Abstract
An important driver of the dynamics of gene regulatory networks is noise generated by transcription and translation processes involving genes and their products. As relatively small numbers of copies of each substrate are involved, such systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distributions for the states defined by the vectors of copy numbers of each substance. Challenges are posed by the large discrete state spaces, and are mainly due to high dimensionality.
In order to address this challenge we propose effective approximation techniques, and, in particular, numerical techniques to solve the master equations. Two theoretical results show that the numerical methods are optimal. The techniques are combined with sparse grids to give an effective method to solve high-dimensional problems.
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References
A. Arkin, J. Ross, and H. H. McAdams. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected escherichia coli. Genetics, 149:1633–1648, 1998.
D. Braess. Finite Elements. Cambridge, second edition, 2005.
G. Dahlquist. Stability and error bounds in the numerical integration of ordinary differential equations. Kungl. Tekn. Högsk. Handl. Stockholm. No., 130:87, 1959.
L. Ferm and P. Lötstedt. Numerical method for coupling the macro and meso scales in stochastic chemical kinetics. Technical Report 2006-001, Uppsala University, January 2006.
J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67(3):225–253, 2001.
D. T. Gillespie. Markov Processes: an introduction for physical scientists. Academic Press, San Diego, USA, 1992.
M. Griebel, M. Schneider, and C. Zenger. A combination technique for the solution of sparse grid problems. In Iterative methods in linear algebra (Brussels, 1991), pages 263–281. North-Holland, Amsterdam, 1992.
M. Hegland. Adaptive sparse grids. ANZIAM J., 44((C)):C335–C353, 2002.
M. Hegland. Additive sparse grid fitting. In Curve and surface fitting (Saint-Malo, 2002), Mod. Methods Math., pages 209–218. Nashboro Press, Brentwood, TN, 2003.
M. Hegland, C. Burden, L. Santoso, S. MacNamara, and H. Booth. A solver for the stochastic master equation applied to gene regulatory networks. J. Comp. Appl. Math., 205:708–724, 2007.
R. A. Horn and C. R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original.
B. Lewin. Genes VIII. Pearson Prentice Hall, 2004.
C. Pflaum and A. Zhou. Error analysis of the combination technique. Numer. Math., 84(2):327–350, 1999.
M. Ptashne and A. Gann. Genes and Signals. Cold Spring Harbor Laboratory Press, 2002.
M. A. Shea and G. K. Ackers. The o r control system of bacteriophage lambda, a physical-chemical model for gene regulation. Journal of Molecular Biology, 181:211–230, 1985.
T. Ström. On logarithmic norms. SIAM J. Numer. Anal., 12(5):741–753, 1975.
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Hegland, M., Burden, C., Santoso, L. (2008). Modelling Gene Regulatory Networks Using Galerkin Techniques Based on State Space Aggregation and Sparse Grids. In: Bock, H.G., Kostina, E., Phu, H.X., Rannacher, R. (eds) Modeling, Simulation and Optimization of Complex Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79409-7_17
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DOI: https://doi.org/10.1007/978-3-540-79409-7_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79408-0
Online ISBN: 978-3-540-79409-7
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