On Random Planar Curves and Their Scaling Limits

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.

Planar Edgeless Detectors for the TOTEM Experiment at the LHC

The TOTEM experiment at the LHC will measure the total proton-proton cross-section with a precision better than 1%, elastic proton scattering over a wide range in momentum transfer -t= p^2 theta^2 up to 10 GeV^2 and diffractive dissociation, including single, double and central diffraction topologies. The total cross-section will be measured with the luminosity independent method that requires the simultaneous measurements of the total inelastic rate and the elastic proton scattering down to four-momentum transfers of a few 10^-3 GeV^2, corresponding to leading protons scattered in angles of microradians from the interaction point. This will be achieved using silicon microstrip detectors, which offer attractive properties such as good spatial resolution (<20 um), fast response (O(10ns)) to particles and radiation hardness up to 10^14 "n"/cm^2. This work reports about the development of an innovative structure at the detector edge reducing the conventional dead width of 0.5-1 mm to 50-60 um, compatible with the requirements of the experiment.

Planar subgraphs without low-degree nodes

We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 1) leads to NP-hard problems.

Thin-Layer Chromatography with Ultraviolet and Mass Spectrometric Detection: From Preparative-Layer to Miniaturized Ultra-Thin-Layer Technique

The present challenge in drug discovery is to synthesize new compounds efficiently in minimal time. The trend is towards carefully designed and well-characterized compound libraries because fast and effective synthesis methods easily produce thousands of new compounds. The need for rapid and reliable analysis methods is increased at the same time. Quality assessment, including the identification and purity tests, is highly important since false (negative or positive) results, for instance in tests of biological activity or determination of early-ADME parameters in vitro (the pharmacokinetic study of drug absorption, distribution, metabolism, and excretion), must be avoided. This thesis summarizes the principles of classical planar chromatographic separation combined with ultraviolet (UV) and mass spectrometric (MS) detection, and introduces powerful, rapid, easy, low-cost, and alternative tools and techniques for qualitative and quantitative analysis of small drug or drug-like molecules. High performance thin-layer chromatography (HPTLC) was introduced and evaluated for fast semi-quantitative assessment of the purity of synthesis target compounds. HPTLC methods were compared with the liquid chromatography (LC) methods. Electrospray ionization mass spectrometry (ESI MS) and atmospheric pressure matrix-assisted laser desorption/ionization MS (AP MALDI MS) were used to identify and confirm the product zones on the plate. AP MALDI MS was rapid, and easy to carry out directly on the plate without scraping. The PLC method was used to isolate target compounds from crude synthesized products and purify them for bioactivity and preliminary ADME tests. Ultra-thin-layer chromatography (UTLC) with AP MALDI MS and desorption electrospray ionization mass spectrometry (DESI MS) was introduced and studied for the first time. Because of the thinner adsorbent layer, the monolithic UTLC plate provided 10 100 times better sensitivity in MALDI analysis than did HPTLC plates. The limits of detection (LODs) down to low picomole range were demonstrated for UTLC AP MALDI and UTLC DESI MS. In a comparison of AP and vacuum MALDI MS detection for UTLC plates, desorption from the irregular surface of the plates with the combination of an external AP MALDI ion source and an ion trap instrument provided clearly less variation in mass accuracy than the vacuum MALDI time-of-flight (TOF) instrument. The performance of the two-dimensional (2D) UTLC separation with AP MALDI MS method was studied for the first time. The influence of the urine matrix on the separation and the repeatability was evaluated with benzodiazepines as model substances in human urine. The applicability of 2D UTLC AP MALDI MS was demonstrated in the detection of metabolites in an authentic urine sample.

