Physical Biology of the Cell. Rob Phillips, Jane Kondev and Julie Theriot

Physical Biology of the Cell Rob Phillips, Jane Kondev and Julie Theriot April 4, 2008

Contents 0.1 Preface................................ 14 I The Facts of Life 21 1 Why: Biology By the Numbers 23 1.1 Physical Biology of the Cell..................... 23 1.2 The Stuff of Life........................... 25 1.3 Model Building in Biology...................... 28 1.3.1 Models as Idealizations................... 28 1.3.2 Cartoons and Models.................... 37 1.4 Quantitative Models and the Power of Idealization........ 41 1.4.1 On the Springiness of Stuff................. 43 1.4.2 The Toolbox of Fundamental Physical Models....... 44 1.4.3 The Role of Estimates.................... 46 1.4.4 On Being Wrong....................... 48 1.4.5 Rules of Thumb: Biology by the Numbers......... 49 1.5 Summary and Conclusions...................... 52 1.6 Further Reading........................... 52 1.7 References............................... 53 2 What and Where: Construction Plans for Cells and Organisms 55 2.1 An Ode to E. coli........................... 56 2.1.1 The Bacterial Standard Ruler................ 57 2.1.2 Taking the Molecular Census................ 59 2.1.3 Looking Inside of Cells.................... 66 2.1.4 Where Does E. coli Fit?................... 67 2.2 Cells and Structures Within Them................. 69 2.2.1 Cells: A Rogue s Gallery................... 69 2.2.2 The Cellular Interior: Organelles.............. 78 2.2.3 Macromolecular Assemblies: The Whole is Greater than the Sum of the Parts..................... 85 2.2.4 Viruses as Assemblies.................... 88 2.2.5 The Molecular Architecture of Cells: From PDB Files to Ribbon Diagrams....................... 92 3

4 CONTENTS 2.3 Telescoping Up in Scale: Cells Don t Go It Alone......... 97 2.3.1 Multicellularity As One of Evolution s Great Inventions. 97 2.3.2 Cellular Structures From Tissues to Nerve Networks... 103 2.3.3 Multicellular Organisms................... 106 2.4 Summary and Conclusions...................... 112 2.5 Problems............................... 113 2.6 Further Reading........................... 115 2.7 References............................... 115 3 When: Stopwatches at Many Scales 119 3.1 The Hierarchy of Temporal Scales................. 119 3.1.1 The Pageant of Biological Processes............ 120 3.1.2 The Evolutionary Stopwatch................ 126 3.1.3 The Cell Cycle and the Standard Clock.......... 131 3.1.4 Three Views of Time in Biology.............. 135 3.2 Procedural Time........................... 136 3.2.1 The Machines (or Processes) of the Central Dogma.... 136 3.2.2 Clocks and Oscillators.................... 140 3.3 Relative Time............................. 145 3.3.1 Checkpoints and the Cell Cycle............... 145 3.3.2 Measuring Relative Time.................. 150 3.3.3 Killing the Cell: The Life Cycles of Viruses........ 153 3.3.4 The Process of Development................ 156 3.4 Manipulated Time.......................... 159 3.4.1 Chemical Kinetics and Enzyme Turnover......... 159 3.4.2 Beating the Diffusive Speed Limit............. 160 3.4.3 Beating the Replication Limit................ 166 3.4.4 Eggs and Spores: Planning for the Next Generation... 168 3.5 Summary and Conclusions...................... 169 3.6 Problems............................... 169 3.7 Further Reading........................... 172 3.8 References............................... 173 4 Who: Bless the Little Beasties 175 4.1 Choosing a Grain of Sand...................... 175 4.1.1 Biochemistry and Genetics................. 177 4.2 Hemoglobin as a Model Protein................... 180 4.2.1 Hemoglobin, Receptor-Ligand Binding and the Other Bohr 182 4.2.2 Hemoglobin and the Origins of Structural Biology.... 184 4.2.3 Hemoglobin and Molecular Models of Disease....... 188 4.2.4 The Rise of Allostery and Cooperativity.......... 188 4.3 Bacteriophage and Molecular Biology............... 189 4.3.1 Bacteriophage and the Origins of Molecular Biology... 190 4.3.2 Bacteriophage and Modern Biophysics........... 195 4.4 A Tale of Two Cells: E. Coli as a Model System......... 199 4.4.1 Bacteria and Molecular Biology............... 199

