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Bifurcation Theory, Symmetry Breaking and Homogenization in Continuum Mechanics Descriptions of DNA

Mathematical Modelling of the Physics of the Double Helix

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A Celebration of Mathematical Modeling

Abstract

The study of DNA minicircles, i.e. closed loops of the double helix with lengths of the order of a few hundred base pairs, is a commonly used experimental technique to probe the sequence-dependent mechanical properties of DNA, such as stiffnesses and intrinsic shape. This article reviews how the mathematical methods of bifurcation theory and symmetry breaking can be used to compute the sequence-dependent equilibrium shapes of mini-circles. The computations yield quite good comparison with experimental data, despite the fact that they assume an isotropic bending law for the DNA, and that at the single base-pair scale the bending response of DNA is almost certainly strongly anisotropic. The effective isotropic behavior can be explained via a two-scale expansion involving the high intrinsic twist parameter of the DNA double helix (one turn per 10.5 base pairs). The effective isotropic stiffnesses can be found in terms of the local anisotropic stiffnesses via a study of the lowest order term in this expansion. However to understand the first correction, symmetry breaking techniques are again required.

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Author information

Authors and Affiliations

  1. Bernoulli Mathematics Institute, Swiss Federal Institute of Technology-Lausanne, CH-1015, Ecublens, Switzerland

    John H. Maddocks

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  1. John H. Maddocks
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Editors and Affiliations

  1. Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa, Israel

    Dan Givoli

  2. Department of Mathematics, University of Basel, Basel, Switzerland

    Marcus J. Grote

  3. Department of Mathematics, Stanford University, Stanford, California, USA

    George C. Papanicolaou

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Maddocks, J.H. (2004). Bifurcation Theory, Symmetry Breaking and Homogenization in Continuum Mechanics Descriptions of DNA. In: Givoli, D., Grote, M.J., Papanicolaou, G.C. (eds) A Celebration of Mathematical Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0427-4_7

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  • DOI: https://doi.org/10.1007/978-94-017-0427-4_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6526-1

  • Online ISBN: 978-94-017-0427-4

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