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6 Conclusion and OutlookThe mean noise intensity over all trajectories leads to a mean diffusion coefficient of D m = 4.5 µm2swhich also is lying in the above range. Furthermore the plots of the considered noise intensitiesversus the single trajectories are showing a wide variance around this mean value. Some of the trajectoriesproduce noise intensities that are leading to diffusion coefficients in the range of 1.0 µm2sto 1.6 µm2s. This suggests that the transducin interactions on the disc membrane may be more complexthan expected and may require more than three states whose biophysical interpretation is yetunclear. This calls for further research in collaboration with the experimentalists.In order to guide future studies, we have proposed a theoretical formulation of an HMM-VARestimation framework that is based on observation data consisting of position differences insteadof positions, while the estimation procedure is identical. We anticipate that this new method willcompensate for some of the deficiencies observed with the methods used in this work.Some options for future studies are performed in the following:1. We have to obtain more experimental trajectory data. This will allow us to accomplishbetter estimation of diffusion constants and also a better determination of the associatedhidden states in the framework of HMM-VAR. Our confidence to reach this aim is justifiedby the estimation results that we have conducted on artificial trajectory data and presentedin section 3.12 on page 46. The good performance of HMM-VAR is shown there, providedthat the considered observation data has a sufficient large length (some 10000 and morethan 100000 observation points).2. The use of global optimization methods like the genetic and the simulated annealing algorithm(see section 3.13 on page 57) is the next step. The application of this algorithms onexisting and additional data may allow better estimation results.3. Finally a direct simulation of the considered biological process in the framework of a generalcell simulation has to be aproached. In our research group an implementation of a socalled ”Cellular Molecular Dynamics” (CMD) has been provided. The CMD is a first approachin order to simulate an arbitrary cell with a solvent and arbitrary definable particleswhich are diffusing in the solvent or on the membrane. Furthermore one is able to define atype of reaction between these particles. The simulation of the retinal rod disc membraneswith integral proteins rhodopsin and the transducin proteins that are diffusing in the solventbetween and on the rod disc membranes should be possible in the framework of the CMD.92
7 Bibliography[1] R. C. Aster, B. Borchers, and C. H. Thurber: Parameter Estimation and Inverse Problems(Elsevier Academic Press, 2005), 1st ed.[2] N. J. T. Bailey: The Elements of Stochastic Processes - with Applications to the NaturalSciences (John Wiley & Sons, 1964), 1st ed.[3] G. P. Basharin, A. N. Langville, and V. A. Naumov: The Life and Work of A. A. Markov,Numerical Solution of Markov Chains (Workshop) pp. 1–22 (1990).[4] H. C. Berg: Random Walks in Biology (Princeton University Press, 1993), new, expanded ed.[5] J. A. Bilmes: A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimationfor Gaussian Mixture and Hidden Markov Models (1998).[6] F. Bruckert, M. Chabre, and T. M. Vuong: Kinetic Analysis of the Activation of Transducinby photoexcited Rhodopsin (Influence of the lateral Diffusion of Transducin and Competitionof GDP and GTP Nucleotide Site), Biophysical Journal 63, 616 (1992).[7] M. E. BURNS and T. D. LAMB: Visual Transduction by Rod and Cone Photoreceptors,in Chalupa LM, Werner LS, editors. The Visual Neurosciences. USA. Bradford Books, MITPress; 2003. pp. 215-233. chapter 16..[8] S. Doose and M. Sauer: Fluoreszenz bringt Licht ins Dunkel - Fluoreszenzmikroskopie ermöglichtdas Studium einzelner Moleküle auch in lebenden Zellen, Physik Journal 6 (2007)Nr. 12, S. 55-60 (2007).[9] A. Einstein: Investigations on the Theory of the Brownian Movement (Dover Publications,Inc., 1956), 2nd ed.[10] A. Einstein: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegungvon in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik pp. 549–560 (1905).[11] S. Felber, H. Breuer, F. Petruccione, J. Honerkamp, and K. Hofmann: Stochastic Simulationof the Transducin GTPase Cycle, Biophysical Journal 71, 3051 (1996).[12] W. Feller: An Introduction to Probabilty Theory and its Applications - Volume I (John Wiley& Sons, 1968), 3rd ed.[13] W. Feller: An Introduction to Probabilty Theory and its Applications - Volume II (John Wiley& Sons, 1968), 3rd ed.[14] J. Herivel: The Man and the Physicist (Oxford University Press, 1975).[15] S. Heyse, O. P. Ernst, Z. Dienes, K. P. Hofmann, and H. Vogel: Incorporation of Rhodopsinin Laterally Structured Supported Membranes: Observation of Transducin Activation withSpatially and Time-Resolved Surface Plasmon Resonance, Biochemistry 37, 507 (1998).93
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Diploma ThesisDepartment for Theore
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AbstractAnomalous diffusion is a ub
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Zusammenfassung 1Anomale Diffusion
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Contents1 Motivation 12 Visual Tran
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List of Figures2.1 Anatomy of the e
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1 Motivation„NEC FASCES, NEC OPES
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Theoretically we profit from enormo
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2 Visual Transduction”The eye owe
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(a)(b)(c)Figure 2.3: (a) The anatom
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effect of generating the photorecep
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3 Theory”The theory is the net, t
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3.2 Markov Chains, Markov Processes
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3.3 Hidden Markov ModelsP[X(t + τ)
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3.4 Maximum Likelihood Principlewit
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3.5 Optimization3.5 OptimizationAcc
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3.7 Baum-Welch-AlgorithmAlgorithm 3
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3.8 Two Approaches on Stochastic Sy
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3.9 DiffusionConsider a basin fille
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3.9 Diffusion169].Rearranging (3.34
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3.9 Diffusionwith τ = t −t ′ .
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3.9 DiffusionD = 2k2 B T 2σ . (3.4
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3.10 Hidden Markov Models with Stoc
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3.10 Hidden Markov Models with Stoc
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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
Page 69 and 70: 3.13 Global Optimization Methods3.1
Page 71 and 72: 3.13 Global Optimization MethodsEve
Page 73 and 74: 4 Fluorescence Tracking Experiments
Page 75 and 76: 4.2 Fluorescence SpectroscopyAfter
Page 77 and 78: 4.3 Single Molecule Tracking via Wi
Page 79 and 80: 4.4 Total Internal Reflection Fluor
Page 81 and 82: 4.4 Total Internal Reflection Fluor
Page 83 and 84: 4.6 The expected range for the Tran
Page 85 and 86: 5 Modeling of the Experiment”The
Page 87 and 88: 5.1 Experimental Data(a)(b)(c)(d)Fi
Page 89 and 90: 5.2 Model Ansatz and Estimation of
Page 91 and 92: 5.2 Model Ansatz and Estimation of
Page 93 and 94: 5.3 Testing the Modelalgorithm was
Page 95 and 96: 5.5 Estimation of the Noise Intensi
Page 97 and 98: 5.6 Estimation on the Basis of Diff
Page 103: 6 Conclusion and OutlookThe main as
Page 107 and 108: [36] C. U. M. Smith: Elements of Mo
Page 109 and 110: Index11-cis retinal isomer, 87TM se
Page 111 and 112: A AppendixA.1 From Copernicus to Ne
Page 113 and 114: A.2 Important Papers on the Rhodops
Page 115 and 116: A.3 Probability TheoryDefinition A.
Page 117 and 118: A.4 Definitions for OptimizationDef
Page 119 and 120: A.4 Definitions for OptimizationDef
Page 121 and 122: A.7 The Fluctuation-Dissipation-The
Page 123 and 124: A.7 The Fluctuation-Dissipation-The
Page 125 and 126: A.8 Kramers-Moyal Forward Expansion
Page 127 and 128: A.9 Deriving the Fokker-Planck Equa
Page 129 and 130: A.9 Deriving the Fokker-Planck Equa
Page 131 and 132: DanksagungenIch möchte folgenden M
Page 133: AffirmationHereby I, Arash Azhand,
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