99 resultados para Complex functions


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Latent transforming growth factor-beta (TGF-beta) binding proteins (LTBPs) -1, -3 and -4 are ECM components whose major function is to augment the secretion and matrix targeting of TGF-beta, a multipotent cytokine. LTBP-2 does not bind small latent TGF-beta but has suggested functions as a structural protein in ECM microfibrils. In the current work we focused on analyzing possible adhesive functions of LTBP-2 as well as on characterizing the kinetics and regulation of LTBP-2 secretion and ECM deposition. We also explored the role of TGF-beta binding LTBPs in endothelial cells activated to mimic angiogenesis as well as in malignant mesothelioma. We found that, unlike most adherent cells, several melanoma cell lines efficiently adhered to purified recombinant LTBP-2. Further characterization revealed that the adhesion was mediated by alpha3beta1 and alpha6beta1 integrins. Heparin also inhibited the melanoma cell adhesion suggesting a role for heparan sulphate proteoglycans. LTBP-2 was also identified as a haptotactic substrate for melanoma cell migration. We used cultured human embryonic lung fibroblasts to analyze the temporal and spatial association of LTBP-2 into ECM. By We found that LTBP-2 was efficiently assembled to the ECM only in confluent cultures following the deposition of fibronectin (FN) and fibrillin-1. In early, subconfluent cultures it remained primarily in soluble form after secretion. LTBP-2 colocalized transiently with FN and fibrillin-1. Silencing of fibrillin-1 expression by lentiviral shRNAs profoundly disrupted the deposition of LTBP-2 indicating that the ECM association of LTBP-2 depends on a pre-formed fibrillin-1 network. Considering the established role of TGF-beta as a regulator of angiogenesis we induced morphological activation of endothelial cells by phorbol 12-myristate 13-acetate (PMA) and followed the fate of LTBP-1 in the endothelial ECM. This resulted in profound proteolytic processing of LTBP-1 and release of latent TGF-beta complexes from the ECM. The processing was coupled with increased activation of MT-MMPs and specific upregulation of MT1-MMP. The major role of MT1-MMP in the proteolysis of LTBP-1 was confirmed by suppressing the expression with lentivirally induced short-hairpin RNAs as well as by various metalloproteinases inhibitors. TGF-beta can promote tumorigenesis of malignant mesothelioma (MM), which is an aggressive tumor of the pleura with poor prognosis. TGF-beta activity was analyzed in a panel of MM tumors by immunohistochemical staining of phosphorylated Smad-2 (P-Smad2). The tumor cells were strongly positive for P-Smad2 whereas LTBP-1 immunoreactivity was abundant in the stroma, and there was a negative correlation between LTBP-1 and P-Smad2 staining. In addition, the high P-Smad2 immunoreactivity correlated with shorter survival of patients. mRNA analysis revealed that TGF-beta1 was the most highly expressed isoform in both normal human pleura and MM tissue. LTBP-1 and LTBP-3 were both abundantly expressed. LTBP-1 was the predominant isoform in established MM cell lines whereas the expression of LTBP-3 was high in control cells. Suppression of LTBP-3 expression by siRNAs resulted in increased TGF-beta activity in MM cell lines accompanied by decreased proliferation. Our results suggest that decreased expression of LTBP-3 in MM could alter the targeting of TGF-beta to the ECM and lead to its increased activation. The current work emphasizes the coordinated process of the assembly and appropriate targeting of LTBPs with distinct adhesive or cytokine harboring properties into the ECM. The hierarchical assembly may have implications in the modulation of signaling events during morphogenesis and tissue remodeling.

