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Discrete-continuous reaction-diffusion model with mobile point-like sources and sinks

  • Svyatoslav Kondrat
  • Olav Zimmermann
  • Wolfgang Wiechert
  • Eric von Lieres
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Abstract.

In many applications in soft and biological physics, there are multiple time and length scales involved but often with a distinct separation between them. For instance, in enzyme kinetics, enzymes are relatively large, move slowly and their copy numbers are typically small, while the metabolites (being transformed by these enzymes) are often present in abundance, are small in size and diffuse fast. It seems thus natural to apply different techniques to different time and length levels and couple them. Here we explore this possibility by constructing a stochastic-deterministic discrete-continuous reaction-diffusion model with mobile sources and sinks. Such an approach allows in particular to separate different sources of stochasticity. We demonstrate its application by modelling enzyme-catalysed reactions with freely diffusing enzymes and a heterogeneous source of metabolites. Our calculations suggest that using a higher amount of less active enzymes, as compared to fewer more active enzymes, reduces the metabolite pool size and correspondingly the lag time, giving rise to a faster response to external stimuli. The methodology presented can be extended to more complex systems and offers exciting possibilities for studying problems where spatial heterogeneities, stochasticity or discreteness play a role.

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Keywords

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Supplementary material

10189_2016_236_MOESM1_ESM.avi (3.7 mb)
Supplementary material
10189_2016_236_MOESM2_ESM.avi (5.7 mb)
Supplementary material

