Variations on a Theme: Graph Homomorphisms

This thesis investigates three areas of the theory of graph homomorphisms: cores of graphs, the homomorphism order, and quantum homomorphisms. A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that, for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets which each induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions. Next we examine the restriction of the homomorphism order of graphs to line graphs. Our main focus is in comparing this restriction to the whole order. The primary tool we use in our investigation is that, as a consequence of Vizing's theorem, this partial order can be partitioned into intervals which can then be studied independently. We denote the line graph of X by L(X). We show that for all n ≥ 2, for any line graph Y strictly greater than the complete graph Kₙ, there exists a line graph X sitting strictly between Kₙ and Y. In contrast, we prove that there does not exist any connected line graph which sits strictly between L(Kₙ) and Kₙ, for n odd. We refer to this property as being ``n-maximal", and we show that any such line graph must be a core and the line graph of a regular graph of degree n. Finally, we introduce quantum homomorphisms as a generalization of, and framework for, quantum colorings. Using quantum homomorphisms, we are able to define several other quantum parameters in addition to the previously defined quantum chromatic number. We also define two other parameters, projective rank and projective packing number, which satisfy a reciprocal relationship similar to that of fractional chromatic number and independence number, and are closely related to quantum homomorphisms. Using the projective packing number, we show that there exists a quantum homomorphism from X to Y if and only if the quantum independence number of a certain product graph achieves |V(X)|. This parallels a well known classical result, and allows us to construct examples of graphs whose independence and quantum independence numbers differ. Most importantly, we show that if there exists a quantum homomorphism from a graph X to a graph Y, then ϑ̄(X) ≤ ϑ̄(Y), where ϑ̄ denotes the Lovász theta function of the complement. We prove similar monotonicity results for projective rank and the projective packing number of the complement, as well as for two variants of ϑ̄. These immediately imply that all of these parameters lie between the quantum clique and quantum chromatic numbers, in particular yielding a quantum analog of the well known ``sandwich theorem". We also briefly investigate the quantum homomorphism order of graphs

Comparison of Pittsburgh compound B and florbetapir in cross-sectional and longitudinal studies.

IntroductionQuantitative in vivo measurement of brain amyloid burden is important for both research and clinical purposes. However, the existence of multiple imaging tracers presents challenges to the interpretation of such measurements. This study presents a direct comparison of Pittsburgh compound B-based and florbetapir-based amyloid imaging in the same participants from two independent cohorts using a crossover design.MethodsPittsburgh compound B and florbetapir amyloid PET imaging data from three different cohorts were analyzed using previously established pipelines to obtain global amyloid burden measurements. These measurements were converted to the Centiloid scale to allow fair comparison between the two tracers. The mean and inter-individual variability of the two tracers were compared using multivariate linear models both cross-sectionally and longitudinally.ResultsGlobal amyloid burden measured using the two tracers were strongly correlated in both cohorts. However, higher variability was observed when florbetapir was used as the imaging tracer. The variability may be partially caused by white matter signal as partial volume correction reduces the variability and improves the correlations between the two tracers. Amyloid burden measured using both tracers was found to be in association with clinical and psychometric measurements. Longitudinal comparison of the two tracers was also performed in similar but separate cohorts whose baseline amyloid load was considered elevated (i.e., amyloid positive). No significant difference was detected in the average annualized rate of change measurements made with these two tracers.DiscussionAlthough the amyloid burden measurements were quite similar using these two tracers as expected, difference was observable even after conversion into the Centiloid scale. Further investigation is warranted to identify optimal strategies to harmonize amyloid imaging data acquired using different tracers

