Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

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Contents4.1 The Physical Principle Of Fluorescence . . . . . . . . . . . . . . . . . . . . . . 624.2 Fluorescence Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Single Molecule Tracking via Wide-Field Fluorescence Spectroscopy . . . . . . 654.4 Total Internal Reflection Fluorescence Microscopy (TIRFM) . . . . . . . . . . . 674.5 Tracking of the Transducin Proteins . . . . . . . . . . . . . . . . . . . . . . . . 704.6 The expected range for the Transducin Diffusion Coefficients . . . . . . . . . . . 715 Modeling of the Experiment 735.1 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Model Ansatz and Estimation of Parameters . . . . . . . . . . . . . . . . . . . . 765.3 Testing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4 Separate Estimation on the Experimental Transducin X- and Y-Components . . . 825.5 Estimation of the Noise Intensities for Single Trajectories . . . . . . . . . . . . . 835.6 Estimation on the Basis of Differentials d x in Place of Positions x . . . . . . . . . 856 Conclusion and Outlook 917 Bibliography 93A Appendix 99A.1 From Copernicus to Newton - A Historical Example for the Role of Observationon Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.2 Important Papers on the Rhodopsin-Transducin Process . . . . . . . . . . . . . . 101A.3 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102A.4 Definitions for Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104A.5 The Auto-Correlation Function and its Spectral Representation . . . . . . . . . . 108A.6 The Langevin Equation as a First Order System . . . . . . . . . . . . . . . . . . 108A.7 The Fluctuation-Dissipation-Theorem . . . . . . . . . . . . . . . . . . . . . . . 109A.8 Kramers-Moyal Forward Expansion . . . . . . . . . . . . . . . . . . . . . . . . 113A.9 Deriving the Fokker-Planck Equation from the Kramers-Moyal Forward Expansion 115X
List of Figures2.1 Anatomy of the eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The layers of the retina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The anatomy of the rod photoreceptor cell (from [7]) . . . . . . . . . . . . . . . 72.4 Scheme of the visual transduction process in the retinal rod discs (from [7]) . . . 82.5 Scheme of the 7TM serpentine receptor (from [36, p.170 , Fig. 8.3]) . . . . . . . 92.6 (A) 11-cis-retinal and (B) all-trans-retinal. C : Carbon, H : Hydrogen, O :Oxygen,N : Nitrogen. The numbers are atom positions. (from [36, p. 190, Fig. 8.23]) . . 103.1 Scheme of a Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Scheme of a Markov chain path (sequence) . . . . . . . . . . . . . . . . . . . . 163.3 Time-Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Scheme of a Hidden Markov Model (HMM) . . . . . . . . . . . . . . . . . . . . 203.5 Global and local optima of a two-dimensional function f (from [46]). This functionis plotted versus two elements X 1 and X 2 of the problem space X. . . . . . . 233.6 Schematic representation of the diffusion phenomenon . . . . . . . . . . . . . . 293.7 (a) Brownian Movement and (b) Stochastic Force (from [34, p. 418]) . . . . . . . 323.8 Stochastic Force Correlation (from [34, p. 419]) . . . . . . . . . . . . . . . . . . 333.9 Three-dimensional free diffusion with 100.000 time steps . . . . . . . . . . . . . 473.10 Three-dimensional free diffusion with 1000 time steps . . . . . . . . . . . . . . 473.11 Three-dimensional diffusion with potential and 1000 time steps . . . . . . . . . . 483.12 Random sequence of three states . . . . . . . . . . . . . . . . . . . . . . . . . . 493.13 2D free diffusion trajectory with 1000 time steps . . . . . . . . . . . . . . . . . 503.14 Likelihood convergence 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.15 Likelihood convergence 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.16 Likelihood landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.17 Energy minimization procedure via Simulated Annealing (SA) . . . . . . . . . . 604.1 Jablonski-Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.2 Stokes-shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Wide-Field Fluorescence Spectroscopy (from [22]) . . . . . . . . . . . . . . . . 654.4 Different types of diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Optical behavior of incident light radiance at the boundary surface between twomedia with different refraction indices . . . . . . . . . . . . . . . . . . . . . . . 684.6 Schematic representation of objective-TIRFM . . . . . . . . . . . . . . . . . . . 694.7 Experimental preparation of disc membranes . . . . . . . . . . . . . . . . . . . 715.1 Transducin trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Several single transducin trajectories in one plot . . . . . . . . . . . . . . . . . . 755.3 Enlargement of the small areas, marked in Figure 5.2 . . . . . . . . . . . . . . . 76XI
Page 1: Diploma ThesisDepartment for Theore
Page 5 and 6: AbstractAnomalous diffusion is a ub
Page 7 and 8: Zusammenfassung 1Anomale Diffusion
Page 9: Contents1 Motivation 12 Visual Tran
Page 13 and 14: 1 Motivation„NEC FASCES, NEC OPES
Page 15 and 16: Theoretically we profit from enormo
Page 17 and 18: 2 Visual Transduction”The eye owe
Page 19 and 20: (a)(b)(c)Figure 2.3: (a) The anatom
Page 21 and 22: effect of generating the photorecep
Page 23 and 24: 3 Theory”The theory is the net, t
Page 25 and 26: 3.2 Markov Chains, Markov Processes
Page 31 and 32: 3.3 Hidden Markov ModelsP[X(t + τ)
Page 33 and 34: 3.4 Maximum Likelihood Principlewit
Page 35 and 36: 3.5 Optimization3.5 OptimizationAcc
Page 37 and 38: 3.7 Baum-Welch-AlgorithmAlgorithm 3
Page 39 and 40: 3.8 Two Approaches on Stochastic Sy
Page 41 and 42: 3.9 DiffusionConsider a basin fille
Page 43 and 44: 3.9 Diffusion169].Rearranging (3.34
Page 45 and 46: 3.9 Diffusionwith τ = t −t ′ .
Page 47 and 48: 3.9 DiffusionD = 2k2 B T 2σ . (3.4
Page 49 and 50: 3.10 Hidden Markov Models with Stoc
Page 55 and 56: 3.11 Hidden Markov Model - Vector A
Page 57 and 58: 3.11 Hidden Markov Model - Vector A
Page 59 and 60: 3.12 Artificial Test Examples for H
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3.12 Artificial Test Examples for H
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3.13 Global Optimization Methods3.1
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3.13 Global Optimization MethodsEve
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4 Fluorescence Tracking Experiments
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4.2 Fluorescence SpectroscopyAfter
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4.3 Single Molecule Tracking via Wi
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4.4 Total Internal Reflection Fluor
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4.4 Total Internal Reflection Fluor
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4.6 The expected range for the Tran
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5 Modeling of the Experiment”The
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5.1 Experimental Data(a)(b)(c)(d)Fi
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5.2 Model Ansatz and Estimation of
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5.2 Model Ansatz and Estimation of
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5.3 Testing the Modelalgorithm was
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5.5 Estimation of the Noise Intensi
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5.6 Estimation on the Basis of Diff
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6 Conclusion and OutlookThe main as
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7 Bibliography[1] R. C. Aster, B. B
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[36] C. U. M. Smith: Elements of Mo
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Index11-cis retinal isomer, 87TM se
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A AppendixA.1 From Copernicus to Ne
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A.2 Important Papers on the Rhodops
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A.3 Probability TheoryDefinition A.
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A.4 Definitions for OptimizationDef
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A.4 Definitions for OptimizationDef
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A.7 The Fluctuation-Dissipation-The
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A.7 The Fluctuation-Dissipation-The
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A.8 Kramers-Moyal Forward Expansion
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A.9 Deriving the Fokker-Planck Equa
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A.9 Deriving the Fokker-Planck Equa
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DanksagungenIch möchte folgenden M
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AffirmationHereby I, Arash Azhand,
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