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Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity

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Abstract

Modern molecular biology has always been a great source of inspiration for computational science. Half a century ago, the challenge from understanding macromolecular dynamics has led the way for computations to be part of the tool set to study molecular biology. Twenty-five years ago, the demand from genome science has inspired an entire generation of computer scientists with an interest in discrete mathematics to join the field that is now called bioinformatics. In this paper, we shall lay out a new mathematical theory for dynamics of biochemical reaction systems in a small volume (i.e., mesoscopic) in terms of a stochastic, discrete-state continuous-time formulation, called the chemical master equation (CME). Similar to the wavefunction in quantum mechanics, the dynamically changing probability landscape associated with the state space provides a fundamental characterization of the biochemical reaction system. The stochastic trajectories of the dynamics are best known through the simulations using the Gillespie algorithm. In contrast to the Metropolis algorithm, this Monte Carlo sampling technique does not follow a process with detailed balance. We shall show several examples how CMEs are used to model cellular biochemical systems. We shall also illustrate the computational challenges involved: multiscale phenomena, the interplay between stochasticity and nonlinearity, and how macroscopic determinism arises from mesoscopic dynamics. We point out recent advances in computing solutions to the CME, including exact solution of the steady state landscape and stochastic differential equations that offer alternatives to the Gilespie algorithm. We argue that the CME is an ideal system from which one can learn to understand “complex behavior” and complexity theory, and from which important biological insight can be gained.

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References

  1. Kurtz T G. The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 1972, 57(7): 2976–2978.

    Article  Google Scholar 

  2. Beard D A, Qian H. Chemical Biophysics: Quantitative Analysis of Cellular Systems. London: Cambridge Univ. Press, 2008.

    MATH  Google Scholar 

  3. Wilkinson D J. Stochastic Modeling for Systems Biology. New York: Chapman & Hall/CRC, 2006.

    Google Scholar 

  4. Schlögl F. Chemical reaction models for non-equilibrium phase transition. Z. Physik., 1972, 253(2): 147–161.

    Article  Google Scholar 

  5. Murray J D. Mathematical Biology: An Introduction. 3rd Ed., New York: Springer, 2002.

    Google Scholar 

  6. Qian H, Saffarian S, Elson E L. Concentration fluctuations in a mesoscopic oscillating chemical reaction system. Proc. Natl. Acad. Sci. USA, 2002, 99(16): 10376–10381.

    Article  MATH  MathSciNet  Google Scholar 

  7. Taylor H M, Karlin S K. An Introduction to Stochastic Modeling. 3rd Ed., New York: Academic Press, 1998.

    MATH  Google Scholar 

  8. Resat H, Wiley H S, Dixon D A. Probability-weighted dynamic Monte Carlo method for reaction kinetics simulations. J. Phys. Chem. B, 2001, 105(44): 11026–11034.

    Article  Google Scholar 

  9. Gardiner C W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. 3rd Ed., New York: Springer, 2004.

    MATH  Google Scholar 

  10. van Kampen N G. Stochastic Processes in Physics and Chemistry. 3rd Ed., Amsterdam: Elsevier Science, 2007.

    Google Scholar 

  11. Vellela M, Qian H. Stochastic dynamics and nonequilibrium thermodynamics of a bistable chemical system: The Schlögl model revisited. J. R. Soc. Interf., 2009, 6(39): 925–940.

    Google Scholar 

  12. Qian H, Qian M, Tang X. Thermodynamics of the general diffusion process: Time-reversibility and entropy production. J. Stat. Phys., 2002, 107(5/6): 1129–1141.

    Article  MATH  MathSciNet  Google Scholar 

  13. Schrödinger E. What Is Life? The Physical Aspect of the Living Cell. New York: Cambridge Univ. Press, 1944.

    Google Scholar 

  14. Nicolis G, Prigogine I. Self-Organization in Nonequilibrium Systems. New York: Wiley-Interscience, 1977.

    MATH  Google Scholar 

  15. Hänggi P, Grabert H, Talkner P, Thomas H. Bistable systems: Master equation versus Fokker-Planck modeling. Phys. Rev. A., 1984, 29(1): 371–378.

    Article  MathSciNet  Google Scholar 

  16. Baras F, Mansour M M, Pearson J E. Microscopic simulation of chemical bistability in homogeneous systems. J. Chem. Phys. 1996, 105(18): 8257–8261.

