Koya Shimokawa

FtsK-dependent XerCD-dif recombination unlinks replication catenanes in a stepwise manner

Significance Newly replicated circular chromosomes are topologically linked. XerC/XerD-dif (XerCD-dif)–FtsK recombination acts in the replication termination region of the Escherichia coli chromosome to remove links introduced during homologous recombination and replication, whereas Topoisomerase IV removes replication links only. Based on gel mobility patterns of the products of recombination, a stepwise unlinking pathway has been proposed. Here, we present a rigorous mathematical validation of this model, a significant advance over prior biological approaches. We show definitively that there is a unique shortest pathway of unlinking by XerCD-dif–FtsK that strictly reduces the complexity of the links at every step. We delineate the mechanism of action of the enzymes at each step along this pathway and provide a 3D interpretation of the results. In Escherichia coli, complete unlinking of newly replicated sister chromosomes is required to ensure their proper segregation at cell division. Whereas replication links are removed primarily by topoisomerase IV, XerC/XerD-dif site-specific recombination can mediate sister chromosome unlinking in Topoisomerase IV-deficient cells. This reaction is activated at the division septum by the DNA translocase FtsK, which coordinates the last stages of chromosome segregation with cell division. It has been proposed that, after being activated by FtsK, XerC/XerD-dif recombination removes DNA links in a stepwise manner. Here, we provide a mathematically rigorous characterization of this topological mechanism of DNA unlinking. We show that stepwise unlinking is the only possible pathway that strictly reduces the complexity of the substrates at each step. Finally, we propose a topological mechanism for this unlinking reaction.

Dehn surgeries on strongly invertible knots which yield lens spaces

In this article we show no Dehn surgery on nontrivial strongly invertible knots can yield the lens space L(2p, 1) for any integer p. In order to do that, we determine band attaches to (2, 2p)-torus links producing the trivial knot.

Bounds for the minimum step number of knots in the simple cubic lattice

Knots are found in DNA as well as in proteins, and they have been shown to be good tools for structural analysis of these molecules. An important parameter to consider in the artificial construction of these molecules is the minimum number of monomers needed to make a knot. Here we address this problem by characterizing, both analytically and numerically, the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Our analytical work is based on improvement of a method introduced by Diao to enumerate conformations of a given knot type for a fixed length. This method allows us to extend the previously known result on the minimum step number of the trefoil knot 31 (which is 24) to the knots 41 and 51 and show that the minimum step numbers for the 41 and 51 knots are 30 and 34, respectively. Using an independent method based on the BFACF algorithm, we provide a complete list of numerical estimates (upper bounds) of the minimum step numbers for prime knots up to ten crossings, which are improvements over current published numerical results. We enumerate all minimum lattice knots of a given type and partition them into classes defined by BFACF type 0 moves.

Rational tangle surgery and Xer recombination on catenanes

The protein recombinase can change the knot type of circular DNA. The action of a recombinase converting one knot into another knot is normally mathematically modeled by band surgery. Band surgeries on a 2-bridge knot N((4mn-1)/(2m)) yielding a (2,2k)-torus link are characterized. We apply this and other rational tangle surgery results to analyze Xer recombination on DNA catenanes using the tangle model for protein-bound DNA.

Finite surgeries on three-tangle pretzel knots

We classify Dehn surgeries on (p, q, r) pretzel knots that result in a manifold of finite funda- mental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are ( 2,3,7) and ( 2,3,9). Agol and Lackenbys 6-theorem reduces the argument to knots with small indices p, q, r. We treat these using the Culler-Shalen norm of the SL(2, C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.

HEEGAARD SPLITTINGS OF THE TRIVIAL KNOT

We show that any Heegaard splitting of the trivial knot in a compact orientable 3-manifold is standard.

