Statistics of End-to-End Distance of a Linear Chain Trapped in a Cubic Lattice of Binding Centers

  • Zbigniew Domański
  • Norbert Sczygiol


Two and three dimensional nanostructured substrates are widely employed in a variety of biomedically-oriented nanodevices as well as in functional devices created with the use of DNA scaffolding. In this context spatial arrangements of binding centers influence the efficiency of these substrates. Here, we concentrate on 3D substrates and we compute and analyze the distribution of distances (q) between binding centers in the case where the centers are localized in nodes of a cubic lattice. We find that for this particular lattice the exact node-to-node probability distribution is a fifth-degree polynomial in q. We merge this polynomial-shaped distribution with an end-to-end distance distribution of a linear chain and we find an excellent agreement between it and the corresponding distribution for a self-avoiding walk in 3D.


Distance distribution DNA scaffolding Micropatterned substrates Polymer adhesion Self-avoiding walks Zigzag path statistics 


  1. 1.
    R. Amin, S. Hwang, S.H. Park, Nanobiotechnology: an Interface Between nanotechnology and biotechnology. Nano: Brief Rep Rev, 6(2), 101–111 (2011)Google Scholar
  2. 2.
    R.J. Kershner, L.D. Bozano, ChM Micheel, A.M. Hung, A.R. Fornof, J.N. Cha, ChT Rettner, M. Bersani, J. Frommer, P.W.K. Rothemund, G.M. Wallraff, Placement and orientation of individual DNA shapes on litographically patterned surfaces. Nat. Nanotechnol. 4(9), 557–561 (2009)CrossRefGoogle Scholar
  3. 3.
    Hh Lin, Y. Liu, S. Rinker, H. Yan, DNA tile based self-assembly: Building complex nanoarchitectures. Chem. Phys. Chem. 7(8), 1641–1647 (2006)Google Scholar
  4. 4.
    S.M. Douglas, A.H. Marblestone, S. Teerapittayanon, A. Vazquez, G.M. Church, W.M. Shih, Rapid prototyping of 3D DNA-origami shapes with caDNAno. Nucl Acids Res 37(15), 5001–5006 (2009)Google Scholar
  5. 5.
    P.W.K. Rothemund, Folding DNA to create nanoscale shapes and patterns. Nature 440, 297–302 (2006)CrossRefGoogle Scholar
  6. 6.
    S. Wang, H. Wang, J.J. Jiao, K.J. Chen, G.E. Owens, K. Kamei, J. Sun, D.J. Sherman, ChP Behrenbruch, H. Wu, H.R. Tseng, Three-dimensional nanostructured substrates toward efficient capture of circulating tumor cells. Angew. Chem. 121, 9132–9135 (2009)CrossRefGoogle Scholar
  7. 7.
    H.J. Yoon, T.H. Kim, Z. Zhang, E. Azizi, T.M. Pham, C. Paoletti, J. Lin, N. Ramnath, M.S. Wicha, D.F. Hayes, D.M. Simeone, S. Ngrath, Sensitive capture of circulating tumor cells by functionalized graphene oxide nanosheets. Nat. Nanotechnol. 8(10), 735–741 (2013)CrossRefGoogle Scholar
  8. 8.
    H. Otsuka, Nanofabrication of nonfouling surfaces for micropatterning of cell and microtissue. Molecules 15(8), 5525–5546 (2010).
  9. 9.
    I. Teraoka, Polymer Solutions: An Introduction to Physical Properties (Wiley, Brooklyn, 2002)CrossRefGoogle Scholar
  10. 10.
    A.D. Sokal, monte carlo methods for the self-avoiding walk, in Monte Carlo and Molecular Dynamics Simulations in Polymer Science, ed. by K. Binder (Oxford University Press, New York, 1995)Google Scholar
  11. 11.
    G. Slade, Self-avoiding walks. Math. Intelligencer 16(1), 29–35 (1994).
  12. 12.
    N.W. Ashroft, N.D. Mermin, Solid State Physics (Harcourt Inc., Orlando, 1976)Google Scholar
  13. 13.
    F.F. Dragan, Estimating all pairs shortest paths in restricted graph families: A unified approach. J. Algorithms 57(1), 1–21 (2005)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Z. Domański, N. Sczygiol, Distribution of the distance between receptors of ordered micropatterned substrates, eds. by H.K. Kim, S.I. Ao, B.B. Rieger. IAENG Transactions on Engineering Technologies (Springer, Dordrecht, 2011) (Special Edition of the World Congress on Engineering and Computer Science)Google Scholar
  15. 15.
    Z. Domański, Geometry-induced transport properties of two dimensional networks, ed. by M. Schmidt. Advances in Computer Science and Engineering (InTech, Rijeka, 2011)Google Scholar
  16. 16.
    Z. Domański, N. Sczygiol, Distribution of node-to-node distance in a cubic lattice of binding centers, lecture notes in engineering and computer science, in Proceedings of The International MultiConference of Engineers and Computer Scientists 2014, IMECS, 12–14 Mar 2014, Hong Kong, pp. 900–903Google Scholar
  17. 17.
    A.Y. Grosberg, A.R. Khoklov, Statistical Physics of Marcomolecules (American Institute of Physics, Woodbury, 1994)Google Scholar
  18. 18.
    F. Valle, M. Favre, P. De Los Rios, A. Rosa, G. Dietler, Scaling exponents and probability distributions of DNA end-to-end distance. Phys. Rev. Lett. 95, 158105 (2005)CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institute of MathematicsCzęstochowa University of TechnologyCzęstochowaPoland
  2. 2.Institute of Computer and Information SciencesCzęstochowa University of TechnologyCzęstochowaPoland

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