Examples & Applications

Examples below demonstrate how to perform the Bayesian model selection on different types of FCS data, including multiple TACF curves and a single intensity trace, using the provided MATLAB package at the download page. The FCS data sets for the following examples can be downloaded here. Unpack the file and put the folder 'example' in the package folder 'FCS_Bayes_package'.

## Example 1 Analysis of multiple TACF curves without raw intensity traces

Proper calculation of model probabilities requires incorporating the noise covariance matrix. When only TACF curves are available instead of raw intensity traces, the noise covariance matrix must be estimated from multiple independent TACF curves measured from the same underlying physical process.

Example file 'FCS_64cur_vary_Dr.mat' contains a cell 'corrFCS_cell' of TACF curves data sets, a lag time vector 't', and the aspect ratio of the focal volume 'ks'. 'corrFCS_cell' contains 15 sets of simulated two-component diffusion TACF curves data with varying D2/D1 ratios. Each data set has 64 individual curves of 2-component diffusion simulated with the same parameters. Please refer to the paper for details about the simulation. The script 'run_FCS_bayes_multi_curves.m' runs the Bayesian model selection on the provided example data.

Typically > 4 TACF curves with 128 data points are needed for the regularized covariance matrix to be non-singular, but the minimum required number of TACFs increases as the number of data points increases.

## Example 2 Analysis of raw intensity traces

When the raw photon-count trace or photon arrival time is available, the noise in a single TACF can be calculated from its underlying photon-count products (see the paper for details).

Example files 'FCS_8int_vary_Dr_D2=#.mat' contains a series simulated photon count traces of two-component diffusion with varying D2/D1 ratios. Variable 'im' is an array of eight simulated 8-bit photon count traces with D2=# and the sampling time 'dt'. Note that the photon count trace contains more than 100 million counts, so a machine equipped with at least 8GB ram is required to run this analysis. The run-script 'run_FCS_bayes_single_trace.m' performs the analysis through the following steps:

(1) Running the blocking analysis to find the minimum block-time (dot) in the blocking curve to generate independent samples required for noise covariance estimation. When the minimum block-time of the blocking curve cannot be found by the algorithm (gray curves), the maximal block-time available in the blocking curve is used to estimate the noise.

(2) Computing the TACF and its covariance matrix using the multi-tau algorithm. If dual detection scheme is used (two photon count traces recorded simultaneously using two detectors), use the function 'compute_CCF2' instead of 'compute_ACF2' for computing TACF.

(3) Running the Bayesian model selection on the computed TACF curves, same as the analysis of multiple curves.

## Example 3 Analysis of fluorescence movies

For imaging FCS in which each pixel acts as a single detector, the intensity trace at each pixel may be analyzed individually in the same manner as in Example 2, yielding maps of model probabilities and diffusion coefficients.

The example file is a simulated 16-bit fluorescence movie of a partitioned microdomain in the membrane with lower diffusivity than the outer, non-domain region (D1 = 4 um2/s and D2 = 0.04 um2/s). Particles undergo free diffusion and switch diffusivity instantaneously upon crossing the microdomain boundary. Two-component diffusion is detected near the domain boundary because the PSF averages intensity from particles inside and outside the domains. The run-script 'run_imagingFCS_bayes.m' performs the analysis through the following steps:

(1) Running the blocking analysis at each pixel to find the minimum block-time (plateau) in the blocking curve to generate independent samples required for noise covariance estimation. When the minimum block-time cannot be found by the algorithm, the pixel is labeled and the maximal block-time available in the blocking curve is used to estimate the noise.

(2) Computing the TACF and its covariance matrix using the multi-tau algorithm and the specified block-time.

(3) Running the Bayesian model selection on the computed TACF curves and generating the map of model probabilities, the map and histogram of diffusion coefficients.