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More About This Textbook
Overview
Life is a chancy proposition: from the movement of molecules to the age at which we die, chance plays a key role in the natural world. Traditionally, biologists have viewed the inevitable "noise" of life as an unfortunate complication. The authors of this book, however, treat random processes as a benefit. In this introduction to chance in biology, Mark Denny and Steven Gaines help readers to apply the probability theory needed to make sense of chance events—using examples from ocean waves to spiderwebs, in fields ranging from molecular mechanics to evolution.
Through the application of probability theory, Denny and Gaines make predictions about how plants and animals work in a stochastic universe. Is it possible to pack a variety of ion channels into a cell membrane and have each operate at nearpeak flow? Why are our arteries rubbery? The concept of a random walk provides the necessary insight. Is there an absolute upper limit to human life span? Could the sound of a cocktail party burst your eardrums? The statistics of extremes allows us to make the appropriate calculations. How long must you wait to see the detail in a moonlit landscape? Can you hear the noise of individual molecules? The authors provide answers to these and many other questions.
After an introduction to the basic statistical methods to be used in this book, the authors emphasize the application of probability theory to biology rather than the details of the theory itself. Readers with an introductory background in calculus will be able to follow the reasoning, and sets of problems, together with their solutions, are offered to reinforce concepts. The use of realworld examples, numerous illustrations, and chapter summaries—all presented with clarity and wit—make for a highly accessible text. By relating the theory of probability to the understanding of form and function in living things, the authors seek to pique the reader's curiosity about statistics and provide a new perspective on the role of chance in biology.
Editorial Reviews
Nature Cell Biology
An excellent introduction to the uses of probability theory for a reader who is more familiar with biology than with mathematics . . . Denny and Gaines have done a valuable service to biologists who are interested in a quantitative approach to life sciences.— Paul Janmey
American Scientist
A lively, wellwritten text. . . . A student who reads this book closely will come away with a much deeper appreciation for the universality of diffusion mechanics in science, the deep connections between the distributions central to inferential statistics, the importance of extreme events and how to deal with them analytically, and, most importantly, the power and limitations inherent in the underpinning of the inferential statistics that the student has learned elsewhere.— Mark R. Patterson
Journal of Experimental Biology
This is a fantastic book. Indeed, one would be hardpressed to find a more readable and lucid introduction to probability theory.— Gary B. Gillis
Nature Cell Biology  Paul Janmey
An excellent introduction to the uses of probability theory for a reader who is more familiar with biology than with mathematics . . . Denny and Gaines have done a valuable service to biologists who are interested in a quantitative approach to life sciences.American Scientist  Mark R. Patterson
A lively, wellwritten text. . . . A student who reads this book closely will come away with a much deeper appreciation for the universality of diffusion mechanics in science, the deep connections between the distributions central to inferential statistics, the importance of extreme events and how to deal with them analytically, and, most importantly, the power and limitations inherent in the underpinning of the inferential statistics that the student has learned elsewhere.Journal of Experimental Biology  Gary B. Gillis
This is a fantastic book. Indeed, one would be hardpressed to find a more readable and lucid introduction to probability theory.From the Publisher
"An excellent introduction to the uses of probability theory for a reader who is more familiar with biology than with mathematics . . . Denny and Gaines have done a valuable service to biologists who are interested in a quantitative approach to life sciences."—Paul Janmey, Nature Cell Biology"A lively, wellwritten text. . . . A student who reads this book closely will come away with a much deeper appreciation for the universality of diffusion mechanics in science, the deep connections between the distributions central to inferential statistics, the importance of extreme events and how to deal with them analytically, and, most importantly, the power and limitations inherent in the underpinning of the inferential statistics that the student has learned elsewhere."—Mark R. Patterson, American Scientist
"This is a fantastic book. Indeed, one would be hardpressed to find a more readable and lucid introduction to probability theory."—Gary B. Gillis, Journal of Experimental Biology
Product Details
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Chance in Biology
Using Probability to Explore NatureBy Mark Denny Steven Gaines
Princeton University Press
Copyright © 2002 Princeton University PressAll right reserved.
