Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Geometry-controlled kinetics

Nature Chemistry volume 2, pages 472–477 (2010)Cite this article

Subjects

Abstract

It has long been appreciated that the transport properties of molecules can control reaction kinetics. This effect can be characterized by the time it takes a diffusing molecule to reach a target—the first-passage time (FPT). Determining the FPT distribution in realistic confined geometries has until now, however, seemed intractable. Here, we calculate this FPT distribution analytically and show that transport processes as varied as regular diffusion, anomalous diffusion, and diffusion in disordered media and fractals, fall into the same universality classes. Beyond the theoretical aspect, this result changes our views on standard reaction kinetics and we introduce the concept of ‘geometry-controlled kinetics’. More precisely, we argue that geometry—and in particular the initial distance between reactants in ‘compact’ systems—can become a key parameter. These findings could help explain the crucial role that the spatial organization of genes has in transcription kinetics, and more generally the impact of geometry on diffusion-limited reactions.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: First-passage time distribution (FPT) and geometry-controlled kinetics.
Figure 2: Universal FPT distribution in the non compact and marginal cases.
Figure 3: Universal FPT distributions in the compact case.
Figure 4: Reaction kinetics as quantified by the survival probability S(t), plotted for different source–target distances r.

Similar content being viewed by others

References

  1. Rice, S. Diffusion-Limited Reactions (Elsevier, 1985).

    Google Scholar 

  2. Kopelman, R. Fractal reaction kinetics. Science 241, 1620–1626 (1988).

    Article  CAS  Google Scholar 

  3. Shlesinger, M. F., Zaslavsky, G. M. & Klafter, J. Strange kinetics. Nature 363, 31–37 (1993).

    Article  CAS  Google Scholar 

  4. Yu, J., Xiao, J., Ren, X., Lao, K. & Xie, X. S. Probing gene expression in live cells, one protein molecule at a time. Science 311, 1600–1603 (2006).

    Article  CAS  Google Scholar 

  5. Elf, J., Li, G.-W. & Xie, X. S. Probing transcription factor dynamics at the single-molecule level in a living cell. Science 316, 1191–1194 (2007).

    Article  CAS  Google Scholar 

  6. Schuss, Z., Singer, A. & Holcman, D. The narrow escape problem for diffusion in cellular microdomains. Proc. Natl Acad. Sci. USA 104, 16098–16103 (2007).

    Article  CAS  Google Scholar 

  7. Loverdo, C., Benichou, O., Moreau, M. & Voituriez, R. Enhanced reaction kinetics in biological cells. Nature Phys. 4, 134–137 (2008).

    Article  CAS  Google Scholar 

  8. Mirny, L. Biophysics: Cell commuters avoid delays. Nature Phys. 4, 93–95 (2008).

    Article  CAS  Google Scholar 

  9. Bénichou, O., Loverdo, C., Moreau, M. & Voituriez, R. Optimizing intermittent reaction paths. Phys. Chem. Chem. Phys. 10, 7059–7072 (2008).

    Article  Google Scholar 

  10. Redner, S. A Guide to First Passage Time Processes (Cambridge University Press, 2001).

  11. Shlesinger, M. F. Mathematical physics: first encounters. Nature 450, 40–41 (2007).

    Article  CAS  Google Scholar 

  12. Condamin, S., Benichou, O. & Moreau, M. First-passage times for random walks in bounded domains. Phys. Rev. Lett. 95, 260601 (2005).

    Article  CAS  Google Scholar 

  13. Condamin, S., Benichou, O., Tejedor, V., Voituriez, R. & Klafter, J. First-passage times in complex scale-invariant media. Nature 450, 77–80 (2007).

    Article  CAS  Google Scholar 

  14. Benichou, O. & Voituriez, R. Narrow-escape time problem: Time needed for a particle to exit a confining domain through a small window. Phys. Rev. Lett. 100, 168105 (2008).

