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Mathematical Biosciences and Engineering

2016, Volume 13, Issue 5: 981-998. doi: 10.3934/mbe.2016026
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Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module

  • 1.
    Institute of Mathematics and Statistics, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-090, São Paulo
  • Received: 01 April 2015 Accepted: 29 June 2018 Published: 01 July 2016

  • MSC : Primary: 82C22, 60J75; Secondary: 37N25.

  • We study an alternative approach to model the dynamical behaviors of biological feedback loop, that is, a type-dependent spin system, this class of stochastic models was introduced by Fernández et. al [13], and are useful since take account to inherent variability of gene expression.We analyze a non-symmetric feedback module being an extension for the repressilator, the first synthetic biological oscillator, invented by Elowitz and Leibler [7]. We consider a mean-field dynamics for a type-dependent Ising model, and then study the empirical-magnetization vector representing concentration of molecules. We apply a convergence result from stochastic jump processes to deterministic trajectories and present a bifurcation analysis for the associated dynamical system. We show that non-symmetric module under study can exhibit very rich behaviours, including the empirical oscillations described by repressilator.

    Citation: Manuel González-Navarrete. Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module[J]. Mathematical Biosciences and Engineering, 2016, 13(5): 981-998. doi: 10.3934/mbe.2016026

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  • We study an alternative approach to model the dynamical behaviors of biological feedback loop, that is, a type-dependent spin system, this class of stochastic models was introduced by Fernández et. al [13], and are useful since take account to inherent variability of gene expression.We analyze a non-symmetric feedback module being an extension for the repressilator, the first synthetic biological oscillator, invented by Elowitz and Leibler [7]. We consider a mean-field dynamics for a type-dependent Ising model, and then study the empirical-magnetization vector representing concentration of molecules. We apply a convergence result from stochastic jump processes to deterministic trajectories and present a bifurcation analysis for the associated dynamical system. We show that non-symmetric module under study can exhibit very rich behaviours, including the empirical oscillations described by repressilator.


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