Effect of knotting on polymer shapes and their enveloping ellipsoids

Millett, Kenneth C.; Plunkett, Patrick; Piatek, Michael; Rawdon, Eric J.; Stasiak, Andrzej

doi:10.1063/1.3117923

Cited by 17 publications

(19 citation statements)

References 65 publications

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“…We note that Kuhn [11] proposed, for entropic reasons, that the shape of random polymer chains at thermodynamic equilibrium should have the shape of a prolate ellipsoid. This has been numerically confirmed [14,16] and experimentally. One consequence of this molecular asymmetry is the spatial complexity encountered in the study of linking within Olympic systems consisting of such ring molecules independent of excluded volume considerations.…”

Section: Introductionsupporting

confidence: 68%

“…It is known that the linear dimensions (span) along the three principle axes of rotation of the cumulative shapes of unknotted polygons using the SBA method, for N = 125, λ 1 = 5.97, λ 2 = 4.09, λ 3 = 2.90 [14,16]. Note that the ellipsoidal reference frame actually has a random orthogonal spatial orientation unaligned with the coordinate axes of the PBC system.…”

Section: Analysis Of Olympic Systemsmentioning

confidence: 99%

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Resolving critical degrees of entanglement in Olympic ring systems

Igram

Millett

Panagiotou

2016

J. Knot Theory Ramifications

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Olympic systems are collections of small ring polymers whose aggregate properties are largely characterized by the extent (or absence) of topological linking in contrast with the topological entanglement arising from physical movement constraints associated with excluded volume contacts or arising from chemical bonds. First, discussed by de Gennes, they have been of interest ever since due to their particular properties and their occurrence in natural organisms, for example, as intermediates in the replication of circular DNA in the mitochondria of malignant cells or in the kinetoplast DNA networks of trypanosomes. Here, we study systems that have an intrinsic one, two, or threedimensional character and consist of large collections of ring polymers modeled using periodic boundary conditions. We identify and discuss the evolution of the dimensional character of the large scale topological linking as a function of density. We identify the critical densities at which infinite linked subsystems, the onset of percolation, arise in the periodic boundary condition systems. These provide insight into the nature of entanglement occurring in such course grained models. This entanglement is measured using Gauss linking number, a measure well adapted to such models. We show that, with increasing density, the topological entanglement of these systems increases in complexity, dimension, and probability.

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Section: Introductionsupporting

confidence: 68%

Section: Analysis Of Olympic Systemsmentioning

confidence: 99%

Resolving critical degrees of entanglement in Olympic ring systems

Igram

Millett

Panagiotou

2016

J. Knot Theory Ramifications

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“…The shape of a polymer chain can be characterized by the inertial properties [16,17,32]: for example, by calculating three principal moments of inertia for a given configuration of the polymer chain. We used an ellipsoid with the same principal moments of inertia to represent the shape of the polymer configuration.…”

Section: Figurementioning

confidence: 99%

“…Recently, the knotted polymersknotted polystyrene (PS) rings have also been synthesized and characterized [11,12]. The universal properties of circular and knotted polymers such as the entropic properties [13,14], the geometrical properties [15][16][17], the scaling behavior [14,18], and the diffusion behavior [19][20][21][22] have been fully investigated.…”

Section: Introductionmentioning

confidence: 99%

Coil to globule transition of homo- and block-copolymer with different topological constraint and chain stiffness

Wang

2015

Sci. China Chem.

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In this paper, we present the coil-to-globule (CG) transitions of homopolymers and multiblock copolymers with different topology and stiffness by using molecular dynamics with integrated tempering sampling method. The sampling method was a novel enhanced method that efficiently sampled the energy space with low computational costs. The method proved to be efficient and precise to study the structural transitions of polymer chains with complex topological constraint, which may not be easily done by using conventional Monte Carlo method. The topological constraint affects the globule shape of the polymer chain, thus further influencing the CG transition. We found that increasing the topological constraint generally decreased CG transition temperature for homopolymers. For semiflexible chains, an additional first-order like symmetry-broken transition emerged. For block copolymers, the topological constraint did not obviously change the transition temperature, but greatly reduced the energy signal of the CG transition.

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“…For example, this accounts for the existence of equilibrium lengths of polygons having fixed knot types. [51][52][53][54] These powerful phenemona appear to account for the principle influences of knotting on the polygonal models of macromolecules. As the length of a walk or polygon of a fixed knot type increases, the average spatial properties converge to those of the unknot.…”

Section: Estimation Of Knot Populationsmentioning

confidence: 99%

Physical Knot Theory: An Introduction to the Study of the Influence of Knotting on the Spatial Characteristics of Polymers

Millett

2011

Introductory Lectures on Knot Theory

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This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.

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Effect of knotting on polymer shapes and their enveloping ellipsoids

Cited by 17 publications

References 65 publications

Resolving critical degrees of entanglement in Olympic ring systems

Resolving critical degrees of entanglement in Olympic ring systems

Coil to globule transition of homo- and block-copolymer with different topological constraint and chain stiffness

Physical Knot Theory: An Introduction to the Study of the Influence of Knotting on the Spatial Characteristics of Polymers

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