Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach

Aleksander Weron

Published: 2018-05-17

Vol. 32 (2018)

Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach

Aleksander Weron

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

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Weron, A. (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

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Vol. 32 (2018)
Published: 2018-08-24

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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