Nov pogled na teorem tri simetrije

It is well-known that, in a Euclidean plane, the product of three reflections is again a reflection, if their axes pass through a common point. For this \u27\u27Three reflections Theorem" (3RT) also non-Euclidean versions exist, see e.g. [4]. This article presents affine versions of it, considering a triplet of skew reflections with axes through a common point. It turns out that the essence of all those cases of 3RT is that the three pairs (axis, reflection direction) of the given (skew) reflections can be observed as an involutoric projectivity. For the Euclidean case and its non-Euclidean counterparts this property is automatically fullled. From the projective geometry point of view a (skew) reflection is nothing but a harmonic homology. In the affine situation a reflection is an indirect involutoric transformation, while \u27\u27direct" or \u27\u27indirect" makes no sense in projective planes. A harmonic homology allows an interpretation both, as an axial reflection and as a point reflection. Nevertheless, one might study products of three harmonic homologies, which result in a harmonic homology again. Some special mutual positions of axes and centres of the given homologies lead to elations or even to the identity, too. A consequence of the presented results are further generalisations of the 3RT, e.g. in planes with Minkowski metric, affine or projective 3-space, or in circle geometries.U euklidskoj ravnini poznato je da je produkt tri simetrije ponovo simetrija ako i samo ako se njihove osi sijeku u jednoj zajedničkoj točki. Također poznat je i neeuklidski analogon \u27\u27teorema tri simetrije" (3RT), vidi npr. [4]. U ovom članku predstavljene su afine verzije tog teorema tako da se proučavaju tri mimosmjerne simetrije kojima se osi sijeku u jednoj točki. Pokazat će se da je važno, u svim verzijama 3RT-a, da se tri para (os, smjer simetrije) danih (mimosmjernih) simetrija mogu proučavati kao involutivni projektivitet. Za euklidski i neeuklidski slučaj ovo svojstvo je automatski ispunjeno. Sa stajališta projektivne geometrije (mimosmjerna) simetrija je harmonička homologija. U afinoj geometriji simetrija je indirektna involutivna transformacija, dok u projektivnoj geometriji nema smisla govoriti o \u27\u27direktnoj" i \u27\u27indirektnoj" transformaciji. Harmonička homologija dopušta interpretaciju i kao osnu simetriju i kao centralnu simetriju. Ipak, može se proučavati produkt triju harmoničkih homologija koji je ponovno harmonička homologija. Nekim posebnim međusobnim položajima centara i osi danih homologija može se dobiti elacija ili čak identitet. Posljedica danih rezultata su daljnje generalizacije 3RT-a, npr. u ravninama s Minkowski metrikom, afinim ili projektivnim 3-dimenzionalnim prostorima ili u geometrijama kružnice

How many points are needed to determine a geometric surface?

A geometric surface can be approximately described using a finite point-sample. The main question of this thesis is the following: how many points, depending on the sampling model and surface genus, are needed to confidently reconstruct the original geometric surface. First we present different sampling models of surface points — the uniform and the random sample. We use topological methods, in particular persistent homology, to process our data. Using Javaplex software package we construct a filtration with Vietoris-Rips simplicial complexes and consider the bar-code diagram of its Betti numbers. Finally we present our computational results for both the sphere and the geometric torus with respect to the two sampling models , and several options for further improvements

