1.3.1 The discrete aspect of light is most apparent at extremely low intensity The photoelectric effect 30

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1 Detailed contents Web resources To the student To the instructor xx xxi xxv Prologue: Preliminaries Signpost: Uncertainty Discrete Probability Distributions A probability distribution summarizes our knowledge about an uncertain situation Conditional probability quantifies the degree to which events are correlated A random variable can be partially described by its expectation and variance Joint distributions Some explicit discrete distributions Dimensional Analysis Continuous Probability Distributions Probability density functions Some explicit continuous distributions More Properties of, and Operations on, Probability Distributions Transformation of a probability density function The sample mean of many independent, identically distributed random variables has lower variance than any one of its constituents Count data are typically Poisson distributed The difference of two noisy quantities can have greater relative standard deviation than either by itself The convolution of two distributions describes the sum of their random variables Thermal Randomness 17 Big Picture 17 Key Formulas 18 Problems 19 PART I Doorways of Light Chapter 1 What Is Light? Signpost: Photons Light Before Basic light phenomena Light displays wavelike behavior in many situations Light Is Lumpy The discrete aspect of light is most apparent at extremely low intensity The photoelectric effect 30 ix

2 x Copyright, Princeton University Press. No part of this book may be Einstein s proposal Light-induced phenomena in biology qualitatively support the Einstein relation Background: Poisson Processes A Poisson process can be defined as a continuous-time limit of repeated Bernoulli trials Blip counts in a fixed interval are Poisson distributed Waiting times are Exponentially distributed A New Physical Model of Light The Light Hypothesis, part The spectrum of light can be regarded as a probability density times an overall rate Light can eject electrons from individual molecules, inducing photochemical reactions Fluorescence and Photoisomerization Can Occur after Photon Absorption The Electron State Hypothesis Atoms have sharp spectral lines Molecules: fluorescence Molecules: photoisomerization Transparent Media Are Unchanged by the Passage of Light, but Slow It Down 47 Big Picture 48 Key Formulas 48 Track Quantum randomness is distinct from classical chaos a The reality of photons b Light also carries momentum c The thermal radiation spectrum d The role of frequency Corrections to Poisson emission and detection a Gamma rays b More about the Light Hypothesis Mechanism of DNA photodamage Dense media a More about atoms and light b A Cauchy distribution in physics a Born-Oppenheimer approximation b Classical approximation for nuclear motion c Debye relaxation Fast conformational changes 58 Problems 59 Chapter 2 Photons and Life Signpost: Seeing and Touching Light-Induced DNA Damage Fluorescence as a Window into Cells Fluorescence can discriminate healthy from diseased tissue during surgery Fluorescence microscopy can reduce background and specifically show only objects of interest Background: Membrane Potential Electric currents involve ion motion An ion imbalance across the cell membrane can create a membrane potential Ion pumps maintain a resting electric potential drop across the cell membrane Ion channels modulate the membrane potential to implement neural signaling 68

3 xi Action potentials can transmit information over long distances Creation and utilization of action potentials More about synaptic transmission Optogenetics Brains are hard to study Channelrhodopsin can depolarize selected neurons in response to light Halorhodopsin can hyperpolarize selected neurons in response to light Other methods Fluorescent Reporters Can Give Real-Time Readout of Cellular Conditions Voltage-sensitive fluorescent reporters Split fluorescent proteins and genetically encoded calcium indicators Two-Photon Excitation Permits Imaging Deep within Living Tissue The problem of imaging thick samples Two-photon excitation depends sensitively on light intensity Multiphoton microscopy can excite a specific volume element of a specimen Fluorescence Resonance Energy Transfer How to tell when two molecules are close to each other A physical model for FRET Some forms of bioluminescence also involve FRET FRET can be used to create a spectroscopic ruler Application of FRET to DNA bending flexibility FRET-based indicators A Glimpse of Photosynthesis It s big Two quantitative puzzles advanced our understanding of photosynthesis Resonance energy transfer resolves both puzzles 96 Big Picture 98 Key Formulas 98 Track More about membrane potentials Other uses for the resting potential The β-squared rule More about two-photon imaging About FRET and its efficiency Other experimental tests of FRET Why reported FRET efficiencies sometimes exceed 100% More details about the photosynthesis apparatus in plants 102 Problems 103 Chapter 3 Color Vision Signpost: A Fifth Dimension Color Vision Confers a Fitness Payoff Newton s Experiments on Color Background: More Properties of Poisson Processes Thinning property Merging property Significance for light Combining Two Beams Corresponds to Summing Their Spectra 112

