Theory and Evaluation of Single-Molecule Signals

Contents......Page 14
Acknowledgment......Page 6
Preface......Page 8
1 Introduction......Page 18
2 General theoretical background: Statistical likelihood......Page 20
2.1. Probability density, likelihood, and information......Page 21
2.2. Statistical tests......Page 24
2.3. Bayesian information criterion and maximum entropy method......Page 26
3 The transition between molecular states occurs much faster than the experimental time resolution — Intermittency......Page 29
4 Experimental time resolution is sufficient to follow the transition between molecular states......Page 32
5 Distributions......Page 34
6 Bursts......Page 38
Acknowledgments......Page 41
References......Page 42
1 Introduction to the Bayesian approach......Page 48
2 Hidden parameters......Page 56
3 Finding quasi-stationary states......Page 58
4 Finding distances between molecules......Page 66
Appendix 1: Product and marginalization rules......Page 71
Appendix 2: Chi-squared distribution......Page 72
Appendix 3: Gaussian integrals......Page 73
Appendix 4: Probability of a histogram......Page 74
Appendix 5: Probability distribution for the distance to the nearest molecule......Page 75
References......Page 76
1 Introduction......Page 78
2 The Poisson process and introduction of generating functions......Page 81
3 Generating functions for photon emission: More complex kinetic models......Page 90
4 Generating functions for photon emission: Quantum treatment......Page 96
5 Quantum dynamics examples......Page 98
6 Conclusion......Page 107
References......Page 108
1 Photon statistics: Factorial moments vs. correlation functions......Page 110
2 Photon statistics in weakly driven systems:Analogy with four wave mixing......Page 116
3 Multipoint correlation functions for slow fluctuations......Page 125
3.1. Markovian dynamics......Page 131
3.2. Gaussian dynamics......Page 134
3.3. NonMarkovian dynamics: Renewal processes, continuous time random walks......Page 141
Acknowledgments......Page 151
References......Page 152
1 Introduction......Page 156
2.1. Theory......Page 158
2.2. Free energy surfaces from a quasi-harmonic approximation......Page 161
2.4. Analysis using weighted histograms......Page 164
2.5. Crooks relation......Page 166
2.7. Practical implementation......Page 167
3.1. Pulling with a constant force......Page 172
3.2. Pulling with time-dependent forces: Force-ramp experiments......Page 174
3.3. Relating constant-force and force-ramp experiments......Page 175
3.4. Effects of anharmonic molecular linkers......Page 176
3.5.1. Bell–Evans model of molecular rupture under time-dependent force......Page 180
3.5.2. General expression for the force-dependent rate of molecular rupture within Kramers theory......Page 181
3.5.3. Microscopic models of force-induced molecular rupture rate k(F )......Page 183
3.5.4. Unified theory of molecular rupture......Page 187
3.6. Analysis of force-ramp experiments......Page 189
4 Concluding remarks......Page 192
References......Page 193
1 Introduction......Page 198
2 General formalism......Page 203
2.1. Rate equations......Page 204
2.2. Dynamics in the absence of detected photons......Page 205
2.3. Distribution of the number of photons in a time interval......Page 207
2.4. Generating function......Page 208
2.5. Moments and the intensity correlation function......Page 211
2.6. Interphoton time distribution......Page 214
2.7. Relation to renewal theory......Page 217
2.8. Poisson statistics......Page 218
3 Fluorescence quenching and conformational dynamics......Page 220
3.1. Two conformational states......Page 221
3.2. Slow conformational dynamics: Two states......Page 222
3.2.1. Distribution of the number of photons in a bin......Page 224
3.2.2. Intensity correlation function......Page 228
3.2.3. Interphoton time distribution......Page 231
3.3. Many conformational states......Page 232
3.4.1. Distribution of the number of photons in a bin......Page 235
3.4.2. Intensity correlation function......Page 237
3.4.3. Interphoton time distribution......Page 238
3.5. Mandel’s formula in the presence of slow conformational fluctuations......Page 239
4 Influence of translational diffusion......Page 242
4.1. Distribution of the number of photons in a time bin......Page 244
4.2. Moments and correlation function......Page 248
4.3. Interphoton time distribution......Page 250
4.4. Diffusing molecules undergoing conformational dynamics......Page 252
5 Concluding remarks......Page 255
References......Page 256
1 Introduction......Page 262
2.1. Definitions......Page 264
2.2. Transfer matrix method......Page 267
2.3.1. Initial time averaging......Page 269
2.3.3. Event correlations......Page 270
2.4. Generating function method......Page 271
2.5. Stochastic rate model......Page 272
3 On–off blinking time series......Page 274
3.1. Definitions......Page 275
3.2. Phenomenological chemical kinetics, average rate, and detailed balance......Page 276
3.3.1. Transfer matrix methods......Page 277
3.3.2. Generating functions......Page 278
3.3.3. Stochastic rate model......Page 279
3.4.1. Cumulant expansion......Page 280
3.4.2. Memory function......Page 281
3.5. Two-state two-channel model: Single molecule echo......Page 282
3.6.2. Two-event number density......Page 285
4 Photon emission time series......Page 286
4.1.1. Photon densities and moments......Page 287
4.1.2. Poisson indicator and renewal indicator......Page 288
4.2.1. Photon-emission time series......Page 290
4.2.2. Transfer matrix and photon number density......Page 291
4.2.3. Renewal processes......Page 292
4.2.4. Non-renewal processes......Page 293
4.3. Diffusion-controlled reactions and Wilemski–Fixman expression......Page 294
5 Data analysis......Page 295
6 Concluding remarks......Page 297
References......Page 298
1 Introduction......Page 304
2 Correlation functions and oscillations......Page 307
3 On–off time distributions and oscillations......Page 321
4 Reaction event statistics and oscillations......Page 324
5 Conclusions......Page 327
References......Page 328
1 Introduction......Page 330
2 Single-molecule experiments......Page 332
3 Theoretical models......Page 333
3.1. Continuum ratchet models......Page 334
3.2. Discrete stochastic models......Page 335
4 Conclusions......Page 346
References......Page 347
1 Introduction......Page 354
2.1. Matrix formulation of the system......Page 357
2.2. Path representation of theWT-PDFs......Page 358
2.3. Relationships between the master equation and the path representation......Page 360
3.1. The rank of φx,y(t1, t2) and its topological
interpretation......Page 361
3.3. Mapping a KS into an RD form......Page 362
3.4. Examples and the utility of RD forms......Page 363
4 Constructing the RD form from the data......Page 365
5 Concluding remarks......Page 369
Appendix A......Page 370
Appendix B......Page 371
References......Page 377
1 Introduction......Page 382
2 Blinking nanocrystals......Page 386
2.1. Distribution of the intensity correlation function......Page 393
2.2. Experimental evidence for weak ergodicity breaking......Page 395
3 Continuous time random walk......Page 397
4 The quenched trap model......Page 400
5 Discussion......Page 404
References......Page 406
About the Editors......Page 410
Index......Page 412