On the Obfuscation Complexity of Planar Graphs

Being motivated by John Tantalo's Planarity Game, we consider straight line plane drawings of a planar graph $G$ with edge crossings and wonder how obfuscated such drawings can be. We define $obf(G)$, the obfuscation complexity of $G$, to be the maximum number of edge crossings in a drawing of $G$. Relating $obf(G)$ to the distribution of vertex degrees in $G$, we show an efficient way of constructing a drawing of $G$ with at least $obf(G)/3$ edge crossings. We prove bounds (\delta(G)^2/24-o(1))n^2 < \obf G <3 n^2 for an $n$-vertex planar graph $G$ with minimum vertex degree $\delta(G)\ge 2$. The shift complexity of $G$, denoted by $shift(G)$, is the minimum number of vertex shifts sufficient to eliminate all edge crossings in an arbitrarily obfuscated drawing of $G$ (after shifting a vertex, all incident edges are supposed to be redrawn correspondingly). If $\delta(G)\ge 3$, then $shift(G)$ is linear in the number of vertices due to the known fact that the matching number of $G$ is linear. However, in the case $\delta(G)\ge2$ we notice that $shift(G)$ can be linear even if the matching number is bounded. As for computational complexity, we show that, given a drawing $D$ of a planar graph, it is NP-hard to find an optimum sequence of shifts making $D$ crossing-free.Comment: 12 pages, 1 figure. The proof of Theorem 3 is simplified. An overview of a related work is adde

Multi feature-rich synthetic colour to improve human visual perception of point clouds

Although point features have shown their usefulness in classification with Machine Learning, point cloud visualization enhancement methods focus mainly on lighting. The visualization of point features helps to improve the perception of the 3D environment. This paper proposes Multi Feature-Rich Synthetic Colour (MFRSC) as an alternative non-photorealistic colour approach of natural-coloured point clouds. The method is based on the selection of nine features (reflectance, return number, inclination, depth, height, point density, linearity, planarity, and scattering) associated with five human perception descriptors (edges, texture, shape, size, depth, orientation). The features are reduced to fit the RGB display channels. All feature permutations are analysed according to colour distance with the natural-coloured point cloud and Image Quality Assessment. As a result, the selected feature permutations allow a clear visualization of the scene's rendering objects, highlighting edges, planes, and volumetric objects. MFRSC effectively replaces natural colour, even with less distorted visualization according to BRISQUE, NIQUE and PIQE. In addition, the assignment of features in RGB channels enables the use of MFRSC in software that does not support colorization based on point attributes (most commercially available software). MFRSC can be combined with other non-photorealistic techniques such as Eye-Dome Lighting or Ambient Occlusion.Xunta de Galicia | Ref. ED481B-2019-061Xunta de Galicia | Ref. ED431F 2022/08Agencia Estatal de Investigación | Ref. PID2019-105221RB-C43Universidade de Vigo/CISU

Kinematic Flexibility Analysis: Hydrogen Bonding Patterns Impart a Spatial Hierarchy of Protein Motion

Elastic network models (ENM) and constraint-based, topological rigidity analysis are two distinct, coarse-grained approaches to study conformational flexibility of macromolecules. In the two decades since their introduction, both have contributed significantly to insights into protein molecular mechanisms and function. However, despite a shared purpose of these approaches, the topological nature of rigidity analysis, and thereby the absence of motion modes, has impeded a direct comparison. Here, we present an alternative, kinematic approach to rigidity analysis, which circumvents these drawbacks. We introduce a novel protein hydrogen bond network spectral decomposition, which provides an orthonormal basis for collective motions modulated by non-covalent interactions, analogous to the eigenspectrum of normal modes, and decomposes proteins into rigid clusters identical to those from topological rigidity. Our kinematic flexibility analysis bridges topological rigidity theory and ENM, and enables a detailed analysis of motion modes obtained from both approaches. Our analysis reveals that collectivity of protein motions, reported by the Shannon entropy, is significantly lower for rigidity theory versus normal mode approaches. Strikingly, kinematic flexibility analysis suggests that the hydrogen bonding network encodes a protein-fold specific, spatial hierarchy of motions, which goes nearly undetected in ENM. This hierarchy reveals distinct motion regimes that rationalize protein stiffness changes observed from experiment and molecular dynamics simulations. A formal expression for changes in free energy derived from the spectral decomposition indicates that motions across nearly 40% of modes obey enthalpy-entropy compensation. Taken together, our analysis suggests that hydrogen bond networks have evolved to modulate protein structure and dynamics

Development of a normal mode-based geometric simulation approach for investigating the intrinsic mobility of proteins