CONTENTS 5 4.4.2 E. coli and The Central Dogma............... 199 4.4.3 The lac Operon as the Hydrogen Atom of Genetic Circuits202 4.4.4 Signaling and Motility: The Case of Bacterial Chemotaxis 205 4.5 Yeast: From Biochemistry to the Cell Cycle............ 207 4.5.1 Yeast and the Rise of Biochemistry............. 208 4.5.2 Dissecting the Cell Cycle.................. 209 4.5.3 Deciding Which Way is Up: Yeast and Polarity...... 209 4.5.4 Dissecting Membrane Traffic................ 212 4.5.5 Genomics and Proteomics.................. 215 4.6 Flies and Modern Biology...................... 218 4.6.1 Flies and the Rise of Modern Genetics........... 218 4.6.2 How the Fly Got His Stripes................ 220 4.7 Of Mice and Men........................... 222 4.8 The Case for Exotica......................... 223 4.8.1 Specialists and Experts................... 224 4.8.2 The Squid Giant Axon and Biological Electricity..... 225 4.8.3 Exotica Toolkit........................ 228 4.9 Summary and Conclusions...................... 229 4.10 Problems............................... 230 4.11 Further Reading........................... 232 4.12 References............................... 234 II Life at Rest 237 5 Mechanical and Chemical Equilibrium in the Living Cell 239 5.1 Energy and the Life of Cells..................... 240 5.1.1 The Interplay of Deterministic and Thermal Forces.... 241 5.1.2 Constructing the Cell: Managing the Mass and Energy Budget of the Cell...................... 244 5.2 Biological Systems as Minimizers.................. 254 5.2.1 Equilibrium Models for Out of Equilibrium Systems... 255 5.2.2 Proteins in Equilibrium.................. 256 5.2.3 Cells in Equilibrium.................... 258 5.2.4 Mechanical Equilibrium From a Minimization Perspective 259 5.3 The Mathematics of Superlatives.................. 265 5.3.1 The Mathematization of Judgement: Functions and Functionals............................. 265 5.3.2 The Calculus of Superlatives................ 267 5.4 Configurational Energy....................... 271 5.4.1 Hooke s Law: Actin to Lipids................ 274 5.5 Structures as Free Energy Minimizers............... 278 5.5.1 Entropy and Hydrophobicity................ 281 5.5.2 Gibbs and the Calculus of Equilibrium........... 285 5.5.3 Structure as a Competition................. 288 5.5.4 An Ode to G........................ 290