The effects of ASR1 protein on salt and osmotic stress in plants

Biljke su izložene djelovanju abiotičkog stresa na dnevnoj bazi. ASR1 je visoko-nabijeni, biljno-specifični protein male molekulske mase (13 kDa), koji sudjeluje u staničnom odgovoru na stres a lokaliziran je u citoplazmi i jezgri stanice. Posjeduje DNA-veznu aktivnost koja ovisi o prisustvu cinka – on potiče prijelaz proteina ASR1 iz nesmotane u smotanu konformaciju. Smatra se da su se današnji oblici gena asr razvili iz jedne porodice koja je kroz vrijeme različito divergirala, ovisno o vrstama u kojima su se geni našli tj. smatra se da su pratili model usklađene evolucije. Dokazi tome su nedavne duplikacije gena asr3 iz asr4 i homologija između asr4 iz rajčice i asr6 iz riže. Iako se o njemu još uvijek zna vrlo malo, smatra se da jezgreni oblik proteina ima funkciju transkripcijskog faktora koji potiče niz signalnih puteva borbe protiv učinka abiotičkog stresa. Za citosolni, nesmotani oblik se još uvijek ne zna kako pridonosi održavanju homeostaze, ali smatra se da bi prijelaz proteina iz jednog oblika u drugi i iz jednog staničnog odjeljka u drugi mogao biti jedan od načina regulacije aktivnosti ovog proteina. Sve u svemu, zna se da njegova ekspresija utječe na povećanje otpornosti biljke na solni (smanjenjem neto nakupljanja natrija u stanici) i osmotski (preko uloge hidrofilina) stres, iako tek slijedi razvoj teorija i metoda pomoću kojih će biti lakše shvatiti njegovu pravu ulogu u biljnim stanicama te konstruirati transgenične biljke koje će biti otpornije na naseljevanje novih staništa i tolerantnije na nove uvjete koje će tamo susresti.Plants are exposed to effects of abiotic stress factors on a daily basis. ASR1 is a small (13 kDa) highly-charged, plant-specific protein, localised in citoplasm and nucleus, which takes part in cell's stress-defending pathways. It has a zinc-dependent DNA-binding activity – zinc enables the protein to transform from unfolded to fully folded form. It is believed that the asr genes, which we know of today, developed from a single gene family, probably through a model of concerned evolution. The evidence for that are gene duplications which took place recently (which state that asr3 originated from asr4) and homology between tomato asr4 and rice asr6 genes. Although we have much to learn about ASR1's functional and biological function, it has become clear that it's nuclear form plays role of a transcription factor, which induces a downstream of genes and signal pathways included in plant's response to abiotic stress. It is not yet known how does the cytosolic form of the protein takes part in homeostasis, but it is thought that it's transformation to folded state and transportation to the nucleus have something to do with the way this protein's activity is regulated. Nevertheless, we do now know that it's expression induces the plant's ability to tolerate damage which took place after plant's been exposed to stress. It does it so by lowering the net uptake of sodium in the cell (contribution to salt stress tolerance) and through it's role as a hydrophilin. Anyway, we've only just began to realise it's biologycal importance as a mean to construct transgenic plants which will be more tolerant to abiotic stress factors and will grow more sucessfully in new habitats

Molecular characterisation of Parainfluenza 3 virus strains of cattle isolated on the theritory of Republic of Serbia

Glavni cilj ove doktorske disertacije je da se izvrši identifikacija i molekularna karakterizacija izolovanih sojeva virusa parainfluence 3 (PI3) goveda sa teritorije Republike Srbije. Izvršena ispitivanja su obuhvatila izolaciju virusa parainfluence 3 iz nosnih briseva goveda na kulturi tkiva i identifikaciju izolovanih sojeva virusa metodom virusneutralizacije, direktne imunofluorescencije i njihovu molekularnu karakterizaciju metodom lanĉane reakcije polimeraze (RT PCR) uz korišćenje prajmera za konzervisane regione HN i F gena virusa PI3 goveda sa sekvenciranjem radi njihove genotipizacije i svrstavanja na filogenetsko stablo izolovanih sojeva pomenutog patogena. Filogenetska analiza izolovanih sojeva izvršena je na osnovu utvrĊivanja homologije i poreĊenja pozicije njihovih nukleotidnih sekvenci sa sekvencama registrovanih sojeva virusa u bazi gena. Na osnovu nukleotidne homologije i pozicije domaćih izolovanih sojeva virusa na filogenetskom stablu, izvršena je njihova genotipizacija. U navedenim ispitivanjima je korišćen referentni soj SF4 virusa PI3 goveda (izolovan u drţavi Tenesi ,SAD). Nazalni brisevi poreklom od 119 goveda razliĉitih rasnih i starosnih kategorija sa ispoljenim simptomima infekcije gornjih partija respiratornog trakta su ispitivani na prisustvo virusa PI3 metodom izolacije na ćelijskoj liniji MDBK (Madin Darby Bovine Kidney). Izolacija virusa vršena je posle najviše tri pasaţe suspektnog materijala na navedenoj ćelijskoj liniji do pojave citopatogenog efekta. Citopatogeni efekat je ustanovljen u ćelijskim linijama posle inokulacije 11 uzoraka ispitivanog materijala, dok je primenom metode virusneutralizacije i direktne imunofluorescencije ustanovljeno prisustvo virusa parainfluece 3 kod 6 inokulisanih ćelijskih linija. U ostalim ćelijskim linijama, pojedinaĉno inokulisanim sa po 5 uzoraka poreklom od ispitivanih nosnih briseva goveda nije dokazano prisustvo virusa parainfluence 3 primenom prethodno navedenih metoda...The main objective of this doctoral thesis was the identification and molecular characterization of isolated strains of parainfluenza 3 (PI3) from cattle on the territory of the Republic of Serbia. The investigations included isolation of parainfluenza 3 virus from nasal swabs from cattle in tissue culture and identification of isolated strains of the virus by method of virusneutralisation, direct imunoflurescense and their molecular caracterisation by method of polymerase chain reaction (RT-PCR) using primers from conserved regions of the HN and F genes of the virus PI3 cattle, with sequencing and genotyping of domestic strains by help of their inclusion on the phylogenetic tree of mentioned pathogens. Phylogenetic analysis of the isolated strains was carried out on the basis of determination of the homology of the nucleotide sequences with the sequences of strains of the virus registered in gene base NCBI . Genotyping of domestic strains was done on the basis of nucleotide homology and position of local isolated virus strains on the phylogenetic tree, of PI3 virus. In these experiments, reference strain SF4 of virus PI3 of cattle was used (isolated in Tennessee, USA). Nasal swabs originating from 119 cattle with symptoms of infection in upper parts of the respiratory tract were examined for the presence of the virus PI3 by method of isolation of virus on cell line MDBK (Madin Darby Bovine Kidney). Virus isolation was done after three passages of suspected material on cell line till the appearance of cytopathic effect. Cytopathic effect was detected in cell lines after inoculation of 11 tested samples, while applying the method of virusneutralisation and direct immunofluorescense, identification of the virus parainfluece 3 were conformed in 6 inoculated cell lines, but not confirmed in other 5 tested samples from nasal swabs of cattle..

