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Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems

Thomas E. Woolley1,2,*, Ruth E. Baker1, Eamonn A. Gaffney1, Philip K. Maini1,3, and Sungrim Seirin-Lee4

  • 1Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, United Kingdom
  • 2Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, United Kingdom
  • 3Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, South Parks Road, Oxford, OX1 3QU, United Kingdom
  • 4Center for Developmental Biology, RIKEN, Minatojima-Minamimachi 2-2-3, Kobe 650-0047, Japan

  • *woolley@maths.ox.ac.uk
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Phys. Rev. E 85, 051914 – Published 22 May, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.051914

Abstract

Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases.

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Phys. Rev. E 85, 051914– Published 22 May, 2012
  • Received 26 February 2012

©2012 American Physical Society

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