993 resultados para Rumen protozoa
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The objective of this study was to evaluate effects of feeding monensin (MON) or a multivalent polyclonal antibody preparation (PAP) against several rumen microorganisms on feedlot performance, carcass characteristics, blood gas profile, and rumenitis of Bos indicus biotype (BT) yearling bulls. The study was designed as a completely randomized design with a 3 x 2 factorial arrangement, replicated 4 times, in which 32 yearling bulls of each of 3 BT evaluated (3-way-cross, TC; Canchim, CC; and Nellore, NE) were fed diets containing either MON at 300 mg.d(-1) or PAP at 10 mL.d(-1) across 3 different periods. No significant (P > 0.10) feed additive (FA) main effects were observed for any of the feedlot performance variables and carcass characteristics with the exception of dressing percentage. Yearling bulls receiving PAP had a decreased (P = 0.047) dressing percentage when compared with yearling bulls receiving MON. Significant (P < 0.05) BT main effects were observed for all feedlot performance variables and carcass characteristics with the exception of kidney-pelvic fat expressed in kilograms (P = 0.49) and LM lipids content (P = 0.45). Crossbred yearling bulls (TC and CC) had greater (P < 0.001) ADG, DMI in kilograms, DMI as % of BW, and improved (P = 0.001) G: F when compared with NE yearling bulls. A tendency (P = 0.072) for a FA main effect was observed for rumenitis scores, in which yearling bulls receiving PAP had lesser rumenitis scores than those receiving MON. When the data were disposed as frequency percentage, 55.6% and 45.7% of the rumens from yearling bulls fed PAP and MON were scored between 0 and 1, respectively (0 = no lesions, 10 = severe lesions). Likewise, a significant BT main effect was observed (P = 0.008), where NE yearling bulls had greater rumenitis scores than those of crossbred yearling bulls (TC and CC). No signifi cant FA main effects were observed (P > 0.10) for any of the fatty acids measured in the subcutaneous adipose tissue, with the exception that yearling bulls receiving MON had greater (P < 0.05) concentrations of palmitic acid (16: 0), margaric acid (17: 0), docosapentaenoic acid (22: 5), and docosahexaenoic acid (22: 6) than those yearling bulls receiving PAP. Feeding PAP tended to decrease incidence of rumen lesions and led to similar feedlot performance compared with feeding MON. Thus, PAP is a new technology that presents a possible alternative for ionophores.
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Tegumentary leishmaniases are caused by approximately 15 species of protozoa of the genus Leishmania. They prevail in tropical and subtropical areas of the Old and New World but human mobility also makes them a medical problem in nonendemic areas. Clinical manifestations may comprise cutaneous and mucocutaneous forms that may be localized, disseminated, or diffuse in distribution and may differ in Old and New World leishmaniases. Diagnosis and treatment vary according to the clinical manifestations, geographic area, and Leishmania species involved. This article highlights the diversity and complexity of tegumentary leishmaniases, which are worsened by human immunodeficiency virus/Leishmania coinfection.
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Rhipicephalus sanguineus is a widely distributed tick species that has adapted to the urban environment, and the dog is its main host. This species is also known as a vector and reservoir of diseases caused by bacteria, protozoa, and viruses. Currently, acaricides of synthetic chemical origin have been widely and indiscriminately used, leading to the development of resistance to these products by ticks and causing damage to the environment. Thus, these issues have made it necessary to seek other forms of controlling these ectoparasites. R. sanguineus was artificially infested in host New Zealand White rabbits, which were divided into four treatment groups: control (CG1 and CG2) and treatment (TG1 and TG2) groups. TG1 and TG2 hosts were provided with feed supplemented with esters of ricinoleic acid from castor oil at a concentration of 5 g/kg of feed for 7 and 15 days. Afterward, the ovaries of the female ticks were removed for analysis by transmission electron microscopy. The results showed ultrastructural changes in the somatic and germ cells of ovaries from TG1 and TG2 females, particularly with respect to chorion deposition, a protective membrane of the oocyte, as well as in the transport process of vitellogenic materials via the hemolymph and pedicel cells. Moreover, the mitochondria were less electron-dense and had cristae that were more disorganized than the mitochondria from CG1 and CG2 individuals. Thus, this study demonstrated the action of esters on the ovaries of R. sanguineus, signaling the prospect of a way to control this ectoparasite without affecting nontarget organisms or the environment. Microsc. Res. Tech., 2012. (c) 2011 Wiley Periodicals, Inc.
