Power spectra methods for a stochastic description of diffusion on deterministically growing domains

Thomas E. Woolley, Ruth E. Baker, Eamonn A. Gaffney, and Philip K. Maini
Phys. Rev. E 84, 021915 – Published 10 August 2011

Abstract

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state.

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  • Received 6 April 2011

DOI:https://doi.org/10.1103/PhysRevE.84.021915

©2011 American Physical Society

Authors & Affiliations

Thomas E. Woolley1,*, Ruth E. Baker1, Eamonn A. Gaffney1, and Philip K. Maini1,2

  • 1Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom
  • 2Department of Biochemistry, Oxford Centre for Integrative Systems Biology, University of Oxford, Oxford, OX1 3QU, United Kingdom

  • *woolley@maths.ox.ac.uk

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Vol. 84, Iss. 2 — August 2011

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