De Witt Sumners

Knots in self-avoiding walks

Discusses the existence of knots in random self-avoiding walks on a lattice. Using Kestens (1963) pattern theorem, it is shown that almost all sufficiently long self-avoiding walks on the three-dimensional simple cubic lattice contain a knot.

Knotting probability of DNA molecules confined in restricted volumes: DNA knotting in phage capsids

When linear double-stranded DNA is packed inside bacteriophage capsids, it becomes highly compacted. However, the phage is believed to be fully effective only if the DNA is not entangled. Nevertheless, when DNA is extracted from a tailless mutant of the P4 phage, DNA is found to be cyclic and knotted (probability of 0.95). The knot spectrum is very complex, and most of the knots have a large number of crossings. We quantified the frequency and crossing numbers of these knots and concluded that, for the P4 tailless mutant, at least half the knotted molecules are formed while the DNA is still inside the viral capsid rather than during extraction. To analyze the origin of the knots formed inside the capsid, we compared our experimental results to Monte Carlo simulations of random knotting of equilateral polygons in confined volumes. These simulations showed that confinement of closed chains to tightly restricted volumes results in high knotting probabilities and the formation of knots with large crossing numbers. We conclude that the formation of the knots inside the viral capsid is driven mainly by the effects of confinement.

DNA–DNA interactions in bacteriophage capsids are responsible for the observed DNA knotting

Recent experiments showed that the linear double-stranded DNA in bacteriophage capsids is both highly knotted and neatly structured. What is the physical basis of this organization? Here we show evidence from stochastic simulation techniques that suggests that a key element is the tendency of contacting DNA strands to order, as in cholesteric liquid crystals. This interaction favors their preferential juxtaposition at a small twist angle, thus promoting an approximately nematic (and apolar) local order. The ordering effect dramatically impacts the geometry and topology of DNA inside phages. Accounting for this local potential allows us to reproduce the main experimental data on DNA organization in phages, including the cryo-EM observations and detailed features of the spectrum of DNA knots formed inside viral capsids. The DNA knots we observe are strongly delocalized and, intriguingly, this is shown not to interfere with genome ejection out of the phage.

Investigation of viral DNA packaging using molecular mechanics models

A simple molecular mechanics model has been used to investigate optimal spool-like packing conformations of double-stranded DNA molecules in viral capsids with icosahedral symmetry. The model represents an elastic segmented chain by using one pseudoatom for each ten basepairs (roughly one turn of the DNA double helix). Force constants for the various terms in the energy function were chosen to approximate known physical properties, and a radial restraint was used to confine the DNA into a sphere with a volume corresponding to that of a typical bacteriophage capsid. When the DNA fills 90% of the spherical volume, optimal packaging is obtained for coaxially spooled models, but this result does not hold when the void volume is larger. When only 60% of the spherical volume is filled with DNA, the lowest energy structure has two layers, with a coiled core packed at an angle to an outer coaxially spooled shell. This relieves bending strain associated with tight curvature near the poles in a model with 100% coaxial spooling. Interestingly, the supercoiling density of these models is very similar to typical values observed in plasmids in bacterial cells. Potential applications of the methodology are also discussed.

Analysis of the mechanism of DNA recombination using tangles

The DNA of all organisms has a complex and essential topology. The three topological properties of naturally occurring DNA are supercoiling, catenation, and knotting. Although these properties are denned rigorously only for closed circular DNA, even linear DNA in vivo can have topological properties because it is divided into topologically separate subdomains (Drlica 1987; Roberge & Gasser, 1992). The essentiality of topological properties is demonstrated by the lethal consequence of interfering with topoisomerases, the enzymes that regulate the level of DNA supercoiling and that unlink DNA during its replication (reviewed in Wang, 1991; Bjornsti, 1991; Drlica, 1992; Ullsperger et al . 1995).

Entanglement complexity of self-avoiding walks

Self-avoiding walks on three-dimensional lattices are flexible linear objects which can be self-entangled. The authors discuss several ways to measure entanglement complexity for n-step walks, and prove that these complexity measures tend to infinity with n. For small n, they use Monte Carlo methods to estimate and compare the n-dependence of two of these complexity measures.

Entanglement complexity of graphs in Z 3

In this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z 3 . What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type? We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z 3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length ( n ) of the embedding goes to infinity, at least linearly in n . One example of a good measure of knot complexity is crossing number. Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.

On Random Knots

In this paper, we consider knotting of Gaussian random polygons in 3-space. A Gaussian random polygon is a piecewise linear circle with n edges in which the length of the edges follows a Gaussian distribution. We prove a continuum version of Kestens Pattern Theorem for these polygons, and use this to prove that the probability that a Gaussian random polygon of n edges in 3-space is knotted tends to one exponentially rapidly as n tends to infinity. We study the properties of Gaussian random knots, and prove that the entanglement complexity of Gaussian random knots gets arbitrarily large as n tends to infinity. We also prove that almost all Gaussian random knots are chiral.

Topological friction strongly affects viral DNA ejection

Significance Bacteriophages are viruses which infect bacteria. Many of these contain double-stranded DNA packed to almost crystalline density and exploit the resulting pressure to trigger DNA ejection into the infected bacterial cell. We show that the ejection kinetics is highly sensitive to the ordering and knotting of the packaged DNA which, in turn, is controlled by DNA self-interactions. The latter favor ordered DNA spools which have a lower effective or topological friction than disordered entangled DNA structures. We also find that torus knots (which can be drawn on the surface of a doughnut) exit the bacteriophage easily; while complex knots or twist knots (which can be formed by linking the ends of a twisted loop) slow down and may stall ejection. Bacteriophages initiate infection by releasing their double-stranded DNA into the cytosol of their bacterial host. However, what controls and sets the timescales of DNA ejection? Here we provide evidence from stochastic simulations which shows that the topology and organization of DNA packed inside the capsid plays a key role in determining these properties. Even with similar osmotic pressure pushing out the DNA, we find that spatially ordered DNA spools have a much lower effective friction than disordered entangled states. Such spools are only found when the tendency of nearby DNA strands to align locally is accounted for. This topological or conformational friction also depends on DNA knot type in the packing geometry and slows down or arrests the ejection of twist knots and very complex knots. We also find that the family of (2, 2k+1) torus knots unravel gradually by simplifying their topology in a stepwise fashion. Finally, an analysis of DNA trajectories inside the capsid shows that the knots formed throughout the ejection process mirror those found in gel electrophoresis experiments for viral DNA molecules extracted from the capsids.

Tangle analysis of Gin site-specific recombination

We use the tangle model to study the action of the site-specific recombinase Gin, an enzyme that can introduce topological changes on circular DNA molecules. Gin and its bound DNA are modelled as a 2-string tangle which undergoes changes during recombination, thereby changing the topology of the DNA substrate. We show that the tangles involved in the analysis are all rational tangles. This technique allows us to prove that, under the model’s assumptions, there is a unique topological description of the enzymatic action. The Gin system is one of the few to date where tangle analysis can be carried out systematically and rigorously, yielding a single, biologically reasonable solution.