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Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. (English) Zbl 1228.93114

Summary: In recent years, both parameter estimation and fractional calculus have attracted a considerable interest. Parameter estimation of the fractional dynamical models is a new topic. In this paper, we consider novel techniques for parameter estimation of fractional nonlinear dynamical models in systems biology. First, a computationally effective fractional predictor-corrector method is proposed for simulating fractional complex dynamical models. Second, we convert the parameter estimation of fractional complex dynamical models into a minimization problem of the unknown parameters. Third, a modified hybrid simplex search (MHSS) and a particle swarm optimization (PSO) is proposed. Finally, these techniques are applied to a dynamical model of competence induction in a cell with measurement error and noisy data. Some numerical results are given that demonstrate the effectiveness of the theoretical analysis.

MSC:

93E10 Estimation and detection in stochastic control theory
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
65L09 Numerical solution of inverse problems involving ordinary differential equations
92C99 Physiological, cellular and medical topics

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[1] Isakov, V., Inverse Problems for Partial Differential Equations (1997), Springer
[2] Milstein, J., The inverse problem: estimation of kinetic parameters, (Ebert, K.; Deuflhard, P.; Jäger, W., Modeling of Chemical Reaction Systems (1981), Springer: Springer Berlin), 93-125
[3] Alcock, J.; Burrage, K., A genetic estimation algorithm for parameters of stochastic ordinary differential equations, Comput. Statist. Data Anal., 47, 2, 255-275 (2004) · Zbl 1429.62362
[4] Kunze, H.; Heidler, K., The collage coding method and its application to an inverse problem for the Lorenz system, Appl. Math. Comput., 186, 124-129 (2007) · Zbl 1114.65086
[5] Barnsley, M. F.; Ervin, V.; Hardin, D.; Lancaster, J., Solution of an inverse problem for fractals and other sets, Proc. Natl. Acad. Sci. USA, 83, 1975-1977 (1985) · Zbl 0613.28008
[6] Deng, X.; Wang, B.; Long, G., The Picard contraction mapping method for the parameter inversion of reaction-diffusion systems, Comput. Math. Appl., 56, 9, 2347-2355 (2008) · Zbl 1165.35399
[7] Kunze, H. E.; Vrscay, E. R., Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems, 15, 745-770 (1999) · Zbl 0978.34013
[8] Kunze, H.; Vrscay, E., Inverse problems for ODEs using contraction maps: sub-optimality of the collage method, Inverse Problems, 20, 977-991 (2004) · Zbl 1067.34010
[9] Süel, G. M.; Garcia-Ojalvo, J.; Liberman, L. M.; Elowitz, M. B., An excitable gene regulatory circuit induces transient cellular differentiation, Nat. Lett., 440, 545-550 (2006)
[10] Fall, C. P.; Marland, E. S.; Wagner, J. M.; Tyson, J. J., Computational Cell Biology (2002), Springer-Verlag · Zbl 1010.92019
[11] Chen, C.; Liu, F.; Turner, I.; Anh, V., Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227, 886-897 (2007) · Zbl 1165.65053
[12] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46, 2, 1079-1095 (2008) · Zbl 1173.26006
[13] Liu, F.; Yang, C.; Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231, 160-176 (2009) · Zbl 1170.65107
[14] Yuste, S. B.; Lindenberg, K., Subdiffusion-limited \(A + A\) reactions, Phys. Rev. Lett., 87, 11, 118301 (2001)
[15] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[16] Yuste, S. B.; Acedo, L.; Lindenberg, K., Reaction front in an \(A + B \to C\) reaction-subdiffusion process, Phys. Rev. E, 69, 036126 (2004)
[17] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019
[18] Berry, H., Monte Carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation, Biophys. J., 83, 4, 1891-1901 (2002)
[19] Martin, D. S.; Forstner, M. B.; Kas, J. A., Apparent subdiffusion inherent to single particle tracking, Biophys. J., 83, 4, 2109-2117 (2002)
