Biological Physics: Energy, Information, Life

Biological Physics: Energy, Information, Life Philip Nelson, www.physics.upenn.edu/?pcn/ With the assistance of Marko Radosavljevi´c (cid:2)c 2000, 2001, 2002 Philip C. Nelson Draft December 8, 2002 2 Not chaos-like together crush’d and bruis’d, But, as the world, harmoniously confus’d: Where order in variety we see, And where, though all things di?er, all agree. – Alexander Pope, 1713 Frontispiece: DNAfromabacteriumthathasbeenlysed(burst),forexamplebyosmoticshock (12200×magni?cation). Thebacterialgenomethatonceoccupiedasmallregioninthecenter of the ?gure now extends in a series of loops from the core structure (arrow). From (Wolfe, 1985). i About this manuscript: Here is the ?rst half of the text. (The second half will come in about a month.) There is some good news and some bad news. The good news is that your text for this course is free, apart from duplication costs. The bad news is that you have to help me write it. No, wait! That’sgoodnews: Here’syourchancetotellanauthorwherealltheconfusingbitsare,what things I’ve wrongly assumed you already know, what questions are phrased ambiguously, and so on. Please pass all these comments, and more, on to me via your professor. Philip Nelson, 12/02 Contents To the student viii To the instructor xii Part I: Mysteries, Metaphors, Models 1 1 What the ancients knew 2 1.1 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Heat is a form of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Just a little history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.3 Preview: The concept of free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 How life generates order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 The puzzle of biological order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 A paradigm for free energy transduction . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Excursion: Commercials, philosophy, pragmatics. . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 How to do better on exams (and discover new physical laws) . . . . . . . . . . . . . . . . . 15 1.4.1 Dimensions and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Using dimensional analysis to catch errors and recall de?nitions . . . . . . . . . . . . 17 1.4.3 Using dimensional analysis to formulate hypotheses . . . . . . . . . . . . . . . . . . 18 1.4.4 Some notational conventions involving ?ux and density . . . . . . . . . . . . . . . . 19 1.5 Other key ideas from physics and chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.1 Molecules are small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.5.2 Molecules are particular spatial arrangements of atoms. . . . . . . . . . . . . . . . . 21 1.5.3 Molecules have de?nite internal energies . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5.4 Low-density gases obey a universal law. . . . . . . . . . . . . . . . . . . . . . . . . . 23 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 What’s inside cells 30 2.1 Cell physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.1.1 Internal gross anatomy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.1.2 External gross anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 The molecular parts list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.1 Small molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.2 Medium-size molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.3 Big molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.2.4 Macromolecular assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Bridging the gap: Molecular devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 The plasma membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.2 Molecular motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.3 Enzymes and regulatory proteins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.4 The overall ?ow of information in cells . . . . . . . . . . . . . . . . . . . . . . . . . . 56 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 ii Contents iii [[Studentversion,December8,2002]] Part II: Di?usion, Dissipation, Drive 64 3 The molecular dance 65 3.1 The probabilistic facts of life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1.1 Discrete distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1.2 Continuous distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1.3 Mean and variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.4 Addition and multiplication rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Decoding the ideal gas law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.1 Temperature re?ects the average kinetic energy of thermal motion . . . . . . . . . . 73 3.2.2 The complete distribution of molecular velocities is experimentally measurable . . . 76 3.2.3 The Boltzmann distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2.4 Activation barriers control reaction rates. . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2.5 Relaxation to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3 Excursion: A lesson from heredity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 Aristotle weighs in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.2 Identifying the physical carrier of genetic information . . . . . . . . . . . . . . . . . 