Stochastic Simulation for Spatial Modelling of Dynamic Processes in a Living Cell

  • Kevin Burrage
  • Pamela M. Burrage
  • André Leier
  • Tatiana Marquez-Lago
  • Dan V. NicolauJr
Chapter

Abstract

One of the fundamental motivations underlying computational cell biology is to gain insight into the complicated dynamical processes taking place, for example, on the plasma membrane or in the cytosol of a cell. These processes are often so complicated that purely temporal mathematical models cannot adequately capture the complex chemical kinetics and transport processes of, for example, proteins or vesicles. On the other hand, spatial models such as Monte Carlo approaches can have very large computational overheads. This chapter gives an overview of the state of the art in the development of stochastic simulation techniques for the spatial modelling of dynamic processes in a living cell.

Keywords

Plasma membrane Chemical kinetics Gene regulation Stochastic simulation algorithm Multiscale stochastic modelling Diffusion Delayed reactions Stochastic simulators 

References

  1. 1.
    Ander M, Beltrao P, Di Ventura B et al (2004) SmartCell, a framework to simulate cellular processes that combines stochastic approximation with diffusion and localisation: analysis of simple networks. Syst Biol 1:129–138CrossRefGoogle Scholar
  2. 2.
    Anderson RG, Jacobson K (2002) A role for lipid shells in targeting proteins to caveolae, rafts, and other lipid domains. Science 296:1821–1825CrossRefGoogle Scholar
  3. 3.
    Andrews SS, Bray D (2004) Stochastic simulation of chemical reactions with spatial resolution and single molecule detail. Phys Biol 1:137–151CrossRefGoogle Scholar
  4. 4.
    Andrews SS, Addy NJ, Brent R, Arkin AP (2010) Detailed simulations of cell biology with Smoldyn 2.1. PLoS Comput Biol 6:e1000705Google Scholar
  5. 5.
    Arjunan SNV, Tomita M (2009) Modeling reaction-diffusion of molecules on surface and in volume spaces with the E-cell system. IJCSIS 3:10060913Google Scholar
  6. 6.
    Arjunan SNV, Tomita M (2010) A new multicompartmental reaction-diffusion modeling method links transient membrane attachment of E. coli MinE to E-ring formation. Syst Synth Biol 4:35–53CrossRefGoogle Scholar
  7. 7.
    Baras F, Mansour MM (1996) Reaction-diffusion master equation: a comparison with microscopic simulations. Phys Rev E 54(6):6139–6148CrossRefGoogle Scholar
  8. 8.
    Barrio M, Burrage K, Leier A, Tian T (2006) Oscillatory regulation of Hes1: discrete stochastic delay modelling and simulation. PLoS Comput Biol 2(9):e117CrossRefGoogle Scholar
  9. 9.
    Boulianne L, Al Assaad S, Dumontier M, Gross WJ (2008) GridCell: a stochastic particle-based biological system simulator. BMC Syst Biol 2:66CrossRefGoogle Scholar
  10. 10.
    Burrage K, Tian T, Burrage PM (2004) A multi-scaled approach for simulating chemical reaction systems. Prog Biophys Mol Biol 85:217–234CrossRefGoogle Scholar
  11. 11.
    Burrage PM, Burrage K (2002) A variable stepsize implementation for stochastic differential equations. SIAM J Sci Comput 24(3):848–864MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Burrage PM, Burrage K, Kurowski K, Lorenc M, Nicolau DV, Swain M, Ragan M (2009) A parallel plasma membrane simulation, In: Guerrero J (ed) Proceedings of 1st international workshop on high performance computational systems biology (HiBi2009), Conference Publishing Services, IEEE Computer Society, Trento, Italy, 14–16 October 2009, pp 105–112, ISBN: 978-0-7695-3809-9Google Scholar
  13. 13.
    Burrage PM, Herdiana R, Burrage K (2004) Adaptive stepsize based on control theory for SDEs. J Comput Appl Math 170:317–336MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Chopard B, Frachebourg L, Droz M (1994) Multiparticle lattice gas automata for reaction diffusion systems. Int J Mod Phys C 5:47–63CrossRefGoogle Scholar
  15. 15.
