# A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Burrage, Kevin, Hegland, M., Macnamara, Shev, & Sidje, Roger
(2006)
A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems. In
Langville, A. N. & Stewart, W. J. (Eds.)
*Proceedings of the Markov 150th Anniversary Conference*, Boston Books, Charleston, South Carolina, pp. 21-38.

## Abstract

Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically diﬃcult to solve, since the state-space involved can be very large or even countably inﬁnite. Recently a ﬁnite state projection method (FSP) that truncates the state-space was suggested and shown to be eﬀective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the ﬁnal time at which the solution was computed, were relatively small. Presented here is a Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger ﬁnal times to be computed in a realistic execution time. Additionally the new method computes the solution at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was ﬁrst demonstrated. Also the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is signiﬁcantly faster and extendable to larger biological models.

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ID Code: | 46148 |
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Item Type: | Conference Paper |

Refereed: | Yes |

ISBN: | 9781932482355 (online ebook) 9781932482348 (print) |

Divisions: | Past > QUT Faculties & Divisions > Faculty of Science and Technology Past > Schools > Mathematical Sciences |

Deposited On: | 27 Sep 2011 01:03 |

Last Modified: | 27 Sep 2011 01:03 |

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