The Generating Function Approach To The Biochemical Reaction Systems And The Stochastic Hodgkin-huxley Neuron

Different kinds of life activities are governed by elegant and complex chemical reaction networks.Modeling the reaction dynamic behaviors of biosystem is an important way to understand life activities.The building block of life is cell.The traditional deterministic approach based on the reaction rate equations is no longer valid to the cellsize systems.Characterizing dynamic processes of the stochastic reaction systems accurately has been one of present research focuses.Information is encoded into the action potentials and transfers in the neuronal network.Generation and propagation of action potentials are governed by currents through ion channels on the neuronal membrane.The HH model characterizes the evolution of action potential.Nevertheless,the further modifications are needed due to the channel noise and the synaptic noise.The stochastic kinetics of biochemical reaction systems and the ion channels for HH neuron could be defined as Markov processes,which can be completely characterized by a master equation,describing the evolution of probabilities.The Gillespie algorithm is one common used Monte Carlo technique for simulating Markov processes exactly.Nonetheless,for most complex systems,the computation is overloaded by Gillespie algorithm.The thesis introduces the generating function approach,which recasts the master equation into a QFT fomulation in which the probability evolution is governed by a“wave equation”,and the field theoretic formulation is equivalent to a generating function approach.We apply it to the nonlinear binary reaction systems and provide several trial solutions.The PDE is reduced into a small number of ODEs by applying a variational technique.The trial solutions lead to reults that overlap significantly better with the exact Gillespie calculation,for different sets of parameter values,and for both the small and large number of molecules.It is computationally more efficient than Gillespie calculation,for the reason that the computation only involves several ODEs,unrelated to the number of molecules.We apply the generation function approach to the stochastic simulation of action potentials.We propose two accelerating algorithms in which only the number of the open ion channels are sampled since the time evolution of the voltage only depends on this number.It is found that in addition to the very high computation effciency,the results match very well with the exact Gillespie computation at any channel number,which fails commonly used Langevin approaches.Henceforth,this finding may see interesting applications in studying noisy action potential propagating in neuronal network.