Motion of chromosomal loci and the mean-squared displacement of a fractional Brownian motion in the presence of static and dynamic errors

2015 ◽  
Author(s):  
Mikael P. Backlund ◽  
W. E. Moerner
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Mean Squared Displacement ◽  
Dynamic Errors ◽  
The Mean ◽  
Chromosomal Loci

ASYMPTOTIC BEHAVIOUR OF A CLASS OF RESOURCE COMPETITION BIOLOGY SPECIES SYSTEM BY THE FRACTIONAL BROWNIAN MOTION

2017 ◽  
Vol 58 (3-4) ◽  
pp. 491-499
Author(s):  
Q. ZHANG ◽  
M. YE ◽  
H. LEI ◽  
Q. JIN
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Fractional Brownian Motion ◽  
Lyapunov Functional ◽  
Resource Competition ◽  
Mortality Rates ◽  
Mean Square ◽  
Chemostat Model ◽  
Extrinsic Noise ◽  
The Mean

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with $n$ species and $k$ resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result.

scholarly journals Random sub-diffusion and capture of genes by the nuclear pore reduces dynamics and coordinates interchromosomal movement

2021 ◽  
Author(s):  
Michael Chas Sumner ◽  
Steven B. Torrisi ◽  
Donna Garvey Brickner ◽  
Jason H. Brickner
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Nuclear Pore Complex ◽  
Nuclear Periphery ◽  
Experimental Models ◽  
Nuclear Pore ◽  
Dynamic Tracking ◽  
Diffusive Movement ◽  
Chromosomal Loci

ABSTRACTHundreds of genes interact with the yeast nuclear pore complex (NPC), localizing at the nuclear periphery and clustering with co-regulated genes. Dynamic tracking of peripheral genes shows that they cycle on and off the NPC and that interaction with the NPC slows their sub-diffusive movement. Furthermore, NPC-dependent inter-chromosomal clustering leads to coordinated movement of pairs of loci separated by hundreds of nanometers. We developed Fractional Brownian Motion simulations for chromosomal loci in the nucleoplasm and interacting with NPCs. These simulations predict the rate and nature of random sub-diffusion during repositioning from nucleoplasm to periphery and match measurements from two different experimental models, arguing that recruitment to the nuclear periphery is due to random subdiffusion, collision, and capture by NPCs. Finally, the simulations do not lead to inter-chromosomal clustering or coordinated movement, suggesting that interaction with the NPC is necessary, but not sufficient, to cause clustering.

Mean-squared-displacement statistical test for fractional Brownian motion

2017 ◽  
Vol 95 (3) ◽  
Author(s):  
Grzegorz Sikora ◽  
Krzysztof Burnecki ◽  
Agnieszka Wyłomańska
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Statistical Test ◽  
Mean Squared Displacement

scholarly journals Linear filtering with fractional Brownian motion in the signal and observation processes

1999 ◽  
Vol 12 (1) ◽  
pp. 85-90 ◽  
Author(s):  
M. L. Kleptsyna ◽  
P. E. Kloeden ◽  
V. V. Anh
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Wiener Process ◽  
Integral Equations ◽  
Hurst Index ◽  
Mean Square ◽  
Linear Filtering ◽  
Observation Process ◽  
The Mean ◽  
Filtering Problem

Integral equations for the mean-square estimate are obtained for the linear filtering problem, in which the noise generating the signal is a fractional Brownian motion with Hurst index h∈(3/4,1) and the noise in the observation process includes a fractional Brownian motion as well as a Wiener process.

Numerics for the fractional Langevin equation driven by the fractional Brownian motion

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peng Guo ◽  
Caibin Zeng ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Order ◽  
Langevin Equation ◽  
Mean Square Displacement ◽  
Hurst Parameter ◽  
Mean Square ◽  
Fractional Langevin Equation ◽  
Normal Diffusion ◽  
The Mean

AbstractWe study analytically and numerically the fractional Langevin equation driven by the fractional Brownian motion. The fractional derivative is in Caputo’s sense and the fractional order in this paper is α = 2 − 2H, where H ∈ ($\tfrac{1} {2} $, 1) is the Hurst parameter (or, index). We give numerical schemes for the fractional Langevin equation with or without an external force. From the figures we can find that the mean square displacement of the fractional Langevin equation has the property of the anomalous diffusion. When the fractional order tends to an integer, the diffusion reduces to the normal diffusion.

scholarly journals Bounds for expected supremum of fractional Brownian motion with drift

2021 ◽  
Vol 58 (2) ◽  
pp. 411-427
Author(s):  
Krzysztof Bisewski ◽  
Krzysztof Dębicki ◽  
Michel Mandjes
Keyword(s):  
Brownian Motion ◽  
Lower Bound ◽  
Fractional Brownian Motion ◽  
Upper Bound ◽  
Numerical Procedure ◽  
Upper And Lower Bounds ◽  
Brownian Motion With Drift ◽  
Similar Shape ◽  
The Mean ◽  
Mean Variance

AbstractWe provide upper and lower bounds for the mean $\mathscr{M}(H)$ of $\sup_{t\geq 0} \{B_H(t) - t\}$ , with $B_H(\!\cdot\!)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$ . We find bounds in (semi-) closed form, distinguishing between $H\in(0,\frac{1}{2}]$ and $H\in[\frac{1}{2},1)$ , where in the former regime a numerical procedure is presented that drastically reduces the upper bound. For $H\in(0,\frac{1}{2}]$ , the ratio between the upper and lower bound is bounded, whereas for $H\in[\frac{1}{2},1)$ the derived upper and lower bound have a strongly similar shape. We also derive a new upper bound for the mean of $\sup_{t\in[0,1]} B_H(t)$ , $H\in(0,\frac{1}{2}]$ , which is tight around $H=\frac{1}{2}$ .

scholarly journals Random sub-diffusion and capture of genes by the nuclear pore reduces dynamics and coordinates interchromosomal movement

eLife ◽  
2021 ◽  
Vol 10 ◽  
Author(s):  
Michael Chas Sumner ◽  
Steven B Torrisi ◽  
Donna G Brickner ◽  
Jason H Brickner
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Nuclear Pore Complex ◽  
Nuclear Periphery ◽  
Experimental Models ◽  
Nuclear Pore ◽  
Dynamic Tracking ◽  
Diffusive Movement ◽  
Chromosomal Loci

Hundreds of genes interact with the yeast nuclear pore complex (NPC), localizing at the nuclear periphery and clustering with co-regulated genes. Dynamic tracking of peripheral genes shows that they cycle on and off the NPC and that interaction with the NPC slows their sub-diffusive movement. Furthermore, NPC-dependent inter-chromosomal clustering leads to coordinated movement of pairs of loci separated by hundreds of nanometers. We developed Fractional Brownian Motion simulations for chromosomal loci in the nucleoplasm and interacting with NPCs. These simulations predict the rate and nature of random sub-diffusion during repositioning from nucleoplasm to periphery and match measurements from two different experimental models, arguing that recruitment to the nuclear periphery is due to random sub-diffusion and transient capture by NPCs. Finally, the simulations do not lead to inter-chromosomal clustering or coordinated movement, suggesting that interaction with the NPC is necessary, but not sufficient, to cause clustering.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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