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Background: Antimicrobial peptides are present in animals, plants and microorganisms and play a fundamental role in the innate immune response. Gomesin is a cationic antimicrobial peptide purified from haemocytes of the spider Acanthoscurria gomesiana. It has a broad-spectrum of activity against bacteria, fungi, protozoa and tumour cells. Candida albicans is a commensal yeast that is part of the human microbiota. However, in immunocompromised patients, this fungus may cause skin, mucosal or systemic infections. The typical treatment for this mycosis comprises three major categories of antifungal drugs: polyenes, azoles and echinocandins; however cases of resistance to these drugs are frequently reported. With the emergence of microorganisms that are resistant to conventional antibiotics, the development of alternative treatments for candidiasis is important. In this study, we evaluate the efficacy of gomesin treatment on disseminated and vaginal candidiasis as well as its toxicity and biodistribution. Results: Treatment with gomesin effectively reduced Candida albicans in the kidneys, spleen, liver and vagina of infected mice. The biodistribution of gomesin labelled with technetium-99 m showed that the peptide is captured in the kidneys, spleen and liver. Enhanced production of TNF-alpha, IFN-gamma and IL-6 was detected in infected mice treated with gomesin, suggesting an immunomodulatory activity. Moreover, immunosuppressed and C. albicans-infected mice showed an increase in survival after treatment with gomesin and fluconazole. Systemic administration of gomesin was also not toxic to the mice Conclusions: Gomesin proved to be effective against experimental Candida albicans infection. It can be used as an alternative therapy for candidiasis, either alone or in combination with fluconazole. Gomesin's mechanism is not fully understood, but we hypothesise that the peptide acts through the permeabilisation of the yeast membrane leading to death and/or releasing the yeast antigens that trigger the host immune response against infection. Therefore, data presented in this study reinforces the potential of gomesin as a therapeutic antifungal agent in both humans and animals.
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Abstract Background Hypoxanthine-guanine phosphoribosyltransferase (HGPRT) (EC 2.4.2.8) is a central enzyme in the purine recycling pathway. Parasitic protozoa of the order Kinetoplastida cannot synthesize purines de novo and use the salvage pathway to synthesize purine bases, making this an attractive target for antiparasitic drug design. Results The glycosomal HGPRT from Leishmania tarentolae in a catalytically active form purified and co-crystallized with a guanosine monophosphate (GMP) in the active site. The dimeric structure of HGPRT has been solved by molecular replacement and refined against data extending to 2.1 Å resolution. The structure reveals the contacts of the active site residues with GMP. Conclusion Comparative analysis of the active sites of Leishmania and human HGPRT revealed subtle differences in the position of the ligand and its interaction with the active site residues, which could be responsible for the different reactivities of the enzymes to allopurinol reported in the literature. The solution and analysis of the structure of Leishmania HGPRT may contribute to further investigations leading to a full understanding of this important enzyme family in protozoan parasites.
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Four castrated crossbred horses were used in a randomized block design to study the use of indigestible internal markers iNDF and iADF obtained in situ (from bovines) or in vivo (from equines). Treatments consisted of determining digestibility by the direct method comprising total feces collection (TC) and by the indirect method comprising internal markers iNDF and iADF obtained by in situ incubation in bovine rumen or in vivo by the mobile nylon bag (MNB) technique with horses. iNDF-IV and iADF-IV resulted in better marker recovery rate (RR) (91.50%), similar to TC. The in situ technique resulted in lower RR values for the two indigestible markers, averaging 86.50% (p < 0.05). Estimates of the nutrient coefficient of digestibility (CD) were adequately predicted by iADF-IV, for horses fed on hay exclusively, with rates 46.41, 48.16, 47.92 and 45.51% for dry matter (DM), organic matter (OM), FDN and gross energy, respectively. Results show that MNB may be used to obtain iADF in horses fed on coast-cross hay exclusively, whereas NDFi and ADFi were selected for horses fed on mixed diets to predict the coefficient of nutrient digestibility.
