Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach
Abstract
In this survey paper we present a systematic methodology of how to identify origins of fractional dynamics. We consider three models leading to it, namely fractional Brownian motion (FBM), fractional Lévy stable motion (FLSM) and autoregressive fractionally integrated moving average (ARFIMA) process. The discrete-time ARFIMA process is stationary, and when aggregated, in the limit, it converges to either FBM or FLSM. In this sense it generalizes both models. We discuss three experimental data sets related to some molecular biology problems described by single particle tracking. They are successfully resolved by means of the universal ARFIMA time series model with various noises. Even if the finer details of the estimation procedures are case specific, we hope that the suggested checklist will still have been of great use as a practical guide. In Appendices A–F we describe useful fractional dynamics identification and validation methods.
Keywords
stochastic processes; fractional particle dynamics; ARFIMA time series; identification and validation algorithms
References
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8. Burnecki K., Gajda J., Sikora G., Stability and lack of memory of the returns of the Hang Seng index, Physica A 390 (2011), 3136–3146.
9. Burnecki K., Kepten E., Garini Y., Sikora G., Weron A., Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors – An alternative approach, Sci. Reports 5 (2015), 11306, 11 pp.
10. Burnecki K., Kepten E., Janczura J., Bronshtein I., Garini Y., Weron A., Universal algorithm for identification of fractional Brownian motion. A case of telomere subdiffusion, Biophys. J. 103 (2012), 1839–1847.
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13. Burnecki K., Sikora G., Estimation of FARIMA parameters in the case of negative memory and stable noise, IEEE Trans. Signal Process. 61 (2013), 2825–2835.
14. Burnecki K., Sikora G.,Weron A., Fractional process as a unified model for subdiffusive dynamics in experimental data, Phys. Rev. E 86 (2012), 041912, 8 pp.
15. Burnecki K.,Weron A., Fractional Lévy stable motion can model subdiffusive dynamics, Phys. Rev. E 82 (2010), 021130, 8 pp.
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23. Crato N., Rothman P., Fractional integration analysis of long-run behavior for US macroeconomic time series, Econom. Lett. 45 (1994), 287–291.
24. Davies R.B., Harte D.S., Tests for Hurst effect, Biometrika 74 (1987), 95–101.
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31. Granger C.W.J., Joyeux R., An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal. 1 (1980), 15–29.
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33. Havlin S., Ben-Avraham D., Diffusion in disordered media, Adv. Phys. 36 (1987), 695–798.
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37. Hosking J.R.M., Fractional differencing, Biometrika 68 (1981), 165–176.
38. Janicki A., Weron A., Simulation and Chaotic Behavior of α-stable Stochastic Processes, Marcel Dekker Inc., New York, 1994.
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42. Kokoszka P.S., Taqqu M.S., Parameter estimation for infinite variance fractional ARIMA, Ann. Statist. 24 (1996), 1880–1913.
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44. Kou S.C., Xie X.S., Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. Rev. Lett. 93 (2004), 180603, 4 pp.
45. Kowarzyk H., Steinhaus H., Szymaniec S., Arrangement of chromosomes in human cells. I. Associations in metaphase figures, Bull. Acad. Polon. Sci., Ser. Sci. Biol. 13 (1965), 321–326.
46. Kowarzyk H., Steinhaus H., Szymaniec S., Arrangement of chromosomes in human cells. 3. Distribution of centromeres and orientation of chromosomes in the metaphase, Bull. Acad. Polon. Sci., Ser. Sci. Biol. 14 (1966), 541–544.
47. Lagarias J.C., Reeds J.A., Wright M.H., Wright P.E., Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim. 9 (1998), 112–147.
48. Lanoiselée Y., Grebenkov D.S., Revealing nonergodic dynamics in living cells from a single particle trajectory, Phys. Rev. E 93 (2016), 052146, 17 pp.
49. Lasota A., Mackey M.C., Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Springer-Verlag, New York, 1994.
50. Lasota A., Mackey M.C.,Ważewska-Czyżewska M., Minimizing therapeutically induced anemia, J. Math. Biol. 13 (1981/82), 149–158.
51. Lim S.C., Muniandy S.V., Self-similar Gaussian processes for modeling anomalous diffusion, Phys. Rev. E 66 (2002), 021114, 14 pp.
52. Ljung G.M., Box G.E.P., On a measure of lack of fit in time series models, Biometrika 65 (1978), 297–303.
53. Loch H., Janczura A., Weron A., Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes, Phys. Rev. E 93 (2016), 043317, 10 pp.
