APh161: The Physics of Biological Structure andFunctionHomework 5Due Date: Thursday, Mar. 1, 2007“Their drills are bloodless battles and their battles are bloody drills.” -JosephusReading: Read chap. 8 of PBOC and write a referee report in the usualway. Please remember as you are doing this that this is a huge help to usas we try to improve the book for its eventual publication. We appreciate itvery much and it is a chance for you to construct a concrete written argument.1. FRAP and diffusion.In class I discussed the problem of two-dimensional FRAP. In this prob-lem, I want you to carry out a full derivation of the concentration as a functionof position and time after photobleaching a cell of radius 25 microns with a“hole” of radius 2 microns. (Looking at the treatment of the one-dimensionalversion of this problem in chap. 12 of PBOC will be helpful.) For simplic-ity, ignore the presence of a nucleus, think of the cell as a perfect circle andimagine the photobleached region as a circle at the center of the circular cell.Consider an initial concentration c0of the fluorescent molecule of interestwhich is uniformly distributed throughout the cell. How many molecules ofthe fluorescent molecule are there - write an equation that gives this number?Before doing any calculations, explain what the final concentration (c∞) willbe after full relaxation and the system has returned to equilibrium. You mayassume that once a molecule has been photobleached it is effectively deadand can be forgotten. Your goal now is to compute the recovery curve. Whatthis means is that you need to work out how many fluorescent molecules arein the photobleached region as a function of time. Make graphs for the casewhere the photobleached region is centered about the origin. Make sure whenyou make your plots you use reasonable values for the diffusion constant -justify your choice. One of the uses of the FRAP technique is to determinethe diffusion constant of various molecules within the cytoplasm of cells. Dis-1cuss how that might work on the basis of the derivation you have given he re.To do this problem you will need the table of zeros of the first derivativeof J0(x) given in the file attached to the homework. Make sure you explainexactly what you are doing and what your results mean. Also, I want you toplot the result for different number of terms kept in the Bessel series. Try itfor N = 10, 20 and 30 and compare the results. Comment on the goodnessof the fit.2. Gaussian Chain.In class, I gave a heuristic “derivation” of the model of the Gaussianchain starting from the binomial distribution for chain configurations. Carryout a complete derivation of the Gaussian chain in one dimension by usingthe Stirling approximation as discussed in class. Make sure to explain howyou turn this into a probability density and explain all of your steps. Whenyou are done, plot the distribution of end-end distances for DNA of length5 Kuhn lengths and 50 Kuhn lengths. Compare the discrete and continuousdistributions. Using the distribution, compute the “size” of the polymer byevaluating < x2> for the one-dimensional chain. Explain your result andcompare it to the intuitive derivation given in
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