The average crossing number of equilateral random polygons

The SKMT Algorithm: A method for assessing and comparing underlying protein entanglement

We present fast and simple-to-implement measures of the entanglement of protein tertiary structures which are appropriate for highly flexible structure comparison. These are performed using the SKMT algorithm, a novel method of smoothing the Cα backbone to achieve a minimal complexity curve representation of the manner in which the protein's secondary structure elements fold to form its tertiary structure. Its subsequent complexity is characterised using measures based on the writhe and crossing number quantities heavily utilised in DNA topology studies, and which have shown promising results when applied to proteins recently. The SKMT smoothing is used to derive empirical bounds on a protein's entanglement relative to its number of secondary structure elements. We show that large scale helical geometries dominantly account for the maximum growth in entanglement of protein monomers, and further that this large scale helical geometry is present in a large array of proteins, consistent across a number of different protein structure types and sequences. We also show how these bounds can be used to constrain the search space of protein structure prediction from small angle x-ray scattering experiments, a method highly suited to determining the likely structure of proteins in solution where crystal structure or machine learning based predictions often fail to match experimental data. Finally we develop a structural comparison metric based on the SKMT smoothing which is used in one specific case to demonstrate significant structural similarity between Rossmann fold and TIM Barrel proteins, a link which is potentially significant as attempts to engineer the latter have in the past produced the former. We provide the SWRITHE interactive python notebook to calculate these metrics.

Pulling-force-induced elongation and alignment effects on entanglement and knotting characteristics of linear polymers in a melt

We employ a primitive path (PP) algorithm and the Gauss linking integral to study the degree of entanglement and knotting characteristics of linear polymer model chains in a melt under the action of a constant pulling force applied to selected chain ends. Our results for the amount of entanglement, the linking number, the average crossing number, the writhe of the chains and their PPs and the writhe of the entanglement strands all suggest a different response at the length scale of entanglement strands than that of the chains themselves and of the corresponding PPs. Our findings indicate that the chains first stretch at the level of entanglement strands and next the PP (tube) gets oriented with the "flow." These two phases of the extension and alignment of the chains coincide with two phases related to the disentanglement of the chains. Soon after the onset of external force the PPs attain a more entangled conformation, and the number of nontrivially linked end-to-end closed chains increases. Next, the chains disentangle continuously to attain an almost unentangled conformation. Using the linking matrix of the chains in the melt, we furthermore show that these phases are accompanied by a different scaling of the homogeneity of the global entanglement in the system. The homogeneity of the end-to-end closed chains first increases to a maximum and then decreases slowly to a value characterizing a completely unlinked system.

The Knot Spectrum of Confined Random Equilateral Polygons

It is well known that genomic materials (long DNA chains) of living organisms are often packed compactly under extreme confining conditions using macromolecular self-assembly processes but the general DNA packing mechanism remains an unsolved problem. It has been proposed that the topology of the packed DNA may be used to study the DNA packing mechanism. For example, in the case of (mutant) bacteriophage P4, DNA molecules packed inside the bacteriophage head are considered to be circular since the two sticky ends of the DNA are close to each other. The DNAs extracted from the capsid without separating the two ends can thus preserve the topology of the (circular) DNAs. It turns out that the circular DNAs extracted from bacteriophage P4 are non-trivially knotted with very high probability and with a bias toward chiral knots. In order to study this problem using a systematic approach based on mathematical modeling, one needs to introduce a DNA packing model under extreme volume confinement condition and test whether such a model can produce the kind of knot spectrum observed in the experiments. In this paper we introduce and study a model of equilateral random polygons con_ned in a sphere. This model is not meant to generate polygons that model DNA packed in a virus head directly. Instead, the average topological characteristics of this model may serve as benchmark data for totally randomly packed circular DNAs. The difference between the biologically observed topological characteristics and our benchmark data might reveal the bias of DNA packed in the viral capsids and possibly lead to a better understanding of the DNA packing mechanism, at least for the bacteriophage DNA. The purpose of this paper is to provide information about the knot spectrum of equilateral random polygons under such a spherical confinement with length and confinement ratios in a range comparable to circular DNAs packed inside bacteriophage heads.

Writhe and mutual entanglement combine to give the entanglement length

We propose a method to estimate N(e), the entanglement length, that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions. This estimate uses the writhe of the chains, the writhe of the primitive paths, and the number of kinks in the chains in a melt. An advantage of this method is that it works for both linear and ring chains, works under all periodic boundary conditions, does not require knowing the contour length of the primitive paths, and does not rely on a smooth set of data. We apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies.

Sedimentation of macroscopic rigid knots and its relation to gel electrophoretic mobility of DNA knots

We address the general question of the extent to which the hydrodynamic behaviour of microscopic freely fluctuating objects can be reproduced by macrosopic rigid objects. In particular, we compare the sedimentation speeds of knotted DNA molecules undergoing gel electrophoresis to the sedimentation speeds of rigid stereolithographic models of ideal knots in both water and silicon oil. We find that the sedimentation speeds grow roughly linearly with the average crossing number of the ideal knot configurations, and that the correlation is stronger within classes of knots. This is consistent with previous observations with DNA knots in gel electrophoresis.