54. Loch-Olszewska H., Sikora G., Janczura J., Weron A., Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length, Phys. Rev. E 94 (2016), 052136, 8 pp.
55. Magdziarz M., Weron A., Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series, Studia Math. 181 (2007), 47–60.
56. Magdziarz M., Weron A., Anomalous diffusion: Testing ergodicity breaking in experimental data, Phys. Rev. E 84 (2011), 051138, 5 pp.
57. Magdziarz M., Weron A., Burnecki K., Klafter J., Fractional Brownian motion versus the continuous-time random walk: a simple test for subdiffusive dynamics, Phys. Rev. Lett. 103 (2009), 180602, 4 pp.
58. Mandelbrot B.B., Wallis J.R., Noah, Joseph, and operational hydrology, Water Resour. Res. 4 (1968), 909–918.
59. Mann M.E., Bradley R.S., Hughes M.K., Global-scale temperature patterns and climate forcing over the past six centuries, Nature 392 (1998), 779–787.
60. Meroz Y., Sokolov I.M., Klafter J., Test for determining a subdiffusive model in ergodic systems from single trajectories, Phys. Rev. Lett. 110 (2013), 090601, 4 pp.
61. Metzler R., Barkai E., Klafter J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys. Rev. Lett. 82 (1999), 3563–3567.
62. Metzler R., Klafter J., The random walk’s guide to anomalous difusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.
63. Nolan J.P., Numerical calculation of stable densities and distribution functions, Commun. Statist.-Stochastic Models 13 (1997), 759–774.
64. Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, 1993.
65. Saxton M.J., Anomalous diffusion due to obstacles: a Monte Carlo study, Biophys. J. 66 (1994), 394–401.
66. Saxton M.J., Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study, Biophys. J. 81 (2001), 2226–2240.
67. Saxton M.J., Wanted: a positive control for anomalous subdiffusion, Biophys. J. 103 (2012), 2411–2422.
68. Schneider W.R., Stable distributions: Fox functions representation and generalization, in: Albeverio S., Casati G., Merlini D. (eds.), Stochastic Processes in Classical and Quantum Systems, Springer, Berlin, 1986, pp. 497–511.
69. Ślęzak J., Drobczyński D., Weron K., Masajada J., Moving average process underlying the holographic-optical-tweezers experiments, Appl. Opt. 53 (2014), B254–B258.
70. Sokolov I.M., Models of anomalous diffusion in crowded environments, Soft Matter 8 (2012), 9043–9052.
71. Stanislavsky A., Memory effects and macroscopic manifestation of randomness, Phys. Rev. E 61 (2000), 4752–4759.
72. Stanislavsky A.A., Burnecki K., Magdziarz M., Weron A., Weron K., FARIMA modeling of solar flare activity from empirical time series of soft X-ray solar emission, Astrophys. J. 693 (2009), 1877–1882.
73. Stanislavsky A., Weron K., Weron A., Anomalous diffusion with transient subordinators: a link to compound relaxation laws, J. Chem. Phys. 140 (2014), 054113, 7 pp.
74. Stoev S., Taqqu M.S., Simulation methods for linear fractional stable motion and FARIMA using the fast Fourier transform, Fractals 12 (2004), 95–121.
75. Weron A., Burnecki K., Mercik S., Weron K., Complete description of all self-similar models driven by Lévy stable noise, Phys. Rev. E 71 (2005), 016113, 10 pp.
76. Weron A., Magdziarz M., Anomalous diffusion and semimartingales, EPL 86 (2009), 60010, 6 pp.
77. Weron A., Weron R., Computer simulation of Lévy α-stable variables and processes, in: Garbaczewski P., Wolf M., Weron A. (eds.), Chaos – the Interplay Between Stochastic and Deterministic Behaviour, Springer, Berlin, 1995, pp. 379–392.
78. Weron R., Levy-stable distributions revisited: tail index > 2 does not exclude the Levystable regime, Int. J. Mod. Phys. C 12 (2001), 209–223.
79. Weron R., Computationally intensive value at risk calculations, in: Gentle J.E., Härdle W., Mori Y. (eds.), Handbook of Computational Statistics. Concepts and Methods, Springer, Berlin, 2004, pp. 911–950.
