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.

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.DNA topology | tangle method | Xer recombination | band surgery | topology simplification T he Escherichia coli chromosome is a 4.6-Mbp circular doublestranded (ds) DNA duplex, in which the two DNA strands are wrapped around each other ∼420,000 times. During replication, DNA gyrase acts to remove the majority of these strand crossings, but those that remain result in two circular sister molecules that are nontrivially linked. This creates the topological problem of separating the two linked sister chromosomes to ensure proper segregation at the time of cell division. Unlinking of replication links in E. coli is largely achieved by Topoisomerase IV (TopoIV), a type II topoisomerase (1, 2). However, Ip et al. demonstrated that XerC/XerD-dif (XerCD-dif) site-specific recombination, coupled with action of the translocase FtsK, could resolve linked plasmid substrates in vitro and hypothesized that this system could work alongside, yet independently of, TopoIV during in vivo unlinking of replicative catenanes in the bacterial chromosome (3). Grainge et al. then demonstrated that increased site-specific recombination could indeed compensate for a loss of TopoIV activity in unlinking chromosomes in vivo (4). When the activity of TopoIV is blocked, the result is cell lethality. We here propose a mathematically rigorous analysis to describe the pathway and mechanisms of unlinking of replication links by XerCD-FtsK. This work places a fundamental biological process within a mathematical context.Site-specific recombination is a process of breakage and reunion at two specific dsDNA duplexes (the recombination sites). When the DNA substrate consists of circular DNA molecules, the recombination sites may occur in a single DNA circle or in separate circles. Two sites are in direct repeat if they are in the same orientation on one DNA circle (Fig. 1). The relative orientation of the sites is harder to characterize when the two sites are on separate DNA circles. In the case of simple torus links with 2m crossings (also called 2m-catenanes, or 2m-cats) for an integer m > 1, the sites are said to be in parallel or antipara...

In Escherichia coli DNA replication yields interlinked chromosomes. Controlling topological changes associated with replication and returning the newly replicated chromosomes to an unlinked monomeric state is essential to cell survival. In the absence of the topoisomerase topoIV, the site-specific recombination complex XerCD- dif-FtsK can remove replication links by local reconnection. We previously showed mathematically that there is a unique minimal pathway of unlinking replication links by reconnection while stepwise reducing the topological complexity. However, the possibility that reconnection preserves or increases topological complexity is biologically plausible. In this case, are there other unlinking pathways? Which is the most probable? We consider these questions in an analytical and numerical study of minimal unlinking pathways. We use a Markov Chain Monte Carlo algorithm with Multiple Markov Chain sampling to model local reconnection on 491 different substrate topologies, 166 knots and 325 links, and distinguish between pathways connecting a total of 881 different topologies. We conclude that the minimal pathway of unlinking replication links that was found under more stringent assumptions is the most probable. We also present exact results on unlinking a 6-crossing replication link. These results point to a general process of topology simplification by local reconnection, with applications going beyond DNA.

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.

We categorize coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterize the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these results to place lower bounds on the minimum number of recombinant events separating DNA configurations, restrict the recombination pathways and determine chirality and/or orientation of the resulting recombinant DNA molecules.