›› 2010, Vol. 25 ›› Issue (1): 154-168.

• Special Issue on Computational Challenges from Modern Molecular Biology • Previous Articles     Next Articles

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

Jie Liang1,2 (梁杰) and Hong Qian3,4(钱纮)   

  1. 1Department of Bioengineering, University of Illinois at Chicago, Chicago, IL 60607, U.S.A.
    2Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai 200240, China
    3Department of Applied Mathematics, University of Washington, Seattle, WA 98195, U.S.A.
    4Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing 100190, China
  • Received:2009-08-29 Revised:2009-11-16 Online:2010-01-05 Published:2010-01-05
  • About author:
    Jie Liang is a professor in the Department of Bioengineering at the University of Illinois at Chicago, and holds a visiting position at Shanghai Jiao Tong University. He received his B.S. degree from Fudan University (1986), MCS and Ph.D. degree from the University of Illinois at Urbana-Champaign (1994). He was an NSF CISE postdoctoral research associate (1994$sim$1996) at the Beckman Institute and National Center for Supercomputing and its Applications (NCSA) in Urbana, Illinois. He was a visiting fellow at the Institute of Mathematics and Applications at Minneapolis, Minnesota in 1996, and an Investigator at SmithKline Beecham Pharmaceuticals in Philadelphia from 1997 to 1999. He was a recipient of the NSF CAREER award in 2003. He is a fellow of American Institute of Medical and Biological Engineering, and served as regular member of the NIH Biological Data Management and Analysis study section. His research interests are in biogeometry, biophysics, computational proteomics, stochastic molecular networks, and cellular pattern formation. His recent work can be accessed at (http://www.uic.edu/$~jliang).
    Hong Qian received his B.S. degree in astrophysics from Peking University. He worked on fluorescence correlation spectroscopy (FCS) and single-particle tracking (SPT) and obtained his Ph.D. degree in biochemistry from Washington University (St. Louis). His research interests turned to theoretical biophysical chemistry and mathematical biology when he was a postdoctoral fellow at the University of Oregon and at the California Institute of Technology. In that period of time, he worked on protein thermodynamics, fluctuations and folding. Between 1994 and 1997, he was with the Department of Biomathematics at the UCLA School of Medicine, where he worked on the theory of motor proteins and single-molecule biophysics. This work led to his current interest in mesoscopic open chemical systems. He joined the University of Washington (Seattle) in 1997 and is now professor of applied mathematics, and an adjunct professor of bioengineering. His current research is in stochastic analysis and statistical physics of cellular systems. His recent book "Chemical Biophysics: Quantitative Analysis of Cellular Systems", co-authored with Daniel A. Beard, has been published by the Cambridge University Press.
  • Supported by:

    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.

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.

