__Biological_Physics___Energy__Information__Life

‫اﻳﻦ ﻓﺎﻳﻞ در ﺷﺎﻣﮕﺎه ‪ 30‬ﻣﻬﺮ و ﺑﺎﻣﺪاد ‪ 1‬آﺑﺎن ‪ 1388‬ﺗﻮﺳﻂ ﺑﻨﺪه ﺣﻘﻴـﺮ‬ ‫ﺟﻬﺖ رﻓﻊ ﻧﻴﺎز دوﺳﺘﺎن و داﻧﺸﺠﻮﻳﺎن و اﺳﺎﺗﻴﺪ ﻋﻠﻲ اﻟﺨﺼـﻮص دوﺳـﺖ‬ ‫ﮔﺮاﻣﻴﻢ اﺣﺴﺎن و ﻧﻴﺰ ﺑﺮاي اﺟﺘﻨﺎب از ﻛﭙﻲﻫـﺎي ﻣﻜـﺮر از ﻛﺘـﺎب اﺻـﻠﻲ‬ ‫)ﻣﺘﻌﻠﻖ ﺑﻪ ﻛﺘﺎﺑﺨﺎﻧﻪ داﻧﺸﮕﺎه ﺗﺤﺼﻴﻼت ﺗﻜﻤﻴﻠﻲ زﻧﺠﺎن( ﻛـﻪ ﻣﻨﺠـﺮ ﺑـﻪ‬ ‫وارد آﻣﺪن ﺧﺴﺎرت ﺑﻪ آن ﻣﻲﺷﺪ‪ ،‬اﺳﻜﻦ ﮔﺮدﻳﺪ‪ .‬اﻣﻴﺪ ﻛﻪ ﻣﻮرد رﺿـﺎﻳﺖ‬ ‫ﺣﻖ ﺗﻌﺎﻟﻲ و ﭘﺴﻨﺪ ﺧﻮاﻧﻨﺪﮔﺎن واﻗﻊ ﮔﺮدد‪.‬‬ ‫اﺑﺮاﻫﻴﻢ دﺳﺘﺠﺮدي‬

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:. Publisher and Acquisitions Editor: Susan Finn emore Brennan Marketing Manager: Mark Santee -,..• <;:;; \"'=:' Project Manager and Text Designer: Leslie Galen, Integre Technical Publishing Company, Inc. '\\ 'r;' Project Editor: Jane O'Neill Cover Designer: Blake Logan ',/ ... \" Illustrations: Sarina Bromberg, Larr y Gonick, Felice Macera Illustration Coordinator: Bill Page Photo Editor : Patricia Marx Production Coordinator: Susan Wein Compo sitio n: lntegre Technical Publishing Company, Inc. Printing and Binding: RR Donnelley & Sons Company Front covet: Purkinj e neuron from rat brain, visualized by two-photon laser scann ing mi- croscopy. The scale bar represents 151lm. The neu ron shown is alive and surrou nded by a dense network of other neurons; a fluo rescent dye has been injected into the cell from the mi- cropipette at lower left, to reveal only the one cell of interest. The dendritic tree of this neuron (top) receives over 100 000 synaptic inpu ts. Dendr itic spines are visible as tiny bump s on the dendrites. A single axon (lower left) sends output signals on to other neurons. [Digital image kindly supplied by K. Svoboda; see also Svoboda et al., 1996.] Title page: DNA from a bacterium that has been lysed (burst) by osmo tic shock. The bacterial genome that once occup ied a small region in the center of the figure now extends in a series of loops from the core structure. Top to bott om, about 1011 m . [Electron micrograph by Ruth Kavenoff.] Libra ry of Co ng ress Cont rol Number : 2003 105929 © 2004 by Philip C. Nelson All rights reserved Print ed in the United States of America First printing 2003 W. H. Freema n an d Co m pany 41 Madison Avenue New York, NY 100I0 HoundmilIs, Basingsroke RG21 6XS, England www.whfreernan.com

- For Dr. Shirley M. Davidso n, 1930-2002

Not chaos-like together crush'd arid bruis'd, But, as the world, harmoniously conflls'd: lVh ere order in variety we see, And where, though all things differ. all agree. -Alexander Pope, 1713

Contents To the Student xv To th e Instructor xx Acknowl edgments xxi v Part I Mysteries, Metaphors, Models Chapter 1 What the Ancients Knew 3 l.l Heat 3 1.1.1 Heat is a form of energy 4 1.1.2 Just a little histor y 6 1.1.3 Preview: The concept of free energy 8 1.2 How life generates order 9 1.2.1 Th e puzzle of biological ord er 9 1.2.2 Osmo tic flow as a paradigm for free energy transduction 12 1.2.3 Preview: Disorder as information 14 1.3 Excursion : Commercials, philo sophy, pra gmatic s 15 1.4 How to do belter on exams (and discover new physical laws) 18 1.4.1 Most physical quantities carry dim ension s 18 1.4.2 Dimensional analysis can help you catch errors and recall definit ions 20 1.4.3 Dim ensiona l analysis can also help you formulate hypotheses 22 1.4.4 Some notational convent ions involving flux and density 22 1.5 Other key ideas from physics and chemistry 23 1.5.1 Molecules are small 23 1.5.2 Molecules are parti cular spatial arr angement s of atoms 25 1.5.3 Molecules have well-defined intern al energies 26 1.5.4 Low-density gases obey a un iversal law 27 The big picture 28 Track 2 30 Problems 31 v

vi Conte nts What's Inside Cells 35 Chapter 2 2.1 Cell physiology 37 59 2.1.1 Internal gross anatomy 40 2.1.2 External gross anatomy 43 2.2 The mo lecular part s list 45 2.2.1 Small molecules 46 2.2.2 Medium-sized mo lecules 48 2.2.3 Big mol ecu les 50 2.2.4 Macromolecular assemblies 54 2.3 Bridging th e gap: Mol ecular devices 54 2.3.1 The plasma membrane 55 2.3.2 Molec ular motors 58 2.3.3 Enzymes an d regulato ry pro teins 58 2.3.4 Th e overa ll flow of inform ation in cells Th e big picture 62 Track 2 63 Prob lems 64 Part II Diffusion, Dissipation, Drive Chapter 3 The Molecular Dance 69 3.1 The probabilistic facts of life 69 3.1.1 Discrete di str ibut ion s 70 3.1.2 Contin uous distr ibution s 71 3.1.3 Mea n and variance 73 3.1.4 Additio n an d multipl icatio n rules 75 3.2 Decoding the ideal gas law 78 3.2.1 Temperature reflects th e average kinetic energy of thermal motion 78 3.2.2 The complete distribu tio n of molecular velocit ies is experimentally measurab le 82 3.2.3 The Boltzma nn distribut ion 83 3.2.4 Activation barriers control reaction rates 86 3.2.5 Relaxat ion to eq uilibriu m 87 3.3 Excursion: A lesson from heredity 89 3.3.1 Aristotle weighs in 89 3.3.2 Identifying th e physical carrier of genetic information 90 3.3.3 Schrodinger's sum ma ry: Genetic information is structural 96 The big picture 101 Track 2 104 Problems 105

Contents vii Chapter 4 Random Walks, Friction, and Diffusion 108 Chapter 5 4. 1 Browni an motion 109 4.1.1 Just a little mo re history 109 4.1.2 Random walks lead to diffusive behavior 110 4.1.3 The diffusion law is model independent 117 4.1.4 Friction is quantit atively related to diffusion 118 4.2 Excursio n: Einstein's role 121 4.3 Other random walks 122 4.3.1 The conformation of polymers 122 4.3.2 Vista: Rando m walks on Wall Street 126 4.4 More about diffusion 127 4.4.1 Diffusion rules the subcellular world 127 4.4.2 Diffusion obeys a simple equatio n 128 4.4.3 Precise statistical prediction of random processes 131 4.5 Function s, derivatives, and snakes under the rug 132 4.5.1 Function s describe the details of quant itative relationships 132 4.5.2 A functio n of two variables can be visualized as a land scape 134 4.6 Biological applications of diffusion 135 4.6.1 The permeab ility of artificial membranes is diffusive 135 4.6.2 Diffusion sets a funda menta l limit on bacterial metabolism 138 4.6.3 The Nernst relation sets the scale of membrane potentials 139 4.6.4 The electrical resistanc e of a solution reflects frictional dissipation 142 4.6.5 Diffusion from a point gives a spreading, Gaussian profile 142 The big picture 144 Track 2 147 Problems 153 Life in the Slow Lane: The Low Reynolds-Number World 158 5.1 Friction in fluids 158 5. 1.1 Sufficiently small particles can rem ain in suspensio n indefinitely 158 5.1.2 The rate of sedimentation depends on solvent viscosity 160 5.1.3 It's hard to m ix a viscous liquid 161 5.2 Low Reynolds nu mber 163 5.2.1 A critical force demarcates the physical regime dominated by friction 164 5.2.2 The Reynolds nu mber qua ntifies the relative importance of frictio n and inertia 166 5.2.3 The tim e-reversal properties of a dynami cal law signal its dissipative character 169

viii Contents 5.3 Biological applications 172 182 Chapter 6 5.3.1 Swimming and pum ping 172 5.3.2 To stir or not to stir? 177 5.3.3 Fo raging, attack, and escape 178 5.3.4 Vascular networks 179 5.3.5 Viscous dra g at the DNA replication fork 5.4 Excursion : The character of physical Laws 184 The big picture 185 Track 2 187 Problems 190 Entropy, Temperature, and Free Energy 195 6.1 How to measure disorder 196 6.2 Entropy 199 6.2.1 The Statistical Postulate 199 6.2.2 Ent ropy is a constant tim es the maximal value of disorder 200 6.3 Temp erature 202 6.3. 1 Heat flows to maximize disorder 202 6.3.2 Temperature is a statistical property of a system in equilibrium 203 6.4 The Second Law 206 6.4.1 Entropy increases spontaneously when a con straint is removed 206 6.4.2 Three remarks 209 6.5 O pen systems 210 6.5.1 The free energy of a subsystem reflects the competition between entropy and energy 210 6.5.2 Entropic forces can be expressed as derivatives of the free energy 213 6.5.3 Free energy transduction is most efficient when it proceeds in small, controlled steps 214 6.5.4 The biosphere as a thermal engine 216 6.6 Microscopic systems 217 6.6.1 The Boltzmann distribution follows from the Stat istical Postu late 218 6.6.2 Kinetic interpretation of the Boltzmann distribution 220 6.6.3 The minimum free energy principle also applies to microscopic subsystems 223 6.6.4 The free energy determi nes the populations of complex two-state systems 225

- Contents lx 6.7 Excur sion : \"RNA foldin g as a two-state system\" by). Liphardt, I. Tinoco, Jr., and C. Busta mante 226 Th e big picture 229 Track 2 232 Problem s 239 Chapt er 7 Entrop ic Forces at Work 245 Chapter 8 7.1 Microscopi c view of entropic forces 246 7.J.l Fixed-volume app roac h 246 7.1.2 Fixed -pressure approach 247 7.2 Osmoti c pressure 248 7.2. 1 Equilibrium osmotic pressure follows th e ideal gas law 248 7.2.2 Osmotic pressure creates a depletion force bet ween large molecules 251 7.3 Beyond equilibrium: Osmo tic flow 254 7.3.1 Osmotic forces arise fro m the rectification of Brownia n moti on 255 7.3.2 Osm otic flow is quantitatively related to forced perm eation 259 7.4 A repu lsive interlude 260 7.4.1 Electrostatic interactions are cr ucial for proper cell functioning 261 7.4.2 Th e Gauss Law 263 7.4.3 Cha rged surfaces are surrounded by neutralizing ion douds 264 7.4.4 Th e repulsion of like-charged surfaces ar ises from com pression of their ion clouds 269 7.4.5 Op positely cha rged surfaces attract by counterion release 272 7.5 Special properti es of water 273 7.5.1 Liquid wate r contains a loose network of hydro gen bo nd s 273 7.5.2 The hydro gen-bond networ k affects the solubility of small m olecules in water 276 7.5.3 Water generates an entropic att raction between nonpolar obj ects 280 Th e big pictu re 28 1 Track 2 283 Probl em s 290 Chemi cal For ces and Self-Assem bly 294 8.1 Chemic al pot ential 294 8. J.l 11 measures the availability of a particle species 295 8.1.2 The Boltzma nn distr ibution has a sim ple generalization acco unting for particle exchange 298