4 xii Copyright, Princeton University Press. No part of this book may be 3.6 Psychophysical Aspects of Color R+G looks like Y Color discrimination is many-to-one Perceptual matching follows quantitative, reproducible, and context-independent rules Color from selective absorption Reflectance and transmittance spectra Subtractive color scheme A Physical Model of Color Vision The color-matching function challenge Available wetware in the eye The trichromatic model The trichromatic model explains why R+G Y Our eyes project light spectra to a 3D vector space A mechanical analogy for color matching Connection between the mechanical analogy and color vision Quantitative comparison to experimentally observed color-matching functions Why the Sky Is Not Violet Direct Imaging of the Cone Mosaic 130 Big Picture 131 Key Formulas 131 Track a Flux, irradiance, and spectral flux irradiance b Combining spectra is a linear operation a Variation of color matching b Colorblindness c Tetrachromacy Perceptual color a Determination of sensitivity functions b Contrast to audition a Enhancement of color contrast in autofluorescence endoscopy b Spectral analysis can discriminate many fluorophores and their combinations a Photoisomerization rate regarded as an inner product b Correction to predicted color matching due to absorption a Relative versus absolute sensitivity b Simplified color space 139 Problems 142 Chapter 4 How Photons Know Where to Go Signpost: Probability Amplitudes Summary of Key Phenomena The Probability Amplitude Reconciling the particle and wave aspects of light requires the introduction of a new kind of physical quantity Background: Complex Numbers Simplify Many Calculations Light Hypothesis, part Basic Interference Phenomena Two-slit interference explained via the Light Hypothesis Newton s rings illustrate interference in a three-dimensional setting 158

5 xiii An objection to the Light Hypothesis The Stationary-Phase Principle The Fresnel integral illustrates the stationary-phase principle The probability amplitude is computed as a sum over all possible photon paths Diffraction through a single, wide aperture Reconciliation of particle and wave aspects 168 Big Picture 170 Key Formulas 170 Track On philosophically repugnant theories a More about the Light Hypothesis b More about uniform media Which slit? More about reflection More objections a The neighborhood of the stationary-phase path b Nonuniform media 176 Problems 177 Chapter 5 Optical Phenomena and Life Signpost: Sorting and Directing Structural Color in Insects, Birds, and Marine Organisms Some animals create color by using nanostructures made from transparent materials An extension of the Light Hypothesis describes reflection and transmission at an interface A single thin, transparent layer reflects with weak wavelength dependence A stack of many thin, transparent layers can generate an optical bandgap Structural color in marine organisms Ray-optics Phenomena The reflection law is a consequence of the stationary-phase principle Transmission and reflection gratings generate non-ray-optics behavior by editing the set of allowed photon paths Refraction arises from the stationary-phase principle applied to a piecewise-uniform medium Total internal reflection provides another tool to enhance signal relative to noise in fluorescence microscopy Refraction is generally wavelength dependent 196 Big Picture 197 Key Formulas 197 Track Transmission and reflection in classical electromagnetism Fine points about reflection and refraction More complicated layers 200 Problems 201

6 xiv PART II Copyright, Princeton University Press. No part of this book may be Human and Superhuman Vision Chapter 6 Direct Image Formation Signpost: Bright Yet Sharp Image Formation Without Lenses Shadow imaging Pinhole imaging suffices for some animals Addition of a Lens Allows Formation of Bright, Yet Sharp, Images The focusing criterion relates object and image distances to lens shape A more general approach Formation of a complete image Aberration degrades image formation outside the paraxial limit The Vertebrate Eye Image formation with an air-water interface Focusing powers add in a compound lens system A deformable lens implements focal accommodation Light Microscopes and Related Instruments Rays of light are a useful idealization in the ray-optics regime Real and virtual images Spherical aberration Dispersion gives rise to chromatic aberration Confocal microscopy suppresses out-of-focus background light Darwin s Difficulty Background: Angles and Angular Area Angles Angular area Diffraction Limit Even a perfect lens will not focus light perfectly Three dimensions: The Rayleigh criterion Animal eyes match their photoreceptor size to the diffraction limit 235 Big Picture 236 Key Formulas 236 Track The retinal pigment epithelium The Abbe criterion 239 Problems 240 Chapter 7 Imaging as Inference Signpost: Information Background: On Inference The Bayes formula tells how to update a probability estimate Inference with a Uniform prior reduces to likelihood maximization Inferring the center of a distribution Parameter estimation and credible intervals Binning data reduces its information content Localization of a Single Fluorophore Localization is an inference problem 251

