×

Modified splitting and composition methods by phase-fitting for simulating biological oscillators. (English) Zbl 1476.65128

Summary: A class of modified splitting and composition methods using the phase-fitting properties of the harmonic oscillators are adapted to the numerical simulation of some biological oscillators. The new phase-fitted splitting and composition methods are furnished with a fitting parameter \(\omega\). In this paper, we present phase-fitted Lie-Trotter and Strang splitting methods and a phase-fitted triple Jump composition method which are generalization of their prototype methods. The result of the experiments on some biological oscillators show the effectiveness and competence of the modified methods over the prototype methods.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
92-10 Mathematical modeling or simulation for problems pertaining to biology
PDFBibTeX XMLCite
Full Text: Link

References:

[1] S. Blanes, F. Casas, A. Murua. Splitting and composition methods in the numerical integration of differential equations.Bol. Soc. Esp. Mat. Apl., 45, (2008), 89-145. · Zbl 1242.65276
[2] S. Blanes, F. Diele, C. Marangi, S. Ragni. Splitting and composition methods for explicit time dependence in separable dynamical systems.J. Comput. Appl. Math. 235, (2010), 646-659. · Zbl 1200.65057
[3] F. Cazals, P. Kornprobst.(Eds.)Modelling in computational biology and biomedicine: A Multidisciplinary Endeavour. Springer. 2013. · Zbl 1255.92001
[4] Z. Chen, X. You, X. Shu, M. Zhang. A new family of phase-fitted and amplification-fitted Runge-Kutta type methods for oscillators.Journal of Applied Mathematics. Volume 2012, Article ID 236281, 27 pages. · Zbl 1268.65091
[5] M.B. Elowitz MB and S. Leibler. A synthetic oscillatory network of transcriptional regulators.Nature, 403, 2000. 335-338.
[6] C. Ge´erard, A. Goldbeter. The cell cycle is a limit cycle.Math. Model. Nat. Phenom. 7, 6, 2012. 126-166. · Zbl 1312.92018
[7] A. Goldbeter. A model for circadian oscillations in the Drosphila period protein (PER).Proc. Biol. Sci.261, 1365, (1995), 319-324.
[8] A. Goldbeter.Biochemical Oscillations and Cellular Rhythms. The molecular bases of periodic and chaotic behaviour. Cambridge University Press, Cambridge, 1996. · Zbl 0837.92009
[9] D. Gonze, W. Abou-Jaoud´e. The Goodwin model: Behind the Hill function.PLOS ONE. 8, 8, (2013) 1-15.
[10] B. C. Goodwin. Oscillatory behaviour in enzymatic control processes.Adv. Enzyme Reg.3, (1965) 425-439.
[11] J. S. Griffith. Mathematics of cellular control processes. I. Negative feedback to one gene.J. Theor. Biol.20, (1968) 202208.
[12] E. Hairer, Ch. Lubich and G. Wanner.Geometric Numerical IntergratorStructure preserving Algorithms for Ordinary Differential Equations, Second Edition, Springer, 2006. · Zbl 1094.65125
[13] I. M. M. van Leeuwen, I. Sanders, O. Staples, S. Lain, and A. J. Munro. Numerical and experimental analysis of the p53-mdm2 regulatory pathway. in Digital Ecosystems, F. A. Basile Colugnati, L. C. Rodrigues Lopes, and S. F. Almeida Barretto, Eds., vol. 67 ofLecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering. (2010) 266-284.
[14] A.J. Lotka, Undamped oscillation derived from the law of mass action.J. Amer. Chem. Soc., 42, (1920) 201-205.
[15] J. D. Murray.Mathematical Biology I: An Introduction. Third Edition. Springer-Verlag, Berlin Heidelberg, 20012.
[16] A. Polynikis, S. J. Hogan, and M. di Bernardo, Comparing different ODE modelling approaches for gene regulatory networks,J. Theor. Biol., vol. 261, no. 4,pp. 511530, 2009. · Zbl 1403.92095
[17] A. D. Raptis, T. E. Simos. A four-step phase-fitted method for the numerical integration of second order initial-value problems.BIT31, (1991), 160-168. · Zbl 0726.65089
[18] G. Strang. On the construction and comparison of difference schemes. SIAM J. Numer. Anal.vol. 5, pp. 506 517, 1968. · Zbl 0184.38503
[19] H. Van de Vyver. Phase-fitted and amplification-fitted two-step hybrid methods fory=f(x, y).J. Comput. Appl. Math. 209, (2007) 33-53. · Zbl 1141.65061
[20] J. Vigo-Aguiar,H. Ramos. On the choice of the frequency in trigonometrically-fitted methods for periodic problems.J. of Comput. and Appl. Math., 277, (2015), 94-105. · Zbl 1302.65161
[21] S. Widder, J. Schicho, and P. Schuster, Dynamic patterns of gene regulation I: simple two-gene systems,J. Theor. Biol., vol. 246, no. 3, pp. 395-419, 2007. · Zbl 1451.92140
[22] M. Xiao, J. Cao, Genetic oscillation deduced from Hopf bifurcation in a genetic regulatory network with delays.Math. Biosci.215(1), (2008) 5563. · Zbl 1156.92031
[23] X. Wu, X. You, B. Wang.Structure preserving algorithms for oscillatory differential equations. Science Press Beijing and Springer-Verlag Heidelberg, 2013. · Zbl 1276.65041
[24] X. You. Limit-Cycle-Preserving simulation of gene regulatory oscillators. Discrete Dyn. in Nat. Soc.Volume 2012, Article ID 673296, 22 pages. · Zbl 1257.92026
[25] X. You, X. Liu, I. H. Musa. Splitting strategy for simulating genetic regulatory networks.Comput Math Methods Med. Vol. 2014. Article ID 683235, (2014), 1-9. · Zbl 1307.92298
[26] R. Zhang, W, Jiang, J.O. Ehigie, Y. Fang, X. You, Novel phase-fitted symmetric splitting methods for chemical oscillators.J. Math. Chem., 55, (2017) 238-258. · Zbl 1357.92035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.