X Contents 8.2 Chemical reaction s 299 8.2.1 Chemical equilibrium occurs when chemical forces balance 299 8.2.2 t:J.G gives a universal criterion for the direction of a che'mical reaction 301 8.2.3 Kinetic interpretation of comp lex equilibria 306 8.2.4 The prim ordial soup was not in chemical equilibrium 307 8.3 Dissociation 308 8.3.1 Ionic and partially ionic bonds dissociate readily in water 308 8.3.2 The strengths of acids and bases reflect their dissociation equilibrium constants 309 8.3.3 The charge o n a protein varies with its environment 3 11 8.3.4 Electrophoresis can give a sensitive measure of protein composition 312 8.4 Self-assembly of amphiphiles 3 t5 8.4. 1 Emulsions fo rm when amphiphilic mol ecules reduce the o il- water interface tension 3 15 8.4.2 Micelles self-assemble suddenly at a critical concentration 3 t7 8.5 Excursion: On filling models to data 321 8.6 Self-assembly in cells 322 8.6. t Bilayers self-assemble from two-tailed amphiphiles 322 8.6.2 Vista: Macromo lecular folding and aggregation 327 8.6.3 Another trip to the kitchen 330 The big picture 332 Track 2 335 Problems 337 Part III Molecules, Machines, Mechanisms Chapter 9 Cooperative Transitions in Macromolecules 341 9.1 Elasticity models of polymers 342 9.1.1 Why physics works (when it does work) 342 9.1.2 Four phenomenological parameters characterize the elasticity o f a long, th in rod 344 9.1.3 Polymers resist stretching with an entropic force 347 9.2 Stretching single macromolecules 350 9.2.1 The force-extension curve can be measured for single DNA mol ecules 350 9.2.2 A two-state system qualitatively explains DNA stretching at low force 352

Conten ts xi 9.3 Eigenvalues for the impatient 354 9.3.1 Matrices and eigenvalues 354 9.3.2 Matri x multiplication 357 9.4 Coope rativity 358 9.4 .1 The transfer matrix technique allows a mo re accu rate treatme nt of bend coopera tivity 358 . 9.4.2 DNA also exhibits linear stretching elasticity at moderate applied force 361 9.4.3 Cooperativity in higher-dimension al systems gives rise to infinitely sharp phase transitions 363 9.5 Thermal, chemical, and mechanical switching 363 9.5.1 The helix-coil transit ion can be observed by using polarized light 364 9.5.2 Three phenomenological parameters describe a given helix-coil transition 366 9.5.3 Calculation of the helix-coil transition 369 9.5.4 DNA also displays a coop erative \"melting\" tran sition 373 9.5.5 Applied mechanical force can induce coo perative structural transitio ns in macromolecules 374 9.6 Allostery 376 9.6.1 Hemoglobin bind s four oxygen molecules coop eratively 376 9.6.2 Alloste ry often invo lves relative m otion of m olecular subunits 379 9.6.3 Vista: Protein substates 380 The big picture 382 Track 2 384 Problem s 396 Chapter 10 Enzymes and Molecular Machines 401 10.1 Survey of molecular devices found in cells 402 10.l.! Terminology 402 10.1.2 Enzymes display satura tion kinetics 403 10.1.3 All eukaryotic cells contain cyclic motors 404 10.1.4 One-shot machines assist in cell locomotion and spatial o rganization 40 7 10.2 Purely mechanical machines 409 10. 2. 1 Macroscopic machines can be described by an energy landscape 409 10.2.2 Microscopic machines can step past ener gy barriers 413 10. 2.3 The Smo luchowski equation gives the rate of a m icrosco pic machine 415

xii Contents 10.3 Molecular implementation of mechanical prin ciples 422 Chapter 11 10.3.1 Three ideas 423 10.3.2 The reaction coo rdinate gives a useful reduced description of a chemical event 423 10.3.3 An enzyme catalyzes a reaction by bindin g to the transition state 425 10.3.4 Mechano chem ical moto rs move by rando m-walking o n a two-dimensional landscape 431 10.4 Kinetics of real enzymes and machin es 432 10.4.1 The Michaelis-Menten rule describes th e kin etics of simple enzymes 433 10.4.2 Modulation of enzyme activity 436 10.4.3 Two- headed kinesin as a tightly coupled, perfect ratchet 437 10.4.4 Mo lecular motors can mo ve even witho ut tight coupling or a power stroke 446 10.5 Vista: Oth er molecular motors 451 The big picture 451 Track 2 455 Problems 464 Machin es in Membranes 469 I l.l Electroosmotic effects 469 I 1.1.1 Before the ancients 469 11.1.2 Ion concentration differences create Ne rnst potenti als 470 11.1.3 Do nnan equilibrium can create a resting membran e potential 474 11.2 Ion pumping 476 11 .2.1 Ob served eukaryotic membrane pot entials imply that these cells are far from Donnan equilibrium 476 11.2.2 Th e Ohmic conductance hypothesis 478 11.2.3 Active pumping maintains steady-s tate membrane pot entials wh ile avo iding large osmotic pressures 48 1 11.3 Mitochon dria as factories 486 11.3.1 Busbars and dr iveshafts distribute energy in factories 487 11.3.2 The biochemical backdrop to respiration 487 11.3.3 The chemios mo tic mechani sm identifies the mitoch ondri al inn er membr ane as a busbar 49 1 11.3.4 Evidence for the chemios mo tic mechanism 492 11.3.5 Vista: Cells use chemios mo tic coupling in many oth er contexts 496

Conte nts xiii 11.4 Excursion : \"Powering up the flagellar motor\" by H. C. Berg and D. Fung 497 The big picture 499 Track 2 50 1 Problems 503 Chapter 12 Nerve Impulses 505 Appendix A 12.1 The problem of nerve impu lses 506 12.1.1 Phenom enology of the action pote nrial 506 12.1.2 The cell membrane can be viewed as an electrical network 509 12.1.3 Membranes with Ohmic conductance lead to a linear cable equation with no traveling wave solutions 514 12.2 Simplified mechanism of the action potent ial 518 12.2.1 The pu zzle 518 12.2.2 A mechanical ana logy 519 12.2.3 Just a little more history 521 12.2.4 The time course of an action potential suggests the hypoth esis of voltage gating 524 12.2.5 Voltage gating leads to a nonlinear cable equation with traveling wave so lutions 527 12.3 The full Hod gkin-Huxley mechanism and its molecular und erpinnings 532 12.3.1 Each ion conductance follows a characteristic time course when the membr ane potent ial changes 532 12.3.2 The patch clamp technique allows the study of single ion channel behavior 536 12.4 Nerve, muscle, synapse 545 12.4.1 Nerve cells are separated by narrow synapses 545 12.4.2 The neuromuscular junction 546 12.4.3 Vista: Neural computation 548 The big picture 549 Track 2 552 Problems 553 Epilogue 557 Global List of Symbols and Units 559 Notation 559 560 Named quantities Dimensions 565 Units 565

xiv Contents Numerical Values 569 Appendix B Fundamental constants 569 575 Magnitudes 569 577 Specialized values 571 591 Credits Bibliography Index

To the Student This is a book for life science students who are willing to use calculus. This is also a book for physical scie nce and eng inee ring students who are willing to think abo ut cells. I believe that in the future every srudent in both group s will need to know the essential core of the o thers' knowledge. In the past few years, I have attended many conferences and seminars. Increas- ingly, I have fou nd myself surrounded not only by physicists, biologists, chemist s, and eng ineers. but also by physicians, mathematicians, and entrepreneurs. These peop le come togeth er to learn from on e anot her, and the traditional academ ic distinction s between their fields are becoming increasingly irrelevant to this exciting work. I want to share som e of the ir exciteme nt with yo u. I began to wo nder how this diverse group man aged to overco me the Tower-o f- Babel syndrome. Slowly I began to realize that, even though each discipline carr ies its immense load of experim ental and theo retical detail, still the headwaters of these rivers are mana geable. and co me from a co mmon spring, a handful of simple, gen eral ideas. Armed with these few ideas, I found that on e can understand an eno rmo us amount of front line research. This book explo res these first co mmo n ideas, ruthlessly suppressing the mo re specialized on es for later. I also realized that my ow n undergraduate education had po stponed the intro- duction of many of the basic ideas to the last year of my degree (or even later) and tha t many programs still have thi s character: We meticu lously build a sophisticated mathematica l edifice 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 start research in their very first year and need the big picture early. Many others create interdiscip linary program s for them selves and may never even get to o ur specialized, advanced co urses. In this book, I hop e to make the big picture accessible to any student who has taken first-year physics and cal- culus (plus a smatter ing of high school chemistr y and biology), and who is willing to stretch. When yo u're don e, you should be in a po sition to read current work in Science and Nature. You won 't get every detail, of co urse. But you w ill get the sweep. Whe n we began to offer this cour se, we were surprised to find that many of ou r graduate students wanted to take it, too. In part this reflected their own compart- mentalized education: The physics students wanted to read the bio logy part and see it integrated with their other knowledge; the biology students wan ted the reverse. To o ur amazeme nt, we found that the cour se 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 boo k have \"Track-S\" adde nda add ressing this more mathemati cally experienced group. xv

I xvi To the Stu den t Physical science versus life science At the dawn of the twentieth century, it was al- read y clear that , chemically speaking, you and I are not much different from can s of soup. And yet we can do many complex and even fun thin gs we do not usually see cans of so up doing. At that time, peopl e had very few co rrect ideas abo ut how living o rgan isms create o rder from food , do wo rk, and even compute thin gs-just a lot of ina ppropriate metapho rs draw n from the technology of the day. By m id-centu ry, it began to be clear that the answe rs to many of these quest ion s would be found in the study o f very big m olecules. Now, as we begin the twenty-fi rst century, iron ically, the situatio n is inverted: The problem is now that we have way too much information abo ut those mo lecu les! We are drowni ng in infor mation ; we need an arma ture, a framework, on which to o rganize all thos e zillio ns of facts. Some life scie ntists di smi ss physics as 'reduc tio nist', tending to strip away all the details that make frogs different from , say, neutron stars. Ot hers believe th at right now so me uni fying framework is esse ntial to see the big picture. I think that the tension between the develo pme ntal/histo rical/com plex sciences and the univ er- sallahisto ricallred uctio nist o nes has been eno rmously fruitful and that the future be- longs to those who can switch fluidly between bo th kinds of bra ins. Setti ng aside philosophy, it's a fact th at the past decade or two has seen a revolu - tion in physical tech niques to get inside th e nanoworld of cells, tweak them in phys- ical ways, and measure qu an titatively th e result s. At last, a lot o f physical ideas lying beh ind th e car toons fou nd in cell biology book s are getting th e precise tests needed to co nfirm o r reject them. At the same time. even so me mech anism s no t necessarily used by Nature have proved to be of immen se techn olo gical value. Why all th e math? I said it in Hebrew, I said it in Dutch, I said it itl German and Greek; BlIt I wholly forgot (and it vexes lIle much ) That English is what yOIl speak! - Lewis Carroll, Th e Hunting ofthe Snark Life science stud ents may wonder whether all the m athematical formulas in this bo ok are really needed. This book's premi se is that the way to be sure that a theory is cor- rect is to m ake quant itative prediction s from a simplified model. then test those pre- dictions exp erim entally. The following chapters supply m any of the tools to do th is. Ultimately, I want you to be able to walk into a room with an un fam iliar problem, pull out the right tool, and solve th e problem. I realize this is not easy, at first. Actuall y, it's true that physicists so me times overdo the mathem atical analysis. In contrast. the po int of view in this book is that beautiful formulas are usually a means, not an end , in our attem pts to und erstand Nature. Usually on ly the sim plest too ls, like dim en sion al analysis, suffice to see what's going on . O nly whe n yo u've been a very good scie ntist. do yo u get the reward o f carrying o ut so me really elabo rate mathematical calcul atio n and seei ng yo ur predictions come to life in an experiment.

