This paper will explore analogs of groups formed by the transformations of two ropes as presented by Minh-Tam Quang Trinh.1 The two operations acting on these ropes are T and S, defined as twists and clockwise turns, respectively. These actions intertwine the ropes, creating tangles. T and S also have the ability to untangle the rope when used in certain combinations. In the two-rope case, these actions expressed as a group presentation were shown to be isomorphic to SL₂(Z). We explored the analog of these opera- tions, T and S, and conjectured the corresponding group to which they are isomorphic. Partial proofs for this conjecture are included and future work would include proving our conjecture as well as generalizing results to k ropes.

Alzheimer’s disease (AD) is one of the most widespread neurodegenerative diseases affecting people averaging over 65 years. It manifests with severe cognitive damage, loss of memory, impairment in performing activities, ventricular expansion and final dementia. AD is associated with the deposit of amyloid β protein (Aβ) plaques, intracellular neurofibrillary tangles (NFTs), progressive inflammation and impairment of synaptic transmission and of mitochondrial function. Due to the poor diagnostic tools for the early stages of the disease, proteomics biomarkers have gained a paramount role because they can monitor the prodromal variations of the disease-linked molecular changes. Further, proteome biomarkers can help the follow up of AD progression over time and aid in setting personalized medical care, before the catastrophic consequences of dementia occurs. Research has focused on the identification of proteome biomarkers in cerebrospinal fluid (CSF) and plasma, which are discussed in the present review, but also in other matrices, as saliva and urine, revealing the high potentiality of proteomics approach and, at the same time, the difficulty to identify, for the different stages of the disease, sensitive and specific biomarkers clinically available.

Synaptic transmission is essential for nervous system function and the loss of synapses is a known major contributor to dementia. Alzheimer’s disease dementia (ADD) is characterized by synaptic loss in the mesial temporal lobe and cerebral neocortex, brain areas associated with memory and cognition. The association of synaptic loss and ADD was established in the late 1980s and it has been estimated that 30-50% of neocortical synaptic protein is lost in ADD, but there has not yet been a quantitative profiling of different synaptic proteins in different brain regions in ADD from the same individuals. Very recently, PET imaging of synapses is being developed, accelerating the focus on the role of synaptic loss in ADD and other conditions. In this study, we quantified densities of two synaptic proteins, the presynaptic protein SNAP25 and the postsynaptic protein PSD95 in human brain using enzyme-linked immunosorbent assays (ELISA). Protein was extracted from the cingulate, hippocampus, frontal, visual, and entorhinal cortex from cognitively unimpaired controls, subjects with mild cognitive impairment (MCI) and demented subjects with different levels of Alzheimer’s pathology. SNAP25 is significantly reduced in ADD when compared to controls in frontal cortex, visual cortex and cingulate; while hippocampus showed a smaller, non-significant reduction and entorhinal cortex concentrations were not different. In contrast, all brain areas showed lower PSD95 concentrations in ADD when compared to non-demented controls, although in hippocampus this failed to reach significance. Interestingly, cognitively unimpaired cases with high levels of AD pathology had higher levels of both synaptic proteins in all brain regions. SNAP25 and PSD95 concentrations significantly correlated with densities of neurofibrillary tangles, amyloid plaques and Mini Mental State Examination scores. Our results suggest that synaptic transmission is affected by ADD in multiple brain regions. Differences were less marked in entorhinal cortex and hippocampus, most likely due to a ceiling effect imposed by the very early development of neurofibrillary tangles in older people in these brain regions.

. The Tau protein is associated with microtubule function in the neuron and is crucial for normal axonal transport. In several different neurodegenerative disorders, Tau misfolding leads to hyper-phosphorylation (p-Tau) and intracellular p-Tau aggregates known as neurofibrillary tangles (NFTs). This is particularly evident in individuals with Down syndrome (DS), who develop Alzheimer’s disease-like (AD) neuropathology early in life with almost complete penetrance, associated with the development of dementia symptoms in their 40s or 50s. Our previous findings have shown that certain forms of p-Tau are present already in childhood in individuals with DS in neuron-derived exosomes, suggesting an early phosphorylation profile in this population that could influence the development of AD pathology. Misfolded p-Tau isoforms are known to be seeding competent and may be responsible for spreading AD pathology in different regions of the brain. This review is focused on the accumulation of p-Tau in the brain of individuals with DS and potential consequences for brain function.