In this dissertation, we construct a general method of multiscale approximations in chemical reaction networks. We apply a continuous time Markov jump process to describe the state of the chemical reactions.
In general chemical reactions, the chemical species numbers and the chemical reaction rate constants will have various orders of magnitude. Therefore, we introduce two different scaling exponents to normalize the numbers of molecules of the chemical species and to scale the chemical reaction rate constants. Applying a time change, we have different time scales for the limiting processes in the reduced subsystems.
A systematic way to select the scaling exponents is suggested to make the normalized system have a nonzero finite limit. This method involves balance equations with the scaling exponents, which we call species and subnetwork balance conditions.
We investigate asymptotic methods used in multiscale approximations. The law of large numbers for Poisson processes is applied to approximate non-integer-valued processes. In each time scale, the slow processes act as constant and the fast processes are averaged out. Then the limit of the intermediate processes is obtained in terms of the averaged fast processes and the initial values of the slow processes.
We introduce a model of the heat shock response and apply the general method of multiscale approximations to this model. We analyze the system and obtain limiting processes in each simplified subsystem, which approximates the normalized processes in the system with different time scales. We obtain error estimates of the difference between the normalized processes and the limiting processes. Simulation results are given to compare the evolution of the processes in the system and the evolution of the approximated processes using the limiting processes in each simplified subsystem.
Applying the martingale central limit theorem and using the averaging, we obtain a central limit theorem for deviation of the normalized processes from their limiting processes in the three species model and in the heat shock response model.
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