To the Student xvii Your other physics and math cour ses will give you the background you'll need for that. Features of tllis book I have tried to adhere to some principles while writing the book. Most of these are boring and technical, but there are four that are worth point- ing ou t here: 1. When possible, relate the ideas to everyday phen omena. 2. Say what's going Oil. Instead of just giving a list of steps, I have tried to explain why we are taking these steps, and how we might have guessed that a step would prove fruitful. This explorator y (or discovery-style) appro ach involves more words than you may be used to in physics texts. Th e goal is to help you make the difficult transition to choosingyour own steps. 3. No black boxes. Th e dreaded ph rase \"it can be shown\" hardly ever appears in Track- I. Almost all mathem atical results menti oned are actually derived here, or taken to the point where you can get them yourself as hom ework prob lems. When I cou ld not obtain a result in a discussion at this level, I usually omitted it alto- gethe r. 4. No fake data. When you see an object that looks like a graph, almost always it really is a graph. That is, the point s are somebody's actual laboratory dat a, usually with a citation. The curves aresome actual mathematical function, usually derived in the text (or in a homework problem). Graphlike sketches are clearly labeled as such. In fact, every figure carries a pedantic little tag giving its logical status, so you can tell which are actual data. which are reconstructions, and which are an artist's sketches. Real data are generally not as pretty as fake data. You need the real thing in order to develop your critical skills. For one thing, some simple theories don't work as well as you might believe just from listening to lectures. On the other hand , some unimp ressive-looking fits of th eory to experiment actually do support strong con- clusions; you need practice looking for the relevant features. Many chapters contain a section titled \"Excursion.\" These sections lie outside the main story line. Some are short articles by leading experimentalists about experi- ments they did. Othersarehistorical or cultural essays. There arealso two appendices. Please take a mom ent now to check them. They include a list of all the symbols used in the text to represent physical quantities, definitions of all the units, and numerical values for many physical quant ities, some of them useful in working the problems. Why the history? This is no t a histor y book, and yet you will find many ancient results discussed. (Many people take \"ancient\" to mean \"before Internet,\" but in this book I use the more classical definition \"before television.\") The old stuff is not there just to give the patina of scholarship. Rath er, a recurring theme of the book is the way in which physical measurements have often disclosed the existence and nature of molecular devices in cells long before traditional biochemical assays nailed down their precise identities. The historical passages document case studies where this has happened; in some cases, the gap has been measured in decades!

xviii To th e Student Even tod ay. with our immensely sophisticated armamentum of molecular bi- ology, the traditi onal knock-out -the-gen e-and-see-what-kind-of-mouse-you -get ex- perim ental strategy can be mu ch slower and more d ifficult to perform and interpret than a more direct, reach -in -and-grab- it approach. In fact, the men u of ingenio us new tools for applying physical stresses to functioning cells or their constit uents (all th e way down to th e single-mo lecule level) and quan titativety measuring th eir re- sponses has grow n rapidly in the last decad e, giving unprecedented oppo rtunities for ind irectly deduci ng wha t must be happen ing at th e molecular level. Scientists who can integrate the lesson s of bot h the biochem ical and biophysical ap proaches will be th e first ones to see the whol e picture. Knowing how it has worked in the past prepares you for your turn. Learning tlris subject If your previou s background in physical science is a first-year undergraduate course in physics or chem istr y, this boo k will have a very di fferent feel from the texts you've read so far. This subject is rapidly evolving; my presentation won't have th at aut ho ritat ive, stone-tablets feeling of a fixed, established subject, nor sho uld it. Instead, I offer you the excitement of a field in flux, a field whe re you per- sonally can make new contr ibut ions witho ut first hackin g thro ugh a ju ngle ofexisting formalism for a decade. If your pre viou s background is in life sciences, you may be accustomed to a writ- ing style in wh ich facts are delivered to you. But in this book, man y of th e assertions. and most of th e formulas, are supposed to follow from the previou s ones, in ways you can and must check. In fact. you will noti ce th e words we. us. our. let's throughout the text. Usually in scientific writing. these words are just pompous ways of saying I, me, my , an d watch me; but in th is boo k, they refer to a team con sisting of you and me. You need to figure o ut which statements arc new information and which are dedu ction s, an d work out the latt er ones. Sometimes, I have flagged especially important logical steps as \"Your Tur n\" quest ions. Most of th ese are sho rt enough tha t you can do th em on the spot before proceeding. It is essential to work th ese out yourself in order to get the skill you need in constructing new physical arguments. Each time th e text introduces a formula, take a moment to look at it and think abo ut its reasonableness. If it says x = yz]w, does it make sense that increasing w sho uld decrease x? How do th e units work out? At first, I'll walk you through the se steps; but from then on , you need to do them automatically. When you find me us- ing an unfamiliar mat hem atical idea, please talk to you r instructor as soon as po ssi- ble instea d of just bleepin g over it. Ano the r helpfu l resource is the bo ok by Sha nkar (Shankar, 1995).' Beyond the questions in the text, you will find pro blem s at th e ends of th e chap- ters. They are not as straightforward as they were in first-year physics; often you will need some common sense, some seat-of-the- pants qualitative judgment, even some advice from your instructor to get off to the right start. Most stude nts are uncomfort- able with thi s approa ch at first- it's not just youl-but in th e end this skill is going to be on e of the most valuable ones yo u'll ever learn, no matter what you do later in life. ' See th e Bibliograph y at the back o f this book.

TO the Student xlx It's a high-technology world out there, and it will be your oyster when you develop the agility to solve open-ended , quantitative problems. The problems also get harder as yo u go o n in the text , so do the early o nes even if they seem easy. IT2 1 Some sections and problems are flagged with this symbo l. These are For Mature Audie nces Onl y. Of co urse, I say it that way to m ake yo u want to read them , whe the r or not yo ur instructor assigns them. These Track-2 sections take the m ath- ematical development a bit further. They forge links to what you are learn ing/will learn in ot he r physics courses. They also advertise some of the cited research liter- ature. The main (Track- I) text does not rely on these sections; it is self-contained . Even Track-2 readers should skip the Track-2 sectio ns on the first readin g. Many students find this course to be a stiff challenge. The physics stud ents have to digest a lot of biological terminology; the biology students have to bru sh up on their math. It's no t easy, but it's worth the effort : Interdisciplinary subjects like this o ne are among the mo st exci ting and fertile. I've noticed that the happi est studen ts are the o nes who team up to work together with ano ther student from a different background and do the problems together, teaching each oth er th ings. Give it a try.

To the Instructor A few years ago, my depart ment asked their und ergraduate students what they need ed but were not gett ing from us. One of th e answers was, «a co urse on biological physics.\" Our stude nts could not help noti cing all th e exciting articles in The New York Times, all the cover articles in Physics Today, and so on; th ey wanted a piece of th e action . Th is book emerged from th eir request. Around the same tim e, ma ny of my friends at other un iversities were begin - nin g to work in th is field and were keenly interested in teach ing a course, bu t th ey felt unco mfortable with the existin g texts. Some were brilliant but decades old; no ne seemed to cover the beau tiful new result s in molecu lar mo tor s, self-assem bly, and single-m olecule man ipu lat ion and ima ging th at were revolutioni zing th e field. My friends and I were also dau nt ed by th e vastness of the literatu re and ou r ow n limited penetration of the field ; we need ed a synthesis. This book is my attempt to answer that need. The book also serves to introduce muc h of the conceptual m aterial u nderlyin g th e young fields of nan ot echn ology and soft materials. It's not surprising- the m olec- ular and supr am olecular machi nes in each of our cells are th e inspiration for mu ch of nanotechnology, and th e polym ers and membrane s from which th ey are construc ted are the inspiration for mu ch of soft- mat erials science. This text was int end ed for use with a wildly diverse audien ce. It is based on a course I have tau ght to a single class conta ining students majoring in physics, biol- ogy, biochemi str y, bio physics, mate rials science, and chemical, mechanica l, an d bio - engineering. I hope the book will prove useful as a main or adjunct text for co urses in any science or engineering department. My students also vary widely in experi- ence , from sop ho mo res to thi rd-year graduate students. You may not want to try such a broa d group. but it works at Pen n. To reach th em all, th e co urse is divided int o two sections; th e graduate section has harder and more mat hematically sop histicated problems and exams. The st ructure of the boo k reflects this division, with num erou s Track-2 sections and problems covering the more advanc ed m aterial. Th ese sections T21·Iare placed at the ends of the chapters and are introduced with a special symbo l: The Track-2 sections are largely indep endent of one another, so you can assign them ala carte. I recommend that all stude nts skip th em on the first read ing. The only prerequisites for th e core, Track- L material are first-year calculus and calculus-based physics, and a distant memor y of high school chem istry and bio logy. The concep ts of calculus are used freely, bu t very little of th e technique; only th e very simplest d ifferen tial equations need to be solved . Mo re impor tant, the student needs to possess or acquire a fluency in th rowing numbers aro und, m akin g estimates. keeping track of un its, and carrying out short derivation s. The Track-2 m aterial and xx

• - ~=.....:::--- - To the Instructor xxi problems sho uld be appropriate for senior physics majors and first-year graduate stude nts . . For a o ne -se me ster class o f less experience d stude nts, yo u will probably want to skip one or bo th of Chapters 9 and 10 (or po ssibly 11 an d 12). 1'or more experienced students, yo u can instead skim the opening chap ters quickly, then spend extra time on th e advanced chapters. Wh en teachin g this course, I also assign suppleme ntary readings from one of the standard cell biology texts. Cell biology inevitably conta ins a lot of nomenclature and iconogra phy; bo th students and instructo r must m ake an investme nt in learning these. The payo ff is clear and im me diate: Not only does this investment allow one to communicate wi th professionals do ing exci ting wo rk in many fields, it is also crucial for seeing what physical problems are relevant to biomedical research. [ have made a special effort to keep the terminology and notation un ified, a diffi- cu lt task wh en spanning several disciplines. App en dix A sum ma rizes all the no tation in o ne place. Appendix B co ntains many useful numerical values. more than are used in the text. (You may find these data useful in making new homework and exam pro blems.) More details abo ut how to get from this book to a full course can be found in the Instructor's Guide, available from W. H. Freeman and Company. The Guide also contains so lutio ns to aU the problems and \"Yo u r Turn\" questions. suggested class demonstratio ns. and the computer co de used to genera te many of the graphs fo und in the text. You can use this code to create computer-ba sed problems, do class demo s, and so on. Errata to this book will appear at http : / /www .whfreeman. c om/b iologicalphysi c s Wh y doesn't my favorite topic appear? A garden isfinis hed when there is nothing left to remove. -Zen aphorism it's probably one of my favorite topics, too. But the text reflects the relentless pursuit of a few maxims: Keep it a co urse, not an en cyclopedia. The book corresponds to what I actually manage to cover (that is, what the students actually manage to learn ) in a typical 42-h our semester, plus abo ut 20% more to allow flexibility. Keep a unifi ed story line. Maintain a balance between recent results and the im portan t classica l to pics. Choose those topic s that open the most doors into physics, biology, chemistry, and engine ering . Make practically no mention of quantum theo ry, which o ur students encounter only after thi s course. Fortunately, a hu ge body of import ant biological physics

xxii To the instructor (including the whole field of soft bioma terials) makes no use of the deep quantum ideas. Restrict the discussion to concrete problemswhere the physical vision leads to falsi- fiable, quantitative predictions and wherelaboratory data areavailable. Every chap- ter presents some real experimental data. But choose prob lems that illumi nate, and are illumin ated by, the big ideas. Students want that- that's why they study science. There arecertainly other topics meeting all these criteria but not covered in this book. I look forward to your suggestions as to which ones to add to the next edition. Underlying the preceding points is a determination to present physical ideas as beautiful and important in their own right. Respect for these foundational ideas has kept me from relegating them to the curren tly fashio nable ut ilitarian status of a mere toolbag to help out with other disciplines. A few apparently d ilatory topics, which pursue the physics beyond the poi nt (currently) needed to explain biological phe- nom ena, reflect this conviction. Statlda rd disclaim ers This is a textbook, not a monograph. I am aware that many subtle subjects are presented in this book with impor tant details burnished off. No attempt has been made to sort out historical priority, except in those sections titled \"history.\" The experiments described here were chosen simply because they fit some pedagogical imperative and seemed to have particularly direct interpretations. The citation of original works is haphazard. except for my own work. which is systemat- ically not cited. No claim is made that anything in th is boo k is original, altho ugh at times I just couldn't stop myself. Is this stuff really phys ics? Should it be taught in a physics department ? If you've come this far, probably you have made up your mind already. But I'll bet you have colleagues who ask this question . The text attem pts to show, not only that many of the foun ders of molecu lar biology had physics background, but conversely that his- toric ally the study of life has fed crucial insights back into physics. It's true at the pedagogical level as well. Many students find the ideas ofstatistical physics to be most vivid in the life science context. In fact. some students take my course after courses in statistical physics or physical chemist ry; they tell me that it puts the pieces together for them in a new and helpful way. More important, I have found a group of students who are interested in studying physics but feel turned away when their physics departments offer no connections to the excitement in the life sciences. It's time to give them what they need. At the same time. your life sciences colleagues may ask, \"Do our students need this much physics?\" The answer is. maybe not in the past. but certainly in the future. You r colleagues may enjoy two recent eloquent articles on this subject (Alberts, 1998; Hopfield , 2002), and the comprehensive NRC report (National Research Council, 2003). This book tries to show that there is a qu antitative, physical sciences approach to problems, and it's versatile. It's not the only toolbox in the well-educated scientist's mind, but it's one of the powerful ones. We need to teach it to everyone, not just to physical science majors. I believe that the recent insularity of physics is only a

To the Inst ructor xxiii tempo rary aberratio n; both sides can o nly stand to prosper by renewing their once- tight linkage. Last I had the great good fortune to see statistical physics for the first time through the beautiful lectures of Sam Treiman (1925- 1999). Treiman was a great scientist and o ne of the spiritual leaders o f a great departm ent. From tim e to tim e, I still go back to my no tes from that co urse. And there he is, just as before.

