Physical Biology of the Cell

Transcription

1 [!_ PH'-{ \to ~UJIJ Physical Biology of the Cell Second Edition Rob Phillips Jane Kondev Julie Theriot Hernan G. Garcia Illustrated by Nigel Orme f'o Garland Science \...:J...:> Taylor & Francis Group LONDON AND NEW YORK

2 Contents Preface Acknowledgments Special Sections Map of the Maps vii xiii xxix XXX Pf.\fn l Chapter 1: Chapter 2: Chapter 3: Chapter 4: THE FACTS OF LIFE Why: Biology by the Numbers What and Where: Construction Plans for Cells and Organisms When: Stopwatches at Many Scales Who: "Bless the Little Beasties" PART 2 LIFE AT REST Chapter 5: Chapter 6: Chapter 7: Chapter 8: Chapter 9: Chapter 10: Chapter 11: Mechanical and Chemical Equilibrium in the Living Cell Entropy Rules! Two-State Systems: From Ion Channels to Cooperative Binding Random Walks and the Structure of Macromolecules Electrostatics for Salty Solutions Beam Theory: Architecture for Cells and Skeletons Biological Membranes: Life in Two Dimensions PART 3 Chapter 12: Chapter 13: Chapter 14: Chapter 15: Chapter 16: Chapter 17: Chapter 18: LIFE IN MOTION The Mathematics of Water A Statistical View of Biological Dynamics Life in Crowded and Disordered Environments Rate Equations and Dynamics in the Cell Dynamics of Molecular Motors Biological Electricity and the Hodgkin-Huxley Model Light and Life CONTENTS xvii

3 PART 4 Chapter 19: Chapter 20: Chapter 21: Chapter 22: Index THE MEANING OF LIFE Organization of Biological Networks Biological Patterns: Order in Space and Time Sequences, Specificity, and Evolution Whither Physical Biology? xviii CONTENTS

4 Contents in Detail Preface Acknowledgments Special Sections Map of the Maps PART 1 THE FACTS OF LIFE 1 Chapter 1 Why: Biology by the Numbers 1.1 BIOLOGICAL CARTOGRAPHY 1.2 PHYSICAL BIOLOGY OF THE CELL Model Building Requires a Substrate of Biological Facts and Physical (or Chemical) Principles 1.3 THE STUFF OF LIFE Organisms Are Constructed from Four Great Classes of Macromolecules Nucleic Acids and Proteins Are Polymer Languages with Different Alphabets vii xiii xxix XXX CELLS AND STRUCTURES WITHIN THEM Cells: A Rogue's Gallery 52 Cells Come in a Wide Variety of Shapes and Sizes and with a Huge Range of Functions 52 Cells from Humans Have a Huge Diversity of Structure and Function The Cellular Interior: Organelles Macromolecular Assemblies: The Whole is Greater than the Sum of the Parts 63 Macromolecules Come Together to Form Assemblies 63 Helical Motifs Are Seen Repeatedly in Molecular Assemblies 64 Macromolecular Assemblies Are Arranged in Superstructures 65 Viruses as Assemblies 66 The Molecular Architecture of Cells: From Protein Data Bank (PDB) Files to Ribbon Diagrams 69 Macromolecular Structure Is Characterized Fundamentally by Atomic Coordinates 69 Chemical Groups Allow Us to Classify Parts of the Structure of Macromolecules MODEL BUILDING IN BIOLOGY Models as Idealizations Biological Stuff Can Be Idealized Using Many Different Physical Models Cartoons and Models Biological Cartoons Select Those Features of the Problem Thought to Be Essential Quantitative Models Can Be Built by Mathematicizing the Cartoons 1.5 QUANTITATIVE MODELS AND THE POWER OF IDEALIZATION On the Springiness of Stuff The Toolbox of Fundamental Physical Models The Unifying Ideas of Biology Mathematical Toolkit The Role of Estimates On Being Wrong Rules of Thumb: Biology by the Numbers 1.6 SUMMARY AND CONCLUSIONS 1.7 FURTHER READING 1.8 REFERENCES Chapter 2 What and Where: Construction Plans for Cells and Organisms 2.1 AN ODE TO E. COLI The Bacterial Standard Ruler The Bacterium E. coli Will Serve as Our Standard Ruler Taking the Molecular Census The Cellular Interior Is Highly Crowded, with Mean Spacings Between Molecules That Are Comparable to Molecular Dimensions Looking Inside Cells Where Does E. coli Fit? Biological Structures Exist Over a Huge Range of Scales TELESCOPING UP IN SCALE: CELLS DON'T GO IT ALONE 72 Multicellularity as One of Evolution's Great Inventions 73 Bacteria Interact to Form Colonies such as Biofilms 73 Teaming Up in a Crisis: Lifestyle of Dictyostelium discoideum 75 Multicellular Organisms Have Many Distinct Communities of Cells 76 Cellular Structures from Tissues to Nerve Networks 77 One Class of Multicellular Structures is the Epithelial Sheets 77 Tissues Are Collections of Cells and Extracellular Matrix Nerve Cells Form Complex, Multicellular Complexes Multicellular Organisms Cells Differentiate During Development Leading to Entire Organisms The Cells of the Nematode Worm, Caenorhabditis Elegans, Have Been Charted, Yielding a Cell-by-Cell Picture of the Organism Higher-Level Structures Exist as Colonies of Organisms SUMMARY AND CONCLUSIONS PROBLEMS FURTHER READING REFERENCES 85 Chapter 3 When: Stopwatches at Many Scales 3.1 THE HIERARCHY OF TEMPORAL SCALES The Pageant of Biological Processes Biological Processes Are Characterized by a Huge Diversity of Time Scales The Evolutionary Stopwatch xix

