Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids
An important driver of the dynamics of gene regulatory networks is noise generated by transcription and translation processes involving genes and their products. As relatively small numbers of copies of each substrate are involved, such systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distributions for the states defined by the vectors of copy numbers of each substance. Challenges are posed by the large discrete state spaces, and are mainly due to high dimensionality.
In order to address this challenge we propose effective approximation techniques, and, in particular, numerical techniques to solve the master equations. Two theoretical results show that the numerical methods are optimal. The techniques are combined with sparse grids to give an effective method to solve high-dimensional problems.
KeywordsMaster Equation Gene Regulatory Network Sparse Grid Promoter Site Chemical Master Equation
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