A solver for the stochastic master equation applied to gene regulatory networks

Journal of Computational and Applied Mathematics

Volumn 205, Issue 2, 2007, Pages 708-724

  Hegland, Markus ^a,b   Burden, Conrad ^c   Santoso, Lucia ^a,b   MacNamara, Shev ^a,b   Booth, Hilary ^d  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^b UNIVERSITY OF QUEENSLAND   (Australia)

^c Biology and Environment   (Australia)

^d NONE

Author keywords

Gene regulatory networks; Master equations; Sparse grids; Systems biology

Indexed keywords

APPROXIMATION THEORY; BIFURCATION (MATHEMATICS); COMPUTER SIMULATION; MULTIRESOLUTION ANALYSIS; PROBLEM SOLVING; RANDOM PROCESSES;

GENE REGULATORY NETWORKS; MASTER EQUATIONS; SPARSE GRIDS; SYSTEMS BIOLOGY;

GENETIC ENGINEERING;

EID: 34248187057     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2006.02.053     Document Type: Article

References (22)

1. Arkin A., Ross J., and McAdams H.H. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli. Genetics 149 (1998) 1633-1648
2. Donachie W.D., and Robinson A. Cell division: parameter values and process. In: Neidhardt F., et al. (Ed). Escherichia coli and Salmonella typhimurium: Cellular and Molecular Biology (1987), ASM Press, Washington DC, USA 1347-1350
3. Elf J., and Ehrenberg M. Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Research 13 (2003) 2475-2484
4. Gillespie D.T. Markov Processes: An Introduction for Physical Scientists (1992), Academic Press, San Diego, USA
5. M. Griebel, M. Schneider, C. Zenger, A combination technique for the solution of sparse grid problems, in: Iterative Methods in Linear Algebra, Brussels, 1991, North-Holland, Amsterdam, 1992, pp. 263-281.
6. Guptasarma P. Does replication-induced transcription regulate the myriad of low copy number proteins of Escherichia coli?. Bioessays 17 (1995) 987-997
7. Haviv M. Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain. SIAM J. Numer. Anal. 24 4 (1987) 952-966
8. M. Hegland, Adaptive sparse grids, in: K. Burrage, R.B. Sidje (Eds.), Proceedings of 10th Computational Techniques and Applications Conference CTAC-2001, vol. 44, April 2003, pp. C335-C353.
9. Horn R.A., and Johnson C.R. Matrix Analysis (1990), Cambridge University Press, Cambridge Corrected reprint of the 1985 original
10. Lipshtat A., Perets H.B., Balaban N.Q., and Biham O. Modeling of negative autoregulated genetic networks in single cells. Gene 347 2 (2005) 265-271
11. Mazurie A., Bottani S., and Vergassola M. An evolutionary and functional assessment of regulatory network motifs. Geonome Biology 6 4 (2005) R35.1-R35.12
12. Meyer C.D. Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev. 31 (1989) 240-272
13. B. Munsky, M. Khammash, The finite state projection algorithm for the solution of the chemical master equation, Technical Report CCEC-05-0505, Center for Control Engineering and Computation, University of California at Santa Barbara, 2005. 〈http://ccec.mee.ucsb.edu/Year/2005.complete.html#CCEC-05-0505〉.
14. Ptashne M. A genetic switch: phage-λ and Higher Organisms (1992), Cell, Cambridge, USA
15. L. Santoso, M. Hegland, H. Booth, C. Burden, Approximation techniques for Markov models of genetic regulatory network, 2005, in preparation.
16. r control system of bacteriophage lambda, a physical-chemical model for gene regulation. J. Mol. Biol. 181 (1985) 211-230
17. Sidje R.B. EXPOKIT: Software package for computing matrix exponentials. ACM Trans. Math. Software 24 1 (1998) 130-156
18. Simon H.A., and Ando A. Aggregation of variables in dynamic systems. Econometrica 29 (1961) 111-138
19. Stewart W. Introduction to the Numerical Solution of Markov Chains (1994), Princeton University Press, Princeton, NJ
20. Tian T., and Burrage K. Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage-λ. J. Theoret. Biol. 227 (2004) 229-237
21. van Kampen N.G. Stochastic Processes in Physics and Chemistry (1981), North-Holland, Amsterdam, the Netherlands
22. C. Zenger, Sparse grids, in: Parallel Algorithms for Partial Differential Equations, Kiel, 1990, Notes Numerical Fluid Mechanics, vol. 31, Vieweg, Braunschweig, 1991, pp. 241-251.