83 3.3.3 Schr¨odinger’s summary: Genetic information is structural . . . . . . . . . . . . . . . 89 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4 Random walks, friction, and di?usion 98 4.1 Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1.1 Just a little more history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1.2 Random walks lead to di?usive behavior . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.1.3 The di?usion law is model-independent . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.1.4 Friction is quantitatively related to di?usion. . . . . . . . . . . . . . . . . . . . . . . 107 4.2 Excursion: What Einstein did and did not do . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.3 Other random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.1 The conformation of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.3.2 Vista: Random walks on Wall Street . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.4 More about di?usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4.1 Di?usion rules the subcellular world . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4.2 Di?usion obeys a simple equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.4.3 Precise statistical prediction of random processes . . . . . . . . . . . . . . . . . . . . 118 4.5 Functions, derivatives, and snakes under the rug . . . . . . . . . . . . . . . . . . . . . . . . 119 4.5.1 Functions describe the details of quantitative relationships. . . . . . . . . . . . . . . 119 4.5.2 A function of two variables can be visualized as a landscape . . . . . . . . . . . . . . 120 4.6 Biological applications of di?usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.6.1 The permeability of arti?cial membranes is di?usive . . . . . . . . . . . . . . . . . . 121 4.6.2 Di?usion sets a fundamental limit on bacterial metabolism . . . . . . . . . . . . . . 123 4.6.3 The Nernst relation sets the scale of membrane potentials . . . . . . . . . . . . . . . 124 4.6.4 The electrical resistance of a solution re?ects frictional dissipation . . . . . . . . . . 127 4.6.5 Di?usion from a point gives a spreading, Gaussian pro?le . . . . . . . . . . . . . . . 128 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5 Life in the slow lane: the low Reynolds-number world 141 5.1 Friction in ?uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.1.1 Sedimentation separates particles by density . . . . . . . . . . . . . . . . . . . . . . 142 5.1.2 The rate of sedimentation depends on solvent viscosity . . . . . . . . . . . . . . . . . 143 5.1.3 It’s hard to mix a viscous liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.2 Low Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.2.1 A critical force demarcates the physical regime dominated by friction. . . . . . . . . 146 5.2.2 The Reynolds number quanti?es the relative importance of friction and inertia . . . 148 5.2.3 The time reversal properties of a dynamical law signal its dissipative character . . . 150 5.3 Biological applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 iv Contents [[Studentversion,December8,2002]] 5.3.1 Swimming and pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.3.2 To stir or not to stir? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.3.3 Foraging, attack, and escape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.3.4 Vascular networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3.5 Viscous drag at the DNA replication fork . . . . . . . . . . . . . . . . . . . . . . . . 161 5.4 Excursion: The character of physical Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6 Entropy, temperature, and free energy 173 6.1 How to measure disorder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2.1 The Statistical Postulate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6.2.2 Entropy is a constant times the maximal value of disorder . . . . . . . . . . . . . . . 177 6.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.3.1 Heat ?ows to maximize disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.3.2 Temperature is a statistical property of a system in equilibrium . . . . . . . . . . . . 180 6.4 The Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.4.1 Entropy increases spontaneously when a constraint is removed . . . . . . . . . . . . 182 6.4.2 Three remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.5 Open systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.5.1 The free energy of a subsystem re?ects the competition between entropy and energy 186 6.5.2 Entropic forces can be expressed as derivatives of the free energy . . . . . . . . . . . 188 6.5.3 Free energy transduction is most e?cient when it proceeds in small, controlled steps 189 6.5.4 The biosphere as a thermal engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.6 Microscopic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.6.1 The Boltzmann distribution follows from the Statistical Postulate . . . . . . . . . . 192 6.6.2 Kinetic interpretation of the Boltzmann distribution . . . . . . . . . . . . . . . . . . 