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Epilysin (MMP-28) is the most recently identified member of the matrix metalloproteinase (MMP) family of extracellular proteases. Together these enzymes are capable of degrading almost all components of the extracellular matrix (ECM) and are thus involved in important biological processes such as development, wound healing and immune functions, but also in pathological processes such as tumor invasion, metastasis and arthritis. MMPs do not act solely by degrading the ECM. They also regulate cell behavior by releasing growth factors and biologically active peptides from the ECM, by modulating cell surface receptors and adhesion molecules and by regulating the activity of many important mediators in inflammatory pathways. The aim of this study was to define the unique role of epilysin within the MMP-family, to elucidate how and when it is expressed and how its catalytic activity is regulated. To gain information on its essential functions and substrates, the specific aim was to characterize how epilysin affects the phenotype of epithelial cells, where it is biologically expressed. During the course of the study we found that the epilysin promoter contains a well conserved GT-box that is essential for the basic expression of this gene. Transcription factors Sp1 and Sp3 bind this sequence and could hence regulate both the basic and cell type and differentiation stage specific expression of epilysin. We cloned mouse epilysin cDNA and found that epilysin is well conserved between human and mouse genomes and that epilysin is glycosylated and activated by furin. Similarly to in human tissues, epilysin is normally expressed in a number of mouse tissues. The expression pattern differs from most other MMPs, which are expressed only in response to injury or inflammation and in pathological processes like cancer. These findings implicate that epilysin could be involved in tissue homeostasis, perhaps fine-tuning the phenotype of epithelial cells according to signals from the ECM. In view of these results, it was unexpected to find that epilysin can induce a stable epithelial to mesenchymal transition (EMT) when overexpressed in epithelial lung carcinoma cells. Transforming growth factor b (TGF-b) was recognized as a crucial mediator of this process, which was characterized by the loss of E-cadherin mediated cell-cell adhesion, elevated expression of gelatinase B and MT1-MMP and increased cell migration and invasion into collagen I gels. We also observed that epilysin is bound to the surface of epithelial cells and that this interaction is lost upon cell transformation and is susceptible to degradation by membrane type-1-MMP (MT1-MMP). The wide expression of epilysin under physiological conditions implicates that its effects on epithelial cell phenotype in vivo are not as dramatic as seen in our in vitro cell system. Nevertheless, current results indicate a possible interaction between epilysin and TGF-b also under physiological circumstances, where epilysin activity may not induce EMT but, instead, trigger less permanent changes in TGF-b signaling and cell motility. Epilysin may thus play an important role in TGF-b regulated events such as wound healing and inflammation, processes where involvement of epilysin has been indicated.

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Poikkijuovaisen luuranko- ja sydänlihaksen supistumisyksikkö, sarkomeeri, koostuu tarkoin järjestyneistä aktiini- ja myosiinisäikeistä. Rakenne eroaa muista solutyypeistä, joissa aktiinisäikeistö muovautuu jatkuvasti ja sen järjestyminen säätelee solun muotoa, solujakautumista, soluliikettä ja solunsisäisten organellien kuljetusta. Myotilin, palladin ja myopalladin kuuluvat proteiiniperheeseen, jonka yhteispiirteenä ovat immunoglobuliinin kaltaiset (Igl) domeenit. Proteiinit liittyvät aktiinitukirankaan ja niiden arvellaan toimivan solutukirangan rakenne-elementteinä ja säätelijöinä. Myotilinia ja myopalladinia ilmennetään poikkijuovaisessa lihaksessa. Sen sijaan palladinin eri silmukointimuotoja tavataan monissa kudostyypeissä kuten hermostossa, ja eri muodoilla saattaa olla solutyypistä riippuvia tehtäviä. Poikkijuovaisessa lihaksessa kaikki perheen jäsenet sijaitsevat aktiinisäikeitä yhdistävässä Z-levyssä ja ne sitovat Z-levyn rakenneproteiinia, -aktiniinia. Myotilingeenin pistemutaatiot johtavat periytyviin lihastauteihin, kun taas palladinin mutaatioiden on kuvattu liittyvän periytyvään haimasyöpään ja lisääntyneeseen sydäninfarktin riskiin. Tässä tutkimuksessa selvitettin myotilinin ja pallainin toimintaa. Kokeissa löydettiin uusia palladinin 90-92kDa alatyyppiin sitoutuvia proteiineja. Yksi niistä on aktiinidynamiikkaa säätelevä profilin. Profilinilla on kahdenlaisia tehtäviä; se edesauttaa aktiinisäikeiden muodostumista, mutta se voi myös eristää yksittäisiä aktiinimolekyylejä ja edistää säikeiden hajoamista. Solutasolla palladinin ja profilinin sijainti on yhtenevä runsaasti aktiinia sisältävillä solujen reuna-alueilla. Palladinin ja profilinin sidos on heikko ja hyvin dynaaminen, joka sopii palladinin tehtävään aktiinisäideiden muodostumisen koordinoijana. Toinen palladinin sitoutumiskumppani on aktiinisäikeitä yhteensitova -aktiniini. -Aktiniini liittää solutukirangan solukalvon proteiineihin ja ankkuroi solunsisäisiä viestintämolekyylejä. Sitoutumista välittävä alue on hyvin samankaltainen palladinissa ja myotilinissa. Luurankolihaksen liiallinen toistuva venytys muuttaa Z-levyjen rakennetta ja muotoa. Prosessin aikana syntyy uusia aktiinifilamenttejä sisältäviä tiivistymiä ja lopulta uusia sarkomeereja. Löydöstemme perusteella myotilinin uudelleenjärjestyminen noudattaa aktiinin muutoksia. Tämä viittaa siihen, että myotilin liittää yhteen uudismuodostuvia aktiinisäikeitä ja vakauttaa niitä. Myotilin saattaa myös ankkuroida viesti- tai rakennemolekyylejä, joiden tehtävänä on edesauttaa Z-levyjen uudismuodostusta. Tulostemme perusteella arvelemme, että myotilin toimii Z-levyjen rakenteen vakaajana ja aktiinisäikeiden säätelijänä. Palladinin puute johtaa sikiöaikaiseen kuolemaan hiirillä, mutta myotilinin puutoksella ei ole samanlaisia vaikutuksia. Tuotettujen myotilin poistogeenisten hiirten todetiin syntyvän ja kehittyvän normaalisti eikä niillä esiintynyt rakenteellisia tai toiminnallisia häiriöitä. Toisaalta aiemmissa kokeissa, joissa hiirille on siirretty ihmisen lihastautia aikaansaava myotilingeeni, nähdään samankaltaisia kuin sairailla ihmisillä. Näin ollen muuntunut myotilin näyttä olevan lihaksen toiminnalle haitallisempi kuin myotilinin puute. Myotilinin ja palladinin yhteisvaikutusta selvittääksemme risteytimme myotilin poistegeenisen hiiren ja hiirilinjan, joka ilmentää puutteellisesti palladinin 200 kDa muotoa. Puutteellisesti 200 kDa palladinia ilmentävien hiirten sydänlihaksessa todettiin vähäisiä hienorakenteen muutoksia, mutta risteytetyillä hiirillä tavattiin rakenteellisia ja toiminnallisia muutoksia myös luurankolihaksessa. Tulosten perusteella voidaan todeta, että palladinin 200 kDa muoto säätelee sydänlihassolujen rakennetta. Luurankolihaksessa sen sijaan myotilinilla ja palladinilla näyttäisi olevan päällekkäisiä tehtäviä.