Specific functions of biological systems often require conformational transitions of macromolecules. Thus, being able to describe and predict conformational changes of biological macromolecules is not only important for understanding their impact on biological function, but will also have implications for the modelling of (macro)molecular complex formation and in structure-based drug design approaches. The “conformational selection model” provides the foundation for computational investigations of conformational fluctuations of the unbound protein state. These fluctuations may reveal conformational states adopted by the bound proteins. The aim of this work is to incorporate directional information in a geometry-based approach, in order to sample biologically relevant conformational space extensively. Interestingly, coarse-grained normal mode (CGNM) approaches, e.g., the elastic network model (ENM) and rigid cluster normal mode analysis (RCNMA), have emerged recently and provide directions of intrinsic motions in terms of harmonic modes (also called normal modes). In my previous work and in other studies it has been shown that conformational changes upon ligand binding occur along a few low-energy modes of unbound proteins and can be efficiently calculated by CGNM approaches. In order to explore the validity and the applicability of CGNM approaches, a large-scale comparison of essential dynamics (ED) modes from molecular dynamics (MD) simulations and normal modes from CGNM was performed over a dataset of 335 proteins. Despite high coarse-graining, low frequency normal modes from CGNM correlate very well with ED modes in terms of directions of motions (average maximal overlap is 0.65) and relative amplitudes of motions (average maximal overlap is 0.73). In order to exploit the potential of CGNM approaches, I have developed a three-step approach for efficient exploration of intrinsic motions of proteins. The first two steps are based on recent developments in rigidity and elastic network theory. Initially, static properties of the protein are determined by decomposing the protein into rigid clusters using the graph-theoretical approach FIRST at an all-atom representation of the protein. In a second step, dynamic properties of the molecule are revealed by the rotations-translations of blocks approach (RTB) using an elastic network model representation of the coarse-grained protein. In the final step, the recently introduced idea of constrained geometric simulations of diffusive motions in proteins is extended for efficient sampling of conformational space. Here, the low-energy (frequency) normal modes provided by the RCNMA approach are used to guide the backbone motions. The NMSim approach was validated on hen egg white lysozyme by comparing it to previously mentioned simulation methods in terms of residue fluctuations, conformational space explorations, essential dynamics, sampling of side-chain rotamers, and structural quality. Residue fluctuations in NMSim generated ensemble is found to be in good agreement with MD fluctuations with a correlation coefficient of around 0.79. A comparison of different geometry-based simulation approaches shows that FRODA is restricted in sampling the backbone conformational space. CONCOORD is restricted in sampling the side-chain conformational space. NMSim sufficiently samples both the backbone and the side-chain conformations taking experimental structures and conformations from the state of the art MD simulation as reference. The NMSim approach is also applied to a dataset of proteins where conformational changes have been observed experimentally, either in domain or functionally important loop regions. The NMSim simulations starting from the unbound structures are able to reach conformations similar to ligand bound conformations (RMSD 0.7) between the RMS fluctuations derived from NMSim generated structures and two experimental structures are observed. Furthermore, intrinsic fluctuations in NMSim simulation correlate with the region of loop conformational changes observed upon ligand binding in 2 out of 3 cases. The NMSim generated pathway of conformational change from the unbound structure to the ligand bound structure of adenylate kinase is validated by a comparison to experimental structures reflecting different states of the pathway as proposed by previous studies. Interestingly, the generated pathway confirms that the LID domain closure precedes the closing of the NMPbind domain, even if no target conformation is provided in NMSim. Hence, the results in this study show that, incorporating directional information in the geometry-based approach NMSim improves the sampling of biologically relevant conformational space and provides a computationally efficient alternative to state of the art MD simulations.Konformationsänderungen von Proteinen sind häufig eine grundlegende Voraussetzung für deren biologische Funktion. Die genaue Charakterisierung und Vorhersage dieser Konformationsänderungen ist für das Verständnis ihres Einflusses auf die Funktion erforderlich. Eines der dafür am häufigsten verwendeten und genauesten computergestützten Verfahren ist die Molekulardynamik-Simulationen (MD Simulationen). Diese sind jedoch nach wie vor sehr rechenintensiv und durchmustern den Konformationsraum nur in begrenztem Maße. Daher wurden Anstrengungen