[1] Kurtz T G. The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys., 1972, 57(7): 2976-2978.
[2] Beard D A, Qian H. Chemical Biophysics: Quantitative Analysis of Cellular Systems. London: Cambridge Univ. Press, 2008.
[3] Wilkinson D J. Stochastic Modeling for Systems Biology. New York: Chapman & Hall/CRC, 2006.
[4] Schl¨ogl F. Chemical reaction models for non-equilibrium phase transition. Z. Physik., 1972, 253(2): 147-161.
[5] Murray J D. Mathematical Biology: An Introduction. 3rd Ed., New York: Springer, 2002.
[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.
[7] Taylor H M, Karlin S K. An Introduction to Stochastic Modeling. 3rd Ed., New York: Academic Press, 1998.
[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.
[9] Gardiner C W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. 3rd Ed., New York: Springer, 2004.
[10] van Kampen N G. Stochastic Processes in Physics and Chemistry. 3rd Ed., Amsterdam: Elsevier Science, 2007.
[11] Vellela M, Qian H. Stochastic dynamics and nonequilibrium thermodynamics of a bistable chemical system: The Schl¨ogl model revisited. J. R. Soc. Interf., 2009, 6(39): 925-940.
[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.
[13] Schr¨odinger E. What Is Life? The Physical Aspect of the Living Cell. New York: Cambridge Univ. Press, 1944.
[14] Nicolis G, Prigogine I. Self-Organization in Nonequilibrium Systems. New York: Wiley-Interscience, 1977.
[15] H¨anggi P, GrabertH, Talkner P, Thomas H. Bistable systems: Master equation versus Fokker-Planck modeling. Phys. Rev. A., 1984, 29(1): 371-378.
[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.
[17] Vellela M, Qian H. A quasistationary analysis of a stochastic chemical reaction: Keizer’s paradox. Bull. Math. Biol., 2007, 69(5): 1727-1746.
[18] Keizer J. Statistical Thermodynamics of Nonequilibrium Processes. New York: Springer-Verlag, 1987.
[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.
[21] Turner B M. Histone acetylation and an epigenetic code. Bioessays, 2000, 22(9): 836-845.
[22] Jones P A, Takai D. The role of DNA methylation in mammalian epigenetics. Science, 2001, 293(5532): 1068-1070.
[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.
[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.
[25] Ptashne M. On the use of the word “epigenetic”. Curr. Biol., 2007, 17(7): R233-R236.
[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.
[27] Fox R F. Stochastic versions of the Hodgkin-Huxley equations. Biophys. J., 1997, 72(5): 2069-2074.
[28] Lamb H. Hydrodynamic. New York: Dover, 1945.
[29] Morton-Firth C J, Bray D. Predicting temporal fluctuations in an intracellular signalling pathway J. Theoret. Biol., 1998, 192(1): 117-128.
[30] Elf J, Ehrenberg M. Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Res., 2003, 13(11): 2475-2484.
[31] Vellela M, Qian H. On Poincar′e-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.
[33] ˇSali A, Shakhnovich E I, Karplus M. How does a protein fold? Nature, 1994, 369(6477): 248-251.
[34] Socci N D, Onuchic J N. Folding kinetics of protein like heteropolymer. J. Chem. Phys., 1994, 101: 1519-1528.
[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.
[36] Klimov D K, Thirumalai D. Criterion that determines the foldability of proteins. Phys. Rev. Lett., 1996, 76(21): 4070- 4073.
[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.
[38] M′elin 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.
[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.
[40] Kachalo S, Lu H, Liang J. Protein folding dynamics via quantification of kinematic energy landscape. Phys. Rev. Lett., 2006, 96(5): 058106.
[41] Chan H S, Dill K A. Compact polymers. Macromolecules, 1989, 22(12): 4559-4573.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[53] Kepler T B, Elston T C. Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations. Biophys. J., 2001, 81(6): 3116-3136.
[54] Schultz D, Onuchic J N, Wolynes P G. Understanding stochastic simulations of the smallest genetic networks. J. Chem. Phys., 2007, 126(24): 245102.
[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.
[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.
[57] Ptashne M. Genetic Switch: Phage Lambda Revisited. New York: Cold Spring Harbor Laboratory Press, 2004.
[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.
[59] Aurell E, Brown S, Johanson J, Sneppen K. Stability puzzles in phage λ. Phys. Rev. E., 2002, 65(5): 051914.
[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.
[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.
[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.
[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.
[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.
[67] Cao Y, Gillespie D T, Petzold L R. The slow-scale stochastic simulation algorithm. J. Chem. Phys., 2005, 122(1): 14116.
[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.
[70] Qian H. Open-system nonequilibrium steady state: Statistical thermodynamics, fluctuations, and chemical oscillations. J. Phys. Chem. B, 2006, 110(31): 15063-15074.
[71] Goutsias J. Classical versus stochastic kinetics modeling of biochemical reaction systems. Biophys. J., 2007, 92(7): 2350- 2365.
[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.
[73] Keizer J. On the macroscopi equivalence of descriptions of fluctuations for chemical reactions. J. Math. Phys., 1977, 18: 1316-1321.
[74] Mitchell M. Complexity: A Guided Tour. London: Oxford Univ. Press, 2009.
[75] Laughlin R B, Pines D, Schmalian J, Stojkovi′c B P,Wolynes P G. The middle way. Proc. Natl. Acad. Sci. USA, 2000, 97(1): 32-37.
[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.

No related articles found!
Full text



No Suggested Reading articles found!

ISSN 1000-9000(Print)

CN 11-2296/TP

Editorial Board
Author Guidelines
Journal of Computer Science and Technology
Institute of Computing Technology, Chinese Academy of Sciences
P.O. Box 2704, Beijing 100190 P.R. China
E-mail: jcst@ict.ac.cn
  Copyright ©2015 JCST, All Rights Reserved