    Article  Google Scholar 

  17. Vellela M, Qian H. A quasistationary analysis of a stochastic chemical reaction: Keizer’s paradox. Bull. Math. Biol., 2007, 69(5): 1727–1746.

    Article  MATH  MathSciNet  Google Scholar 

  18. Keizer J. Statistical Thermodynamics of Nonequilibrium Processes. New York: Springer-Verlag, 1987.

    Google Scholar 

  19. Bishop L, Qian H. Stochastic bistability and bifurcation in a mesoscopic signaling system with autocatalytic kinase. Biophys. J., 2010. (in the press)

  20. Kussell E, Kishony R, Balaban N Q, Leibler S. Bacterial persistence: A model of survival in changing environments. Genetics, 2005, 169(4): 1804–1807.

    Article  Google Scholar 

  21. Turner B M. Histone acetylation and an epigenetic code. Bioessays, 2000, 22(9): 836–845.

    Article  Google Scholar 

  22. Jones P A, Takai D. The role of DNA methylation in mammalian epigenetics. Science, 2001, 293(5532): 1068–1070.

    Article  Google Scholar 

  23. Dodd I B, Micheelsen M A, Sneppen K, Thon G. Theoretical analysis of epigenetic cell memory by nucleosome modification. Cell, 2007, 129(4): 813–822.

    Article  Google Scholar 

  24. Zhu X M, Yin L, Hood L, Ao P. Robustness, stability and efficiency of phage λ genetic switch: Dynamical structure analysis. J. Bioinf. Compt. Biol., 2004, 2(4): 785–817.

    Article  Google Scholar 

  25. Ptashne M. On the use of the word “epigenetic”. Curr. Biol., 2007, 17(7): R233–R236.

    Article  Google Scholar 

  26. Mino H, Rubinstein J T, White J A. Comparison of algorithms for the simulation of action potentials with stochastic sodium channels. Ann. Biomed. Eng., 2002, 30(4): 578–587.

    Article  Google Scholar 

  27. Fox R F. Stochastic versions of the Hodgkin-Huxley equations. Biophys. J., 1997, 72(5): 2069–2074.

    Article  Google Scholar 

  28. Lamb H. Hydrodynamic. New York: Dover, 1945.

    Google Scholar 

  29. Morton-Firth C J, Bray D. Predicting temporal fluctuations in an intracellular signalling pathway J. Theoret. Biol., 1998, 192(1): 117–128.

    Article  Google Scholar 

  30. Elf J, Ehrenberg M. Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res., 2003, 13(11): 2475–2484.

    Article  Google Scholar 

  31. Vellela M, Qian H. On Poincaré-Hill cycle map of rotational random walk: Locating stochastic limit cycle in reversible Schnakenberg model. Proc. Roy. Soc. A: Math. Phys. Engr. Sci., 2009. (in the press)

  32. Dill K A, Bromberg S, Yue K, Fiebig K M, Yee D P, Thomas P D, Chan H S. Principles of protein-folding — A perspective from simple exact models. Prot. Sci., 1995, 4(4): 561–602.

    Article  Google Scholar 

  33. Šali A, Shakhnovich E I, Karplus M. How does a protein fold? Nature, 1994, 369(6477): 248–251.

    Article  Google Scholar 

  34. Socci N D, Onuchic J N. Folding kinetics of protein like heteropolymer. J. Chem. Phys., 1994, 101: 1519–1528.

    Article  Google Scholar 

  35. Shrivastava I, Vishveshwara S, Cieplak M, Maritan A, Banavar J R. Lattice model for rapidly folding protein-like heteropolymers. Proc. Natl. Acad. Sci. U.S.A, 1995, 92(20): 9206–9209.

    Article  Google Scholar 

  36. Klimov D K, Thirumalai D. Criterion that determines the foldability of proteins. Phys. Rev. Lett., 1996, 76(21): 4070–4073.

    Article  Google Scholar 

  37. Cieplak M, Henkel M, Karbowski J, Banavar J R. Master equation approach to protein folding and kinetic traps. Phys. Rev. Lett., 1998, 80(16): 3654–3657.

    Article  Google Scholar 

  38. Mélin R, Li H, Wingreen N, Tang C. Designability, thermodynamic stability, and dynamics in protein folding: A lattice model study. J. Chem. Phys., 1999, 110(2): 1252–1262.

