Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains

PLoS Computational Biology

Volumn 10, Issue 3, 2014, Pages

  Kazeev, Vladimir ^a   Khammash, Mustafa ^a   Nip, Michael ^b   Schwab, Christoph ^a  

^a ETH ZURICH   (Switzerland)

^b Santa Barbara   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIOLOGY; CHEMICAL ANALYSIS; COMPLEX NETWORKS; COMPUTATIONAL CHEMISTRY; GALERKIN METHODS; NUMERICAL METHODS; STATISTICAL MECHANICS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS; TENSORS;

ANALYSIS AND SIMULATION; CHEMICAL MASTER EQUATION; CURSE OF DIMENSIONALITY; DIRECT SOLUTION; MODELS OF BIOCHEMICAL REACTIONS; SIMULATION OF MODELS; STOCHASTIC ANALYSIS; STOCHASTIC SIMULATIONS; TENSOR TRAINS; TIME STEP;

QUANTUM CHEMISTRY;

ALGORITHM; ARTICLE; CHEMICAL MASTER EQUATION; CONTROLLED STUDY; LINEAR SYSTEM; MATHEMATICAL PARAMETERS; MATHEMATICS; QUANTIZED TENSOR TRAIN; QUANTUM CHEMISTRY; STATISTICAL ANALYSIS; STATISTICAL MODEL; SYSTEMS BIOLOGY;

ALGORITHMS; BIOCHEMISTRY; COMPUTATIONAL BIOLOGY; COMPUTER SIMULATION; MODELS, STATISTICAL; MONTE CARLO METHOD; PROBABILITY; STOCHASTIC PROCESSES;

EID: 84897420039     PISSN: 1553734X     EISSN: 15537358     Source Type: Journal    
DOI: 10.1371/journal.pcbi.1003359     Document Type: Article

References (77)

