### Abstract

We propose an efficient and accurate numerical scheme for solving probability generating functions arising in stochastic models of general first-order reaction networks by using the characteristic curves. A partial differential equation derived by a probability generating function is the transport equation with variable coefficients. We apply the idea of characteristics for the estimation of statistical measures, consisting of the mean, variance, and marginal probability. Estimation accuracy is obtained by the Newton formulas for the finite difference and time accuracy is obtained by applying the fourth order Runge-Kutta scheme for the characteristic curve and the Simpson method for the integration on the curve. We apply our proposed method to motivating biological examples and show the accuracy by comparing simulation results from the characteristic method with those from the stochastic simulation algorithm.

Original language | English |
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Pages (from-to) | 316-337 |

Number of pages | 22 |

Journal | Journal of Mathematical Chemistry |

Volume | 51 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 Jan 1 |

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### Keywords

- Characteristic method
- First-order partial differential equation
- First-order reaction network
- Monte Carlo method

### ASJC Scopus subject areas

- Chemistry(all)
- Applied Mathematics

### Cite this

**A numerical characteristic method for probability generating functions on stochastic first-order reaction networks.** / Lee, Chang Hyeong; Shin, Jaemin; Kim, Junseok.

Research output: Contribution to journal › Article

*Journal of Mathematical Chemistry*, vol. 51, no. 1, pp. 316-337. https://doi.org/10.1007/s10910-012-0085-8

}

TY - JOUR

T1 - A numerical characteristic method for probability generating functions on stochastic first-order reaction networks

AU - Lee, Chang Hyeong

AU - Shin, Jaemin

AU - Kim, Junseok

PY - 2013/1/1

Y1 - 2013/1/1

N2 - We propose an efficient and accurate numerical scheme for solving probability generating functions arising in stochastic models of general first-order reaction networks by using the characteristic curves. A partial differential equation derived by a probability generating function is the transport equation with variable coefficients. We apply the idea of characteristics for the estimation of statistical measures, consisting of the mean, variance, and marginal probability. Estimation accuracy is obtained by the Newton formulas for the finite difference and time accuracy is obtained by applying the fourth order Runge-Kutta scheme for the characteristic curve and the Simpson method for the integration on the curve. We apply our proposed method to motivating biological examples and show the accuracy by comparing simulation results from the characteristic method with those from the stochastic simulation algorithm.

AB - We propose an efficient and accurate numerical scheme for solving probability generating functions arising in stochastic models of general first-order reaction networks by using the characteristic curves. A partial differential equation derived by a probability generating function is the transport equation with variable coefficients. We apply the idea of characteristics for the estimation of statistical measures, consisting of the mean, variance, and marginal probability. Estimation accuracy is obtained by the Newton formulas for the finite difference and time accuracy is obtained by applying the fourth order Runge-Kutta scheme for the characteristic curve and the Simpson method for the integration on the curve. We apply our proposed method to motivating biological examples and show the accuracy by comparing simulation results from the characteristic method with those from the stochastic simulation algorithm.

KW - Characteristic method

KW - First-order partial differential equation

KW - First-order reaction network

KW - Monte Carlo method

UR - http://www.scopus.com/inward/record.url?scp=84871709683&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871709683&partnerID=8YFLogxK

U2 - 10.1007/s10910-012-0085-8

DO - 10.1007/s10910-012-0085-8

M3 - Article

AN - SCOPUS:84871709683

VL - 51

SP - 316

EP - 337

JO - Journal of Mathematical Chemistry

JF - Journal of Mathematical Chemistry

SN - 0259-9791

IS - 1

ER -