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Basidiomycetous white-rot fungi are the only organisms that can efficiently decompose all the components of wood. Moreover, white-rot fungi possess the ability to mineralize recalcitrant lignin polymer with their extracellular, oxidative lignin-modifying enzymes (LMEs), i.e. laccase, lignin peroxidase (LiP), manganese peroxidase (MnP), and versatile peroxidase (VP). Within one white-rot fungal species LMEs are typically present as several isozymes encoded by multiple genes. This study focused on two effi cient lignin-degrading white-rot fungal species, Phlebia radiata and Dichomitus squalens. Molecular level knowledge of the LMEs of the Finnish isolate P. radiata FBCC43 (79, ATCC 64658) was complemented with cloning and characterization of a new laccase (Pr-lac2), two new LiP-encoding genes (Pr-lip1, Pr-lip4), and Pr-lip3 gene that has been previously described only at cDNAlevel. Also, two laccase-encoding genes (Ds-lac3, Ds-lac4) of D. squalens were cloned and characterized for the first time. Phylogenetic analysis revealed close evolutionary relationships between the P. radiata LiP isozymes. Distinct protein phylogeny for both P. radiata and D. squalens laccases suggested different physiological functions for the corresponding enzymes. Supplementation of P. radiata liquid culture medium with excess Cu2+ notably increased laccase activity and good fungal growth was achieved in complex medium rich with organic nitrogen. Wood is the natural substrate of lignin-degrading white-rot fungi, supporting production of enzymes and metabolites needed for fungal growth and the breakdown of lignocellulose. In this work, emphasis was on solid-state wood or wood-containing cultures that mimic the natural growth conditions of white-rot fungi. Transcript analyses showed that wood promoted expression of all the presently known LME-encoding genes of P. radiata and laccase-encoding genes of D. squalens. Expression of the studied individual LME-encoding genes of P. radiata and D. squalens was unequal in transcript quantities and apparently time-dependent, thus suggesting the importance of several distinct LMEs within one fungal species. In addition to LMEs, white-rot fungi secrete other compounds that are important in decomposition of wood and lignin. One of these compounds is oxalic acid, which is a common metabolite of wood-rotting fungi. Fungi produce also oxalic-acid degrading enzymes of which the most widespread is oxalate decarboxylase (ODC). However, the role of ODC in fungi is still ambiguous with propositions from regulation of intra and extracellular oxalic acid levels to a function in primary growth and concomitant production of ATP. In this study, intracellular ODC activity was detected in four white-rot fungal species, and D. squalens showed the highest ODC activity upon exposure to oxalic acid. Oxalic acid was the most common organic acid secreted by the ODC-positive white-rot fungi and the only organic acid detected in wood cultures. The ODC-encoding gene Ds-odc was cloned from two strains of D. squalens showing the first characterization of an odc-gene from a white-rot polypore species. Biochemical properties of the D. squalens ODC resembled those described for other basidiomycete ODCs. However, the translated amino acid sequence of Ds-odc has a novel N-terminal primary structure with a repetitive Ala-Ser-rich region of ca 60 amino acid residues in length. Expression of the Ds-odc transcripts suggested a constitutive metabolic role for the corresponding ODC enzyme. According to the results, it is proposed that ODC may have an essential implication for the growth and basic metabolism of wood-decaying fungi.

