Novel techniques in parameter estimation for fractional dynamical models arising from biological systems

Computers and Mathematics with Applications

Volumn 62, Issue 3, 2011, Pages 822-833

  Liu, F ^a   Burrage, K ^a,b  

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^b UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Competence induction; Fractional dynamical models; Fractional PredictorCorrector method; Hybrid simplex search and particle swarm optimization; Parameter estimation

Indexed keywords

COMPETENCE INDUCTION; DYNAMICAL MODEL; FRACTIONAL CALCULUS; MINIMIZATION PROBLEMS; NOISY DATA; NONLINEAR DYNAMICAL MODELS; NOVEL TECHNIQUES; NUMERICAL RESULTS; PREDICTOR-CORRECTOR METHODS; SIMPLEX SEARCH; SYSTEMS BIOLOGY; UNKNOWN PARAMETERS;

BIOLOGICAL SYSTEMS; ESTIMATION; PARTICLE SWARM OPTIMIZATION (PSO);

PARAMETER ESTIMATION;

EID: 79961005041     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2011.03.002     Document Type: Article

References (46)

1. V. Isakov Inverse Problems for Partial Differential Equations 1997 Springer
2. J. Milstein The inverse problem: estimation of kinetic parameters K. Ebert, P. Deuflhard, W. Jger, Modeling of Chemical Reaction Systems 1981 Springer Berlin 93 125
3. J. Alcock, and K. Burrage A genetic estimation algorithm for parameters of stochastic ordinary differential equations Comput. Statist. Data Anal. 47 2 2004 255 275
4. H. Kunze, and K. Heidler The collage coding method and its application to an inverse problem for the Lorenz system Appl. Math. Comput. 186 2007 124 129 (Pubitemid 46435605)
5. M.F. Barnsley, V. Ervin, D. Hardin, and J. Lancaster Solution of an inverse problem for fractals and other sets Proc. Natl. Acad. Sci. USA 83 1985 1975 1977
6. X. Deng, B. Wang, and G. Long The Picard contraction mapping method for the parameter inversion of reactiondiffusion systems Comput. Math. Appl. 56 9 2008 2347 2355
7. H.E. Kunze, and E.R. Vrscay Solving inverse problems for ordinary differential equations using the Picard contraction mapping Inverse Problems 15 1999 745 770 (Pubitemid 129561612)
8. H. Kunze, and E. Vrscay Inverse problems for ODEs using contraction maps: sub-optimality of the collage method Inverse Problems 20 2004 977 991
9. G.M. Sel, J. Garcia-Ojalvo, L.M. Liberman, and M.B. Elowitz An excitable gene regulatory circuit induces transient cellular differentiation Nat. Lett. 440 2006 545 550
10. C.P. Fall, E.S. Marland, J.M. Wagner, and J.J. Tyson Computational Cell Biology 2002 Springer-Verlag
11. C. Chen, F. Liu, I. Turner, and V. Anh Fourier method for the fractional diffusion equation describing sub-diffusion J. Comput. Phys. 227 2007 886 897
12. P. Zhuang, F. Liu, V. Anh, and I. Turner New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation SIAM J. Numer. Anal. 46 2 2008 1079 1095
13. F. Liu, C. Yang, and K. Burrage Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term J. Comput. Appl. Math. 231 2009 160 176
14. S.B. Yuste, and K. Lindenberg Subdiffusion-limited A + A reactions Phys. Rev. Lett. 87 11 2001 118301
15. R. Metzler, and J. Klafter The random walk's guide to anomalous diffusion: a fractional dynamics approach Phys. Rep. 339 2000 1 77
16. S.B. Yuste, L. Acedo, and K. Lindenberg Reaction front in an A + B → C reactionsubdiffusion process Phys. Rev. E 69 2004 036126
17. F. Liu, V. Anh, and I. Turner Numerical solution of the space fractional FokkerPlanck equation J. Comput. Appl. Math. 166 2004 209 219 (Pubitemid 38342127)
18. H. Berry Monte Carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation Biophys. J. 83 4 2002 1891 1901
19. D.S. Martin, M.B. Forstner, and J.A. Kas Apparent subdiffusion inherent to single particle tracking Biophys. J. 83 4 2002 2109 2117
20. R. Kopelman, S. Parus, and J. Prasad Fractal-like exciton kinetics in porous glasses, organic membranes, and filter papers Phys. Rev. Lett. 56 1986 1742 1745