References

  1. 1.
    G.C. Bond, Heterogeneous Catalysis: Principles and Applications (Claredon, Oxford, 1987)Google Scholar
  2. 2.
    G. Oshanin, A. Blumen, J. Chem. Phys. 108, 1140 (1998)CrossRefADSGoogle Scholar
  3. 3.
    G. Oshanin, M.N. Popescu, S. Dietrich, Phys. Rev. Lett. 93, 020602 (2004)CrossRefADSGoogle Scholar
  4. 4.
    P.M. Voorhees, R.J. Schaefer, Acta Metall. 35, 327 (1987)CrossRefGoogle Scholar
  5. 5.
    D.A. Lauffenburger, J.J. Linderman, Receptors: Models for Binding, Trafficking and Signaling (Oxford University Press, 1993)Google Scholar
  6. 6.
    L. Stryer, J.L. Tymoczko, J.M. Berg, Biochemistry: A short course (W.H. Freeman & Company, 2011)Google Scholar
  7. 7.
    D.T. Gillespie, J. Phys. Chem. 81, 2340 (1977)CrossRefGoogle Scholar
  8. 8.
    S. Andrews, D. Bray, Phys. Biology 1, 137 (2004)CrossRefADSGoogle Scholar
  9. 9.
    J. Hattne, D. Fange, J. Elf, Bioinformatics 21, 2923 (2005)CrossRefGoogle Scholar
  10. 10.
    R. Erban, S.J. Chapman, Phys. Biology 4, 16 (2007)CrossRefADSGoogle Scholar
  11. 11.
    R. Erban, S.J. Chapman, Phys. Biology 6, 046001 (2009)CrossRefADSGoogle Scholar
  12. 12.
    J. Murray, Mathematical Biology (Springer, 2002)Google Scholar
  13. 13.
    N. McDonald, W. Strieder, J. Chem. Phys. 118, 4598 (2003)CrossRefADSGoogle Scholar
  14. 14.
    I. Oppenheim, K.E. Shuler, G.H. Weiss, J. Chem. Phys. 50, 460 (1969)CrossRefADSGoogle Scholar
  15. 15.
    T.G. Kurtz, J. Chem. Phys. 57, 2976 (1972)CrossRefADSGoogle Scholar
  16. 16.
    T. Karapiperis, B. Blankleider, Physica D 78, 30 (1994)CrossRefADSGoogle Scholar
  17. 17.
    D.T. Gillespie, J. Chem. Phys. 113, 297 (2000)CrossRefADSGoogle Scholar
  18. 18.
    M. Flegg, J. Chapman, R. Erban, J. R. Soc., Interface 9, 859 (2011)CrossRefGoogle Scholar
  19. 19.
    M. Robinson, M. Flegg, R. Erban, J. Chem. Phys. 140, 124109 (2014)CrossRefADSGoogle Scholar
  20. 20.
    P. Bauler, G.A. Huber, J.A. McCammon, J. Chem. Phys. 136, 164107 (2012)CrossRefADSGoogle Scholar
  21. 21.
    E.L. Haseltine, J.B. Rawlings, J. Chem. Phys. 117, 6959 (2002)CrossRefADSGoogle Scholar
  22. 22.
    C.V. Rao, A.P. Arkin, J. Chem. Phys. 118, 4999 (2003)CrossRefADSGoogle Scholar
  23. 23.
    J. Puchałka, A.M. Kierzek, Biophys. J. 86, 1357 (2004)CrossRefADSGoogle Scholar
  24. 24.
    Y. Cao, D. Gillespie, L. Petzold, J. Comput. Phys. 206, 395 (2005)CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    M. Griffith, T. Courtney, J. Peccoud, W.H. Sanders, Bioinformatics 22, 2782 (2006)CrossRefGoogle Scholar
  26. 26.
    G. Kalantzis, Comput. Biol. Chem. 33, 205 (2009)CrossRefMathSciNetGoogle Scholar
  27. 27.
    K.-H. Chiam, C.M. Tan, V. Bhargava, G. Rajagopal, Phys. Rev. E 74, 051910 (2006)CrossRefADSGoogle Scholar
  28. 28.
    D.C. Wylie, Y. Hori, A.R. Dinner, A.K. Chakraborty, J. Phys. Chem. B 110, 12749 (2006)CrossRefGoogle Scholar
  29. 29.
    J.L. Doob, Trans. Am. Math. Soc. 58, 455 (1945)MathSciNetGoogle Scholar
  30. 30.
    M.A. Gibson, J. Bruck, J. Phys. Chem. A 104, 1876 (2000)CrossRefGoogle Scholar
  31. 31.
    D.T. Gillespie, J. Chem. Phys. 115, 1716 (2001)CrossRefADSGoogle Scholar
  32. 32.
    Y. Cao, D.T. Gillespie, L.R. Petzold, J. Chem. Phys. 124, 044109 (2006)CrossRefADSGoogle Scholar
  33. 33.
    A. Lopez-Campistrous, P. Semchuk, L.B.T. Palmer-Stone, S.J. Brokx, G. Broderick, D. Bottorff, S. Bolch, J.H. Weiner, M.J. Ellison, Mol Cell Proteomics 4, 1205 (2005)CrossRefGoogle Scholar
  34. 34.
    Q. Zheng, J. Ross, J. Chem. Phys. 94, 3644 (1991)CrossRefADSGoogle Scholar
  35. 35.
    G. Lente, J. Chem. Phys. 137, 164101 (2012)CrossRefADSGoogle Scholar
  36. 36.
    B.D. Bennett, E.H. Kimball, M. Gao, R. Osterhout, S.J.V. Dien, J.D. Rabinowitz, Nat. Chem. Biol. 5, 593 (2009)CrossRefGoogle Scholar
  37. 37.
    S.S. Andrews, N.J. Addy, R. Brent, A. Arkin, PLoS Comp. Biol. 6, e1000705 (2010)CrossRefADSGoogle Scholar
  38. 38.
    S.S. Andrews, “Spatial and stochastic cellular modeling with the Smoldyn simulator”, (2012) Chapt. 26, pp. 519--542Google Scholar
  39. 39.
  40. 40.
    N.M. Shnerb, Y. Louzoun, E. Bettelheim, S. Solomon, Proc. Natl. Acad. Sci. U.S.A. 97, 10322 (2000)CrossRefADSGoogle Scholar
  41. 41.
    Y. Togashi, K. Kaneko, Phys. Rev. E 70, 020901(R) (2004)CrossRefADSGoogle Scholar
  42. 42.
    J. Schöneberg, F. Noé, Plos ONE 8, e74261 (2013)CrossRefADSGoogle Scholar
  43. 43.
    P. Hunter, A. Pullan, FEM/BEM Notes (The University of Auckland, New Zealand, 2001)Google Scholar
  44. 44.
    D.J. Wilkinson, Nat. Rev. Gen. 10, 122 (2009)CrossRefGoogle Scholar
  45. 45.
    A.A. Lee, S. Kondrat, G. Oshanin, A.A. Kornyshev, Nanotechnology 25, 315401 (2014)CrossRefADSGoogle Scholar
  46. 46.
    A.A. Lee, S. Kondrat, D. Vella, A. Goriely, Phys. Rev. Lett. 115, 106101 (2015)CrossRefADSGoogle Scholar
  47. 47.
    C. Sanford, M.L. Yip, C. White, J. Parkinson, Bioinformatics 22, 2918 (2006)CrossRefGoogle Scholar
  48. 48.
    P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, O. Sander, Computing 82, 103 (2008)CrossRefMathSciNetGoogle Scholar
  49. 49.
    P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, O. Sander, Computing 82, 121 (2008)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Distributed and Unified Numerics Environment, http://www.dune-project.org
  51. 51.
    An adaptive hierarchical finite element toolbox, http://www.alberta-fem.de/
  52. 52.
  53. 53.
    Y. Yang, C.-W. Shu, Numer. Math. 124, 753 (2013)CrossRefMathSciNetGoogle Scholar
  54. 54.
    A.L. Hanharta, M.K. Gobberta, L.T. Izub, J. Comput. Appl. Math. 169, 431 (2004)CrossRefADSMathSciNetGoogle Scholar
  55. 55.
    M. Blatt, P. Bastian, in Applied Parallel Computing. State of the Art in Scientific Computing (Springer, 2007)Google Scholar
  56. 56.
    P. Bastian, M. Blatt, Int. J. Comput. Sci. Engin. 4, 56 (2008)CrossRefGoogle Scholar
  57. 57.
    P.R. Amestoy, I.S. Duff, J.-Y. L’Excellent, Comput. Methods Appl. Mech. Engin. 184, 501 (2000)CrossRefADSGoogle Scholar
  58. 58.
    P.R. Amestoy, A. Guermouche, J.-Y. L’Excellent, S. Pralet, Parallel Comput. 32, 136 (2006)CrossRefMathSciNetGoogle Scholar
  59. 59.
    A MUltifrontal Massively Parallel sparse direct Solver, http://mumps.enseeiht.fr/
  60. 60.
    M. Raju, S. Khaitan, J. Appl. Fluid Mech. 5, 123 (2012)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Svyatoslav Kondrat
    • 1
  • Olav Zimmermann
    • 2
  • Wolfgang Wiechert
    • 1
  • Eric von Lieres
    • 1
  1. 1.IBG-1: BiotechnologyForschungszentrum JülichJülichGermany
  2. 2.Jülich Supercomputing CenterForschungszentrum JülichJülichGermany

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