1. Elowitz M, Levine A, Siggia E, Swain P, (2002) Stochastic gene expression in a single cell. Nature 297: 1183-1186.
2. McAdams HH, Arkin A, (1997) Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences of the United States of America 94: 814-819.
3. Gillespie DT, (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics 22: 403-434.
4. Jahnke T, Huisinga W, (2007) Solving the chemical master equation for monomolecular reaction systems analytically. Journal of Mathematical Biology 54: 1-26.
5. Munsky B, Khammash M, (2006) The finite state projection algorithm for the solution of the chemical master equation. The Journal of Chemical Physics 124: 044104.
6. Henzinger T, Mateescu M, Wolf V (2009) Sliding window abstraction for infinite markov chains. In: Bouajjani A, Maler O, editors, Computer Aided Verification, Springer Berlin/Heidelberg, volume 5643 of Lecture Notes in Computer Science. pp. 337-352. URL http://dx.doi.org/10.1007/978-3-642-02658-422 7. 10.1007/978-3-642-02658-4-27.
7. Cao Y, Liang J, (2008) Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability. BMC Systems Biology 2: 30.
8. Cao Y, Lu HM, Liang J, (2010) Probability landscape of heritable and robust epigenetic state of lysogeny in phage lambda. Proceedings of the National Academy of Sciences of the United States of America 107: 18445-18450.
9. Gibson MA, Bruck J, (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. The Journal of Physical Chemistry A 104: 1876-1889.
10. Gillespie DT, (2001) Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics 115: 1716-1733.
11. Hespanha JP, Singh A, (2005) Stochastic models for chemically reacting systems using polynomial stochastic hybrid systems. Int J on Robust Control, Special Issue on Control at Small Scales Issue 1 15: 669-689.
12. Gomez-Uribe CA, Verghese GC, (2007) Mass fluctuation kinetics: Capturing stochastic effects in systems of chemical reactions through coupled mean-variance computations. The Journal of Chemical Physics 126: 024109.
13. van Kampen NG (1992) Stochastic Processes in Physics and Chemistry. Amsterdam and New York: North-Holland.
14. Gillespie DT, (2000) The chemical langevin equation. The Journal of Chemical Physics 113: 297-306.
15. Ethier SN, Kurtz TG (2005) Markov Processes: Characterization and Convergence. New York: Wiley-Interscience.
16. Puchalka J, Kierzek AM, (2004) Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks. Biophysical Journal 86: 1357-1372.
17. Haseltine EL, Rawlings JB, (2002) Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. The Journal of Chemical Physics 117: 6959-6969.
18. Salis H, Kaznessis Y, (2005) Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. The Journal of Chemical Physics 122: 054103.
19. Hellander A, Lotstedt P, (2007) Hybrid method for the chemical master equation. Journal of Computational Physics 227: 100-122.
20. Jahnke T, (2011) On reduced models for the chemical master equation. Multiscale Modeling and Simulation 9: 1646.
21. Nip M, Hespanha J, Khammash M (2012) A spectral methods-based solution of the chemical master equation for gene regulatory networks. In: Decision and Control (CDC), 2012 IEEE 51st Annual Conference on. pp. 5354-5360. doi:10.1109/CDC.2012.6425804.
22. Engblom S, (2009) Spectral approximation of solutions to the chemical master equation. Journal of Computational and Applied Mathematics 229: 208-221.
23. Deuflhard P, Huisinga W, Jahnke T, Wulkow M, (2008) Adaptive discrete Galerkin methods applied to the chemical master equation. SIAM Journal on Scientific Computing 30: 2990-3011.
24. Hegland M, Burden C, Santoso L, MacNamara S, Booth H, (2007) A solver for the stochastic master equation applied to gene regulatory networks. Journal of Computational and Applied Mathematics 205: 708-724.
25. Jahnke T, Udrescu T, (2010) Solving chemical master equations by adaptive wavelet compression. Journal of Computational Physics 229: 5724-5741.
26. Bellman R, (1961) Adaptive Control Processes: A Guided Tour. Princeton, NJ: Princeton University Press.
27. Hitchcock FL, (1926) The expression of a tensor or a polyadic as a sum of products. Journal of Mathematics and Physics 6: 164-189.
28. Caroll JD, Chang JJ, (1970) Analysis of individual differences in multidimensional scaling via an n-way generalization of "Eckart-Young" decomposition. Psychometrika 35: 283-319.
29. Hegland M, Garcke J, (2011) On the numerical solution of the chemical master equation with sums of rank one tensors. ANZIAM Journal 52: C628-C643.
30. de Silva V, Lim LH, (2008) Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM Journal on Matrix Analysis and Applications 30: 1084-1127.
31. Håstad J, (1990) Tensor rank is NP-complete. Journal of Algorithms 11: 644-654.
32. Hillar C, Lim LH (2009) Most tensor problems are NP hard. arXiv abs/0911.1393.
33. Jahnke T, Huisinga W, (2008) A dynamical low-rank approach to the chemical master equation. Bulletin of Mathematical Biology 70: 2283-2302.
34. Oseledets IV, Tyrtyshnikov EE, (2009) Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM Journal on Scientific Computing 31: 3744-3759.
35. Oseledets I, (2009) Approximation of matrices with logarithmic number of parameters. Doklady Mathematics 80: 653-654.
36. Oseledets IV, (2010) Approximation of 2d×2d matrices using tensor decomposition. SIAM Journal on Matrix Analysis and Applications 31: 2130-2145.
37. Khoromskij BN, (2011) <$>\scale 80%\raster="rg4"\<$> (d log n)-quantics approximation of n-d tensors in high-dimensional numerical modeling. Constructive Approximation 34: 257-280.
38. Kazeev V, Reichmann O, Schwab C (2012) hp-DG-QTT solution of high-dimensional degenerate diffusion equations. Research Report 11, Seminar for Applied Mathematics, ETH Zürich. URL http://www.sam.math.ethz.ch/reports/2012/11.
39. Kolda TG, Bader BW, (2009) Tensor decompositions and applications. SIAM Review 51: 455-500.
40. Hackbusch W (2012) Tensor Spaces and Numerical Tensor Calculus, volume 42 of Springer Series in Computational Mathematics. Springer. doi:10.1007/978-3-642-28027-6. URL http://www.springerlink.com/content/l62t86.
