Skip to main content
SpringerLink
Log in

Model of Fractal Organization of Chromatin in Two-Dimensional Space

  • STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS
  • Published:
Journal of Experimental and Theoretical Physics Aims and scope Submit manuscript

Abstract

Chromatin, consisting of a meter-long DNA strand and associated proteins, is packed into the nucleus of a biological cell tightly but without entanglement. There is a hypothesis, confirmed by experiments involving the chromatin conformation capture technology [1], that curves densely filling the space (Peano or Hilbert curves) provide a good theoretical model to describe the chromatin packing into the nucleus. However, small-angle neutron scattering (SANS) experiments show a bifractal organization of chromatin in the interphase nucleus, thus demonstrating the presence of a logarithmic fractal on larger scales and a volume fractal on smaller scales [2]. In this paper, numerical Fourier analysis in the two-dimensional space is applied to simulate neutron scattering, and a model of a unified bifractal object is presented. It is shown that, in numerical radiation scattering experiments in the two-dimensional space, the mass and logarithmic fractals are significantly different from space-filling curves and from nonfractal objects. For instance, for a logarithmic fractal with a Hausdorff dimension of 2, scattering intensity decreases with increasing Fourier coordinate q by the power law q–2. For curves filling the two-dimensional space, the intensity decreases by the power law q–3, just as for nonfractal objects with sharp boundary in the plane. Thus, first, it is demonstrated that the model of space-filling curves is inadequate to describe the chromatin packing into the nucleus of a biological cell; second, a model of a unified bifractal object is proposed that combines logarithmic and mass fractals on different scales; and, third, a model of chromatin packing is proposed that can describe the data of both small-angle neutron scattering experiments and experiments involving chromatin conformation capture technology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.

REFERENCES

  1. E. Lieberman-Aiden, N. L. van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B. R. Lajoie, P. J. Sabo, M. O. Dorschner, R. Sandstrom, B. Bernstein, M. A. Bender, M. K. Groudine, A. Gnirke, et al., Science (Washington, DC, U. S.) 326, 289 (2009).

    Article  ADS  Google Scholar 

  2. E. G. Iashina and S. V. Grigoriev, J. Exp. Theor. Phys. 129, 455 (2019).

    Article  ADS  Google Scholar 

  3. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983).

    Book  Google Scholar 

  4. H. O. Peitgen and P. H. Richter, The Beauty of Fractals (Springer, Berlin, 1986).

    Book  MATH  Google Scholar 

  5. L. S. Liebovitch, Fractals and Chaos Simplified for the Life Science (Oxford Univ. Press, New York, 1998).

    MATH  Google Scholar 

  6. I. C. Andronache, H. Ahammer, H. F. Jelineck, D. Peptenatu, A.-M. Ciobotaru, C. C. Draghici, R. D. Pintilii, A. G. Simion, and C. Teodorescu, Chaos, Solitons Fract. 91, 310 (2016).

    Article  ADS  Google Scholar 

  7. D. V. Lebedev, M. V. Filatov, A. I. Kuklin, A. K. Isla-mov, E. Kentzinger, R. A. Pantina, B. P. Toperverg, and V. V. Isaev-Ivanov, FEBS Lett. 579, 1465 (2005).

    Article  Google Scholar 

  8. E. G. Iashina, E. V. Velichko, M. V. Filatov, W. G. Bouwman, C. P. Duif, A. Brulet, and S. V. Grigoriev, Phys. Rev. E 96, 012411 (2017).

  9. E. G. Iashina, M. V. Filatov, R. A. Pantina, E. Yu. Varfolomeeva, W. G. Bouwman, Ch. P. Duif, D. Honecker, V. Pipich, and S. V. Grigoriev, J. Appl. Crystallogr. 52, 844 (2019).

    Article  Google Scholar 

  10. S. V. Grigoriev, E. G. Iashina, V. Yu. Bairamukov, V. Pipich, A. Radulescu, M. V. Filatov, R. A. Pantina, and E. Yu. Varfolomeeva, Phys. Rev. E 102, 032415 (2020).

  11. S. V. Grigoriev, E. G. Iashina, B. Wu, V. Pipich, Ch. Lang, A. Radulescu, V. Yu. Bairamukov, M. V. Filatov, R. A. Pantina, and E. Yu. Varfolomeeva, Phys. Rev. E 104, 044404 (2021).

  12. E. G. Iashina, E. Yu. Varfolomeeva, R. A. Pantina, V. Yu. Bairamukov, R. A. Kovalev, N. D. Fedorova, V. Pipich, A. Radulescu, and S. V. Grigoriev, Phys. Rev. E 104, 064409 (2021).

