Transposition Products Research Articles

Upper embeddability of graphs and products of transpositions associated with edges

Upper embeddability of graphs and products of transpositions associated with edges

Catalytic 1,3-Proton Transfer in Alkenes Enabled by Fe═NR Bond Cooperativity: A Strategy for pKa-Dictated Regioselective Transposition of C═C Double Bonds.

Transition metal catalyzed alkene double bond transposition usually involves metal hydride intermediates. Despite significant advances in the design of catalysts that dictate product selectivity, control over substrate selectivity is less advanced and transition metal catalysts that selectively transpose double bonds in substrates containing multiple 1-alkene functionalities are rare. Herein, we report that the three-coordinate high spin (S = 2) Fe(II) imido complex [Ph2B(tBuIm)2Fe═NDipp][K(18-C-6)THF2] (1-K(18-C-6)) catalyzes 1,3-proton transfer from 1-alkene substrates to afford 2-alkene transposition products. Mechanistic investigations involving kinetics, competition, and isotope labeling studies, supported by experimentally calibrated DFT computations, strongly support an unusual nonhydridic mechanism for alkene transposition that is enabled by the cooperative action of the iron center and basic imido ligand. As dictated by the pKa of the allylic protons, this catalyst enables the regioselective transposition of C═C double bonds in substrates containing multiple 1-alkenes. The high spin (S = 2) state of the complex allows a wide scope of functional groups to be tolerated, including those that are typical catalyst poisons, such as amines, N-heterocycles, and phosphines. These results demonstrate a new strategy for metal-catalyzed alkene transposition with predictable substrate regioselectivity.

Reduced Word Enumeration, Complexity, and Randomization

A reduced word of a permutation w is a minimal length expression of w as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that the Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, is typically exponentially large. This implies a result of B. Pawlowski, that it has exponentially growing expectation. Our result is established by a formal run-time analysis of A. Lascoux and M. P. Schützenberger's transition algorithm. The more general problem of Hecke word enumeration, and its closely related question of counting set-valued standard Young tableaux, is also investigated. The latter enumeration problem is further motivated by work on Brill-Noether varieties due to M. Chan-N. Pflueger and D. Anderson-L. Chen-N. Tarasca.

A geometric representation of fragmentation processes on stable trees

We provide a new geometric representation of a family of fragmentation processes by nested laminations which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation, obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous–Pitman fragmentation of the Brownian continuum random tree and minimal factorizations of the n-cycle, that is, factorizations of the permutation (12⋯n) into a product of (n−1) transpositions, proving this way a conjecture of Féray and Kortchemski. We discuss various properties of these new lamination-valued processes, and we notably show that they can be coded by explicit Lévy processes.

Unbiased profiling of CRISPR RNA-guided transposition products by long-read sequencing

Bacterial transposons propagate through either non-replicative (cut-and-paste) or replicative (copy-and-paste) pathways, depending on how the mobile element is excised from its donor source. In the well-characterized E. coli transposon Tn7, a heteromeric TnsA-TnsB transposase directs cut-and-paste transposition by cleaving both strands at each transposon end during the excision step. Whether a similar pathway is involved for RNA-guided transposons, in which CRISPR-Cas systems confer DNA target specificity, has not been determined. Here, we apply long-read, population-based whole-genome sequencing (WGS) to unambiguously resolve transposition products for two evolutionarily distinct transposon types that employ either Cascade or Cas12k for RNA-guided DNA integration. Our results show that RNA-guided transposon systems lacking functional TnsA primarily undergo copy-and-paste transposition, generating cointegrate products that comprise duplicated transposon copies and genomic insertion of the vector backbone. Finally, we report natural and engineered transposon variants encoding a TnsAB fusion protein, revealing a novel strategy for achieving RNA-guided transposition with fewer molecular components.

The Parity Theorem Shuffle

The Parity Theorem states that any permutation can be written as a product of transpositions, but no permutation can be written as a product of both an even number and an odd number of transpositions. Most proofs of the Parity Theorem take several pages of mathematical formalism to complete. This article presents an alternative but equivalent statement of the Parity Theorem, replacing “transposition” with “adjacent transposition,” as well as a proof based on an inversion count technique. To build an intuitive understanding of the concept of inversion and to help students discover the Parity Theorem on their own, this article presents a series of five classroom activities based on the group-building initiative “Telephone Pole Shuffle.”

