In this paper we prove that given two sets E_1,E_2 \subset \mathbb{Z} of positive density, there exists k \geq 1 which is bounded by a number depending only on the densities of E_1 and E_2 such that ...k\mathbb{Z} \subset (E_1-E_1)\cdot (E_2-E_2). As a corollary of the main theorem we deduce that if \alpha ,\beta > 0, then there exist N_0 and d_0 which depend only on \alpha and \beta such that for every N \geq N_0 and E_1,E_2 \subset \mathbb{Z}_N with \vert E_1\vert \geq \alpha N, \vert E_2\vert \geq \beta N there exists d \leq d_0 a divisor of N satisfying d \, \mathbb{Z}_N \subset (E_1-E_1)\cdot (E_2-E_2).
In this paper we show that any Bohr-zero non-periodic set 𝐵 of traceless integer-valued matrices, denoted by Λ, intersects non-trivially the conjugacy class of any matrix from Λ. As a corollary, we ...obtain that the family of characteristic polynomials of 𝐵 contains all characteristic polynomials of matrices from Λ. The main ingredient used in this paper is an equidistribution result for an 𝑆𝐿𝑑(ℤ) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work J. Amer. Math. Soc. 24 (2011), 231–280.
Quantitative twisted patterns in positive density subsets, Discrete Analysis 2024:1, 17 pp. A major theme of arithmetic combinatorics is the structures that can be found inside difference sets of ...dense sets of integers, or dense subsets of more general groups. Many results in this direction concern subsets of finite groups, but there are also interesting results about infinite groups such as $\mathbb Z^d$, with an appropriately chosen notion of density. One such result, which is in a similar spirit to the results of this paper, concerns the question of what distances can be found in difference sets of dense sets. To be more precise, let $A$ be a subset of $\mathbb Z^d$ with positive upper Banach density $d^*(A)$ (defined to be the lim sup of $N^{-d}|A\cap0,N)^d$). Then what can we say about the set $D(A)$ of all positive integers of the form $\|x-y\|_2^2$ with $x,y\in A$? If $d=1$, then $D(A)$ is the set of squares of elements of $A-A$, so the question is not asking anything more interesting than "What can we say about $A-A$?" In general, the elements of $D(A)$ are sums of $d$ squares, so for $d\leq 3$ we have a restriction on the integers that can appear in $D(A)$, whereas by Lagrange's theorem we have no such restriction for $d\geq 4$. Another obvious remark is that if all the coordinates of all the points in $A$ are divisible by $m$, then so are all the elements of $D(A)$. When $d\geq 5$, a theorem of Magyar shows that this is almost the only restriction. It says that for every $d\geq 5$ and every $\epsilon>0$ there exists a positive integer $k$ such that for every set $A\subset\mathbb Z^d$ of upper Banach density at least $\epsilon$ the set $D(A)$ contains all sufficiently large multiples of $k$. Note that the "sufficiently large" depends on $A$ but $k$ depends only on the density of $A$. In the light of this result, it is tempting to ask similar results about the images of distance sets under other functions, and in particular other polynomials. That is the theme of this paper. In fact, it was also the theme of some earlier papers, where results of the above type were proved, but with one important difference: the $k$ that was obtained in those results depended on the set $A$. This paper aims to remedy that defect by obtaining uniform versions of the results, so that they match better the result of Magyar. A wide class of polynomials was defined in a paper of Björklund and the second author, and a non-uniform Magyar-type result was proved for all the polynomials in that class. In this paper that result is upgraded to a uniform one. Two examples that the authors focus on are the polynomials $x^2+y^2-z^2$ and $xy-z^2$. If $F$ is one or other of these two polynomials, then for every $\epsilon>0$ there exists $k$ such that for every set $A\subset\mathbb Z^3$ of upper Banach density at least $\epsilon$ we have that $k\mathbb Z\subset F(A-A)$. The polynomial $xy-z^2$ is the determinant of the matrix $\begin{pmatrix}z&-x\\ y&-z\\ \end{pmatrix}$, and the set of all such matrices forms an additive subgroup of the group of all $2\times 2$ integer matrices of trace 0. The group $SL_2(\mathbb Z)$ acts on this group by conjugation, which preserves the determinant, and the set of matrices just defined is an orbit of the action. The wider class of polynomials comes from generalizing this observation. In this paper, the authors show that under certain additional conditions, one can prove a uniform version of this result. In particular, they obtain uniform statements for the polynomials $x^2+y^2-z^2$ and $xy-z^2$. One of the main tools in the proof is a generalization of the Furstenberg-Sárközy theorem that applies to certain polynomials that do not necessarily have a zero constant term. An appealing corollary of the results in this paper is that if $R$ is an integer polynomial of degree at least 2 with zero constant term and we set $P(x,y)$ to be $x+R(y)$, then for every $\epsilon>0$ there exists $k$, depending only on $\epsilon$ and $R$, such that if $A\subset\mathbb Z^2$ is any set of upper Banach density at least $\epsilon$, then $k\mathbb Z\subset P(A-A)$.
