DOCSLIB.ORG

Algorithms in Rigidity Theory with Applications to Protein Flexibility and Mechanical Linkages

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algorithms in Rigidity Theory with Applications to Protein Flexibility and Mechanical Linkages ALGORITHMS IN RIGIDITY THEORY WITH APPLICATIONS TO PROTEIN FLEXIBILITY AND MECHANICAL LINKAGES ADNAN SLJOKA A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS YORK UNIVERSITY, TORONTO, ONTARIO August 2012 ALGORITHMS IN RIGIDITY THEORY WITH APPLICATIONS TO PROTEIN FLEXIBILITY AND MECHANICAL LINKAGES by Adnan Sljoka a dissertation submitted to the Faculty of Graduate Studies of York University in partial fulfillment of the requirements for the degree of Doctor of Philosophy c 2012 Permission has been granted to: a) YORK UNIVERSITY LIBRARIES to lend or sell copies of this thesis in paper, microform or electronic formats, and b) LIBRARY AND ARCHIVES CANADA to reproduce, lend, distribute, or sell copies of this thesis any- where in the world in microform, paper or electronic formats and to authorize or pro- cure the reproduction, loan, distribution or sale of copies of this thesis anywhere in the world in microform, paper or electronic formats. The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the authors written permission. ii .......... iii Abstract A natural question was asked by James Clark Maxwell: Can we count vertices and edges in a framework in order to make predictions about its rigidity and flexibility? The first complete combinatorial generalization for (generic) bar and joint frameworks in dimension 2 was confirmed by Laman in 1970. The 6jV j − 6 counting condition for 3-dimensional molecular structures, and a fast `pebble game' algorithm which tracks this count in the multigraph have led to the development of the program FIRST for rapid predictions of the rigidity and flexibility of proteins. In this thesis we develop the mathematical models, algorithms and theory using the concepts from combinatorial rigidity theory that have various important applications in protein science and mechanical engineering. Using our extensions of the pebble game algorithm, we have developed a novel protein hinge prediction algo- rithm. The hinge predictions are tested on numerous hinge bending proteins. We have also introduced several rigidity-based allostery models with corresponding allostery- detection algorithms that can detect transmission of rigidity (change in shape) be- tween distant sites. Various examples and theoretical results on the rigidity-based allosteric communication will be provided. We will apply our algorithm on some allosteric proteins, including the important class of signalling proteins known as G- protein couple receptors. The results obtained show that rigidity-based modelling of allostery and our algorithms are promising tools in studying allostery in proteins. We iv also present a novel FIRST-ensemble method which extends FIRST to give a predic- tion of rigidity/flexibility of structural protein ensembles, and when combined with ensemble solvent accessibility data, it gives a good prediction of hydrogen-deuterium exchange. In the area of mechanical linkages, we have developed novel techniques and a pinned version of the pebble game algorithm that can decompose linkages into important d-Assur graphs. v To my parents vi Acknowledgments I would like to express my deep sense of gratitude and heartfelt thanks to my super- visor, Professor Walter J. Whiteley. Without his invaluable guidance and countless support this work would have been impossible. I feel very grateful and privileged to have worked with Professor Whiteley and to have him as my supervisor during my graduate studies. He has taught me a lot about research, to not hesitate to ask questions and make new conjectures, and his friendly teaching approach, enthusiasm and expertise on many subjects has made my research and studies a lot more enjoy- able. He has always been generous with his time and made himself available during the countless (sometimes very long) stimulating meetings, and has made numerous suggestions on this thesis. The nurturing spirit and support from Professor Whiteley during the difficult times will always be remembered. I am indebted to Professor Whiteley for nourishing my personal, intellectual and academic growth, from which I will benefit for a long time. As my work spanned different fields, this resulted in several critical collabo- rations. I would like to express special thanks to Professor Derek Wilson, who was also on my supervisory committee, for his patience and numerous helpful discussions about protein flexibility and how it relates to my work. I am also grateful for many vii fruitful discussions and e-mail exchanges with Professor Offer Shai and his encour- agement to pursue and extend my work to problems and applications in mechanical engineering. I also want to thank Professor Huaxiong Huang and Professor Jorg Grigull for taking their time to serve on my supervisory committee. Their input, advice and connections