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[11] J M Bower and H Bolouri, eds Computational Modeling of Genetic and Biochemical Networks MIT Press, Cambridge, MA, 2001 [12] P Br maud Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues e Springer, New York, 1999 [13] L A Breyer and A G Hart Approximations of quasi-stationary distributions for Markov chains Mathematical and Computer Modelling, 31: 69 79, 2000 [14] K Burrage, T Tian, and P Burrage A multi-scaled approach for simulating chemical reaction systems Progress in Biophysics and Molecular Biology, 85: 217 234, 2004 [15] H Busch, W Sandmann, and V Wolf A numerical aggregation algorithm for the enzyme-catalyzed substrate conversion In Proceedings of the 2006 International Conference on Computational Methods in Systems Biology, Lecture Notes in Computer Science 4210, pp 298 311 Springer, Berlin/Heidelberg/New York, 2006 [16] W Cai, M H Kalos, M de Koning, and V V Bulatov Importance sampling of rare transition events in Markov processes Physical Review E , 66: 046703, 2002 [17] Y Cao, D T Gillespie, and L R Petzold The slow-scale stochastic simulation algorithm Journal of Chemical Physics, 122(1): 014116/1 014116/18, 2005 [18] Y Cao, D T Gillespie, and L R Petzold Ef cient stepsize selection for the tauleaping simulation Journal of Chemical Physics, 124: 044109/1 144109/11, 2006 [19] Y Cao, D T Gillespie, and L R Petzold The adaptive explicit-implicit tau-leaping method with automatic tau selection Journal of Chemical Physics, 126(22): 224101/1 224101/9, 2007 [20] Y Cao, H Li, and L R Petzold Ef cient formulation of the stochastic simulation algorithm for chemically reacting systems Journal of Chemical Physics, 121(9): 4059 4067, 2004 [21] Y Cao and L R Petzold Trapezoidal tau-leaping formula for the stochastic simulation of biochemical systems In Proceedings of the 1st Conference on Foundations of Systems Biology in Engineering, pp 149 152, 2005 [22] Y Cao and L R Petzold Accuracy limitations and the measurement of errors in the stochastic simulation of chemically reacting systems Journal of Computational Physics, 212(1): 6 24, 2006 [23] Y Cao and L R Petzold Slow-scale tau-leaping method Computer Methods in Applied Mechanics and Engineering, 197(43 44): 3472 3479, 2008 [24] T H Cormen, C E Leiserson, and R L Rivest Introduction to Algorithms MIT Press, Cambridge, MA, 1990 [25] P-J Courtois Decomposability, Queueing and Computer Applications Academic Press, New York, 1977 [26] P-J Courtois On time and space decomposition of complex structures Communications of the ACM , 28(6): 590 603, 1985 [27] J N Darroch and E Seneta On quasistationary distributions in absorbing continuous-time nite Markov chains Journal of Applied Probability, 4: 192 196, 1967 [28] M de Koning, W Cai, B Sadigh, M H Kalos, and V V Bulatov Adaptive importance sampling Monte Carlo simulation of rare transition events Journal of Chemical Physics, 122(7): 074103, 2005
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[29] M Delbr ck Statistical uctuations in autocatalytic reactions Journal of Chemical u Physics, 8: 120 124, 1940 [30] C Dellago, P G Bolhuis, and P L Geissler Transition path sampling Advances in Chemical Physics, 123: 1 78, 2002 [31] R Elber Long-timescale simulation methods Current Opinion in Structural Biology, 15: 151 156, 2005 [32] N Fedoroff and W Fontana Small numbers of big molecules Science, 297: 1129 1131, 2002 [33] M A Gibson and J Bruck Ef cient exact stochastic simulation of chemical systems with many species and many channels Journal of Physical Chemistry A, 104: 1876 1889, 2000 [34] D T Gillespie A general method for numerically simulating the time evolution of coupled chemical reactions Journal of Computational Physics, 22: 403 434, 1976 [35] D T Gillespie Exact stochastic simulation of coupled chemical reactions Journal of Physical Chemistry, 71(25): 2340 2361, 1977 [36] D T Gillespie A rigorous derivation of the chemical master equation Physica A, 188: 404 425, 1992 [37] D T Gillespie Approximate accelerated stochastic simulation of chemically reacting systems Journal of Chemical Physics, 115: 1716 1732, 2001 [38] D T Gillespie and L R Petzold Improved leap-size selection for accelerated stochastic simulation Journal of Chemical Physics, 119: 8229 8234, 2003 [39] L A Harris and P Clancy A partitioned leaping approach for multiscale modeling of chemical reaction dynamics Journal of Chemical Physics, 125(14): 144107, 2006 [40] E L Haseltine and J B Rawlings Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics Journal of Chemical Physics, 117(15): 6959 6969, 2002 [41] H Kitano Foundations of Systems Biology MIT Press, Cambridge, MA, 2001 [42] E Klipp, R Herwig, A Kowald, C Wierling, and H Lehrach Systems Biology in Practice Wiley-VCH, Weinheim, 2005 [43] T G Kurtz The relationship between stochastic and deterministic models for chemical reactions Journal of Chemical Physics, 57(7): 2976 2978, 1972 [44] J Ledoux, G Rubino, and B Sericola Exact aggregation of absorbing Markov processes using the quasi-stationary distribution Journal of Applied Probability, 31: 626 634, 1994 [45] H Li and L R Petzold Logarithmic direct method for discrete stochastic simulation of chemically reacting systems Technical Report, 2006 http://wwwengineeringucsbedu/ cse/Files/ldm0513pdf [46] H H McAdams and A Arkin Stochastic mechanisms in gene expression Proceedings of the National Academy of Science USA, 94: 814 819, 1997 [47] H H McAdams and A Arkin It s a noisy business! Trends in Genetics, 15(2): 65 69, 1999 [48] J M McCollum, G D Peterson, C D Cox, M L Simpson, and N F Samatova The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior Computational Biology and Chemistry, 30(1): 39 49, 2006
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