Computational Studies on Bimetallic Clusters

The chemical and physical properties of bimetallic clusters have attracted considerable attention due to the potential technological applications of mixed-metal systems. It is of fundamental interests to study clusters because they are the link between atomic surface and bulk properties. More information of metal-metal bond in small clusters can be hence released. The studies in my thesis mainly focus on the two different kinds of bimetallic clusters: the clusters consisting of extraordinary shaped all metal four-membered rings and a series of sodium auride clusters. As described in most general organic chemistry books nowadays, a group of compounds are classified as aromatic compounds because of their remarkable stabilities, particular geometrical and energetic properties and so on. The notation of aromaticity is essentially qualitative. More recently, the connection has been made between aromaticity and energetic and magnetic properties. Also, the discussions of the aromatic nature of molecular rings are no longer limited to organic compounds obeying the Hückel’s rule. In our research, we mainly applied the GIMIC method to several bimetallic clusters at the CCSD level, and compared the results with those obtained by using chemical shift based methods. The magnetically induced ring currents can be generated easily by employing GIMIC method, and the nature of aromaticity for each system can be therefore clarified. We performed intensive quantum chemical calculations to explore the characters of the anionic sodium auride clusters and the corresponding neutral clusters since it has been fascinating in investigating molecules with gold atom involved due to its distinctive physical and chemical properties. As small gold clusters, the sodium auride clusters seem to form planar structures. With the addition of a negative charge, the gold atom in anionic clusters prefers to carry the charge and orients itself away from other gold atoms. As a result, the energetically lowest isomer for an anionic cluster is distinguished from the one for the corresponding neutral cluster. Mostly importantly, we presented a comprehensive strategy of ab initio applications to computationally implement the experimental photoelectron spectra.

Distortion of Dimension under Quasiconformal Mappings

Quasiconformal mappings are natural generalizations of conformal mappings. They are homeomorphisms with 'bounded distortion' of which there exist several approaches. In this work we study dimension distortion properties of quasiconformal mappings both in the plane and in higher dimensional Euclidean setting. The thesis consists of a summary and three research articles. A basic property of quasiconformal mappings is the local Hölder continuity. It has long been conjectured that this regularity holds at the Sobolev level (Gehring's higher integrabilty conjecture). Optimal regularity would also provide sharp bounds for the distortion of Hausdorff dimension. The higher integrability conjecture was solved in the plane by Astala in 1994 and it is still open in higher dimensions. Thus in the plane we have a precise description how Hausdorff dimension changes under quasiconformal deformations for general sets. The first two articles contribute to two remaining issues in the planar theory. The first one concerns distortion of more special sets, for rectifiable sets we expect improved bounds to hold. The second issue consists of understanding distortion of dimension on a finer level, namely on the level of Hausdorff measures. In the third article we study flatness properties of quasiconformal images of spheres in a quantitative way. These also lead to nontrivial bounds for their Hausdorff dimension even in the n-dimensional case.

Seeing and hearing speech, sounds, and signs: functional magnetic resonance studies on fluent and dyslexic readers

Speech has both auditory and visual components (heard speech sounds and seen articulatory gestures). During all perception, selective attention facilitates efficient information processing and enables concentration on high-priority stimuli. Auditory and visual sensory systems interact at multiple processing levels during speech perception and, further, the classical motor speech regions seem also to participate in speech perception. Auditory, visual, and motor-articulatory processes may thus work in parallel during speech perception, their use possibly depending on the information available and the individual characteristics of the observer. Because of their subtle speech perception difficulties possibly stemming from disturbances at elemental levels of sensory processing, dyslexic readers may rely more on motor-articulatory speech perception strategies than do fluent readers. This thesis aimed to investigate the neural mechanisms of speech perception and selective attention in fluent and dyslexic readers. We conducted four functional magnetic resonance imaging experiments, during which subjects perceived articulatory gestures, speech sounds, and other auditory and visual stimuli. Gradient echo-planar images depicting blood oxygenation level-dependent contrast were acquired during stimulus presentation to indirectly measure brain hemodynamic activation. Lip-reading activated the primary auditory cortex, and selective attention to visual speech gestures enhanced activity within the left secondary auditory cortex. Attention to non-speech sounds enhanced auditory cortex activity bilaterally; this effect showed modulation by sound presentation rate. A comparison between fluent and dyslexic readers' brain hemodynamic activity during audiovisual speech perception revealed stronger activation of predominantly motor speech areas in dyslexic readers during a contrast test that allowed exploration of the processing of phonetic features extracted from auditory and visual speech. The results show that visual speech perception modulates hemodynamic activity within auditory cortex areas once considered unimodal, and suggest that the left secondary auditory cortex specifically participates in extracting the linguistic content of seen articulatory gestures. They are strong evidence for the importance of attention as a modulator of auditory cortex function during both sound processing and visual speech perception, and point out the nature of attention as an interactive process (influenced by stimulus-driven effects). Further, they suggest heightened reliance on motor-articulatory and visual speech perception strategies among dyslexic readers, possibly compensating for their auditory speech perception difficulties.