Band Surgeries between Knots and Links with Small Crossing Numbers

In DNA site-specific recombination, an enzyme attaches to a pair of DNA sites, and recombines the sites to different ends. During site-specific recombination, the topology of circular DNA can change, forming knots and links. We model Xer recombination as band surgery. By regarding Xer recombinations as band surgeries and applying mathematical results on band surgery, we confirm experimental results of Xer recombination acting on circular DNA. In particular, our motivation is the unlinking of DNA catenanes by Xer-dif-FtsK recombinations reported by Grainge et al.,6) and Xer recombination at the psi-site on DNA catenanes with 2k crossings which yields DNA knots with 2k + 1 crossings reported by Bath, Sherratt and Colloms.1) The main result of this paper is summarized in Table II of band surgeries between knots with 7 and fewer crossings and catenanes with 8 and fewer crossings. In §2, we relate band surgeries to site-specific recombinations. In §3, we give the table which characterize band surgeries. In §4, we give a table for band surgery between knots. For knots with 7 and fewer crossings, we use the classical notation as in the book by Rolfsen (see Appendix C in 14)). For a knot or link K, we denote by K! the mirror image of K throughout this paper. We use several invariants of knots and links to construct Table II (see Theorems 3.4, 3.5, 3.7, 3.9, 3.11, and 3.13).

Bounds for the minimum step number of knots confined to slabs in the simple cubic lattice

Volume confinement is a key determinant of the topology and geometry of a polymer. For instance recent experimental studies have shown that the knot distribution observed in bacteriophage P4 cosmids is different from that found in full bacteriophages. In particular it was observed that the complexity of the knots decreases sharply when the length of packed genome decreases. However it is not well understood exactly how the volume confinement affects the topology and geometry of a polymer. This problem is the motivation of this paper. Here a polymer is modeled as a self-avoiding polygon on the simple cubit lattice and the confining condition is such that the polygon is bounded between two parallel planes (i.e., bounded within a slab). We estimate the minimum length required for such a polygon to realize a knot type. Our numerical simulations show that in order to realize a prime knot (with up to 10 crossings) in a 1-slab (i.e. a slab of height one), one needs a polygon with length longer than the minimum length needed to realize the same knot when there is no confining condition. In the case of the trefoil knot, we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1-slab is 26, which is greater than 24, the known minimum length required to tie a trefoil without a confinement condition. Additionally, we find that in the 1-slab not all geometrical realizations of a given knot type are equivalent to each other, suggesting that the topology of a polymer in confined volume alone is not enough to describe the state of a polymer. AMS classification scheme numbers: Primary 57M25, secondary 92B99. Submitted to: J. Phys. A: Math. Gen. ‖ Current address: Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. § To whom correspondence should be addressed (kshimoka@rimath.saitama-u.ac.jp.) Minimum step knots in confined cubic lattice regions 2

Signed Unknotting Number and Knot Chirality Discrimination via Strand Passage

Which chiral knots can be unknotted in a single step by a + to − (+−) crossing change, and which by a − to + (−+) crossing change? Numerical results suggest that if a knot with 6 or fewer crossings can be unknotted by a +− crossing change then it cannot be unknotted by a −+ one, and vice versa. However, we exhibit one chiral 8-crossing knot and one chiral 9-crossing knot which can be unknotted by either crossing change. Furthermore, we address the question analytically using results of Taniyama and Traczyk. We apply Taniyama’s classification of unknotting operations to chiral rational knots and fully classify all those which, in a single step, can be unknotted by either type of crossing change; the first of these is 813. As a corollary, we obtain Stoimenow’s result that all chiral twist knots can be unknotted by only one of the two crossing change types, +− or −+. Thus, as was observed numerically, all chiral knots with unknotting number one, and seven or fewer crossings, can be unknotted by only one of the two crossing change types. Traczyk’s results allow us to address the question for some non-rational chiral unknotting number one knots with 9 or fewer crossings, however, for others the question remains open. We propose a numerical approach for investigating the latter type of knot. We also discuss the implications of our work in the context of DNA topology.

ON NON-SIMPLE REFLEXIVE LINKS

We consider Dehn surgeries on some non-simple 2-component links in S3 which yield S3. As corollary we give a class of 2-component links which are determined by their complements.