unternommen, alternative geometriebasierte Methoden (wie etwa CONCOORD oder FRODA) zu entwickeln, die auf einer reduzierten Darstellung von Proteinen beruhen. Das Ziel dieser Arbeit ist es, Richtungsinformationen in einen geometriebasierten Ansatz zu integrieren, und so den biologisch relevanten Konformationsraum erschöpfend zu durchmustern. Diese Idee führte kürzlich zur Entwicklung von „coarse-grained normal mode“ (CGNM) Methoden, wie zum Beispiel dem „elastic network model“ (ENM) und der von mir in vorangegangenen Arbeiten entwickelte „rigid cluster normal mode analysis“ (RCNMA). Beide Methoden liefern die gewünschte Richtungsinformation der intrinsischen Bewegungen eines Proteins in Form von harmonischen Moden (auch Normalmoden). Um die Aussagekraft, Robustheit und breite Anwendbarkeit solcher CGNM Verfahren zu untersuchen, wurde im Rahmen dieser Dissertation ein umfangreicher Vergleich zwischen „essential dynamics“ (ED) Moden aus MD Simulationen und Normalmoden aus CGNM Berechnungen durchgeführt. Der zugrundeliegende Datensatz enthielt 335 Proteine. Obwohl die CGNM Verfahren eine stark vereinfachte Darstellung für Proteine verwenden, korrelieren die niederfrequenten Moden dieser Verfahren bezüglich ihrer Bewegungs-Richtung (durchschnittliche maximale Überschneidung: 0,65) und -Amplitude (durchschnittliche maximale Überschneidung: 0,73) sehr gut mit ED Moden. Im Durchschnitt beschreibt das erste Viertel der Normalmoden 85 % des Raumes, der durch die ersten fünf ED Moden aufgespannt wird. Um die Leistungsfähigkeit von CGNM Verfahren genauer zu bestimmen, wurde im Rahmen der vorliegenden Studie eine dreistufige Methode zur Untersuchung der intrinsischen Dynamik von Proteinen entwickelt. Die ersten beiden Stufen basieren auf neusten Entwicklungen in der Rigiditäts-Theorie und der Beschreibung von elastischen Netzwerken. Diese sind im RCNMA Ansatz verwirklich und ermöglichen die Bestimmung der Normalmoden. Im letzten Schritt werden die Bewegungen des Proteinrückgrates entlang der mittels RCNMA erzeugten niederenergetischen Normalmoden ausgerichtet. Die Seitenkettenkonformrationen werden dabei durch Diffusionsbewegungen hin zu energetisch günstigen Rotameren erzeugt. Dies ist ein iterativer Prozess, bestehend aus mehreren kleineren Schritten, in denen jeweils intermediäre Konformationen erzeugt werden. Zur Validierung des NMSim Ansatzes wurde dieser mit den anderen zuvor genannten Simulationsmethoden am Beispiel von Lysozym verglichen. Die Fluktuationen der Aminosäurereste aus dem mit NMSim erzeugten Ensemble stimmen mit berechneten Fluktuationen aus der MD Simulation gut überein (Korrelationskoeffizient R = 0,79). Ein Vergleich der unterschiedlichen geometriebasierten Simulationsansätze zeigt, dass bei FRODA die Durchmusterung des Konformationsraumes des Proteinrückrates unzureichend ist. Bei CONCOORD ist hingegen die Durchmusterung des Konformationsraumes der Seitenketten unzureichend. NMSim hingegen durchmustert sowohl den Konformationsraum des Proteinrückrates als auch den der Seitenketten angemessen, wenn man die experimentell und mittels MD Simulationen erzeugten Konformationen als Referenz verwendet. Der NMSim Ansatz wurde ebenfalls auf einen Datensatz von Proteinen angewendet, für die Konformationsänderungen in Domänen oder in funktionell wichtigen Schleifenregionen experimentell beobacht wurden. In Übereinstimmung mit dem Konformations-Selektions-Modell ist der NMSim Ansatz bei vier von fünf Proteinen, die eine Domänenbewegung aufweisen, in der Lage, ausgehend von der ungebundenen Struktur neue Konformationen zu erzeugen, die der ligandgebundenen Konformation entsprechen (RMSD 0,7) zwischen der RMS Fluktuation der durch NMSim erzeugten Konformationen und jeweils zwei experimentellen Strukturen erreicht. Hingegen korrelieren die intrinischen Fluktuationen der NMSim Simulation in zwei von drei Fällen mit dem Bereich der ligandinduzierten Konformationsänderung in den Schleifen. Der mit NMSim generierte Pfad für die Konformationsänderungen von der ungebundenen Struktur zur ligandgebundenen Struktur der Adenylat-Kinase wurde durch den Vergleich zu experimentellen Strukturen validiert, die verschiedene Zustände des Pfades widerspiegeln. Die unterschiedlichen Kristallstrukturen, die entlang der Konformationsänderungen von der ungebundenen zur ligandgebundenen Struktur liegen, werden auf dem von NMSim erzeugten Pfad durchmustert. Interessanterweise bestätigt der generierte Pfad, dass die Schließbewegung der LID Domäne derjenigen der NMPbind Domäne vorangeht, sogar wenn keine Zielkonformation für die NMSim Simulation verwendet wurde

Graph Theoretical Modelling of Electrical Distribution Grids

This thesis deals with the applications of graph theory towards the electrical distribution networks that transmit electricity from the generators that produce it and the consumers that use it. Specifically, we establish the substation and bus network as graph theoretical models for this major piece of electrical infrastructure. We also generate substation and bus networks for a wide range of existing data from both synthetic and real grids and show several properties of these graphs, such as density, degeneracy, and planarity. We also motivate future research into the definition of a graph family containing bus and substation networks and the classification of that family as having polynomial expansion