CHAPTER 1 - SYSTEM OPERATIONZEISS Left Tool Area and Hardware Control Tools LSM 880196 000000-2071-464 10/2014 V_015.3.4.3 Model EquationsIn the following, the mathematical equations of the correlation functions and the fit equations will bemore detailed. The acquired correlation functions must be fitted to models in order to retrievemeaningful parameters. It depends on the process, which model is the most appropriate. If theunderlying process is known, the model can be chosen a priori. For example, if one studies diffusion in amembrane, a 2-D diffusion model should be applied. In other cases, the process is not known, forexample, if one deals with free or anomalous diffusion. In this case, one can screen different potentialmodels and look for the best fit taken into account the Χ2 value. Often two models work nearly thesame, for example, a two component free diffusion model can give you as satisfactorily a fit as a onecomponent anomalous diffusion model and without prior knowledge on the system it will be impossibleto decide, which is the better one. In principle, models can be ruled out, if the fit does not work,however, a working model is only a potential candidate but does not signify it to be the correct one. Careshould be taken to minimize the free parameters as much a possible to improve on the fit quality. It doesnot make too much sense to fit to three components without fixing parameters of at least one of them.For example, if the diffusion time of a free ligand can be determined in a pre-experiment, that valueshould be fixed to reduce the number of floating parameters for the evaluation of the bindingexperiment to its receptor.The FCS software is designed to be flexible. That means that the user can define or assemble equationsthat do not make sense. Care should therefore be exerted and used formulas matched with the onesfrom literature to obtain meaningful results. Also, the presence of a model does not automatically mean,that the recorded data are of a quality that allows its usage. For example, anti-bunching requires a lot ofcare in data acquisition like long measurement times and cross-correlation to reduce dead times of thedetectors and elimination of after-pulsing artefacts. It is in the responsibility of the user to set up hisexperiments accordingly.(1) The correlation functionThe auto-correlation function is defined as follows:( ) ( )( )( ) ( )( )( )( )( ) ( )( )( )( )∫∫∫∫ +⋅⋅=⋅+⋅⋅=+⋅= TTTTIdttIdttItITdttITdttItITtItItIG02002202 11)(dtdddtdddtddtd (1a)or( ) ( )( )( ) ( )( )( )( )( ) ( )( )( )( )∫∫∫∫ +⋅⋅=⋅+⋅⋅=+⋅= TTTTIdttIdttItITdttITdttItITtItItIG02002202 11)(tttt (1b)where denotes the time average and ( ) ( ) ( )tItItI −=d describes the fluctuations around the meanintensity.For long time average of I (no bleaching) the following relation exists:( ) ( )ttdII GG += 1 (1c)
