Fundamental Concepts in Biophysics : Volume 1

1 Mathematical Methods in Biophysics Rajiv R.P. Singh 1.1. Functions of One Variable and Ordinary Differential Equations ........................................ 1 1.2. Functions of Several Variables: Diffusion Equation in One Dimension ............................. 2 1.3. Random Walks and Diffusion ............................................................................................. 4 1.4. Random Variables, Probability Distribution, Mean, and Variance ..................................... 7 1.5. Diffusion Equation in Three Dimensions............................................................................ 8 1.6. Complex Numbers, Complex Variables, and Schrödinger's Equation ................................ 9 1.7. Solving Linear Homogeneous Differential Equations......................................................... 10 1.8. Fourier Transforms.............................................................................................................. 13 1.9. Nonlinear Equations: Patterns, Switches and Oscillators .................................................... 14 2 Quantum Mechanics Basic to Biophysical Methods William Fink 2.1. Quantum Mechanics Postulates........................................................................................... 17 2.2. One-Dimensional Problems................................................................................................. 23 2.3. The Harmonic Oscillator ..................................................................................................... 27 2.4. The Hydrogen Atom............................................................................................................ 30 2.5. Approximate Methods......................................................................................................... 33 2.6. Many Electron Atoms and Molecules ................................................................................. 36 2.7. The Interaction of Matter and Light .................................................................................... 38 3 Computational Modeling of Receptor–Ligand Binding and Cellular Signaling Processes Subhadip Raychaudhuri, Philippos Tsourkas, and Eric Willgohs 3.1. Introduction ......................................................................................................................... 41 3.2. Differential Equation-Based Mean-Field Modeling ............................................................ 42 3.3. Application: Clustering of Receptor–Ligand Complexes.................................................... 45 3.4. Modeling Membrane Deformation as a Result of Receptor–Ligand Binding ................................................................................................... 46 3.5. Limitations of Mean-Field Differential Equation-Based Modeling .................................... 47 3.6. Master Equation: Calculating the Time Evolution of a Chemically Reacting System.................................................................................................................. 47 viii CONTENTS 3.7. Stochastic Simulation Algorithm (SSA) of Gillespie .......................................................... 48 3.8. Application of the Stochastic Simulation Algorithm (SSA)................................................ 49 3.9. Free Energy-Based Metropolis Monte Carlo Simulation .................................................... 49 3.10. Application of Metropolis Monte Carlo Algorithm............................................................. 50 3.11. Stochastic Simulation Algorithm with Reaction and Diffusion: Probabilistic Rate Constant–Based Method ........................................................................ 51 3.12. Mapping Probabilistic and Physical Parameters.................................................................. 51 3.13. Modeling Binding between Multivalent Receptors and Ligands......................................... 53 3.14. Multivalent Receptor–Ligand Binding and Multimolecule Signaling Complex Formation............................................................................................................. 54 3.15. Application of Stochastic Simulation Algorithm with Reaction and Diffusion........................................................................................................ 55 3.16. Choosing the Most Efficient Simulation Method ................................................................ 55 3.17. Summary ............................................................................................................................. 56 4 Fluorescence Spectroscopy Yin Yeh, Samantha Fore, and Huawen Wu 4.1. Introduction ......................................................................................................................... 63 4.2. Fundamental Process of Fluorescence................................................................................. 64 4.3. Fluorescence Microscopy.................................................................................................... 71 4.4. Types of Biological Fluorophores ....................................................................................... 76 4.5. Application of Fluorescence in Biophysical Research ........................................................ 86 4.6. Dynamic Processes Probed by Fluorescence....................................................................... 96 5 Electrophysiological Measurements of Membrane Proteins Tsung-Yu Chen, Yu-Fung Lin, and Jie Zheng 5.1. Membrane Bioelectricity ..................................................................................................... 111 5.2. Electrochemical Driving Force............................................................................................ 112 5.3. Voltage Clamp versus Current Clamp................................................................................. 114 5.4. Principles of Silver Chloride Electrodes.............................................................................. 115 5.5. Capacitive Current and Ionic Current.................................................................................. 115 5.6. Gating and Permeation Functions of Ion Channels ............................................................. 116 5.7. Two-Electrode Voltage Clamp for Xenopus Oocyte Recordings ........................................ 118 5.8. Patch-Clamp Recordings ..................................................................................................... 122 5.9. Patch-Clamp Fluorometry ................................................................................................... 137 HANDBOOK OF MODERN BIOPHYSICS, VOLUME 1 ix 6 Single-Particle Tracking Michael J. Saxton 6.1. Introduction ......................................................................................................................... 147 6.2. The Broader Field................................................................................................................ 148 6.3. Labeling the Dots ................................................................................................................ 150 6.4. Locating the Dots ................................................................................................................ 153 6.5. Connecting the Dots ............................................................................................................ 158 6.6. Interpreting the Dots: Types of Motion............................................................................... 158 6.7. Is It Really a Single Particle? .............................................................................................. 163 6.8. Enhancing z-Resolution....................................................................................................... 164 6.9. Can a Single Fluorophore Be Seen in a Cell?...................................................................... 165 6.10. Colocalization..................................................................................................................... 165 6.11. Example: Motion in the Plasma Membrane Is More Complicated than is Often Assumed ................................................................................... 167 6.12. Example: From DNA to Protein.......................................................................................... 168 6.13. Example: Infection of a Cell by a Virus .............................................................................. 169 7 NMR Measurement of Biomolecule Diffusion Thomas Jue 7.1. Introduction ......................................................................................................................... 181 7.2. Relaxation and Field Gradient Measurement of Diffusion.................................................. 182 7.3. Frequency Encoding of Spatial Position with Field Gradient ............................................. 183 7.4. Phase Encoding by the Field Gradient................................................................................. 184 7.5. Diffusion and Pulsed Field Gradient Signal Intensity ......................................................... 187 7.6. Fick’s Laws of Diffusion..................................................................................................... 188 7.7. Biomolecule Diffusion in the Cell....................................................................................... 189 7.8. Stimulated Echo and Biomolecule Diffusion in the Cell..................................................... 190 7.9. Myoglobin Function in the Cell........................................................................................... 190 7.10. Perfused Heart Model.......................................................................................................... 191 7.11. O2 Diffusion in Muscle Cell ................................................................................................ 192 7.12. Translational Diffusion of Mb in Vitro ............................................................................... 193 7.13. Translational Diffusion of Mb In Vivo................................................................................ 193 7.14. Mb Contribution to O2 transport in Vivo............................................................................. 193 7.15. Mb-Facilitated Diffusion and Myocardial Function............................................................ 195 7.16. Mb-Facilitated Diffusion and Skeletal Muscle Function..................................................... 195 7.17. Cytoplasmic Properties and Architecture ............................................................................ 196 7.18. Summary ............................................................................................................................. 196 Problem Solutions...................................................................................................... 201 Index........................................................................................................................... 233