21. R. Kopelman Fractal reaction kinetics Science 241 1988 1620 1626
22. M.J. Saxton Anomalous diffusion due to obstacle: a Monte Carlo study Biophys. J. 66 1994 394 401 (Pubitemid 24058459)
23. P.R. Smith, I.E.G. Morrison, K.M. Wilson, N. Fernández, and R.J. Cherry Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking Biophys. J. 76 1999 3331 3344 (Pubitemid 29269481)
24. T. Jovin, and W.L.C. Vaz Rotational and translational diffusion in membranes measured by fluorescence and phosphorescence methods Methods Enzymol. 172 1989 471 573
25. M. Weiss, M. Elsner, F. Kartberg, and T. Nilsson Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells Biophys. J. 87 2004 3518 3524 (Pubitemid 40468604)
26. Y. Shav-Tal, X. Darzacq, S.M. Shenoy, D. Fusco, S.M. Janicki, D.L. Spector, and R.H. Singer Dynamics of single mRNPs in nuclei of living cells Science 304 5678 2004 1797 1800 (Pubitemid 38787895)
27. M. Wachsmuth, W. Waldeck, and J. Langowski Amomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially resolved fluorescence correlation spectroscopy J. Mol. Biol. 298 2000 677 698
28. P. Schwile, J. Korlach, and W.W. Webb Fluorescence correlation spectroscopy with single-molecule sensitivity on cell and model membranes Cytometry 36 1999 176 182 (Pubitemid 29291466)
29. Schnell, and T.E. Turner Reaction kinetics intracellular environments with macromolecular crowding: simulations and rate laws Prog. Biophys. Mol. Biol. 85 2004 235 260 (Pubitemid 38629911)
30. T.E. Turner, S. Schnell, and K. Burrage Stochastic approaches for modeling in vivo reactions Comput. Biol. Chem. 28 2004 165 178
31. D.V. Nicolau Jr., and K. Burrage Stochastic simulation of chemical reactions in spatially complex media Comput. Math. Appl. 55 5 2008 1007 1018
32. D.V. Nicolau Jr., J.F. Hancock, and K. Burrage Sources of anomalous diffusion on cell membranes: a Monte Carlo study Biophys. J. 92 2007 1975 1987 (Pubitemid 46495261)
33. K. Seki, M. Wojcik, and M. Tachiya Fractional reactiondiffusion equation J. Chem. Phys. 119 2003 2165 2174
34. B.I. Henry, and S.L. Wearne Fractional reactiondiffusion Physica A 276 2000 448 455
35. I. Podlubny Fractional Differential Equations 1999 Academic Press New York
36. R. Malti, S. Victor, and A. Oustaloup Advances in system identification using fractional models J. Comput. Nonlinear Dyn. 3 2 2008 021401
37. T. Point, and J. Trigeassou Identification of fractional systems using an output-error technique Nonlinear Dynam. 38 2004 133 154 (Pubitemid 40400612)
38. E. Hairer, and G. Wanner Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems 1991 Springer Berlin
39. K. Diethelm, N.J. Ford, and D. Freed Alen Detailed error analysis for a fractional Adams method Numerical Algorithms 2004 Kluwer Academic Publishers 31 52 (Pubitemid 39072748)
40. C. Yang, and F. Liu A computationally effective predictor-corrector method for simulating fractional order dynamical control system ANZIAM J. 47 2006 168 184
41. J.A. Nelder, and R.A. Mead Simplex method for function minimization Comput. J. 7 1965 308 313
42. P.S. Shelokar, P. Siarry, V.K. Jayaraman, and B.D. Kulkarni Particle swarm and colony algorithms hybridized for improved continuous optimization Appl. Math. Comput. 188 2007 129 142 (Pubitemid 46686707)
43. S. Fan, Y. Liang, and E. Zahara Hybrid simplex seaplex and particle swarm optimization for the global optimization of multimodal functions Eng. Optim. 36 4 2004 401 418
44. G.E. Uhlenbeck, and L.S. Ornstein On the theory of Brownian motion Phys. Rev. 36 1930 823 841
45. J.L. Doob The Brownian movement and stochastic equations Ann. of Math. 43 1942 351 369
46. D.T. Gillespie Exact numerical simulation of the OrnseinUhlenbeck process and its integral Phys. Rev. E 54 1996 2084 2091 (Pubitemid 126589550)