    Article  CAS  Google Scholar 

  15. Benichou, O., Meyer, B., Tejedor, V. & Voituriez, R. Zero constant formula for first-passage observables in bounded domains. Phys. Rev. Lett. 101, 130601 (2008).

    Article  CAS  Google Scholar 

  16. Bouchaud, J.-P. & Georges, A. Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990).

    Article  Google Scholar 

  17. Kurzynski, M., Palacz, K. & Chelminiak, P. Time course of reactions controlled and gated by intramolecular dynamics of proteins: Proc. Natl Acad. Sci. USA 95, 11685–11690 (1998).

    Article  CAS  Google Scholar 

  18. Kolesov, G., Wunderlich, Z., Laikova, O. N., Gelfand, M. S. & Mirny, L. A. How gene order is inuenced by the biophysics of transcription regulation. Proc. Natl Acad. Sci. USA 104, 13948–13953 (2007).

    Article  CAS  Google Scholar 

  19. Fraser, P. & Bickmore, W. Nuclear organization of the genome and the potential for gene regulation. Nature 447, 413–417 (2007).

    Article  CAS  Google Scholar 

  20. Zimmerman, S. B. & Minton, A. P. Macromolecular crowding: Biochemical, biophysical, and physiological consequences. Ann. Rev. Biophys. Biom. 22 (1993).

  21. Zhou, H.-X., Rivas, G. & Minton, A. P. Macromolecular crowding and confinement: Biochemical, biophysical, and potential physiological consequences. Ann. Rev. Biophys. 37, 375–397 (2008).

    Article  CAS  Google Scholar 

  22. Ben-Avraham, D. & Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems (Cambridge Univ. Press, 2000).

    Book  Google Scholar 

  23. de Gennes, P. G. Kinetics of diffusion-controlled processes in dense polymer systems. i. nonentangled regimes. J. Chem. Phys. 76, 3316–3321 (1982).

    Article  CAS  Google Scholar 

  24. Condamin, S., Benichou, O. & Moreau, M. Random walks and brownian motion: a method of computation for first-passage times and related quantities in confined geometries. Phys. Rev. E 75, 021111 (2007).

    Article  CAS  Google Scholar 

  25. O'Shaughnessy, B. & Procaccia, I. Diffusion on fractals. Phys. Rev. A 32, 3073–3083 (1985).

    Google Scholar 

  26. Montroll, E. W. Random walks on lattices. iii. calculation of first-passage times with ap- plication to exciton trapping on photosynthetic units. J. Math. Phys. 10, 753–765 (1969).

    Article  CAS  Google Scholar 

  27. Kozak, J. J. & Balakrishnan, V. Analytic expression for the mean time to absorption for a random walker on the sierpinski gasket. Phys. Rev. E 65, 021105 (2002).

    Article  Google Scholar 

  28. Agliari, E. Exact mean first-passage time on the t-graph. Phys. Rev. E 77, 011128 (2008).

    Article  CAS  Google Scholar 

  29. Haynes, C. P. & Roberts, A. P. Global first-passage times of fractal lattices. Phys. Rev. E 78, 041111 (2008).

    Article  CAS  Google Scholar 

  30. Tejedor, V., Bénichou, O. & Voituriez, R. Global mean first-passage times of random walks on complex networks. Phys. Rev. E 80, 065104 (2009).

    Article  CAS  Google Scholar 

  31. Metzler, R. & Klafter, J. The random walk's guide to anomalous diffusion: a fractionnal dynamics approach. Phys. Rep. 339, 1–77 (2000).

    Article  CAS  Google Scholar 

  32. Wachsmuth, M., Waldeck, W. & Langowski, J. Anomalous diffusion of uorescent probes inside living cell nuclei investigated by spatially-resolved uorescence correlation spectroscopy. J. Mol. Biol. 298, 677–689 (2000).