Algebraic properties of spectral invariants in Floer Homology.

U ovom radu definisemo i analiziramo spektralne invarijante u Florovoj homologiji za konormalno raslojenje i Florovoj homologiji za otvoren skup. Konstrukcija preslikavanja tipa Piunikin Salamon Xvarc, koju mi ovde izvodimo, neophodan je korak za dobru definisanost ovih invarijanti. Postojanje dodatnih algebarskih struktura, kao xto je proizvod na spomenutim Florovim homologijama, omoguava nam da izvedemo razne nejednakosti medju spektralnim invarijantama. Posma- tranjem perturbovano holomorfnih Rimanovih povrxi sa granicom mo- emo da uporedimo invarijante iz raznih Florovih homologija...In this doctoral dissertation we dene and investigate spectral invariants in Floer homology for conormal bundle and Floer homology of an open sub- set. As a key step to well dened spectral invariants we give a construction of Piunikhin-Salamon-Schwarz isomorphism in both of these homologies. Ad- ditional algebraic structures, such as a product on Floer homology, give us various inequalities between spectral invariants. We can compare spectral in- variants from dierent Floer homologies by observing appropriate perturbed holomorphic Riemmanian surfaces with boundary..

Topological Approach to Analyses of Omics Data

Genes expression is often a good indicator for prediction of patient's clinical results. In diseases such as cancer is inevitable to identify subcategories of phenotype. The goal of the Thesis is to use persistent homology on cancer tissue gene expression to identify new subgroups and try to predict the survival of patients in corresponding groups. We analyse the date from the International Consortium for Cancer Research. Simplicial complexes were built different resolutions using Vietoris-Rips algorithm. We counted the persistent homology and draw persistent diagrams. We developed a method for calculating confidence interval on persistent diagrams to precisely divide cancer subcategories. This method gave us promising results by discovering new subcategories and was accurate in prediction of patient clinical results. Results were obtained on data of different cancer types. Results were compared with different unsupervised learning methods

