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Neurotrophins, such as nerve growth factor (NGF) and brain derived neurotrophic factor (BDNF), are key factors in neuronal survival and function. In Alzheimer's disease (AD), there is a change in the normal level of these neurotrophins and their precursors (proNGF and proBDNF). The mechanism/s underlying changes in the levels of these neurotrophins in AD is not fully understood. According to the amyloid cascade hypothesis, amyloid-β is the original insult in AD and tau pathology is a downstream event. Amyloid-β interferes with axonal transport and reduces BDNF levels. However, it is not clear if amyloid-β affects neurotrophin levels directly or through tau hyperphosphorylation. If tau is responsible for changes in the level of neurotrophins in AD, we expect to observe the same alteration in neurotrophin levels in other diseases with tau dysfunction such as tauopathies. We tested the levels of BDNF mRNA and proNGF protein in subjects with tauopathies. We observed significant decrease in BDNF mRNA levels in subjects with Corticobasal degeneration. Our result suggests that BDNF may be down-regulated by tau hyperphosphorylation. Moreover, we showed that there was a significant increase in the level of proNGF in Pick's disease (PiD). Interestingly, AD and PiD share common tau modifications. Our result demonstrates a role for tau dysfunction in changes in the level of proNGF. Therefore, study of the levels of NGF and BDNF in non-AD tauopathies has shed light on the mechanisms underlying neurotrophin dysregulation in AD. How do increased levels of proNGF impact the brain in AD or PiD? Is neuronal degeneration in AD or PiD due to the lack of neurotrophic support of proNGF or do increased levels of apoptotic proNGF cause neurodegeneration? Lee et al. (2001b) and Fahnestock et al. (2004a) produced two different cleavage-resistant proNGFs with opposite activities (apoptotic versus neurotrophic). Structural and procedural differences between the two cleavage resistant proNGFs and different bioassays can cause opposite activities. We showed that proNGF from Lee's lab was neurotrophic when it was expressed in the expression system used by Fahnestock et al. or when it was purified. ProNGF expressed in a different expression system was also neurotrophic. ProNGF was neurotrophic in all bioassays except the serum withdrawal assay. We conclude that proNGF is normally neurotrophic but may be apoptotic when cell survival is already compromised. We propose that in AD, cells undergo degeneration due to the lack of neurotrophic support of proNGF (impaired transport). Moreover, TrkA is downregulated in AD which compromises cell survival and may lead to apoptosis induced by increased levels of proNGF. / Thesis / Doctor of Philosophy (PhD)

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2017-03-20T12:17:35Z No. of bitstreams: 2 Dissertação - Wesley da Silva Carvalho - 2016.pdf: 2739140 bytes, checksum: 009cb3705c3ac6a28927493419d88e0c (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-03-20T14:06:03Z (GMT) No. of bitstreams: 2 Dissertação - Wesley da Silva Carvalho - 2016.pdf: 2739140 bytes, checksum: 009cb3705c3ac6a28927493419d88e0c (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-03-20T14:06:03Z (GMT). No. of bitstreams: 2 Dissertação - Wesley da Silva Carvalho - 2016.pdf: 2739140 bytes, checksum: 009cb3705c3ac6a28927493419d88e0c (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-12-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we first state Euler's polyhedral formula for a set of points with Euler characteristic 2. We address the two known ways to prove Euler's Theorem: beginning with the classical proof by using Euclidian Geometry and afterwards we take the advantage of Spherical Geometry to give another proof. Furthermore, we address a version of Euler's formula for planar polyhedron, as well as, Pick's formula and the equivalence between Euler and Pick's formula. In the end, we provide application of Euler and Pick's formula, via two pedagogy tools Geoplano and GeoGebra, by giving examples to teach in classroom. / Esta dissertação trata inicialmente da Fórmula de Euler e de sua validade para os conjuntos de pontos com característica de Euler igual a 2. São feitas duas demonstrações da Fórmula de Euler, uma utilizando conceitos de Geometria Euclidiana e uma outra via Geometria Esférica, além da apresentação de uma versão para poliedros planos da Fórmula de Euler. Posteriormente, é apresentada a Fórmula de Pick para o cálculo de áreas de polígonos simples reticulados e sua relação de equivalência com a Fórmula de Pick para poliedros planos. Finalmente mostramos duas possibilidades de trabalho com a Fórmula de Pick, no Geoplano e no software GeoGebra.

Lattices are an easy and clean class of periodic arrangements that are not only discrete but associated with algebraic structures. We will specifically discuss applying lattices theory to computing the area of polygons in the plane and some optimization problems. This thesis will details information about Pick's Theorem and the higher-dimensional cases of Ehrhart Theory. Closely related to Pick's Theorem and Ehrhart Theory is the Frobenius Problem and Integer Knapsack Problem. Both of these problems have higher-dimension applications, where the difficulties are similar to those of Pick's Theorem and Ehrhart Theory. We will directly relate these problems to optimization problems and operations research.

Chapter 1 presents the definitions and basic properties of trigonometric functions including: Sum Identities, Difference Identities, Product-Sum Identities and Sum-Product Identities. These formulas provide effective tools to solve the problems in trigonometry. Chapter 2 handles the most important two theorems in trigonometry: The laws of sines and cosines and show how they can be applied to derive many well known theorems including: Ptolemy¡¦s theorem, Euler Triangle Formula, Ceva¡¦s theorem, Menelaus¡¦s Theorem, Parallelogram Law, Stewart¡¦s theorem and Brahmagupta¡¦s Formula. Moreover, the formulas of computing a triangle area like Heron¡¦s formula and Pick¡¦s theorem are also discussed. Chapter 3 deals with the method of superposition, inverse trigonometric functions, polar forms and De Moivre¡¦s Theorem.