Mind Switches in Futurama and Stargate

SummaryWe generalize two mind-switching problems that arise in connection with the popular sci-fi television series Futurama and Stargate SG-1. Optimal solutions to these problems are found by answering the following question about a permutation σ expressed as a product of nontrivial disjoint cycles. “When writing σ as a product of distinct transpositions, none occurring as factors in the disjoint cycle representation of σ, what is the smallest number of transpositions that can be used?”

Keeler's Theorem and Products of Distinct Transpositions

An episode of the television series Futurama features a two-body mind-switching machine, which will not work more than once on the same pair of bodies. After the Futurama community engages in a mind-switching spree, the question is asked, “Can the switching be undone so as to restore all minds to their original bodies?” Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two “outsiders.” We refine Keeler's result by providing a more efficient algorithm that uses the smallest possible number of switches. We also present best possible algorithms for undoing two natural sequences of switches, each sequence effecting a cyclic mind-scrambling permutation in the symmetric group Sn. Finally, we give necessary and sufficient conditions on m and n for the identity permutation to be expressible as a product of m distinct transpositions in Sn.

TED, an Autonomous and Rare Maize Transposon of theMutatorSuperfamily with a High Gametophytic Excision Frequency

Mutator (Mu) elements, one of the most diverse superfamilies of DNA transposons, are found in all eukaryotic kingdoms, but are particularly numerous in plants. Most of the present knowledge on the transposition behavior of this superfamily comes from studies of the maize (Zea mays) Mu elements, whose transposition is mediated by the autonomous Mutator-Don Robertson (MuDR) element. Here, we describe the maize element TED (for Transposon Ellen Dempsey), an autonomous cousin that differs significantly from MuDR. Element excision and reinsertion appear to require both proteins encoded by MuDR, but only the single protein encoded by TED. Germinal excisions, rare with MuDR, are common with TED, but arise in one of the mitotic divisions of the gametophyte, rather than at meiosis. Instead, transposition-deficient elements arise at meiosis, suggesting that the double-strand breaks produced by element excision are repaired differently in mitosis and meiosis. Unlike MuDR, TED is a very low-copy transposon whose number and activity do not undergo dramatic changes upon inbreeding or outcrossing. Like MuDR, TED transposes mostly to unlinked sites and can form circular transposition products. Sequences closer to TED than to MuDR were detected only in the grasses, suggesting a rather recent evolutionary split from a common ancestor.

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

In this paper, we prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say: 1. the classical simple compact Lie groups: special orthogonal groups SO(n), special unitary groups SU(n) and compact symplectic groups USp(n); 2. the real, complex and quaternionic Grassmannian varieties (including the real spheres, and the complex or quaternionic projective spaces when q = 1): SO(p + q)/(SO(p)×SO(q)), SU(p + q)/S(U(p)×U(q)) and USp(p + q)/(USp(p)×USp(q)); 3. the spaces of real, complex and quaternionic structures: SU(n)/SO(n), SO(2n)/ U(n), SU(2n)/USp(n) and USp(n)/UU(n). Denoting μt the law of the Brownian motion at time t, we give explicit lower bounds for dTV(μt,Haar) if \(t t_{\text{cut-of\/f}}\). This provides in particular an answer to some questions raised in recent papers by Chen and Saloff-Coste. Our proofs are inspired by those given by Rosenthal and Porod for products of random rotations in SO(n), and by Diaconis and Shahshahani for products of random transpositions in \(\mathfrak{S}_{n}\).

A transgenic system for generation of transposon Ac/Ds-induced chromosome rearrangements in rice

The maize Activator (Ac)/Dissociation (Ds) transposable element system has been used in a variety of plants for insertional mutagenesis. Ac/Ds elements can also generate genome rearrangements via alternative transposition reactions which involve the termini of closely linked transposons. Here, we introduced a transgene containing reverse-oriented Ac/Ds termini together with an Ac transposase gene into rice (Oryza sativa ssp. japonica cv. Nipponbare). Among the transgenic progeny, we identified and characterized 25 independent genome rearrangements at three different chromosomal loci. The rearrangements include chromosomal deletions and inversions and one translocation. Most of the deletions occurred within the T-DNA region, but two cases showed the loss of 72 kilobase pairs (kb) and 79 kb of rice genomic DNA flanking the transgene. In addition to deletions, we obtained chromosomal inversions ranging in size from less than 10 kb (within the transgene DNA) to over 1 million base pairs (Mb). For 11 inversions, we cloned and sequenced both inversion breakpoints; in all 11 cases, the inversion junctions contained the typical 8 base pairs (bp) Ac/Ds target site duplications, confirming their origin as transposition products. Together, our results indicate that alternative Ac/Ds transposition can be an efficient tool for functional genomics and chromosomal manipulation in rice.Electronic supplementary materialThe online version of this article (doi:10.1007/s00122-012-1925-4) contains supplementary material, which is available to authorized users.