Structure and function of holoenzyme exemplified by bacterial RNA-polymerase

RNA-polimeraza zajedno sa σ-podjedinicom čini holoenzim sposoban za specifično prepoznavanje promotorske sekvence te započinjanje važnog procesa u genskoj ekspresiji, transkripcije. Međusobno vezanje promotorskih sljedova, polimeraze i σ-podjedinice dovodi do niza alosteričkih promjena – otkrivanje veznih sljedova, stvaranje novih ionskih mostova, formiranje aktivnih mjesta, de/stabilizacija određenih struktura, itd. koje osiguravaju specifičnost holoenzima za određeni promotor. Također brojne međumolekulske interakcije s bazama promotora održavaju visoku konzerviranost određenih nukleotida (A-11, T-7) u svim domenama. Jedno od najvažnijih otkrića sigurno je bilo da se σ-podjedinica zadržava nakon same inicijacije jer je time složenost regulacije postala još veća, s obzirom da isti kompleks sada odjednom mora prelaziti između nekoliko stanja u kojima afinitet za nekoliko konstantnih sljedova u određenom trenu mora rasti ili padati, kako bi se postigao prijelaz s jednog procesa na drugi. Današnje tehnike obilježavanja fluorescentnim probama, kristalografije i masene spektrometrije otvaraju put preciznijim saznanjima o građi i interakcijama ovakvih kompleksnih struktura kao što su holoenzimi te će u budućnosti vjerojatno mnoge dosadašnje dogme također biti poljuljane.RNA-polymerase together with the σ-subunit creates a holoenzyme capable of specific promoter sequence recognition. Thus it can commence important step of gene expression – transcription. Interwoven binding of promoter sequence, polymerase and σ-subunit leads to a number of allosteric modifications – exposure of binding sequences, salt bridge establishments, active site formation, specific structure de/stabilization , etc., all of which guarantee promoter-specificity of holoenzyme. Also, numerous intermolecular interactions with promoter bases maintain high conserved nucleotide sequence (A-11, T-7) in all living domains. One of the biggest discoveries certainly is the retaining of σ-subunit after the initiation. The complexity of regulation has increased even more since the same complex has to recognize the same sequences but with different affinity at different point in time, so the transition from one state to another can be achieved. Nowadays, techniques such as fluorescence probing, crystallography or mass spectrometry, lead to more precise organization or interaction determination inside the complex structures such as holoenzymes that will probably lead to many reconsiderations of what was thought to be dogmatic

Paralogs of aminoacyl-tRNA synthetases

Aminoacil-tRNA-sintetaze ključni su enzimi u biosintetskom putu proteina i jedni od evolucijski najstarijih enzima. Analizama njima homolognih genskih sekvenci i proteinskih struktura utvrđeno je postojanje cijelog spektra tzv. krnjih paraloga, koji su funkcionalno uglavnom divergirali od svojih originala u više različitih smjerova. Ovim radom napravljen je osvrt na gotovo sve skupine otkrivenih paraloga te je izložena i komentirana funkcija, struktura i evolucija najzanimljivijih od njih. Paralozi su svrstani prema svojstvu homologije koju nose u dvije veće skupine, ovisno o tome sadrže li sekvencu homolognu katalitičkoj domeni odgovarajuće aaRS ili ne. Poseban je naglasak stavljen na nedavno otkrivene paraloge specifičnog metanogenog tipa seril-tRNA-sintetaza, čije su detaljne biokemijske i strukturalne analize upravo u tijeku.Aminoacyl-tRNA synthetases are one of the key enzymes in protein biosynthesis and one of the evolutionary oldest enzymes as well. By analyses of their homologic sequences and protein structures, the existance of a wide spectrum of socalled truncated paralogs, which have mostly functionaly diverged from their originals into several different new pathways, has been shown. In this work a critical review of almost all groups of determined paralogs has been made. Also, function, structure and evolution of the most interesting ones has been established and commented. Paralogs have been divided into two larger subgroups based on their homology characteristics, depending on the homologous catalytic domain possesion. A special emphasis has been put on recently discovered methanogenic seryl-tRNA synthetases paralogs, whose detailed biochemical and structural analyses have just been in progress

Topological Approach to Analyses of Omics Data

Genes expression is often a good indicator for prediction of patient's clinical results. In diseases such as cancer is inevitable to identify subcategories of phenotype. The goal of the Thesis is to use persistent homology on cancer tissue gene expression to identify new subgroups and try to predict the survival of patients in corresponding groups. We analyse the date from the International Consortium for Cancer Research. Simplicial complexes were built different resolutions using Vietoris-Rips algorithm. We counted the persistent homology and draw persistent diagrams. We developed a method for calculating confidence interval on persistent diagrams to precisely divide cancer subcategories. This method gave us promising results by discovering new subcategories and was accurate in prediction of patient clinical results. Results were obtained on data of different cancer types. Results were compared with different unsupervised learning methods