Large Scale Gene Expression Profiles of Regenerating Inner Ear Sensory Epithelia

Loss of inner ear sensory hair cells (HC) is a leading cause of human hearing loss and balance disorders. Unlike mammals, many lower vertebrates can regenerate these cells. We used cross-species microarrays to examine this process in the avian inner ear. Specifically, changes in expression of over 1700 transcription factor (TF) genes were investigated in hair cells of auditory and vestibular organs following treatment with two different damaging agents and regeneration in vitro. Multiple components of seven distinct known signaling pathways were clearly identifiable: TGFβ, PAX, NOTCH, WNT, NFKappaB, INSULIN/IGF1 and AP1. Numerous components of apoptotic and cell cycle control pathways were differentially expressed, including p27KIP and TFs that regulate its expression. A comparison of expression trends across tissues and treatments revealed identical patterns of expression that occurred at identical times during regenerative proliferation. Network analysis of the patterns of gene expression in this large dataset also revealed the additional presence of many components (and possible network interactions) of estrogen receptor signaling, circadian rhythm genes and parts of the polycomb complex (among others). Equal numbers of differentially expressed genes were identified that have not yet been placed into any known pathway. Specific time points and tissues also exhibited interesting differences: For example, 45 zinc finger genes were specifically up-regulated at later stages of cochlear regeneration. These results are the first of their kind and should provide the starting point for more detailed investigations of the role of these many pathways in HC recovery, and for a description of their possible interactions

Inhibition of the Mitochondrial Enzyme ABAD Restores the Amyloid-β-Mediated Deregulation of Estradiol

Alzheimer's disease (AD) is a conformational disease that is characterized by amyloid-β (Aβ) deposition in the brain. Aβ exerts its toxicity in part by receptor-mediated interactions that cause down-stream protein misfolding and aggregation, as well as mitochondrial dysfunction. Recent reports indicate that Aβ may also interact directly with intracellular proteins such as the mitochondrial enzyme ABAD (Aβ binding alcohol dehydrogenase) in executing its toxic effects. Mitochondrial dysfunction occurs early in AD, and Aβ's toxicity is in part mediated by inhibition of ABAD as shown previously with an ABAD decoy peptide. Here, we employed AG18051, a novel small ABAD-specific compound inhibitor, to investigate the role of ABAD in Aβ toxicity. Using SH-SY5Y neuroblastoma cells, we found that AG18051 partially blocked the Aβ-ABAD interaction in a pull-down assay while it also prevented the Aβ42-induced down-regulation of ABAD activity, as measured by levels of estradiol, a known hormone and product of ABAD activity. Furthermore, AG18051 is protective against Aβ42 toxicity, as measured by LDH release and MTT absorbance. Specifically, AG18051 reduced Aβ42-induced impairment of mitochondrial respiration and oxidative stress as shown by reduced ROS (reactive oxygen species) levels. Guided by our previous finding of shared aspects of the toxicity of Aβ and human amylin (HA), with the latter forming aggregates in Type 2 diabetes mellitus (T2DM) pancreas, we determined whether AG18051 would also confer protection from HA toxicity. We found that the inhibitor conferred only partial protection from HA toxicity indicating distinct pathomechanisms of the two amyloidogenic agents. Taken together, our results present the inhibition of ABAD by compounds such as AG18051 as a promising therapeutic strategy for the prevention and treatment of AD, and suggest levels of estradiol as a suitable read-out

Common genetic variants in the CLDN2 and PRSS1-PRSS2 loci alter risk for alcohol-related and sporadic pancreatitis

Pancreatitis is a complex, progressively destructive inflammatory disorder. Alcohol was long thought to be the primary causative agent, but genetic contributions have been of interest since the discovery that rare PRSS1, CFTR, and SPINK1 variants were associated with pancreatitis risk. We now report two significant genome-wide associations identified and replicated at PRSS1-PRSS2 (1×10-12) and x-linked CLDN2 (p < 1×10-21) through a two-stage genome-wide study (Stage 1, 676 cases and 4507 controls; Stage 2, 910 cases and 4170 controls). The PRSS1 variant affects susceptibility by altering expression of the primary trypsinogen gene. The CLDN2 risk allele is associated with atypical localization of claudin-2 in pancreatic acinar cells. The homozygous (or hemizygous male) CLDN2 genotype confers the greatest risk, and its alleles interact with alcohol consumption to amplify risk. These results could partially explain the high frequency of alcohol-related pancreatitis in men – male hemizygous frequency is 0.26, female homozygote is 0.07

Modes of Aβ toxicity in Alzheimer’s disease

Alzheimer’s disease (AD) is reaching epidemic proportions, yet a cure is not yet available. While the genetic causes of the rare familial inherited forms of AD are understood, the causes of the sporadic forms of the disease are not. Histopathologically, these two forms of AD are indistinguishable: they are characterized by amyloid-β (Aβ) peptide-containing amyloid plaques and tau-containing neurofibrillary tangles. In this review we compare AD to frontotemporal dementia (FTD), a subset of which is characterized by tau deposition in the absence of overt plaques. A host of transgenic animal AD models have been established through the expression of human proteins with pathogenic mutations previously identified in familial AD and FTD. Determining how these mutant proteins cause disease in vivo should contribute to an understanding of the causes of the more frequent sporadic forms. We discuss the insight transgenic animal models have provided into Aβ and tau toxicity, also with regards to mitochondrial function and the crucial role tau plays in mediating Aβ toxicity. We also discuss the role of miRNAs in mediating the toxic effects of the Aβ peptide

