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Optimal shapes of compact strings

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Abstract

Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines1,2,3,4,5. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest packing fraction; only recently has it been proved1,2 that the answer for infinite systems is a face-centred-cubic lattice. This simply stated problem has had a profound impact in many areas3,4,5, ranging from the crystallization and melting of atomic systems, to optimal packing of objects and the sub-division of space. Here we study an analogous problem—that of determining the optimal shapes of closely packed compact strings. This problem is a mathematical idealization of situations commonly encountered in biology, chemistry and physics, involving the optimal structure of folded polymeric chains. We find that, in cases where boundary effects6 are not dominant, helices with a particular pitch-radius ratio are selected. Interestingly, the same geometry is observed in helices in naturally occurring proteins.

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Figure 1: Examples of optimal strings.
Figure 2: Optimal string with local constraint.
Figure 3: Packing of collagen helices.

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Acknowledgements

This work was supported by INFN, NASA and The Donors of the Petroleum Research Fund administered by the American Chemical Society. A.T. thanks the Physics Department of Università degli Studi di Padova, Padova, Italy, for its hospitality.

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Authors and Affiliations

  1. International School for Advanced Studies (SISSA), Via Beirut 2–4, 34014 Trieste, Istituto Nazionale per la di Fisica della Materia (INFM) and the Abdus Salam International Center for Theoretical Physics, Trieste, Italy

    Amos Maritan, Cristian Micheletti & Antonio Trovato

  2. Department of Physics and Center for Materials Physics, 104 Davey Laboratory, The Pennsylvania State University, University Park, 16802, Pennsylvania, USA

    Jayanth R. Banavar

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  1. Amos Maritan
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  2. Cristian Micheletti
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  3. Antonio Trovato
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  4. Jayanth R. Banavar
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Maritan, A., Micheletti, C., Trovato, A. et al. Optimal shapes of compact strings. Nature 406, 287–290 (2000). https://doi.org/10.1038/35018538

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