 Author
 Year
 2015
 Title
 Solution of the chemical master equation by radial basis functions approximation with interface tracking
 Journal
 BMC Systems Biology
 Volume
 9
 Article number
 67
 Number of pages
 17
 Document type
 Article
 Faculty
 Faculty of Science (FNWI)
 Institute
 Van 't Hoff Institute for Molecular Sciences (HIMS)
 Abstract

Background: The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differentialdifference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is highdimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.
Results: We introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multimodal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way
that the solution is approximated only on the essential support instead of the whole state space.
Discussion: In order to demonstrate the effectiveness of the method, we consider four application examples: a) the selfregulating gene model, b) the 2dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3dimensional tristable problem, and d) a 3dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.
Conclusions: The proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multidimensional distributions is recovered. The method is especially attractive when dealing with models that yield solutions of a complex structure, for instance, featuring multistability.
 URL
 go to publisher's site
 Language
 English
 Note
 With additional flles
 Permalink
 http://hdl.handle.net/11245/1.512805
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