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267778 Efficient Generation of Moment-Equations for Stochastic Simulation and Moment-Closure

The use of stochastic mathematics has become common in chemical reaction simulation, especially for biological models, but current methods have severe computational drawbacks. These probability-based approaches are necessary when a system exists far from the thermodynamic limit, a common condition in biological systems. Unlike deterministic simulations, however, the most computationally effective way to produce simulation results is a kinetic Monte Carlo method which samples the underlying Chemical Master Equation (CME) dynamics. Unfortunately, such a method makes complete system analysis computationally expensive.

An alternative proposed by McQuarrie and others in the 1960s was a probability moment-based approach that reduces the CME to a set of linear ordinary differential equations (ODEs).^{1}While this transform is effective in limited cases, such a reduction results in an infinite set of linear ODEs, or equivalently an infinite dimensional matrix, when 2nd-order reactions are involved. Despite this drawback the production of analytical expressions for the time-derivative of moments for an arbitrary chemical network is an active area of research. While these analytical expressions are useful for simple system, there is a need for a streamlined algorithmic approach when considering complex systems and the potential for moment-closure.

In the case where a moment-closure scheme can be applied it is obvious that all moments up to an order-M (for an M+1 moment closure) will be needed. Analytical approaches waste computation time in producing this potentially massive generate matrix. Additionally, the matrix is nearly lower-diagonal in nature and can require enormous memory allocation in the case of complex systems. The work presented focuses on the efficient production of all moment equations up to an order-M. The approach is based on applying differentials to the Z-transform (discrete Laplace transform) of the CME (Z-CME). It is shown that in all cases the modified basis set will conserve memory storage and produce a banded matrix rather than a nearly-lower diagonal matrix as is found with polynomial moments.

The advantages of this novel and efficient algorithm are that: 1) It is regressive in nature such that all moment equations are produced with little wasted computation; 2) It produces a banded matrix that necessitates the allocation of relatively little memory. A discussion of current work concerning moment-closure of arbitrary chemical networks, including the cumulant-zero techniques and System-Size Expansion, will be provided. Promising future paths will be presented in the context of applications utilizing the Z-CME moment generating algorithm.

^{1} D. A. McQuarrie, C. J. Jachimowski, and M. E. Russell, Journal of Chemical Physics 40 (1964).

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