to new projects and applications was very valuable. Thanks also to Professor Mike Thorpe, Professor Suprakash Datta and Pro- fessor Ada Chan for agreeing to be on my examining committee. I want to thank Professor Thorpe for his expertise and several valuable discussions about my work and topics on protein flexibility. I benefitted a lot from other students and friends who were very helpful through- out my studies. I especially thank Alexandr Bezginov for his valuable help in the initial implementations, with Pymol scripts and continued interest in the rigidity ap- plications. I want to thank Dr. Naveen Vaidya, a former graduate student, for his assistance and new collaborative projects and his friendship over the years. I also thank Dr. Bernd Schulze for many discussions on the rigidity and his friendship and interest in collaboration. I also want to thank Primrose Miranda for her very helpful administrative support. I want to give a special thanks to Cora for all the support and encouragement through the long writing process. Last but not least, I want to thank my parents for their love and continued support. They were always there for me and made many sacrifices through this long and sometimes challenging journey. Puno hvala na svemu ˇstoste mi pruˇzili,za podrˇskui veliko stpljenje, voli vas vaˇssin. I also thank my brother for his support (voli te tvoj brat) and his family for their understanding. viii Contents Abstract :::::::::::::::::::::::::::::::::::::: iv Acknowledgments :::::::::::::::::::::::::::::::: vii 1 Introduction :::::::::::::::::::::::::::::::::: 1 1.1 Overview . .1 1.2 Protein rigidity and flexibility . .4 1.2.1 Protein flexibility is related to protein function . .5 1.3 Protein flexibility can be studied using Rigidity Theory . .8 1.3.1 FIRST method overview . .8 1.4 Thesis Outline . 15 2 Rigidity Theory: Background and Searching for the Counts :::: 20 2.1 Overview . 20 2.2 Rigidity Theory { introduction . 21 2.2.1 Basic concepts . 21 2.2.2 Definitions, Rigidity and Infinitesimal Rigidity . 27 2.3 Counting for rigidity in plane graphs and Laman's Theorem . 39 2.3.1 Counting is not sufficient in dimension 3 . 45 2.4 Counting in Body-Bar and Molecular Structures . 48 3 The Pebble Game Algorithm and Relevant Regions ::::::::: 61 3.1 Overview . 61 3.2 The Pebble Game Algorithm . 62 3.2.1 Pebble game illustration and properties . 70 3.3 The Relevant Regions . 81 3.3.1 Quantifying relative flexibility of the core . 82 3.3.2 Relevant region detection . 89 ix 3.3.3 Properties of the Relevant regions . 101 4 Hinge motion predictions ::::::::::::::::::::::::: 103 4.1 Overview . 103 4.2 Hinge motions and background . 105 4.3 Hinge motion detection via Relevant Regions . 110 4.4 Hinge-Prediction algorithm for proteins . 124 4.5 Application of hinge-prediction algorithm on proteins . 129 4.6 Concluding remarks and future work . 152 5 Rigidity model of Protein Allostery ::::::::::::::::::: 157 5.1 Overview . 157 5.2 Allostery background . 159 5.3 A Rigidity approach to allostery . 165 5.4 Rigidity models of allostery . 175 5.5 Methods and Algorithms to detect rigidity-based allosteric communi- cation . 181 5.5.1 Algorithms . 182 5.5.2 Examples . 190 5.5.3 Properties of Allostery type 2 . 203 5.5.4 Cyclic communication in Allostery type 2 . 206 5.6 Applications to protein allostery . 214 5.6.1 Rigidity-based allostery in calmodulin . 214 5.6.2 Applications to G-Protein Coupled Receptors . 222 5.7 Concluding remarks and future work . 252 6 Predictions of rigidity and flexibility of Protein ensembles and Hydrogen- Deuterium exchange ::::::::::::::::::::::::::::: 260 6.1 Overview . 260 6.2 Introduction . 262 6.2.1 Extending FIRST to Protein ensembles . 262 6.2.2 Hydrogen/Deuterium exchange is dependent on Rigidity and Solvent Accessibility . 267 6.3 Methods and Algorithms . 270 6.3.1 FIRST-ensemble rigidity/flexibility predictions . 270 6.3.2 FIRST-ensemble algorithm . 276 x 6.3.3 Solvent Accessibility Calculations . 282 6.3.4 Combining FIRST-ensemble and Solvent Accessibility on en- semble as a Predictor of HD-exchange . 284 6.4 Results and Discussions . 287 6.4.1 Experimental HDX profile of Sso AcP . 287 6.4.2 FIRST rigidity/flexibility predictions using individual NMR mod- els ................................. 290 6.4.3 FIRST-ensemble rigidity/flexibility predictions . 292 6.4.4 Solvent Accessibility ensemble data . 299 6.4.5 Combined rigidity and solvent accessibility predictions of HDX 304 6.5 Concluding remarks and future work . 308 7 Decompositions of Linkages, Assur graphs and the pinned pebble game algorithm :::::::::::::::::::::::::::::::: 311 7.1 Overview . 311 7.2 Introduction . 313 7.3 Decomposition of Pinned Directed Graphs . 320 7.3.1 Strongly connected component decomposition . 321 7.3.2 Equivalent orientations of a graph . 324 7.4 Decomposition of Pinned d-isostatic Graphs . 327 7.4.1 Pinned d-isostatic graphs and pinned rigidity matrix . 327 7.4.2 Directed graph decomposition of the pinned rigidity matrix . 332 7.4.3 d-Assur graphs and minimal components .