Monte Carlo Simulations of Molecular Clusters in Nucleation

A better understanding of the limiting step in a first order phase transition, the nucleation process, is of major importance to a variety of scientific fields ranging from atmospheric sciences to nanotechnology and even to cosmology. This is due to the fact that in most phase transitions the new phase is separated from the mother phase by a free energy barrier. This barrier is crossed in a process called nucleation. Nowadays it is considered that a significant fraction of all atmospheric particles is produced by vapor-to liquid nucleation. In atmospheric sciences, as well as in other scientific fields, the theoretical treatment of nucleation is mostly based on a theory known as the Classical Nucleation Theory. However, the Classical Nucleation Theory is known to have only a limited success in predicting the rate at which vapor-to-liquid nucleation takes place at given conditions. This thesis studies the unary homogeneous vapor-to-liquid nucleation from a statistical mechanics viewpoint. We apply Monte Carlo simulations of molecular clusters to calculate the free energy barrier separating the vapor and liquid phases and compare our results against the laboratory measurements and Classical Nucleation Theory predictions. According to our results, the work of adding a monomer to a cluster in equilibrium vapour is accurately described by the liquid drop model applied by the Classical Nucleation Theory, once the clusters are larger than some threshold size. The threshold cluster sizes contain only a few or some tens of molecules depending on the interaction potential and temperature. However, the error made in modeling the smallest of clusters as liquid drops results in an erroneous absolute value for the cluster work of formation throughout the size range, as predicted by the McGraw-Laaksonen scaling law. By calculating correction factors to Classical Nucleation Theory predictions for the nucleation barriers of argon and water, we show that the corrected predictions produce nucleation rates that are in good comparison with experiments. For the smallest clusters, the deviation between the simulation results and the liquid drop values are accurately modelled by the low order virial coefficients at modest temperatures and vapour densities, or in other words, in the validity range of the non-interacting cluster theory by Frenkel, Band and Bilj. Our results do not indicate a need for a size dependent replacement free energy correction. The results also indicate that Classical Nucleation Theory predicts the size of the critical cluster correctly. We also presents a new method for the calculation of the equilibrium vapour density, surface tension size dependence and planar surface tension directly from cluster simulations. We also show how the size dependence of the cluster surface tension in equimolar surface is a function of virial coefficients, a result confirmed by our cluster simulations.

Theoretical and computational approaches on heterogeneous nucleation

Nucleation is the first step of a first order phase transition. A new phase is always sprung up in nucleation phenomena. The two main categories of nucleation are homogeneous nucleation, where the new phase is formed in a uniform substance, and heterogeneous nucleation, when nucleation occurs on a pre-existing surface. In this thesis the main attention is paid on heterogeneous nucleation. This thesis wields the nucleation phenomena from two theoretical perspectives: the classical nucleation theory and the statistical mechanical approach. The formulation of the classical nucleation theory relies on equilibrium thermodynamics and use of macroscopically determined quantities to describe the properties of small nuclei, sometimes consisting of just a few molecules. The statistical mechanical approach is based on interactions between single molecules, and does not bear the same assumptions as the classical theory. This work gathers up the present theoretical knowledge of heterogeneous nucleation and utilizes it in computational model studies. A new exact molecular approach on heterogeneous nucleation was introduced and tested by Monte Carlo simulations. The results obtained from the molecular simulations were interpreted by means of the concepts of the classical nucleation theory. Numerical calculations were carried out for a variety of substances nucleating on different substances. The classical theory of heterogeneous nucleation was employed in calculations of one-component nucleation of water on newsprint paper, Teflon and cellulose film, and binary nucleation of water-n-propanol and water-sulphuric acid mixtures on silver nanoparticles. The results were compared with experimental results. The molecular simulation studies involved homogeneous nucleation of argon and heterogeneous nucleation of argon on a planar platinum surface. It was found out that the use of a microscopical contact angle as a fitting parameter in calculations based on the classical theory of heterogeneous nucleation leads to a fair agreement between the theoretical predictions and experimental results. In the presented cases the microscopical angle was found to be always smaller than the contact angle obtained from macroscopical measurements. Furthermore, molecular Monte Carlo simulations revealed that the concept of the geometrical contact parameter in heterogeneous nucleation calculations can work surprisingly well even for very small clusters.