5 3.1.3 The Cell Cycle and the Standard Clock 99 Structural Biology Has Its Roots in the The E. coli Cell Cycle Will Serve as Our Standard Determination of the Structure of Hemoglobin 145 Stopwatch Hemoglobin and Molecular Models of Disease Three Views of Time in Biology The Rise of Allostery and Cooperativity PROCEDURAL TIME BACTERIOPHAGES AND MOLECULAR BIOLOGY The Machines (or Processes) of the Central Dogma Bacteriophages and the Origins of Molecular Biology 148 The Central Dogma Describes the Processes Bacteriophages Have Sometimes Been Called the Whereby the Genetic Information Is Expressed "Hydrogen Atoms of Biology" 148 Chemically 107 Experiments on Phages and Their Bacterial Hosts The Processes of the Central Dogma Are Carried Out Demonstrated That Natural Selection Is Operative in by Sophisticated Molecular Machines 108 Microscopic Organisms Developing Embryos Divide on a Regular Schedule Nature of Genetic Material and Elucidated One of the Dictated by an Internal Clock 111 Mechanisms of Viral DNA Entry into Cells 149 Clocks and Oscillators 110 The Hershey-Chase Experiment Both Confirmed the Diurnal Clocks Allow Cells and Organisms to Be on Experiments on Phage T4 Demonstrated the Time Everyday 111 Sequence Hypothesis of Collinearity of DNA and Proteins RELATIVE TIME 114 The Triplet Nature of the Genetic Code and DNA Checkpoints and the Cell Cycle 115 Sequencing Were Carried Out on Phage Systems 150 The Eukaryotic Cell Cycle Consists of Four Phases Phages Were Instrumental in Elucidating the 151 Involving Molecular Synthesis and Organization 115 Existence of mrna Measuring Relative Time 117 General Ideas about Gene Regulation Were Learned Genetic Networks Are Collections of Genes from the Study of Viruses as a Model System 152 Whose Expression Is Interrelated Bacteriophages and Modern Biophysics 153 The Formation of the Bacterial Flagellum Is Many Single- Molecule Studies of Molecular Motors Intricately Organized in Space and Time 119 Have Been Performed on Motors from Bacteriophages Killing the Cell: The Life Cycles of Viruses A TALE OF TWO CELLS: E. COLI AS A MODEL SYSTEM 154 Viral Life Cycles Include a Series of Self-Assembly Processes Bacteria and Molecular Biology E. coli and the Central Dogma The Process of Development 122 The Hypothesis of Conservative Replication Has 3.4 MANIPULATED TIME 125 Falsifiable Consequences Chemical Kinetics and Enzyme Turnover 125 Extracts from E. coli Were Used to Perform In Vitro Beating the Diffusive Speed Limit 126 Synthesis of DNA, mrna, and Proteins 157 Diffusion Is the Random Motion of Microscopic The lac Operon as the "Hydrogen Atom" of Genetic Particles in Circuits 157 Solution 127 Gene Regulation in E. coli Serves as a Model for Diffusion Times Depend upon the Length Scale 127 Genetic Circuits in General 157 Diffusive Transport at the Synaptic Junction Is the The lac Operon Is a Genetic Network That Controls Dynamical Mechanism for Neuronal Communication 128 the Production of the Enzymes Responsible for Digesting the Sugar Lactose 158 Molecular Motors Move Cargo over Large Distances in a Directed Way Signaling and Motility: The Case of Bacterial Chemotaxis 159 Membrane-Bound Proteins Transport Molecules E. coli Has Served as a Model System for the from One Side of a Membrane to the Other 130 Analysis of Cell Motility Beating the Replication Limit Eggs and Spores: Planning for the Next 4.5 YEAST: FROM BIOCHEMISTRY TO THE CELL CYCLE 161 Generation 132 Yeast Has Served as a Model System Leading to 3.5 SUMMARY AND CONCLUSIONS 133 Insights in Contexts Ranging from Vitalism to the Functioning of Enzymes to Eukaryotic Gene 3.6 PROBLEMS 133 Regulation FURTHER READING Yeast and the Rise of Biochemistry REFERENCES Dissecting the Cell Cycle Deciding Which Way Is Up: Yeast and Polarity 164 Chapter 4 Who: "Bless the Little Beasties" Dissecting Membrane Traffic Genomics and Proteomics CHOOSING A GRAIN OF SAND 137 Modern Genetics Began with the Use of Peas as a 4.6 FLIES AND MODERN BIOLOGY 170 Model System Flies and the Rise of Modern Genetics Biochemistry and Genetics 138 Drosophila melanogaster Has Served as a Model System for Studies Ranging from Genetics to 4.2 HEMOGLOBIN AS A MODEL PROTEIN 143 Development to the Functioning of the Brain and Hemoglobin, Receptor-Ligand Binding, and the Even Behavior 170 Other Bohr How the Fly Got His Stripes 171 The Binding of Oxygen to Hemoglobin Has Served as a Model System for Ligand- Receptor Interactions 4.7 OF MICE AND MEN 173 More Generally THE CASE FOR EXOTICA 174 Quantitative Analysis of Hemoglobin Is Based upon Specialists and Experts 174 Measuring the Fractional Occupancy of the The Squid Giant Axon and Biological Electricity 175 Oxygen-Binding Sites as a Function of Oxygen There Is a Steady-State Potential Difference Across Pressure 144 the Membrane of Nerve Cells Hemoglobin and the Origins of Structural Biology 144 Nerve Cells Propagate Electrical Signals and Use The Study of the Mass of Hemoglobin Was Central in Them to Communicate with Each Other 176 the Development of Centrifugation Exotica Toolkit 178 XX

6 p SUMMARY AND CONCLUSIONS PROBLEMS FURTHER READING REFERENCES J:' PART 2 LIFE AT REST Chapter 5 Mechanical and Chemical Equilibrium in the Living Cell ENERGY AND THE LIFE OF CELLS The Interplay of Deterministic and Thermal Forces 189 Thermal Jostling of Particles Must Be Accounted for in Biological Systems Constructing the Cell: Managing the Mass and Energy Budget of the Cell BIOLOGICAL SYSTEMS AS MINIMIZERS Equilibrium Models for Out of Equilibrium Systems 200 Equilibrium Models Can Be Used for Nonequilibrium Problems if Certain Processes Happen Much Faster Than Others Proteins in "Equilibrium" 202 Protein Structures are Free-Energy Minimizers Cells in "Equilibrium" Mechanical Equilibrium from a Minimization Perspective 204 The Mechanical Equilibrium State is Obtained by Minimizing the Potential Energy THE MATHEMATICS OF SUPERLATIVES The Mathematization of Judgement: Functions and Functionals 209 Functionals Deliver a Number for Every Function They Are Given The Calculus of Superlatives 211 Finding the Maximum and Minimum Values of a Function Requires That We Find Where the Slope of the Function Equals Zero CONFIGURATIONAL ENERGY 214 In Mechanical Problems, Potential Energy Determines the Equilibrium Structure Hooke's Law: Actin to Lipids 216 There is a Linear Relation Between Force and Extension of a Beam 216 The Energy to Deform an Elastic Material is a Quadratic Function of the Strain STRUCTURES AS FREE-ENERGY MINIMIZERS 219 The Entropy is a Measure of the Microscopic Degeneracy of a Macroscopic State Entropy and Hydrophobicity 222 Hydrophobicity Results from Depriving Water Molecules of Some of Their Configurational Entropy 222 Amino Acids Can Be Classified According to Their Hydrophobicity 224 When in Water, Hydrocarbon Tails on Lipids Have an Entropy Cost Gibbs and the Calculus of Equilibrium 225 Thermal and Chemical Equilibrium are Obtained by Maximizing the Entropy Departure from Equilibrium and Fluxes Structure as a Competition 228 Free Energy Minimization Can Be Thought of as an Alternative Formulation of Entropy Maximization An Ode to llg 230 The Free Energy Reflects a Competition Between Energy and Entropy SUMMARY AND CONCLUSIONS APPENDIX: THE EULER-LAGRANGE EQUATIONS, FINDING THE SUPERLATIVE Finding the Extrema of Functionals Is Carried Out Using the Calculus of Variations The Euler-Lagrange Equations Let Us Minimize Functionals by Solving Differential Equations PROBLEMS FURTHER READING REFERENCES Chapter 6 Entropy Rules! THE ANALYTICAL ENGINE OF STATISTICAL MECHANICS 237 The Probability of Different Microstates Is Determined by Their Energy A First Look at Ligand-Receptor Binding The Statistical Mechanics of Gene Expression: RNA Polymerase and the Promoter 244 A Simple Model of Gene Expression Is to Consider the Probability of RNA Polymerase Binding at the Promoter 245 Most Cellular RNA Polymerase Molecules Are Bound to DNA 245 The Binding Probability of RNA Polymerase to Its Promoter Is a Simple Function of the Number of Polymerase Molecules and the Binding Energy Classic Derivation of the Boltzmann Distribution 248 The Boltzmann Distribution Gives the Probability of Microstates for a System in Contact with a Thermal Reservoir Boltzmann Distribution by Counting 250 Different Ways of Partitioning Energy Among Particles Have Different Degeneracies Boltzmann Distribution by Guessing 253 Maximizing the Entropy Corresponds to Making a Best Guess When Faced with Limited Information 253 Entropy Maximization Can Be Used as a Tool for Statistical Inference 25 S The Boltzmann Distribution is the Maximum Entropy Distribution in Which the Average Energy is Prescribed as a Constraint ON BEING IDEAL Average Energy of a Molecule in a Gas 259 The Ideal Gas Entropy Reflects the Freedom to Rearrange Molecular Positions and Velocities Free Energy of Dilute Solutions 262 The Chemical Potential of a Dilute Solution Is a Simple Logarithmic Function of the Concentration Osmotic Pressure as an Entropic Spring 264 Osmotic Pressure Arises from Entropic Effects 264 Viruses, Membrane-Bound Organelles, and Cells Are Subject to Osmotic Pressure 265 Osmotic Forces Have Been Used to Measure the lnterstrand Interactions of DNA THE CALCULUS OF EQUILIBRIUM APPLIED: LAW OF MASS ACTION Law of Mass Action and Equilibrium Constants 267 Equilibrium Constants are Determined by Entropy Maximization APPLICATIONS OF THE CALCULUS OF EQUILIBRIUM A Second Look at Ligand-Receptor Binding Measuring Ligand-Receptor Binding Beyond Simple Ligand-Receptor Binding: The Hill Function ATP Power 274 The Energy Released in ATP Hydrolysis Depends Upon the Concentrations of Reactants and Products 275 xxi

7 6.5 SUMMARY AND CONCLUSIONS 276 The Probability of a Given Macromolecular State 6.6 PROBLEMS 276 Depends Upon Its Microscopic Degeneracy FURTHER READING 278 Entropy Determines the Elastic Properties of Polymer Chains 6.8 REFERENCES The Persistence Length Is a Measure of the Length Scale Over Which a Polymer Remains Roughly Chapter 7 Two-State Systems: From lon Straight How Big Is a Genome? 321 Channels to Cooperative Binding The Geography of Chromosomes MACROMOLECULES WITH MULTIPLE STATES 281 Genetic Maps and Physical Maps of Chromosomes The Internal State Variable Idea 281 Describe Different Aspects of Chromosome The State of a Protein or Nucleic Acid Can Be Structure 322 Characterized Mathematically Using a State Different Structural Models of Chromatin Are Variable 282 Characterized by the Linear Packing Density ion Channels as an Example of Internal State of DNA 323 Variables 286 Spatial Organization of Chromosomes Shows The Open Probability (IT) of an ion Channel Can Be Elements of Both Randomness and Order 324 Computed Using Statistical Mechanics 287 Chromosomes Are Tethered at Different Locations 325 Chromosome Territories Have Been Observed 7.2 STATE VARIABLE DESCRIPTION OF BINDING 289 in Bacterial Cells The Gibbs Distribution: Contact with a Particle Chromosome Territories in Vibrio cholerae Can Be Reservoir 289 Explored Using Models of Polymer Confinement The Gibbs Distribution Gives the Probability of and Tethering 328 Microstates for a System in Contact with a Thermal DNA Looping: From Chromosomes to Gene and Particle Reservoir 289 Regulation Simple Ligand-Receptor Binding Revisited 291 The Lac Repressor Molecule Acts Mechanistically Phosphorylation as an Example of Two Internal by Forming a Sequestered Loop in DNA 334 State Variables 292 Looping of Large DNA Fragments Is Dictated Phosphorylation Can Change the Energy Balance by the Difficulty of Distant Ends Finding Each Other 334 Between Active and Inactive States 293 Chromosome Conformation Capture Reveals Two-Component Systems Exemplify the Use of the