Skip to main content
SpringerLink
Log in

The Influence of Spatial Variation in Chromatin Density Determined by X-Ray Tomograms on the Time to Find DNA Binding Sites

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

In this work, we examine how volume exclusion caused by regions of high chromatin density might influence the time required for proteins to find specific DNA binding sites. The spatial variation of chromatin density within mouse olfactory sensory neurons is determined from soft X-ray tomography reconstructions of five nuclei. We show that there is a division of the nuclear space into regions of low-density euchromatin and high-density heterochromatin. Volume exclusion experienced by a diffusing protein caused by this varying density of chromatin is modeled by a repulsive potential. The value of the potential at a given point in space is chosen to be proportional to the density of chromatin at that location. The constant of proportionality, called the volume exclusivity, provides a model parameter that determines the strength of volume exclusion. Numerical simulations demonstrate that the mean time for a protein to locate a binding site localized in euchromatin is minimized for a finite, nonzero volume exclusivity. For binding sites in heterochromatin, the mean time is minimized when the volume exclusivity is zero (the protein experiences no volume exclusion). An analytical theory is developed to explain these results. The theory suggests that for binding sites in euchromatin there is an optimal level of volume exclusivity that balances a reduction in the volume searched in finding the binding site, with the height of effective potential barriers the protein must cross during the search process.

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

Similar content being viewed by others

Notes

  1. Amira, Visualization Sciences Group

References

  • Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., & Walter, P. (2007). Molecular biology of the cell (5th ed.). New York: Garland Science.

    Google Scholar 

  • Bancaud, A., Huet, S., Daigle, N., Mozziconacci, J., Beaudouin, J., & Ellenberg, J. (2009). Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin. EMBO J., 28, 3785.

    Article  Google Scholar 

  • Berg, O. G., Winter, R. B., & von Hippel, P. H. (1981). Diffusion-driven mechanisms of protein translocation on nucleic acids. 1. Models and theory. Biochemistry, 20, 6929.

    Article  Google Scholar 

  • Bortz, A. B., Kalos, M. H., & Lebowitz, J. L. (1975). A new algorithm for Monte Carlo simulation of Ising spin systems. J. Comput. Phys., 17, 10.

    Article  Google Scholar 

  • Cheviakov, A. F., & Ward, M. J. (2011). Optimizing the principal eigenvalue of the Laplacian in a sphere with interior traps. Math. Comput. Model., 53, 1394.

    Article  MathSciNet  MATH  Google Scholar 

  • Clowney, E. J., Le Gros, M. A., Mosley, C. P., Clowney, F. G., Markenskoff-Papadimitriou, E. C., Myllys, M., Barnea, G., Larabell, C. A., & Lomvardas, S. (2012). Nuclear aggregation of olfactory receptor genes governs their monogenic expression. Cell, 151, 724.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Gibson, M. A., & Bruck, J. (2000). Effcient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. A, 104, 1876.

    Article  Google Scholar 

  • Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical-reactions. J. Phys. Chem., 81, 2340.

    Article  Google Scholar 

  • Halford, S. (2009). An end to 40 years of mistakes in DNA-protein association kinetics? Biochem. Soc. Trans., 37, 343.

    Article  Google Scholar 

  • Hammar, P., Leroy, P., Mahmutovic, A., Marklund, E. G., Berg, O. G., & Elf, J. (2012). The lac repressor displays facilitated diffusion in living cells. Science, 336, 1595.

    Article  Google Scholar 

  • Isaacson, S. A., McQueen, D. M., & Peskin, C. S. (2011). The influence of volume exclusion by chromatin on the time required to find specific DNA binding sites by diffusion. Proc. Natl. Acad. Sci. USA, 108, 3815.

    Article  Google Scholar 

  • Kevorkian, J., & Cole, J. D. (1996). Multiple scale and singular perturbation methods. In Applied mathematical sciences (Vol. 114). New York: Springer.

    Google Scholar 

  • Le Gros, M. A., Clowney, E. J., Magklara, A., Yen, A., Markenscoff-Papadimitriou, E., Colquitt, B., Smith, E. A., Myllys, M., Kellis, M., Lomvardas, S., & Larabell, C. A. (2013) Gradual chromatin compaction and reorganization during neurogenesis in vivo. Submitted.

  • Li, G. W., Berg, O. G., & Elf, J. (2009). Effects of macromolecular crowding and DNA looping on gene regulation kinetics. Nat. Phys., 5, 294.