6 CONTENTS 5.6 Summary and Conclusions...................... 291 5.7 Appendix: The Euler-Lagrange Equations, Finding the Superlative292 5.8 Problems............................... 294 5.9 Further Reading........................... 298 5.10 References............................... 299 6 Entropy Rules! 301 6.1 The Analytical Engine of Statistical Mechanics.......... 301 6.1.1 A First Look at Ligand-Receptor Binding......... 307 6.1.2 The Statistical Mechanics of Gene Expression: RNA Polymerase and the Promoter.................. 312 6.1.3 Classic Derivation of the Boltzmann Distribution..... 318 6.1.4 Boltzmann Distribution by Counting............ 321 6.1.5 Boltzmann Distribution by Guessing............ 325 6.2 On Being Ideal............................ 331 6.2.1 Average Energy of a Molecule in a Gas.......... 332 6.2.2 Free Energy of Dilute Solutions............... 335 6.2.3 Osmotic Pressure as an Entropic Spring.......... 337 6.3 The Calculus of Equilibrium Applied: Law of Mass Action.... 341 6.3.1 Law of Mass Action and Equilibrium Constants...... 343 6.4 Applications of the Calculus of Equilibrium............ 345 6.4.1 A Second Look at Ligand-Receptor Binding........ 345 6.4.2 Measuring Ligand-Receptor Binding............ 347 6.4.3 Beyond Simple Ligand-Receptor Binding: The Hill Function347 6.4.4 ATP Power.......................... 350 6.5 Summary and Conclusions...................... 352 6.6 Problems............................... 353 6.7 Further Reading........................... 355 6.8 References............................... 355 7 Two-State Systems: From Ion Channels to Cooperative Binding 357 7.1 Macromolecules With Multiple States............... 358 7.1.1 The Internal State Variable Idea.............. 358 7.1.2 Ion Channels as an Example of Internal State Variables. 361 7.2 State Variable Description of Binding............... 366 7.2.1 The Gibbs Distribution: Contact with a Particle Reservoir 367 7.2.2 Simple Ligand-Receptor Binding Revisited........ 369 7.2.3 Phosphorylation as an Example of Two Internal State Variables........................... 371 7.2.4 Hemoglobin as a Case Study in Cooperativity....... 375 7.3 Summary and Conclusions...................... 386 7.4 Problems............................... 386 7.5 Further Reading........................... 388 7.6 References............................... 389

CONTENTS 7 8 Random Walks and the Structure of Macromolecules 391 8.1 What is a Structure: PDB or R G?................. 391 8.1.1 Deterministic vs. Statistical Descriptions of Structure.. 392 8.2 Macromolecules as Random Walks................. 393 8.2.1 A Mathematical Stupor................... 394 8.2.2 How Big is a Genome?.................... 403 8.2.3 The Geography of Chromosomes.............. 405 8.2.4 DNA Looping: From Chromosomes to Gene Regulation. 421 8.2.5 PCR, DNA Melting and DNA Bubbles........... 425 8.3 The New World of Single Molecule Mechanics........... 430 8.3.1 Force-Extension Curves: A New Spectroscopy....... 431 8.3.2 Random Walk Models for Force-Extension Curves.... 433 8.4 Proteins as Random Walks..................... 438 8.4.1 Compact Random Walks and the Size of Proteins..... 438 8.4.2 Hydrophobic and Polar Residues: The HP Model..... 439 8.4.3 HP Models of Protein Folding................ 443 8.5 Summary and Conclusions...................... 446 8.6 Problems............................... 447 8.7 Further Reading........................... 450 8.8 References............................... 451 9 Electrostatics for Salty Solutions 453 9.1 Water as Life s Aether........................ 453 9.2 The Chemistry of Water....................... 455 9.2.1 ph and the Equilibrium Constant............. 455 9.2.2 The Charge on DNA and Proteins............. 456 9.2.3 Salt and Binding....................... 458 9.3 Electrostatics for Salty Solutions.................. 460 9.3.1 An Electrostatics Primer.................. 460 9.3.2 The Charged Life of a Protein............... 471 9.3.3 The Notion of Screening: Electrostatics in Salty Solutions 473 9.3.4 The Poisson-Boltzmann Equation.............. 478 9.3.5 Viruses as Charged Spheres................. 482 9.4 Summary and Conclusion...................... 486 9.5 Problems............................... 486 9.6 Further Reading........................... 492 9.7 References............................... 492 10 Beam Theory: Architecture for Cells and Skeletons 495 10.1 Beams are Everywhere: From Flagella to the Cytoskeleton.... 496 10.2 Geometry and Energetics of Beam Deformation.......... 497 10.2.1 Stretch, Bend and Twist................... 497 10.2.2 Beam Theory and the Persistence Length: Stiffness is Relative.............................. 503 10.2.3 Elasticity and Entropy: The Worm-like Chain....... 506 10.3 The Mechanics of Transcriptional Regulation: DNA Looping Redux508

