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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 673296, 22 pages
http://dx.doi.org/10.1155/2012/673296
Research Article

Limit-Cycle-Preserving Simulation of Gene Regulatory Oscillators

1Department of Applied Mathematics, Nanjing Agricultural University, Nanjing 210095, China
2State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, China

Received 9 September 2012; Accepted 18 November 2012

Academic Editor: Xiang Ping Yan

Copyright © 2012 Xiong You. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. de Jong, “Modeling and simulation of genetic regulatory systems: a literature review,” Journal of Computatational Biology, vol. 9, no. 1, pp. 67–103, 2002.
  2. S. Widder, J. Schicho, and P. Schuster, “Dynamic patterns of gene regulation. I. Simple two-gene systems,” Journal of Theoretical Biology, vol. 246, no. 3, pp. 395–419, 2007. View at Publisher · View at Google Scholar
  3. A. Polynikis, S. J. Hogan, and M. di Bernardo, “Comparing different ODE modelling approaches for gene regulatory networks,” Journal of Theoretical Biology, vol. 261, no. 4, pp. 511–530, 2009.
  4. A. Altinok, D. Gonze, F. Lévi, and A. Goldbeter, “An automaton model for the cell cycle,” Interface Focus, vol. 1, no. 1, pp. 36–47, 2011.
  5. C. Gérard and A. Goldbeter, “A skeleton model for the network of cyclin-dependent kinases driving the mammalian cell cycle,” Interface Focus, vol. 1, no. 1, pp. 24–35, 2011.
  6. J. C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, 2nd edition, 2008. View at Publisher · View at Google Scholar
  7. J. C. Butcher and G. Wanner, “Runge-Kutta methods: some historical notes,” Applied Numerical Mathematics, vol. 22, no. 1–3, pp. 113–151, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer, Berlin, Germany, 1993.
  9. D. G. Bettis, “Numerical integration of products of Fourier and ordinary polynomials,” Numerische Mathematik, vol. 14, no. 5, pp. 421–434, 1970.
  10. W. Gautschi, “Numerical integration of ordinary differential equations based on trigonometric polynomials,” Numerische Mathematik, vol. 3, no. 1, pp. 381–397, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. P. Martín and J. M. Ferrándiz, “Multistep numerical methods based on the Scheifele G-functions with application to satellite dynamics,” SIAM Journal on Numerical Analysis, vol. 34, no. 1, pp. 359–375, 1997. View at Publisher · View at Google Scholar
  12. A. D. Raptis and T. E. Simos, “A four-step phase-fitted method for the numerical integration of second order initial value problems,” BIT Numerical Mathematics, vol. 31, no. 1, pp. 160–168, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. D. G. Bettis, “Runge-Kutta algorithms for oscillatory problems,” Zeitschrift für Angewandte Mathematik und Physik, vol. 30, no. 4, pp. 699–704, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. B. Paternoster, “Runge-Kutta(-Nyström) methods for ODEs with periodic solutions based on trigonometric polynomials,” Applied Numerical Mathematics, vol. 28, no. 2–4, pp. 401–412, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. J. M. Franco, “Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators,” Computer Physics Communications, vol. 147, no. 3, pp. 770–787, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. J. M. Franco, “Runge-Kutta methods adapted to the numerical integration of oscillatory problems,” Applied Numerical Mathematics, vol. 50, no. 3-4, pp. 427–443, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Z. A. Anastassi and T. E. Simos, “A dispersive-fitted and dissipative-fitted explicit Runge-Kutta method for the numerical solution of orbital problems,” New Astronomy, vol. 10, no. 1, pp. 31–37, 2004.
  18. Z. Chen, X. You, W. Shi, and Z. Liu, “Symmetric and symplectic ERKN methods for oscillatory Hamiltonian systems,” Computer Physics Communications, vol. 183, no. 1, pp. 86–98, 2012. View at Publisher · View at Google Scholar
  19. X. You, Y. Zhang, and J. Zhao, “Trigonometrically-fitted Scheifele two-step methods for perturbed oscillators,” Computer Physics Communications, vol. 182, no. 7, pp. 1481–1490, 2011. View at Publisher · View at Google Scholar
