Writhe and mutual entanglement combine to give the entanglement length

Panagiotou, Eleni; Kröger, Martin; Millett, Kenneth C.

doi:10.1103/physreve.88.062604

Cited by 35 publications

(42 citation statements)

References 79 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The effective potentials induced by the virtual tubes of knots are similar to those induced by entanglements in multichain systems [36,52]. However, the interchain entanglements are more abundant than self-entanglements in entangled polymer melts [53,54].…”

Section: Prl 114 037801 (2015) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 67%

Origin of Metastable Knots in Single Flexible Chains

2015

View full text Add to dashboard Cite

Recent theoretical progress has explained the physics of knotting of semiflexible polymers, yet knotting of flexible polymers is relatively unexplored. We herein develop a new theory for the size distribution of knots on a flexible polymer and the existence of metastable knots. We show the free energy of a flexible molecule in a tube can be mapped to quantitatively reproduce the free energy distribution of a knot on a flexible chain. The size distribution of knots on flexible chains is expected to be universal and might be observed at a macroscopic scale, such as a string of hard balls. DOI: 10.1103/PhysRevLett.114.037801 PACS numbers: 61.41.+e, 02.10.Kn, 87.10.Rt, 87.15.-v Long polymer molecules can find themselves in knotted conformations, akin to those seen in jumbled strings [1] or agitated chains [2][3][4], and these knots become increasingly likely as the length of the polymer is increased [5]. Knotted conformations are frequently found in polymers in poor solvents [6] or confined within small volumes, such as a viral capsid [7,8], and a number of knotted protein conformations have been discovered [9][10][11]. To study the properties of knots on polymers more closely, knots have been tied on actin filaments [12] and DNA molecules [13] with optical tweezers, and knots have been generated on DNA in microfluidic devices through collisions with defects [14] or the application of electric fields [15] These developments have motivated fundamental study of knots on polymers. The knotting probability and the size distribution of knots have been investigated for chains under various conditions [29][30][31][32][33][34][35]. An intriguing finding from simulations is that the cores of knots often localize to small portions of semiflexible [36] or flexible chain [6,[37][38][39][40] [ Fig. 1(a)]. Grosberg and Rabin posited a theory that predicts a metastable knot size for thin semiflexible polymers, such as double-stranded DNA [41]. The Grosberg-Rabin (GR) theory considers the bending and confining energies within the knot, which tend to swell and shrink the knot, respectively, leading to a metastable size. We extended the GR theory to incorporate the effect of a finite chain width and demonstrated quantitative agreement between these theories and simulation results [36]. These theories, however, cannot address knotting in flexible chains, such as single-stranded DNA and most common synthetic polymers, yet recent simulations observed a metastable knot size in these flexible molecules [40].In this Letter, we present a new theory that predicts a metastable knot size for flexible polymers. We show results of simulations of trefoil knots in long, flexible polymer molecules exhibit a metastable knot size. We develop a theory that considers a flexible knot to be effectively confined in a virtual tube, and we perform simulations to understand the free energy of confinement of flexible chains in tubes. Finally, we show that the simulations of flexible chains in tubes can be directly mapped to the free energies of knots by t...

show abstract

Section: Prl 114 037801 (2015) P H Y S I C a L R E V I E W L E T T Ementioning

confidence: 67%

Origin of Metastable Knots in Single Flexible Chains

2015

View full text Add to dashboard Cite

show abstract

“…Finally, our focus has been self-entanglements of a single chain produced by flow. The entanglement of different chains in solution is a topic of interest [49,50] and is a possible direction for future research using the approach developed in this paper. is shown in figure 4(a) and denoted as = I a 0 .…”

Section: Discussionmentioning

confidence: 99%

Dynamics and topology of a flexible chain: knots in steady shear flow

et al. 2015

View full text Add to dashboard Cite

We use numerical simulations of a bead-spring model chain to investigate the evolution of the conformations of long and flexible elastic fibers in a steady shear flow. In particular, for rather open initial configurations, and by varying a dimensionless elastic parameter, we identify two distinct conformational modes with different final size, shape, and orientation. Through further analysis we identify slipknots in the chain. Finally, we provide examples of initial configurations of an 'open' trefoil knot that the flow unknots and then knots again, sometimes repeating several times. and the influence of electric fields on knots [31]. Also, the untying of a knot in an extensional flow has been studied recently [32], and has the spirit of the present paper where we ask about the effects of flow. In contrast, for non-Brownian systems other kinds of forcing, such as vibration, can produce knotting of chains of linked spheres [13,14]. We are not aware of any similar studies, experimental or numerical simulations, highlighting conditions that lead to long non-Brownian chains forming knots in flows.Thus, while significant steps have been taken towards understanding the detailed dynamics of flexible non-Brownian fibers, to understand topology is much more difficult: it requires three-dimensional detail and very flexible, long and thin fibers as, in theory (when there is no flow), the probability of knotting in a random walk increases exponentially with the length of the walk [33]. Moreover, the initial conditions may matter and an operational definition is needed for a knot in an open fiber.In this paper, we focus on the dynamics and topology of long, flexible, non-Brownian chains of beads in a steady shear flow and use numerical simulations to investigate whether knots in the fiber occur as a result of the flow. We will find that unknotted fibers are capable of forming (open) knots, and in some cases we document a sequence of unknotting-knotting transitions. The numerical solution of this kind of problem requires tracking the motion of N beads, where hydrodynamic interactions between the beads mean that the dynamics of the topology of the object involve N 3 coupled nonlinear ordinary differential equations. Not surprisingly, such dynamics should be expected to be chaotic though we have not attempted in this paper to link our study of complex shapes and the unknotting-knotting transition to the underlying chaotic dynamics. Our results highlight new aspects of the dynamics of flexible filaments in flow.

show abstract

“…1b), the result would be the (integer) linking number [27]. By applying the method to open chains [9,23,[28][29][30], the GE provides a real number that quantifies the mutual arXiv:1809.02173v3 [q-bio.BM] 31 May 2019 FIG. 2.…”

mentioning

confidence: 99%

Sequence and structural patterns detected in entangled proteins reveal the importance of co-translational folding

Baiesi

Orlandini

Seno

et al. 2019

Sci Rep

View full text Add to dashboard Cite

Proteins must fold quickly to acquire their biologically functional three-dimensional native structures. Hence, these are mainly stabilized by local contacts, while intricate topologies such as knots are rare. Here, we reveal the existence of specific patterns adopted by protein sequences and structures to deal with backbone self-entanglement. A large scale analysis of the Protein Data Bank shows that loops significantly intertwined with another chain portion are typically closed by weakly bound amino acids. Why is this energetic frustration maintained? A possible picture is that entangled loops are formed only toward the end of the folding process to avoid kinetic traps. Consistently, these loops are more frequently found to be wrapped around a portion of the chain on their N-terminal side, the one translated earlier at the ribosome. Finally, these motifs are less abundant in natural native states than in simulated protein-like structures, yet they appear in 32% of proteins, which in some cases display an amazingly complex intertwining.

show abstract

Writhe and mutual entanglement combine to give the entanglement length

Cited by 35 publications

References 79 publications

Origin of Metastable Knots in Single Flexible Chains

Origin of Metastable Knots in Single Flexible Chains

Dynamics and topology of a flexible chain: knots in steady shear flow

Sequence and structural patterns detected in entangled proteins reveal the importance of co-translational folding

Contact Info

Product

Resources

About