Brief contents. Chapter 1 Virus Dynamics 33. Chapter 2 Physics and Biology 52. Randomness in Biology. Chapter 3 Discrete Randomness 59

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1 Brief contents I First Steps Chapter 1 Virus Dynamics 33 Chapter 2 Physics and Biology 52 II Randomness in Biology Chapter 3 Discrete Randomness 59 Chapter 4 Some Useful Discrete Distributions 96 Chapter 5 Continuous Distributions 125 Chapter 6 Model Selection and Parameter Estimation 152 Chapter 7 Poisson Processes 184 III Control in Cells Chapter 8 Randomness in Cellular Processes 213 Chapter 9 Negative Feedback Control 238 Chapter 10 Genetic Switches in Cells 278 Chapter 11 Cellular Oscillators 315 Epilog 336 7

2 8 Appendix A Global List of Symbols 340 Appendix B Units and Dimensional Analysis 346 Appendix C Numerical Values 353 Acknowledgments 355 Credits 358 Bibliography 360 Index 370 Printed August 19, 2014; Contents Index Notation

3 Detailed contents Web resources 18 To the student 19 To the instructor 23 Prolog: A breakthrough on HIV 27 I First Steps Chapter 1 Virus Dynamics First signpost Modeling the course of HIV infection Biological background An appropriate graphical representation can bring out key features of data Physical modeling begins by identifying the key actors and their main interactions Mathematical analysis yields a family of predicted behaviors Most models must be fitted to data Overconstraint versus overfitting Just a few words about modeling 41 Key Formulas 43 Track Exit from the latency period a Informal criterion for a falsifiable prediction b More realistic viral dynamics models c Eradication of HIV 47 Problems 48 Chapter 2 Physics and Biology Signpost The intersection Dimensional analysis 54 Key Formulas 55 Problems 56 II Randomness in Biology 9

4 10 Chapter 3 Discrete Randomness Signpost Avatars of randomness Five iconic examples illustrate the concept of randomness Computer simulation of a random system Biological and biochemical examples False patterns: Clusters in epidemiology Probability distribution of a discrete random system A probability distribution describes to what extent a random system is, and is not, predictable A random variable has a sample space with numerical meaning The addition rule The negation rule Conditional probability Independent events and the product rule Crib death and the prosecutor s fallacy The Geometric distribution describes the waiting times for success in a series of independent trials Joint distributions The proper interpretation of medical tests requires an understanding of conditional probability The Bayes formula streamlines calculations involving conditional probability Expectations and moments The expectation expresses the average of a random variable over many trials The variance of a random variable is one measure of its fluctuation The standard error of the mean improves with increasing sample size 82 Key Formulas 84 Track a Extended negation rule b Extended product rule c Extended independence property Generalized Bayes formula a Skewness and kurtosis b Correlation and covariance c Limitations of the correlation coe cient 88 Problems 90 Chapter 4 Some Useful Discrete Distributions Signpost Binomial distribution Drawing a sample from solution can be modeled in terms of Bernoulli trials The sum of several Bernoulli trials follows a Binomial distribution Expectation and variance 99 Printed August 19, 2014; Contents Index Notation

5 4.2.4 How to count the number of fluorescent molecules in a cell Computer simulation Poisson distribution The Binomial distribution becomes simpler in the limit of sampling from an infinite reservoir The sum of many Bernoulli trials, each with low probability, follows a Poisson distribution Computer simulation Determination of single ion-channel conductance The Poisson distribution behaves simply under convolution The jackpot distribution and bacterial genetics It matters Unreproducible experimental data may nevertheless contain an important message Two models for the emergence of resistance The Luria-Delbrück hypothesis makes testable predictions for the distribution of survivor counts Perspective 114 Key Formulas 115 Track On resistance More about the Luria-Delbrück experiment a Analytical approaches to the Luria-Delbrück calculation b Other genetic mechanisms c Non-genetic mechanisms c Direct confirmation of the Luria-Delbrück hypothesis 118 Problems 119 Chapter 5 Continuous Distributions Signpost Probability density function The definition of a probability distribution must be modified for the case of a continuous random variable Three key examples: Uniform, Gaussian, and Cauchy distributions Joint distributions of continuous random variables Expectation and variance of the example distributions Transformation of a probability density function Computer simulation More about the Gaussian distribution The Gaussian distribution arises as a limit of Binomial The central limit theorem explains the ubiquity of Gaussian distributions When to use/not use a Gaussian More on long-tail distributions 140 Key Formulas 141 Track Notation used in mathematical literature c 2010,2011,2012,2013,2014 Philip Nelson; Contents

6 Interquartile range a Terminology b The movements of stock prices 144 Problems 147 Chapter 6 Model Selection and Parameter Estimation Signpost Maximum likelihood How good is your model? Decisions in an uncertain world The Bayes formula gives a consistent approach to updating our degree of belief in the light of new data A pragmatic approach to likelihood Parameter estimation Intuition The maximally likely value for a model parameter can be computed on the basis of a finite dataset The credible interval expresses a range of parameter values consistent with the available data Summary Biological applications Likelihood analysis of the Luria-Delbrück experiment Superresolution microscopy On seeing Fluorescence imaging at one nanometer accuracy Localization microscopy: PALM/FPALM/STORM An extension of maximum likelihood lets us infer functional relationships from data 168 Key Formulas 170 Track Cross-validation a Binning data reduces its information content b Odds a The role of idealized distribution functions b Improved estimator a Credible interval for the expectation of Gaussian-distributed data b Confidence intervals in classical statistics c Asymmetric and multivariate credible intervals More about FIONA More about superresolution What to do when data points are correlated 177 Problems 180 Chapter 7 Poisson Processes Signpost 184 Printed August 19, 2014; Contents Index Notation

