The Group of Disjoint 2-Spheres in 4-Space

We compute the group of link homotopy classes of link maps of two 2-spheres into 4-space. It turns out to be free abelian, generated by geometric constructions applied to the Fenn-Rolfsen link map and detected by two self-intersection invariants introduced by Paul Kirk in this setting. As a corollary, we show that any link map with one topologically embedded component is link homotopic to the unlink. Our proof introduces a new basic link homotopy, which we call a Whitney homotopy, that shrinks an embedded Whitney sphere constructed from four copies of a Whitney disk. Freedman's disk embedding theorem is applied to get the necessary embedded Whitney disks, after constructing sufficiently many accessory spheres as algebraic duals for immersed Whitney disks. To construct these accessory spheres and immersed Whitney disks we use the algebra of metabolic forms over the group ring Z[Z], and introduce a number of new 4-dimensional constructions, including maneuvers involving the boundary arcs of Whitney disks.Comment: This version significantly reorganized to account for referee's report. Published version: Annals of Mathematics, November 2019 issue (Volume 190, no. 3) 76 pages, 54 figure

Untwisting information from Heegaard Floer homology

The unknotting number of a knot is the minimum number of crossings one must change to turn that knot into the unknot. We work with a generalization of unknotting number due to Mathieu-Domergue, which we call the untwisting number. The p-untwisting number is the minimum number (over all diagrams of a knot) of full twists on at most 2p strands of a knot, with half of the strands oriented in each direction, necessary to transform that knot into the unknot. In previous work, we showed that the unknotting and untwisting numbers can be arbitrarily different. In this paper, we show that a common route for obstructing low unknotting number, the Montesinos trick, does not generalize to the untwisting number. However, we use a different approach to get conditions on the Heegaard Floer correction terms of the branched double cover of a knot with untwisting number one. This allows us to obstruct several 10 and 11-crossing knots from being unknotted by a single positive or negative twist. We also use the Ozsv\'ath-Szab\'o tau invariant and the Rasmussen s invariant to differentiate between the p- and q-untwisting numbers for certain p,q > 1.Comment: 21 pages, 11 figures; final version, accepted for publication in Algebraic & Geometric Topolog

USING SINGLE MOLECULE TECHNIQUES TO DETERMINE THE MECHANISM OF DNA TOPOLOGY SIMPLIFICATION BY TYPE IIA TOPOISOMERASES

Type IIA topoisomerases are essential, universally conserved proteins that modify DNA topology by passing one segment of duplex DNA (the transfer, or T-segment) through a transient double strand break in a second segment of DNA (the gate, or G-segment) in an ATP-dependent reaction. Type IIA topoisomerases decatenate, unknot, and relax supercoiling in DNA to levels below equilibrium, resulting in global topology simplification. The mechanism underlying non-equilibrium topology simplification remains speculative, though several plausible models have been proposed. This thesis tests two of these, the bend angle and kinetic proofreading models, using single-molecule techniques. The bend angle model postulates that non-equilibrium topology simplification scales with the bend angle imposed on the G-segment DNA by a type IIA topoisomerase. To test this model, we used atomic force microscopy and single molecule Förster resonance energy transfer to measure the extent of bending imposed on DNA by three type IIA topoisomerases that span the range of topology simplification activity. We found that all proteins bent DNA, but the imposed bends are similar and cannot account for the differences among the enzymes. These data do not support the bend angle model and suggest that DNA bending is not the sole determinant of non-equilibrium topology simplification. Based on the assumption that the rates of collision between DNA segments is higher in knotted, linked, and supercoiled DNA than in topologically free or relaxed DNA, the kinetic proofreading model proposes that two successive binding events between a G-segment bound topoisomerase and a putative T-segment are required to initiate strand passage. As a result of the two step process, the overall rate of strand passage should scale with the square of the collision probability of two DNA segments. To test this model, we used magnetic tweezers to manipulate a paramagnetic bead tethered to the surface by two DNA molecules. By rotating the bead, we varied the proximity, and thus collision rate, of the two molecules to determine the relationship between collision probability and rate of strand passage. Our data indicate that the strand passage rate scales linearly with the collision probability, which is inconsistent with the kinetic proofreading model

A load-sharing architecture for high performance optimistic simulations on multi-core machines

In Parallel Discrete Event Simulation (PDES), the simulation model is partitioned into a set of distinct Logical Processes (LPs) which are allowed to concurrently execute simulation events. In this work we present an innovative approach to load-sharing on multi-core/multiprocessor machines, targeted at the optimistic PDES paradigm, where LPs are speculatively allowed to process simulation events with no preventive verification of causal consistency, and actual consistency violations (if any) are recovered via rollback techniques. In our approach, each simulation kernel instance, in charge of hosting and executing a specific set of LPs, runs a set of worker threads, which can be dynamically activated/deactivated on the basis of a distributed algorithm. The latter relies in turn on an analytical model that provides indications on how to reassign processor/core usage across the kernels in order to handle the simulation workload as efficiently as possible. We also present a real implementation of our load-sharing architecture within the ROme OpTimistic Simulator (ROOT-Sim), namely an open-source C-based simulation platform implemented according to the PDES paradigm and the optimistic synchronization approach. Experimental results for an assessment of the validity of our proposal are presented as well