    Article  Google Scholar 

  • Malherbe, G., & Holcman, D. (2008). The search kinetics of a target inside the cell nucleus. arXiv:0712.3467v1 [q-bio.BM]

  • McDermott, G., Le Gros, M. A., Knoechel, C. G., Uchida, M., & Larabell, C. A. (2009). Soft X-ray tomography and cryogenic light microscopy: the cool combination in cellular imaging. Trends Cell Biol., 19, 587.

    Article  Google Scholar 

  • Mirny, L., Slutsky, M., Wunderlich, Z., Tafvizi, A., Leith, J., & Kosmrlj, A. (2009). How a protein searches for its site on DNA: the mechanism of facilitated diffusion. J. Phys. A, Math. Theor., 42, 434013.

    Article  MathSciNet  MATH  Google Scholar 

  • Normanno, D., Dahan, M., & Darzacq, X. (2012). Intra-nuclear mobility and target search mechanisms of transcription factors: a single-molecule perspective on gene expression. Biochim. Biophys. Acta, 1819, 482.

    Article  Google Scholar 

  • Schermelleh, L., Carlton, P. M., Haase, S., Shao, L., Winoto, L., Kner, P., Burke, B., Cardoso, M. C., Agard, D. A., Gustafsson, M. G. L., Leonhardt, H., & Sedat, J. W. (2008). Subdiffraction multicolor imaging of the nuclear periphery with 3D structured illumination microscopy. Science, 320, 1332.

    Article  Google Scholar 

  • Slutsky, M., & Mirny, L. A. (2004). Kinetics of protein-DNA interaction: facilitated target location in sequence-dependent potential. Biophys. J., 87, 4021.

    Article  Google Scholar 

  • Smoluchowski, M. V. (1917). Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem., 92, 129.

    Google Scholar 

  • Svetlov, V., & Nudler, E. (2013). Looking for a promoter in 3D. Nat. Struct. Mol. Biol., 20, 141.

    Article  Google Scholar 

  • Tsuchiyama, A., Uesugi, K., Nakano, T., & Ikeda, S. (2005). Quantitative evaluation of attenuation contrast of X-ray computed tomography images using monochromatized beams. Am. Mineral., 90, 132.

    Article  Google Scholar 

  • Vargas, D. Y., Raj, A., Marras, S. A. E., Kramer, F. R., & Tyagi, S. (2005). Mechanism of mRNA transport in the nucleus. Proc. Natl. Acad. Sci. USA, 102, 17008.

    Article  Google Scholar 

  • Veksler, A., & Kolomeisky, A. B. (2013). Speed-selectivity paradox in the protein search for targets on DNA: is it real or not? J. Phys. Chem. B. doi:10.1021/jp311466f.

    Google Scholar 

Download references

Acknowledgements

S.A. Isaacson, D.M. McQueen, and C.S. Peskin were supported by the Systems Biology Center New York (National Institutes of Health Grant P50GM071558). S.A. Isaacson was also supported by National Science Foundation Grant DMS-0920886. M.A. Le Gros and C.A. Larabell were supported by the Department of Energy Office of Biological and Environmental Research Grant DE-AC02-05CH11231, the NIH National Center for Research Resources (5P41 RR019664-08), and the National Institute of General Medical Sciences (8P41 GM103445-08) from the National Institutes of Health.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Samuel A. Isaacson

  2. Department of Anatomy, University of California, San Francisco, CA, USA

    Carolyn A. Larabell & Mark A. Le Gros

  3. Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Carolyn A. Larabell & Mark A. Le Gros

  4. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    David M. McQueen & Charles S. Peskin

Authors
  1. Samuel A. Isaacson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carolyn A. Larabell
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mark A. Le Gros
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. David M. McQueen
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Charles S. Peskin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel A. Isaacson.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(MOV 4.3 MB)

Appendix: Soft X-ray Tomography Measurement Error

Appendix: Soft X-ray Tomography Measurement Error

The X-ray microscope employs monochromatic X-rays and, therefore, the values obtained from computed tomography measurements are equal to the LAC values calculated from the atomic composition of the specimen. The SXT technique avoids the beam hardening effects commonly found in polychromatic tomographic imaging (see Tsuchiyama et al. 2005). The measurement error for each pixel of a single projection image is of order 3 %, determined by photon shot noise. The LAC value of each 32 nm voxel is obtained from tomographic reconstruction of many such projections and is typically less than 1 %. LAC measurement errors are insignificant compared to the observed cell-to-cell variation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Isaacson, S.A., Larabell, C.A., Le Gros, M.A. et al. The Influence of Spatial Variation in Chromatin Density Determined by X-Ray Tomograms on the Time to Find DNA Binding Sites. Bull Math Biol 75, 2093–2117 (2013). https://doi.org/10.1007/s11538-013-9883-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-013-9883-9

Keywords

Search

Navigation