Universal Approach to FRAP Analysis of Arbitrary Bleaching Patterns

The original approach to calculating diffusion coefficients of a fluorescent probe from Fluorescence Recovery After Photobleaching (FRAP) measurements assumes bleaching with a circular laser beam of a Gaussian intensity profile. This method was used without imaging the bleached cell. An empirical equation for calculating diffusion coefficients from a rectangular bleaching geometry, created in a confocal image, was later published, however a single method allowing the calculation of diffusion coefficients for arbitrary geometry does not exist. Our simulation approach allows computation of diffusion coefficients regardless of bleaching geometry used in the FRAP experiment. It accepts a multiple-frame TIFF file, representing the experiment as input, and simulates the (pure) diffusion of the fluorescent probes (2D random walk) starting with the first post-bleach frame of the actual data. It then fits the simulated data to the real data and extracts the diffusion coefficient. We validate our approach using a well characterized diffusing molecule (DiIC18) against well-established analytical procedures. We show that the algorithm is able to calculate the absolute value of diffusion coefficients for arbitrary bleaching geometries, including exaggeratedly large ones. It is provided freely as an ImageJ plugin, and should facilitate quantitative FRAP measurements for users equipped with standard fluorescence microscopy setups.


Supplementary Material Simulation Approach to FRAP Computation and Analysis
Our simulation approach allows computation of diffusion coefficients regardless of bleaching geometry used in the FRAP series. The method is based on fitting a computer-simulated recovery to actual recovery data of a FRAP series. The algorithm accepts a multiple-frame TIFF file, representing the experiment as input, and simulates the (pure) diffusion of the fluorescent probes (2D random walk) starting with the first post-bleach frame of the actual data. Once the simulated recovery is finished, the algorithm fits the simulated data to the real one and extracts the diffusion coefficient.
The algorithm iteratively creates a series of simulated images, where each frame corresponds to a single iteration. The intensity values are extracted from the (user indicated) bleached area of the simulated frames, thus determining the general shape of the recovery curve. The "time" axis at this stage is in arbitrary units (iterations). To extract the diffusion coefficient, the simulated recovery curve needs to be fitted to the real recovery curve, by appropriately stretching the "time" axis. The time between frames in the actual data set is obviously known, thus once overlapping optimally the simulated curve with the real one, the duration of one iteration, in real time units, is determined. The diffusion coefficient of the simulated series is then calculated using (1) = where D s is the simulation-extracted diffusion coefficient, is the step of a molecule in each iteration of the simulation, corresponding to one pixel in the image (the pixel size is calibrated previously, by imaging a known calibration sample), and is the time interval between steps (determined as explained).
Technically, the simulation proceeds until a plateau is reached (equilibration of the fluorescence intensity in the bleached area). The number of data points in the simulated recovery is typically different (larger) than the number of experimental points. In addition, the real experimental data may not have been acquired until equilibration of fluorescence. For this reason, in order to determine , the algorithm scans a range of possible values for the total duration represented by the simulation and calculates a value ( 2 ) for the goodness-of-fit between the simulated data and the real FRAP data. Total simulation duration is selected as the one that that produces the minimal 2 . An example of thus simulated data overlapped on real data is shown in Supplementary figure S3a. Supplementary figure S3b shows the goodness-of-fit value as a function of total duration represented by the whole simulation. The sharp minimum occurs for 11.7 seconds (this is the total time represented by the blue set in Supplementary figure S3a). The value of is then calculated by dividing the total simulation duration (determined as above) by the number of iterations.

Traditional Analysis of FRAP Data
Circular Bleaching Pattern Cells were bleached with beam diameters between 2.4 6.0 and fluorescence recovery was imaged at rates of 5 -20 frames per second. A recovery curve was extracted by quantifying the intensity of the bleached area along the stack of images and normalized by dividing each value by the intensity of the same area before bleaching. From the recovery plot, a typical recovery time, , was extracted by fitting the data with a model in the form of 1 : where ( ) is the fluorescence intensity of the bleached area divided by the pre-bleaching intensity, is the extent of bleaching, is the typical recovery time and n is a positive integer. Fitting was done with a series of n=6. To calculate the diffusion coefficient, ( ) we used 1 : where , the radius of the Gaussian bleaching profile at 1/ height, was extracted from the frame following the bleaching pulse (Supplementary figure S4e,f), and was from fitting equation (2) to the data.
The most important cause for variation in the calculated value of D is the difficulty to extract a reliable value for from the Gaussian fit to the bleached spot. The problem in resolving is caused by bleached spots that are asymmetric in both geometry and intensity, as well as random noise in the imaging system. To minimize these, we performed angular averaging of the data around the center of the bleached spot, as described in 2 , and shown in Supplementary figure S4 Rectangular Bleaching Pattern Cells were bleached with rectangular patterns with dimensions between 2.5X2.5 µm 2 to 4X4 µm 2 . Calculating the diffusion coefficient for rectangular areas was done by fitting the recovery data to a previously reported model in the form of: where ( ) is the fluorescence intensity as a function of time, ( ) is the intensity for and is the "width" (somewhat obscure definition, especially for high aspect ratios) of the rectangle. To avoid the confusion associated with the definition of , we avoided bleaching high aspect-ratio rectangles.

Calculating Error Bars
Circular bleaching pattern According to equation (3), the error in the diffusion coefficient (D c ) stems from two sources. First, error in the estimation of the bleach spot radius -reported by the fitting algorithm that fits the bleached spot intensity profile to a Gaussian and second, the error in the estimation of extracted from fitting FRAP data to equation (2). Rectangular Bleaching Pattern D c was extracted directly from fitting the experimental data with the empirical equation (4). The error is reported by the fitting algorithm.

Simulation
The simulation diffusion coefficient (D s ) was calculated using equation (1). Similarly to the circular bleaching pattern, the error here stems from two sources. First, error in pixel size ( ) (provided as input by the user), which was estimated as 10% of the value. Second, error in the estimation of resulting from fitting different iteration times for the simulation and choosing the one which best fits the real FRAP data. The error range was estimated as the time difference between the values of adjacent (on the left and right) to the best (chosen) one.

Experimental guidelines for choosing the correct bleached area
First, it is important to acquire high contrast data expressed in a high signal to noise ratio. Particularly important is the background, please make sure that your subject has more fluorescent intensity than the background. Next, bleaching should be carried out to an extent that allowing differentiation between bleached and unbleached areas. If, after taking all these into consideration, there is still doubt on the exact bleached area, it is possible to use an image analysis software (such as ImageJ) to create an intensity profile along the axis of the bleached area, using it to find the exact edge location.

SUPPLEMETARY FIGURES
Supplementary figure S1 -Lack of correlation between bleached area size and diffusion coefficient Diffusion coefficient extracted from simulation is not correlated (r=0.283, p=0.17) with bleached area dimensions. The different groups, distinguished by their bleaching geometry are marked with different colors. Circular (Gaussian) spots in purple, Box geometry in green and arbitrary shapes in orange.
Supplementary figure S2 -Bleaching an exaggeratedly large diameter circular area (a,b) Images of a cell prior to, and immediately after bleaching an exaggeratedly large diameter circular laser beam, respectively. (c) Diffusion coefficients resulting from attempting to use the classical, closed-form solution for increasingly large circular areas. When the bleached area is greater than ~25%, the calculation results in significant errors.