TY - JOUR
T1 - A closure scheme for chemical master equations
AU - Smadbeck, Patrick
AU - Kaznessis, Yiannis
PY - 2013/8/27
Y1 - 2013/8/27
N2 - Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higherorder moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kineticMonte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
AB - Probability reigns in biology, with random molecular events dictating the fate of individual organisms, and propelling populations of species through evolution. In principle, the master probability equation provides the most complete model of probabilistic behavior in biomolecular networks. In practice, master equations describing complex reaction networks have remained unsolved for over 70 years. This practical challenge is a reason why master equations, for all their potential, have not inspired biological discovery. Herein, we present a closure scheme that solves the master probability equation of networks of chemical or biochemical reactions. We cast the master equation in terms of ordinary differential equations that describe the time evolution of probability distribution moments. We postulate that a finite number of moments capture all of the necessary information, and compute the probability distribution and higherorder moments by maximizing the information entropy of the system. An accurate order closure is selected, and the dynamic evolution of molecular populations is simulated. Comparison with kineticMonte Carlo simulations, which merely sample the probability distribution, demonstrates this closure scheme is accurate for several small reaction networks. The importance of this result notwithstanding, a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
KW - Entropy maximization
KW - Information theory
KW - Statistical mechanics
KW - Stochastic models
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U2 - 10.1073/pnas.1306481110
DO - 10.1073/pnas.1306481110
M3 - Article
C2 - 23940327
AN - SCOPUS:84883337952
SN - 0027-8424
VL - 110
SP - 14261
EP - 14265
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
IS - 35
ER -