[20] Kopelman, R.; Parus, S.; Prasad, J., Fractal-like exciton kinetics in porous glasses, organic membranes, and filter papers, Phys. Rev. Lett., 56, 1742-1745 (1986)
[21] Kopelman, R., Fractal reaction kinetics, Science, 241, 1620-1626 (1988)
[22] Saxton, M. J., Anomalous diffusion due to obstacle: a Monte Carlo study, Biophys. J., 66, 394-401 (1994)
[23] Smith, P. R.; Morrison, I. E.G.; Wilson, K. M.; Fernández, N.; Cherry, R. J., Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking, Biophys. J., 76, 3331-3344 (1999)
[24] Jovin, T.; Vaz, W. L.C., Rotational and translational diffusion in membranes measured by fluorescence and phosphorescence methods, Methods Enzymol., 172, 471-573 (1989)
[25] Weiss, M.; Elsner, M.; Kartberg, F.; Nilsson, T., Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells, Biophys. J., 87, 3518-3524 (2004)
[26] Shav-Tal, Y.; Darzacq, X.; Shenoy, S. M.; Fusco, D.; Janicki, S. M.; Spector, D. L.; Singer, R. H., Dynamics of single mRNPs in nuclei of living cells, Science, 304, 5678, 1797-1800 (2004)
[27] Wachsmuth, M.; Waldeck, W.; Langowski, J., Amomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially resolved fluorescence correlation spectroscopy, J. Mol. Biol., 298, 677-698 (2000)
[28] Schwile, P.; Korlach, J.; Webb, W. W., Fluorescence correlation spectroscopy with single-molecule sensitivity on cell and model membranes, Cytometry, 36, 176-182 (1999)
[29] Schnell; Turner, T. E., Reaction kinetics intracellular environments with macromolecular crowding: simulations and rate laws, Prog. Biophys. Mol. Biol., 85, 235-260 (2004)
[30] Turner, T. E.; Schnell, S.; Burrage, K., Stochastic approaches for modeling in vivo reactions, Comput. Biol. Chem., 28, 165-178 (2004) · Zbl 1087.92035
[31] Nicolau, D. V.; Burrage, K., Stochastic simulation of chemical reactions in spatially complex media, Comput. Math. Appl., 55, 5, 1007-1018 (2008) · Zbl 1137.92015
[32] Nicolau, D. V.; Hancock, J. F.; Burrage, K., Sources of anomalous diffusion on cell membranes: a Monte Carlo study, Biophys. J., 92, 1975-1987 (2007)
[33] Seki, K.; Wojcik, M.; Tachiya, M., Fractional reaction-diffusion equation, J. Chem. Phys., 119, 2165-2174 (2003)
[34] Henry, B. I.; Wearne, S. L., Fractional reaction-diffusion, Physica A, 276, 448-455 (2000)
[35] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[36] Malti, R.; Victor, S.; Oustaloup, A., Advances in system identification using fractional models, J. Comput. Nonlinear Dyn., 3, 2, 021401 (2008)
[37] Point, T.; Trigeassou, J., Identification of fractional systems using an output-error technique, Nonlinear Dynam., 38, 133-154 (2004) · Zbl 1134.93324
[38] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (1991), Springer: Springer Berlin · Zbl 0729.65051
[39] Diethelm, K.; Ford, N. J.; Freed Alen, D., Detailed error analysis for a fractional Adams method, (Numerical Algorithms (2004), Kluwer Academic Publishers), 31-52 · Zbl 1055.65098
[40] Yang, C.; Liu, F., A computationally effective predictor-corrector method for simulating fractional order dynamical control system, ANZIAM J., 47, 168-184 (2006)
[41] Nelder, J. A.; Mead, R. A., Simplex method for function minimization, Comput. J., 7, 308-313 (1965) · Zbl 0229.65053
[42] Shelokar, P. S.; Siarry, P.; Jayaraman, V. K.; Kulkarni, B. D., Particle swarm and colony algorithms hybridized for improved continuous optimization, Appl. Math. Comput., 188, 129-142 (2007) · Zbl 1114.65334
[43] Fan, S.; Liang, Y.; Zahara, E., Hybrid simplex seaplex and particle swarm optimization for the global optimization of multimodal functions, Eng. Optim., 36, 4, 401-418 (2004)
[44] Uhlenbeck, G. E.; Ornstein, L. S., On the theory of Brownian motion, Phys. Rev., 36, 823-841 (1930) · JFM 56.1277.03
[45] Doob, J. L., The Brownian movement and stochastic equations, Ann. of Math., 43, 351-369 (1942) · Zbl 0063.01145
[46] Gillespie, D. T., Exact numerical simulation of the Ornsein-Uhlenbeck process and its integral, Phys. Rev. E, 54, 2084-2091 (1996)
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