Tensors and Stochastic Automata Networks with Application to Chemical Kinetics
Often, given a system of biochemical reactions, it is useful to be able to predict the system’s future state from the initial quantities of the involved molecules. There are many methodologies for developing such predictions, ranging from simple approaches such as Monte Carlo simulations to more sophisticated higher-order tensors and stochastic automata networks. Many revolve around solving the chemical master equation that arises in the modeling of the underlying biochemical kinetics. Traditionally, the chemical master equation models states consisting of the population counts present of each molecular species. This work considers the case of dealing with the resulting high dimensional data and shows how tensor representations allow us to cope with the “curse of dimensionality” that significantly complicates such problems. One key outcome in this work is the demonstration of the inherent differences and similarities between two prominent modeling methods, by computational examples on one hand and a mathematical proof on the other hand. The second key outcome is the development of a tensor-based representation of the chemical master equation modeling states as the number of times a given reaction has fired. This means that states are reaction counts as opposed to the traditional population counts. The tensor-based solutions considered in this thesis may have applications in dealing with many other high dimensional data outside of strictly chemical reaction systems.
Chemical Master Equation, Stochastic Automata, Tensor Train Decomposition, Tensors
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