  20. J. Vigo-Aguiar and H. Ramos, “Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations,” Journal of Computational and Applied Mathematics, vol. 158, no. 1, pp. 187–211, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. J. Vigo-Aguiar and J. M. Ferrándiz, “A general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems,” SIAM Journal on Numerical Analysis, vol. 35, no. 4, pp. 1684–1708, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. T. E. Simos and J. Vigo-Aguiar, “An exponentially-fitted high order method for long-term integration of periodic initial-value problems,” Computer Physics Communications, vol. 140, no. 3, pp. 358–365, 2001.
  23. A. Tocino and J. Vigo-Aguiar, “Symplectic conditions for exponential fitting Runge-Kutta-Nyström methods,” Mathematical and Computer Modelling, vol. 42, no. 7-8, pp. 873–876, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. Vigo-Aguiar and T. E. Simos, “An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems,” Astronomical Journal, vol. 122, no. 3, pp. 1656–1660, 2001.
  25. G. Avdelas, T. E. Simos, and J. Vigo-Aguiar, “An embedded exponentially-fitted Runge-Kutta method for the numerical solution of the Schrödinger equation and related periodic initialvalue problems,” Computer Physics Communications, vol. 131, no. 1-2, pp. 52–67, 2000.
  26. T. E. Simos and J. Vigo-Aguiar, “A symmetric high order method with minimal phase-lag for the numerical solution of the Schrodinger equation,” International Journal of Modern Physiscs C, vol. 12, no. 7, pp. 1035–1042, 2001.
  27. T. E. Simos and J. Vigo-Aguiar, “A dissipative exponentially-fitted method for the numerical solution of the Schrödinger equation and related problems,” Computer Physics Communications, vol. 152, no. 3, pp. 274–294, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. H. Van de Vyver, “Phase-fitted and amplification-fitted two-step hybrid methods for y=f(x,y),” Journal of Computational and Applied Mathematics, vol. 209, no. 1, pp. 33–53, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. P. J. van der Houwen and B. P. Sommeijer, “Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 595–617, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. T. E. Simos, “An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions,” Computer Physics Communications, vol. 115, no. 1, pp. 1–8, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. G. Vanden Berghe, H. De Meyer, M. Van Daele, and T. Van Hecke, “Exponentially fitted Runge-Kutta methods,” Journal of Computational and Applied Mathematics, vol. 125, no. 1-2, pp. 107–115, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. G. Vanden Berghe, L. Gr. Ixaru, and M. Van Daele, “Optimal implicit exponentially fitted Runge-Kutta methods,” Computer Physics Communications, vol. 140, no. 3, pp. 346–357, 2001.
  33. G. Vanden Berghe, L. Gr. Ixaru, and H. De Meyer, “Frequency determination and step-length control for exponentially-fitted Runge-Kutta methods,” Journal of Computational and Applied Mathematics, vol. 132, no. 1, pp. 95–105, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. L. Gr. Ixaru, G. Vanden Berghe, and H. De Meyer, “Frequency evaluation in exponential fitting multistep algorithms for ODEs,” Journal of Computational and Applied Mathematics, vol. 140, no. 1-2, pp. 423–434, 2002.
  35. J. Vigo-Aguiar, T. E. Simos, and J. M. Ferrándiz, “Controlling the error growth in long-term numerical integration of perturbed oscillations in one or several frequencies,” Proceedings of The Royal Society of London A, vol. 460, no. 2042, pp. 561–567, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. H. Van de Vyver, “Frequency evaluation for exponentially fitted Runge-Kutta methods,” Journal of Computational and Applied Mathematics, vol. 184, no. 2, pp. 442–463, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. H. Ramos and J. Vigo-Aguiar, “On the frequency choice in trigonometrically fitted methods,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1378–1381, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH