“main” page i Physical Models of Living Systems Philip Nelson University of Pennsylvania with the assistance of Sarina Bromberg, Ann Hermundstad, and Jason Prentice W. Freeman and Company New York “main” page ii Publisher: Kate Parker Acquisitions Editor: Alicia Brady Senior Development Editor: Blythe Robbins Assistant Editor: Courtney Lyons Editorial Assistant: Nandini Ahuja Marketing Manager: Taryn Burns Senior Media and Supplements Editor: Amy Thorne Director of Editing, Design, and Media Production: Tracey Kuehn Managing Editor: Lisa Kinne Project Editor: Kerry O’Shaughnessy Production Manager: Susan Wein Design Manager and Cover Designer: Vicki Tomaselli Illustration Coordinator: Matt McAdams Photo Editors: Christine Buese, Richard Fox Composition: codeMantra Printing and Binding: RR Donnelley Cover: [Two-color, superresolution optical micrograph.] Two specific structures in a mammalian cell have been tagged with fluorescent molecules via immunostaining: microtubules (false-colored green) and clathrin-coated pits, cellular structures used for receptor-mediated endocytosis (false-colored red). See also Figure 6. The magnification is such that the height of the letter “o” in the title corresponds to about 1.
[Image courtesy Mark Bates, Dept. of NanoBiophotonics, Max Planck Institute for Biophysical Chemistry, published in Bates et al. Reprinted with permission from AAAS.] Inset: The equation known today as the “Bayes formula” first appeared in recognizable form around 1812, in the work of Pierre Simon de Laplace. In our notation, the formula appears as Equation 3.
P (The letter “S” in Laplace’s original formulation is an obsolete notation for sum, now written as .) This formula forms the basis of statistical inference, including that used in superresolution microscopy. Title page: Illustration from James Watt’s patent application. The green box encloses a centrifugal governor. [From A treatise on the steam engine: Historical, practical, and descriptive (1827) by John Farey.] Library of Congress Preassigned Control Number: 2014949574 ISBN-13: 978-1-4641-4029-7 ISBN-10: 1-4641-4029-4 ©2015 by Philip C.
Nelson All rights reserved Printed in the United States of America First printing W. Freeman and Company, 41 Madison Avenue, New York, NY 10010 Houndmills, Basingstoke RG21 6XS, England www.com “main” page iii For my classmates Janice Enagonio, Feng Shechao, and Andrew Lange. “main” page iv Whose dwelling is the light of setting suns, And the round ocean and the living air, And the blue sky, and in the mind of man: A motion and a spirit, that impels All thinking things, all objects of all thought, And rolls through all things. – William Wordsworth “main” page v Brief Contents Prolog: A breakthrough on HIV 1 PART I First Steps Chapter 1 Virus Dynamics 9 Chapter 2 Physics and Biology 27 PART II Randomness in Biology Chapter 3 Discrete Randomness 35 Chapter 4 Some Useful Discrete Distributions 69 Chapter 5 Continuous Distributions 97 Chapter 6 Model Selection and Parameter Estimation 123 Chapter 7 Poisson Processes 153 Jump to Contents Jump to Index v “main” page vi vi Brief Contents PART III Control in Cells Chapter 8 Randomness in Cellular Processes 179 Chapter 9 Negative Feedback Control 203 Chapter 10 Genetic Switches in Cells 241 Chapter 11 Cellular Oscillators 277 Epilog 299 Appendix A Global List of Symbols 303 Appendix B Units and Dimensional Analysis 309 Appendix C Numerical Values 315 Acknowledgments 317 Credits 321 Bibliography 323 Index 333 Jump to Contents Jump to Index “main” page vii Detailed Contents Web Resources xvii To the Student xix To the Instructor xxiii Prolog: A breakthrough on HIV 1 PART I First Steps Chapter 1 Virus Dynamics 9 1.2 Modeling the Course of HIV Infection 10 1.2 An appropriate graphical representation can bring out key features of data 12 1.3 Physical modeling begins by identifying the key actors and their main interactions 12 1.4 Mathematical analysis yields a family of predicted behaviors 14 1.5 Most models must be fitted to data 15 1.6 Overconstraint versus overfitting 17 1.3 Just a Few Words About Modeling 17 Key Formulas 19 Track 2 21 1.4′ Exit from the latency period 21 1.6′ a Informal criterion for a