Most of people have fatal attitude to Mathematics: some of them are capable to learn it form nature, but the others are not. So is their fate – to suffer from it for the whole of life… But it is a rude though natural mistake, as it results from means of mathematical education and its content. Most of parents and teachers are directed on these aspects both in kindergarten and at primary school. Of course, parents are different. Nevertheless so many parents can’t possibly but speak about achievements of their children. Some start making their own children learn better by the example of success of the others. They make their children learn long chains of figures with no understanding. It is even more sad to see how a mom asks her 4-year old son: “How much is two plus three?..’ But he replies just because he learned the answer but not calculated. Not only parents but also kindergarten tutors don’t want to understand that drilling for arithmetic has no sense. For a specialist it would take two days only…But teach him how to think logically – is a goal demanding from him, reached by different means.

Modern Internet technologies open new possibilities in a wide spectrum of traditional methods, used in mathematical education. One of the areas, where these technologies can be efficiently used, is an organization of mathematical competitions. Contestants can stay in their schools or universities in different cities and even different countries and try to solve as many mathematical problems as possible and then submit their solutions to organizers through the Internet. Simple Internet technologies supply audio and video connection between participants and organizers in a time of the competitions.

Didactic proposals on modelling in mathematics education mostly give priority to models which describe, explain as well as partially forecast and provide mathematical solutions to real situations. A view of the modelling concept of informatics, which also initiates rapidly generalised deliberations of models, can also make a contribution to the spectrum of models, which are treated in a meaningful sense in mathematics lessons so as to expand some interesting aspects. In this paper, this is illustrated by means of conceptual design models – and, here, especially of process models – using the example of elevator organisation in a multi-storey construction.

On the basis of experiences and studies developed in the last ten years, the contribution aims to discuss some different peculiarities between Cooperative Learning and Peer Tutoring models in Mathematics lesson. These models are specific interpretations of a way of conducting Mathematics lessons which requires the activity of students, their personal participation in the construction of knowledge. In the description of the two teaching-learning models, the analysis will deal in particular with the social aspects these models involve. Describing these two modalities of cooperation, also the importance of the care for the choice of suitable mathematical tasks and for different pedagogical setting they require will appear clearly. The issues described, together with the analogies and differences between the two models, could contribute to suggest more adequate didactical projects for teachers and deeper studies about students’ collaboration based models for researchers.

Certainly one of the most powerful and important modeling languages of our time is the Calculus. But research consistently shows that students do not understand how the variables in calculus-based mathematical models relate to aspects of the systems that those models are supposed to represent. Because of this, students never access the true power of calculus: its suitability to model a wide variety of real-world systems across domains. In this paper, we describe the motivation and theoretical foundations for the DeltaTick and HotLink Replay applications, an effort to address these difficulties by a) enabling students to model a wide variety of systems in the world that change over time by defining the behaviors of that system, and b) making explicit how a system\''s behavior relates to the mathematical trends that behavior creates. These applications employ the visualization and codification of behavior rules within the NetLogo agent-based modeling environment (Wilensky, 1999), rather than mathematical symbols, as their primary building blocks. As such, they provide an alternative to traditional mathematical techniques for exploring and solving advanced modeling problems, as well as exploring the major underlying concepts of calculus.