80. Wigner E.P., The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math. 13 (1960), 1–14.
2. Barndorff-Nielsen O.E., Normal//inverse Gaussian Processes and the Modelling of Stock Returns, Research Report 300, Department of Theoretical Statistics, University of Aarhus, 1995.
3. Beran J., Statistics for Long-Memory Processes, Chapman and Hall, New York, 1994.
4. Brcich R.F., Iskander D.R., Zoubir A.M., The stability test for symmetric alpha-stable distributions, IEEE Trans. Signal Process. 53 (2005), 977–986.
5. Brockwell P.J., Davis R.A., Introduction to Time Series and Forecasting, Springer-Verlag, New York, 2002.
6. Burnecki K., FARIMA processes with application to biophysical data, J. Stat. Mech. 2012, P05015, 18 pp.
7. Burnecki K., Identification, Validation and Prediction of Fractional Dynamical Systems, Wroclaw University of Technology Press, Wrocław, 2012.
8. Burnecki K., Gajda J., Sikora G., Stability and lack of memory of the returns of the Hang Seng index, Physica A 390 (2011), 3136–3146.
9. Burnecki K., Kepten E., Garini Y., Sikora G., Weron A., Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors – An alternative approach, Sci. Reports 5 (2015), 11306, 11 pp.
10. Burnecki K., Kepten E., Janczura J., Bronshtein I., Garini Y., Weron A., Universal algorithm for identification of fractional Brownian motion. A case of telomere subdiffusion, Biophys. J. 103 (2012), 1839–1847.
11. Burnecki K., Magdziarz M., Weron A., Identification and validation of fractional subdiffusion dynamics, in: Klafter J., Lim S.C., Metzler R. (eds.), Fractional Dynamics. Recent Advances, World Scientific, Singapore, 2012, pp. 331–351.
12. Burnecki K., Muszkieta M., Sikora G., Weron A., Statistical modelling of subdiffusive dynamics in the cytoplasm of living cells: A FARIMA approach, EPL 98 (2012), 10004, 6 pp.
13. Burnecki K., Sikora G., Estimation of FARIMA parameters in the case of negative memory and stable noise, IEEE Trans. Signal Process. 61 (2013), 2825–2835.
14. Burnecki K., Sikora G.,Weron A., Fractional process as a unified model for subdiffusive dynamics in experimental data, Phys. Rev. E 86 (2012), 041912, 8 pp.
15. Burnecki K.,Weron A., Fractional Lévy stable motion can model subdiffusive dynamics, Phys. Rev. E 82 (2010), 021130, 8 pp.
16. Burnecki K.,Weron A., Algorithms for testing of fractional dynamics: a practical guide to ARFIMA modelling, J. Stat. Mech. 2014, P10036, 26 pp.
17. Burnecki K., Wyłomańska A., Beletskii A., Gonchar V., Chechkin A., Recognition of stable distribution with Lévy index α close to 2, Phys. Rev. E 85 (2012), 056711, 8 pp.
18. Caccia D.C., Percival D., Cannon M.J., Raymond G., Bassingthwaighte J.B., Analyzing exact fractal time series: evaluating dispersional analysis and rescaled range methods, Physica A 246 (1997), 609–632.
19. Cambanis S., Podgórski K.,Weron A., Chaotic behavior of infinitely divisible processes, Studia Math. 115 (1995), 109–127.
20. Chan T.F., Vese L.A., Active contours without edges, IEEE Trans. Image Process. 10 (2001), 266–277.
21. Chang T., Sauer T., Schiff S.J., Tests for nonlinearity in short stationary time series, Chaos 5 (1995), 118–126.
22. Clark S., The dark side of the Sun, Nature 441 (2006), 402–404.
23. Crato N., Rothman P., Fractional integration analysis of long-run behavior for US macroeconomic time series, Econom. Lett. 45 (1994), 287–291.
24. Davies R.B., Harte D.S., Tests for Hurst effect, Biometrika 74 (1987), 95–101.
25. Fleck L., Kowarzyk H., Steinhaus H., La distribution des leucocytes dans les préparations du sang, J. Suisse de Médecine 77 (1947), 1283.
26. Fouskitakis G.N., Fassois S.D., Pseudolinear estimation of fractionally integrated ARMA (ARFIMA) models with automotive application, IEEE Trans. Signal Process. 47 (1999), 3365–3380.
27. Fuliński A., Communication: How to generate and measure anomalous diffusion in simple systems, J. Chem. Phys. 138 (2013), 021101, 4 pp.
