Saturation numbers in tripartite graphs

Given graphs $H$ and $F$, a subgraph $G\subseteq H$ is an $F$-saturated subgraph of $H$ if $F\nsubseteq G$, but $F\subseteq G+e$ for all $e\in E(H)\setminus E(G)$. The saturation number of $F$ in $H$, denoted $\text{sat}(H,F)$, is the minimum number of edges in an $F$-saturated subgraph of $H$. In this paper we study saturation numbers of tripartite graphs in tripartite graphs. For $\ell\ge 1$ and $n_1$, $n_2$, and $n_3$ sufficiently large, we determine $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell})$ and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-1})$ exactly and $\text{sat}(K_{n_1,n_2,n_3},K_{\ell,\ell,\ell-2})$ within an additive constant. We also include general constructions of $K_{\ell,m,p}$-saturated subgraphs of $K_{n_1,n_2,n_3}$ with few edges for $\ell\ge m\ge p>0$.Comment: 18 pages, 6 figure

Unlocking an HDAC Toolbox: Methods toward Understanding Isozyme-specific Activity.

Lysine acetylation is a dynamic post-translational modification occurring ubiquitously in cells. Histone deacetylases (HDACs) catalyze the enzymatic hydrolysis of acetyllysine. There are 18 HDACs, tasked with the negative regulation of acetylation of thousands of proteins. It is therefore critical to understand the reactivity and specificity of these enzymes. To probe the substrate specificity of HDAC isozymes, we developed and optimized a real-time enzyme-coupled assay that measures deacetylation via the formation of acetate and two mass spectrometric assays that measure the mass change associated with deacetylation. These assays were used to measure the reactivity of HDAC8 and HDAC11 with a variety of acetylated peptides and inhibitors, providing insight into the substrate selectivity. To characterize HDAC11, we recombinantly expressed a SUMO-HDAC11 fusion protein in E. coli in the presence of the molecular chaperone trigger factor. Interestingly, HDAC11 expressed in bacteria is inactive, in contrast to the catalytically active HDAC11 expressed and purified from eukaryotic cells. HDAC11 purified from insect cells catalyzes deacetylation of unlabeled peptide substrates and demonstrate unique selectivity that differs from HDAC8. To identify protein substrates, we introduced a chip-based proteomics method to screen the reactivity of HDAC isozymes with thousands of full-length human proteins. This work identified 44 and 25 potential HDAC8 and HDAC11 substrates, respectively. These substrates were then validated using peptide mimics with the enzyme-coupled and mass spectrometric assays mentioned above, determining rate constants spanning three orders of magnitude. Based on these results, isocitrate dehydrogenase 1 (IDH1) variants containing single, biologically relevant acetyllysine side chains were expressed and purified. HDAC8 catalyzes the deacetylation of these full-length protein substrates in vitro with high efficiency. Furthermore, the acetyllysine modifications decrease the catalytic activity of IDH1. Finally, the first HDAC prodrug, SAHA-TAP, is shown to covalently modify a conserved cysteine residue, Cys153 in HDAC8, leading to irreversible inactivation and simultaneous release of the competitive inhibitor SAHA. Overall, this work provides new methods for the characterization of HDAC reactivity, the identification of novel HDAC8 and HDAC11 substrates, and analysis of HDAC-inhibition. This work provides a foundation for understanding disease states that arise from aberrant acetylation and deacetylation.PHDChemical BiologyUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/135788/1/eds_1.pd

Minimum saturated subgraphs of tripartite graphs

Let F and H be graphs. A subgraph G of H is an F-saturated subgraph of H if F is not a subgraph of G and F is a subgraph of G+e for any edge e in E(H) E(G). The saturation number of F in H is the minimum number of edges in a F-saturated subgraph of H. We denote the saturation number of F in H as sat(H,F). In this thesis we review the history of saturated subgraphs, and prove new results on saturated subgraphs of tripartite graphs. Let Ka,b,c be a compete tripartite graph, with partite sets of size a, b, and c. Specifically, we determine sat(Kn1,n2,n3,Kl,l,l), for n1≥ n2≥ n3, when n2 bounded by a linear function of n3. We also examine the special case when l=1 and determine sat(Kn1,n2,n3,K3)$ for n1≥ n2≥ n3, and n_3 sufficiently large. We also consider two natural variants of saturated subgraphs that arise in the tripartite setting. We examine the behavior of these extensions using illustrative examples to highlight the differences between these variations and the original problem

Being Sisyphus: A writing pedagogy for at-risk students

This thesis discussess the limitations of the standards-based movement and suggests that some schools, especially those whose mission it is to work exclusively with at-risk students, need to be allowed to set local behavioral standards before any consideration can be given to setting and teaching academic standards. It mainly focuses on Phoenix High School, a community day school in the Corona-Norco Unified School District, and discusses how the standards based movement is not suited to meet the needs of its students