The spectrum of filament entanglement complexity and an entanglement phase transition

DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes, cosmic strings and extension cords; filaments come in all sizes and with diverse qualities. Filaments tangle, with profound results: DNA replication is halted, field energy is stored, polymer materials acquire their remarkable properties, textiles are created and shoes stay on feet. We classify entanglement patterns by the rate with which entanglement complexity grows with the length of the filament. We show which rates are possible and which are expected in arbitrary circumstances. We identify a fundamental phase transition between linear and nonlinear entanglement rates. We also find (perhaps surprising) relationships between total curvature, bending energy and entanglement.

On the mean and variance of the writhe of random polygons

We here address two problems concerning the writhe of random polygons. First, we study the behavior of the mean writhe as a function length. Second, we study the variance of the writhe. Suppose that we are dealing with a set of random polygons with the same length and knot type, which could be the model of some circular DNA with the same topological property. In general, a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons; if the mean writhe is non-zero then the knot is chiral. How accurate is this method? For example, if for a specific knot type K the mean writhe decreased to zero as the length of the polygons increased, then this method would be limited in the case of long polygons. Furthermore, we conjecture that the sign of the mean writhe is a topological invariant of chiral knots. This sign appears to be the same as that of an "ideal" conformation of the knot. We provide numerical evidence to support these claims, and we propose a new nomenclature of knots based on the sign of their expected writhes. This nomenclature can be of particular interest to applied scientists. The second part of our study focuses on the variance of the writhe, a problem that has not received much attention in the past. In this case, we focused on the equilateral random polygons. We give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons (of length n) behaves as a linear function of the length of the equilateral random polygon.

Effect of knotting on polymer shapes and their enveloping ellipsoids

We simulate freely jointed chains to investigate how knotting affects the overall shapes of freely fluctuating circular polymeric chains. To characterize the shapes of knotted polygons, we construct enveloping ellipsoids that minimize volume while containing the entire polygon. The lengths of the three principal axes of the enveloping ellipsoids are used to define universal size and shape descriptors analogous to the squared radius of gyration and the inertial asphericity and prolateness. We observe that polymeric chains forming more complex knots are more spherical and also more prolate than chains forming less complex knots with the same number of edges. We compare the shape measures, determined by the enveloping ellipsoids, with those based on constructing inertial ellipsoids and explain the differences between these two measures of polymer shape.

Efficient chain moves for Monte Carlo simulations of a wormlike DNA model: excluded volume, supercoils, site juxtapositions, knots, and comparisons with random-flight and lattice models

We develop two classes of Monte Carlo moves for efficient sampling of wormlike DNA chains that can have significant degrees of supercoiling, a conformational feature that is key to many aspects of biological function including replication, transcription, and recombination. One class of moves entails reversing the coordinates of a segment of the chain along one, two, or three axes of an appropriately chosen local frame of reference. These transformations may be viewed as a generalization, to the continuum, of the Madras-Orlitsky-Shepp algorithm for cubic lattices. Another class of moves, termed T+/-2, allows for interconversions between chains with different lengths by adding or subtracting two beads (monomer units) to or from the chain. Length-changing moves are generally useful for conformational sampling with a given site juxtaposition, as has been shown in previous lattice studies. Here, the continuum T+/-2 moves are designed to enhance their acceptance rate in supercoiled conformations. We apply these moves to a wormlike model in which excluded volume is accounted for by a bond-bond repulsion term. The computed autocorrelation functions for the relaxation of bond length, bond angle, writhe, and branch number indicate that the new moves lead to significantly more efficient sampling than conventional bead displacements and crankshaft rotations. A close correspondence is found in the equilibrium ensemble between the map of writhe computed for pair of chain segments and the map of site juxtapositions or self-contacts. To evaluate the more coarse-grained freely jointed chain (random-flight) and cubic lattice models that are commonly used in DNA investigations, twisting (torsional) potentials are introduced into these models. Conformational properties for a given superhelical density sigma may then be sampled by computing the writhe and using White's formula to relate the degree of twisting to writhe and sigma. Extensive comparisons of contact patterns and knot probabilities of the more coarse-grained models with the wormlike model show that the behaviors of the random-flight model are similar to that of DNA molecules in a solution environment with high ionic strengths, whereas the behaviors of the cubic lattice model with excluded volume are akin to that of DNA molecules under low ionic strengths.

Topologically driven swelling of a polymer loop

Numerical studies of the average size of trivially knotted polymer loops with no excluded volume were undertaken. Topology was identified by Alexander and Vassiliev degree 2 invariants. Probability of a trivial knot, average gyration radius, and probability density distributions as functions of gyration radius were generated for loops of up to N = 3,000 segments. Gyration radii of trivially knotted loops were found to follow a power law similar to that of self-avoiding walks consistent with earlier theoretical predictions.