Load more

Recommended publications
  • On Rigidity of Graphs
    On Rigidity of Graphs Bachelor Thesis of Oliver Suchan at the Department of Informatics Institute of Theoretical Informatics Reviewers: Prof. Dr. Dorothea Wagner Prof. Dr. Peter Sanders Advisors: Dr. Torsten Ueckerdt Marcel Radermacher, M. Sc. Time Period: 05.11.2018 – 04.03.2019 KIT – The Research University in the Helmholtz Association www.kit.edu Acknowledgement I would like to thank my advisors Dr. Torsten Ueckerdt and Marcel Radermacher for many hours of sharing and discussing ideas. Statement of Authorship I hereby declare that this thesis has been composed by myself and describes my own work, unless otherwise acknowledged in the text. Karlsruhe, March 4, 2019 iii Abstract A rigid framework refers to an embedding of a graph, in which each edge is represented by a straight line. Additionally, the only continuous displacements of the vertices, that maintain the lengths of all edges, are isometries. If the edge lengths are allowed to only be perturbed in first order, the framework is called infinitesimally rigid. In this thesis the degrees of freedom of a certain, new type of rigidity, called edge-x- rigidity, for a given framework in the 2-dimensional Euclidean space are examined. It does not operate on the length of an edge, but on the intersection of an edge’s support line and the x-axis. This may then help to determine the class of graphs where every edge intersection with the x-axis can be repositioned along the x-axis, such that there still exists a framework that satisfies these position constraints. Deutsche Zusammenfassung Ein starres Fachwerk bezeichnet eine Einbettung eines Graphen - bei der jede Kante durch ein gerade Linie repräsentiert wird - und es keine stetige nicht-triviale Verschiebung der Knoten gibt, welche die Längen aller Kanten gleich lässt.
    [Show full text]
  • Avis, David; Katoh, Naoki; Ohsaki, Makoto; Streinu, Ileana; Author(S) Tanigawa, Shin-Ichi
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Kyoto University Research Information Repository Enumerating Constrained Non-crossing Minimally Rigid Title Frameworks Avis, David; Katoh, Naoki; Ohsaki, Makoto; Streinu, Ileana; Author(s) Tanigawa, Shin-ichi Citation Discrete and Computational Geometry (2008), 40(1): 31-46 Issue Date 2008-07 URL http://hdl.handle.net/2433/84862 Right The original publication is available at www.springerlink.com. Type Journal Article Textversion author Kyoto University Enumerating Constrained Non-crossing Minimally Rigid Frameworks David Avis 1 Naoki Katoh 2 Makoto Ohsaki 2 Ileana Streinu 3 Shin-ichi Tanigawa 2 April 25, 2007 Abstract In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all the constrained non-crossing Laman frameworks on a given point set is connected by flips which preserve the Laman property. Key words: geometric enumeration; rigidity; constrained non-crossing minimally rigid frameworks; constrained Delaunay triangulation. 1 Introduction Let G be a graph with vertices {1,...,n} and m edges. G is a Laman graph if m = 2n − 3 and every subset of n′ ≤ n vertices spans at most 2n′ − 3 edges.
