Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach
Journal Title: Annales Mathematicae Silesianae - Year 2018, Vol 32, Issue
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.
Authors and Affiliations
Aleksander Weron
Keywords
Related Articles
- EP ID EP524574
- DOI 10.1515/amsil-2017-0019
- Views 143
- Downloads 0
Numerical solution of time fractional Schrödinger equation by using quadratic B-spline finite elements
In this article, quadratic B-spline Galerkin method has been employed to solve the time fractional order Schrödinger equation. Numerical solutions and error norms $L_2$ and $L_∞$ are presented in tables.
Some new Ostrowski’s inequalities for functions whose nth derivatives are logarithmically convex
Some new Ostrowski’s inequalities for functions whose n-th derivative are logarithmically convex are established.
On Popoviciu-Ionescu functional equation
We study a functional equation first proposed by T. Popoviciu [15] in 1955. It was solved for the easiest case by Ionescu [9] in 1956 and, for the general case, by Ghiorcoiasiu and Roscau [7] and Radó [17] in 1962. Our s...
Complementary results to Heuvers’s characterization of logarithmic functions
Based on a characterization of logarithmic functions due to Heuvers we develop analogous results for multiplicative, exponential and additive functions, respectively.
Mixed type of additive and quintic functional equations
In this paper, we investigate the general solution and Hyers–Ulam–Rassias stability of a new mixed type of additive and quintic functional equation of the form $$f(3x + y) - 5f(2x + y) + f(2x - y) + 10f(x + y) - 5f(x - y...