Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group: We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.Comment: 36 pages, 10 figure

The Automorphism Group of the Infinite-Rank Free Group is Coarsely Bounded

We prove that the full automorphism group and the outer automorphism group of the free group of countably infinite rank are coarsely bounded. That is, these groups admit no continuous actions on a metric space with unbounded orbits, and have the quasi-isometry type of a point.Comment: 4 pages. v2: Incorporated referee's comment. To appear in New York Journal of Mathematic

Finding and Combining Indicable Subgroups of Big Mapping Class Groups

We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, right-angled Artin groups, free groups, Baumslag-Solitar groups, mapping class groups of other surfaces, and a large collection of wreath products. For each such subgroup $H$ and surface $S$, we show that there are countably many non-conjugate embeddings of $H$ into $\text{Map}(S)$; in certain cases, there are uncountably many such embeddings. The images of each of these embeddings cannot lie in the isometry group of $S$ for any hyperbolic metric and are not contained in the closure of the compactly supported subgroup of $\text{Map}(S)$. In this sense, our construction is new and does not rely on previously known techniques for constructing subgroups of mapping class groups. Notably, our embeddings of $\text{Map}(S')$ into $\text{Map}(S)$ are not induced by embeddings of $S'$ into $S$. Our main tool for all of these constructions is the utilization of special homeomorphisms of $S$ called shift maps, and more generally, multipush maps.Comment: 31 pages, 19 figures. Results have been improved to show countably many non-conjugate embeddings of each subgroup we construc

Genotype-phenotype correlation at codon 1740 ofSETD2

The SET domain containing 2, histone lysine methyltransferase encoded by SETD2 is a dual-function methyltransferase for histones and microtubules and plays an important role for transcriptional regulation, genomic stability, and cytoskeletal functions. Specifically, SETD2 is associated with trimethylation of histone H3 at lysine 36 (H3K36me3) and methylation of α-tubulin at lysine 40. Heterozygous loss of function and missense variants have previously been described with Luscan-Lumish syndrome (LLS), which is characterized by overgrowth, neurodevelopmental features, and absence of overt congenital anomalies. We have identified 15 individuals with de novo variants in codon 1740 of SETD2 whose features differ from those with LLS. Group 1 consists of 12 individuals with heterozygous variant c.5218C>T p.(Arg1740Trp) and Group 2 consists of 3 individuals with heterozygous variant c.5219G>A p.(Arg1740Gln). The phenotype of Group 1 includes microcephaly, profound intellectual disability, congenital anomalies affecting several organ systems, and similar facial features. Individuals in Group 2 had moderate to severe intellectual disability, low normal head circumference, and absence of additional major congenital anomalies. While LLS is likely due to loss of function of SETD2, the clinical features seen in individuals with variants affecting codon 1740 are more severe suggesting an alternative mechanism, such as gain of function, effects on epigenetic regulation, or posttranslational modification of the cytoskeleton. Our report is a prime example of different mutations in the same gene causing diverging phenotypes and the features observed in Group 1 suggest a new clinically recognizable syndrome uniquely associated with the heterozygous variant c.5218C>T p.(Arg1740Trp) in SETD2

Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs

We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir--Bestvina and the work of Mann--Rafi on the large-scale geometry of mapping class groups of infinite-type surfaces. Using the framework of Rosendal for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact). Our techniques also give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and allow us to compute the asymptotic dimension of all of the locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.Comment: 50 pages, 14 figures. Added a new section 9 computing asymptotic dimension for locally CB pure mapping class groups. Also made several edits throughout the document and included new figure

Dihydrodipicolinate synthase from Thermotoga maritima

DHDPS (dihydrodipicolinate synthase) catalyses the branch point in lysine biosynthesis in bacteria and plants and is feedback inhibited by lysine. DHDPS from the thermophilic bacterium Thermotoga maritima shows a high level of heat and chemical stability. When incubated at 90 °C or in 8 M urea, the enzyme showed little or no loss of activity, unlike the Escherichia coli enzyme. The active site is very similar to that of the E. coli enzyme, and at mesophilic temperatures the two enzymes have similar kinetic constants. Like other forms of the enzyme, T. maritima DHDPS is a tetramer in solution, with a sedimentation coefficient of 7.2 S and molar mass of 133 kDa. However, the residues involved in the interface between different subunits in the tetramer differ from those of E. coli and include two cysteine residues poised to form a disulfide bond. Thus the increased heat and chemical stability of the T. maritima DHDPS enzyme is, at least in part, explained by an increased number of inter-subunit contacts. Unlike the plant or E. coli enzyme, the thermophilic DHDPS enzyme is not inhibited by (S)-lysine, suggesting that feedback control of the lysine biosynthetic pathway evolved later in the bacterial lineage

Genotype–phenotype correlation at codon 1740 of SETD2

The SET domain containing 2, histone lysine methyltransferase encoded by SETD2 is a dual-function methyltransferase for histones and microtubules and plays an important role for transcriptional regulation, genomic stability, and cytoskeletal functions. Specifically, SETD2 is associated with trimethylation of histone H3 at lysine 36 (H3K36me3) and methylation of α-tubulin at lysine 40. Heterozygous loss of function and missense variants have previously been described with Luscan-Lumish syndrome (LLS), which is characterized by overgrowth, neurodevelopmental features, and absence of overt congenital anomalies. We have identified 15 individuals with de novo variants in codon 1740 of SETD2 whose features differ from those with LLS. Group 1 consists of 12 individuals with heterozygous variant c.5218C>T p.(Arg1740Trp) and Group 2 consists of 3 individuals with heterozygous variant c.5219G>A p.(Arg1740Gln). The phenotype of Group 1 includes microcephaly, profound intellectual disability, congenital anomalies affecting several organ systems, and similar facial features. Individuals in Group 2 had moderate to severe intellectual disability, low normal head circumference, and absence of additional major congenital anomalies. While LLS is likely due to loss of function of SETD2, the clinical features seen in individuals with variants affecting codon 1740 are more severe suggesting an alternative mechanism, such as gain of function, effects on epigenetic regulation, or posttranslational modification of the cytoskeleton. Our report is a prime example of different mutations in the same gene causing diverging phenotypes and the features observed in Group 1 suggest a new clinically recognizable syndrome uniquely associated with the heterozygous variant c.5218C>T p.(Arg1740Trp) in SETD2