ISBN: 0691094942
Chapter One
The Nature of Chance1.1 Silk, Strength, and Statistics
Spider silk is an amazing material. Pound for pound it is four times as strong as steel and can absorb three times as much energy as the KEVLAR from which bulletproof vests are made (Gosline et al. 1986). Better yet, silk doesn't require a blast furnace or a chemical factory for its production; it begins as a viscous liquid produced by small glands in the abdomen of a spider, and is tempered into threads as the spider uses its legs to pull the secretion through small spigots. Silk threads, each a tenth the diameter of a human hair, are woven into webs that allow spiders to catch prey as large as hummingbirds. Incredible stuff!
The strength and resilience of spiders' silks have been known since antiquity. Indeed, anyone who has walked face first into a spiderweb while ambling down a woodland path has firsthand experience in the matter. Furthermore, the basic chemistry of silk has been known since early in this century: it is a protein, formed from the same amino acids that make our skin and muscles. But how does a biological material, produced at room temperature, get to be as strong as steel and as energyabsorbing as KEVLAR? Therein lies a mystery that has taken years to solve.
The first clues came in the 1950s with the application of Xray crystallography to biological problems. The information provided by this technique, used so successfully by James Watson and Francis Crick to deduce the structure of DNA, allowed physical chemists to search for an orderly arrangement of the amino acids in silksand order they found. Spider silk is what is known as a crystalline polymer. As with any protein, the aminoacid building blocks of silk are bound together in long chains. But in silks, portions of these chains are held in strict alignmentfrozen parallel to each other to form crystalsand these long, thin crystals are themselves aligned with the axis of the thread. The arrangement is reminiscent of other biological crystalline polymers (such as cellulose), and in fact can account for silk's great strength.
It cannot, however, account for the KEVLARlike ability of spider silk to absorb energy before it breaks. Energy absorption has two requirements: strength (how much force the material can resist) and extensibility (how far the material can stretch). Many crystalline polymers have the requisite strength; cellulose, for example, is almost as strong as silk. But the strength of crystalline polymers is usually gained at the loss of extensibility. Like a chain aligned with its load, a protein polymer in a crystal can extend only by stretching its links, and these do not have much give. Cellulose fibers typically can extend by only about 5% before they break. In contrast, spider silk can extend by as much as 30% (Denny 1980). As a result of this difference in extensibility, spider silk can absorb ten times more energy than cellulose. Again, how does silk do it?
This mystery went unsolved for 30 years. As powerful a tool as Xray crystallography is, it only allows one to "see" the ordered (aligned) parts of a molecule, and the ordered parts of silk (the crystals) clearly don't allow for the requisite extension. If silk is to be extensible, some portion of its molecular structure must be sufficiently disordered to allow for rearrangement without stretching the bonds between amino acids in the protein chain. But how does one explore the structure of these amorphous molecules?
The answer arrived one dank and dreary night in Vancouver, British Columbia, as John Gosline performed an elegant, if somewhat bizarre, experiment (Gosline et al. 1984). He glued a short length of spider silk to a tiny glass rod, and, like a sinker on a fishing line, glued an even smaller piece of glass to the thread's loose end. Pole, line, and sinker were then placed in a small vial of water so that the spider silk was vertical, held taut by the weight at its end (fig. 1.1). The vial was stoppered and placed in a second container of water in front of a microscope. By circulating water through this second bath, Gosline could control the temperature of the silk without otherwise disturbing it, and by watching the weight through the microscope he could keep track of the thread's length. The stage was set.
Slowly Gosline raised the temperature. If silk behaved like a "normal" material, its length would increase as the temperature rose. Heat a piece of steel, for instance, and it will expand. In contrast, if the molecules in the amorphous portions of the silk are sufficiently free to rearrange, they should behave differently. In that case, as the temperature rose, the amorphous protein chains would be increasingly rattled by thermal agitation and should become increasingly contorted. As with a piece of string, the more contorted the molecules, the closer together their ends should be. In other words, if the amorphous proteins in silk were free to move around, the silk should get shorter as the temperature was raised. This strange effect could be predicted on the basis of statistics and thermodynamics (that's how Gosline knew to try the experiment) and had already been observed in manmade rubbery materials. Was spider silk built like steel or like rubber?