Monte Carlo simulations of molecular clusters in nucleation

A better understanding of the limiting step in a first order phase transition, the nucleation process, is of major importance to a variety of scientific fields ranging from atmospheric sciences to nanotechnology and even to cosmology. This is due to the fact that in most phase transitions the new phase is separated from the mother phase by a free energy barrier. This barrier is crossed in a process called nucleation. Nowadays it is considered that a significant fraction of all atmospheric particles is produced by vapor-to liquid nucleation. In atmospheric sciences, as well as in other scientific fields, the theoretical treatment of nucleation is mostly based on a theory known as the Classical Nucleation Theory. However, the Classical Nucleation Theory is known to have only a limited success in predicting the rate at which vapor-to-liquid nucleation takes place at given conditions. This thesis studies the unary homogeneous vapor-to-liquid nucleation from a statistical mechanics viewpoint. We apply Monte Carlo simulations of molecular clusters to calculate the free energy barrier separating the vapor and liquid phases and compare our results against the laboratory measurements and Classical Nucleation Theory predictions. According to our results, the work of adding a monomer to a cluster in equilibrium vapour is accurately described by the liquid drop model applied by the Classical Nucleation Theory, once the clusters are larger than some threshold size. The threshold cluster sizes contain only a few or some tens of molecules depending on the interaction potential and temperature. However, the error made in modeling the smallest of clusters as liquid drops results in an erroneous absolute value for the cluster work of formation throughout the size range, as predicted by the McGraw-Laaksonen scaling law. By calculating correction factors to Classical Nucleation Theory predictions for the nucleation barriers of argon and water, we show that the corrected predictions produce nucleation rates that are in good comparison with experiments. For the smallest clusters, the deviation between the simulation results and the liquid drop values are accurately modelled by the low order virial coefficients at modest temperatures and vapour densities, or in other words, in the validity range of the non-interacting cluster theory by Frenkel, Band and Bilj. Our results do not indicate a need for a size dependent replacement free energy correction. The results also indicate that Classical Nucleation Theory predicts the size of the critical cluster correctly. We also presents a new method for the calculation of the equilibrium vapour density, surface tension size dependence and planar surface tension directly from cluster simulations. We also show how the size dependence of the cluster surface tension in equimolar surface is a function of virial coefficients, a result confirmed by our cluster simulations.

Generation of Flat and Curved Membranes by Inverse-BAR Domain Proteins

Biological membranes are tightly linked to the evolution of life, because they provide a way to concentrate molecules into partially closed compartments. The dynamic shaping of cellular membranes is essential for many physiological processes, including cell morphogenesis, motility, cytokinesis, endocytosis, and secretion. It is therefore essential to understand the structure of the membrane and recognize the players that directly sculpt the membrane and enable it to adopt different shapes. The actin cytoskeleton provides the force to push eukaryotic plasma membrane in order to form different protrusions or/and invaginations. It has now became evident that actin directly co-operates with many membrane sculptors, including BAR domain proteins, in these important events. However, the molecular mechanisms behind BAR domain function and the differences between the members of this large protein family remain largely unresolved. In this thesis, the structure and functions of the I-BAR domain family members IRSp53 and MIM were thoroughly analyzed. By using several methods such as electron microscopy and systematic mutagenesis, we showed that these I-BAR domain proteins bind to PI(4,5)P2-rich membranes, generate negative membrane curvature and are involved in the formation of plasma membrane protrusions in cells e.g. filopodia. Importantly, we characterized a novel member of the BAR-domain superfamily which we named Pinkbar. We revealed that Pinkbar is specifically expressed in kidney and epithelial cells, and it localizes to Rab13-positive vesicles in intestinal epithelial cells. Remarkably, we learned that the I-BAR domain of Pinkbar does not generate membrane curvature but instead stabilizes planar membranes. Based on structural, mutagenesis and biochemical work we present a model for the mechanism of the novel membrane deforming activity of Pinkbar. Collectively, this work describes the mechanism by which I-BAR domain proteins deform membranes and provides new information about the biological roles of these proteins. Intriguingly, this work also gives evidence that significant functional plasticity exists within the I-BAR domain family. I-BAR proteins can either generate negative membrane curvature or stabilize planar membrane sheets, depending on the specific structural properties of their I-BAR domains. The results presented in this thesis expand our knowledge on membrane sculpting mechanisms and shows for the first time how flat membranes can be generated in cells.