Acknowledgments I think I wrote the first draft- it's hard to rem ember-but th e book yo u no w hold was shape d by man y peopl e, including many stud ents. Like th e cells in your body, nearly eve ry senten ce of this book has turned ove r at least once, thanks in part to the help of these readers. 1 am deeply grate ful to all of them. The book grew out of Cha pters 7 and 9, which themselves grew out of a set of lectures I gave at th e ln stitut d' Etud es Scientifiques in Cargese, on the island of Co rsica. I thank Bertrand Fourcade for inviting me to give these lectures. Of co urse, there is a certai n distance between a few lectures and a boo k; Rama- m urti Shankar and Joseph Dan pitile ssly explained to m e why 1 had to go this who le distance. Throu gh out the journey, I benefited from N ily Dan's inc isive sugges tio ns, which covered every issue of stra tegy and indeed the book's wh ole purpose. The road wou ld also have been mu ch darker wi thout the constant suppo rt and insight o f my friends Gin o Segre and Scott Weinstein . In an intense collaboration lasting several months, Sarina Bromberg made extensive improvements to both the science and the presentation, including the permutation of entire chapters and sections. Dr. Bromberg's background in bio- chemistry saved me from many missteps, small and large; she also created several of the boo k's most comp lex graphics. The book owes a very di rect debt to th e authors who contributed th e Excur- sions in Chapters 6 and 11, to Marko Rado savljevic, who wrote many of the prob- lem solutions, and to many othe r colleagues wh o cont rib uted, or helped m e find , striking graphics: How ard Berg, Paul Bian cani ello, Scott Brad y, David Deamer, David Derosier, Tony Dinsmore. Dennis Discher. Ken Down ing. Deborah Kuchnir Fygenson, David Goodsell, Julian Heath, John Heuser, Nobuta ka Hirok awa, A. James Hudspeth, Sir And rew Hu xley, Milo slav Kalab, Trevor Lam b, Jan Liphardt, Berenike Maier, Elisha Moses, Steve Nielsen, lwan Schaa p, Ch ristoph Sch midt, Corn elis Storm, Karel Svoboda, and Iun Zhang. Amo ng the many readers, I'd especially like to mention major con tribu- tions from David Busch, Mich ael Farr ies, Rodrigo Guerra, Rob Ph illips, and Tom Pologruto, each of whom generously committed their time and zeal to making this a better book . Beyond helping me to catch errors. these readers also made many suggestions abo ut the bo ok's pedagogy. Many other colleagues answered my endless question s, supplied their ex- perimental data. explained their research to me. and gave me helpful sugges- tion s or criticism. including Ralph Amado, Charles Asbury. Howard Berg. Steven Block. Robijn Bruinsma, Vincent Croquette. David Deamer. Dennis Discher. David Fung, Raym ond Goldstein, A. James Hudspeth, Wolfgang Junge, Randall Kamien, xxiv

Acknowledgments XXV David Keller, Matthew Lan g, Tom Lubensky, Ma rcelo Magnasco, John Marko, Simo n Mochrie, Alan Perelson, Charles Peskin, Dan Rot h man, Jeffery Saven, Mark Schnitzer, Udo Seifert , Cornelis Storm, Kim Sharp, Edwin Taylor, Koen Visscher, Don ald Voet, Michelle Wang, Eric Weeks, and John Weeks. . Still oth er colleagues reviewed one or more chapters and pounced on false- hoods, typos. scams, sloppiness, obfuscation. and missed opportunities. includ- ing Clay Armstro ng, John Broadhurst, Russell Co mpos to, M. Fevzi Daldal, Isard Dunietz, Bret Flanders, Jeff Gelles, Mark Goulian, Tho mas Gruhn, David Hackney, Steve Hagen, Donald Jacobs, Pon zy Lu, Kristina Lynch, John Marko, Eugene Mele, Tom Moody, John Nagle, Lee Peachey, Scott Poethi g, Tom Powers, Steven Qu ake, M. Thomas Record, [r. , Sam Safran, Brian Salzberg, Mark Schn itzer, Paul Selvin, Peter Sterling, Steven Vogel, Roy Wood, Michael Wortis, and Sally Zigmo nd. Th is book has been extensively tested on live organisms. I am grateful to those colleagues who taught courses using draft versions of the book and, ofcourse, to their long-sufferin g students as well: Anjum Ansari (University of Illinois at Chicago) , Thomas Duke (Un iversity of Cambridge), Michael Fisher (University of Maryla nd ), Bret Fland ers (Oklaho ma State), Erwin Frey (Hahn- Meitner Institut, Freie Uni- versitat Berlin ), Steve Hagen (Un iversity of Florida ), Gus Hart (Northern Arizon a University), John Hegseth (University of New Orleans), Jane Kondev (Brandeis University), Serge Lem ay (Delft University of Tech nology), Bob Martinez (Univer- sity of Texas at Austin ), Car l Michal (University of British Colum bia), John Nagle (Carn egie Mellon University), David Nelson (Harvard University), Rob Phillips (Californ ia Inst itute of Technology), Tom Powers (Brown University), Daniel Reich (Johns Hopkins University), Michael Schick (Un iversity of Washington ), Ulrich Schwarz (Max-Planck-Institut fu r Kolloid - und Grenzflachenforsch ung), Har vey Shepard (Un iversity of New Hampshire ), Ho lger Stark (Universitat Konstanz), Koen Visscher (University of Arizona), Z. Jane Wan g (Corne ll University) , Shimon Weiss (University of Californ ia, Berkeley), Ch ris Wiggins (New York University), Charles Wolgemuth (University of Connecticut Health Center), and [un Zha ng (Courant Institute). I'd especially like to thank my own students at Penn , partic ularly Cristian Dobre, Thomas Pologruto, and Kathleen Vernovsky, and my class assistants David Busch, Alper Corlu, Corey O'Hern, and Marko Radosavljevic for their enthusiasm and spirit of collaboration. This book could not have been written without the support of several institu- tion s. Mygreatest debt is to the University of Pen nsylvania. Besides being a nonstop circus of terrific science, Penn has for sixteen years supported my ideas about teach- ing and research, in very tan gible ways. Certa inly the com plet ion of this book has de- pend ed on having colleagues and two depar tm en t Chairs who supported a vision that involved my droppi ng many ot her balls over the last three years. The cru cial earliest stage of the project also benefited from the warm hospi tality of the Weizman n Insti- tute of Science; I am grateful to Elisha Moses for creating thi s opportunity. Finally, some of the most arduous revision was don e in the infinite calm of the Aspen Center for Physics, and the infinite hu bbu b of the Philadelphia Museum of Art. Partial financial support for this work was provided by the National Science Foundation's Division of Undergraduate Education (Course, Curriculum, and lab- oratory Improvement Program) and by the Division of Materials Research. I'd

xxvi Acknowledgm ents particul arly like to thank G. Bruce Taggart and Herbert Levitan at NSF for their initiative in supporting me at two tricky junctures in my ever-unpredictable career. A large team of ded icated profession als have helped make the book a reality, right from its very beginnin gs. At W. H. Freeman and Compa ny, I'm partic ularly grateful to Susan Finnemore Brennan, for her constant advice and support. (The reader will thank Susan for her gentle but persistent reminders that a shorter book is a better book.) I'd also like to th ank Ellen Cash, Kathleen Civetta, Julia DeRosa, Brian Donnellan, Blake Logan, Patricia Marx, Philip McCaffrey, Eileen McGinnis. Jane O'Neill, Bill Page, Nancy Walker, Susan Wein, and Tobi Zausner at Freeman for their expert help. Finally, Jodi Simpson's contribution just can't be reduced to the words copy editor. Words like mastermind or nerve center spring to mind when I recall her unifying influence on practically every sentence. At Penn, it's been a great pleasureworking with Felice Macera as he created many of the drawin gs. Annette Day and Jean O'Boyle relentlessly tracked down countless references and copyright permissions. Steven Nelson offered his expertise in color reproduction. Last, I've been fortun ate to have the help ofLeslie Galen, who created the book's graceful design. Donald DeLand, Leslie Galen, and the rest of the Int egre Technical Publishing team then carr ied out the intri cate typo graphy on a crushi ng sched ule, despite all my attempts to derail them with endless revisions. Of course, my biggest debt is to the people who invented all these beautiful sto- ries and to those friends and strangers who first told them to me, through theirbooks, articles, and talks; in their classrooms; while watching meteor showers or riding on the subway; and so on. Surely most of these people don't even realize how happy th ey have made me. Thank you. Philip Nelson Philadelphia, March 2003

PART I Mysteries, Metaphors, Models ~- - +.. t f t.,, t Transduction of free energy. [Drawing by Eric Sloane, from Eric Sloane. Diary ofan early American boy (Funk and Wagna lls, New York , 1962).)



1CHAPTER What the Ancients Knew Although thereis no direct connection between beerand the First Law of thermodynamics, the influence of Joule's professional expertisein brewingtechnologyon his scientific work is clearly discernible. - Hans Christian von Baeyer, Warmth disperses and tim e passes The mo dest goal of th is book is to take you from the mid- nineteenth century, where first-year physics co urses often end, to the science headlin es you read this morning. It's a long road. To get to OUf destin ation on tim e, we'll need to fo cus tightly on just a few core issues involving the interplay between energy, information, and life. We will eventua lly erect a framew ork, based on on ly a few prin ciples, in which to begi n addressing these issues. It's not enough sim ply to enunciate a handful of key ideas, of course. If it were, then this book could have been publi shed on a single wallet card. The pleasure, the depth, the craft of our subject lie in th e details of how living organisms work out the solutions to their challenges within the framework of physical law. The aim of the book is to show you a few of th ese details. Each chapter of this boo k opens with a biological question, and a terse slogan encapsulating a physical idea relevant to the question. Think about these as you read th e chapter. Biological question: How can living organisms be so highly ordered? Physical idea: The flow of energy can leave beh ind increased order. 1.1 HEAT Living organisms eat, grow, reproduce, and compute. They do these things in ways th at appear totally different from man -made machines. One key difference involves the role of temperature. For example, if you chill your vacuum cleaner, or even your television, to a degree above freezing, these appliances continue to work fine. But try this with a grasshopper, or even a bacterium, and you find that life processes practi- cally stop. (After all, that's why you own a freezer in the first place.) Unde rstanding th e interplay of heat and work will become a central obsession of this book. This chapter will develop some plausible but preliminary ideas abou t this interplay; Part 11 of the book will sharpen these ideas into precise, quantitative tools. 3