194 6.6.3 The minimum free energy principle also applies to microscopic subsystems. . . . . . 197 6.6.4 The free energy determines the populations of complex two-state systems . . . . . . 198 6.7 Excursion: “RNAfoldingasatwo-statesystem”byJ.Liphardt,I.Tinoco,Jr.,andC.Bus- tamante . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7 Entropic forces at work 215 7.1 Microscopic view of entropic forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.1.1 Fixed-volume approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.1.2 Fixed-pressure approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.2 Osmotic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.2.1 Equilibrium osmotic pressure obeys the ideal gas law . . . . . . . . . . . . . . . . . . 218 7.2.2 Osmotic pressure creates a depletion force between large molecules . . . . . . . . . . 220 7.3 Beyond equilibrium: Osmotic ?ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.3.1 Osmotic forces arise from the recti?cation of Brownian motion . . . . . . . . . . . . 224 7.3.2 Osmotic ?ow is quantitatively related to forced permeation . . . . . . . . . . . . . . 226 7.4 A repulsive interlude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.4.1 Electrostatic interactions are crucial for proper cell functioning . . . . . . . . . . . . 227 7.4.2 The Gauss Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.4.3 Charged surfaces are surrounded by neutralizing ion clouds . . . . . . . . . . . . . . 230 7.4.4 The repulsion of like-charged surfaces arises from compressing their ion clouds . . . 234 7.4.5 Oppositely charged surfaces attract by counterion release . . . . . . . . . . . . . . . 235 7.5 Special properties of water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.5.1 Liquid water contains a loose network of hydrogen bonds . . . . . . . . . . . . . . . 236 7.5.2 The hydrogen-bond network a?ects the solubility of small molecules in water . . . . 239 7.5.3 Water generates an entropic attraction between nonpolar objects . . . . . . . . . . . 241 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Contents v [[Studentversion,December8,2002]] Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 8 Chemical forces and self-assembly 259 8.1 Chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 8.1.1 µ measures the availability of a particle species . . . . . . . . . . . . . . . . . . . . . 260 8.1.2 TheBoltzmanndistributionhasasimplegeneralizationaccountingforparticleexchange262 8.2 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 8.2.1 Chemical equilibrium occurs when chemical forces balance . . . . . . . . . . . . . . . 263 8.2.2 ?G gives a universal criterion for the direction of a chemical reaction . . . . . . . . 265 8.2.3 Kinetic interpretation of complex equilibria . . . . . . . . . . . . . . . . . . . . . . . 269 8.2.4 The primordial soup was not in chemical equilibrium . . . . . . . . . . . . . . . . . . 270 8.3 Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 8.3.1 Ionic and partially ionic bonds dissociate readily in water . . . . . . . . . . . . . . . 271 8.3.2 The strengths of acids and bases re?ect their dissociation equilibrium constants . . . 272 8.3.3 The charge on a protein varies with its environment . . . . . . . . . . . . . . . . . . 273 8.3.4 Electrophoresis can give a sensitive measure of protein composition . . . . . . . . . . 275 8.4 Self-assembly of amphiphiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.4.1 Emulsions form when amphiphilic molecules reduce the oil-water interface tension . 277 8.4.2 Micelles self-assemble suddenly at a critical concentration . . . . . . . . . . . . . . . 279 8.5 Excursion: On ?tting models to data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8.6 Self-assembly in cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 8.6.1 Bilayers self-assemble from two-tailed amphiphiles . . . . . . . . . . . . . . . . . . . 283 8.6.2 Vista: Macromolecular folding and aggregation . . . . . . . . . . . . . . . . . . . . . 287 8.6.3 Another trip to the kitchen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 Part III: Molecules, Machines, Mechanisms 298 9 Cooperative transitions in macromolecules 299 9.1 Elasticity models of polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 9.1.1 Why physics works (when it does work) . . . . . . . . . . . . . . . . . . . . . . . . . 300 9.1.2 Four phenomenological parameters characterize the elasticity of a long, thin rod . . 302 9.1.3 Polymers resist stretching with an entropic force . . . . . . . . . . . . . . . . . . . . 304 9.2 Stretching single macromolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 9.2.1 The force–extension curve can be measured for single DNA molecules . . . . . . . . 306 9.2.2 A simple two-state system qualitatively explains DNA stretching at low force . . . . 308 9.3 Eigenvalues for the impatient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.3.1 Matrices and eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.3.2 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.4 Cooperativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.4.1 The transfer matrix technique allows a more accurate treatment of bend cooperativity313 9.4.2 DNA also exhibits linear stretching elasticity at moderate applied force . . . . . . . 