7 7.2 The kinetics of a single-molecule machine Random processes Geometric distribution revisited A Poisson process can be defined as a continuous-time limit of repeated Bernoulli trials Continuous waiting times are Exponentially distributed Distribution of counts Useful properties of Poisson processes Thinning property Merging property Significance of thinning and merging properties More examples Enzyme turnover at low concentration Neurotransmitter release Convolution and multistage processes Myosin-V is a processive molecular motor whose stepping times display a dual character The randomness parameter can be used to reveal substeps in a kinetic scheme Computer simulation Simple Poisson process Poisson processes with multiple event types 200 Key Formulas 201 Track More about motor stepping a More detailed models of enzyme turnovers b More detailed models of photon arrivals 204 Problems III Control in Cells Chapter 8 Randomness in Cellular Processes Signpost Random walks and beyond Situations studied so far Periodic stepping in random directions Irregularly timed, unidirectional steps A more realistic model of Brownian motion includes both random step times and random step directions Molecular population dynamics as a Markov process The birth-death process describes population fluctuations of a chemical species in a cell In the continuous, deterministic approximation, a birth-death process approaches a steady population level The Gillespie algorithm The birth-death process undergoes fluctuations in its steady state 220 c 2010,2011,2012,2013,2014 Philip Nelson; Contents

8 Gene expression Exact mrna populations can be monitored in living cells mrna is produced in bursts of transcription Perspective Vista: Randomness in protein production 228 Key Formulas 228 Track The master equation More about gene expression a The role of cell division b Stochastic simulation of a transcriptional bursting experiment c Analytical results on the bursting process 235 Problems 236 Chapter 9 Negative Feedback Control Signpost Mechanical feedback and phase portraits The problem of cellular homeostasis Negative feedback can bring a system to a stable setpoint and hold it there Wetware available in cells Many cellular state variables can be regarded as inventories The birth-death process includes a simple form of feedback Cells can control enzyme activities via allosteric modulation Transcription factors can control a gene s activity Artificial control modules can be installed in more complex organisms Dynamics of molecular inventories Transcription factors stick to DNA by the collective e ect of many weak interactions The probability of binding is controlled by two rate constants The repressor binding curve can be summarized by its equilibrium constant and cooperativity parameter The gene regulation function quantifies the response of a gene to a transcription factor Dilution and clearance oppose gene transcription Synthetic biology Network diagrams Negative feedback can stabilize a molecule inventory, mitigating cellular randomness A quantitative comparison of regulated- and unregulated-gene homeostasis A natural example: The trp operon Some systems overshoot on their way to their stable fixed point Two-dimensional phase portraits The chemostat Perspective 268 Printed August 19, 2014; Contents Index Notation

9 15 Key Formulas 269 Track a Contrast to electronic circuits b Permeability Other control mechanisms a More about transcription in bacteria b More about activators Gene regulation in eukaryotes a More general gene regulation functions b Cell cycle e ects a Simplifying approximations b The Systems Biology Graphical Notation Exact solution Taxonomy of fixed points 275 Problems 276 Chapter 10 Genetic Switches in Cells Signpost Bacteria have behavior Cells can sense their internal state and generate switch-like responses Cells can sense their external environment and integrate it with internal state information Novick and Weiner characterized induction at the single-cell level The all-or-none hypothesis Quantitative prediction for Novick-Weiner experiment Direct evidence for the all-or-none hypothesis Summary Positive feedback can lead to bistability Mechanical toggle Electrical toggles Positive feedback leads to neural excitability The latch circuit A 2D phase portrait can be partitioned by a separatrix A synthetic toggle switch network in E. coli Two mutually repressing genes can create a toggle The toggle can be reset by pushing it through a bifurcation Perspective Natural examples of switches The lac switch The lambda switch 301 Key Formulas 302 Track More details about the Novick-Weiner experiments a Epigenetic e ects b Mosaicism 304 c 2010,2011,2012,2013,2014 Philip Nelson; Contents

10 a A compound operator can implement more complex logic b A single-gene toggle Adiabatic approximation DNA looping Randomness in cellular networks 312 Problems 313 Chapter 11 Cellular Oscillators Signpost Some single cells have diurnal or mitotic clocks Synthetic oscillators in cells Negative feedback with delay can give oscillatory behavior Three repressors in a ring arrangement can also oscillate Mechanical clocks and related devices can also be represented by their phase portraits Adding a toggle to a negative feedback loop can improve its performance Synthetic-biology realization of the relaxation oscillator Natural oscillators Protein circuits The mitotic clock in Xenopus laevis 324 Key Formulas 328 Track a Attractors in phase space b Deterministic chaos a Linear stability analysis b Noise-induced oscillation Analysis of Xenopus mitotic oscillator 331 Problems 334 Epilog 336 Appendix A Global List of Symbols 340 A.1 Mathematical notation 340 A.2 Graphical notation 341 A.2.1 Phase portraits 341 A.2.2 Network diagrams 341 A.3 Named quantities 342 Appendix B Units and Dimensional Analysis 346 B.1 Base units 347 B.2 Dimensions versus units 347 B.3 Dimensionless quantities 349 B.4 About graphs 349 B.4.1 Arbitrary units 349 Printed August 19, 2014; Contents Index Notation

11 17 B.5 About angles 350 B.6 Payo 350 Appendix C Numerical Values 353 C.1 Fundamental constants 353 Acknowledgments 355 Credits 358 Bibliography 360 Index 370 c 2010,2011,2012,2013,2014 Philip Nelson; Contents