Synthesis and Antiproliferative Effects of [3]Ferrocenophane Transposition Products and Pinacols Obtained from McMurry Cross-Coupling Reactions

We here report the synthesis and antiproliferative activities of two new series of ferrocenophanes obtained from McMurry cross-coupling reactions of [3]ferrocenophan-1-one with benzophenone, 4-hydroxybenzophenone, 4,4′-dihydroxybenzophenone, and 4,4′-diacetylaminobenzophenone. In addition to the main formation of olefins at reflux, tetrahedral transposition products, resulting from a pinacolic rearrangement, were also isolated in about 10% yields. Lowering the temperature of the reaction to 0 °C allowed the isolation of pinacols, which could be transformed into transposition compounds in good yields. Three ferrocenophane compounds have been characterized by X-ray crystallography: 1-(p-hydroxyphenyl)-1-phenyl-2-oxo[4]ferrocenophane (7b), 1,1-diphenyl-2-oxo[4]ferrocenophane (7c), and 1-hydroxy-1-[1-hydroxy-1-[3]ferrocenophanyl][3]ferrocenophane (12) crystallize in monoclinic P21/n, triclinic P1, and monoclinic P21/c space groups, respectively. The antiproliferative effects on hormone-independent breast can...

ISPpy1, a novel mobile element of the ancient Psychrobacter maritimus permafrost strain: Translocations in Escherichia coli K-12 cells and formation of composite transposons

It was shown that IS element ISPpyl isolated earlier in the permafrost strain Psychrobacter maritimus MR29-12 has a high level of functional activity in cells of the heterologous host Escherichia coli K-12. ISPpyl can be translocated in E. coli cells by itself and mobilize adjacent genes and can also form composite transposons flanked by two copies of this element. Apart from translocations between different plasmids, the composite ISPpyl-containing transposon Tn5080a is capable of translocation from the plasmid into the E. coli chromosome with high frequency and from the chromosome into the plasmid. Among products of Tn5080a transposition into plasmid R388, simple insertions were predominantly formed together with cointegrates. Upon mobilization of adjacent genes with the use of one ISPpyl copy, only cointegrates arise.

On the cut-off phenomenon for the transitivity of randomly generated subgroups

Consider K ≥ 2 independent copies of the random walk on the symmetric group SN starting from the identity and generated by the products of either independent uniform transpositions or independent uniform neighbor transpositions. At any time \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $n\in \mathbb{N}$ \end{document} **image** , let Gn be the subgroup of SN generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut-off phenomenon at time N ln(N)/(2K) for the non-existence of fixed point of Gn and for the transitivity of Gn, thus showing that these properties occur before the chains have reached equilibrium. In the uniform neighbor transposition model, a transition for the non-existence of a fixed point of Gn appears at time of order \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} $N^{1+\frac{2}{K}}$ \end{document} **image** (at least for K ≥ 3), but there is no cut-off phenomenon. In the latter model, we recover a cut-off phenomenon for the non-existence of a fixed point at a time proportional to N by allowing the number K to be proportional to ln(N). The main tools of the proofs are spectral analysis and coupling techniques. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.

Asymptotics of characters of symmetric groups related to Stanley character formula

We prove an upper bound for characters of the symmetric groups. Namely, we show that there exists a constant a>0 with a property that for every Young diagram \lambda with n boxes, r(\lambda) rows and c(\lambda) columns |Tr \rho^\lambda(\pi) / Tr \rho^\lambda(e)| < [a max(r(\lambda)/n, c(\lambda)/n,|\pi|/n) ]^{|\pi|}, where |\pi| is the minimal number of factors needed to write \pi\in S_n as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov's and Biane's character formulas and give a new formula for free cumulants of the transition measure.

