Effect of Knotting on the Shape of Polymers

Abstract

Momentary configurations of long polymers at thermal equilibrium usually deviate from spherical symmetry and can be better described, on average, by a prolate ellipsoid. The asphericity and nature of asphericity (or prolateness) that describe these momentary ellipsoidal shapes of a polymer are determined by specific expressions involving the three principal moments of inertia calculated for configurations of the polymer. Earlier theoretical studies and numerical simulations have established that as the length of the polymer increases, the average shape for the statistical ensemble of random configurations asymptotically approaches a characteristic universal shape that depends on the solvent quality. It has been established, however, that these universal shapes differ for linear, circular, and branched chains. We investigate here the effect of knotting on the shape of cyclic polymers modeled as random isosegmental polygons. We observe that random polygons forming different knot types reach asymptotic shapes that are distinct from the ensemble average shape. For the same chain length, more complex knots are, on average, more spherical than less complex knots.

Large-Scale Conformational Transitions in Supercoiled DNA Revealed by Coarse-Grained Simulation

“…The asphericity is a scale-independent shape descriptor that measures the extent to which the principal moments of gyration of a polymer deviate from one another, and possesses a value of 0 for an isotropic polymer conformation, and a value of 1 for a perfect rod (57).…”

Monte Carlo simulations of densely-packed athermal polymers in the bulk and under confinement

“…Image created with the VMD software (Humphrey et al, 1996). Milner, 2011;Rawdon et al, 2008aRawdon et al, , 2008b. It is thus tempting to try to correlate the two different forms of topological constraints: entanglements, of inter-chain origin, with knots, an intramolecular topological quantity.…”

Conformation of Ring Polymers in 2D Constrained Environments

“…Thus we show further that spatially confined molecules fold into specific conformations close to those found for linear chains, and strongly dependent on the size of the confining box. DOI: 10.1103/PhysRevLett.106.248301 PACS numbers: 82.35.Gh, 87.64.Dz, 36.20.Ey, 87.14.gk Ring closure of a polymer is one of the important factors influencing its statistical mechanical properties [1], e.g., scaling [2,3], shape [4,5], and diffusion [6][7][8], because it restrains the accessible phase space. The theoretical description of circular chains (knots or catenanes) is a challenging problem, owing to the difficulties inherent to a systematic theoretical analysis of such objects constrained to a unique topology.…”

“…Ring closure of a polymer is one of the important factors influencing its statistical mechanical properties [1], e.g., scaling [2,3], shape [4,5], and diffusion [6][7][8], because it restrains the accessible phase space. The theoretical description of circular chains (knots or catenanes) is a challenging problem, owing to the difficulties inherent to a systematic theoretical analysis of such objects constrained to a unique topology.…”

“…From our results for the correlation function, ring polymers in classes I and II are clearly described very well by the LSF model, and thus we pursue our comparison with the results of Fisher and coworkers who extensively studied the shape parameters of vesicles. As the correlation function, these shape parameters are convenient measures characterizing the different polymer classes because they depend on the type of polymer (Gaussian, self-avoiding), its dimensionality, its topology, and its rigidity [4,5,27,28]. Typical shape measures include the anisotropy AE and the asphericity A, which are defined as combinations of R 2 G1 and R 2 G2 , the small and large principal axes of the radius-of-gyration tensor R G , by AE ¼ hR 2 G1 =R 2 G2 i and…”