Fractal globule

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A fractal globule also sometimes called a crumpled globule is a name used to describe polymers that have compact local and global scaling.[1] They can be modeled through a Hamiltonian Walk, a lattice walk in which every point is only visited once and no paths intersect, this prevents knot formation.A hamiltonian walk of a graph is a shortest closed walk that passes through every vertex at least once, and the length of a hamiltonian walk is the total number of edges traversed by the walk.[2] A crumpled globule forms through local regions crumpling, i.e. collapsing in on themselves and this iteratively occurring over the whole polymer. This process follows the Space Filling Peano Curve.

A Hamiltonian walk is a closed walk of minimal length which visits every vertex of a graph (and may visit vertices and edges multiple times). For example, a Hamiltonian walk on the above 3-pan graph is given by the vertex sequence 4, 3, 1, 2, 3, 4 and hence is of length 5. The length of a Hamiltonian walk in a graph is called the Hamiltonian number . For a Hamiltonian graph, , where is the vertex count.[3]

Conformations of the fractal. The fractal globule presents a territorial organization. Source:Lieberman-Aiden E, Van Berkum NL, Williams L et al (2009)

The fractal globule is a compact polymer state that emerges during polymer condensation as a result of topological constraints, which prevent one region of the chain from passing across another one.This long polymer folded into a fractal globule is extraordinarily dense, but completely unknotted. Experimental evidence indicates that the human genome may fold into fractal globules. This structure keeps genetic information compact, yet highly accessible.

The fractal globule architecture, while proposed as a theoretical possibility more than 20 years ago, has never previously been observed. The polymer was first introduced in 1988 (Grosberg et al. 1988)[4] and has not been observed in experiments or simulations until recently by Lieberman Aiden, 2009.[5] Recent characterization of human chromatin using a novel chromosome conformational capture technique brought the fractal globule into the spotlight as a structural model of human chromosome on the scale of up to 10 Mb.See [6] for more details. In the past, many scientists had thought that DNA was compressed into a different architecture called an equilibrium globule, a configuration that is problematic because it can become densely knotted(see the second figure on the right).

The equilibrium globule presents a mixed organization.Source:Lieberman-Aiden E, Van Berkum NL, Williams L et al (2009)

References[edit]

  1. ^ Dewey, T. Gregory (1998) Fractals in Biophysics. Oxford University Press
  2. ^ A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, MA, 1974).
  3. ^ http://mathworld.wolfram.com/HamiltonianWalk.html
  4. ^ Grosberg AY, Nechaev SK, Shakhnovich EI (1988) The role of topological constraints in the kinetics of collapse of macromolecules. J Phys 49:2095–2100
  5. ^ Lieberman-Aiden E, Van Berkum NL, Williams L et al (2009) Comprehensive mapping of long-range interactions reveals folding principles of the human genome. Science 326:289–293
  6. ^ Imakaev M and Mirny L. A (2010) Fractal globule as model of chromatin architectute: biophysical properties to be published.