Approximating the solution of the chemical master equation by aggregation

Abstract

The chemical master equation is a continuous time discrete space Markov model of chemical reactions. The chemical master equation is derived mathematically and it is shown that the corresponding initial value problem has a unique solution. Conditions are given under which this solution is a probability distribution. We present finite state and aggregation-disaggregation approximations and provide error bounds for the case of piecewise constant disaggregation. The aggregation-disaggregation approximation allows the solution of the chemical master equation for larger state spaces and is also an important tool for the solution of multidimensional problems. References Adam Arkin, John Ross, and Harley H. McAdams. {Stochastic Kinetic Analysis of Developmental Pathway Bifurcation in Ph age lambda-Infected Escherichia coli Cells}. Genetics, 149(4):1633--1648, 1998. http://www.genetics.org/cgi/content/abstract/149/4/1633 P. Deuflhard, W. Huisinga, T. Jahnke, and M. Wulkow. Adaptive discrete Galerkin methods applied to the chemical master equation. Technical Report ZIB-Report 07-04, Konrad--Zuse Zentrum fuer Informationstechnik Berlin, 2007. http://www.zib.de/Publications/Reports/ZR-07-04.pdf Stefan Engblom. Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput., 180(2):498--515, 2006. doi:10.1016/j.amc.2005.12.032 Lars Ferm and Per Loetstedt. Numerical method for coupling the macro and meso scales in stochastic chemical kinetics. BIT Numerical Mathematics, 47:735--762, 2007. doi:10.1007/s10543-007-0150-z Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81(25):2340--2361, December 1977. Markus Hegland, Conrad Burden, Lucia Santoso, Shev MacNamara, and Hilary Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comput. Appl. Math., 205(2):708--724, 2007. doi:10.1016/j.cam.2006.02.053 Leo Huber and Markus Hegland. Dimension adaptive sparse grids for the chemical master equations. submitted, 2008. B. Munsky and M. Khammash. The finite state projection algorithm for the solution of the chemical master equation. The Journal of Chemical Physics, 124:044104--1--044104--1, 2006. doi:10.1063/1.2145882 Roger B. Sidje. EXPOKIT: Software package for computing matrix exponentials. ACM Transactions on Mathematical Software, 24(1):130--156, March 1998. doi:10.1145/285861.285868 T. Tian and K. Burrage. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage-$\lambda $. Journal of Theoretical Biology, 227:229--237, 2004. doi:10.1016/j.jtbi.2003.11.003 N. G. van Kampen. Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam, the Netherlands, 1981.