4 Cha pte r 1 Wh a t the Ancie nts Knew 1.1 .1 Heat is a form o f ene rgy Whe n a ro ck of mass 111 falls freely. its altit ude z an d veloci ty v cha nge togeth er in ju st suc h a way as to ens ure that the quant ity E = mgz + ~ mv2 stays constant, where g is the acceleration of gravity at Earth's surface. Example: Show this. Solut ion : We need to show that th e time deri vative ~ equals O. Taking v to be the velocit y in th e upward direction i, we have v = ~ . Applying the cha in rule from calculus th en gives ~ = mu(g + ~). But th e accelerati o n, ~. is always eq ual to -g in free fall. Hence. ~ = 0 throughout the motion: Th e energy is a constant. Go ttfried Leib nit z o btai ned this result in 1693. We call th e first ter m of E (that is. mgz) the poten tia l energy of the rock, and the second term ( i mv2) its ki netic ene rgy. We'll call th eir sum th e mechanical energy of the rock. We exp ress the con sta ncy of E by saying th at \"energy is conserved.\" Now suppose our rock lands in so me mud at z = O. The instant before it lands, its kin etic energy is non zero, so E is nonzero, too. An instant later, th e rock is at rest in the mud and its total mechanical energy is zero. Apparently, mechanical energy is not conserved in the presen ce of mud! Every first-year ph ysics stude nt learn s why: A mys- teriou s \"frictional\" effect in the m ud d rain ed off th e mec hanical energy of the rock. Th e genius of Isaac Newto n lay in part in his realizing th at th e laws of motion wer e best stud ied in the context of the motions of can no nba lls and planets, where compli- cations like frict iona l effects are tiny: Here th e conserva tio n of ene rgy. so ap pare ntly false on Earth, is mo st clearly seen . It took another two centuries before others would ar rive at a precise statement of the more subtle idea tha t Friction converts m echanical energy into therm al form. Wh en ther- ( 1.1) mal energy is properly accounted for, the energy accounts balance. Th at is, the actua l co nserved qu an tity is not th e mechanical ene rgy, but th e total ene rgy, th e sum of the mechanical energy plus heat. But wha t is friction? What is heat? On a practical level, if energy is conserved. if it cannot be created o r destroyed. the n why m ust we be ca reful no t to \"waste\" it? Indeed, wha t co uld \"was te\" mean ? We' ll need to loo k a bit mor e deeply befor e we really un de rstand Idea i . i .' Idea 1.1 says that frictio n is no t a process of energy loss bu t rather of ene rgy conversion. just as the fall of a ro ck co nver ts potenti al to kinetic en ergy. You may have seen an illustrat io n ofene rgy conversio n in a grammar school exercise exp loring .l{ the pathways that could take energy from th e Sun and convert it to useful work, for exa mple, a trip up a hill (Figure I. I ). A po int your schoolteache r may not have men tioned is th at, in princip le. all the energy conversions in Figure 1.1 are two-way: Light from th e Sun can generate electr icity in a solar cell, that energy can be part ially conver ted back to light with a \"Througho ut this boo k, the references Equation 1I.m, Idea /1.111, and Reaction 11.1n all refer to a single sequence of num bered items. Thus Equatio n 1.2 comes after Idea 1.1; there is no Idea 1.2.

1.1 Heat 5 Figu re 1.1 : (Diagram.) Various ways to get up a hill. Each arrow represents an energy-conversion process. light bu lb, and so on . The key word here is partially. We never get all the orig inal energy back in th is way: Some is lost as heat, in both the solar cell and the light bulb . The word lost doesn't imply that energy disappears, but rather that some of it makes a one-way conversion to heat. 0 '.r The same idea holds for the falling rock. We could let it down on a pulley, taking so me of its gravitational potential energy to run a la\\vniilt),~er. But if we just let it plop into the mud , its mechani cal energy is lost. Nobody has ever seen a rock sitting in warm mud suddenly fly up into space, leaving cold mud beh ind , even though such a process is perfectly compatible with the conservation of energy! l./ So, even though energy is strictly conserved, somethinghas been wasted when we let the rock plop. To make a scientific theory of th is something, we'd like to find an independent, measurable quantity describing the \"quality\" or \"usefulness\"of energy; then we could assert that sunlight, or the potential energy of a rock, has high qual- ity, whereas therm al energy (heat) has poo r quality. We could also try to argue that the quality o f energy always degrades in any transaction , and thus explain why the co nversio ns indicated by arrows in Figure 1.1 are so much easier than those moving against the arrows. Before doi ng these things. though . it's wo rthwhile to recall how the ancie nts arrived at Idea 1.1.

r 6 Chapte r 1 What the Ancients Knew 1.1 .2 Just a little histo ry k. Physicists like a tidy world with as few irreducib le concepts as possible. If mechanical energy can be converted to thermal energy, and (par tially) reconverted back again, and the sum of these forms of energy is always constant, then it's attractive to suppose that in some sense these two forms of energy are really the same thing. But we can't build scientific theories on eestb etic, culturally dependent judgments-Nature cares ~ ( -~-P' ., -\" .J.... little for OUf prejudices. a\"n..;d'.o--ther eras have had different prejudices. Instead. we must p-Pranchor Idea Li on some firmer ground. An example may help to underscore this point. We remember Benjamin Franklin as the great scientist who developed a theory of electricity as an invisible fluid. FrankJin proposed that a positively charged body had \"too mu ch\" of this fluid' and a negative body \"too little.\" When such bodies were placed in contact, the fluid flowed from one to the other, much like joining a cylinder of compressed air to a balloon and opening the valve. What's less well~emem bered is that Franklin, and most of his ~c~ntem pora ries) had a similar vision of heat. In this view, heat also was an invisible fluid. Hot bodies had \"too much;' cold bodi es \"too little.\" When one placed such bodies in contact, the fluid flowed until the fluid was under the same \"pressure\" in each-or in ot her words, until both were at the same temp erature. The fluid theory of heat made some fUPerficial sense. A large body would need more heat fluid to increase its temperature by on e degree than would a small body, just as a large balloo n needs mo re air than does a small one to increase its internal pressure to, say. 1.1 times atmo spheric pressure. Nevertheless. today we believe that Franklin's theory of electricity was exactly correct. but the fluid theory ofheat was dead wrong. How did this change in attitudes come abo ut? Franklin's contemporary Benjamin Thomp son was also intrigued by the prob- lem of heat. After leaving the American colonies in a hurry in 1775 (he was a spy for the British), Thompson eventually became a major general in the court of the Duke of Bavaria. In the course of his duties, Thompson arranged for the manufacture of weapons. A curious phenom enon in the boring (drilling) of cannon barrels aroused his cur iosity. Drilling takes a lot ofwork, at that time supplied by horses. It also gener- ates a lot of frictional heat. Ifh eat were a fluid, one might expect that rubbing would transfer som e of it from one body to another.just as brushing your cat leaves cat and bru sh with opposite electrical charges. But the drill bit doesn't grow cold while the cannon barrel becomes hot! Bothbecome hot. Moreover, the fluid theory of heat seems to imply that eventually the cannon barrel would become depleted of heat fluid and that no more heat could be gener- ated by additi on al friction . This is not what Thom pson observed. On e barrel could generate eno ugh heat to boil a surrounding bath of water. The bath could be replaced by cool water, which would also eventu ally boil, ad infinit um. A fresh cannon barrel proved neither better nor worse at heating water than one that had already boiled many liters. Thompson also weighed the metal chips cut out of the barrel and found \"Pranklin's convention for the sign of charge was unfortunate. Today we know that the main carriers of charge-electrons-each carry a negative quantity of charge in his co nventio n. Thus, it's more accurate to say that a positively charged body has too few electrons, and a negatively charged body too many.

- 1.1 Heat 7 their ma ss plus that of the bar rel to be equal to the origina l ma ss of the barrel: No material substance had been lost. Wha t Tho m pson no ticed instead was that heat production f rom fr iction ceases the moment we stop doing mechanical work on the system. This was a suggestive ob - servation . But later work, present ed independentl y in 1847 by James Joul e and Her - mann von Helmholtz, went mu ch fur ther. Joul e and Helmholt z up graded Thomp- son's qualitative observatio n to a quan tita tive law: The heat produced by fricti on is a constant times the mechanical work done against that friction, or (heat produced ) = (mechan ical ene rgy in put) x (0.24caIfJ ). (1.2) Let's pau se to sor t out the sho rthand in th is for m ula. We measure hea t in calorie s: One calorie is ro ughly th e amo u nt of heat needed to warm a gram of water by one degree Celsius.' The mechan ical energy input, or work done, is th e force applied (in Thompson's case, by the horse), tim es th e distance (walked by the horse); we measure it in joules ju st as in first-year physics. Multiplying work by the constant 0.24 cal / J creates a quant ity with un its of calories. The formula asserts that thi s quantity is the amoun t of heat created . Equation 1.2 sharpens Idea 1.1 into a qu antitative assertion. It also succinctly predicts the outcom es of several di fferent kinds of experiments: It says that the horse will boil twice as many liters of wate r if it walks twice as far, or walks equally far wh ile exerting twice the force, and so on. It thu s conta ins vastly more info rma tion than the precise bu t limited stateme nt that hea t ou tput stops when work inpu t stops. Scientists like hypoth eses that mak e such a sweeping web of int erlocking predi ction s, because the success of such a hypothesis is hard to brush aside as a mere fluke. We say that such hypo theses are highly falsifi ab le, becau se any on e of the many pr edic- tion s of Equation 1.2, if disproved experime ntally, wou ld kill the whole thi ng. Th e fluid theo ry of heat ma de no comparably broad , correc t predi ctions. Indeed , as we have seen, it does make some wrong qualitative pr edictions. This sor t of reasoni ng ultimately led to the dem ise of th e fluid th eory, despite the stren uo us efforts of its powerful adheren ts to save it. Suppose that we use a very dull d rill bit , so th at in one revolution we mak e little progress in dri lling; tha t is, the can no n barrel (and the drill itself) are not changed very mu ch. Equation 1.2 says that th e net work do ne on the system equals th e net heat given off. More generally, Suppose that a system undergoes a process that leaves it in its original ( 1.3) state (that is, a cyclic process). Then the net of the m echanical work done on the system, and by the system, equals the n et of the heat it gives o ff an d takes in, once we convert the work into calories using Equation 1.2. -'The modern defin ition of the calori c acknowledges the mechanical equiva lent of heat: One calorie is now defined as the quantity of thermal energy created by convert ing exactly 4.184 J of mechanical work. (The \"Calorie\" appearing on nutritional statements is actually o ne tho usand of the physical scient ist's calories, or one kllocalorie.)

8 Cha pte r 1 Wh at the Ancie nt s Knew It do esn't matt er whe ther the mech anic al wo rk was done by a horse. or by a co iled spring, or even by a flywheel th at was initi ally spi nning . What about processes that do change the system und er study? In this case, we'll need to am end Idea 1.3 to account for the energy th at was sto red in (or released from ) the system. For example, the heat released when a match burn s represen ts energy ini- tially sto red in ch em ical fo rm. A trem endo us amount of nineteenth-centu ry research by Joule and Helm holt z (a mo ng many others) convinced scientists that whe n every form of ene rgy is properly incl uded, the acco unts balance fo r all the arrows in Fig- ure 1.1, and for every o ther thermal/mechanical/chem ical process. This generalized form of Idea 1.3 is now called the First Law of th ermodynami cs. 1.1.3 Preview : The concept of free energy This subsectio n is just a preview of ideas to be mad e precise later. Don't wo rry if these ideas don't see m firm yet. The goal is to build up so me intu ition . so me expecta- tion s, abo ut the interplay of order and energy. Chapters 3- 5 w ill give many con crete exam ples of thi s interp lay, to get us ready for th e abst ract formulat ion in Chapter 6. Th e quantitative co nnect ion between heat and work lent strong suppo rt to an old idea (Ne wton had d iscussed it in th e seventeenth century) th at heat really is noth- ing but a particul ar form o f mechanical en ergy, namel y, the kineti c ene rgy of the in - dividua l molecules constituting a body. In this view, a ho t body has a lot of ene rgy stored in an (im per ceptible) jiggling of its (invisible) molecules. Ce rtainly we'll have to work hard to justify claim s about th e imperceptible and the invisible. But befor e do ing this , we m ust dea l with a more direct problem. Equa tion I.2 is some times called the \"mechan ical equivalent of heat.\" The dis- cussio n in Section 1.1.1 m akes it clear, however, that this phrase is a slight mi sno m er: Heat is not fully equivalent to mech anica l wo rk, beca use one cannot be fully con - verted to the ot her. Chapter 3 will explore the view that slowly emerged in the late nineteenth cent ury, wh ich is that therma l energy is the portion of th e total energy at- tributable to random m ole cular motion (all mol ecu les jiggling in random direction s) and so is distin ct from the organized kin etic en ergy of a fallin g rock (all molecules have the same average velocity). Thu s, the random characte r of th erm al mo tion mu st be th e key to its low qu ality. In oth er wo rds. we are propos ing that the distinction between high- and low-quality energy is a matter of organizatio n. Everyone knows that an o rderly system tend s to degrade in to a disorgani zed , random mess. Sorting it back o ut again always see ms to take work. bo th in the co lloq uial se nse (so rting a big pile of co ins into penn ies, nick- els. and so on is a lot o f work) and in the strict se nse. For exam ple. an air co nditio ner consumes elec trical ene rgy to supp ress random molecular motio n in the air of your room; hen ce, it hea ts the outside wor ld more than it co ols your room . The idea in the preceding pa ragraph may be inte rest ing, but it hardly qual ifies as a testa ble physical hypothesis. We need a quantitative meas ure of the IIseflll ene rgy of a system, th e part of the to tal that can actually be harnessed to do useful work. A major goal of Chap ter 6 will be to find such a measu re, whic h we will call free ene rgy and denote by the symbol F. But we can already see what to expect. T he idea we are consi de ring is that F is less than the total energy E by an amount related to