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Four crossbred geldings were used in a randomized blocks experimental design. The objective was to study the use of the internal markers indigestible cellulose (iCEL) and indigestible lignin (iLIG), obtained in situ (cattle) or in vivo (equine) to predict nutrient apparent digestibility in horses. Treatments consisted of different methodologies to determine digestibility: direct method with total feces collection (TC), and indirect method using internal markers iCEL and iLIG obtained either by in situ incubation in bovine rumen or in vivo (IV) using the mobile nylon bag (MNB) technique in horses. Feces production was 2.80 kg in DM, and average recovery rate (p > 0.05) was 101%. Nutrient digestibility coefficient (p > 0.05) estimates were adequately predicted by iCEL and iLIG, obtained in situ or in vivo, with average values of 52.63, 54.17, 64.90, 43.73 and 98.28% for dry matter, organic matter, crude protein, neutral detergent fiber and starch, respectively. It can be concluded that iCEL and iLIG may be obtained in vivo by MNB in horses consuming a forage-concentrate diet, to predict nutrient digestibility coefficients.
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Some non-pathogenic trypanosomatids maintain a mutualistic relationship with a betaproteobacterium of the Alcaligenaceae family. Intensive nutritional exchanges have been reported between the two partners, indicating that these protozoa are excellent biological models to study metabolic co-evolution. We previously sequenced and herein investigate the entire genomes of five trypanosomatids which harbor a symbiotic bacterium (SHTs for Symbiont-Haboring Trypanosomatids) and the respective bacteria (TPEs for Trypanosomatid Proteobacterial Endosymbiont), as well as two trypanosomatids without symbionts (RTs for Regular Trypanosomatids), for the presence of genes of the classical pathways for vitamin biosynthesis. Our data show that genes for the biosynthetic pathways of thiamine, biotin, and nicotinic acid are absent from all trypanosomatid genomes. This is in agreement with the absolute growth requirement for these vitamins in all protozoa of the family. Also absent from the genomes of RTs are the genes for the synthesis of pantothenic acid, folic acid, riboflavin, and vitamin B6. This is also in agreement with the available data showing that RTs are auxotrophic for these essential vitamins. On the other hand, SHTs are autotrophic for such vitamins. Indeed, all the genes of the corresponding biosynthetic pathways were identified, most of them in the symbiont genomes, while a few genes, mostly of eukaryotic origin, were found in the host genomes. The only exceptions to the latter are: the gene coding for the enzyme ketopantoate reductase (EC:1.1.1.169) which is related instead to the Firmicutes bacteria; and two other genes, one involved in the salvage pathway of pantothenic acid and the other in the synthesis of ubiquinone, that are related to Gammaproteobacteria. Their presence in trypanosomatids may result from lateral gene transfer. Taken together, our results reinforce the idea that the low nutritional requirement of SHTs is associated with the presence of the symbiotic bacterium, which contains most genes for vitamin production.