Rare coding variants in PLCG2, ABI3, and TREM2 implicate microglial-mediated innate immunity in Alzheimer's disease

We identified rare coding variants associated with Alzheimer’s disease (AD) in a 3-stage case-control study of 85,133 subjects. In stage 1, 34,174 samples were genotyped using a whole-exome microarray. In stage 2, we tested associated variants (P<1×10-4) in 35,962 independent samples using de novo genotyping and imputed genotypes. In stage 3, an additional 14,997 samples were used to test the most significant stage 2 associations (P<5×10-8) using imputed genotypes. We observed 3 novel genome-wide significant (GWS) AD associated non-synonymous variants; a protective variant in PLCG2 (rs72824905/p.P522R, P=5.38×10-10, OR=0.68, MAFcases=0.0059, MAFcontrols=0.0093), a risk variant in ABI3 (rs616338/p.S209F, P=4.56×10-10, OR=1.43, MAFcases=0.011, MAFcontrols=0.008), and a novel GWS variant in TREM2 (rs143332484/p.R62H, P=1.55×10-14, OR=1.67, MAFcases=0.0143, MAFcontrols=0.0089), a known AD susceptibility gene. These protein-coding changes are in genes highly expressed in microglia and highlight an immune-related protein-protein interaction network enriched for previously identified AD risk genes. These genetic findings provide additional evidence that the microglia-mediated innate immune response contributes directly to AD development

Genomic Relationships, Novel Loci, and Pleiotropic Mechanisms across Eight Psychiatric Disorders

Genetic influences on psychiatric disorders transcend diagnostic boundaries, suggesting substantial pleiotropy of contributing loci. However, the nature and mechanisms of these pleiotropic effects remain unclear. We performed analyses of 232,964 cases and 494,162 controls from genome-wide studies of anorexia nervosa, attention-deficit/hyper-activity disorder, autism spectrum disorder, bipolar disorder, major depression, obsessive-compulsive disorder, schizophrenia, and Tourette syndrome. Genetic correlation analyses revealed a meaningful structure within the eight disorders, identifying three groups of inter-related disorders. Meta-analysis across these eight disorders detected 109 loci associated with at least two psychiatric disorders, including 23 loci with pleiotropic effects on four or more disorders and 11 loci with antagonistic effects on multiple disorders. The pleiotropic loci are located within genes that show heightened expression in the brain throughout the lifespan, beginning prenatally in the second trimester, and play prominent roles in neurodevelopmental processes. These findings have important implications for psychiatric nosology, drug development, and risk prediction.Peer reviewe

Brain volumetric deficits in MAPT mutation carriers: a multisite study

Objective: MAPT mutations typically cause behavioral variant frontotemporal dementia with or without parkinsonism. Previous studies have shown that symptomatic MAPT mutation carriers have frontotemporal atrophy, yet studies have shown mixed results as to whether presymptomatic carriers have low gray matter volumes. To elucidate whether presymptomatic carriers have lower structural brain volumes within regions atrophied during the symptomatic phase, we studied a large cohort of MAPT mutation carriers using a voxelwise approach. Methods: We studied 22 symptomatic carriers (age 54.7 ± 9.1, 13 female) and 43 presymptomatic carriers (age 39.2 ± 10.4, 21 female). Symptomatic carriers’ clinical syndromes included: behavioral variant frontotemporal dementia (18), an amnestic dementia syndrome (2), Parkinson’s disease (1), and mild cognitive impairment (1). We performed voxel-based morphometry on T1 images and assessed brain volumetrics by clinical subgroup, age, and mutation subtype. Results: Symptomatic carriers showed gray matter atrophy in bilateral frontotemporal cortex, insula, and striatum, and white matter atrophy in bilateral corpus callosum and uncinate fasciculus. Approximately 20% of presymptomatic carriers had low gray matter volumes in bilateral hippocampus, amygdala, and lateral temporal cortex. Within these regions, low gray matter volume