316 9.4.3 Cooperativityinhigher-dimensionalsystemsgivesrisetoin?nitelysharpphasetran- sitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 9.5 Thermal, chemical, and mechanical switching . . . . . . . . . . . . . . . . . . . . . . . . . . 318 9.5.1 The helix–coil transition can be observed using polarized light . . . . . . . . . . . . 318 9.5.2 Three phenomenological parameters describe a given helix–coil transition . . . . . . 321 9.5.3 Calculation of the helix-coil transition . . . . . . . . . . . . . . . . . . . . . . . . . . 323 9.5.4 DNA also displays a cooperative “melting” transition . . . . . . . . . . . . . . . . . 326 9.5.5 Applied mechanical force can induce cooperative structural transitions in macro- molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 9.6 Allostery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 9.6.1 Hemoglobin binds four oxygen molecules cooperatively . . . . . . . . . . . . . . . . . 328 9.6.2 Allostery involves relative motion of molecular subunits . . . . . . . . . . . . . . . . 331 9.6.3 Vista: Protein substates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 vi Contents [[Studentversion,December8,2002]] Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 10 Enzymes and molecular machines 349 10.1 Survey of molecular devices found in cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 10.1.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 10.1.2 Enzymes display saturation kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 10.1.3 All eukaryotic cells contain cyclic motors . . . . . . . . . . . . . . . . . . . . . . . . 352 10.1.4 One-shot motors assist in cell locomotion and spatial organization . . . . . . . . . . 354 10.2 Purely mechanical machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 10.2.1 Macroscopic machines can be described by an energy landscape . . . . . . . . . . . . 357 10.2.2 Microscopic machines can step past energy barriers . . . . . . . . . . . . . . . . . . . 360 10.2.3 The Smoluchowski equation gives the rate of a microscopic machine . . . . . . . . . 362 10.3 Molecular implementation of mechanical principles . . . . . . . . . . . . . . . . . . . . . . . 368 10.3.1 Three ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 10.3.2 The reaction coordinate gives a useful reduced description of a chemical event. . . . 368 10.3.3 An enzyme catalyzes a reaction by binding to the transition state. . . . . . . . . . . 370 10.3.4 Mechanochemical motors move by random-walking on a two-dimensional landscape. 375 10.4 Kinetics of real enzymes and machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 10.4.1 The Michaelis–Menten rule describes the kinetics of simple enzymes . . . . . . . . . 377 10.4.2 Modulation of enzyme activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 10.4.3 Two-headed kinesin as a tightly coupled, perfect ratchet . . . . . . . . . . . . . . . . 380 10.4.4 Molecular motors can move even without tight coupling or a power stroke . . . . . . 388 10.5 Vista: Other molecular motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 11 Molecular machines in membranes 406 11.1 Electro-osmotic e?ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 11.1.1 Before the ancients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 11.1.2 Ion concentration di?erences create Nernst potentials . . . . . . . . . . . . . . . . . 407 11.1.3 Donnan equilibrium can create a resting membrane potential . . . . . . . . . . . . . 410 11.2 Ion pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 11.2.1 ObservedeukaryoticmembranepotentialsimplythatthesecellsarefarfromDonnan equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 11.2.2 The Ohmic conductance hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 11.2.3 Active pumping maintains steady-state membrane potentials while avoiding large osmotic pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 11.3 Mitochondria as factories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 11.3.1 Busbars and driveshafts distribute energy in factories . . . . . . . . . . . . . . . . . 421 11.3.2 The biochemical backdrop to respiration . . . . . . . . . . . . . . . . . . . . . . . . . 422 11.3.3 Thechemiosmoticmechanismidenti?esthemitochondrialinnermembraneasabusbar425 11.3.4 Evidence for the chemiosmotic mechanism . . . . . . . . . . . . . . . . . . . . . . . . 426 11.3.5 Vista: Cells use chemiosmotic coupling in many other contexts . . . . . . . . . . . . 429 11.4 Excursion: “Powering up the ?agellar motor” by H. C. Berg and D. Fung . . . . . . . . . 430 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 12 Nerve impulses 437 12.1 The problem of nerve impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 12.1.1 Phenomenology of the action potential . . . . . . . . . . . . . . . . . . . . . . . . . . 438 12.1.2 The cell membrane can be viewed as an electrical network . . . . . . . . . . . . . . . 441 12.1.3 MembraneswithOhmicconductanceleadtoalinearcableequationwithnotraveling- wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 12.2 Simpli?ed mechanism of the action potential . . . . . . . . . . . . . . . . . . . . . . . . . . 