As the temperature slowly drifted up10ºC, 12ºC, and higherthe silk slowly shortened (fig. 1.2). The amorphous portions of silk are a rubber! From this simple experiment, we now know a great deal about how the noncrystalline portions of the silk molecules are arranged (fig. 1.3), and we can indeed account for silk's great extensibility. Knowing the basis for both silk's strength and extensibility, we can in turn explain its amazing capacity to absorb energyknowledge that can potentially help us to design manmade materials that are lighter, stronger, and tougher.
There are two morals to this story. First, it is not the orderly part of spiders' silk that makes it special. Instead, it is the molecular disorder, the random, evershifting, stochastic arrangement of amorphous protein chains that gives the material its unique properties. Second, it was a knowledge of probability and statistics that allowed Gosline to predict the consequences of this disorder and thereby perform the critical experiment. As we will see, the theory of probability can be an invaluable tool.
Probability theory was originally devised to predict the outcome in games of chance, but its utility has been extended far beyond games. Life itself is a chancy proposition, a fact apparent in our daily lives. Some days you are luckyevery stoplight turns green as you approach and you breeze in to work. Other days, just by chance, you are stopped by every light. The probability of rain coinciding with weddings, picnics, and parades is a standard worry. On a more profound level, many of the defining moments of our lives (when we are born, whom we marry, when we die) have elements of chance associated with them. However, as we have seen with spider silk, the role of chance in biology extends far beyond the random events that shape human existence. Chance is everywhere, and its role in life is the subject of this book.
1.2 What Is Certain?
As an instructive example, imagine yourself sitting with a friend beside a mountain stream, the afternoon sun shining through the trees overhead, the water babbling as it flows by. What can you say with absolute certainty about the scene in front of you? Well, yes, the light will get predictably dimmer as the afternoon progresses toward sunset, but if you look at one spot on the river bank you notice that there is substantial shortterm variation in light intensity as well. As sunlight propagates through the foliage on its way to the ground, the random motion of leaves modulates the rays, and the intensity of light on the bank varies unpredictably both in space and in time. Yes, a leaf falling off a tree will accelerate downward due to the steady pull of gravity, but even if you knew exactly where the leaf started its fall, you would be hardpressed to predict exactly where it would end up. Turbulent gusts of wind and the leaf's own tumbling will affect its trajectory. A close look into the stream reveals a pair of trout spawning, doing their instinctive best to reproduce. But even with the elaborate rituals and preparations of spawning, and even if all the eggs are properly fertilized, there is chance involved. Which of the parents' genes are incorporated into each gamete is a matter of chance, and which of the millions of sperm actually fertilize the hundreds of eggs is impossible to predict with precision.
Even the act of talking to a friend is fraught with chance when done next to a mountain stream. The babbling sound of the brook is pleasing because it is so unpredictable, but this lack of predictability can make communication difficult. Somehow your ears and your brain must extract from this background noise the information in speech.
So, chance in life is unavoidable. Given this fact, how should a biologist react? In many disciplines, the traditional reaction is to view the random variations of life as a necessary evil that can be exorcised (or at least tamed) through the application of clever ideas and (as a last resort) inferential statistics. Even then we are taught in our statistics classes to abhor unexplained variation. In a welldesigned experiment, the less chance involved in the outcome, the better!
There is an alternative, however: the approach taken by Gosline in his experiment on spider silk. If chance is a given in life, why not use it to our advantage? In other words, if we know that a system will behave in a random fashion in the short term and at small scale (as with the random thermal motions of protein chains in silk), we can use this information to make accurate predictions as to how the system will behave in the long run and on a larger scale. Therein lies the thread of our tale. The diffusion of molecules, the drift of genes in a population, the longevity of phytoplankton, all include a large element of chance, and we will see why. How soft can a sound be before no animal can detect it? How fast must a mouse move in the moonlight before no owl can see it? We will be able to make predictions.
Before we embark on this exploration, we need to discuss briefly the nature of variation and which of nature's variations will (and will not) be included here.
1.3 Determinism versus Chance
One of Sir Isaac Newton's grand legacies is the idea that much about how the universe works can be precisely known. For example, if we know the exact mass of the moon and Earth and their current speed relative to each other, Newtonian mechanics and the law of gravitation should be able to tell us exactly where the moon is relative to Earth at any future time. As a practical matter, this is very close to being true. We can, for instance, predict solar and lunar eclipses with reasonable accuracy centuries in advance. Processes such as the moon's orbital mechanics are said to be deterministic, implying that, given sufficient knowledge of the initial state of a system, its future is determined exactly.
