Stochastic pattern formation and spontaneous polarisation: The linear noise approximation and beyond

Alan J. McKane, Tommaso Biancalani, Tim Rogers

Research output: Contribution to journalArticle

39 Citations (Scopus)

Abstract

We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.
LanguageEnglish
Pages895-921
JournalBulletin of Mathematical Biology
Volume76
Issue number4
Early online date8 Mar 2013
DOIs
StatusPublished - Apr 2014

Fingerprint

Turing Instability
Stochasticity
Polarity
Biological systems
Pattern Formation
Biological Systems
Amplification
Noise
Governing equation
Time Scales
Polarization
polarization
Cell Polarity
Generalise
Cell
stochasticity
Approximation
Modeling
amplification
timescale

Cite this

Stochastic pattern formation and spontaneous polarisation : The linear noise approximation and beyond. / McKane, Alan J.; Biancalani, Tommaso; Rogers, Tim.

In: Bulletin of Mathematical Biology, Vol. 76, No. 4, 04.2014, p. 895-921.

Research output: Contribution to journalArticle

@article{7a5556bb734c4eb99c916e40407a9b7c,
title = "Stochastic pattern formation and spontaneous polarisation: The linear noise approximation and beyond",
abstract = "We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.",
author = "McKane, {Alan J.} and Tommaso Biancalani and Tim Rogers",
year = "2014",
month = "4",
doi = "10.1007/s11538-013-9827-4",
language = "English",
volume = "76",
pages = "895--921",
journal = "Bulletin of Mathematical Biology",
issn = "0092-8240",
publisher = "Springer New York",
number = "4",

}

TY - JOUR

T1 - Stochastic pattern formation and spontaneous polarisation

T2 - Bulletin of Mathematical Biology

AU - McKane, Alan J.

AU - Biancalani, Tommaso

AU - Rogers, Tim

PY - 2014/4

Y1 - 2014/4

N2 - We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

AB - We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.

UR - http://www.scopus.com/inward/record.url?scp=84874641364&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1007/s11538-013-9827-4

UR - http://arxiv.org/abs/1211.0462

U2 - 10.1007/s11538-013-9827-4

DO - 10.1007/s11538-013-9827-4

M3 - Article

VL - 76

SP - 895

EP - 921

JO - Bulletin of Mathematical Biology

JF - Bulletin of Mathematical Biology

SN - 0092-8240

IS - 4

ER -