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Problem statement: The aim of the present study was to characterize and differentiate the effects of addition of flavomycin or monensin on ruminal fermentation and degradability as well as on total digestibility in bovine. Approach: Twelve non-pregnant and non-lactating cows (736 kg of BW) were randomly assigned to three treatments: control, flavomycin (20 mg animal-1 day-1) and monensin (300 mg animal-1 day-1). The trial lasted 21 days. The last 10 days were used for external marker administration (15 g of chromic oxide animal-1 day-1). The last 5 days of the trial were used for feces collection and evaluation of corn grain, soybean meal or sugarcane ruminal degradability and the 21st day was used for ruminal fluid sampling. Results: Monensin increased 27.2%, on average, propionate molar proportion at 0, 4, 6, 8, 10 and 12 h after feeding, compared to control and flavomycin groups. When compared to control, flavomycin reduced the degradation rate of soybean meal CP in 31.0%, decreasing the effective degradability when passage rates of 5 and 8% h-1 were used. Dry matter intake, pH, total Short Chain Fatty Acids (tSCFA) or ammoniacal Nitrogen (NH3-N) concentration were not influenced by the addition of either antibiotics. Effective degradability of sugarcane NDF was not influenced by the use of either antibiotic; neither were the TDN nor the digestibility of DM, CP, EE, NFE, ADF, NDF, GE or starch of the diet. Conclusion/Recommendations: In the present study, it was possible to show the beneficial effects of monensin but not of flavomycin, on rumen fermentation
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In den letzten fünf Jahren hat sich mit dem Begriff desspektralen Tripels eine Möglichkeit zur Beschreibungdes an Spinoren gekoppelten Gravitationsfeldes auf(euklidischen) nichtkommutativen Räumen etabliert. Die Dynamik dieses Gravitationsfeldes ist dabei durch diesogenannte spektrale Wirkung, dieSpur einer geeigneten Funktion des Dirac-Operators,bestimmt. Erstaunlicherweise kann man die vollständige Lagrange-Dichtedes (an das Gravitationsfeld gekoppelten) Standardmodellsder Elementarteilchenphysik, also insbesondere auch denmassegebenden Higgs-Sektor, als spektrale Wirkungeines entsprechenden spektralen Tripels ableiten. Diesesspektrale Tripel ist als Produkt des spektralenTripels der (kommutativen) Raumzeit mit einem speziellendiskreten spektralen Tripel gegeben. In der Arbeitwerden solche diskreten spektralen Tripel, die bis vorKurzem neben dem nichtkommutativen Torus die einzigen,bekannten nichtkommutativen Beispiele waren, klassifiziert. Damit kannnun auch untersucht werden, inwiefern sich dasStandardmodell durch diese Eigenschaft gegenüber anderenYang-Mills-Higgs-Theorien auszeichnet. Es zeigt sichallerdings, dasses - trotz mancher Einschränkung - eine sehr große Zahl vonModellen gibt, die mit Hilfe von spektralen Tripelnabgeleitet werden können. Es wäre aber auch denkbar, dass sich das spektrale Tripeldes Standardmodells durch zusätzliche Strukturen,zum Beispiel durch eine darauf ``isometrisch'' wirkendeHopf-Algebra, auszeichnet. In der Arbeit werden, um dieseFrage untersuchen zu können, sogenannte H-symmetrischespektrale Tripel, welche solche Hopf-Isometrien aufweisen,definiert.Dabei ergibt sich auch eine Möglichkeit, neue(H-symmetrische) spektrale Tripel mit Hilfe ihrerzusätzlichen Symmetrienzu konstruieren. Dieser Algorithmus wird an den Beispielender kommutativen Sphäre, deren Spin-Geometrie hier zumersten Mal vollständig in der globalen, algebraischen Sprache der NichtkommutativenGeometrie beschrieben wird, sowie dem nichtkommutativenTorus illustriert.Als Anwendung werden einige neue Beipiele konstruiert. Eswird gezeigt, dass sich für Yang-Mills Higgs-Theorien, diemit Hilfe von H-symmetrischen spektralen Tripeln abgeleitetwerden, aus den zusätzlichen Isometrien Einschränkungen andiefermionischen Massenmatrizen ergeben. Im letzten Abschnitt der Arbeit wird kurz auf dieQuantisierung der spektralen Wirkung für diskrete spektraleTripel eingegangen.Außerdem wird mit dem Begriff des spektralen Quadrupels einKonzept für die nichtkommutative Verallgemeinerungvon lorentzschen Spin-Mannigfaltigkeiten vorgestellt.