    Article  CAS  Google Scholar 

  33. Golding, E. & Cox, E. Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96, 981102 (2006).

    Article  Google Scholar 

  34. Condamin, S., Tejedor, V., Voituriez, R., Benichou, O. & Klafter, J. Probing microscopic origins of confined subdiffusion by first-passage observables. Proc. Natl Acad. Sci. USA 105, 5675–5680 (2008).

    Article  CAS  Google Scholar 

  35. Montroll, E. W. & Weiss, G. H. Random walks on lattices. ii. J. Math. Phys. 6, 167–181 (1965).

    Article  Google Scholar 

  36. Shlesinger, M. F. Asymptotic solutions of continuous-time random walks. J. Stat. Phys. 10, 421–434 (1974).

    Article  Google Scholar 

  37. Condamin, S., Benichou, O. & Klafter, J. First-passage time distributions for subdiffusion in confined geometry. Phys. Rev. Lett. 98, 250602 (2007).

    Article  CAS  Google Scholar 

  38. Parson, R. P. & Kopelman, R. Percolative versus homogenous energy transport kinetics: time-resolved donor and acceptor uoresence of isotopic mixed naphthalene crystals. Chem. Phys. Lett. 87, 528–532 (1982).

    Article  CAS  Google Scholar 

  39. Saxton, M. J. A biological interpretation of transient anomalous subdiffusion. ii. reaction kinetics. Biophys. J. 94, 760–771 (2008).

    Article  CAS  Google Scholar 

  40. Malchus, N. & Weiss, M. Elucidating anomalous protein diffusion in living cells with fluorescence correlation spectroscopy—facts and pitfalls. J. Fluoresc. (2009).

  41. Monson, E. & Kopelman, R. Observation of laser speckle effects and nonclassical kinetics in an elementary chemical reaction. Phys. Rev. Lett. 85, 666–669 (2000).

    Article  CAS  Google Scholar 

  42. Markstein, M., Markstein, P., Markstein, V. & Levine, M. S. Genome-wide analysis of clustered dorsal binding sites identifies putative target genes in the drosophila embryo. Proc. Natl Acad. Sci. USA 99, 763–768 (2002).

    Article  CAS  Google Scholar 

  43. Platani, M., Goldberg, I., Lamond, A. I. & Swedlow, J. R. Cajal body dynamics and association with chromatin are atp-dependent. Nat. Cell Biol. 4, 502–508 (2002).

    Article  CAS  Google Scholar 

  44. Bancaud, A. et al. Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin. EMBO J. 28, 3785–3798 (2009).

    Article  CAS  Google Scholar 

  45. Lebedev, D. V. et al. Fractal nature of chromatin organization in interphase chicken erythro- cyte nuclei: Dna structure exhibits biphasic fractal properties. FEBS Lett. 579, 1465–1468 (2005).

    Article  CAS  Google Scholar 

  46. Lebedev, D. et al. Structural hierarchy of chromatin in chicken erythrocyte nuclei based on small-angle neutron scattering: Fractal nature of the large-scale chromatin organization. Crystallogr. Rep. 53, 110–115 (2008).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

Support from ANR grants DYOPTRI and DYNAFT is acknowledged.

Author information

Authors and Affiliations

  1. UPMC Univ Paris 06, CNRS-UMR 7600 Laboratoire de Physique Théorique de la Matière Condensée, 4 Place Jussieu, Paris, F-75005, France

    O. Bénichou, C. Chevalier, B. Meyer & R. Voituriez

  2. School of Chemistry, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel; and Institute for Advanced Studies (FRIAS), University of Freiburg, Freiburg, 79104, Germany

    J. Klafter

Authors
  1. O. Bénichou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Chevalier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Klafter
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B. Meyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. R. Voituriez
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to O. Bénichou.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary information

Supplementary information (PDF 416 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bénichou, O., Chevalier, C., Klafter, J. et al. Geometry-controlled kinetics. Nature Chem 2, 472–477 (2010). https://doi.org/10.1038/nchem.622

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nchem.622

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing