The ideal trefoil knot

The most tight conformation of the trefoil knot found by the SONO algorithm is presented. Structure of the set of its self-contact points is analyzed.Comment: 11 pages, 8 figure

Numerical study of linear and circular model DNA chains confined in a slit: metric and topological properties

Advanced Monte Carlo simulations are used to study the effect of nano-slit confinement on metric and topological properties of model DNA chains. We consider both linear and circularised chains with contour lengths in the 1.2--4.8 $\mu$m range and slits widths spanning continuously the 50--1250nm range. The metric scaling predicted by de Gennes' blob model is shown to hold for both linear and circularised DNA up to the strongest levels of confinement. More notably, the topological properties of the circularised DNA molecules have two major differences compared to three-dimensional confinement. First, the overall knotting probability is non-monotonic for increasing confinement and can be largely enhanced or suppressed compared to the bulk case by simply varying the slit width. Secondly, the knot population consists of knots that are far simpler than for three-dimensional confinement. The results suggest that nano-slits could be used in nano-fluidic setups to produce DNA rings having simple topologies (including the unknot) or to separate heterogeneous ensembles of DNA rings by knot type.Comment: 12 pages, 10 figure

The Shapes of Tight Composite Knots

We present new computations of tight shapes obtained using the constrained gradient descent code RIDGERUNNER for 544 composite knots with 12 and fewer crossings, expanding our dataset to 943 knots and links. We use the new data set to analyze two outstanding conjectures about tight knots, namely that the ropelengths of composite knots are at least 4\pi-4 less than the sums of the prime factors and that the writhes of composite knots are the sums of the writhes of the prime factors.Comment: Summary text file of tight knot lengths and writhing numbers stored in anc/ropelength_data.txt. All other data freely available at http:://www.jasoncantarella.com/ and through Data Conservanc

Repulsive Forces Between Looping Chromosomes Induce Entropy-Driven Segregation

One striking feature of chromatin organization is that chromosomes are compartmentalized into distinct territories during interphase, the degree of intermingling being much smaller than expected for linear chains. A growing body of evidence indicates that the formation of loops plays a dominant role in transcriptional regulation as well as the entropic organization of interphase chromosomes. Using a recently proposed model, we quantitatively determine the entropic forces between chromosomes. This Dynamic Loop Model assumes that loops form solely on the basis of diffusional motion without invoking other long-range interactions. We find that introducing loops into the structure of chromatin results in a multi-fold higher repulsion between chromosomes compared to linear chains. Strong effects are observed for the tendency of a non-random alignment; the overlap volume between chromosomes decays fast with increasing loop number. Our results suggest that the formation of chromatin loops imposes both compartmentalization as well as order on the system without requiring additional energy-consuming processes

Reduced P300 amplitude during retrieval on a spatial working memory task in a community sample of adolescents who report psychotic symptoms.

BACKGROUND: Deficits in working memory are widely reported in schizophrenia and are considered a trait marker for the disorder. Event-related potentials (ERPs) and imaging data suggest that these differences in working memory performance may be due to aberrant functioning in the prefrontal and parietal cortices. Research suggests that many of the same risk factors for schizophrenia are shared with individuals from the general population who report psychotic symptoms. METHODS: Forty-two participants (age range 11--13 years) were divided into those who reported psychotic symptoms (N = 17) and those who reported no psychotic symptoms, i.e. the control group (N = 25). Behavioural differences in accuracy and reaction time were explored between the groups as well as electrophysiological correlates of working memory using a Spatial Working Memory Task, which was a variant of the Sternberg paradigm. Specifically, differences in the P300 component were explored across load level (low load and high load), location (positive probe i.e. in the same location as shown in the study stimulus and negative probe i.e. in a different location to the study stimulus) and between groups for the overall P300 timeframe. The effect of load was also explored at early and late timeframes of the P300 component (250-430 ms and 430-750 ms respectively). RESULTS: No between-group differences in the behavioural data were observed. Reduced amplitude of the P300 component was observed in the psychotic symptoms group relative to the control group at posterior electrode sites. Amplitude of the P300 component was reduced at high load for the late P300 timeframe at electrode sites Pz and POz. CONCLUSIONS: These results identify neural correlates of neurocognitive dysfunction associated with population level psychotic symptoms and provide insights into ERP abnormalities associated with the extended psychosis phenotype

Thickness of knots

AbstractClassical knot theory studies one-dimensional filaments; in this paper we model knots as more physically “real”, e.g., made of some “rope” with nonzero thickness. A motivating question is: How much length of unit radius rope is needed to tie a nontrivial knot?For a smooth knot K, the “injectivity radius” R(K) is the supremum of radii of embedded tubular neighborhoods. The “thickness” of K, a new measure of knot complexity, is the ratio of R(K) to arc-length. We relate thickness to curvature, self-distance, distortion, and (for knot types) edge-number

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