Geometry of Packing of Entire Genomes Phosphorylation in Signal Transduction 295 in Cells Hemoglobin as a Case Study in Cooperativity 298 The Binding Affinity of Oxygen for Hemoglobin 8.3 THE NEW WORLD OF SINGLE-MOLECULE Depends upon Whether or Not Other Oxygens Are MECHANICS 337 Already Bound 298 Single-Molecule Measurement Techniques Lead to A Toy Model of a Dime ric Hemoglobin (Dimoglobin) Force Spectroscopy 337 Illustrate the Idea of Cooperativity Force-Extension Curves: A New Spectroscopy 339 The Monod-Wyman-Changeux (MWC) Model Different Macromolecules Have Different Force Provides a Simple Example of Cooperative Binding 300 Signatures When Subjected to Loading 339 Statistical Models of the Occupancy of Hemoglobin Random Walk Models for Force-Extension Curves 340 Can Be Written Using Occupation Variables 301 The Low-Force Regime in Force-Extension Curves There is a Logical Progression of Increasingly Can Be Understood Using the Random Walk Model 340 Complex Binding Models for Hemoglobin PROTEINS AS RANDOM WALKS 344 ION CHANNELS REVISITED: LIGAND-GATED Compact Random Walks and the Size of Proteins 345 CHANNELS AND THE MWC MODEL 305 The Compact Nature of Proteins Leads to an 7.4 SUMMARY AND CONCLUSIONS 308 Estimate of Their Size PROBLEMS Hydrophobic and Polar Residues: The HP Model FURTHER READING 310 The HP Model Divides Amino Acids into Two 7.7 REFERENCES Classes: Hydrophobic and Polar HP Models of Protein Folding 348 Chapter 8 Random Walks and the 8.5 SUMMARY AND CONCLUSIONS 351 Structure of Macromolecules PROBLEMS FURTHER READING WHAT IS A STRUCTURE: PDB OR Rc? Deterministic versus Statistical Descriptions of 8.8 REFERENCES 353 Structure 312 PDB Files Reflect a Deterministic Description of Chapter 9 Electrostatics for Salty Macromolecular Structure 312 Statistical Descriptions of Structure Emphasize Solutions 355 Average Size and Shape Rather Than Atomic 9.1 WATER AS LIFE'S AETHER 355 Coordinates THE CHEMISTRY OF WATER MACROMOLECULES AS RANDOM WALKS ph and the Equilibrium Constant 358 Random Walk Models of Macromolecules View Dissociation of Water Molecules Reflects a Them as Rigid Segments Connected by Hinges 312 Competition Between the Energetics of Binding A Mathematical Stupor 313 and the Entropy of Charge Liberation 358 In Random Walk Models of Polymers, Every The Charge on DNA and Proteins 359 Macromolecular Configuration Is Equally Probable 313 The Charge State of Biopolymers Depends The Mean Size of a Random Walk Macromolecule upon the ph of the Solution 3S9 Scales as the Square Root of the Number of Different Amino Acids Have Different Charge States 359 Segments,..fN Salt and Binding 360 xxii

8 ELECTROSTATICS FOR SALTY SOLUTIONS DNA PACKING: FROM VIRUSES TO EUKARYOTES An Electrostatics Primer 361 The Packing of DNA in Viruses and Cells Requires A Charge Distribution Produces an Electric Field Enormous Volume Compaction 398 Throughout Space The Problem of Viral DNA Packing 400 The Flux of the Electric Field Measures the Density Structural Biologists Have Determined the Structure of Electric Field Lines 363 of Many Parts in the Viral Parts List 400 The Electrostatic Potential Is an Alternative Basis The Packing of DNA in Viruses Results in a for Describing the Electrical State of a System 364 Free-Energy Penalty 402 There Is an Energy Cost Associated With Assembling A Simple Model of DNA Packing in Viruses Uses the a Collection of Charges 367 Elastic Energy of Circular Hoops 403 The Energy to Liberate Ions from Molecules Can DNA Self-Interactions Are also Important in Be Comparable to the Thermal Energy 368 Establishing the Free Energy Associated with DNA The Charged Life of a Protein 369 Packing in Viruses The Notion of Screening: Electrostatics in Salty DNA Packing in Viruses Is a Competition Between Solutions 370 Elastic and Interaction Energies 406 Ions in Solution Are Spatially Arranged to Shield Constructing the Nucleosome 407 Charged Molecules Such as DNA 370 Nucleosome Formation Involves Both Elastic The Size of the Screening Cloud Is Determined Deformation and Interactions Between Histones by a Balance of Energy and Entropy of the and DNA 408 Surrounding Ions Equilibrium Accessibility of Nucleosomal DNA The Poisson-Boltzmann Equation 374 The Equilibrium Accessibility of Sites within the The Distribution of Screening Ions Can Be Found Nucleosome Depends upon How Far They Are by Minimizing the Free Energy 374 from the Unwrapped Ends 409 The Screening Charge Decays Exponentially Around Macromolecules in Solution THE CYTOSKELETON AND BEAM THEORY Viruses as Charged Spheres 377 Eukaryotic Cells Are Threaded by Networks of Filaments SUMMARY AND CONCLUSION The Cellular Interior: A Structural Perspective PROBLEMS 380 Prokaryotic Cells Have Proteins Analogous to the 9.6 FURTHER READING 382 Eukaryotic Cytoskeleton REFERENCES Stiffness of Cytoskeletal Filaments 416 The Cytoskeleton Can Be Viewed as a Collection of Elastic Beams 416 Chapter 10 Beam Theory: Architecture Cytoskeletal Buckling 419 A Beam Subject to a Large Enough Force Will Buckle for Cells and Skeletons Estimate of the Buckling Force BEAMS ARE EVERYWHERE: FROM FLAGELLA TO THE Beam Buckling Occurs at Smaller Forces for Longer CYTOSKELETON 383 Beams 420 One-Dimensional Structural Elements Are the Basis of Much of Macromolecular and Cellular 10.6 SUMMARY AND CONCLUSIONS 421 Architecture APPENDIX: THE MATHEMATICS OF THE WORM-LIKE CHAIN GEOMETRY AND ENERGETICS OF BEAM 10.8 PROBLEMS 424 DEFORMATION Stretch, Bend, and Twist FURTHER READING 426 Beam Deformations Result in Stretching, Bending, REFERENCES 426 and