Several mechanisms have been proposed for the synthesis of substrate‐linked ubiquitin chains. HECT ligases directly catalyse protein ubiquitination and have been found to non‐covalently interact with ...ubiquitin. We report crystal structures of the Nedd4 HECT domain, alone and in complex with ubiquitin, which show a new binding mode involving two surfaces on ubiquitin and both subdomains of the HECT N‐lobe. The structures suggest a model for HECT‐to‐substrate ubiquitin transfer, in which the growing chain on the substrate is kept close to the catalytic cysteine to promote processivity. Mutational analysis highlights differences between the processes of substrate polyubiquitination and self‐ubiquitination.
Analysis of ubiquitin binding to the HECT domain of Nedd4 suggests that the ubiquitin chain being elongated is kept close to the catalytic cysteine to promote processivity. Together with the accompanying paper by the Huibregtse group, this study shows the catalysis of polyubiquitin chains by HECT E3 ligases.
Faithful chromosome segregation depends on the ability of sister kinetochores to attach to spindle microtubules. The outer layer of kinetochores transiently expands in early mitosis to form a fibrous ...corona, and compacts following microtubule capture. Here we show that the dynein adaptor Spindly and the RZZ (ROD-Zwilch-ZW10) complex drive kinetochore expansion in a dynein-independent manner. C-terminal farnesylation and MPS1 kinase activity cause conformational changes of Spindly that promote oligomerization of RZZ-Spindly complexes into a filamentous meshwork in cells and in vitro. Concurrent with kinetochore expansion, Spindly potentiates kinetochore compaction by recruiting dynein via three conserved short linear motifs. Expanded kinetochores unable to compact engage in extensive, long-lived lateral microtubule interactions that persist to metaphase, and result in merotelic attachments and chromosome segregation errors in anaphase. Thus, dynamic kinetochore size regulation in mitosis is coordinated by a single, Spindly-based mechanism that promotes initial microtubule capture and subsequent correct maturation of attachments.
BRCA1-BARD1-catalyzed ubiquitination of histone H2A is an important regulator of the DNA damage response, priming chromatin for repair by homologous recombination. However, no specific ...deubiquitinating enzymes (DUBs) are known to antagonize this function. Here we identify ubiquitin specific protease-48 (USP48) as a H2A DUB, specific for the C-terminal BRCA1 ubiquitination site. Detailed biochemical analysis shows that an auxiliary ubiquitin, an additional ubiquitin that itself does not get cleaved, modulates USP48 activity, which has possible implications for its regulation in vivo. In cells we reveal that USP48 antagonizes BRCA1 E3 ligase function and in BRCA1-proficient cells loss of USP48 results in positioning 53BP1 further from the break site and in extended resection lengths. USP48 repression confers a survival benefit to cells treated with camptothecin and its activity acts to restrain gene conversion and mutagenic single-strand annealing. We propose that USP48 promotes genome stability by antagonizing BRCA1 E3 ligase function.