    [Show full text]
  • Rigid and Generic Structures
    RIGID AND GENERIC STRUCTURES by Garn Cooper B.Sc.(Hons) Submitted in fulfilment of the requirements for the degree of Master of Science University of Tasmania February, 1993 This thesis contains no material which has been accepted for the award of any other degree or diploma in any tertiary institution, and to the best of my knowledge and belief this thesis contains no material previously published or written by another person, except when due reference is made in the text of the thesis. /(9 Abstract This thesis is an examination of infinitesimal rigidity in generic structures using linear algebra and matroid theory techniques. The structures examined are bar and joint structures in 1, 2 and 3 dimensions, and hinged panel structures. The focus of this work is a conjectured environment for higher dimensional analogues of Laman's theorem, and some light is consequently shed on the quest for a combinatorial characterisation of (generic) rigid bar and joint structures in three dimensions. Towards completing this thesis Don helped as supervisor, Betty helped type it, Kym helped computer it, and Robyn helped illustrate it. Contents Introduction 1 Preamble 12 Chapter 1 — Defining Rigidity 16 Chapter 2 — Rigidity in 012 32 Chapter 3 — Genericity 40 Chapter 4 — Generic Rigidity in R 2 47 Chapter 5 — Rigidity in 013 63 Chapter 6 — Hinged Panel Structures 72 Conclusions 86 Reference List 93 Introduction. Consider a triangular framework and a square framework in a plane, for which the edges are inflexible rods which are joined at the vertices by universal joints. The latter is flexible in the plane since it can deform into the shape of a rhombus.
    [Show full text]
  • The Maximum Matroid for a Graph
    The Maximum Matroid for a Graph Meera Sitharam and Andrew Vince University of Florida Gainesville, FL, USA [email protected] Abstract The ground set for all matroids in this paper is the set of all edges of a complete graph. The notion of a maximum matroid for a graph is introduced. The maximum matroid for K3 (or 3-cycle) is shown to be the cycle (or graphic) matroid. This result is persued in two directions - to determine the maximum matroid for the m-cycle and to determine the maximum matroid for the complete graph Km. While the maximum matroid for an m-cycle is determined for all m ≥ 4; the determination of the maximum matroids for the complete graphs is more complex. The maximum matroid for K4 is the matroid whose bases are the Laman graphs, related to structural rigidity of frameworks in the plane. The maximum matroid for K5 is related to a famous 153 year old open problem of J. C. Maxwell. An Algorithm A is provided, whose input is a given graph G and whose output is a matroid MA(G), defined in terms of its closure operator. The matroid MA(G) is proved to be the maximum matroid for G, implying that every graph has a unique maximum matroid. Keywords: graph, matroid, framework Mathematical subject codes: 05B35, 05C85, 52C25 1 Introduction Fix n and let Kn be the complete graph on n vertices. A graph in this paper is a subset E of the edge set E(Kn) of Kn. The number of edges of a graph E is denoted jEj and graph isomorphism by ≈.
    [Show full text]
  • Dos Problemas De Combinatoria Geométrica: Triangulaciones
    Dos problemas de combinatoria geometrica:´ Triangulaciones eficientes del hipercubo; Grafos planos y rigidez. (Two problems in geometric combinatorics: Efficient triangulations of the hypercube; Planar graphs and rigidity). David Orden Mart´ın Universidad de Cantabria Tesis doctoral dirigida por el Profesor Francisco Santos Leal. Santander, Marzo de 2003. 2 ´Indice Agradecimientos iii Pre´ambulo v Preamble vii 1 Introducci´on y Preliminares 1 1.1 Triangulaciones y subdivisiones de politopos . 1 1.1.1 Preliminares sobre Teor´ıa de Politopos . 2 1.1.2 Triangulaciones de cubos . 6 1.2 Teor´ıa de Grafos . 11 1.2.1 Preliminares y relaciones con politopos . 11 1.2.2 Inmersiones planas de grafos: Teorema de Tutte . 15 1.3 Teor´ıa de Rigidez . 20 1.3.1 La matriz de rigidez . 20 1.3.2 Tensiones y grafos rec´ıprocos . 23 1.3.3 Rigidez infinitesimal y grafos isost´aticos . 26 1.4 Pseudo-triangulaciones de cuerpos convexos y puntos . 29 1.4.1 Pseudo-triangulaciones de cuerpos convexos . 29 1.4.2 Pseudo-triangulaciones de puntos . 31 1.5Resultados y Problemas abiertos . 35 1.5.1 Triangulaciones eficientes de cubos . 35 1.5.2 Grafos sin cruces, rigidez y pseudo-triangulaciones . 36 1 Introduction and Preliminaries 41 1.1 Triangulations and subdivisions of polytopes . 41 1.1.1 Preliminaries about Polytope Theory . 42 1.1.2 Triangulations of cubes . 46 1.2 Graph Theory . 50 1.2.1 Preliminaries and relations with polytopes . 51 1.2.2 Plane embeddings of graphs: Tutte’s Theorem . 54 1.3 Rigidity Theory . 59 1.3.1 The rigidity matrix .
    [Show full text]