Twisting 385 A Bent Beam Can Be Analyzed as a Collection of Chapter 11 Biological Membranes: Life in Stretched Beams 385 Two Dimensions 427 The Energy Cost to Deform a Beam Is a Quadratic Function of the Strain THE NATURE OF BIOLOGICAL MEMBRANES Beam Theory and the Persistence Length: Stiffness Cells and Membranes 427 is Relative 389 Cells and Their Organelles Are Bound by Complex Thermal Fluctuations Tend to Randomize the Membranes 427 Orientation of Biological Polymers 389 Electron Microscopy Provides a Window on Cellular The Persistence Length Is the Length Over Which a Membrane Structures 429 Polymer Is Roughly Rigid The Chemistry and Shape of Lipids 431 The Persistence Length Characterizes the Membranes Are Built from a Variety of Molecules Correlations in the Tangent Vectors at Different That Have an Ambivalent Relationship with Water 431 Positions Along the Polymer 390 The Shapes of Lipid Molecules Can Induce The Persistence Length Is Obtained by Averaging Spontaneous Curvature on Membranes 436 Over All Configurations of the Polymer The Liveliness of Membranes Elasticity and Entropy: The Worm-Like Chain 392 Membrane Proteins Shuttle Mass Across Membranes 437 The Worm-Like Chain Model Accounts for Both Membrane Proteins Communicate Information the Elastic Energy and Entropy of Polymer Across Membranes 439 Chains 392 Specialized Membrane Proteins Generate ATP 439 Membrane Proteins Can Be Reconstituted in Vesicles THE MECHANICS OF TRANSCRIPTIONAL REGULATION: DNA LOOPING REDUX ON THE SPRINGINESS OF MEMBRANES The lac Operon and Other Looping Systems An Interlude on Membrane Geometry 440 Transcriptional Regulation Can Be Effected Membrane Stretching Geometry Can Be Described by DNA Looping 395 by a Simple Area Function Energetics of DNA Looping 395 Membrane Bending Geometry Can Be Described by Putting It All Together: The J-Factor 396 a Simple Height Function, h(x, y) 441 xxiii

9 Membrane Compression Geometry Can Be Described by a Simple Thickness Function, w(x,y) 444 Membrane Shearing Can Be Described by an Angle Variable, IJ Free Energy of Membrane Deformation 445 There Is a Free-Energy Penalty Associated with Changing the Area of a Lipid Bilayer 445 There Is a Free-Energy Penalty Associated with Bending a Lipid Bilayer 446 There Is a Free-Energy Penalty for Changing the Thickness of a Lipid Bilayer 446 There Is an Energy Cost Associated with the Gaussian Curvature Water as a Continuum Though Fluids Are Composed of Molecules It Is Possible to Treat Them as a Continuous Medium What Can Newton Tell Us? Gradients in Fluid Velocity Lead to Shear Forces F = rna for Fluids The Newtonian Fluid and the Navier-Stokes Equations The Velocity of Fluids at Surfaces Is Zero THE RIVER WITHIN: FLUID DYNAMICS OF BLOOD Boats in the River: Leukocyte Rolling and Adhesion STRUCTURE, ENERGETICS, AND FUNCTION OF VESICLES Measuring Membrane Stiffness 448 Membrane Elastic Properties Can Be Measured by Stretching Vesicles Membrane Pulling Vesicles in Cells 453 Vesicles Are Used for a Variety of Cellular Transport Processes 453 There Is a Fixed Free-Energy Cost Associated with Spherical Vesicles of All Sizes 455 Vesicle Formation Is Assisted by Budding Proteins 456 There Is an Energy Cost to Disassemble Coated Vesicles FUSION AND FISSION Pinching Vesicles: The Story of Dynamin MEMBRANES AND SHAPE The Shapes of Organelles 462 The Surface Area of Membranes Due to Pleating Is So Large That Organelles Can Have Far More Area than the Plasma Membrane The Shapes of Cells 465 The Equilibrium Shapes of Red Blood Cells Can Be Found by Minimizing the Free Energy THE ACTIVE MEMBRANE Mechanosensitive ion Channels and Membrane Elasticity 467 Mechanosensitive ion Channels Respond to Membrane Tension Elastic Deformations of Membranes Produced by Proteins 468 Proteins Induce Elastic Deformations in the Surrounding Membrane 468 Protein-Induced Membrane Bending Has an Associated Free-Energy Cost One-Dimensional Solution for MscL 470 Membrane Deformations Can Be Obtained by Minimizing the Membrane Free Energy 470 The Membrane Surrounding a Channel Protein Produces a Line Tension SUMMARY AND CONCLUSIONS PROBLEMS FURTHER READING REFERENCES 479 PART 3 LIFE IN MOTION 481 Chapter 12 The Mathematics of Water PUTIING WATER IN ITS PLACE HYDRODYNAMICS OF WATER AND OTHER FLUIDS THE LOW REYNOLDS NUMBER WORLD Stokes Flow: Consider a Spherical Bacterium Stokes Drag in Single-Molecule Experiments 498 Stokes Drag Is Irrelevant for Optical Tweezers Experiments Dissipative Time Scales and the Reynolds Number Fish Gatta Swim, Birds Gatta Fly, and Bacteria Gatta Swim Too 500 Reciprocal Deformation of the Swimmer's Body Does Not Lead to Net Motion at Low Reynolds Number Centrifugation and Sedimentation: Spin It Down SUMMARY AND CONCLUSIONS PROBLEMS FURTHER READING REFERENCES 507 Chapter 13 A Statistical View of Biological Dynamics DIFFUSION IN THE CELL Active versus Passive Transport Biological Distances Measured in Diffusion Times 511 The Time It Takes a Diffusing Molecule to Travel a Distance L Grows as the Square of the Distance 51 2 Diffusion Is Not Effective Over Large Cellular Distances 5 I Random Walk Redux CONCENTRATION FIELDS AND DIFFUSIVE DYNAMICS 515 Fick's Law Tells Us How Mass Transport Currents Arise as a Result of Concentration Gradients 51 7 The Diffusion Equation Results from Fick's Law and Conservation of Mass Diffusion by Summing Over Microtrajectories Solutions and Properties of the Diffusion Equation 524 Concentration Profiles Broaden Over Time in a Very Precise Way FRAP and FCS Drunks on a Hill: The Smoluchowski Equation The Einstein Relation DIFFUSION TO CAPTURE Modeling the Cell Signaling Problem 532 Perfect Receptors Result in a Rate of Uptake 4rrOc 0 a 533 A Distribution of Receptors Is Almost as Good as a Perfectly Absorbing Sphere 534 Real Receptors Are Not Always Uniformly Distributed A "Universal" Rate for Diffusion-Limited Chemical Reactions SUMMARY AND CONCLUSIONS PROBLEMS xxiv

