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Using FRAP to determine a diffusion coefficient

Fluorescence recovery after photobleaching (FRAP) is an important technique for measuring properties of macromolecules such as the diffusion coefficient of a protein. In this problem, you'll collaborate to determine a diffusion coefficient from some (fake) experimental data. You don't have to understand how to solve the diffusion equation, but you have to understand how length and time scale in diffusional processes.

We start with a large number of fluorescently tagged biomolecules all of the same species. The technique takes advantage of the fact that if subject to a high enough intensity of light, fluorescent molecules lose their ability to fluoresce, a phenomenon known as "photobleaching".

You don't have to read the whole thing, but understand the basic idea of the technique. A person with bio background should try to help with this part of this problem.

 

  • 2. We want to determine the diffusion coefficient of a protein bound to the membrane but free to move across it so that all motion is effectively two dimensional. We assume that initially a circular region of radius 2 microns is completely photobleached, so right after the photobleaching, no molecules in that circle are able to fluoresce. As time goes on, fluorescing molecules (that haven't been bleached) will move into the circular region by diffusion. To simplify the analysis further, we'll measure the fluorescent intensity at the center of the circle as a function of time (though integrating it over the circular region is what is normally done.)

Go to the directory (i.e. folder) in the homework tree hw3/frap and look at  measurements.txt. It's the (fake) data that we obtain from our (fake) frap experiment. The first column is the time measured in seconds. The second columns is the normalized fluorescent intensity. It is normalized so that it will fully recover, for long times, to a value of 1.

We also simulate the system in 2d_frap.py. The initial radius is arbitrarily set to 8. To see this in the code, the last argument in

set_circle(T,0.0,(nx/2,ny/2),8)

is 8, which is the radius. When you run the script you'll see a 2d image of the fluorescence (called T in the code). And then the value of the fluorescence in the center as a function of time is displayed, and also saved in "grap_comparison2.txt".

You can use this simulation and the measured data to calculate the diffusion coefficient of the protein, if you understand scaling. This is quite tricky conceptually but does not require much math. Don't get bogged down in trying to solve partial differential equations. It's not the way to solve this problem. This is all about understanding scaling which is something that should be emphasized more because it's so powerful and can give you very  precise results if you understand how it works. So read what's written below carefully, probably a few times, and draw some pictures. This illustrates a very powerful and useful method of solving seemingly intractable problems.

Useful analogy with cooking

Consider the problem of boiling a spherical potato (though any shape will actually do!). Suppose it takes 10 minutes to boil a potato of 4cm radius, how long will it take to boil a potato of radius 8cm? Map this problem onto the FRAP situation discussed above. Do not try to make the radius of the circle 2 because that will lead to inaccurate results. Instead make it 8 and compare that result with when the radius is 16. You'll see that you can rescale the time axis to make the two curves superimpose. The way you rescale is crucial to solving this problem.

Now suppose you have an onion of the same shape and size as a potato, but it takes twice of long for the center of the onion to reach the same temperature as the potato. How are the two diffusion coefficients related? (I doubt that onions have a particularly different diffusion coefficient. Assuming this is just to illustrate a point.)

Strategy to solve problem

You'll also need to know the diffusion coefficient of the 2d FRAP simulation. You can use 2d_diff.py to do that. The same equation is evolved with a different initial condition. All the particles start at a single point and diffuse from there. The mean square displacement is calculated as a function of time. From that you can get the diffusion coefficient of the simulation. See Berg Links to an external site. (Random Walks in Biology) Eq 1.13.

  • 3 Briefly explain the different stages of FRAP with diagrams for each stage accompanying your explanations. Sketch qualitatively the recovery of fluorescence intensity as a function of time for an spot of radius R, and how this plot is changed (by rescaling) when the radius is changed to R' . Figure out how to rescale the time axis to make the two curves coincide. How can you represent this succinctly in terms of a mathematical expression? That is, figure out how to write the measured intensity at the center, as a function of time t, and R.
  • 4 Putting this altogether, you should obtain the diffusion coefficient of the protein from measurements.txt. The three parts to this analysis are understanding the FRAP technique, understanding scaling, and rescaling the simulation results to match the measurements.