28. Gajda J., Sikora G., Wyłomańska A., Regime variance testing – a quantile approach, Acta Phys. Pol. B 44 (2013), 1015–1035.
29. Gil-Alana L., A fractionally integrated model for the Spanish real GDP, Econ. Bull. 3 (2004), 1–6.
30. Golding I., Cox E.C., Physical nature of bacterial cytoplasm, Phys. Rev. Lett. 96 (2006), 098102, 4 pp.
31. Granger C.W.J., Joyeux R., An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal. 1 (1980), 15–29.
32. Guigas G., Kalla C., Weiss M., Probing the nanoscale viscoelasticity of intracellular fluids in living cells, Biophys. J. 93 (2007), 316–323.
33. Havlin S., Ben-Avraham D., Diffusion in disordered media, Adv. Phys. 36 (1987), 695–798.
34. Hellmann M., Klafter J., Heermann D.W., Weiss M., Challenges in determining anomalous diffusion in crowded fluids, J. Phys.: Condens. Matter 23 (2011), 234113, 7 pp.
35. Hoffmann-Jørgensen J., Stable densities, Theory Probab. Appl. 38 (1993), 350–355.
36. Höfling F., Franosch T., Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76 (2013), 046602, 50 pp.
37. Hosking J.R.M., Fractional differencing, Biometrika 68 (1981), 165–176.
38. Janicki A., Weron A., Simulation and Chaotic Behavior of α-stable Stochastic Processes, Marcel Dekker Inc., New York, 1994.
39. Kehr K.W., Kutner R., Random walk on a random walk, Physica A 110 (1982), 535–549.
40. Kokoszka P.S., Prediction of infinite variance fractional ARIMA, Probab. Math. Statist. 16 (1996), 65–83.
41. Kokoszka P.S., Taqqu M.S., Fractional ARIMA with stable innovations, Stochastic Process. Appl. 60 (1995), 19–47.
42. Kokoszka P.S., Taqqu M.S., Parameter estimation for infinite variance fractional ARIMA, Ann. Statist. 24 (1996), 1880–1913.
43. Kolmogorov A.N., Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum, C. R. (Doklady) Acad. Sci. URSS (N.S.) 26 (1940), 115–118.
44. Kou S.C., Xie X.S., Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. Rev. Lett. 93 (2004), 180603, 4 pp.
45. Kowarzyk H., Steinhaus H., Szymaniec S., Arrangement of chromosomes in human cells. I. Associations in metaphase figures, Bull. Acad. Polon. Sci., Ser. Sci. Biol. 13 (1965), 321–326.
46. Kowarzyk H., Steinhaus H., Szymaniec S., Arrangement of chromosomes in human cells. 3. Distribution of centromeres and orientation of chromosomes in the metaphase, Bull. Acad. Polon. Sci., Ser. Sci. Biol. 14 (1966), 541–544.
47. Lagarias J.C., Reeds J.A., Wright M.H., Wright P.E., Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM J. Optim. 9 (1998), 112–147.
48. Lanoiselée Y., Grebenkov D.S., Revealing nonergodic dynamics in living cells from a single particle trajectory, Phys. Rev. E 93 (2016), 052146, 17 pp.
49. Lasota A., Mackey M.C., Chaos, Fractals, and Noise. Stochastic Aspects of Dynamics, Springer-Verlag, New York, 1994.
50. Lasota A., Mackey M.C.,Ważewska-Czyżewska M., Minimizing therapeutically induced anemia, J. Math. Biol. 13 (1981/82), 149–158.
51. Lim S.C., Muniandy S.V., Self-similar Gaussian processes for modeling anomalous diffusion, Phys. Rev. E 66 (2002), 021114, 14 pp.
52. Ljung G.M., Box G.E.P., On a measure of lack of fit in time series models, Biometrika 65 (1978), 297–303.
53. Loch H., Janczura A., Weron A., Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes, Phys. Rev. E 93 (2016), 043317, 10 pp.
54. Loch-Olszewska H., Sikora G., Janczura J., Weron A., Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length, Phys. Rev. E 94 (2016), 052136, 8 pp.
55. Magdziarz M., Weron A., Fractional Langevin equation with α-stable noise. A link to fractional ARIMA time series, Studia Math. 181 (2007), 47–60.
56. Magdziarz M., Weron A., Anomalous diffusion: Testing ergodicity breaking in experimental data, Phys. Rev. E 84 (2011), 051138, 5 pp.