449 12.2.1 A mechanical analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 12.2.2 Just a little more history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 Contents vii [[Studentversion,December8,2002]] 12.2.3 The time course of an action potential suggests the hypothesis of voltage gating . . 454 12.2.4 Voltage gating leads to a nonlinear cable equation with traveling-wave solutions. . . 457 12.3 The full Hodgkin–Huxley mechanism and its molecular underpinnings . . . . . . . . . . . . 461 12.3.1 Eachionconductancefollowsacharacteristictimecoursewhenthemembranepoten- tial changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 12.3.2 The patch-clamp technique allows the study of single ion channel behavior . . . . . 464 12.4 Nerve, muscle, synapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 12.4.1 Nerve cells are separated by narrow synapses . . . . . . . . . . . . . . . . . . . . . . 472 12.4.2 The neuromuscular junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 12.4.3 Vista: Neural computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 The big picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 Track 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 Proble.m.s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 13 Epilogue 482 Acknowledgments 484 A Global list of symbols and units 485 B Numerical values 492 Bibliography 497 Credits 506 viii To the student To the student This is a book for life-science students who are willing to use calculus. This is also a book for physical-science and engineering students who are willing to think about cells. I believe that in the future every student in either group will need to know the essential core of the others’ knowledge. In the past few years I have attended many conferences and seminars. Increasingly I have found myself surrounded not only by physicists, biologists, chemists, and engineers, but also by physicians, mathematicians, and entrepreneurs. At these conferences nobody ever stands up and says, “Hey, isthisnanotechnologyorbiomathematics?” becausenobodyreallycares. Thesepeople come together to learn from each other, and the traditional academic distinctions between their ?elds are becoming increasingly irrelevant to this exciting work. In this book I want to share some of their excitement. I began to wonder how this diverse group managed to overcome the Tower-of-Babel syndrome. Slowly I began to realize that while each discipline carries its immense load of experimental and theoreticalmachinery,stilltheheadwatersoftheseriversaremanageable,andcomefromacommon spring, a handful of simple, general ideas. Armed with these few ideas, I found that one can understand an enormous amount of front-line research. In this book I want to explore these ?rst common ideas, ruthlessly suppressing the more specialized ones for later. I also realized that my own undergraduate education had postponed the introduction of many of these ideas to the last year of my degree (or even later), and that many programs still have this character: We meticulously build a sophisticated mathematical edi?ce before introducing many of the Big Ideas. My colleagues and I became convinced that this approach did not serve the needs of our students. Many of our undergraduate students get started on research in their very ?rst year and need the big picture early. Many others create interdisciplinary programs for themselves and mayneverevengettoourspecialized,advancedcourses. InthisbookIwanttopresentsomeofthe big picture in a way accessible to any student who has taken ?rst-year physics and calculus (plus a smattering of high-school chemistry and biology), and who is willing to stretch. When you’re done youshouldbeinapositiontoreadcurrentworkinScience andNature. Youwon’tgeteverydetail, of course. But you will get the sweep. Whenwebegantoo?erthiscourse,weweresurprisedto?ndthatmanyofourgraduatestudents wanted to take it too. In part this re?ected their own compartmentalized education: The physics students wanted to read the biology part and see it integrated with their other knowledge, the biology students wanted the reverse, and so on. To our amazement, we found that the course became popular with students at all levels from sophomore to third-year graduate, with the latter digging more deeply into the details. Accordingly, many sections in this book have “Track–2” addenda addressing this more mathematically experienced group. Physical science vs life science At the dawn of the twentieth century it was already clear that, chemically speaking, you and I are not much di?erent from cans of soup. And yet we can do many complex and even fun things we do not usually see cans of soup doing. At that time people had basically no correct ideas for how living organisms create order from food, do work, and even compute things—just a lot of inappropriate metaphors drawn from the technology of the day. Bymid-centuryitbegantobeclearthattheanswerstomanyofthesequestionswouldbefound inthestudyofverybigmolecules. Now,aswebeginthetwenty-?rstcentury,ironically,thesituation is inverted: The problem is now that we have way too much information about those molecules!