Analysis of FRAP Data: An exercise using quantitative imaging results and spatial modeling to explore principles of diffusion.
Introduction
The exercise is based on a series of Fluorescence Redistribution After Photobleaching
(FRAP) experiments designed to demonstrate how FRAP experiments are done and some basic principles of diffusion. Sample data is provided from a student laboratory, but if you have the resources available you can do the experiment as well as the data analysis. There are a number of FRAP simulations on the web, my favorite is
The Experiment
In the sample experiment, allophycocanine dye (APC; Invitrogen SKU#A-803l) was microinjected into B104 cells. APC is soluble, primarily monomer and generally is large enough that it does not enter the nucleus. APC was microinjected at ~ 0.8 ug/ul in distilled water, keeping the injection volume to less than 10% of the cell volume. Imaging was done using a
Zeiss LSM510 confocal microscope with a 40X 0.95NA lens (dry), using the timeseries module that allows you to bleach with distinct laser lines/intensity any region of interest you select. With the N.A. used, the bleaching beam will approximate a column throughout the height of the cell
(about 10 um at the nucleus, much less in the periphery) The protocol for using the Zeiss software is included in the CD, but a similar experiment can be done with any microscope that allows you to rapidly switch laser intensity in a defined region of interest. The imaging is set up to collect fluorescence from the entire depth of the specimen in a single plane – i.e. in a widefield configuration. To do this the pinhole aperture is opened to its maximum diameter. The five prebleach images are collected, and the bleach is accomplished with a high intensity bleaching beam. A time series is collected beginning as rapidly as possible after the bleaching (in this case, about 3 ms). The time per image and the total number of images collected are determined empirically; you are trying to achieve a redistribution curve (return of fluorescence in the bleached region) that yields at least several points in the linear early phase, and collect for a long enough time that the redistribution is essentially complete. In the sample data we performed 4 different experiments on the same cell, using different sized bleached regions. (Note that for a real experiment, it is not advisable to repeat the photobleaching on the same cell).

The Data
The Excel spreadsheet entitled ““APC_FRAP Data”” contains an analysis of FRAP image data. The Experimental Data sheet shows the average fluorescence intensity (after subtracting background) for each time t (in secs) for 1) the entire bleach region; 2) the center of the bleach region, and the whole cell (average of pixels within the entire cell selected by thresholding the image).
To approximate the diffusion coefficient from the individual plots of f(t) vs t, the curves can be fit to the function
I(t) = I f
(1-f
B
) – [I f
(1-f
B
) – I
0
] [r (1+t/τ
1
) -1 + 1 - r] , where:
I
0
equals the initial value of I(t),
I f
equals the final value of I(t), f
B
equals the fraction of the cell area bleached, r equals the mobile fraction,
using an appropriate curve fitting program. For the purposes of this exercise, however, we will compare the data to results of a spatial simulation. Thus we will simply normalize the data to the average prebleach intensity to directly compare with the results of spatial simulations.
Normalize and Plot Data
Calculate the average prebleach intensity in each region by averaging the data from the first 5 images. For each of the intensity values, divide by the average prebleach value, and create a plot of Normalized Intensity vs Time
Estimate Diffusion Coefficient
For 2 dimensional diffusion, assuming a small fraction of the cell is bleached,

D
= w 2 / 4D; you can estimate 
D
(a characteristic time) as the time it takes to reach ~1/2 the maximum intensity value. Estimate 
D from your plot and calculate D based on this value.

Creating a Spatial Simulation using Virtual Cell based on the FRAP tutorial
Setting Model Geometry
Use the Virtual Cell to compare your “guess” for D to your actual results. Within Virtual Cell,
File -> Open the FRAP tutorial [Open-> BioModels -> Shared Folder ->FRAP tutorial]. We will ignore the nucleus for this exercise. Save the tutorial as a new model under your username. You need to change the geometry to use the image of the cell we actually photobleached. Use the
Change Geometry button on the BioModel Application to access the geometry in the Geometry
Database. You will find it in Shared Geometry->Education\Public 2D image Geometry “FRAP excercise” and it is also included in the data files on the CD. The total size of the geometry is
237 X169 pixels at 0.44 um/pixel.
Set Initial Conditions
The simulation will start immediately following the bleach. We will use an initial concentration of APC of 10 uM. In the Initial Conditions you need to set the concentration of APC in the bleach region to be 20% of the initial value.
The initial conditions for APC will be the equation: (2 + 8.0 *((x < 73.92) || (x > 85.36) ||
(y < 38.72) || (y >47.08)))
In this equation, the expression “((x < 73.92) || (x > 85.36) || (y < 38.72) || (y >47.08))” evaluates to 0 for all x values from 73.92 – 85.36, with any y value from 38.72 – 47.08, and evaluates to 1 for all positions of x,y. Inside the bleach region, the entire expression will evaluate to 2 uM (i.e.
2 + 0), outside the bleach region, the entire expression will evaluate to 10 (i.e. 2 + 8*1)
Also Set the Diffusion Coefficient to correspond to your initial estimate of D.
Set Simulation Tasks
For the simulations, click the Edit Button and select the Task page, keep the Start Time at 0, set the time step to 0.01 and set End Time to 22.0 seconds; also set keep every 10 time samples.