7 xv Formulation of a probabilistic model Maximum-likelihood analysis of image data Results for molecular motor stepping Localization Microscopy Defocused Orientation Imaging 258 Big Picture 260 Key Formulas 261 Track a Airy point spread function b Anisotropic point spread function c Other tacit assumptions Advantages of the maximum likelihood method Background estimation Interferometric PALM imaging More about anisotropy 266 Problems 267 Chapter 8 Imaging by X-Ray Diffraction Signpost: Inversion It s Hard to See Atoms Diffraction Patterns A periodic array of narrow slits creates a diffraction pattern of sharp lines Generalizations to the setup needed to handle x-ray crystallography An array of slits with substructure gives a diffraction pattern modulated by a form factor A 2D crystal yields a 2D diffraction pattern D crystals can be analyzed by similar methods The Diffraction Pattern of DNA Encodes Its Double-Helical Character The helical pitch, base pair rise, helix offset, and diameter of DNA can be obtained from its diffraction pattern Accurate determination of size parameters led to a breakthrough on the puzzle of DNA structure and function 282 Big Picture 283 Key Formulas 283 Track Factorization of amplitudes a How to treat fiber samples of DNA b The phase problem 286 Problems 287 Chapter 9 Vision in Dim Light Signpost: Construction The Edge of Vision Many ecological niches are dimly lit The single-photon challenge Measures of detector performance Psychophysical Measurements of Human Vision The probabilistic character of vision is most evident under dim-light conditions Rod cells must be able to respond to individual photon absorptions 295

8 xvi Copyright, Princeton University Press. No part of this book may be The eigengrau hypothesis states that true photon signals are merged with a background of spontaneous events Forced-choice experiments characterize the dim-light response Questions raised by psychophysical experiments Single-Cell Measurements Vertebrate photoreceptors can be monitored via the suction pipette method Determination of threshold, quantum catch, and spontaneous signaling rate Direct confirmation that the rod cell imposes no threshold Additional single-cell results Questions raised by the single-cell measurements 308 Big Picture 308 Key Formulas 309 Track a The fraction of light absorbed by a sample depends exponentially on thickness b The quantum yield for rod signaling c Quantum catch for a single human rod cell under axial illumination d The whole-retina quantum catch is the product of several factors 313 Problems 315 Chapter 10 The Mechanism of Visual Transduction Signpost: Dynamic Range Photoreceptors Photoreceptors are a specialized class of neurons Each rod cell simultaneously monitors one hundred million rhodopsin molecules Background: Cellular Control and Transduction Networks Cells can control enzyme activities via allosteric modulation Single-cell organisms can alter their behavior in response to environmental cues, including light The two-component signaling pathway motif Network diagrams summarize complex reaction networks Cooperativity can increase the sensitivity of a network element Photon Response Events Localized to One Disk Step 1: photoisomerization of rhodopsin in the disk membrane Step 2: activation of transducin in the disk membrane Steps 3 4: activation of phosphodiesterase in the disk membrane, and hydrolysis of cyclic GMP in the cytosol Events Elsewhere in the Rod Outer Segment Ion pumps in the rod cell plasma membrane maintain nonequilibrium ion concentrations Step 5: ion channel closing in the plasma membrane Events at the Synaptic Terminal Step 6: hyperpolarization of the plasma membrane Step 7: modulation of neurotransmitter release into the synaptic cleft Summary of the Visual Cascade 337 Big Picture 338 Key Formulas 340 Track Higher light intensities a More about two-component signaling pathways b Phototaxis 341