falsifiable prediction 21 Jump to Contents Jump to Index vii “main” page viii viii Detailed Contents 1.6′ b More realistic viral dynamics models 21 1.6′ c Eradication of HIV 22 Problems 23 Chapter 2 Physics and Biology 27 2.3 Dimensional Analysis 29 Key Formulas 30 Problems 31 PART II Randomness in Biology Chapter 3 Discrete Randomness 35 3.2 Avatars of Randomness 36 3.1 Five iconic examples illustrate the concept of randomness 36 3.2 Computer simulation of a random system 40 3.3 Biological and biochemical examples 40 3.4 False patterns: Clusters in epidemiology 41 3.3 Probability Distribution of a Discrete Random System 41 3.1 A probability distribution describes to what extent a random system is, and is not, predictable 41 3.2 A random variable has a sample space with numerical meaning 43 3.3 The addition rule 44 3.4 The negation rule 44 3.1 Independent events and the product rule 45 3.1 Crib death and the prosecutor’s fallacy 47 3.2 The Geometric distribution describes the waiting times for success in a series of independent trials 47 3.3 The proper interpretation of medical tests requires an understanding of conditional probability 50 3.4 The Bayes formula streamlines calculations involving conditional probability 52 3.5 Expectations and Moments 53 3.1 The expectation expresses the average of a random variable over many trials 53 3.2 The variance of a random variable is one measure of its fluctuation 54 3.3 The standard error of the mean improves with increasing sample size 57 Key Formulas 58 Track 2 60 Jump to Contents Jump to Index ‘‘main’’ page ix Detailed Contents ix 3.1′ a Extended negation rule 60 3.1′ b Extended product rule 60 3.1′ c Extended independence property 60 3.4′ Generalized Bayes formula 60 3.2′ a Skewness and kurtosis 60 3.2′ b Correlation and covariance 61 3.2′ c Limitations of the correlation coefficient 62 Problems 63 Chapter 4 Some Useful Discrete Distributions 69 4.1 Drawing a sample from solution can be modeled in terms of Bernoulli trials 70 4.2 The sum of several Bernoulli trials follows a Binomial distribution 71 4.3 Expectation and variance 72 4.4 How to count the number of fluorescent molecules in a cell 72 4.1 The Binomial distribution becomes simpler in the limit of sampling from an infinite reservoir 74 4.2 The sum of many Bernoulli trials, each with low probability, follows a Poisson distribution 75 4.4 Determination of single ion-channel conductance 78 4.5 The Poisson distribution behaves simply under convolution 79 4.4 The Jackpot Distribution and Bacterial Genetics 81 4.2 Unreproducible experimental data may nevertheless contain an important message 81 4.3 Two models for the emergence of resistance 83 4.4 The Luria-Delbrück hypothesis makes testable predictions for the distribution of survivor counts 84 4.5 Perspective 86 Key Formulas 87 Track 2 89 4.3 More about the Luria-Delbrück experiment 89 4.5′ a Analytical approaches to the Luria-Delbrück calculation 89 ′ 4.5 b Other genetic mechanisms 89 4.5′ c Non-genetic mechanisms 90 4.5′ d Direct confirmation of the Luria-Delbrück hypothesis 90 Problems 91 Jump to Contents Jump to Index “main” page x x Detailed Contents Chapter 5 Continuous Distributions 97 5.2 Probability Density Function 98 5.1 The definition of a probability distribution must be modified for the case of a continuous random variable 98 5.2 Three key examples: Uniform, Gaussian, and Cauchy distributions 99 5.3 Joint distributions of continuous random variables 101 5.4 Expectation and variance of the example distributions 102 5.5 Transformation of a probability density function 104 5.3 More About the Gaussian Distribution 106 5.1 The Gaussian distribution arises as a limit of Binomial 106 5.2 The central limit theorem explains the ubiquity of Gaussian distributions 108 5.3 When to use/not use a Gaussian 109 5.4 More on Long-tail Distributions 110 Key Formulas 112 Track 2 114 5.1′ Notation used in mathematical literature 114 5.4′ b The movements of stock prices 115 Problems 118 Chapter 6 Model Selection and Parameter Estimation 123 6.1 How good is your model? 