Select the Mesh page, and set the number of mesh elements at x = 237, y =169. Close the Edit: menu
Run the Simulation
Click the run button to run the simulation.
Results
To view results of the simulation, click the Results button. By selecting the point button in the bottom right of the panel, you can create a single point over time graph with the VCell analysis tools. Click within the geometry, then click Show Time Plot. The results can also be viewed as a table of data and exported using the export tab. Compare your data to the data from the “Center of Bleach region” data in the Normalized Data worksheet. Change the diffusion coefficient in the initial conditions and rerun the simulation to best match your simulation results with your experimental curve.
The Simulation 1 worksheet in the Excel file contains shows results from this simulation compared to the experimental results, using a D = 8 um 2 /s. The fit is good in the early phase of the redistribution, but not very good at later times.
Additional Exercises
1. Predict the diffusion coefficient of APC in cytoplasm given:
MW of APC = 104,000
Equivalent radius of sphere r ~ 1.5 nm = 1.5 X 10 -7
T (Room Temperature) = 300  K
η water
= .01 poise k (Boltzmann constant) = 1.38 X 10 -16 gcm 2 /s 2 K
cm
From Stokes-Einstein equation:
D

6 kT
 r
= 146 um
If η cytosol
= (4-11)* η water
, D in cytosol ~ 13 - 36 um 2 /s
2 /s

2.
The diffusion coefficient we obtained from the simulations (8um 2 /s) was less than we calculated for cytosol 13 - 36 um 2 /s. Can you think of 1 or 2 possible reasons for the discrepancy? a) The molecule may not be spherical and thus have a larger hydrodynamic radius. b) The geometry we used for the bleach region assumed no diffusion during bleaching
(we used a rectangle). The actual geometry does not have precise edges, because diffusion occurs during the time it took to bleach the molecules.
(There are actually several ways to improve the simulation to account for diffusion during bleaching. It is possible to simulate the bleach itself by including a reaction – a destruction of the fluorophore – that occurs on the time scale that it took to bleach the molecule. You can find a method to do this in the “FRAP with binding” tutorial on the website; go to www.vcell.org
and from the VCell Software page click on Virtual Cell
User Documentation ( http://vcell.org/login/documentation.html
). Alternatively, you can create starting conditions by importing a normalized form of the first postbleach image.
The software can use the values in the image as the starting values for the APC concentration. Instructions for this can be found in the user guide.)
c) The APC molecule may exhibit rapid binding to other immobile components. What you measure by FRAP would be the time-averaged diffusion between the time spent freely diffusing and the time spent bound (D = 0).
3.
In theory for diffusion, it takes 4X as long to go 2X as far. Try this with your simulations.
Use the same cell geometry and increase the diameter of the bleach are by factors of 2. Plot the curves and calculate the τ
1/2
. It doesn’t quite work. Why not?
It doesn’t work because there is not an infinite source of molecules for the redistribution.
4. What happens to the redistribution curve if APC reversibly binds to an immobile cytosolic component?
This can be simulated by adding a new species to the tutorial with D = 0, and creating a reaction where APC binds to the new species. The FRAP with binding tutorial
( http://vcell.org/login/documentation.html
) takes you through this process step by step.
4.
What happens to the redistribution curve if APC is transported into and out of the nucleus?
You will add a new feature (nucleus) to the simulation, with another APC species
(APC_nucleus) in the nucleus (in Virtual Cell, a species must be defined in the context of each compartment). Add a flux reaction between the nucleus and cytosol. Appropriate flux rates can be obtained from the literature.