Identification and molecular characterization of equine herpesviruses

Cilj ove doktorske disertacije je identifikacija i molekularna karakterizacija sojeva herpesvirusa konja poreklom iz uzoraka od konja sa teritorije Republike Srbije i Republike Srpske (BiH) kao i arhiviranih sojeva EHV-1 Katedre za mikrobiologiju Fakulteta veterinarske medicine Univerziteta u Beogradu. Arhivirani sojevi EHV-1 su liofilizati pomenutog virusa izolovani osamdesetih godina prošlog veka iz uzoraka pobačenih fetusa konja, kao i iz uginule novorođene ždrebadi sa ergele „Ljubičevo“. Ispitivani su uzorci poreklom od ukupno 137 nevakcinisanih konja sa ergela i iz privatnog sektora koji su obuhvatali 112 uzoraka nosnih briseva i 100 uzoraka organa - submandibularnih limfnih čvorova, slezine, produžene moždine i kičmene moždine prikupljenih od 25 konja. Prikupljeni uzorci nosnih briseva i organa konja su primenom klasičnih i molekularnih metoda virusološke dijagnostike ispitivani na prisustvo konjskih herpesvirusa 1, 4, 2 i 5 (EHV-1, EHV-4, EHV-2 i EHV-5), dok su arhivirani sojevi ispitivani primenom molekularnih metoda u cilju molekularne karakterizacije. Posle izolacije i identifikacije virusa na kulturi tkiva primenom testa virus neutralizacije, direktne imunofluorescencije i metode Nested multiplex PCR od ukupno 212 ispitivanih uzoraka organa i nosnih briseva konja prisustvo konjskog herpesvirusa 1 je utvrđeno u 72 uzorka, dok je metodom izolacije virusa na kulturi ćelija sa identifikacijom primenom metode Nested multiplex PCR iz prethodno navedenih uzoraka konjski herpesvirus 5 detektovan u 7 uzoraka, a konjski herpesvirusi 4 i 2 nisu izolovani ni iz jednog uzorka nosnih briseva i organa konja. Primenom metode Nested multiplex PCR za ispitivanje prisustva nukleinskih kiselina konjskih herpesvirusa 1, 4, 2 i 5 direktno u uzorcima ispitivanog materijala (organa i nosnih briseva konja) detektovano je ukupno 162 uzorka pri čemu je EHV-1 detektovan u 153 uzorka, EHV-4 u 11, EHV-2 u 4, a EHV-5 u 23 uzorka ispitivanog materijala. Od ukupnog broja pozitivnih uzoraka mešovite infekcije sa dva ili više konjska herpesvirusa utvrđene su u 16,67% uzoraka, pri čemu su EHV-2 i EHV-4 identifikovani isključivo u okviru mešovitih infekcija sa EHV-1 i/ili EHV-5...The aim of this PhD thesis was the identification and molecular characterization of equine herpesviruses from horses originating from the Republic of Serbia and The Republic of Srpska (Bosnia and Herzegovina), as well as archivated EHV-1 strains in the possession of The Department of Microbiology, Faculty of Veterinary Medicine, University of Belgrade. Archivated EHV-1 strains were isolated during abortion storms that occured on „Ljubičevo“ stud farm during the 1980s. Samples originating from 137 unvaccinated horses from stud farms and private breeders, i.e. 112 nasal swab samples and 100 organ samples from 25 horses (submandibular lymph nodes, spleen, spinal cord and medulla) were examined for the presence of equine herpesviruses 1, 4, 2 and 5 (EHV-1, EHV-4, EHV-2 and EHV-5) using standard and molecular virological methods. Archivated EHV-1 samples were examined by molecular methods in order to perform their genetic characterization. Equine herpesvirus 1 was isolated and identified by virus neutralization test, direct immunofluorescence and Nested multiplex PCR in 72 samples. Equine herpesvirus 5 was isolated from 7 examined samples and identified by Nested multiplex PCR, whilst the isolation of EHV-4 and EHV-2 was not successful. The direct examination of the presence of equine herpesviruses 1, 4, 2 and 5 was performed by Nested multiplex PCR directly from the samples of organs and nasal swabs and confirmed 162 positive samples of which 16.67% accounted for mixed infections with multiple equine herpesviruses. Equine herpesvirus 1 was confirmed in 153 samples, EHV-4 in 11, EHV-2 in 4 and EHV-5 in 23 samples, whilst EHV-2 and EHV-4 were only detected in mixed infections with EHV-1 and/or EHV-5. Partial gB gene nucleotide sequences of identified EHV-1 strains were 98 to 100% homologous amongst each other and with sequences from GenBank whilst their phylogenetic analysis showed grouping with strains from Turkey, UK, USA and Japan..