Numerical Solution of the Fokker- Planck Approximation of - DiVA

104 Pages·2005·3.27 MB·English

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DepartmentofInformationTechnology
DepartmentofInformationTechnology,UppsalaUniversity,Sweden

Numerical Solution of the Fokker-Planck
Approximation of the
Chemical Master Equation
BY
PAULSJO¨BERG
December2005
DIVISIONOFSCIENTIFICCOMPUTING
DEPARTMENTOFINFORMATIONTECHNOLOGY
UPPSALAUNIVERSITY
UPPSALA
SWEDEN
DissertationforthedegreeofLicentiateofTechnologyinScientificComputing
atUppsalaUniversity2005
Numerical Solution of the Fokker-Planck Approximation of the
Chemical Master Equation
PaulSjo¨berg
[email protected]
DivisionofScientificComputing
DepartmentofInformationTechnology
UppsalaUniversity
Box337
SE-75105 Uppsala
Sweden
http://www.it.uu.se/
(cid:176)c PaulSjo¨berg2005
ISSN1404-5117
PrintedbytheDepartmentofInformationTechnology,UppsalaUniversity,Sweden
Abstract
The chemical master equation (CME) describes the probability for the
discrete molecular copy numbers that define the state of a chemical sys-
tem. Each molecular species in the chemical model adds a dimension to
the state space. The CME is a difference-differential equation which can
be solved numerically if the state space is truncated at an upper limit
of the copy number in each dimension. The size of the truncated CME
suffers from an exponential growth for an increasing number of chemical
species.
In this thesis the CME is approximated by a continuous Fokker-
Planck equation (FPE) which makes it possible to use sparser computa-
tional grids than for CME. FPE on conservative form is used to compute
steady state solutions by computation of an extremal eigenvalue and the
corresponding eigenvector as well as time-dependent solutions by an im-
plicit time-stepping scheme.
The performance of the numerical solution is compared to a standard
Monte Carlo algorithm. The computational work for solutions with the
sameestimatederroriscomparedforthetwomethods. Dependingonthe
problem, FPE or the Monte Carlo algorithm will be more efficient. FPE
is well suited for problems in low dimensions, especially if high accuracy
is desirable.
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List of Papers
A J. Elf, P. L¨otstedt and P. Sj¨oberg. Problems of high dimension
in molecular biology. Proceedings of the 19th GAMM-Seminar,
Leipzig 2003.
B L. Ferm, P. L¨otstedt and P. Sj¨oberg. Conservative solution of the
Fokker-Planck equation for stochastic chemical equations. Techni-
cal Report 2004-054, Department of Information Technology, Up-
psala University, 2005, revised version.
C P. Sj¨oberg, P. L¨otstedt and J. Elf. Fokker-Planck approximation
of the master equation in molecular biology. Technical Report,
Department of Information Technology, Uppsala University, 2005,
to appear, will be available at:
http://www.it.uu.se/research/publications/reports/.
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Acknowledgments
I would like to thank my advisor Per L¨otstedt for his support and advice
and also Johan Elf and M˚ans Ehrenberg for discussions and guidance in
the microcosmos of molecular biology.
This work has been supported by the National Graduate School in
ScientificComputing,theSwedishFoundationforStrategicResearchand
the Swedish Research Council.
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