41. Oseledets IV, (2011) Tensor Train decomposition. SIAM Journal on Scientific Computing 33: 2295-2317.
42. Oseledets I, (2013) Constructive representation of functions in low-rank tensor formats. Constructive Approximation 37: 1-18.
43. Hackbusch W, Kühn S, (2009) A new scheme for the tensor representation. Journal of Fourier Analysis and Applications 15: 706-722.
44. Grasedyck L, (2010) Hierarchical Singular Value Decomposition of tensors. SIAM Journal on Matrix Analysis and Applications 31: 2029-2054.
45. Grasedyck L (2010) Polynomial approximation in Hierarchical Tucker Format by vectortensorization. Preprint 308, Institut für Geometrie und Praktische Mathematik, RWTH Aachen. URL http://www.igpm.rwth-aachen.de/Download/reports/pdf/I GP M 308k.pdf.
46. White SR, (1993) Density-matrix algorithms for quantum renormalization groups. Phys Rev B 48: 10345-10356.
47. Verstraete F, Porras D, Cirac JI, (2004) Density matrix renormalization group and periodic boundary conditions: A quantum information perspective. Phys Rev Lett 93: 227205.
48. Vidal G, (2003) Efficient classical simulation of slightly entangled quantum computations. Phys Rev Lett 91: 147902.
49. Ballani J, Grasedyck L, (2013) A projection method to solve linear systems in tensor format. Numerical Linear Algebra with Applications 20: 27-43.
50. Kressner D, Tobler C, (2011) Preconditioned low-rank methods for high-dimensional elliptic PDE eigenvalue problems. Computational Methods in Applied Mathematics 11: 363-381.
51. Dolgov S, Khoromskij B, Oseledets I, (2012) Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the fokker-planck equation. SIAM Journal on Scientific Computing 34: A3016-A3038.
52. Kressner D, Tobler C (2010) Low-rank tensor Krylov subspace methods for parametrized lin- ear systems. Research Report 16, Seminar for Applied Mathematics, ETH Zürich. URL http://www.sam.math.ethz.ch/reports/2010/16.
53. Khoromskij BN, Schwab C, (2011) Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM Journal on Scientific Computing 33: 364-385.
54. Tyrtyshnikov EE, (2003) Tensor approximations of matrices generated by asymptotically smooth functions. Sbornik: Mathematics 194: 941-954.
55. Oseledets IV, (2010) QTT decomposition of the characteristic function of a simplex. Personal communication.
56. Schotzau D, Schwab C, (2000) An hp a priori error analysis of the DG time-stepping method for initial value problems. Calcolo 37: 207-232.
57. Saad Y, Schultz M, (1986) Gmres: A generalized minimal residual algorithm for solving nonsym- metric linear systems. SIAM Journal on Scientific and Statistical Computing 7: 856-869.
58. Saad Y, (1992) Analysis of some krylov subspace approximations to the matrix exponential operator. SIAM Journal on Numerical Analysis 29: 209-228.
59. MacNamara S, Burrage K, Sidje R, (2008) Multiscale modeling of chemical kinetics via the master equation. Multiscale Modeling & Simulation 6: 1146-1168.
60. Sidje RB, (1998) Expokit: a software package for computing matrix exponentials. ACM Trans Math Softw 24: 130-156.
61. Gardner TS, Cantor CR, Collins JJ, (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403: 339-342.
62. Munsky B, Khammash M, (2008) The finite state projection approach for the analysis of stochastic noise in gene networks. Automatic Control, IEEE Transactions on 53: 201-214.
63. Sjöberg P, Lötstedt P, Elf J, (2009) Fokker-Planck approximation of the master equation in molecular biology. Computing and Visualization in Science 12: 37-50.
64. Schwender J, Ohlrogge J, Shachar-Hill Y, (2004) Understanding flux in plant metabolic networks. Current Opinion in Plant Biology 7: 309-317.
65. Samoilov M, Plyasunov S, Arkin AP, (2005) Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. Proceedings of the National Academy of Sciences of the United States of America 102: 2310-2315.
66. Jahnke T, (2010) An adaptive wavelet method for the chemical master equation. SIAM Journal on Scientific Computing 31: 4373.
67. Kazeev V, Schwab C (2013) Tensor approximation of stationary distributions of chemical reaction networks. Research Report 18, Seminar for Applied Mathematics, ETH Zürich. URL http://www.sam.math.ethz.ch/reports/2013/18.
68. Anderson DF, Craciun G, Kurtz TG, (2010) Product-form stationary distributions for deficiency zero chemical reaction networks. Bulletin of Mathematical Biology 72: 1947-1970.
69. Feinberg M (1979). Lectures on chemical reaction networks.URL http://www.chbmeng.ohio-state.edu/feinberg/LecturesOnReactionNetworks.
70. Dolgov SV, Savostyanov DV (2013) Alternating minimal energy methods for linear sys- tems in higher dimensions. Part I: SPD systems. arXiv preprint 1301.6068. URL http://arxiv.org/abs/1301.6068.
71. Verstraete F, Cirac JI, Murg V (2009) Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems. arXiv preprint 0907.2796. URL http://arxiv.org/abs/0907.2796.
72. Cirac JI, Verstraete F, (2009) Renormalization and tensor product states in spin chains and lattices. Journal of Physics A: Mathematical and Theoretical 42: 504004.
73. Dolgov SV, Khoromskij BN (2012) Tensor-product approach to global time- space-parametric discretization of chemical master equation. Preprint 68, Max-Planck-Institut für Mathematik in den Naturwissenschaften. URL http://www.mis.mpg.de/publications/preprints/2012/prepr2012 - 68.html.
74. Oseledets I, Dolgov S, (2012) Solution of linear systems and matrix inversion in the tt-format. SIAM Journal on Scientific Computing 34: A2718-A2739.
75. Rohwedder T, Uschmajew A (2012) Local convergence of alternating schemes for optimization of convex problems in the TT format. Preprint 112, DFG-Schwerpunktprogramm 1324. URL http://www.dfg-spp1324.de/download/preprints/preprint112.pdf.
76. Holtz S, Rohwedder T, Schneider R, (2012) The alternating linear scheme for tensor optimization in the Tensor Train format. SIAM Journal on Scientific Computing 34: A683-A713.
77. Sheppard PW, Rathinam M, Khammash M, (2013) Spsens: a software package for stochastic parameter sensitivity analysis of biochemical reaction networks. Bioinformatics 29: 140-142.