124 6.2 Decisions in an uncertain world 125 6.3 The Bayes formula gives a consistent approach to updating our degree of belief in the light of new data 126 6.4 A pragmatic approach to likelihood 127 6.2 The maximally likely value for a model parameter can be computed on the basis of a finite dataset 129 6.3 The credible interval expresses a range of parameter values consistent with the available data 130 6.1 Likelihood analysis of the Luria-Delbrück experiment 133 6.2 Fluorescence imaging at one nanometer accuracy 133 Jump to Contents Jump to Index “main” page xi Detailed Contents xi 6.3 Localization microscopy: PALM/FPALM/STORM 136 6.5 An Extension of Maximum Likelihood Lets Us Infer Functional Relationships from Data 137 Key Formulas 141 Track 2 142 6.4′ a Binning data reduces its information content 142 6.2′ a The role of idealized distribution functions 143 6.3′ a Credible interval for the expectation of Gaussian-distributed data 144 6.3′ b Confidence intervals in classical statistics 145 6.3′ c Asymmetric and multivariate credible intervals 146 6.2′ More about FIONA 146 ′ 6.3 More about superresolution 147 6.5′ What to do when data points are correlated 147 Problems 149 Chapter 7 Poisson Processes 153 7.2 The Kinetics of a Single-Molecule Machine 153 7.1 Geometric distribution revisited 156 7.2 A Poisson process can be defined as a continuous-time limit of repeated Bernoulli trials 157 7.1 Continuous waiting times are Exponentially distributed 158 7.2 Distribution of counts 160 7.3 Useful Properties of Poisson processes 161 7.3 Significance of thinning and merging properties 163 7.1 Enzyme turnover at low concentration 164 7.5 Convolution and Multistage Processes 165 7.1 Myosin-V is a processive molecular motor whose stepping times display a dual character 165 7.2 The randomness parameter can be used to reveal substeps in a kinetic scheme 168 7.1 Simple Poisson process 168 7.2 Poisson processes with multiple event types 168 Jump to Contents Jump to Index “main” page xii xii Detailed Contents Key Formulas 169 Track 2 171 7.2′ More about motor stepping 171 7.1′ a More detailed models of enzyme turnovers 171 7.1′ b More detailed models of photon arrivals 171 Problems 172 PART III Control in Cells Chapter 8 Randomness in Cellular Processes 179 8.2 Random Walks and Beyond 180 8.1 Situations studied so far 180 8.1 Periodic stepping in random directions 180 8.2 Irregularly timed, unidirectional steps 180 8.2 A more realistic model of Brownian motion includes both random step times and random step directions 180 8.3 Molecular Population Dynamics as a Markov Process 181 8.1 The birth-death process describes population fluctuations of a chemical species in a cell 182 8.2 In the continuous, deterministic approximation, a birth-death process approaches a steady population level 184 8.3 The Gillespie algorithm 185 8.4 The birth-death process undergoes fluctuations in its steady state 186 8.1 Exact mRNA populations can be monitored in living cells 187 8.2 mRNA is produced in bursts of transcription 189 8.4 Vista: Randomness in protein production 193 Key Formulas 194 Track 2 195 8.4′ The master equation 195 ′ 8.4 More about gene expression 197 8.2′ a The role of cell division 197 8.2′ b Stochastic simulation of a transcriptional bursting experiment 198 8.2′ c Analytical results on the bursting process 199 Problems 200 Chapter 9 Negative Feedback Control 203 9.2 Mechanical Feedback and Phase Portraits 204 9.1 The problem of cellular homeostasis 204 Jump to Contents Jump to Index “main” page xiii Detailed Contents xiii 9.2 Negative feedback can bring a system to a stable setpoint and hold it there 204 9.3 Wetware Available in Cells 206 9.1 Many cellular state variables can be regarded as inventories 206 9.2 The birth-death process includes a simple form of feedback 207 9.3 Cells can control enzyme activities via allosteric modulation 207 9.4 Transcription factors can control a gene’s activity 208 9.5 Artificial control modules can be installed in more complex organisms 211 9.4 Dynamics of Molecular Inventories 212 9.1 Transcription factors stick to DNA by the collective effect of many weak interactions 212 9.2 The probability of binding is controlled by two rate constants 213 9.3 The repressor binding curve can be summarized by its equilibrium constant and cooperativity parameter 214 9.