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 .