Abstract
The dynamics of how the constituent components of a natural system interact defines the spatio-temporal response of the system to stimuli. Modeling the kinetics of the processes that represent a biophysical system has long been pursued with the aim of improving our understanding of the studied system. Due to the unique properties of biological systems, in addition to the usual difficulties faced in modeling the dynamics of physical or chemical systems, biological simulations encounter difficulties that result from intrinsic multi-scale and stochastic nature of the biological processes.
This chapter discusses the implications for simulation of models involving interacting species with very low copy numbers, which often occur in biological systems and give rise to significant relative fluctuations. The conditions necessitating the use of stochastic kinetic simulation methods and the mathematical foundations of the stochastic simulation algorithms are presented. How the well-organized structural hierarchies often seen in biological systems can lead to multi-scale problems and the possible ways to address the encountered computational difficulties are discussed. We present the details of the existing kinetic simulation methods and discuss their strengths and shortcomings. A list of the publicly available kinetic simulation tools and our reflections for future prospects are also provided.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Gillespie DT (1976) A general method for numerically simulating stochastic time evolution of coupled chemical-reactions. Journal of Computational Physics 22, 403–434.
Gillespie DT (1977) Exact stochastic simulation of coupled chemical-reactions. Journal of Physical Chemistry 81, 2340–2361.
Gillespie DT (1977) Concerning validity of stochastic approach to chemical-kinetics. Journal of Statistical Physics 16, 311–318.
Gillespie DT (1992) A rigorous derivation of the chemical master equation. Physica A 188, 404–425.
Arkin A, Ross J, and McAdams HH (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. Genetics 149, 1633–1648.
McAdams HH and Arkin A (1997) Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences, USA 94, 814–819.
McAdams HH and Arkin A (1999) It's a noisy business! Genetic regulation at the nanomolar scale. Trends in Genetics 15, 65–69.
Bortz AB, Kalos MH, and Lebowitz JL (1975) New algorithm for Monte-Carlo simulation of Ising spin systems. Journal of Computational Physics 17, 10–18.
Endy D and Brent R (2001) Modelling cellular behaviour. Nature 409, 391–395.
Gibson MA and Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. Journal of Physical Chemistry A 104, 1876–1889.
Goss PJE and Peccoud J (1998) Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. Proceedings of the National Academy of Sciences, USA 95, 6750–6755.
Kastner J, Solomon J, and Fraser S (2002) Modeling a Hox gene network in silico using a stochastic simulation algorithm. Developmental Biology 246, 122–131.
Kepler TB and Elston TC (2001) Stochasticity in transcriptional regulation: Origins, consequences, and mathematical representations. Biophysical Journal 81, 3116–3136.
Rao CV and Arkin AP (2003) Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. Journal of Chemical Physics 118, 4999–5010.
Simpson ML, Cox CD, and Sayler GS (2003) Frequency domain analysis of noise in autoregulated gene circuits. Proceedings of the National Academy of Sciences, USA 100, 4551–4556.
Smolen P, Baxter DA, and Byrne JH (1999) Effects of macromolecular transport and stochastic fluctuations on dynamics of genetic regulatory systems. American Journal of Physiology-Cell Physiology 277, C777–C790.
Levchenko A (2003) Dynamical and integrative cell signaling: challenges for the new biology. Biotechnol Bioeng 84, 773–782.
Haseltine EL and Rawlings JB (2002) Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. Journal of Chemical Physics 117, 6959–6969.
Haseltine EL and Rawlings JB (2005) On the origins of approximations for stochastic chemical kinetics. Journal of Chemical Physics 123, 164115.
Puchalka J and 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.
Fricke T and Wendt D (1995) The Markov Automaton – A new algorithm for simulating the time-evolution of large stochastic dynamic-systems. International Journal of Modern Physics C-Physics and Computers 6, 277–306.
Elf J, Doncic A, and Ehrenberg M (2003) Mesoscopic reaction-diffusion in intracellular signaling. In: SPIE's First International Symposium on Fluctuations and Noise, pp. 114–124.
Elf J and Ehrenberg M (2003) Fast evaluation of fluctuations in biochemical networks with the linear noise approximation. Genome Research 13, 2475–2484.
Resat H, Wiley HS, and Dixon DA (2001) Probability-weighted dynamic Monte Carlo method for reaction kinetics simulations. Journal of Physical Chemistry B 105, 11026–11034.
Resat H, Ewald JA, Dixon DA, and Wiley HS (2003) An integrated model of epidermal growth factor receptor trafficking and signal transduction. Biophysical Journal 85, 730–743.
Gillespie DT and Petzold LR (2003) Improved leap-size selection for accelerated stochastic simulation. Journal of Chemical Physics 119, 8229–8234.