1.2 How life generales order 9 the randomness. or disorder, of the system. More precisely. Chapter 6 will show how to characterize this disorder by using a quan tity called entrop y and denoted by the letter S. The free energy will turn out to be given by the simple formul a F=E- TS. ( 1.4) where T is the tem perature of the system. We can now state the propo sal that F mea- sures the \"useful\" ener~y of a system a bit more clearly: A sys tem held at a fixed tem pera ture T can spontaneo usly drive a ( 1.5) p rocess jf the net effect of the process is to redu ce the system 's free energy F. Thus, lEthe system 's free energy is already at a minimum , no spon taneo us change will occur. According to Equation lA , a decrease in free energy can come about either by lower- ing the energy E (rocks tend to fall) or by increasing the entropy S (diso rder tends to increase). We can also use Equat ion 1.4 to clarify ou r idea of the \"quality\" of energy: A system's free energy is always less than its mechan ical energy. If the disorder is small, tho ugh, so that T S is much smaller than E, then F '\" E; we then say that the system's energy con tent is of \"high quali ty.\" (More precisely still, we should discuss changes of ene rgy and ent ropy; see Sectio n 6.5.4.) Again, Equatio n 1.4 and Idea 1.5 are provision al- we haven't even defined the quantity S yet. Nev ertheless, they sho uld at least seem reaso nable. In particular, it makes sense that the seco nd term o n the right side of Equation 1.4 sho uld be mu l- tipli ed by T, because hotter systems have mo re thermal motio n and so sho uld be eve n more strongly influenced by the tend ency to maximi ze disorder than co ld o nes. Chapters 6 and 7 will make these ideas precise. Chapter 8 will extend the idea of free energy to include chemical form s of energy; these are also of high quality. 1.2 HOW LIFE GENERATES ORDER 1.2.1 The puzzle of biolog ical order The ideas of the previo us sectio n have a certain intuitive appeal. When we put a drop of ink in a glass of water, the ink eventually mixes, a process we will study in great detail in Chapter 4. We never see an ink- water mi xtu re spontaneo usly un m ix. Chap- ter 6 will make th is intuition precise, formulating a principle called the Second Law of thermo dynamics. Roughly speaking, it says that in an isolated system molecular disorder never decreases spo ntaneo usly. But now we are in a bit o f a bind. We have just co ncluded that a mi xture o f hy- drogen, carbo n, oxygen , nitrogen , phosphoru s, and traces o f a few o ther elements, left alone and isolated in a beaker, will never o rganize spontaneo usly to make a liv- ing organi sm. After all, even the lowliest bacterium is full o f exquisite structure (see Chapter 2), whereas physical systems tend relentlessly toward greater disorder. Yet the Earth is teemi ng with life, even thou gh lo ng ago it was barren. How indeed doe s any o rganism manage to remain alive. let alone create progeny, and even evolve to more

10 Chapter 1 What the Ancients Knew sophisticated organisms? Stated bluntl y, o ur pu zzle is, Must we suppose that living organisms somehow lie olltside thejurisdiction ofphysical law? At the end of the nineteenth century, many respected scientists still answered \"yes\" to this que stion . Their doctrine was called \"vitalism.\" Today vitalism has go ne the way of the fluid theor y of heat, as answers to the paradox of how living things generate order have em erged. Sketching a few of the details of these answers, along with their precise quantitative tests, is the goal of this book. It will take so me time to reach that goal. But we can already propose the o utlines of an answe r in the language developed so far. It's encouraging to notice that living creatures obey at least some of the same physical laws as inanim ate matter, even tho se invo lving heat. For examp le, we can measure the heat given off by a mou se, and add the wo rk it do es on its exercise wheel by using the conversion formula (Equation 1.2). Over the course of a few days, the mouse doesn't cha nge. The First Law of therm odynamics, Idea 1.3, then says that the total energy out put must be proportional to the food intake of the mouse, and indeed it's roughly tru e. (The bookk eeping can get a bit tricky-see Problem 1.7.) Thu s, livin g o rganisms don 't manage to create energy from nothi ng. Still, when we loo k around, it seems o bvio us that life is co nstantly generating order from noth- ing (that is, from diso rder). To escape from vitalism, then, we mu st reconcile this common place observation with the Seco nd Law of thermodynam ics. Such a reco nciliatio n is easier than it at first so unds. After all, a sealed jar full of dense water vapo r changes spo ntaneo usly into a jar with a pudd le of water at the bottom and very little vapo r. After this transfo rmation , the inside o f the jar is mo re organized than before: Most of the water molecules are stuck in a very thi n layer instead of moving freely througho ut the interior of the jar. But nobody would be tempted to believe that an unphysical, occult influence ord ered the water mo lecules! To see what is hap pening, we must recall that the Second Law app lies on ly to an isolated system . Even tho ugh the jar with water vapor is sealed, it gave off heat to its surro undings as the water co ndensed; so it's not isolated. And there is no thing para- doxical about a subsystem of the wo rld spo ntaneo usly increasing its order. lndeed, Jl Section 1.1.3 proposed that a system (in this case, the contents of the jar) will tend spo ntaneo usly to move toward lower free energy F, whic h is not necessarily the same as moving toward higher disord er. Accordin g to our proposed formula for F (Equa- tion 1.4), the subsystem's entropy S can indeed decrease (the water can co ndense ) without raising F, if the internal energy E also decreases by a large enou gh amo unt (via heat loss). The Earth, like o ur jar, is not an isolated system. To see whether the increase in the ordering of molecules on Earth as life began to develop really contradicts the Second Law, then, we must look globally at what flows int o and out of the Earth. Figure 1.2a depicts the stream of solar energy impinging o n Earth. Because Earth's temp erature is roughly stable over the lon g term, all o f this energy must also leave the Earth (along with a bit of geothermal energy generated here). Some of this energy is just reflected into space. The rest leaves when the Earth radiates it away as thermal energy to the rest o f the Universe. Thu s, Earth co nstantly accepts energy from the Sun, a very hot bod y, and expo rts it as radiation at its own surface temper ature. On

' .2 How life ge ne rates order 11 a hi gh-quali ty solar energy in Earth radiates low-quality heat + [life] b waste heat, 0 2 plant s : sugar, fat , p la nt t issue . . . light H20 CO2 . c waste heat, CO 2 , H20 animals : animal ti ssue , s ugar, 9th Symphony ... fat, O 2 Figu re 1.2 : (D iagram. ) (a) Energy budget of Earth's biosphere. Most of the incident high- quality energy is degraded to thermal energy and radiated into space, but some gets captured and used to create the order we see in life. (b) What plants do with energy: High-quality so- lar energy is partly used to upgrade low-energy molecules to high-energy molecules and the ordered structures they form; the rest is released in thermal form. (c) What animals do wi- energy: The high-q uality energy in food molecules is partly used to do mechanical work and create ordered structures; the rest is released in thermal form. a dead rock like the Moon, this is the whole story. But , as depicted symbolically in Figure l .zb.c, there is a more interestin g possibility. Suppose that the incom ing energy is of higher \"quality\" than the outgoing energy and hence represent s a net flow of order into the Eart h (Chap ter 6 will sharpe n thi s statement ). Then we can imagine some enterpr ising middleman inserting itself in the middle of this process and skin;;;'ing off some of the incoming flow oforder, using it to create more and better middl emen! Looki ng only at the middle layer, it would seem as though ord er were magically increasing. That is, The flow of energy through a system can leave behind increased order. (1.6)

Whallhe Ancients Knew This is life's big tri ck. The middle zon e is our biosphere ; we are th e middlemen.' Gree n plant s ingest a high-qua lity form of energy (sunligh t) , passing it th rough th eir bodies to exit as th ermal energy (Figu re 1.2b). Th e plant needs some of thi s energy just to resist the de gradi ng tendency of thermal disorder to turn its tissues int o well- m ixed chemica l solut ions. By processing even more ene rgy th rough its bod y th an thi s min imum, th e plant can grow and do some \"useful work;' for exam ple, up grad ing some of its input matt er from a low-energy form (carbon dioxide and water) to a high-energy form (carbohyd rate). Plants consume order, not energy. Closer to hom e, each of us m ust constantly process abo ut 100 joules per second ( lOOW) of high-qual ity energy th rou gh o ur bodies (for example, by eatin g the car- bohydrate mo lecules m anu factured by plan ts), even at rest. If we eat more th an th at , we can generate some excess mechan ical (ordered) energy to bu ild our hom es and so on . As shown in Figure 1.2c, th e input energy again leaves in a low-quality form (heat) . Animals, too, consume order, not ellergy. Again . life doesn't really create order from nowhe re. Life captures order. ulti- mately from the Sun. This order then tric kles through the bio sphere in an intricate set of pro cesses th at we will refer to gene rically as free energy transductions. Looking onl y at the biosph ere, it seems as th ou gh life has created orde r. 1.2.2 Osmotic flow as a paradigm for free energy transduction If the trick described in Section 1.2.1 were uni qu e to living organi sms, then we might still feel th at th ey sat outside th e ph ysical world. But nonli vin g systems can tr ansduce free ene rgy. too: Th e dr awin g on page I shows a machine th at pro cesses solar energy and performs useful work. Unfortunately, th is sort of mach ine is not a very preci se metaphor for the processes d riving living cells. Figure 1.3 sketches anothe r sort of machine. mo re closely related to wha t we are looking for. A sealed tank of water has two freely slid ing pistons. When on e piston moves to th e left, so does th e ot he r, because th e water between th em is pr actically in compressible (and un stretch able). Across the middl e of th e cham ber, we place a membran e permeab le to water but not to d issolved sugar molecules. Th e who le sys- tem is kept at roo m temperat ure: Any heat that mu st be added or removed to hold it at this temp era ture com es from (or goes into ) the surround ing room. Initially, a lum p of suga r is unc overed on the righ t side. What happens? At first, nothing seems to happ en at all. But as th e suga r dissolves and spreads th rou ghout the right -hand chamber, a mysterio us force begins to pu sh th e piston s to the right . This is an honest, mechan ical force; we cou ld use it to lift a weight . as shown in Figu re l.3a . The process is called osmotic flow. Where did th e energy to lift the weight come from? Th e only possi ble source of energy is the ou tside world. Indeed, careful m easurements show that the system ab - sorbs heat from its surround ings; some how th is th ermal energy gets converted to mech an ical work. Did n't Sect ion 1.1.3 arg ue that it is impo ssible to convert heat 4A second, largely independ ent, biosphere exists in hot ocean vents, fueled not by the Sun but by high- energy chemicals escaping from inside the Earth.

1.2 How life generates o rder 13 a semi pe rmea ble membran e sma ll to __ . .~.._ IIlot ion o f lo a d pistons b sugar big loa d Rg ure 1.3 : (Schematic.) A machine tr ansducing free energy. A cylinder filled with water is separated into two chambers by a semipermeable membran e. Th e membra ne is anchored to the cylinder. Two pistons slide freely, thu s allowing the volumes of the two chambers to cha nge as water molecules (solid dots) cross th e memb rane. The dista nce between th e pisto ns stays fixed , however, because the water between them is incompressible. Sugar molecules (open circles) remain co nfined to th e right-hand chamber. (a ) Osmotic flow: As long as the weight is no t too heavy, when we release the pistons, wat er crosses the membrane, thereby forcin g both pisto ns to the right and lifting th e weight. The suga r molecules th en spread out in to the increased volume of wate r on the right. ( b) Reverse os mosis: If we pull hard eno ugh, ho wever, the pisto ns will mo ve to the left. th ereby increasing th e concentratio n o f the sugar soluti o n in th e righ t-h and chamber and generati ng heat. completely back into mechanical work? Yes, but we are paying for th is transaction; something is gettin g used up. That something is order. Initially, the sugar molecules are partially confined: Each one moves freely, and randomly, thro ughout the region between the membra ne and the right-h and piston. As water flows through the mem- brane. forcing the pistons to the right. the sugar molecules lose some of their order (or gain some disorder), being no longer confin ed to just one-half of the total volum e of water. When finally the left side has shrunk to zero, the sugar molecules have free run of the entire volum e of water between the piston s; their disord er can't increase any more . Ou r device then stops and will yield no more work. even thou gh there's plenty ofthermal energy left in the surrounding world. Osmotic flow sacrifices molec- ular order to organize rand om therm al motion into gross mechanical motion against a load.

