A solver for the stochastic master equation applied to gene regulatory networks

  • Authors:

  • Markus Hegland;Conrad Burden;Lucia Santoso;Shev MacNamara;Hilary Booth

  • Affiliations:

  • Centre for Mathematics and Its Application, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia and ARC Centre in Bioinformatics, The University of Queen ...;Centre for Bioinformation Science, John Curtin School of Medical Research & Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia;ARC Centre in Bioinformatics, The University of Queensland, St Lucia, Qld 4072, Australia and Centre for Bioinformation Science, Mathematical Sciences Institute, Australian National University, Ca ...;Centre for Mathematics and Its Application, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia and ARC Centre in Bioinformatics, The University of Queen ...;-

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2007

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Abstract

An important driver of gene regulatory networks is noise arising from the stochastic nature of interactions of genes, their products and regulators. Thus, such systems are stochastic and can be modelled by the chemical master equations. A major challenge is the curse of dimensionality which occurs when one attempts to integrate these equations. While stochastic simulation techniques effectively address the curse, many repeated simulations are required to provide precise information about stationary points, bifurcation phenomena and other properties of the stochastic processes. An alternative way to address the curse of dimensionality is provided by sparse grid approximations. The sparse grid methodology is applied and the application demonstrated to work efficiently for up to 10 proteins. As sparse grid methods have been developed for the approximation of smooth functions, a variant for infinite sequences had to be developed together with a multiresolution analysis similar to Haar wavelets. Error bounds are provided which confirm the effectiveness of sparse grid approximations for smooth high-dimensional probability distributions.