Rathinam M, Petzold LR, Cao Y, and Gillespie DT (2003) Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method. Journal of Chemical Physics 119, 12784–12794.
Cao Y, Gillespie DT, and Petzold LR (2006) Efficient step size selection for the tau-leaping simulation method. Journal of Chemical Physics 124, 044109.
Chatterjee A, Mayawala K, Edwards JS, and Vlachos DG (2005) Time accelerated Monte Carlo simulations of biological networks using the binomial tau-leap method. Bioinformatics 21, 2136–2137.
Chatterjee A, Vlachos DG, and Katsoulakis MA (2005) Binomial distribution based tau-leap accelerated stochastic simulation. Journal of Chemical Physics 122, 024112.
Tian TH and Burrage K (2004) Binomial leap methods for simulating stochastic chemical kinetics. Journal of Chemical Physics 121, 10356–10364.
Pettigrew MF and Resat H (2007) A multinomial tau-leaping method for stochastic kinetic simulations. Journal of Chemical Physics 126, 084101.
Stundzia AB and Lumsden CJ (1996) Stochastic simulation of coupled reaction-diffusion processes. Journal of Computational Physics 127, 196–207.
Burke P, Schooler K, and Wiley HS (2001) Regulation of epidermal growth factor receptor signaling by endocytosis and intracellular trafficking. Molecular Biology of the Cell 12, 1897–1910.
Ozcelik S, Orr G, Hu D, Chii-Shiarng C, Resat H, Harms GS, Opresko LK, Wiley HS, and Colson SD (2004) FRET measurements between small numbers of molecules identifies subtle changes in receptor interactions. Proceedings of the International Society of Optical Engineering 5323, 119–127.
Viollier PH, Thanbichler M, McGrath PT, West L, Meewan M, McAdams HH, and Shapiro L (2004) Rapid and sequential movement of individual chromosomal loci to specific subcellular locations during bacterial DNA replication. Proceedings of the National Academy of Sciences, USA 101, 9257–9262.
McAdams HH and Shapiro L (2003) A bacterial cell-cycle regulatory network operating in time and space. Science 301, 1874–1877.
Judd EM, Ryan KR, Moerner WE, Shapiro L, and McAdams HH (2003) Fluorescence bleaching reveals asymmetric compartment formation prior to cell division in Caulobacter. Proceedings of the National Academy of Sciences, USA 100, 8235–8240.
Gardiner CW, McNeil KJ, Walls DF, and Matheson IS (1976) Correlations in stochastic theories of chemical reactions. Journal of Statistical Physics 14, 307–331.
Chaturvedi S, Gardiner CW, Matheson IS, and Walls DF (1977) Stochastic analysis of a chemical reaction with spatial and temporal structures. Journal of Statistical Physics 17, 469–489.
Gillespie DT (1992) Markov Processes: An Introduction for Physical Scientists. Academic Press, San Diego, California.
Pettigrew MF and Resat H (2005) Modeling signal transduction networks: a comparison of two stochastic kinetic simulation algorithms. Journal of Chemical Physics 123, 114707.
Ascher UM and Petzold LR (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia.
McQuarrie DA (1967) Stochastic approach to chemical kinetics. Journal of Applied Probability 4, 413–478.
Gibson MA and Bruck J (1998) An efficient algorithm for generating trajectories of stochastic gene regulation reactions. California Institute of Technology Report ETR026 .
Cao Y, Li H, and Petzold LR (2004) Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. Journal of Chemical Physics 121, 4059–4067.
Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. Journal of Chemical Physics 115, 1716–1733.
Cao Y, Gillespie DT, and Petzold LR (2005) Avoiding negative populations in explicit Poisson tau-leaping. Journal of Chemical Physics 123, 054104.
Gillespie DT (2000) The chemical Langevin equation. Journal of Chemical Physics 113, 297–306.
Gillespie DT (2002) The chemical Langevin and Fokker-Planck equations for the reversible isomerization reaction. Journal of Physical Chemistry A 106, 5063–5071.
Gillespie DT (1996) The multivariate Langevin and Fokker-Planck equations. American Journal of Physics 64, 1246–1257.
Morton-Firth CJ (1998) Ph.D. thesis. Stochastic simulation of cell signalling pathways. University of Cambridge, Cambridge, UK.
Blinov ML, Faeder JR, Goldstein B, and Hlavacek WS (2004) BioNetGen: software for rule-based modeling of signal transduction based on the interactions of molecular domains. Bioinformatics 20, 3289–3291.
Sauro HM, Hucka M, Finney A, Wellock C, Bolouri H, Doyle J, and Kitano H (2003) Next generation simulation tools: the Systems Biology Workbench and BioSPICE integration. Omics 7, 355–372.
