A DYNAMIC ROD MODEL TO SIMULATE MECHANICS OF CABLES AND DNA by Sachin Goyal A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering and Scientific Computing) in The University of Michigan 2006 Doctoral Committee: Professor Noel Perkins, Chair Associate Professor Edgar Meyhofer, Co-Chair Assistant Professor Ioan Andricioaei Assistant Professor Krishnakumar Garikipati Assistant Professor Jens-Christian Meiners Dedication To my parents ii Acknowledgements I am extremely grateful to my advisor Prof. Noel Perkins for his highly conducive and motivating research guidance and overall mentoring. I sincerely appreciate many engaging discussions of DNA mechanics and supercoiling with Prof. E. Meyhofer and Prof. K. Garikipati in Mechanical Engineering Program, Prof. C. Meiners in the Biophysics Program, Prof. I. Andricioaei in the Biochemistry/ Biophysics Program and and Dr. S. Blumberg in Medical school at the University of Michigan and Dr. C. L. Lee at the Lawrence Livermore National Laboratory, California. I also appreciate Professor Jason D. Kahn (Department of Chemistry and Biochemistry, University of Maryland) for providing the sequences used in his experiments, and Professor Wilma K. Olson (Department of Chemistry and Chemical Biology, Rutgers University) for suggesting alternative binding topologies in DNA-protein complexes. I also acknowledge recent graduate students T. Lillian in Mechanical Engineering Program, D. Wilson in Physics Program and J. Wereszczynski in Chemistry Program to extend my work and collaborate on ongoing studies beyond the scope of dissertation. I also gratefully acknowledge the research support provided by the U. S. Office of Naval Research, Lawrence Livermore National Laboratories, and the National Science Foundation. iii Table of Contents Dedication........................................................................................................................... ii Acknowledgements............................................................................................................iii List of Figures.................................................................................................................... vi List of Tables ..................................................................................................................... xi List of Appendices ............................................................................................................ xii Abstract............................................................................................................................xiii Chapter................................................................................................................................1 1. Introduction............................................................................................................. 1 1.1 Underwater (Marine) Cable Applications............................................................. 4 1.2 Mechanics of DNA: Looping and Supercoiling ................................................... 8 1.3 Research Objective ............................................................................................. 13 1.4 Scope of Dissertation and Previous Rod Theories.............................................. 15 1.5 Summary of Research Contributions.................................................................. 21 2. The Rod Model – Theoretical Formulation .......................................................... 30 2.1 Definitions and Assumptions.............................................................................. 30 2.2 Equations of Motion ........................................................................................... 33 2.3 Constraints and Summary................................................................................... 34 3. The Rod Model – Computational Formulation..................................................... 37 3.1 The Generalized-α Method ................................................................................. 37 3.2 Space-Time Discretization.................................................................................. 40 3.3 Kinematics of Cross-Section Rotation................................................................ 45 3.4 Summary of Numerical Enhancements .............................................................. 47 4. Benchmarking and Extensions of Prior Studies: Equilibria and Dynamic Transitions..................................................................................................................... 49 4.1 Input Parameters ................................................................................................. 50 4.1.1 Constitutive Law.......................................................................................... 50 4.1.2 Distributed Loading ..................................................................................... 52 4.1.3 Initial and Boundary Conditions.................................................................. 54 4.2 Equilibria Benchmarking (Slow Loading d& → 0) ............................................. 56 4.2.1 Case 1: R/ = 0, Planar buckling .................................................................... 57 4.2.2 Case 2: R/ = 0, Spatial buckling.................................................................... 59 4.2.3 Case 3: R/ =1 Planar and spatial buckling.................................................. 60 4.3 Dynamics and Hysteresis (Fast Loading - Finite d& ).......................................... 62 5. Tension-Torque Coupling..................................................................................... 67 5.1 Modified Constitutive Law................................................................................. 67 iv 5.2 Modified Buckling Condition (Linear)............................................................... 70 5.3 Influence on Loop Topology and Bifurcations (Nonlinear) ............................... 72 5.4 Summary of Effects of Tension-Torque Coupling ............................................. 77 6. Dynamics of Self-Contact and Intertwining ......................................................... 79 6.1 Numerical Model of Dynamic Self-Contact....................................................... 79 6.2 Torsional Buckling Leading To Intertwining ..................................................... 81 6.3 Topological Changes .......................................................................................... 87 7. Protein-Mediated DNA Looping .......................................................................... 91 7.1 Introduction to LacR-DNA Modeling ................................................................ 91 7.2 Methods............................................................................................................... 95 7.3 Results............................................................................................................... 103 7.4 Discussion and Conclusions ............................................................................. 110 8. Summary, Conclusions and Future Work........................................................... 123 8.1 Summary and Major Conclusions..................................................................... 123 8.2 Future Work on DNA ....................................................................................... 127 8.2.1 Structural Characterization of DNA ......................................................... 127 8.2.2 Modeling Entropic Effects......................................................................... 129 8.2.3 Coupling of Protein Flexibility/ Dynamics................................................ 130 8.2.4 Histone Unwrapping .................................................................................. 130 Appendices...................................................................................................................... 133 v List of Figures Figure 1.1 Low tension cable forming loops and tangles on the sea floor. ........................ 1 Figure 1.2 Electron micrographs of a DNA polymer in two different conformations (Courtesy: Lehninger et al. [2]). The interwound conformation (lower image) is an example of intertwining that is topologically equivalent to tangles in underwater cables.................................................................................................... 2 Figure 1.3 A twisted cable collapses under slack conditions and ultimately forms a loop or hockle as well as an intertwined ‘snarl’ (Courtesy: Goss et al. [4])................... 5 Figure 1.4 S-tether mooring collapses into a loop (‘hockle’) under torsion (due to yawing) in a low-tension zone. The symbol ρ represents density. ....................................... 6 Figure 1.5 DNA shown on three length scales. Smallest scale (left) shows double-helix structure (sugar-phosphate chains and base-pairs). Intermediate scale (middle) shows how several double-helices form a continuous piece of double-stranded DNA. Largest scale (right) shows how the strand ultimately curves and twists in forming supercoils (one interwound or plectonemic, and one solenoidal). (Courtesy: Branden and Tooze [10] and Lehninger et al. [2])................................ 9 Figure 2.1 Free body diagram of an infinitesimal element of a Kirchhoff rod................. 31 Figure 3.1 Space-time disretization grid (Method of Lines)............................................