    Chopard B, Droz M (1998) Cellular automata modeling of physical systems. Cambridge University Press, Cambridge, UKMATHCrossRefGoogle Scholar
  16. 16.
    Collins FC, Kimball GE (1949) Diffusion-controlled reaction rates. J Colloid Sci 4:425–437CrossRefGoogle Scholar
  17. 17.
    Crampin E, Smith N, Hunter P (2004) Multi-scale modelling and the IUPS Physiome Project. J Mol Histol 35(7):707–714CrossRefGoogle Scholar
  18. 18.
    Dobrzyński M, Rodríguez JV, Kaandorp JA, Blom JG (2007) Computational methods for diffusion-influenced biochemical reactions. Bioinformatics 23:1969–1977CrossRefGoogle Scholar
  19. 19.
    Drawert B, Lawson MJ, Petzold L, Khammash M (2010) The diffusive finite state projection algorithm for efficient simulation of the stochastic reaction-diffusion master equation. J Chem Phys 132:074101. doi:10.1063/1.3310809CrossRefGoogle Scholar
  20. 20.
    Edidin M (2003) The state of lipid rafts: from model membranes to cells. Annu Rev Biophys Biomol Struct 32:257–283CrossRefGoogle Scholar
  21. 21.
    Elf J, Doncic A, Ehrenberg M (2003) Mesoscopic reaction-diffusion in intracellular signaling. In: Bezrukov SM, Frauenfelder H, Moss F (eds) Fluctuations and noise in biological, biophysical, and biomedical systems, Proceedings of the SPIE 5110, pp 114–125Google Scholar
  22. 22.
    Elf J, Ehrenberg M (2004) Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. Syst Biol 1:230–236CrossRefGoogle Scholar
  23. 23.
    Engblom S (2009) Galerkin spectral method applied to the chemical master equation. Commun Comput Phys v5(i5):871–896Google Scholar
  24. 24.
    Engblom S, Ferm L, Hellander A, Loetstedt P (2009) Simulation of stochastic reaction-diffusion processes on unstructured meshes. SIAM J Sci Comput 31:1774–1797MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Erban R, Chapman SJ, Maini PK (2007) A practical guide to stochastic simulations of reaction–diffusion processes. arXiv:0704.1908Google Scholar
  26. 26.
    Erban R, Chapman SJ (2009) Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions. Phys Biol 6:046001CrossRefGoogle Scholar
  27. 27.
    Fange D, Elf J (2006) Noise-induced Min phenotypes in E. coli. PLoS Comput Biol 2:e80Google Scholar
  28. 28.
    Gibson MA, Bruck J (2000) Efficient exact atochastic simulation of chemical systems with many species and many channels. J Phys Chem 104(9):1876–1889Google Scholar
  29. 29.
    Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361CrossRefGoogle Scholar
  30. 30.
    Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115(4):1716–1733CrossRefGoogle Scholar
  31. 31.
    Goodwin BC (1965) Oscillatory behavior in enzymatic control processes. Adv Enzyme Regul 3:425–438CrossRefGoogle Scholar
  32. 32.
    Hattne J, Fange D, Elf J (2005) Stochastic reaction-diffusion simulation with MesoRD. Bioinformatics 21:2923–2924CrossRefGoogle Scholar
  33. 33.
    Hedley W, Nelson MR, Bullivant DP, Nielsen PF (2001) A short introduction to CellML. Philos Trans R Soc Lond A 359:1073–1089MATHCrossRefGoogle Scholar
  34. 34.
    Isaacson SA (2009) The reaction-diffusion master equation as an asymptotic approximation of diffusion to a small target. SIAM J Appl Math 70:77–111MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Jahnke T, Galan S (2008) Solving chemical master equations by an adaptive wavelet method. In: Simos TE, Psihoyios G, Tsitouras C (eds) Numerical analysis and applied mathematics: international conference on numerical analysis and applied mathematics 2008, vol. 1048 of AIP Conference Proceedings, Psalidi, Kos, Greece, 16–20 September 2008, pp. 290–293Google Scholar
  36. 36.
    Kerr RA, Bartol TM, Kaminski B et al (2008) Fast Monte Carlo simulation methods for biological reaction-diffusion systems in solution and on surfaces. SIAM J Sci Comput 30:3126MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Springer-Verlag, BerlinMATHGoogle Scholar