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Proteins are complex biomacromolecules playing fundamental roles in the physiological processes of all living organisms. They function as structural units, enzymes, transporters, process regulators, and signal transducers. Defects in protein functions often derive from genetic mutations altering the protein structure, and impairment of essential protein functions manifests itself as pathological conditions. Proteins operate through interactions, and all protein functions depend on protein structure. In order to understand biological mechanisms at the molecular level, one has to know the structures of the proteins involved. This thesis covers structural and functional characterization of human filamins. Filamins are actin-binding and -bundling proteins that have numerous interaction partners. In addition to their actin-organizing functions, filamins are also known to have roles in cell adhesion and locomotion, and to participate in the logistics of cell membrane receptors, and in the coordination of intracellular signaling pathways. Filamin mutations in humans induce severe pathological conditions affecting the brain, bones, limbs, and the cardiovascular system. Filamins are large modular proteins composed of an N-terminal actin-binding domain and 24 consecutive immunoglobulin-like domains (IgFLNs). Nuclear magnetic resonance (NMR) spectroscopy is a versatile method of gaining insight into protein structure, dynamics and interactions. NMR spectroscopy was employed in this thesis to study the atomic structure and interaction mechanisms of C-terminal IgFLNs, which are known to house the majority of the filamin interaction sites. The structures of IgFLN single-domains 17 and 23 and IgFLN domain pairs 16-17 and 18-19 were determined using NMR spectroscopy. The structures of domain pairs 16 17 and 18 19 both revealed novel domain domain interaction modes of IgFLNs. NMR titrations were employed to characterize the interactions of filamins with glycoprotein Ibα, FilGAP, integrin β7 and dopamine receptors. Domain packing of IgFLN domain sextet 16 21 was further characterized using residual dipolar couplings and NMR relaxation analysis. This thesis demonstrates the versatility and potential of NMR spectroscopy in structural and functional studies of multi-domain proteins.

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Structural biology is a branch of science that concentrates on the relationship between the structure and function of biological macromolecules. The prevalence of a large number of three dimensional structures offers effective tools for bio-scientists to understand the living world. Actin is the most abundant cellular protein and one of its main functions is to produce movement in living cells. Actin forms filaments that are dynamic and which are regulated by a number of different proteins. A class of these regulatory proteins contains actin depolymerizing factor homology (ADF-H) domains. These directly interact with actin through their ADF-H domains. Although ADF-H domains possess very similar three dimensional structures to one another, they vary in their functional properties. One example of this is the ability to bind to actin monomers or filaments. During the work for this thesis two structures of ADF-H domains were solved by nuclear magnetic resonance spectroscopy (NMR). The elucidated structures help us understand the binding specificities of the ADF-H family members.