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Symbiotische Mikroorganismen aus dem Termitendarm Es ist eine bekannte Tatsache, dass die Hauptaufgabe der Darmflora der niederen Termite im Abbau von Holz liegt. Im Laufe der Millionen Jahre alten Entwicklung der Termiten hat sich in ihrem Darm ein Ökosystem aufgebaut, das aus Protozoen, Archaeen, Bakterien und Hefen besteht. Ziel der vorliegenden Arbeit war die teilweise Erforschung der symbiotischen Zusammenhänge in diesem ökologischen System. Es wurden dabei zwei Gebiete genauer untersucht: Cellulolytische Bakterien im Darm von niederen Termiten Die bisher vorherrschende Meinung sah in den Protozoen die Hauptkomponenten des Celluloseabbaus in Termiten. In dieser Arbeit gelang es 164 cellulolytische Bakterienstämme aus sieben niederen Termitenarten zu isolieren und zu identifizieren. Diese Vielzahl cellulolytischer Bakterienarten könnte ein Indiz dafür sein, dass Bakterien beim Holzabbau von Termiten eine effizientere Rolle spielen als angenommen. Oberflächenbakterien von Mixotricha paradoxa, einem Flagellaten aus dem Darm der niederen Termite Mastotermes darwiniensisMixotricha paradoxa ist ein Beispiel der seltenen Form einer Bewegungssymbiose zwischen Protozoen und Bakterien. Der Flagellat wird von Spirochäten, die auf seiner Oberfläche befestigt sind, vorangetrieben. Zusätzlich leben noch stäbchenförmige Bakterien auf der Hülle. Drei Spirochätenarten und das stäbchenförmige Bakterium konnten identifiziert und lokalisiert werden. Es wird angenommen, dass alle drei Spirochätenarten Mitglieder der Bewegungssymbiose sind.
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Sandwich-Singularitäten sind die Singularitäten auf derNormalisierung von Aufblasungen eines regulärenFlächenkeimes. In der Arbeit wird ein enger Zusammenhangzwischen Topologie und Deformationstheorie vonSandwich-Singularitäten einerseits und ebenenKurvensingularitäten andererseits dargestellt. NeueErgebnisse betreffen u.a. Deformationen vonnulldimensionalen komplexen Räumen in der Ebene, die durchvollständige Ideale beschrieben werden, z.B. wann'simultanes Aufblasen' der Fasern einer solchen Deformationmöglich ist. Zudem werden Glättungskomponenten und dieKollar-Vermutung für Sandwich-Singularitäten untersucht undim Zusammenhang damit numerische Kriterien für die Frage, obdie symbolische Algebra einer Raumkurve endlich erzeugt ist.
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Two Amerindian populations from the Peruvian Amazon (Yanesha) and from rural lowlands of the Argentinean Gran Chaco (Wichi) were analyzed. They represent two case study of the South American genetic variability. The Yanesha represent a model of population isolated for long-time in the Amazon rainforest, characterized by environmental and altitudinal stratifications. The Wichi represent a model of population living in an area recently colonized by European populations (the Criollos are the population of the admixed descendents), whose aim is to depict the native ancestral gene pool and the degree of admixture, in relation to the very high prevalence of Chagas disease. The methods used for the genotyping are common, concerning the Y chromosome markers (male lineage) and the mitochondrial markers (maternal lineage). The determination of the phylogeographic diagnostic polymorphisms was carried out by the classical techniques of PCR, restriction enzymes, sequencing and specific mini-sequencing. New method for the detection of the protozoa Trypanosoma cruzi was developed by means of the nested PCR. The main results show patterns of genetic stratification in Yanesha forest communities, referable to different migrations at different times, estimated by Bayesian analyses. In particular Yanesha were considered as a population of transition between the Amazon basin and the Andean Cordillera, evaluating the potential migration routes and the separation of clusters of community in relation to different genetic bio-ancestry. As the Wichi, the gene pool analyzed appears clearly differentiated by the admixed sympatric Criollos, due to strict social practices (deeply analyzed with the support of cultural anthropological tools) that have preserved the native identity at a diachronic level. A pattern of distribution of the seropositivity in relation to the different phylogenetic lineages (the adaptation in evolutionary terms) does not appear, neither Amerindian nor European, but in relation to environmental and living conditions of the two distinct subpopulations.
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The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.
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The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.