  38. 38.
    Kurtz TG (1972) The relationship between stochastic and deterministic models for chemical reactions. J Chem Phys 57(7):2976–2978CrossRefGoogle Scholar
  39. 39.
    Kusumi A, Koyama-Honda I, Suzuki K (2004) Molecular dynamics and interactions for creation of stimulation-induced stabilized rafts from small unstable steady-state rafts. Traffic 5:213–230CrossRefGoogle Scholar
  40. 40.
    Lampoudi S, Gillespie DT, Petzold L (2009) The multinomial simulation algorithm for discrete stochastic simulation of reaction-diffusion systems. J Chem Phys 130:094104CrossRefGoogle Scholar
  41. 41.
    Leier A, Marquez-Lago TT (2011) Correction factors for boundary diffusion and bimolecular reactions in reaction-diffusion master equations. To be submittedGoogle Scholar
  42. 42.
    Loew LM, Schaff JC (2001) The virtual cell: a software environment for computational cell biology. Trends Biotechnol 19(10):401–406CrossRefGoogle Scholar
  43. 43.
    MacNamara S, Bersani AM, Burrage K, Sidje RB (2008) Stochastic chemical kinetics and the total quasi-steady-state assumption: application to the stochastic simulation algorithm and chemical master equation. J Chem Phys 129(9):095105CrossRefGoogle Scholar
  44. 44.
    MacNamara S, Burrage K, Sidje RB (2008) Multiscale modeling of chemical kinetics via the master equation. SIAM J Multiscale Model Simul 6(4):1146–1168MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Marquez-Lago TT, Burrage K (2007) Binomial tau-leap spatial stochastic simulation algorithm for applications in chemical kinetics. J Chem Phys 127:104101CrossRefGoogle Scholar
  46. 46.
    Marquez-Lago TT, Leier A, Burrage K (2010) Probability distributed time delays: integrating spatial effects into temporal models. BMC Syst Biol 4:19CrossRefGoogle Scholar
  47. 47.
    Marsh BJ (2006) Toward a ‘visible cell’ … and beyond. Aust Biochemist 37:5–10Google Scholar
  48. 48.
    Mélykúti B, Burrage K, Zygalakis KC (2010) Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation. J Chem Phys 132:1Google Scholar
  49. 49.
    Morton-Firth CJ, Bray D (1998) Predicting temporal fluctuations in an intracellular signalling pathway. J Theor Biol 192:117–128CrossRefGoogle Scholar
  50. 50.
    Murase K, Fujiwara T, Umemura TY (2004) Ultrafine membrane compartments for molecular diffusion as revealed by single molecule techniques. Biophys J 86:4075–4093CrossRefGoogle Scholar
  51. 51.
    Nicolau Jr, DV, Burrage K, Parton RG et al (2006) Identifying optimal lipid raft characteristics required to promote nanoscale protein-protein interactions on the plasma membrane. Mol Cell Biol 26(1):313–323CrossRefGoogle Scholar
  52. 52.
    Nicolau Jr, DV, Hancock JF, Burrage K (2007) Sources of anomalous diffusion on cell membranes: a Monte Carlo study. Biophys J 92:1975–1987CrossRefGoogle Scholar
  53. 53.
    Nicolau Jr, DV, Burrage K (2008) Stochastic simulation of chemical reactions in spatially complex media. Comput Math Appl 55(5):1007–1018MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Oppelstrup T, Bulatov VV, Donev A et al (2006) First-passage kinetic Monte Carlo method. Phys Rev Lett 97:230602CrossRefGoogle Scholar
  55. 55.
    Peleš S, Munsky B, Khammash M (2006) Reduction and solution of the chemical master equation using time scale separation and finite state projection. J Chem Phys 125:204104–1–13Google Scholar
  56. 56.
    Plimpton SJ, Slepoy A (2003) ChemCell: a particle-based model of protein chemistry and diffusion in microbial cells. Sandia National Laboratory Technical Report 2003, Albuquerque, NMGoogle Scholar
  57. 57.
    Plimpton SJ, Slepoy A (2005) Microbial cell modeling via reacting diffusing particles. J Physiol 16:305Google Scholar
  58. 58.
    Prior IA, Muncke C, Parton RG et al (2003) Direct visualization of Ras proteins in spatially distinct cell surface microdomains. J Cell Biol 160:165–170CrossRefGoogle Scholar