8 CONTENTS 10.3.1 The Lac Operon and Other Looping Systems....... 509 10.3.2 Energetics of DNA Looping................. 511 10.3.3 Putting it all together: The J Factor............ 511 10.4 DNA Packing: From Viruses to Eukaryotes............ 513 10.4.1 The Problem of Viral DNA Packing............ 516 10.4.2 Constructing the Nucleosome................ 525 10.4.3 Equilibrium Accessibility of Nucleosomal DNA...... 528 10.5 The Cytoskeleton and Beam Theory................ 533 10.5.1 The Cellular Interior: A Structural Perspective...... 535 10.5.2 Stiffness of Cytoskeletal Filaments............. 538 10.5.3 Cytoskeletal Buckling.................... 542 10.5.4 Estimate of the Buckling Force............... 543 10.6 Beams and Biotechnology...................... 545 10.6.1 Biofunctionalized Cantilevers and Molecular Recognition. 546 10.7 Summary and Conclusions...................... 549 10.8 Appendix: The Mathematics of the Worm-Like Chain...... 550 10.9 Problems............................... 552 10.10Further Reading........................... 556 10.11References............................... 558 11 Biological Membranes: Life in Two Dimensions 561 11.1 The Nature of Biological Membranes................ 561 11.1.1 Cells and Membranes.................... 561 11.1.2 The Chemistry and Shape of Lipids............ 566 11.1.3 The Liveliness of Membranes................ 570 11.2 On the Springiness of Membranes.................. 576 11.2.1 An Interlude on Membrane Geometry........... 577 11.2.2 Free Energy of Membrane Deformation.......... 582 11.3 Structure, Energetics and Function of Vesicles........... 587 11.3.1 Measuring Membrane Stiffness............... 587 11.3.2 Membrane Pulling...................... 591 11.3.3 Vesicles in Cells........................ 595 11.3.4 Fusion and Fission...................... 602 11.4 Membranes and Shape........................ 602 11.4.1 The Shapes of Organelles.................. 604 11.4.2 The Shapes of Cells..................... 607 11.5 The Active Membrane........................ 611 11.5.1 Mechanosensitive Ion Channels and Membrane Elasticity 611 11.5.2 Elastic Deformations of Membranes Produced by Proteins 611 11.5.3 One-Dimensional Solution for MscL............ 614 11.6 Summary and Conclusions...................... 621 11.7 Problems............................... 622 11.8 Further Reading........................... 627 11.9 References............................... 629

CONTENTS 9 III Life in Motion 633 12 The Mathematics of Water 635 12.1 Putting Water in its Place...................... 635 12.2 Hydrodynamics of Water and Other Fluids............ 636 12.2.1 Water as a continuum.................... 636 12.2.2 What Can Newton Tell Us?................. 637 12.2.3 F = ma For Fluids...................... 639 12.2.4 The Newtonian Fluid and the Navier-Stokes Equations.. 643 12.3 The River Within: Fluid Dynamics of Blood........... 644 12.3.1 Boats in the River: Leukocyte Rolling and Adhesion... 647 12.4 The Low-Reynolds Number World................. 648 12.4.1 Stokes Flow: Consider a Spherical Bacterium....... 648 12.4.2 Stokes Drag in Single Molecule Experiments........ 653 12.4.3 Dissipative Time Scales and the Reynolds Number.... 654 12.4.4 Fish Gotta Swim, Birds Gotta Fly and Bacteria Gotta Swim Too........................... 656 12.4.5 Centrifugation and Sedimentation: Spin it Down..... 659 12.5 Summary and Conclusions...................... 663 12.6 Problems............................... 663 12.7 Further Reading........................... 666 12.8 References............................... 668 13 A Statistical View of Biological Dynamics 669 13.1 Diffusion in the Cell......................... 669 13.1.1 Active versus Passive Transport............... 670 13.1.2 Biological Distances Measured in Diffusion Times..... 672 13.1.3 Random Walk Redux.................... 676 13.2 Concentration Fields and Diffusive Dynamics........... 678 13.2.1 Diffusion by Summing Over Microtrajectories....... 682 13.2.2 Solutions and properties of the diffusion equation..... 689 13.2.3 FRAP and FCS....................... 690 13.2.4 Drunks on a Hill: The Smoluchowski Equation...... 695 13.2.5 The Einstein Relation.................... 696 13.3 Diffusion to capture......................... 698 13.3.1 Modeling the cell signaling problem............ 699 13.3.2 A Universal Rate for Diffusion-Limited Chemical Reactions.............................. 704 13.4 Summary and Conclusions...................... 705 13.5 Problems............................... 706 13.6 Further Reading........................... 707 13.7 References............................... 709

