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

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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.

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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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Authors and Affiliations

  1. Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK

    Alan J. McKane, Tommaso Biancalani & Tim Rogers

  2. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Tim Rogers

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  1. Alan J. McKane
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  2. Tommaso Biancalani
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  3. Tim Rogers
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Correspondence to Alan J. McKane.

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McKane, A.J., Biancalani, T. & Rogers, T. Stochastic Pattern Formation and Spontaneous Polarisation: The Linear Noise Approximation and Beyond. Bull Math Biol 76, 895–921 (2014). https://doi.org/10.1007/s11538-013-9827-4

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