Late endosomes and lysosomes are dynamic organelles that constantly move and fuse to acquire cargo from early endosomes, phagosomes and autophagosome. Defects in lysosomal dynamics cause severe ...neurodegenerative and developmental diseases, such as Niemann-Pick type C disease and ARC syndrome, yet little is known about the regulation of late endosomal fusion in a mammalian system. Mammalian endosomes destined for fusion need to be transported over very long distances before they tether to initiate contact. Here, we describe that lysosomal tethering and transport are combined processes co-regulated by one multi-protein complex: RAB7-RILP-ORP1L. We show that RILP directly and concomitantly binds the tethering HOPS complex and the p150(Glued) subunit of the dynein motor. ORP1L then functions as a cholesterol-sensing switch controlling RILP-HOPS-p150(Glued) interactions. We show that RILP and ORP1L control Ebola virus infection, a process dependent on late endosomal fusion. By combining recruitment and regulation of both the dynein motor and HOPS complex into a single multiprotein complex, the RAB7-RILP-ORP1L complex efficiently couples and regulates the timing of microtubule minus-end transport and fusion, two major events in endosomal biology.
Optical fault injection is a type of attack vector targeting cryptographic circuits where the adversary injects faults during system operation to bypass defenses or reveal secret information. Since ...preventing this kind of attack is generally impractical, most known countermeasures focus on indirect (logic based) or direct detection. Indirect detection mechanisms monitor the effects of optical fault injections in a circuit, whereas direct sensors track the illumination itself. In this paper, we present a compact <inline-formula> <tex-math notation="LaTeX">1.29{\mu }\text{m}\times 1.8{\mu }\text{m} </tex-math></inline-formula> direct optical sensor implemented in 65nm CMOS technology located inside the digital logic fabric. Because it is based on standard CMOS technology, it can be implemented using standard design flow. Measurements on four dedicated chips showed high sensitivity to fault injection attacks: the sensor was 2 to 6 times more sensitive than the combinational logic it protects. As a result of the sub-Vt operation of the transistors, these sensors exhibited post-attack self-recovery ability and high reliability, with a false positive rate under PVT of less than 10−7.
With its noncatalytic domains, DNA-binding regions, and a catalytic core targeting the histone tails, LSD1-CoREST (lysine-specific demethylase 1; REST corepressor) is an ideal model system to study ...the interplay between DNA binding and histone modification in nucleosome recognition. To this end, we covalently associated LSD1-CoREST to semisynthetic nucleosomal particles. This enabled biochemical and biophysical characterizations of nucleosome binding and structural elucidation by small-angle X-ray scattering, which was extensively validated through binding assays and site-directed mutagenesis of functional interfaces. Our results suggest that LSD1-CoREST functions as an ergonomic clamp that induces the detachment of the H3 histone tail from the nucleosomal DNA to make it available for capture by the enzyme active site. The key notion emerging from these studies is the inherently competitive nature of the binding interactions because nucleosome tails, chromatin modifiers, transcription factors, and DNA represent sites for multiple and often mutually exclusive interactions.
Significance The correct and regulated readout of epigenetic marks on chromatin is essential to modulate gene expression in living cells. The regulation of chromatin accessibility is ensured by such epigenetic tags, which form a platform for the binding of specific enzymatic modules. A clear example of this mechanism is represented by the histone demethylase LSD1-CoREST, which removes methylation marks from lysine 4 of histone protein H3. We developed a crosslinking technology to capture this histone demethylase in contact with the nucleosome and used this methodology to explore the structural and biophysical properties of this complex. This is one of the very few successful attempts to visualize the molecular mechanism underlying the recognition of the nucleosomal substrate by a histone-modifying enzyme complex.
DNA mismatch repair detects and removes mismatches from DNA by a conserved mechanism, reducing the error rate of DNA replication by 100- to 1,000-fold. In this process, MutS homologs scan DNA, ...recognize mismatches and initiate repair. How the MutS homologs selectively license repair of a mismatch among millions of matched base pairs is not understood. Here we present four cryo-EM structures of Escherichia coli MutS that provide snapshots, from scanning homoduplex DNA to mismatch binding and MutL activation via an intermediate state. During scanning, the homoduplex DNA forms a steric block that prevents MutS from transitioning into the MutL-bound clamp state, which can only be overcome through kinking of the DNA at a mismatch. Structural asymmetry in all four structures indicates a division of labor between the two MutS monomers. Together, these structures reveal how a small conformational change from the homoduplex- to heteroduplex-bound MutS acts as a licensing step that triggers a dramatic conformational change that enables MutL binding and initiation of the repair cascade.
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