10 CONTENTS 14 Life in Crowded and Disordered Environments 711 14.1 Crowding, Linkage and Entanglement............... 711 14.1.1 The Cell is Crowded..................... 712 14.1.2 Macromolecular Networks: The Cytoskeleton and Beyond 713 14.1.3 Crowding on Membranes.................. 715 14.1.4 Consequences of Crowding.................. 717 14.2 Equilibria in Crowded Environments................ 721 14.2.1 Crowding and binding.................... 721 14.2.2 Osmotic Pressures in Crowded Solutions.......... 725 14.2.3 Depletion Forces: Order from Disorder........... 729 14.2.4 Excluded Volume and Polymers............... 736 14.3 Crowded Dynamics.......................... 740 14.3.1 Crowding and Reaction Rates................ 740 14.3.2 Diffusion in Crowded Environments............ 742 14.4 Summary and Conclusions...................... 745 14.5 Problems............................... 745 14.6 Further Reading........................... 747 14.7 References............................... 748 15 Rate Equations and Dynamics in the Cell 751 15.1 Biological Statistical Dynamics: A First Look........... 751 15.1.1 Cells as Chemical Factories................. 752 15.1.2 Dynamics of the Cytoskeleton................ 753 15.2 A Chemical Picture of Biological Dynamics............ 758 15.2.1 The Rate Equation Paradigm................ 758 15.2.2 All Good Things Must End................. 760 15.2.3 A Single Molecule View of Degradation: Statistical Mechanics Over Trajectories.................. 762 15.2.4 Bimolecular Reactions.................... 768 15.2.5 Dynamics of Ion Channels as a Case Study........ 771 15.2.6 Rapid equilibrium...................... 776 15.2.7 Michaelis-Menten and Enzyme Kinetics.......... 783 15.3 The Cytoskeleton is Always Under Construction......... 786 15.3.1 The Eukaryotic Cytoskeleton................ 788 15.3.2 The Curious Case of the Bacterial Cytoskeleton...... 789 15.4 Simple Models of Cytoskeletal Polymerization........... 792 15.4.1 The Equilibrium Polymer.................. 795 15.4.2 Rate Equation Description of Cytoskeletal Polymerization 800 15.4.3 Nucleotide Hydrolysis and Cytoskeletal Polymerization. 806 15.4.4 Dynamic Instability: A Toy Model of the Cap....... 808 15.5 Summary and Conclusions...................... 814 15.6 Problems............................... 815 15.7 Further Reading........................... 818 15.8 References............................... 819

