TY - JOUR
T1 - Fractal and statistical properties of large compact polymers
T2 - A computational study
AU - Lua, Rhonald
AU - Borovinskiy, Alexander L.
AU - Grosberg, Alexander Yu
N1 - Funding Information:
Authors acknowledge the useful discussions with Deguchi T and Moore N. Computations for the present work were performed using Minnesota Supercomputing Institute facilities. This work was supported in part by the MRSEC Program of the National Science Foundation under Award Number DMR-0212302.
PY - 2004/1/15
Y1 - 2004/1/15
N2 - We propose a novel combinatorial algorithm for efficient generation of Hamiltonian walks and cycles on a cubic lattice, modeling the conformations of lattice toy proteins. Through extensive tests on small lattices (allowing complete enumeration of Hamiltonian paths), we establish that the new algorithm, although not perfect, is a significant improvement over the earlier approach by Ramakrishnan et al. [J Chem Phys 103(17) 7592 (1995)], as it generates the sample of conformations with dramatically reduced statistical bias. Using this method, we examine the fractal properties of typical compact conformations. In accordance with Flory theorem celebrated in polymer physics, chain pieces are found to follow Gaussian statistics on the scale smaller than the globule size. Cross-over to this Gaussian regime is found to happen at the scales which are numerically somewhat larger than previously believed. We further used Alexander and Vassiliev degrees 2 and 3 topological invariants to identify the trivial knots among the Hamiltonian loops. We found that the probability of being knotted increases with loop length much faster than it was previously thought, and that chain pieces are consistently more compact than Gaussian if the global loop topology is that of a trivial knot.
AB - We propose a novel combinatorial algorithm for efficient generation of Hamiltonian walks and cycles on a cubic lattice, modeling the conformations of lattice toy proteins. Through extensive tests on small lattices (allowing complete enumeration of Hamiltonian paths), we establish that the new algorithm, although not perfect, is a significant improvement over the earlier approach by Ramakrishnan et al. [J Chem Phys 103(17) 7592 (1995)], as it generates the sample of conformations with dramatically reduced statistical bias. Using this method, we examine the fractal properties of typical compact conformations. In accordance with Flory theorem celebrated in polymer physics, chain pieces are found to follow Gaussian statistics on the scale smaller than the globule size. Cross-over to this Gaussian regime is found to happen at the scales which are numerically somewhat larger than previously believed. We further used Alexander and Vassiliev degrees 2 and 3 topological invariants to identify the trivial knots among the Hamiltonian loops. We found that the probability of being knotted increases with loop length much faster than it was previously thought, and that chain pieces are consistently more compact than Gaussian if the global loop topology is that of a trivial knot.
KW - Compact Hamiltonian walks on a lattice
KW - Knots
KW - Polymer statistics
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U2 - 10.1016/j.polymer.2003.10.073
DO - 10.1016/j.polymer.2003.10.073
M3 - Article
AN - SCOPUS:0346492907
SN - 0032-3861
VL - 45
SP - 717
EP - 731
JO - Polymer
JF - Polymer
IS - 2
ER -