14 Chapter 1 What the Ancients Knew We can rephrase the above argument in the language introduced in Section 1.1.3. Idea 1.5 introduced the idea that the osmoti c machine will spontaneously move in the direction that lowers its free energy F. According to Equation lA, F can decrease even if the potential energy of the weight increases, as long as the entropy increases by a com pensating amount. But the previou s para graph argu ed that, as the pisto ns move to the right , th e disord er (and hence the entropy) increases. So, ind eed, Idea 1.5 predicts that the piston s will move to the right , as lon g as the weight is not too heavy. Now suppose we pull very hard on the left piston, as in Figure 1.3b. Th is time, a right ward mo veme nt of the piston would increase the potential energy of the weight so mu ch that F increases, despite the second term of Equation 104. Instead, th e pis- ton s will mo ve to the lef t, the region of concentrated solution will shrink and becom e more concentrated, and the system will gain ord er. This really works-it's a com - mon indu strial pro cess called reverse osmosis (or ultra filtration ). You cou ld use it to purify water before drinking it. Reverse osmosis (Figure l .3b ) is just the sort of pro cess we were looki ng for. An input of high-quality energy (in this case, mechanical work ) suffices to upgrade the order of our system . Th e energy input mu st go somewhere, accord ing to th e First Law (Idea 1.3), and indeed it do es: Th e system gives off heat in the pro cess. Wepassed energy through our system, which degraded the energyfrom mechanicaLform to thermaL form while increasing its own order. We could even mak e our ma chin e cyclic. After pulling th e piston s all th e way to the left, we dump out the contents of each side, move the pisto ns all th e way to the right (lifting the weight ), refill the right side with sugar solution, and repea t everything. Th en our machine cont inuously accep ts hig h- quality (mechanical) energy, degrad es it into th ermal ener gy, and creates mo lecular order (by separ ating the sugar solution into sugar and pure water ). But that's the same trick we ascribed to living or ganisms, as sum ma rized in Fig- ure 1.21 It's not pr ecisely the same- in Eart h's biosph ere, the input stream of high - quality energ y is sunlight, whereas our reverse-osmo sis machine runs on externally supplied mechan icai wor k. Nevert heless, much of thi s book will be devot ed to show- ing that at a deep level these proc esses, one from the living and one from the nonl iving world , are essent ially the same. In particular, Chapters 6, 7, and 10 will pick up this story and parlay ou r understandin g into a view of biomol ecular machines. The mo- tors fou nd in living cells differ from our osmotic mac hine by bein g single molecules, or collections of a few molecules. But we'll argue that these \"molecular motors\" are again just free energy tran sdu cers, essentially like Figure 1.3. They work better than simple mach ines because evolution has engineered them to work better, not because of some f unda mental exempt ion f rom physical law. 1.2.3 Preview : Disord er as information Th e osmotic ma chine illustrate s anot her key idea, on which Chapter 6 will bu ild, namely, the connection between disord er and information . To introduce this con- cept, consider again th e case of a small load (Figure 1.3a). Suppose that we mea sure experimen tally the maximum work don e by th e piston, by integrating the maxi m um force the pisto n call exert over the distance it tra vels. Doing this experiment at room temp eratu re yields an empirical ob servat ion :

1.3 Excu rsion : Commercials, philosophy, pra g matics 15 (ma ximum work) \"\" N x (4. 1 X 10- 21 J x y ). ( 1.7) Here N is the number of disso lved sugar molecu les. (y is a numer ical co nstant whos e value is not important right now ; you will find it in Your Turn 7B.) In fact. Eq uation 1.7 holds for any dilu te solution at ro om temperature, not just sug ar disso lved in wate r, regardless of th e details of the size or shape of the con tainer and the number of molecules. Such a universal law m ust hav e a deep m ean ing. To in- ter pret it, we retu rn to Equation 104. We get the m axim um work whe n we let the pis- to ns move grad ually, always app lying the biggest possible load. Acco rd ing to Idea 1.5, the largest load we can apply without stalling the m achine is the one for which the free energy F hardly decreases at all. In thi s case, Equation 1.4 cla im s that th e cha nge in potential ene rgy of th e weight (that is, th e m echani cal work done) ju st equals th e tem peratu re times the cha nge of entro py, Writing tJ.S for the entropy change, Equa- tio n 1.7 says TtJ.S \"\" N x (4. 1 X 10- 21 J x y ). We alread y have the expectation that en tropy involves diso rder, an d indeed, so me o rder does disappear whe n the pistons m ove all th e way to th e righ t in Figure l.3a . In itially, each suga r m olecul e was confine d to half th e total volume, wh ereas in the end th ey are not so co nfined . Thus, wh at's lost as the pistons move is a knowledge of wh ich half of the chambe r each sugar molecule was in-a binary choice. If th ere are N sugar m olecules in all, we need to spec ify N b ina ry digits (bits) of in form ation to specify where each o ne sits in the final state, to the same accuracy th at we knew it originally. Co mbining thi s remark with the resu lt of the previo us paragraph gives tJ.S = co nstant x (num ber of bit s lost). Thus, the entropy, whi ch we have been thin king of qualitatively as a m easur e of disor- der, tu rn s out to have a qua nt itative int er pretatio n . If we find that biom olecular mo- tors also obey so me version of Equation 1.7, wit h the same numerical constant, then we will be on firm gro u nd whe n we asse rt th at th ey really are free energy transdu ction devices; and we can make a fair claim to have learned som ething fundament al about how th ey work. Chapter 10 will develop thi s idea, 1.3 EXCURSION : COMMERCIALS, PHILOSOPHY, PRAGMATICS And oftentimes, to winne us to c nr hnrme Cell and tissue. shell and bone, leafand flower. areso many The Instruments of Darkness tell ti S Truths portionsof matter, and it is in obedience to the laws of physics that their particles have been moved. moulded, Winne us with honest trifles, to betray's and conformed. ItJdeepest consequence. - D'Arcy Thompson, On growth and form. 1917 - Shakespeare, Macbeth Section 1.2 do ve di rectly into the technical issues that we'll wres tle with throu ghout th is book. But befo re we begin o ur explorat ion in earnest, a very few words are in o rder abo ut th e relation between phy sical science and biolo gy.

16 Cha pte r 1 What th e Ancients Knew t est a ble Fig ure 1.4 : (Vision .) One approach to understanding natural phenom ena. Th e quo tes above were chosen to highlight a fru itful tension between the two cultures: The physical scientist's impulse is to look for the for est, not the trees, to see that which is universal and simple in any system. Traditionally, life scientists have been more likely to emphasize that. in the inher- ently complex living world, frozen accidents of history often dominate what we see. not universal laws. In such a world, often it's the details that really matter most. The views are complementary; one needs the agility to use whichever approach is appropriate at any given mom ent and a w illin gness to entertain the po ssibilit y that the other one is valuable, too. How can one synthesize these two approaches? Figure 1.4 shows the essential stra tegy. The first step is to look around at the rich fabri c of the pheno me na around us. Next, we selectively ignore nearly everything about these phenomena, snipping the fabric down to just a few th reads. Th is process involves (3) selecting a simplified but real model system for detailed study and (b) representing the simple system by an equally simple mathematical model, with as few independent constructs and re- lations as possible. The steps (a) and (b) are not dedu ctive; words like mystery and insight apply to this process.

1.3 Excur sion : Co mm er cia ls, philo sophy, pragm at ics 17 The last step is to (c ) deduce from the mathem atical model some non obviou s, quanti tative, and experimentally testable pred iction s. If a model makes many such successful prediction s, we gain co nvictio n that we have found the few key ingredients in our sim plifying steps (a) and (b) . Words like hygiene and technique apply to step (c) . Even tho ugh this step is dedu ctive, again imagination is needed to find tho se con - sequences of th e model th at are both nont rivial and practical to test. The best, most striking results are those for which th e right side of th e figure op ens up to embrace phenom ena that had previo usly seemed unrelated. We have already fo reshadowed an exam ple of such a global linkage of ideas: The physics of osmotic flow is linked to the biology of molecu lar machines. In the best case, th e results of step (c) give the sense of getting som et hing for noth ing: The mod el generates more structure than was present in its bare stateme nt (the middl e part of Figure 1.4), a structure, moreover, that is usually buried in the m ass of raw ph enomena we bega n wit h (left end of Figure 1.4). In add itio n, we may in the process find that the mo st satisfactory physical model involves some threads, or postul ated physical entities (mid dle part of the figu re), whose very ex;stence wasn't obv;ons from the observed phenomena (left part ) but can be substantiated by m aking and testing quant itat ive pred iction s (right part). One famo us exam ple of this process is Max Delbruck's dedu ction of the existence ofa hereditary mo lecule, to be discussed in Chapter 3. We'll see ano the r example in Cha pters I I and 12, nam ely, the discovery of ion pump s and channels in cells. Physics stud ent s are heavily trained on the right end of the figure, th e techniqu es for working through the co nsequences of a mathema tical mod el. But this techn ical expertise is no t eno ugh. Uncritically accept ing sorneone's mod el can easily lead to a large bod y of bo th theo ry and experiment culm inating in irrelevant results. Similarly, biology studen ts are heavily tra ined in th e left side, th e amass ing of m any details of a system. For them , the risk is that of becoming an archivist who mi sses the big picture. To avoid both these fates, one mu st usually know all the details of a biolo gical system, then tra nscend them with an appropriate simple m od el. Is the physicist's insistence on sim plicity, co ncreteness, and quantitative tests on model systems just an immature craving for certainty in an uncertain wo rld? Cer- tainly, at times. But at o ther times . this approach lets us perceive conn ection s not visible \"on the ground\" by viewing th e world \"from above.\" When we find such uni- versality. we get a sense o f having explained so me thing. We can also get more prag- matic benefits: Often, whe n we forge such a lin k, we find th at powerfu l th eoretical tools useful to solve o ne problem have already been created in the con text of another. An ex- amp le is the mathematical solutio n of the helix-co il transitio n model discussed in Cha pter 9. Simi larly, we can carryover powerful existing experimentaltechniq ues as well. For exam ple. the realizatio n that DNA and proteins we re molecules led Max Perut z, Linus Pauling. Maurice Wilkins, and others to study the struc ture of these mole- cules wi th X-ray diffraction , a technique invented to find the structure of simple, nonbiological crystals like quartz.