Adalsteinsson D, McMillen D, and Elston TC (2004) Biochemical Network Stochastic Simulator (BioNetS): Software for stochastic modeling of biochemical networks. BMC Bioinformatics 5, 24.
Funahashi A, Tanimura N, Morohashi M, and Kitano H (2003) CellDesigner: A process diagram editor for gene-regulatory and biochemical networks. Bio Silico 1, 159–162.
Kitano H (2003) A graphical notation for biochemical networks. Bio Silico 1, 169–176.
Shapiro BE, Levchenko A, Meyerowitz EM, Wold BJ, and Mjolsness ED (2003) Cellerator: Extending a computer algebra system to include biochemical arrows for signal transduction simulations. Bioinformatics 19, 677–678.
Hoops S, Sahle S, Gauges R, Lee C, Pahle J, Simus N, Singhal M, Xu L, Mendes P, and Kummer U (2006) COPASI – A COmplex PAthway SImulator. Bioinformatics 22, 3067–3074.
Ramsey S, Orrell D, and Bolouri H (2005) Dizzy: Stochastic simulation of large-scale genetic regulatory networks. Journal of Bioinformatics and Computational Biology 3, 415–436.
Takahashi K, Ishikawa N, Sadamoto Y, Sasamoto H, Ohta S, Shiozawa A, Miyoshi F, Naito Y, Nakayama Y, and Tomita M (2003) E-Cell 2: Multi-platform E-Cell simulation system. Bioinformatics 19, 1727–1729.
Takahashi K, Kaizu K, Hu B, and Tomita M (2004) A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics 20, 538–546.
Dhar PK, Meng TC, Somani S, Ye L, Sakharkar K, Krishnan A, Ridwan AB, Wah SH, Chitre M, and Hao Z (2005) Grid cellware: The first grid-enabled tool for modelling and simulating cellular processes. Bioinformatics 21, 1284–1287.
Stiles JR and Bartol TM (2001) Monte Carlo methods for simulating realistic synaptic microphysiology using MCell. In: Computational Neuroscience: Realistic Modeling for Experimentalists. De Schutter E, ed., CRC Press, Boca Raton, pp. 87–127.
Hattne J, Fange D, and Elf J (2005) Stochastic reaction-diffusion simulation with MesoRD. Bioinformatics 21, 2923–2924.
Ander M, Beltrao P, Di Ventura B, Ferkinghoff-Borg J, Foglierini M, Kaplan A, Lemerle C, Tomas-Oliveira I, and Serrano L (2004) SmartCell, a framework to simulate cellular processes that combines stochastic approximation with diffusion and localisation: Analysis of simple networks. System Biology (Stevenage) 1, 129–138.
Cao Y and Petzold LR 2005) Trapezoidal tau-leaping formula for the stochastic simulation of bio-chemical systems. In: Proceedings of Foundations of Systems Biology in Engineering (FOSBE 2005), pp. 149–152.
Schmidt H and Jirstrand M (2006) Systems biology toolbox for MATLAB: A computational platform for research in systems biology. Bioinformatics 22, 514–515.
Slepchenko BM, Schaff JC, Macara I, and Loew LM (2003) Quantitative cell biology with the virtual cell. Trends Cell Biology 13, 570–576.
Moraru, II, Schaff JC, Slepchenko BM, and Loew LM (2002) The virtual cell: An integrated modeling environment for experimental and computational cell biology. Annals of the New York Academy of Sciences 971, 595–596.
Isaacson SA and Peskin CS (2004) Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations. Courant Institute of Mathematical Sciences Report.
Mattheyses T and Simmons M (2004) Hybrid simulation of cellular behavior. Bioinformatics 20, 316–322.
Cao Y, Gillespie DT, and Petzold L (2005) The slow-scale stochastic simulation algorithm. Journal of Chemical Physics 122, 014116.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Humana Press, a part of Springer Science+Business Media, LLC
About this protocol
Cite this protocol
Resat, H., Petzold, L., Pettigrew, M.F. (2009). Kinetic Modeling of Biological Systems. In: Ireton, R., Montgomery, K., Bumgarner, R., Samudrala, R., McDermott, J. (eds) Computational Systems Biology. Methods in Molecular Biology, vol 541. Humana Press. https://doi.org/10.1007/978-1-59745-243-4_14
Download citation
DOI: https://doi.org/10.1007/978-1-59745-243-4_14
Published:
Publisher Name: Humana Press
Print ISBN: 978-1-58829-905-5
Online ISBN: 978-1-59745-243-4
eBook Packages: Springer Protocols