    Article  Google Scholar 

  39. Ozkan S B, Bahar I, Dill K A. Transition states and the meaning of ϕ-values in protein folding kinetics. Nature Struct. Biol., 2001, 8(9): 765–769.

    Article  Google Scholar 

  40. Kachalo S, Lu H, Liang J. Protein folding dynamics via quantification of kinematic energy landscape. Phys. Rev. Lett., 2006, 96(5): 058106.

    Article  Google Scholar 

  41. Chan H S, Dill K A. Compact polymers. Macromolecules, 1989, 22(12): 4559–4573.

    Article  Google Scholar 

  42. Chan H S, Dill K A. The effects of internal constraints on the configurations of chain molecules. J. Chem. Phys., 1990, 92(5): 3118–3135.

    Article  Google Scholar 

  43. Liang J, Zhang J, Chen R. Statistical geometry of packing defects of lattice chain polymer from enumeration and sequential Monte Carlo method. J. Chem. Phys., 2002, 117(7): 3511–3521.

    Article  Google Scholar 

  44. Zhang J, Chen Y, Chen R, Liang J. Importance of chirality and reduced flexibility of protein side chains: A study with square and tetrahedral lattice models. J. Chem. Phys., 2004, 121(1): 592–603.

    Article  Google Scholar 

  45. Williams P D, Pollock D D, Goldstein R A. Evolution of functionality in lattice proteins. J. Mole. Graph. Modelling, 2001, 19(1): 150–156.

    Article  Google Scholar 

  46. Bloom J D, Wilke C O, Arnold F H, Adami C. Stability and the evolvability of function in a model protein. Biophys. J., 2004, 86(5): 2758–2764.

    Article  Google Scholar 

  47. Lu H M, Liang J. A model study of protein nascent chain and cotranslational folding using hydrophobic-polar residues. Prot. Struct. Funct. Bioinf., 2008, 70(2): 442–449.

    Article  Google Scholar 

  48. Cao Y, Liang J. Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability. BMC Syst. Biol., 2008, 2: 30.

    Article  Google Scholar 

  49. Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, van der Vorst H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. 2nd Ed., Philadelphia, PA: SIAM, 1994.

    Google Scholar 

  50. Lehoucq R, Sorensen D, Yang C. Arpack Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Philadelphia, PA: SIAM, 1998.

    Google Scholar 

  51. Cao Y, Lu H M, Liang J. Stochastic probability landscape model for switching efficiency, robustness, and differential threshold for induction of genetic circuit in phage λ. In Proc. the 30th Annual Int. Conf. IEEE Eng. Med. Biol. Soc., Vancouver, Canada, Aug. 20–24, 2008, pp.611–614.

  52. Gardner T S, Canter C R, Collins J J. Construction of a genetic toggle switch in Escherichia coli. Nature, 2000, 403(6767): 339–342.

    Article  Google Scholar 

  53. Kepler T B, Elston T C. Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations. Biophys. J., 2001, 81(6): 3116–3136.

    Article  Google Scholar 

  54. Schultz D, Onuchic J N, Wolynes P G. Understanding stochastic simulations of the smallest genetic networks. J. Chem. Phys., 2007, 126(24): 245102.

    Article  Google Scholar 

  55. Kim K Y, Wang, J. Potential energy landscape and robustness of a gene regulatory network: Toggle Switch. PLoS Comput. Biol., 2007, 3(3): e60.

    Article  MathSciNet  Google Scholar 

  56. Wang J, Xu L, Wang E. Potential landscape and flux framework of nonequilibrium networks: Robustness, dissipation, and coherence of biochemical oscillations. Proc. Natl. Acad. Sci. U.S.A., 2008, 105(34): 12271–12276.

    Article  Google Scholar 

  57. Ptashne M. Genetic Switch: Phage Lambda Revisited. New York: Cold Spring Harbor Laboratory Press, 2004.

    Google Scholar 

  58. Arkin A, Ross J, McAdams H H. Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected Escherichia coli cells. Genetics, 1998, 149(44): 1633–1648.

    Google Scholar 

  59. Aurell E, Brown S, Johanson J, Sneppen K. Stability puzzles in phage λ. Phys. Rev. E., 2002, 65(5): 051914.

    Article  Google Scholar 

  60. Munsky B, Khammash M. The finite state projection algorithm for the solution of the chemical master equation. J. Chem. Phys., 2006, 124(4): 044104.