Architecture of the Tn7 Posttransposition Complex: An Elaborate Nucleoprotein Structure

Four transposition proteins encoded by the bacterial transposon Tn7, TnsA, TnsB, TnsC, and TnsD, mediate its site- and orientation-specific insertion into the chromosomal site attTn7. To establish which Tns proteins are actually present in the transpososome that executes DNA breakage and joining, we have determined the proteins present in the nucleoprotein product of transposition, the posttransposition complex (PTC), using fluorescently labeled Tns proteins. All four required Tns proteins are present in the PTC in which we also find that the Tn7 ends are paired by protein–protein contacts between Tns proteins bound to the ends. Quantification of the relative amounts of the fluorescent Tns proteins in the PTC indicates that oligomers of TnsA, TnsB, and TnsC mediate Tn7 transposition. High-resolution DNA footprinting of the DNA product of transposition attTn7∷Tn7 revealed that about 350 bp of DNA on the transposon ends and on attTn7 contact the Tns proteins. All seven binding sites for TnsB, the component of the transposase that specifically binds the ends and mediates 3′ end breakage and joining, are occupied in the PTC. However, the protection pattern of the sites closest to the Tn7 ends in the PTC are different from that observed with TnsB alone, likely reflecting the pairing of the ends and their interaction with the target nucleoprotein complex necessary for activation of the breakage and joining steps. We also observe extensive protection of the attTn7 sequences in the PTC and that alternative DNA structures in substrate attTn7 that are imposed by TnsD are maintained in the PTC.

Fusion of reverse-oriented Ds termini following abortive transposition in Arabidopsis: implications for the mechanism of Ac/Ds transposition

We studied the products of alternative transposition reactions that utilize reverse-oriented Ds termini as substrates. In this configuration, Ds transposition can generate genome rearrangements including deletions, inversions, and reciprocal translocations. In approximately half of the transposition products recovered in Arabidopsis, the termini of the reversed ends Ds element were ligated together. The sequences at these fused-end junctions suggest that the excised transposon termini form covalently closed hairpin structures. These results shed new light on the mechanism of Ac/Ds transposition.

Control of directionality in the DNA strand-exchange reaction catalysed by the tyrosine recombinase TnpI

In DNA site-specific recombination catalysed by tyrosine recombinases, two pairs of DNA strands are sequentially exchanged between separate duplexes and the mechanisms that confer directionality to this theoretically reversible reaction remain unclear. The tyrosine recombinase TnpI acts at the internal resolution site (IRS) of the transposon Tn4430 to resolve intermolecular transposition products. Recombination is catalysed at the IRS core sites (IR1–IR2) and is regulated by adjacent TnpI-binding motifs (DR1 and DR2). These are dispensable accessory sequences that confer resolution selectivity to the reaction by stimulating synapsis between directly repeated IRSs. Here, we show that formation of the DR1–DR2-containing synapse imposes a specific order of activation of the TnpI catalytic subunits in the complex so that the IR1-bound subunits catalyse the first strand exchange and the IR2-bound subunits the second strand exchange. This ordered pathway was demonstrated for a complete recombination reaction using a TnpI catalytic mutant (TnpI-H234L) partially defective in DNA rejoining. The presence of the DR1- and DR2-bound TnpI subunits was also found to stabilize transient recombination intermediates, further displacing the reaction equilibrium towards product formation. Implication of TnpI/IRS accessory elements in the initial architecture of the synapse and subsequent conformational changes taking place during strand exchange is discussed.

Generating restricted classes of involutions, Bell and Stirling permutations

We present a recursive generating algorithm for unrestricted permutations which is based on both the decomposition of a permutation as a product of transpositions and that as a union of disjoint cycles. It generates permutations at each recursive step and slight modifications of it produce generating algorithms for Bell permutations and involutions. Further refinements yield algorithms for these classes of permutations subject to additional restrictions: a given number of cycles or/and fixed points. We obtain, as particular cases, generating algorithms for permutations counted by the Stirling numbers of the first and second kind, even permutations, fixed-point-free involutions and derangements. All of these algorithms run in constant amortized time.

On the baker’s map and the simplicity of the higher dimensional Thompson groups &lt;I&gt;nV&lt;/I&gt;

We show that the baker's map is a product of transpositions (particularly pleasant involutions), and conclude from this that an existing very short proof of the simplicity of Thompson's group V applies with equal brevity to the higher dimensional Thompson groups nV.