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EPR spectroscopy of site-directed spin labeled proteins is extremely informative in the studies of protein dynamics; however, it is difficult to interpret the spectra in terms of the conformational dynamics in atomic detail.In the present work we aimed to investigate the site-specific structural dynamics of proteins by using MD simulations upon analyzing and interpreting the EPR data. The major goal of this work is to know how far the computer simulations can meet the experiments. As a first step, MD simulations are performed to identify the location and orientation of the tyrosine radical in the R2 subunit of ribonucleotide reductase. The MD results show that the tyrosine is moving away from the diiron center in its radical state. This data is in agreement with EPR results and suggests reorientation of the tyrosine radical when compared to its neutral state. In further studies, the behavior of a methanethiosulfonate spin label, R1, in various environments of the protein is characterized by using MD simulations. RMSD analysis and angle ß distributions of the nitroxide show that R1 in buried sites in a protein helix is significantly immobile and in surface exposed sites it is highly mobile. Analyses of MD data suggest that internal rotations of x4 and x5 dihedrals of R1 are dominant in the R1 dynamics.Our studies also show that interaction with the surrounding residues show significant influence on the dynamics of R1. MD simulations data of the vinculin tail protein, both in water and in vacuo, are compared to the experimental results for further analysis of 12 different R1 sites in various environments.In a study on the photosynthetic reaction center(RC),MD is used to identify the location of the R1 binding site (H156)and thereby exploring the conformational dynamics in the RC protein upon light activation. The distance between the primary quinone, QA, and H156R1 determined from MD is in reasonable agreement with that measured by EPR.

Zellen benutzen Signalwege ein, um auf Änderungen in ihrer unmittelbaren Umgebung zu reagieren. Der Signalweg des transformierenden Wachstumsfaktors β (TGFβ) spielt eine entscheidende Rolle bei der Regulierung vieler zellulärer Prozesse, einschließlich Zellproliferation, Differenzierung und Migration. Obwohl die Hauptkomponenten der TGFβ-Signalgebung in den letzten Jahrzehnten identifiziert und erforscht wurden, ist das Verständnis ihres dynamischen Verhaltens durch das Fehlen von Methoden eingeschränkt, die die Steuerung der TGFβ-Signalgebung mit hoher räumlicher und zeitlicher Auflösung ermöglichen. In dieser Arbeit wurde ein optogenetisches System (das optoTGFβ-System) entwickelt, bei dem Licht dazu verwendet wird, die TGFβ-Signalgebung zeitlich und räumlich präzise zu steuern. Erstens wurde die Funktionalität des optoTGFβ-Systems durch Vergleich mit dem endogenen TGFβ-Signalsystem überprüft. Zweitens wurde durch das gleichzeitige Überwachen der subzellulären Translokation der Rezeptoren und der Smad-Proteine mittels „Live Cell Imaging“ gezeigt, dass die TGFβ-Signalgebung durch Modulation der Lichtstimulationen in einzelnen Zellen spezifisch aktiviert werden kann. Drittens wurde in Kombination mit der mathematischen Modellierung die Dynamik der TGFβ-Signalgebung im optoTGFβ-System quantitativ bestimmt. Die räumliche und zeitliche Präzision der optischen Kontrolle machen das optoTGFβ-System zu einem neuartigen und leistungsfähigen Methode für die quantitative Analyse und Manipulation von TGFβ-Signalen auf Einzelzellebene. / Cells employ signaling pathways to make decisions in response to changes in their immediate environment. Transforming Growth Factor β (TGFβ) signaling pathway plays pivotal roles in regulating many cellular processes, including cell proliferation, differentiation and migrations. Although the principal components of TGFβ signaling have been identified and explored in recent decades, understanding its dynamic behavior is limited by the lack of tools that allow the control of TGFβ signaling at high spatiotemporal resolution. In this thesis, we developed an optogenetic system (the optoTGFβ system), in which light is used to control TGFβ signaling precisely in time and space. First, we validated the functionality of the optoTGFβ system by comparing it with the endogenous TGFβ signaling system. Second, by simultaneously monitoring the subcellular translocation of the receptors and Smad proteins using live cell imaging, we showed that TGFβ signaling can be specifically activated in single cells through modulating the light stimulations. Third, in combination with mathematical modeling, we quantitatively characterized the dynamics of TGFβ signaling in the optoTGFβ system. The spatial and temporal precision of optical control makes the optoTGFβ system a novel and powerful tool for quantitative analyses and manipulation of TGFβ signaling at the single cell level.