9 xvii More about adaptation in chemotaxis Cone and cone bipolar cells Recently discovered vertebrate photoreceptors Glutamate removal a Termination of the photoreceptor response b Negative feedback implements adaptation and standardizes signals from rod cells c Recycling of retinal 348 Problems 350 Chapter 11 The First Synapse and Beyond Signpost: False Positives Transmission at the First Synapse The synapse from rod to rod bipolar cells inverts its signal via another G protein cascade The first synapse also rejects rod signals below a transmission breakpoint Synthesis of Psychophysics and Single-Cell Physiology Why does vision require several captured photons? Review of the eigengrau hypothesis Single-rod measurements constrain the fit parameters in the eigengrau model Processing beyond the first synapse is highly efficient A Multistep Relay Sends Signals on to the Brain The classical rod pathway implements the single-photon response Other signaling pathways Optogenetic retinal prostheses Evolution and Vision Darwin s difficulty, revisited Parallels between vision, olfaction, and hormone reception 364 Big Picture 366 Key Formulas 367 Track a Instrumental noise b Quantal release noise c Mechanism of discrimination at the first synapse d Why discrimination at the first synapse is advantageous e Thresholding at later stages of processing Psychophysics with single photon stimuli a ON and OFF pathways b Image processing in the retina Rhabdomeric photoreceptors 373 Problems 375 PART III Advanced Topics Chapter 12 Electrons, Photons, and the Feynman Principle Signpost: Universality Electrons From paths to trajectories The action functional singles out classical trajectories as its stationary points 382

10 xviii Copyright, Princeton University Press. No part of this book may be The Feynman principle expresses probability amplitudes as sums over trajectories States and operators arise from partial summation over trajectories Stationary states are invariant under time evolution A confined-electron problem Light absorption by ring-shaped molecules The Schrödinger equation emerges in the limit of an infinitesimal time step Photons The action functional for photon trajectories The special case of a monochromatic light source reduces to our earlier formulation Vista: reflection, transmission, and the index of refraction 394 Big Picture 397 Chapter 13 Field Quantization, Polarization, and the Orientation of a Single Molecule Signpost: Fields A Single Molecule Emits Photons in a Dipole Distribution Classical Field Theory of Light Quantization Replaces Field Variables by Operators Photon States Basis states can be formed by applying creation operators to the vacuum state Coherent states mimic classical states in the limit of large occupation numbers Interaction with Electrons Classical interactions involve adding source terms to the field equations Electromagnetic interactions can be treated perturbatively The dipole emission pattern Electrons and positrons can also be created and destroyed Vistas Connection to the approach used in earlier chapters Some invertebrates can detect the polarization of light Invertebrate photoreceptors have a different morphology from vertebrates Polarized light must be used for single photoreceptor measurements Some transitions are far more probable than others Lasers exploit a preference for emission into an already occupied state Fluorescence polarization anisotropy 413 Big Picture 414 Chapter 14 Quantum-Mechanical Theory of FRET Signpost: Decoherence Two-state Systems FRET displays both classical and quantum aspects An isolated two-state system oscillates in time Environmental effects modify the behavior of a two-state system in solution The density operator summarizes the effect of the environment Time development of the density operator FRET The weakly coupled, strongly incoherent limit displays first-order kinetics Förster s formula arises in the electric dipole approximation More realistic treatment of the role of FRET in photosynthesis 421 Big Picture 422

11 xix Epilogue 423 Acknowledgments 427 Appendix A Global List of Symbols 431 A.1 Mathematical Notation 431 A.2 Network Diagrams 433 A.3 Named Quantities 433 Appendix B Units and Dimensional Analysis 439 B.1 Base Units 439 B.2 Dimensions Versus Units 440 B.3 About Graphs 441 B.3.1 Arbitrary units 442 B.3.2 Angles 442 B.4 Payoff 442 Appendix C Numerical Values 446 C.1 Fundamental Constants 446 C.2 Optics 446 C.2.1 Index of refraction for visible light 446 C.2.2 Miscellaneous 447 C.3 Eyes 447 C.3.1 Geometric 447 C.3.2 Rod cells 447 C.3.3 Cone cells 448 C.3.4 Beyond photoreceptors 448 C.4 B-Form DNA 448 Appendix D Complex Numbers 449 Bibliography 453 Credits 467 Index 473

Visual pigments. Neuroscience, Biochemistry Dr. Mamoun Ahram Third year, 2019

Visual pigments. Neuroscience, Biochemistry Dr. Mamoun Ahram Third year, 2019 Visual pigments Neuroscience, Biochemistry Dr. Mamoun Ahram Third year, 2019 References Webvision: The Organization of the Retina and Visual System (http://www.ncbi.nlm.nih.gov/books/nbk11522/#a 127) The