  13. A. Yu. Grosberg, S. K. Nechaev, and E. I. Shakhnovich, J. Phys. France 49, 2095 (1988).

    Article  Google Scholar 

  14. A. Grosberg, Y. Rabin, S. Havlin, and A. Neer, Europhys. Lett. 23, 373 (1993).

    Article  ADS  Google Scholar 

  15. N. R. Batullin, V. S. Fishman, A. A. Khabarova, M. Yu. Pomaznoi, T. A. Shnaider, D. A. Afonnikov, and O. L. Serov, Vavilov. Zh. Genet. Selekts. 18, 338 (2014).

    Google Scholar 

  16. A. Zlotina, A. Maslova, N. Kosyakova, A. B. H. Al-Rikabi, T. Liehr, and A. Krasikova, Chromosome Res. 27, 253 (2019).

    Article  Google Scholar 

  17. L. A. Mirny, Chromosome Res. 19, 37 (2011).

    Article  Google Scholar 

  18. J. D. Halverson, W. B. Lee, G. S. Grest, A. Y. Grosberg, and K. Kremer, J. Chem. Phys. 134, 204904 (2011).

  19. J. D. Halverson, J. Smrek, K. Kremer, and A. Y. Grosberg, Rep. Prog. Phys. 77, 022601 (2014).

  20. M. V. Imakaev, K. M. Tchourine, S. K. Nechaev, and L. A. Mirny, Soft Matter 11, 665 (2015).

    Article  ADS  Google Scholar 

  21. E. G. Iashina and S. V. Grigor’ev, J. Surf. Invest.: X‑ray, Synchrotron Neutron Tech. 11, 897 (2017).

    Article  Google Scholar 

  22. J. O. Indekeu and G. Fleerackers, Phys. A (Amsterdam, Neth.) 261, 294 (1998).

  23. J. P. Richter and R. C. Bell, The Notebooks of Leonardo da Vinci (Dover, New York, 1970).

    Google Scholar 

  24. J. E. Martin and A. J. Hurd, J. Appl. Crystallogr. 20, 61 (1987).

    Article  Google Scholar 

  25. J. Teixeira, J. Appl. Crystallogr. 21, 781 (1988).

    Article  Google Scholar 

  26. D. I. Svergun, M. H. J. Koch, P. A. Timmins, and R. P. May, Small Angle X-ray and Neutron Scattering from Solutions of Biological Macromolecules (Oxford Univ. Press, Oxford, 2013).

    Book  Google Scholar 

  27. T. Ficker, A. Len, and P. Nemec, J. Phys. D: Appl. Phys. 40, 4055 (2007).

    Article  ADS  Google Scholar 

  28. P. M. Pustovoit, E. G. Yashina, K. A. Pshenichnyi, and S. V. Grigoriev, J. Surf. Invest.: X-ray, Synchrotron Neutron Tech. 14, 1232 (2020).

    Article  Google Scholar 

  29. J. W. Goodman, Introduction to Fourier Optics (W. H. Freeman, New York, 2017).

    Google Scholar 

  30. A. N. Matveev, Optics (Vysshaya Shkola, Moscow, 1985; Mir, Moscow, 1988).

  31. A. A. Zinchik, Ya. B. Muzychenko, A. V. Smirnov, and S. K. Stafeev, Nauch.-Tekh. Vestn. SPbGU ITMO 60 (2), 17 (2009).

    Google Scholar 

  32. https://github.com/tre3k/fractal.

Download references

Funding

This work was supported by the Russian Science Foundation, project no. 20-12-00188.

Author information

Authors and Affiliations

  1. Petersburg Nuclear Physics Institute, National Research Center “Kurchatov Institute”, 188300, Gatchina, Leningrad oblast, Russia

    S. V. Grigoriev, O. D. Shnyrkov, K. A. Pshenichnyi, P. M. Pustovoit & E. G. Yashina

  2. St. Petersburg State University, 198504, St. Petersburg, Russia

    S. V. Grigoriev & E. G. Yashina

Authors
  1. S. V. Grigoriev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. D. Shnyrkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. A. Pshenichnyi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. M. Pustovoit
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. E. G. Yashina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. V. Grigoriev.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by I. Nikitin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grigoriev, S.V., Shnyrkov, O.D., Pshenichnyi, K.A. et al. Model of Fractal Organization of Chromatin in Two-Dimensional Space. J. Exp. Theor. Phys. 136, 378–388 (2023). https://doi.org/10.1134/S1063776123030123

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063776123030123

Search

Navigation