10 13.6 I 3.7 FURTHER READING S40 Decay of One Species Corresponds to Growth in the REFERENCES 540 Number of a Second Species 585 I Bimolecular Reactions 586 Chemical Reactions Can Increase the Concentration Chapter 14 Life in Crowded and of a Given Species 586 Disordered Environments 543 Equilibrium Constants Have a Dynamical Interpretation in Terms of Reaction Rates CROWDING, LINKAGE, AND ENTANGLEMENT 543 I Dynamics of lon Channels as a Case Study I The Cell Is Crowded 544 Rate Equations for lon Channels Characterize the Macromolecular Networks: The Cytoskeleton Time Evolution of the Open and Closed Probability 590 and Beyond I Rapid Equilibrium 591 I Crowding on Membranes 547 I Michaelis-Menten and Enzyme Kinetics I.4 Consequences of Crowding Crowding Alters Biochemical Equilibria 548 I 5.3 THE CYTOSKELETON IS ALWAYS UNDER Crowding Alters the Kinetics within Cells 548 CONSTRUCTION 599 I 5.3. I The Eukaryotic Cytoskeleton EQUILIBRIA IN CROWDED ENVIRONMENTS 550 The Cytoskeleton Is a Dynamical Structure That Is Crowding and Binding 550 Always Under Construction 599 Lattice Models of Solution Provide a Simple I The Curious Case of the Bacterial Cytoskeleton 600 Picture of the Role of Crowding in Biochemical Equilibria 5SO I 5.4 SIMPLE MODELS OF CYTOSKELETAL POLYMERIZATION 602 I Osmotic Pressures in Crowded Solutions 552 The Dynamics of Polymerization Can Involve Many Osmotic Pressure Reveals Crowding Effects 552 Distinct Physical and Chemical Effects Depletion Forces: Order from Disorder 554 I 5.4. I The Equilibrium Polymer 604 The Close Approach of Large Particles Excludes Equilibrium Models of Cytoskeletal Filaments Smaller Particles Between Them, Resulting in an Describe the Distribution of Polymer Lengths for Entropic Force 554 Simple Polymers 604 Depletion Forces Can Induce Entropic Ordering! 559 An Equilibrium Polymer Fluctuates in Time 606 I Excluded Volume and Polymers Rate Equation Description of Cytoskeletal Excluded Volume Leads to an Effective Repulsion Polymerization 609 Between Molecules 559 Polymerization Reactions Can Be Described by Rate Self-avoidance Between the Monomers of a Polymer Equations 609 Leads to Polymer Swelling 561 The Time Evolution of the Probability Distribution Case Study in Crowding: How to Make a Helix 563 Pn(t) Can Be Written Using a Rate Equation Crowding at Membranes 565 Rates of Addition and Removal of Monomers Are Often Different on the Two Ends of Cytoskeletal 14.3 CROWDED DYNAMICS 566 Filaments I Crowding and Reaction Rates 566 I Nucleotide Hydrolysis and Cytoskeletal Enzymatic Reactions in Cells Can Proceed Faster Polymerization 614 than the Diffusion Limit Using Substrate ATP Hydrolysis Sculpts the Molecular Interface, Channeling 566 Resulting in Distinct Rates at the Ends of Protein Folding Is Facilitated by Chaperones 567 Cytoskeletal Filaments 614 I Diffusion in Crowded Environments Dynamic Instability: A Toy Model of the Cap 615 A Toy Model of Dynamic Instability Assumes That I 4.4 SUMMARY AND CONCLUSIONS 569 Catastrophe Occurs When Hydrolyzed Nucleotides I 4.5 PROBLEMS 569 Are Present at the Growth Front FURTHER READING REFERENCES 571 I 5.5 SUMMARY AND CONCLUSIONS 618 I 5.6 PROBLEMS 619 Chapter 15 Rate Equations and 15.7 FURTHER READING 621 Dynamics in the Cell REFERENCES BIOLOGICAL STATISTICAL DYNAMICS: A FIRST Chapter 16 Dynamics of Molecular LOOK S73 Motors 623 I 5. I. I Cells as Chemical Factories 574 I Dynamics of the Cytoskeleton THE DYNAMICS OF MOLECULAR MOTORS: LIFE IN THE NOISY LANE A CHEMICAL PICTURE OF BIOLOGICAL DYNAMICS I. I Translational Motors: Beating the Diffusive Speed I The Rate Equation Paradigm 579 Limit 625 Chemical Concentrations Vary in Both Space and The Motion of Eukaryotic Cilia and Flagella Is Driven Time 580 by Translational Motors 628 Rate Equations Describe the Time Evolution of Muscle Contraction Is Mediated by Myosin Motors 630 Concentrations Rotary Motors All Good Things Must End Polymerization Motors: Pushing by Growing 637 Macromolecular Decay Can Be Described by a Translocation Motors: Pushing by Pulling 638 Simple, First-Order Differential Equation A Single-Molecule View of Degradation: Statistical 16.2 RECTIFIED BROWNIAN MOTION AND Mechanics Over Trajectories 582 MOLECULAR MOTORS 639 Molecules Fall Apart with a Characteristic Lifetime 582 I The Random Walk Yet Again 640 Decay Processes Can Be Described with Two-State Molecular Motors Can Be Thought of as Random 583 Walkers Trajectories 640 xxv