18 Chapter 1 What the Ancients Knew Finally. perceiving a link between two circles of ideas can lead us to ask new ques- tions that later prove to be imp ortant. For examp le, even after James Watson and Francis Crick's discovery that the DNA mo lecule was a very long sentence written in an alphabet with four letters (see Chapter 3), attention did not focus at once on the importance of finding the dictionary. or code. relating sequences of those letters to the 20-lette r alphabet of amino acids that const itute pro teins. Thinkin g abo ut the problem as one in info rmation transfer led Geo rge Camow, a physicist interested in bio logy, to write an influential paper in 1954 asking this question and suggesting that answering it might not be so difficult as it at first seemed. It may seem that we need no longer co ntent o urselves with sim ple model s. Can't massive computers now follow the fine details o f any process? Yes and no . Many 10\\'1- level processes can now be followed in mo lecular detail. Nevertheless, our ability to get a detailed picture of even simple systems is sur prisingly limited , in part by th e rapid increase of co mput ational co mp lexity when we study large numb ers of par- ticles. Surprisingly, though, man y physical systems have simple \"emergent proper- ties\" not visible in the co mplex dyna mic s o f their individual mole cules. The simple equations we'll study seek to encapsu late these properties and ofte n manage to cap- ture the important features of the whole complex system. Exampl es in this boo k will include the po werful property of hydrodynami c scale invariance to be explored in Chapter 5, the mean -field behavior of ions in Chap ter 7, and th e elasticity of macro- molecules in Chapter 9. The need to exploit such simplicity and regularity in the collective behavior of many simila r actors becomes even more acute when we begi n to study even larger systems than the o nes discussed in this book. 1.4 HOW TO DO BETTER ON EXAMS (AND DISCOVER NEW PHYSICAL LAWS) Equation 1.2 and the discussio n followin g it made use of some simple ideas invo lv- ing units. Stude nts often see uni ts, and the associated ideas of dimens ion al ana lysis, presented with a brush -your -teeth attitude. This is regrettable. Dimension al analy- sis is more than just hygiene. It's a shortcut to insight, a way to organi ze and clas- sify nu mbers and situatio ns, and even to guess new physical laws. Wo rking scientists event ually realize that, when faced with an unfamiliar situatio n, dimension al analysis is always step one. 1.4.1 Most physical quantities carry dimensions A physical quantity generally has abst ract dimensions that tell us what kind of thing it represents. Each kind of dim ension can be measu red by using a variety o f different units. The cho ice of units is arbitrary. People o nce used the size of the king's foo t. This book will instead use pr imarily the Systerne Intern ation al d'Un ites, or SI units. In this

1.4 How to Do Better on Exams 19 system, len gth s are mea sured in m eters, ma sses in kilograms, time in seconds, and electric cha rge in coulombs. The distinction between dimen sions and units becom es clearer when we look at some examples: 1. Length has dimen sions of R.., by defin ition. In 51 units, we m easu re it in me ters, abbreviated in this book as m. 2. Mass has dimen sion s of M, by definit ion . In 51 un its, we m easur e it in kilograms, abbreviated as kg. 3. Time has d imension s of T , by definition. In 51 uni ts, we measure it in seconds, abbreviated as s. 4. Veloc ity has dim ensio ns ofII.:Ir- I . In 51 un its, we measur e it in m 5- 1 (pro nounced \"meters per second\"). 5. Acceleration has d imen sions of IL1I'- 2. In 51 units, we m easure it in m 5- 2. 6. Force has dim en sion s of MlI.....Ir-2. In 51 un its, we m easure it in kg m 5 - 2, wh ich we also call newton s and abbreviate as N. 7. Ene rgy ha s dimen sion s of MIL2'lI' -2 . In 51 units, we mea sur e it in kg m2 5- 2 , whic h we also call joules and abbreviate as J. 8. Electric cha rge ha s dimen sion s of Q\\ by definiti on . In 51 un its, we measur e it in cou lombs, abbreviated in th is book as co ul to avoid con fusion with th e sym bol C. The flow rate of cha rge, or electriccurrent, th en mu st have dimen sion s of Q1r- I . In 51 units, we m easu re it in coulombs per second, or cou l 5- 1, also called amperes , abbrevia ted as A. 9. We defer a discu ssion of temperature units to Section 6.3.2. No tice th at in thi s book all units are set in a special typeface, to help you distin guish th em from nam ed quant ities (such as m for th e ma ss of an obj ect ). We also create related units by attaching prefixes giga (= 109 , or billion ), m ega (= 106, or million ), kilo (= 103, or th ou san d ), milli (= 10- 3, or thousandth ), m icro (= 10-6, or m illionth ), nano (= 10- 9 , or billionth ), pico (= 10- 12) . In writing, we ab - breviate th ese prefixes to G, M, k, m, /1, n, and p, respectively. Thus, 1 Gy is a billion years, 1 pN is a billionth of a newton, and so on . Forces in cells are usua lly in th e pN ra n ge. A few non-51 units, like em and kca l, are so traditional that we'll occasionally use th em as well. You will consta ntly find these units in th e research literature, so you mi ght as well get good at intercon verting th em now. See Appe ndix A for a list of all th e units in thi s book; Appe ndix B p resents th e hiera rchy of length , ti me, and en ergy scales of interest to cell biology and pu lls together the numeri cal values of man y useful consta nts. In any qu antitative prob lem , it is absolutely crucia l to kee p units in mind at all times. Students some times don 't take dim ensional analysis too seriously because it seems tri vial, but it's a very powerfu l m ethod for catch ing algeb raic errors. A few physical qu antities are dimensionless (they are also called \"p ure num- bers\" ). For examp le, a geometrical angle is dim ensionless; it expr esses th e circumfer- ence of a part of a circle divided by th e circle's radius. Neverthe less, we some times use d imen sionl ess units to desc ribe such quantities. A dim en sionle ss unit is just an

20 Chapter 1 What the Ancients Knew abbrevia tion for some pure nu mber. Thu s the degree of angle, repr esented by the symbol 0, denotes the numb er 2rr/ 360. From this point ofview, the \"radian\" is not h- ing but the pure number I and may be dropped from formulas; we sometimes retain it just to emphasize that a particular quantity is an angle. A quantity with dimension s is sometimes called dimension al. It's important to understand that the units are an integral part of such a quantity. Thus, when we use a named variable for a physical quantity, the units are partofwhat the name represents. For example. we don't say. \"A force equal to f newtons\" but rather, «A force equal to f\" where, say,f = 5 N. In fact, a dimensional quantity should be thou ght of as th e product of a \"nu- merical part\" times some units; this viewpoint makes it dear that the numerical part depends on the units chosen. For example. the quantity 1 m is equal to the quan- tity 1000 mm. Similarly, the phrase \"ten square micrometers:' or \"10 11 m2,\" refers to 10 x (j1m)' = 10- 11 m' , not (10 I'm )' = 10- 10 m'. To convert from one unit to another, we take any equivalence between units, for example 1 ineh = 2.54 em, and recxpress it as .,.1.:.i..n::c:::h:... = 1. 2.54 em Then, we take any expression and multiply or divide by I, canceling the undesired units. For example, we can convert the acceleration of gravity to inch 5- 2 by writing 91' 100qt\\ l inch inch g = 9.8- x - 91-' x = 386-. 2.54 qt\\ 5' 5' Finally, no dimension al quantit y can be called \"large\" in any absolute sense. Thus, a speed of 1cm 5- 1 may seem slow to you, but it's impo ssibly fast to a bac- terium. In contrast, dimensionless quantities do have an absolute meaning: When we say that they are \"large\" or \"small,\" we implicitly mean \"compared with I.\" Finding relevant dimensionless combinations of parameters is often a key step to classifying the qualitative properties of a system. Section 5.2 will illustrate this idea, defining th e \"Reynolds num ber\" to classify fluid flows. 1.4.2 Dimension al analysis can help you catch errors and recall definitions Isn't this a lot of pedantic fuss oversomething trivial? Not really. Things can get com- plicated pretty quickly; for exam pie, on an exam. Training yourself to carr yall the units explicitly, through every calculation, can save you from many errors. Suppose you need to compute a force. You write down a formula that con- tains various quantities. To check your work, write down the dimensions of each of the quantities in your answer, cancel whatevercancels, and make sure the result is MIL11'-'. If it's not, yo u probably forgot to copy something from on e step to the next. It's easy, and it's amazing how many errors you can find in this way. (You can also catch yo ur instructors' errors.) When you multiply two quantiti es, the dimensions just pile up: force (MILlI'- ' ) times length (IL) has dimensions of energy (Mll.,'lI'-'). But yo u can never add or sub-

1.4 How to Do Better on Exam s 21 tract terms with different dimensio ns in a valid equatio n. any mo re than you can add do llars to kilograms. You can add euros to rupees, with the appropriate con version factor, and similarly meters to mil es. Meters and m iles are different units that both carry the same dimension, nam ely, length (IL ). Anoth er useful rule of thumb involving dime nsion s is that you can only take the exponential of a dimensionless number. The same thin g hold s for o ther familiar function s, such as sin. cos, and In. One way to understand this rule is to recall that tx'expx = I + x + + ... .According to the previous paragraph , th is sum makes no sense unless x is dimen sionless. (Recall also that the sine function's argument is an angle, and angles are dim ensionless.) Suppo se you run into a new con stant in a formu la. For examp le, the force be- tween two poi nt charges ql and q2 in vacuum, separated by distance r, is f = _ I_ q ,q, . ( 1.8) 4JrEo r2 What are the dim ension s of th e co nstant EO? Just co mpare: In this fo rmul a, the no tation [Eo] means \"the dime nsions of EO\"; it's so me co m bina- tion of lL, MJ, 11', <Q that we want to find. Remem ber that numbers like 411 have no d imen sion s. (After all, 11 is the rat io of two lengths, the circumference and the diam - eter of a cirde.) So right away, we find [EO] = <Q'1I\"1L - ' M - 1 which you can then use > to check o ther formul as conta ining EO . Finally, dimensional analysis help s yo u rem ember thing s. Suppose you're faced with an obscure SI un it, say, \"farad\" (abbreviated F). Yo u don't remem ber its defi- nition . You know it measures capaci tance. and yo u have so me formul a invo lving it, say, E = t q'j C, where E is the stored electro static energ y, q is the stored cha rge, and C is the capacitance. Starting from the dim ensions o f energy and charge, yo u find that the dim ensions of Care [C] = 1I\" <Q' M -' 1L- 2 Substituting the SI uni ts sec- o nd, coul omb, kilogram , and meter, we find that the natural SI uni t for capacitance is s'coul'kg-'m- '. That's what a farad really is. Example: App endix B lists the un its of the per m ittivity of empty space Eo as F/m. Check this statement. Solution: You could use Equation 1.8, but here's another way. Th e electrostatic po- tential VCr ) a distance r away from a point charge q is V( r) = -q-. ( 1.9) 4Jr Eo r The pot enti al energy of another charge q sitt ing at r equals qV(r) . Becau se we know the dim ensions of energy, charge , and distance, we work o ut [EO] = 1I\"<Q'MJ- 11L- 3, as we already found. Also using what we fou nd earlier for the dimensio ns o f capaci tance gives [Eo] = [CIIIL, so the SI uni ts for EO are the same as those for capacitance per length, or F m\" .

22 Chapter 1 What the Ancients Knew 1.4.3 Dimensional analysis can also help you fo rmulate hypotheses Dimensional analysis has oth er uses. Fo r exam ple, it can actually help us to guess new physical laws. sChapter 4 will discuss the \"visco us friction co efficient\" for an objec t immersed in a fluid. This parameter equals the force applied to the object, divided by its re- suIting speed; so its dimension s are MIT. We will also discu ss another quantity, the \"diffusion constant\" D of the same object, which has dim ensions IL' nL Both I; and D depend in very complicated ways on the temperature, the shape and size of the object, and the nature of th e fluid. Suppose now that someone tells you th at, despite this great compl exity, the prod- uct SD is very simple: This product dep ends only on the temp erature, not on the na- ture of the object no r even on the kind of fluid it's in. What could the relation be? You work out the dim ension s of th e product to be M IL'/T o. That's an energy. What so rt of en ergy scales are relevant to our prob lem ? It o ccurs to you that the energy of therm al mo tion , Ethermal (to be discussed in Chapter 3), is relevant to the physics of friction , because friction makes heat. So you could guess that if there is any funda- menta l relation, it mu st have the form , ( l. to) { D == Ethcrmal. You win. You have just guessed a true law of Nature, one that we will derive in Chapter 4. In this case, Albert Einstein got there ahead of you, but maybe next tim e you'll have pr iorit y. As we'll see, Einstein had a specific goal: By measuring bo th I; and D experimentally, he realized, on e could find Ethcrmal. We'll see how this gave Einstein a way to measure how big atoms are, without ever needin g to manipula te them individually. And .. . atoms really are rha t size! What did we really accomplish here? This isn't th e end, it's the beginn ing: We didn't find any explanation of frictional drag, nor of diffusion , yet. But we know a lorabo ut how that theory should work. It has to give a relatio n that looks like Equa- tion 1.10. This result helps in figuring out the real theory. 1.4.4 Some notational conventions involving flux and density To illustrate how units help us disentangle related concepts, consider a family of re- lated qua nt ities tha t will be used throughout the book. (See Append ix A for a com - plete list of symbols used in the book .) • We will often use the symbols N to denote the numb er of discrete things (a dim en- sionless integer), V to denot e volume (with 51un its rn\"), and q to denote a quanti ty of electric charge (with 51un it co ul). • The rates of change of these quanti ties will generally be writte n dN / dt (with unit s 5- \\ ) , Q (the volume flow rate, with unit s m3 5- J ) , and I (the electric current, with units caul 5 - 1) , respecti vely. If we have five balls in a room of volum e 1000 rrr ' , we say that the nu mbe r density (or concentration ) of ba lls in the room is c = 0.005 m- 3 . Densities of dimensional quantit ies are traditionally denoted by the symbol p; a subscript will indic ate what