In fact, good examples of realworld deterministic processes are difficult to find. As our example of an afternoon spent observing a mountain stream is meant to convey, many of the processes that seem simple when described in the abstract (the variation in light intensity with the position of the sun, the downward acceleration of an object falling from a height) are exceedingly complex in reality. Details (rustling leaves and atmospheric turbulence) inevitably intrude, bringing with them an element of unpredictability. In some cases, the amount of variability associated with a process is sufficiently small that we are willing to view the system as being deterministic, and accept as fact predictions regarding its behavior. The physics of a pendulum clock, for instance, is so straightforward that we are content to use these machines as an accurate means of measuring time. In biology, few systems are so reliable, and deterministic behavior can be viewed at best as a polite fiction. As you might expect from the title of this book, deterministic processes will have no place here.
If a system or process is not deterministic, it is by definition stochastic. Even if we know exactly the state of a stochastic system at one time, we can never predict exactly what its state will be in the future. Some element of chance is involved. Unlike pregnancy and perfection, stochasticity can manifest itself to a variable degree. Many stochastic processes are approximately predictable with just a minor overlay of random behavior. The light intensity at our mountain stream is an example. Yes, there are minor random shortterm fluctuations, but if we were to take 5minute averages of the light level at the forest floor, they would closely follow predictions based on knowing the elevation of the sun. In other cases, the predictability of a system is negligible, and chance alone governs its behavior. The movement of molecules in a roomtemperature gas is a good example. Both types of systems will be included in our exploration.
As a practical matter, the dividing line between "deterministic" and "stochastic" is open to interpretation. For example, it is common practice (both in sports and introductory texts on probability theory) to accept the flip of a coin as a chance proposition, a stochastic process. But if you know enough about the height above the ground at which the coin is flipped, the angular velocity initially imparted to the coin, and the effects of air resistance, it should be possible to decide in advance whether the coin will land heads up. Indeed, much of what we accept as stochastic may well be deterministic given sufficient understanding of the mechanism involved. In this respect, the line between "deterministic" and "stochastic" is often drawn as a matter of convenience. If the precise predictions that are possible in theory are too difficult to carry out in practice, we shift the line a bit and think of the process as being stochastic.
This is not to imply that all processes are deterministic, however. As far as physicists have been able to divine, there are aspects of nature, encountered at very small scales of time and space, that are unpredictable even in theory. For example, there are limits to the precision with which you can know both the velocity and the location of an object (this is a rough statement of Heisenberg's uncertainty principle). In other words, if you could know exactly where an electron is at some point in time, you couldn't know what its velocity is. Conversely, if you know exactly what its velocity is, you can't know its position. In either case, you can't predict exactly where the electron will be even a short time in the future.
Continues...
Table of Contents
Preface xi
1 The Nature of Chance 3
1.1 Silk, Strength, and Statistics 3
1.2 What Is Certain? 7
1.3 Determinism versus Chance 8