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Aminopolykarboksyylaatteja, kuten etyleenidiamiinitetraetikkahappoa (EDTA), on käytetty useiden vuosikymmenien ajan erinomaisen metalli-ionien sitomiskyvyn vuoksi kelatointiaineena lukuisissa sovelluksissa sekä analytiikassa että monilla teollisisuuden aloilla. Näiden yhdisteiden biohajoamattomuus on kuitenkin herättänyt huolta viime aikoina, sillä niiden on havaittu olevan hyvin pysyviä luonnossa. Tämä työ on osa laajempaa tutkimushanketta, jossa on tavoitteena löytää korvaavia kelatointiaineita EDTA:lle. Tutkimuksen aiheena on kuuden kelatointiaineen metalli-ionien sitomiskyvyn kartoitus. EDTA:a paremmin luonnossa hajoavina nämä ovat ympäristöystävällisiä ehdokkaita korvaaviksi kelatointiaineiksi useisiin sovelluksiin. Työssä tutkittiin niiden kompleksinmuodostusta useiden metalli-ionien kanssa potentiometrisella titrauksella. Metalli-ionivalikoima vaihteli hieman kelatointiaineesta riippuen sisältäen magnesium-, kalsium-, mangaani-, rauta-, kupari-, sinkki-, kadmium-, elohopea-, lyijy- ja lantaani-ionit. Tutkittavat metallit oli valittu tähtäimessä olevien sovellusten, synteesissä ilmenneiden ongelmien tai ympäristönäkökohtien perusteella. Tulokset osoittavat näiden yhdisteiden metallinsitomiskyvyn olevan jonkin verran heikompi kuin EDTA:lla, mutta kuitenkin riittävän useisiin sovelluksiin kuten sellunvalkaisuprosessiin. Myrkyllisten raskasmetallien, kadmiumin, elohopen ja lyijyn kohdalla EDTA:a heikompi sitoutuminen on eduksikin, koska se yhdistettynä parempaan biohajoavuuteen saattaa alentaa tutkittujen yhdisteiden kykyä mobilisoida kyseisiä metalleja sedimenteistä. Useimmilla tutkituista yhdisteistä on ympäristönäkökulmasta etuna myös EDTA:a pienempi typpipitoisuus.

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This study aims at improving understanding of the interactions of livelihoods and the environment focusing on both socio-economic and biodiversity implications of land use change in the context of population pressure, global and local markets, climate change, cultural and regional historical factors in the highlands of East Africa. The study is based on three components (1) two extensive livelihood surveys, one on Mt. Kilimanjaro in Tanzania and the other in the Taita Hills of Kenya, (2) a land use change study of the southern slopes of Mt. Kilimanjaro focusing on land use trends between 1960s and 1980s and 1980s and 2000 and (3) a bird diversity study focusing on the potential impacts of the future land use change on birds in the main land use types on the slopes and the adjacent plains of Mt. Kilimanjaro. In addition, information on the highlands in Embu and the adjacent lowlands in Mbeere of Kenya are added to the discussion. Some general patterns of livelihood, land use and environment interactions can be found in the three sites. However, the linkages are very complex. Various external factors at different times in history have influenced most of the major turning points. Farmers continually make small adaptations to their farming practices, but the locally conceived alternatives are too few. Farmers lack specific information and knowledge on the most suitable crops, market opportunities and the quality requirements for growing the crops for markets. Population growth emerges as the most forceful driver of land use and environmental change. The higher altitudes have become extremely crowded with population densities in some areas higher than typical urban population densities. Natural vegetation has almost totally been replaced by farmland. Decreasing farm size due to population pressure is currently threatening the viability of whole farming systems. In addition, capital-poor intensification has lead to soil fertility depletion. Agricultural expansion to the agriculturally marginal lowlands has created a new and distinct group of farmers struggling constantly with climate variability causing frequent crop failures. Extensification to the fragile drylands is the major cause of fragmentation and loss of wildlife habitat. The linkages between livelihoods, land use and the environment generally point to degradation of the environment leading to reduced environmental services and ecosystem functions. There is no indication that the system is self-regulating in this respect. Positive interventions will be needed to maintain ecosystem integrity.

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A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

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Let X be a topological space and K the real algebra of the reals, the complex numbers, the quaternions, or the octonions. The functions form X to K form an algebra T(X,K) with pointwise addition and multiplication. We study first-order definability of the constant function set N' corresponding to the set of the naturals in certain subalgebras of T(X,K). In the vocabulary the symbols Constant, +, *, 0', and 1' are used, where Constant denotes the predicate defining the constants, and 0' and 1' denote the constant functions with values 0 and 1 respectively. The most important result is the following. Let X be a topological space, K the real algebra of the reals, the compelex numbers, the quaternions, or the octonions, and R a subalgebra of the algebra of all functions from X to K containing all constants. Then N' is definable in , if at least one of the following conditions is true. (1) The algebra R is a subalgebra of the algebra of all continuous functions containing a piecewise open mapping from X to K. (2) The space X is sigma-compact, and R is a subalgebra of the algebra of all continuous functions containing a function whose range contains a nonempty open set of K. (3) The algebra K is the set of reals or the complex numbers, and R contains a piecewise open mapping from X to K and does not contain an everywhere unbounded function. (4) The algebra R contains a piecewise open mapping from X to the set of the reals and function whose range contains a nonempty open subset of K. Furthermore R does not contain an everywhere unbounded function.

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The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.