Geometric properties of crumpled wires and the condensed non-solid packing state of very long molecular chains

Abstract

Aspectos geométricos associados com o empacotamento hierárquico e heterogêneo de arames amassados são revistos. O fenômeno recentemente descoberto de condensação de energia elástica de curvatura nessas estruturas é discutido e novos resultados são apresentados, com ênfase em leis de escala robustas. Examina-se a possível relevância destas leis nas propriedades conformacionais de cadeias moleculares longas e densamente empacotadas em estado não-sólido como observado, por exemplo, para o DNA nos cromossomas ou nas cápsulas virais. Em particular, argumentos de campo médio são usados para estimar a dependência entre o número de laços e o comprimento dessas cadeias.Geometric aspects associated with the hierarchical and heterogeneous packing of crumpled wires are reviewed. The recently discovered phenomenon of condensation of elastic energy of curvature in these structures is discussed, and new results are presented with emphasis on robust scaling laws. It is examined the possible relevance of these laws in the conformational properties of long molecular chains densely packaged in a non-solid state, as e.g. in the packing of DNA strands in chromosomes, or in virus capsids. In particular, mean field arguments are used to estimate the dependence of the number of loops in the dense non-solid packed three-dimensional configurations of a very long polymer strand as a function of the number of monomers or the chain length.

Unpacking of a Crumpled Wire from Two-Dimensional Cavities

“…If we proceed in our analogy we should expect that the energy E is proportional to the square of the charge, here E ∼ l 2 . This last expression is in agreement with the self-exclusion energy commonly used to model this experiment [17,25,26]. We conjecture that the study of the packing of a wire in a two-dimensional cavity as a capacitor could be useful for the problem of delivery of biopolymers or polymers in biological tissues with the aid of natural or artificial nano carriers [7][8][9][10][11][12].…”

Tight packing of a flexible rod in two-dimensional cavities

“…We study the influence of the available area in rectangular cavities by three different ways: (i) by controlling the total area of the cavity keeping fixed the aspect ratio; (ii) by controlling the width of the cavity keeping fixed its depth; and (iii) by inserting a number of circular discs. The first and second settings can be seen as a study of the finite size effect for this system and can be used to establish a mass-size relationship [16], while the latter corresponds to a model of an elastic one-dimensional object embedded in an environment partially occupied by a competing "fluid".…”

“…Although the problem studied here have an intrinsic interest, it could have some additional interest in a number of technological devices and natural processes aiming at accommodating, manipulating, and storing an object with one-dimensional topology in confined environments [14,15,18,19,20,21]. The optimal injection of a long rod of length L into a finite two-dimensional cavity gives origin to a cascade of N loops which are, in general, heterogeneously distributed along the cavity [11,12,16,22,31]. Here, the experimental study of this process is extensively investigated in 27 types of cavities of several sizes and topologies.…”

“…In these circumstances, the morphologies of the structures formed have a larger surface area as compared to the bulk content, and they can be easily accessed for image recording. Beyond the intrinsic theoretical interests, the problem discussed here has connections with some important issues in the life sciences, as the packing of DNA molecules in viral capsids [14,15,16,17], the physics of an unbranched polymer in confined geometries [18,19], and the problem of packing and delivery of a polymer-like drug with the aid of nano-capsids [20,21]. The packing process studied here generates a planar pattern with a set of loops that is somewhat similar to the classical two-dimensional packing of discs, a problem that has a wide range of applications in technology and science, including the structures of glasses, crystals, granular piles, molecular films and foams [22].…”

A mechanical model of bacteriophage DNA ejection

“…The desktop experiment described here is only meant to illustrate the part of the DNA-phage system behavior relating to the capstan mechanism and is not macroscopic mechanical models to represent the phage-DNA system [28,29,30].…”