 
Summary: The finite state projection algorithm for the solution of the chemical
master equation
Brian Munskya
and Mustafa Khammashb
Mechanical and Environmental Engineering, University of CaliforniaSanta Barbara, Santa Barbara,
California 93106
Received 28 March 2005; accepted 7 November 2005; published online 25 January 2006
This article introduces the finite state projection FSP method for use in the stochastic analysis of
chemically reacting systems. One can describe the chemical populations of such systems with
probability density vectors that evolve according to a set of linear ordinary differential equations
known as the chemical master equation CME . Unlike Monte Carlo methods such as the stochastic
simulation algorithm SSA or leaping, the FSP directly solves or approximates the solution of the
CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP
method provides an exact analytical solution. When an infinite or extremely large number of
population variations is possible, the state space can be truncated, and the FSP method provides a
certificate of accuracy for how closely the truncated space approximation matches the true solution.
The proposed FSP algorithm systematically increases the projection space in order to meet
prespecified tolerance in the total probability density error. For any system in which a sufficiently
accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP
is utilized to solve two examples taken from the field of systems biology, and comparisons are made
