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

5 Modeling of the Experiment”The Begin of all sciences is the astonishmentthat the things are, like they are.”(Aristotle) 1In the last chapter the experimental methods which were used, in order to track single transducinmolecules in a solvent above the membrane were presented.As we pointed out in section 4.5, the experiments on the movement of the transducin molecules canbe separated in two different experimental setups, namely the ”Dark State”, denoting the situationwhen the rhodopsin molecules on the membrane are not activated, and the situation when therhodopsin molecules are activated, denoted by ”Active State”. First we will investigate the darkstate in the following, and afterwards we will come to the active state.In the first section of this chapter we will present the related trajectories graphically. This mayhelp us to understand what type of observation we have and how we can model it.In the next section (section 5.2) a model ansatz will be introduced, discussed and a parameterestimation on the basis of this model will be accomplished. Discussion of the results will also bedelivered here, while testing of the model will be performed in section 5.3.Related to the transition matrices we will use the term ”holding probability” when we mean thediagonl entries of the considered matrices. This holding probabilities denote the probabilities thata certain state makes a transition to itself and not to another state.Further analysis of the data based on the separate estimation of the x- and y-components andseparate estimation of the single trajectories will be delivered in the sections 5.4 and 5.5.A theoretical development of an estimation framework based on Hidden Markov Model - StochasticDifferential Equation with differentials d x in place of positions x will be given in section 5.6.The associated implementation in the framework of an object oriented programming language inorder to estimate data for model parameters will not be delivered. This is a task for future works.1 German translation: ”Der Beginn aller Wissenschaften ist das Erstaunen, daß die Dinge sind, wie sie sind” (Aristoteles).73