57. Magdziarz M., Weron A., Burnecki K., Klafter J., Fractional Brownian motion versus the continuous-time random walk: a simple test for subdiffusive dynamics, Phys. Rev. Lett. 103 (2009), 180602, 4 pp.
58. Mandelbrot B.B., Wallis J.R., Noah, Joseph, and operational hydrology, Water Resour. Res. 4 (1968), 909–918.
59. Mann M.E., Bradley R.S., Hughes M.K., Global-scale temperature patterns and climate forcing over the past six centuries, Nature 392 (1998), 779–787.
60. Meroz Y., Sokolov I.M., Klafter J., Test for determining a subdiffusive model in ergodic systems from single trajectories, Phys. Rev. Lett. 110 (2013), 090601, 4 pp.
61. Metzler R., Barkai E., Klafter J., Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach, Phys. Rev. Lett. 82 (1999), 3563–3567.
62. Metzler R., Klafter J., The random walk’s guide to anomalous difusion: a fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.
63. Nolan J.P., Numerical calculation of stable densities and distribution functions, Commun. Statist.-Stochastic Models 13 (1997), 759–774.
64. Samko S.G., Kilbas A.A., Marichev O.I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Amsterdam, 1993.
65. Saxton M.J., Anomalous diffusion due to obstacles: a Monte Carlo study, Biophys. J. 66 (1994), 394–401.
66. Saxton M.J., Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study, Biophys. J. 81 (2001), 2226–2240.
67. Saxton M.J., Wanted: a positive control for anomalous subdiffusion, Biophys. J. 103 (2012), 2411–2422.
68. Schneider W.R., Stable distributions: Fox functions representation and generalization, in: Albeverio S., Casati G., Merlini D. (eds.), Stochastic Processes in Classical and Quantum Systems, Springer, Berlin, 1986, pp. 497–511.
69. Ślęzak J., Drobczyński D., Weron K., Masajada J., Moving average process underlying the holographic-optical-tweezers experiments, Appl. Opt. 53 (2014), B254–B258.
70. Sokolov I.M., Models of anomalous diffusion in crowded environments, Soft Matter 8 (2012), 9043–9052.
71. Stanislavsky A., Memory effects and macroscopic manifestation of randomness, Phys. Rev. E 61 (2000), 4752–4759.
72. Stanislavsky A.A., Burnecki K., Magdziarz M., Weron A., Weron K., FARIMA modeling of solar flare activity from empirical time series of soft X-ray solar emission, Astrophys. J. 693 (2009), 1877–1882.
73. Stanislavsky A., Weron K., Weron A., Anomalous diffusion with transient subordinators: a link to compound relaxation laws, J. Chem. Phys. 140 (2014), 054113, 7 pp.
74. Stoev S., Taqqu M.S., Simulation methods for linear fractional stable motion and FARIMA using the fast Fourier transform, Fractals 12 (2004), 95–121.
75. Weron A., Burnecki K., Mercik S., Weron K., Complete description of all self-similar models driven by Lévy stable noise, Phys. Rev. E 71 (2005), 016113, 10 pp.
76. Weron A., Magdziarz M., Anomalous diffusion and semimartingales, EPL 86 (2009), 60010, 6 pp.
77. Weron A., Weron R., Computer simulation of Lévy α-stable variables and processes, in: Garbaczewski P., Wolf M., Weron A. (eds.), Chaos – the Interplay Between Stochastic and Deterministic Behaviour, Springer, Berlin, 1995, pp. 379–392.
78. Weron R., Levy-stable distributions revisited: tail index > 2 does not exclude the Levystable regime, Int. J. Mod. Phys. C 12 (2001), 209–223.
79. Weron R., Computationally intensive value at risk calculations, in: Gentle J.E., Härdle W., Mori Y. (eds.), Handbook of Computational Statistics. Concepts and Methods, Springer, Berlin, 2004, pp. 911–950.
80. Wigner E.P., The unreasonable effectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math. 13 (1960), 1–14.
WeronA. (2018). Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach. Annales Mathematicae Silesianae, 32, 5-41. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/13911
Aleksander Weron
aleksander.weron@pwr.edu.pl
Centrum im. Hugona Steinhausa, Wydział Matematyki, Politechnika Wrocławska Poland
Centrum im. Hugona Steinhausa, Wydział Matematyki, Politechnika Wrocławska Poland
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