11 The One-State Model 641 Voltage-Gated Channels Result in a Nonlinear The Dynamics of a Molecular Motor Can Be Written Current-Voltage Relation for the Cell Membrane 699 Using a Master Equation 642 A Patch of Membrane Acts as a Bistable Switch 700 The Driven Diffusion Equation Can Be Transformed The Dynamics of Voltage Relaxation Can Be into an Ordinary Diffusion Equation 644 Modeled Using an RC Circuit Motor Stepping from a Free-Energy Perspective The Cable Equation The Two-State Model Depolarization Waves 705 The Dynamics of a Two-State Motor Is Described Waves of Membrane Depolarization Rely on by Two Coupled Rate Equations 651 Sodium Channels Switching into the Open State 705 Internal States Reveal Themselves in the Form Spikes 710 of the Waiting Time Distribution 654 Hodgkin-Huxley and Membrane Transport More General Motor Models Coordination of Motor Protein Activity Inactivation of Sodium Propagating Spikes Channels Leads to Rotary Motors SUMMARY AND CONCLUSIONS 714 POLYMERIZATION AND TRANSLOCATION AS 17.6 PROBLEMS 714 MOTOR ACTION FURTHER READING The Polymerization Ratchet 663 REFERENCES 715 The Polymerization Ratchet Is Based on a Polymerization Reaction That Is Maintained Out of Equilibrium 666 Chapter 18 light and life 717 The Polymerization Ratchet Force-Velocity Can Be 18.1 INTRODUCTION 718 Obtained by Solving a Driven Diffusion Equation PHOTOSYNTHESIS Force Generation by Growth 670 Organisms From All Three of the Great Domains Polymerization Forces Can Be Measured Directly 670 of Life Perform Photosynthesis 720 Polymerization Forces Are Used to Center Cellular Quantum Mechanics for Biology 724 Structures 672 Quantum Mechanical Kinematics Describes The Translocation Ratchet 673 States of the System in Terms of Wave Functions 725 Protein Binding Can Speed Up Translocation Quantum Mechanical Observables Are Represented through a Ratcheting Mechanism 674 by Operators 728 The Translocation Time Can Be Estimated by The Time Evolution of Quantum States Can Be Solving a Driven Diffusion Equation 676 Determined Using the Schrodinger Equation 729 SUMMARY AND CONCLUSIONS The Particle-in-a-Box Model Solutions for the Box of Finite Depth Do Not Vanish 16.5 PROBLEMS 677 at the Box Edges FURTHER READING Exciting Electrons With Light REFERENCES 679 Absorption Wavelengths Depend Upon Molecular Size and Shape 735 Chapter 17 Biological Electricity Moving Electrons From Hither to Yon 737 and the Hodgkin-Huxley Model 681 Excited Electrons Can Suffer Multiple Fates 737 Electron Transfer in Photosynthesis Proceeds by 17.1 THE ROLE OF ELECTRICITY IN CELLS 681 Tunneling THE CHARGE STATE OF THE CELL 682 Electron Transfer Between Donor and Acceptor Is The Electrical Status of Cells and Their Membranes 682 Gated by Fluctuations of the Environment Electrochemical Equilibrium and the Nernst Equation 683 Resonant Transfer Processes in the Antenna ion Concentration Differences Across Membranes Complex Efficiently Deliver Energy to the Reaction Lead to Potential Differences 683 Center Bioenergetics of Photosynthesis 748 MEMBRANE PERMEABILITY: PUMPS AND Electrons Are Transferred from Donors to Acceptors CHANNELS 685 Within and Around the Cell Membrane 748 A Nonequilibrium Charge Distribution Is Set Up Water, Water Everywhere, and Not an Electron to Between the Cell Interior and the External World 685 Drink Signals in Cells Are Often Mediated by the Presence Charge Separation across Membranes Results in a of Electrical Spikes Called Action Potentials 686 Proton-Motive Force 751 ion Channels and Membrane Permeability Making Sugar 752 ion Permeability Across Membranes Is Mediated Destroying Sugar 757 by lon Channels Photosynthesis in Perspective 758 A Simple Two-State Model Can Describe Many of the Features of Voltage Gating of I on Channels THE VISION THING Maintaining a Nonequilibrium Charge State Bacterial "Vision" 760 Ions Are Pumped Across the Cell Membrane Microbial Phototaxis and Manipulating Cells with Against an Electrochemical Gradient 691 Light Animal Vision THE ACTION POTENTIAL 693 There Is a Simple Relationship between Eye Membrane Depolarization: The Membrane as a Geometry and Resolution 765 Bistable Switch 693 The Resolution of Insect Eyes Is Governed by Coordinated Muscle Contraction Depends Upon Both the Number of Ommatidia and Diffraction Membrane Depolarization 694 Effects 768 A Patch of Cell Membrane Can Be Modeled as an The Light-Driven Conformational Change of Retinal Electrical Circuit 696 Underlies Animal Vision 769 The Difference Between the Membrane Potential and Information from Photon Detection Is Amplified the Nernst Potential Leads to an Ionic Current by a Signal Transduction Cascade in the Across the Cell Membrane 698 Photoreceptor Cell 773 xxvi

12 p The Vertebrate Visual System Is Capable of 19.3 REGULATORY DYNAMICS 83S Detecting Single Photons The Dynamics of RNA Polymerase and the Sex, Death, and Quantum Mechanics 781 Promoter 835 Let There Be Light: Chemical Reactions Can Be Used The Concentrations of Both RNA and Protein Can Be to Make Light 784 Described Using Rate Equations Dynamics of mrna Distributions SUMMARY AND CONCLUSIONS 785 Unregulated Promoters Can Be Described By a 18.S APPENDIX: SIMPLE MODEL OF ELECTRON TUNNELING 785 Poisson Distribution PROBLEMS Dynamics of Regulated Promoters 843 The Two-State Promoter Has a Fano Factor Greater 18.7 FURTHER READING 795 Than One REFERENCES 796 Different Regulatory Architectures Have Different Fano Factors Dynamics of Protein Translation 8S4 1 PART 4 THE MEANING OF LIFE Genetic Switches: Natural and Synthetic Genetic Networks That Oscillate 870 Chapter 19 Organization of Biological Networks CELLULAR FAST RESPONSE: SIGNALING Bacterial Chemotaxis 873 The MWC Model Can Be Used to Describe Bacterial 19.1 CHEMICAL AND INFORMATIONAL ORGANIZATION Chemotaxis 878 IN THE CELL 801 Precise Adaptation Can Be Described by a Simple Many Chemical Reactions in the Cell are Linked in Balance Between Methylation and Demethylation 881 Complex Networks Biochemistry on a Leash 883 Genetic Networks Describe the Linkages Between Tethering Increases the Local Concentration of a Different Genes and Their Products 802 Ligand 884 Developmental Decisions Are Made by Regulating Signaling Networks Help Cells Decide When and Genes 802 Where to Grow Their Actin Filaments for Motility 884 