    Article  Google Scholar 

  61. Munsky B, Khammash M. A multiple time interval finite state projection algorithm for the solution to the chemical master equation. J. Comput. Phys., 2007, 226(1): 818–835.

    Article  MATH  MathSciNet  Google Scholar 

  62. Macnamara S, Bersani A M, Burrage K, Sidje R B. Stochastic chemical kinetics and the total quasi-steady-state assumption: Application to the stochastic simulation algorithm and chemical master equation. J. Chem. Phys., 2008, 129(9): 095105.

    Article  Google Scholar 

  63. Datta B N. Numerical Linear Algebra and Applications. Brooks/Cole Pub. Co., 1995.

  64. Golub G H, van Loan C F. Matrix Computations. Johns Hopkins Univ. Press, 1996.

  65. Sidje R B. Expokit: A software package for computing matrix exponentials. ACM Trans. Math. Softw., 1998, 24(1): 130–156.

    Article  MATH  Google Scholar 

  66. Lu H M, Liang J. Perturbation-based Markovian transmission model for probing allosteric dynamics of large macromolecular assembling: A study of GroEL-GroES. PLoS Comput. Biol., 2009, 5(10): e1000526.

    Article  Google Scholar 

  67. Cao Y, Gillespie D T, Petzold L R. The slow-scale stochastic simulation algorithm. J. Chem. Phys., 2005, 122(1): 14116.

    Article  Google Scholar 

  68. Cao Y, Liang J. Nonlinear coupling for improved stochastic network model: A study of Schnakenberg model. In Proc. the 3rd Symp. Optimiz. Syst. Biol., Zhangjiajie, China, Sept. 20–22, 2009, pp.379–386.

  69. Schnakenberg J. Simple chemical reaction systems with limit cycle behaviour. J. Theoret. Biol., 1979, 81(3): 389–400.

    Article  MathSciNet  Google Scholar 

  70. Qian H. Open-system nonequilibrium steady state: Statistical thermodynamics, fluctuations, and chemical oscillations. J. Phys. Chem. B, 2006, 110(31): 15063–15074.

    Article  Google Scholar 

  71. Goutsias J. Classical versus stochastic kinetics modeling of biochemical reaction systems. Biophys. J., 2007, 92(7): 2350–2365.

    Article  Google Scholar 

  72. Uribe C A, Verghese G C. Mass fluctuation kinetics: Capturing stochastic effects in systems of chemical reactions through coupled mean-variance computations. J. Chem. Phys., 2007, 126(2): 024109.

    Article  Google Scholar 

  73. Keizer J. On the macroscopi equivalence of descriptions of fluctuations for chemical reactions. J. Math. Phys., 1977, 18: 1316–1321.

    Article  Google Scholar 

  74. Mitchell M. Complexity: A Guided Tour. London: Oxford Univ. Press, 2009.

    Google Scholar 

  75. Laughlin R B, Pines D, Schmalian J, Stojković B P, Wolynes P G. The middle way. Proc. Natl. Acad. Sci. USA, 2000, 97(1): 32–37.

    Article  Google Scholar 

  76. Qian H, Shi P Z, Xing J. Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity. Physical Chemistry Chemical Physics, 2009, 11(24): 4861–4870.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Bioengineering, University of Illinois at Chicago, Chicago, IL, 60607, U.S.A.

    Jie Liang

  2. Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China

    Jie Liang

  3. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195, U.S.A.

    Hong Qian

  4. Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing, 100190, China

    Hong Qian

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  1. Jie Liang
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  2. Hong Qian
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Correspondence to Jie Liang.

Additional information

This work is supported by US NIH under Grant Nos. GM079804, GM081682, GM086145, GM068610, NSF of USA under Grant Nos. DBI-0646035 and DMS-0800257, and ‘985’ Phase II Grant of Shanghai Jiao Tong University under Grant No. T226208001.

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Liang, J., Qian, H. Computational Cellular Dynamics Based on the Chemical Master Equation: A Challenge for Understanding Complexity. J. Comput. Sci. Technol. 25, 154–168 (2010). https://doi.org/10.1007/s11390-010-9312-6

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  • DOI: https://doi.org/10.1007/s11390-010-9312-6

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