CONTENTS 11 16 Dynamics of Molecular Motors 821 16.1 The Dynamics of Molecular Motors: Life in the Noisy Lane... 821 16.1.1 Translational Motors: Beating the Diffusive Speed Limit. 824 16.1.2 Rotary Motors........................ 835 16.1.3 Polymerization Motors: Pushing By Growing....... 839 16.1.4 Translocation Motors: Pushing by Pulling......... 841 16.2 Rectified Brownian Motion and Molecular Motors......... 843 16.2.1 The Random Walk Yet Again................ 844 16.2.2 The One-state Model.................... 847 16.2.3 Motor stepping from a free energy perspective...... 856 16.2.4 The Two-state model.................... 861 16.2.5 More General Motor Models................ 868 16.2.6 Coordination of Motor Protein Activity.......... 869 16.2.7 Rotary Motors........................ 875 16.3 Polymerization and Translocation as Motor Action........ 878 16.3.1 The Polymerization Ratchet................. 878 16.3.2 Force Generation by Growth................ 887 16.3.3 The Translocation Ratchet................. 890 16.4 Summary and Conclusions...................... 895 16.5 Problems............................... 895 16.6 Further Reading........................... 898 16.7 References............................... 900 17 Biological Electricity and the Hodgkin-Huxley Model 903 17.1 The Role of Electricity in Cells................... 903 17.2 The Charge State of the Cell.................... 905 17.2.1 The Electrical Status of Cells and Their Membranes... 905 17.2.2 Electrochemical Equilibrium and the Nernst Equation.. 905 17.3 Membrane Permeability: Pumps and Channels.......... 908 17.3.1 Ion Channels and Membrane Permeability......... 910 17.3.2 Maintaining a Nonequilibrium Charge State........ 915 17.4 The Action Potential......................... 918 17.4.1 Membrane Depolarization: The Membrane as a Bistable Switch............................. 918 17.4.2 The Cable Equation..................... 931 17.4.3 Depolarization waves..................... 933 17.4.4 Spikes............................. 937 17.4.5 Hodgkin-Huxley and Membrane Transport........ 939 17.5 Summary and Conclusions...................... 941 17.6 Problems............................... 941 17.7 Further Reading........................... 944 17.8 References............................... 945

12 CONTENTS IV The Meaning of Life 947 18 Sequences, Specificity and Evolution 949 18.1 Biological Information........................ 950 18.1.1 Why Sequences?....................... 951 18.1.2 Genomes and Sequences by the Numbers......... 952 18.2 Sequence Alignment and Homology................. 954 18.2.1 The HP Model as a Coarse-Grained Model for Bioinformatics............................. 960 18.2.2 Scoring Success........................ 962 18.3 Sequences and Evolution....................... 973 18.3.1 Evolution by the Numbers: Hemoglobin as a Case Study in Sequence Alignment.................... 975 18.3.2 Evolution and Drug Resistance............... 979 18.3.3 The Evolution of Viruses.................. 982 18.3.4 Phylogenetic Trees...................... 984 18.4 The Molecular Basis of Fidelity................... 987 18.4.1 Keeping it Specific: Beating Thermodynamic Specificity. 988 18.5 Summary and Conclusions...................... 995 18.6 Problems............................... 996 18.7 Further Reading........................... 1000 18.8 References............................... 1002 19 Network Organization in Space and Time 1005 19.1 Chemical and Informational Organization in the Cell....... 1005 19.2 Genetic Networks: Doing the Right Thing at the Right Time.. 1012 19.2.1 The Molecular Implementation of Regulation: Promoters, Activators and Repressors.................. 1013 19.2.2 The Mathematics of Recruitment and Rejection...... 1016 19.2.3 Transcriptional Regulation By the Numbers: Binding Energies and Equilibrium Constants.............. 1026 19.2.4 A Simple Statistical Mechanics Model of Positive and Negative Regulation..................... 1027 19.2.5 The lac Operon........................ 1028 19.3 Regulatory Dynamics........................ 1038 19.3.1 The Dynamics of RNA Polymerase and the Promoter.. 1038 19.3.2 Genetic Switches: Natural and Synthetic......... 1039 19.3.3 Genetic Networks That Oscillate: The Repressilator... 1045 19.3.4 Putting Space in the Model: Reaction-Diffusion Models. 1051 19.4 Cellular Fast Response: Signaling.................. 1052 19.4.1 Bacterial Chemotaxis.................... 1053 19.4.2 Biochemistry on a Leash................... 1058 19.5 Summary and Conclusions...................... 1065 19.6 Appendix: Stability Analysis for the Genetic Switch....... 1066 19.7 Problems............................... 1068 19.8 Further Reading........................... 1072

CONTENTS 13 19.9 References............................... 1073 20 Whither Physical Biology? 1075 20.1 Quantitative Data Demands Quantitative Models......... 1075 20.2 Wrong Again............................. 1079 20.3 Order-of-Magnitude Biology and Beyond.............. 1080 20.4 Difficulties on Theory....................... 1081 20.5 A Charge to the Reader....................... 1086 20.6 Further Reading........................... 1087 20.7 References............................... 1088