Schnell, S., & Maini, P. K. (2000). Clock and induction model for somitogenesis. Developmental Dynamics, 217(4), 415–420.CrossRefGoogle Scholar
  9. 9.
    Collier, J. R., Mcinerney, D., Schnell, S., Maini, P. K., Gavaghan, D. J., Houston, P., et al. (2000). A cell cycle model for somitogenesis: Mathematical formulation and numerical simulation. Journal of Theoretical Biology, 207(3), 305–316.CrossRefGoogle Scholar
  10. 10.
    Kerszberg, M., & Wolpert, L. (2000). A clock and trail model for somite formation, specialization and polarization. Journal of Theoretical Biology, 205(3), 505–510.CrossRefGoogle Scholar
  11. 11.
    Herrgen, L., Ares, S., Morelli, L. G., Schroter, C., Julicher, F., & Oates, A. C. (2010). Intercellular coupling regulates the period of the segmentation clock. Current Biology, 20, 1244–1253.CrossRefGoogle Scholar
  12. 12.
    Baker, R. E., Schnell, S., & Maini, P. K. (2006). A mathematical investigation of a clock and wavefront model for somitogenesis. Journal of Mathematical Biology, 52, 458–482.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Baker, R. E., Schnell, S., & Maini, P. K. (2006). A clock and wavefront mechanism for somite formation. Development Biology, 293, 116–126.CrossRefGoogle Scholar
  14. 14.
    Polezhaev, A. A. (1995). Mathematical modelling of the mechanism of vertebrate somitic segmentation. Journal of Biological Systems, 3(04), 1041–1051.CrossRefGoogle Scholar
  15. 15.
    Lin, G., & Slack, J. M. (2008). Requirement for Wnt and FGF signaling in Xenopus tadpole tail regeneration. Development Biology, 316, 323–335.CrossRefGoogle Scholar
  16. 16.
    Polezhayev, A. A. (1995). Mathematical model of segmentation in somitogenesis in vertebrates. Biophysics, 3(40), 583–589.Google Scholar
  17. 17.
    Murray, P. J., Maini, P. K., & Baker, R. E. (2011). The clock and wavefront model revisited. Journal of Theoretical Biology, 283, 227–238.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Munoz Alicea, R. (2011). Introduction to bifurcations and the Hopf bifurcation theorem for planar systems. Opreation for Mathematics, 640, 11.Google Scholar
  19. 19.
    Kuznetsov, Y. (2006). Andronov-Hopf bifurcation. Scholarpedia, 1(10), 1858.CrossRefGoogle Scholar
  20. 20.
    Dhooge, A., Govaerts, W., & Kuznetsov, Y. A. (2003). MATCONT: A Matlab package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 29, 141–164.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kuehn, C. (2012). From first lyapunov coefficients to maximal canards. International Journal of Bifurcation and Chaos, 20(5), 1467–1475.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Sarmah, H. K., Baishya, T. K., & Das, M. C. (2014). Hopf-bifurcation in a two dimensional nonlinear differential equation. International Journal of Modern Engineering and Research Technology, 4, 2249–6645.Google Scholar
  23. 23.
    Das D., Banerjee, D. & Bhattacharjee, J. K. (2013). Super-critical and Sub-Critical Hopf bifurcations in two and three dimensions. arXiv:1309.5470v1.
  24. 24.
    Chen, K. W., Liao, K. L., & Shih, C. W. (2018). The kinetics in mathematical models on segmentation clock genes in zebrafish. Journal of Mathematical Biology, 76(1–2), 97–150.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Shimono, Y., Yamaguchi, I., Ishimatsu, K., Akao, A., Ogawa, Y., Jimbo, Y., & Kotani, K. (2018) Evaluation of heuristic reductions of a model for the segmentation clock in zebrafish. In IEEJ transactions on electrical and electronic engineering.Google Scholar
  26. 26.
    Ajima, R., Suzuki, E., & Saga, Y. (2017). Pofut1 point-mutations that disrupt O-fucosyltransferase activity destabilize the protein and abolish Notch1 signaling during mouse somitogenesis. PLoS ONE, 12(11), e0187248.CrossRefGoogle Scholar
  27. 27.
    Gallagher, T. L., Tietz, K. T., Morrow, Z. T., McCammon, J. M., Goldrich, M. L., Derr, N. L., et al. (2017). Pnrc2 regulates 3’UTR-mediated decay of segmentation clock-associated transcripts during zebrafish segmentation. Developmental Biology, 429(1), 225–239.CrossRefGoogle Scholar
  28. 28.
    Uriu, K., Bhavna, R., Oates, A. C., & Morelli, L. G. (2017). A framework for quantification and physical modeling of cell mixing applied to oscillator synchronization in vertebrate somitogenesis. Biology Open, 6, 1235.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina
  2. 2.Institute of Applied MathematicsXuchang UniversityXuchangChina

Personalised recommendations