Gene Expression Is Measured Quantitatively in Synthetic Signaling Networks Permit a Dissection of Terms of How Much, When, and Where 804 Signaling Pathways GENETIC NETWORKS: DOING THE RIGHT THING AT 19.5 SUMMARY AND CONCLUSIONS 888 THE RIGHT TIME PROBLEMS 889 Promoter Occupancy Is Dictated by the Presence of Regulatory Proteins Called Transcription Factors The Molecular Implementation of Regulation: Promoters, Activators, and Repressors 808 Repressor Molecules Are the Proteins That Implement Negative Control 808 Activators Are the Proteins That Implement Positive Control 809 Genes Can Be Regulated During Processes Other 19.7 FURTHER READING REFERENCES 892 Chapter 20 Biological Patterns: Order in Space and Time INTRODUCTION: MAKING PATIERNS Patterns in Space and Time Rules for Pattern-Making 89S Than Transcription MORPHOGEN GRADIENTS The Mathematics of Recruitment and Rejection The French Flag Model 896 Recruitment of Proteins Reflects Cooperativity How the Fly Got His Stripes 898 Between Different DNA-Binding Proteins 810 Bicoid Exhibits an Exponential Concentration The Regulation Factor Dictates How the Bare RNA Gradient Along the Anterior-Posterior Axis of Fly Polymerase Binding Probability Is Altered by Embryos 898 Transcription Factors 812 A Reaction-Diffusion Mechanism Can Give Rise to Activator Bypass Experiments Show That Activators an Exponential Concentration Gradient 899 Work by Recruitment Precision and Scaling 905 Repressor Molecules Reduce the Probability Morphogen Patterning with Growth in Anabaena 912 Polymerase Will Bind to the Promoter Transcriptional Regulation by the Numbers: Binding 20.3 REACTION-DIFFUSION AND SPATIAL PATIERNS 914 Energies and Equilibrium Constants Putting Chemistry and Diffusion Together: Turing Equilibrium Constants Can Be Used To Determine Patterns 914 Regulation Factors How Bacteria Lay Down a Coordinate System A Simple Statistical Mechanical Model of Positive Phyllotaxis: The Art of Flower Arrangement 926 and Negative Regulation The lac Operon 822 The lac Operon Has Features of Both Negative and 20.4 TURNING TIME INTO SPACE: TEMPORAL OSCILLATIONS IN CELL FATE SPECIFICATION 931 Positive Regulation Somitogenesis 932 The Free Energy of DNA Looping Affects the Seashells Forming Patterns in Space and Time 93S Repression of the lac Operon 824 Inducers Tune the Level of Regulatory Response Other Regulatory Architectures 829 The Fold-Change for Different Regulatory Motifs Depends Upon Experimentally Accessible Control Parameters 830 Quantitative Analysis of Gene Expression in Eukaryotes Can Also Be Analyzed Using Thermodynamic Models S PATIERN FORMATION AS A CONTACT SPORT S.1 The Notch-Delta Concept Drosophila Eyes SUMMARY AND CONCLUSIONS PROBLEMS FURTHER READING REFERENCES 950 xxvii

13 Chapter 21 Sequences, Specificity, Evolution and Drug Resistance Viruses and Evolution 1000 and Evolution The Study of Sequence Makes It Possible to Trace 21.1 BIOLOGICAL INFORMATION 952 the Evolutionary History of HIV Why Sequences? 953 The Luria-Delbruck Experiment Reveals the Genomes and Sequences by the Numbers 957 Mathematics of Resistance SEQUENCE ALIGNMENT AND HOMOLOGY Phylogenetic Trees 1008 Sequence Comparison Can Sometimes Reveal Deep 21.5 THE MOLECULAR BASIS OF FIDELITY 1010 Functional and Evolutionary Relationships Between Keeping It Specific: Beating Thermodynamic Genes, Proteins, and Organisms 961 Specificity loll The HP Model as a Coarse-Grained Model The Specificity of Biological Recognition Often Far for Bioinformatics 964 Exceeds the Limit Dictated by Free-Energy Scoring Success 966 Differences 1011 A Score Can Be Assigned to Different Alignments High Specificity Costs Energy 1015 Between Sequences 966 Comparison of Full Amino Acid Sequences Requires 21.6 SUMMARY AND CONCLUSIONS 1016 a 20-by-20 Scoring Matrix PROBLEMS 1017 Even Random Sequences Have a Nonzero Score FURTHER READING 1020 The Extreme Value Distribution Determines the Probability That a Given Alignment Score Would Be 21.9 REFERENCES 1021 Found by Chance 971 False Positives Increase as the Threshold for Acceptable Expect Values (also Called E-Values) Is Chapter 22 Whither Physical Biology? 1023 Made Less Stringent DRAWING THE MAP TO SCALE 1023 Structural and Functional Similarity Do Not Always Guarantee Sequence Similarity NAVIGATING WHEN THE MAP IS WRONG THE POWER OF SEQUENCE GAZING INCREASING THE MAP RESOLUTION Binding Probabilities and Sequence 977 Position Weight Matrices Provide a Map Between 22.4 "DIFFICULTIES ON THEORY" 1030 Sequence and Binding Affinity 978 Modeler's Fantasy 1031 Frequencies of Nucleotides at Sites Within a Is It Biologically Interesting? 1031 Sequence Can Be Used to Construct Position Weight Uses and Abuses of Statistical Mechanics 1032 Matrices 979 Out-of-Equilibrium and Dynamic Using Sequence to Find Binding Sites 983 Uses and Abuses of Continuum Mechanics Do Nucleosomes Care About Their Positions on Too Many Parameters 1033 Genomes? 988 Missing Facts DNA Sequencing Reveals Patterns of Nucleosome 1033 Occupancy on Genomes 989 Too Much Stuff 1033 A Simple Model Based Upon Self-Avoidance Leads to Too Little Stuff 1034 a Prediction for Nucleosome Positioning 990 The Myth of "THE" Cell 1034 Not Enough Thinking SEQUENCES AND EVOLUTION Evolution by the Numbers: Hemoglobin and 22.5 THE RHYME AND REASON OF IT ALL 1035 Rhodopsin as Case Studies in Sequence Alignment FURTHER READING 1036 Sequence Similarity Is Used as a Temporal Yardstick to Determine Evolutionary Distances REFERENCES 1037 Modern-Day Sequences Can Be Used to Reconstruct the Past 996 Index 1039 xxviii