  59. 59.
    Ridgway D, Broderick G, Lopez-Campistrous A et al (2008) Coarse-grained molecular simulation of diffusion and reaction kinetics in a crowded virtual cytoplasm. Biophys J 94:3748–3759CrossRefGoogle Scholar
  60. 60.
    Rodríguez JV, Kaandorp JA, Dobrzyński M, Blom JG (2006) Spatial stochastic modelling of the phosphoenolpyruvate-dependent phosphotransferase (PTS) pathway in Escherichia coli. Bioinformatics 22:1895–1901CrossRefGoogle Scholar
  61. 61.
    Sanford C, Yip MLK, White C, Parkinson J (2006) \(\mathrm{Cell} + +\)–simulating biochemical pathways. Bioinformatics 22:2918–2925CrossRefGoogle Scholar
  62. 62.
    Séguis J-C, Burrage K, Erban R, Kay D (2010) Efficient numerical model for lipid rafts and protein interactions on a cell membrane, in preparationGoogle Scholar
  63. 63.
    Sharma P, Varma R, Sarasij RC et al (2004) Nanoscale organization of multiple GPI-anchored proteins in living cell membranes. Cell 116:577–589CrossRefGoogle Scholar
  64. 64.
    Simons K, Toomre D (2000) Lipid rafts and signal transduction. Nat Rev Mol Cell Biol 1:31–39CrossRefGoogle Scholar
  65. 65.
    Singer SJ, Nicolson GL (1972) The fluid mosaic model of the structure of cell membranes. Science 175:720–731CrossRefGoogle Scholar
  66. 66.
    Stiles JR, Bartol TM (2001) Monte Carlo methods for simulating realistic synaptic microphysiology using MCell. CRC Press, Boca Raton, FLGoogle Scholar
  67. 67.
    Stundzia AB, Lumsden CJ (1996) Stochastic simulation of coupled reaction-diffusion processes. J Comp Physiol 127:196–207MATHGoogle Scholar
  68. 68.
    Takahashi K, Ishikawa N, Sadamoto Y et al (2003) E-Cell 2: multi-platform E-Cell simulation system. Bioinformatics 19:1727–1729CrossRefGoogle Scholar
  69. 69.
    Takahashi K, Kaizu K, Hu B, Tomita M (2004) A multi-algorithm, multi-timescale method for cell simulation. Bioinformatics 20:538–546CrossRefGoogle Scholar
  70. 70.
    Takahashi K, Arjunan SNV, Tomita M (2005) Space in systems biology of signaling pathways – towards intracellular molecular crowding in silico. FEBS Lett 579:1783–1788CrossRefGoogle Scholar
  71. 71.
    Takahashi K, Tănase-Nicola S, ten Wolde PR (2010) Spatio-temporal correlations can drastically change the response of a MAPK pathway. PNAS 107(6):2473–2478CrossRefGoogle Scholar
  72. 72.
    Tian T, Burrage K (2004) Binomial leap methods for simulating stochastic chemical kinetics. J Chem Phys 121:10356–10364CrossRefGoogle Scholar
  73. 73.
    Tian T, Harding A, Westbury E, Hancock J (2007) Plasma membrane nano-switches generate robust high-fidelity Ras signal transduction. Nat Cell Biol 9:905–914CrossRefGoogle Scholar
  74. 74.
    Tomita M, Hashimoto K, Takahashi K et al (1999) E-CELL: software environment for whole-cell simulation. Bioinformatics 15:72–84CrossRefGoogle Scholar
  75. 75.
    Turner TE, Schnell S, Burrage K (2004) Stochastic approaches for modelling in vivo reactions. Comput Biol Chem 28:165–178MATHCrossRefGoogle Scholar
  76. 76.
    van Zon JS, ten Wolde PR (2005) Green’s-function reaction dynamics: a particle-based approach for simulating biochemical networks in time and space. J Chem Phys 123: 1–16Google Scholar
  77. 77.
    van Zon JS, ten Wolde PR (2005) Simulating biochemical networks at the particle level and in time and space: Green’s function reaction dynamics. Phys Rev Lett 94:128103CrossRefGoogle Scholar
  78. 78.
    Wils S, De Schutter E (2009) STEPS: modeling and simulating complex reaction-diffusion systems with Python. Front Neuroinform 3:15CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Kevin Burrage
    • 1
    • 2
  • Pamela M. Burrage
  • André Leier
  • Tatiana Marquez-Lago
  • Dan V. NicolauJr
  1. 1.Computing LaboratoryOxfordUK
  2. 2.Department of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia

Personalised recommendations