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Bulletin of Mathematical Biology

, Volume 76, Issue 4, pp 895–921 | Cite as

Stochastic Pattern Formation and Spontaneous Polarisation: The Linear Noise Approximation and Beyond

  • Alan J. McKane
  • Tommaso Biancalani
  • Tim Rogers
Original Article

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.

Keywords

Stochastic models and the master equation Stochastic patterns and cell polarity 

Notes

Acknowledgements

This work was supported in part under EPSRC Grant No. EP/H02171X/1 (A.J.M. and T.R.). T.B. also wishes to thank the EPSRC for partial support.

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Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  • Alan J. McKane
    • 1
  • Tommaso Biancalani
    • 1
  • Tim Rogers
    • 1
    • 2
  1. 1.Theoretical Physics Division, School of Physics and AstronomyUniversity of ManchesterManchesterUK
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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