Journal of Mathematical Biology

, Volume 71, Issue 4, pp 921–959 | Cite as

Mean field analysis of a spatial stochastic model of a gene regulatory network

  • M. SturrockEmail author
  • P. J. Murray
  • A. Matzavinos
  • M. A. J. Chaplain


A gene regulatory network may be defined as a collection of DNA segments which interact with each other indirectly through their RNA and protein products. Such a network is said to contain a negative feedback loop if its products inhibit gene transcription, and a positive feedback loop if a gene product promotes its own production. Negative feedback loops can create oscillations in mRNA and protein levels while positive feedback loops are primarily responsible for signal amplification. It is often the case in real biological systems that both negative and positive feedback loops operate in parameter regimes that result in low copy numbers of gene products. In this paper we investigate the spatio-temporal dynamics of a single feedback loop in a eukaryotic cell. We first develop a simplified spatial stochastic model of a canonical feedback system (either positive or negative). Using a Gillespie’s algorithm, we compute sample trajectories and analyse their corresponding statistics. We then derive a system of equations that describe the spatio-temporal evolution of the stochastic means. Subsequently, we examine the spatially homogeneous case and compare the results of numerical simulations with the spatially explicit case. Finally, using a combination of steady-state analysis and data clustering techniques, we explore model behaviour across a subregion of the parameter space that is difficult to access experimentally and compare the parameter landscape of our spatio-temporal and spatially-homogeneous models.


Gene regulatory network Feedback loop Spatial stochastic model Mean field Data clustering 

Mathematics Subject Classification




MS would like to thank the support from Mathematical Biosciences Institute at The Ohio State University and NSF grant DMS0931642. MAJC gratefully acknowledges the support of the ERC Advanced Investigator Grant no. 227619, “M5CGS: From Mutations to Metastases: Multiscale Mathematical Modelling of Cancer Growth and Spread”.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • M. Sturrock
    • 1
    Email author
  • P. J. Murray
    • 2
  • A. Matzavinos
    • 3
  • M. A. J. Chaplain
    • 2
  1. 1.Mathematical Biosciences InstituteThe Ohio State UniversityColumbusUSA
  2. 2.Division of MathematicsUniversity of DundeeDundeeUK
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

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