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### A probability generating function method for stochastic reaction networks

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Affiliations:
1 Ulsan National Institute of Science and Technology (UNIST), Ulsan Metropolitan City 689-798, South Korea
a) Email: pwkim@unist.ac.kr.
b) Author to whom correspondence should be addressed. Electronic mail: chlee@unist.ac.kr.
J. Chem. Phys. 136, 234108 (2012)
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### References

• Pilwon Kim and Chang Hyeong Lee
• Source: J. Chem. Phys. 136, 234108 ( 2012 );
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4. C. Gadgil, C. H. Lee, and H. G. Othmer, “A stochastic analysis of first-order reaction networks,” Bull. Math. Biol. 67, 901946 (2005).
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7. E. L. Haseltine and J. B. Rawlings, “Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics,” J. Chem. Phys. 117(15), 69596969 (2002).
http://dx.doi.org/10.1063/1.1505860
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8. G. A. Baker and P. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University Press, 1996).
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9. D. Barik, M. R. Paul, W. T. Baumann, Y. Cao, and J. J. Tyson, “Stochastic simulation of enzyme-catalyzed reactions with disparate timescales,” Biophys. J. 95, 35633574 (2008).
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journal-id:
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## Figures

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FIG. 1.

The graphs of for a binding reaction in Sec. IV A are illustrated. denotes difference between the approximations of consecutive orders N and N−2. It gradually diminishes as N increases.

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FIG. 2.

Upper three figures are mean, variance, and marginal probability of the number of species A obtained from the PGF method for binding/unbinding model. In the figure of probability distribution p, each p(i) denotes the marginal probability that the number of A is i. Lower three figures illustrate the errors in log10 generated from the PGF method, that is, log10 |exact solution − approximate solution|. The initial condition a 0 = 20, b 0 = 10, c 0 = 0, and parameters c 1 = 1, c −1 = 0.1 (arbitrary units) are assumed. The exact probability, mean, and variance are obtained directly by solving Kolmogorov equation dp/dt = Kp.3 Some parts of the error graphs are not properly marked because the corresponding errors are lower than the machine precision.

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FIG. 3.

Upper three figures are mean, variance, and marginal probability of the number of enzyme E obtained from the PGF method for the enzyme-substrate model. In the figure of probability distribution p, each p(i) denotes the marginal probability that the number of enzyme is i. Lower three figures illustrate the errors in log10 generated from the PGF method, that is, log10 |exact solution − approximate solution|. In the figures, the condition n 1(0) = 5, n 2(0) = 10, and a 0 = 5, and parameters c 1 = c 2 = 0.1, c 3 = 1 are assumed. The exact probability, mean and variance are obtained directly by solving Kolmogorov equation dp/dt = Kp. According to the figures, the error gets bigger as the system approaches its steady state, but it is still very small, compared to the magnitudes of mean, variance, and probability.

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FIG. 4.

Mean, standard deviation, and probability obtained from the PGF method and the SSA (stochastic simulation algorithm) under the initial condition n 1(0) = 10, n 2(0) = 10, n 3(0) = 40, n 4(0) = n 5(0) = n 6(0) = 0. μ i and σ i , i = 1, 2, 3 denote the mean and standard deviation of n 1, n 2, and n 3, respectively. In the figure of probability, each curve p(n 1 = i) denotes the time-dependent probability solution that n 1 = i, i = 1, 3, 5. The results by the SSA are based on 30 000 realizations.

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FIG. 5.

Mean, standard deviation, and probability obtained from the PGF method and SSA under the initial condition n 1(0) = 10, n 2(0) = 9, n 3(0) = 5, n 4(0) = 4, n 5(0) = 4, n 6(0) = 5, n 7(0) = n 8(0) = n 9(0) = n 10(0) = 0. μ i and σ i , i = 1, 2, 3 denote the mean and standard deviation of n 1, n 2, and n 3, respectively. In the figure of probability, each curve p(n 2 = i) denotes the time-dependent probability solution that n 2 = i, i = 1, 3, 5. The results by the SSA are based on 30 000 realizations.

/content/aip/journal/jcp/136/23/10.1063/1.4729374
2012-06-20
2013-12-06

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