The Characterisation of DNA Topoisomerase VI from Methanosarcina mazei Using Ensemble and Single-Molecule Methods

DNA topoisomerase VI (topo VI) is a type IIB DNA topoisomerase, found predominantly in archaea but with eukaryotic presence in various plant and algal species, and possibly plasmodia. In Arabidopsis thaliana, topo VI is an indispensable nuclear protein, essential for endoreduplication, despite the presence of other type IIA topoisomerases, capable of performing the same reactions. Whilst topo VI has been proposed to be a DNA decatenase since its discovery, robust evidence and a mechanism for its preferential decatenation activity was lacking. In this research, the activity of topo VI from the mesophilic archaeon Methanosarcina mazei (MmT6) was characterised using ensemble biochemistry, single-molecule magnetic tweezers and next-generation DNA sequencing (NGS). This demonstrated that MmT6 activity was significantly enhanced in the presence of catenated or braided DNA substrates, as opposed to supercoiled, through a strong preference for DNA crossings with angles close to 90°. In addition, MmT6 behaved as a true crossing sensor with dramatic increases in ATPase activity, DNA binding and the rate of strand-passage, with increasing DNA writhe. Taken together, these results strongly suggest that MmT6 has evolved an intrinsic preference for the unknotting and decatenation of interlinked chromosomes, simultaneously disfavouring the relaxation of supercoils, by sensing DNA crossings directly with geometries close to 90°. This provided an explanation for why topo VI homologues persist in higher eukaryotes during situations in which the genome undergoes rapid duplication, potentially as a dedicated DNA decatenase that cannot be substituted by other topoisomerases. It was also shown, using an NGS-based technique, that MmT6 cleaved DNA with the sequence preference 5ʹ-[G/C][G/C]^[A/C/T], generating 2-base overhangs. The in vitro plasmid-based NGS technique developed to attain this result was extended to begin exploring the in vitro DNA cleavage preferences of other type IIA topoisomerases, namely E. coli DNA topoisomerase IV and E. coli DNA gyrase

Predicting Knot and Catenane Type of Products of Site-specific Recombination on Twist Knot Substrates

Site-specific recombination on supercoiled circular DNA molecules can yield a variety of knots and catenanes. Twist knots are some of the most common conformations of these products and they can act as substrates for further rounds of site-specific recombination. They are also one of the simplest families of knots and catenanes. Yet, our systematic understanding of their implication in DNA and important cellular processes like site-specific recombination is very limited. Here we present a topological model of site-specific recombination characterising all possible products of this reaction on twist knot substrates, extending previous work of Buck and Flapan. We illustrate how to use our model to examine previously uncharacterised experimental data. We also show how our model can help determine the sequence of products in multiple rounds of processive recombination and distinguish between products of processive and distributive recombination. This model studies generic site- specific recombination on arbitrary twist knot substrates, a subject for which there is limited global understanding. We also provide a systematic method of applying our model to a variety of different recombination systems.Comment: 17 pages, 13 figures. One figure correction. To appear in the Journal of Molecular Biology. See also arXiv:1007.2115v1 math.GT for topological proofs for the model presented her

Two-Dimensional Gel Electrophoresis to Study the Activity of Type IIA Topoisomerases on Plasmid Replication Intermediates

DNA topoisomerases are the enzymes that regulate DNA topology in all living cells. Since the discovery and purification of ω (omega), when the first were topoisomerase identified, the function of many topoisomerases has been examined. However, their ability to relax supercoiling and unlink the pre-catenanes of partially replicated molecules has received little attention. Here, we used two-dimensional agarose gel electrophoresis to test the function of three type II DNA topoisomerases in vitro: the prokaryotic DNA gyrase, topoisomerase IV and the human topoisomerase 2α. We examined the proficiency of these topoisomerases on a partially replicated bacterial plasmid: pBR-TerE@AatII, with an unidirectional replicating fork, stalled when approximately half of the plasmid had been replicated in vivo. DNA was isolated from two strains of Escherichia coli: DH5αF’ and parE10. These experiments allowed us to assess, for the first time, the efficiency of the topoisomerases examined to resolve supercoiling and pre-catenanes in partially replicated molecules and fully replicated catenanes formed in vivo. The results obtained revealed the preferential functions and also some redundancy in the abilities of these DNA topoisomerases in vitro