1.4 Chaos 9
1.5 A Road Map 11
2 Rules of Disorder 12
2.1 Events, Experiments, and Outcomes 12
2.1.1 Sarcastic Fish 13
2.1.2 Bipolar Smut 14
2.1.3 Discrete versus Continuous 17
2.1.4 Drawing Pictures 18
2.2 Probability 19
2.3 Rules and Tools 20
2.3.1 Events Are the Sum of Their Parts 20
2.3.2 The Union of Sets 21
2.3.3 The Probability of a Union 23
2.3.4 Probability and the Intersection of Sets 24
2.3.5 The Complement of a Set 25
2.3.6 Additional Information and Conditional Probabilities 27
2.3.7 Bayes' Formula 29
2.3.8 AIDS and Bayes' Formula 30
2.3.9 The Independence of Sets 32
2.4 Probability Distributions 34
2.5 Summary 37
2.6 Problems 37
3 Discrete Patterns of Disorder 40
3.1 Random Variables 40
3.2 Expectations Defined 42
3.3 The Variance 46
3.4 The Trials of Bernoulli 48
3.5 Beyond 0's and 1's 50
3.6 Bernoulli = Binomial 51
3.6.1 Permutations and Combinations 53
3.7 Waiting Forever 60
3.8 Summary 65
3.9 Problems 66
4 Continuous Patterns of Disorder 68
4.1 The Uniform Distribution 69
4.1.1 The Cumulative Probability Distribution 70
4.1.2 The Probability Density Function 71
4.1.3 The Expectation 74
4.1.4 The Variance 76
4.2 The Shape of Distributions 77
4.3 The Normal Curve 79
4.4 Why Is the Normal Curve Normal? 82
4.5 The Cumulative Normal Curve 84
4.6 The Standard Error 86
4.7 A Brief Detour to Statistics 89
4.8 Summary 92
4.9 Problems 93
4.10 Appendix 1: The Normal Distribution 94
4.11 Appendix 2: The Central Limit Theorem 98
5 Random Walks 106
5.1 The Motion of Molecules 106
5.2 Rules of a Random Walk 110
5.2.1 The Average 110
5.2.2 The Variance 112
5.2.3 Diffusive Speed 115
5.3 Diffusion and the Real World 115
5.4 A Digression on the Binomial Theorem 117
5.5 The Biology of Diffusion 119
5.6 Fick's Equation 123
5.7 A Use of Fick's Equation: Limits to Size 126
5.8 Receptors and Channels 130
5.9 Summary 136
5.10 Problems 137
6 More Random Walks 139
6.1 Diffusion to Capture 139
6.1.1 Two Absorbing Walls 142
6.1.2 One Reflecting Wall 144
6.2 Adrift at Sea: Turbulent Mixing of Plankton 145
6.3 Genetic Drift 148
6.3.1 A Genetic Diffusion Coefficient 149
6.3.2 Drift and Fixation 151
6.4 Genetic Drift and Irreproducible Pigs 154
6.5 The Biology of Elastic Materials 156
6.5.1 Elasticity Defined 156
6.5.2 Biological Rubbers 157
6.5.3 The Limits to Energy Storage 161
6.6 Random Walks in Three Dimensions 163
6.7 Random Protein Configurations 167
6.8 A Segue to Thermodynamics 169
6.9 Summary 173
6.10 Problems 173
7 The Statistics of Extremes 175
7.1 The Danger of Cocktail Parties 175
7.2 Calculating the Maximum 182
7.3 Mean and Modal Maxima 185
7.4 Ocean Waves 186
7.5 The Statistics of Extremes 189
7.6 Life and Death in Rhode Island 194
7.7 Play Ball! 196
7.8 A Note on Extrapolation 204
7.9 Summary 206
7.10 Problems 206
8 Noise and Perception 208
8.1 Noise Is Inevitable 208
8.2 Dim Lights and Fuzzy Images 212
8.3 The Poisson Distribution 213
8.4 Bayes' Formula and the Design of Rods 218
8.5 Designing ErrorFree Rods 219
8.5.1 The Origin of Membrane Potentials 220
8.5.2 Membrane Potential in Rod Cells 222
8.6 Noise and Ion Channels 225
8.6.1 An Electrical Analog 226
8.6.2 Calculating the Membrane Voltage 227
8.6.3 Calculating the Size 229
8.7 Noise and Hearing 230
8.7.1 Fluctuations in Pressure 231
8.7.2 The Rate of Impact 232
8.7.3 Fluctuations in Velocity 233
8.7.4 Fluctuations in Momentum 235
8.7.5 The Standard Error of Pressure 235
8.7.6 Quantifying the Answer 236
8.8 The Rest of the Story 239
8.9 Stochastic Resonance 239
8.9.1 The Utility of Noise 239
8.9.2 Nonlinear Systems 242
8.9.3 The History of Stochastic Resonance 244
8.10 Summary 245
8.11 A Word at the End 246
8.12 A Problem 247
8.13 Appendix 248
9 The Answers 250
9.1 Chapter 2 250
9.2 Chapter 3 256
9.3 Chapter 4 262
9.4 Chapter 5 266
9.5 Chapter 6 269
9.6 Chapter 7 271
9.7 Chapter 8 273
Symbol Index 279
Author Index 284
Subject Index 286