HW_3_2011_v2

Such that your answer has 5 accuracy in the region of

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such that your answer has 5% accuracy in the region of interest, namely, the FRAPed region, and tell us how many terms you used. Here is some stuff that will come in handy when thinking about this problem. First, you should try to obtain this equation d 2 ρ dz 2 + 1 z dz + ρ = 0 . (1) The only solution to this equation that does not diverge for z 0 is the zero order Bessel function J 0 ( z ). Next, the boundary conditions at the edge of the cell will lead to a condition of the form J 0 0 ( kR ) = 0 . (2) Interestingly, the roots of J 0 0 are just the roots of J 1 because of the identity J 0 0 ( z ) = - J 1 . The full solution you are looking for will emerge as (make sure you demonstrate this clearly and convincingly) c ( r,t ) = a 0 + X i =1 a i e - DK 2 i t J 0 ( K i r ) . (3) We can determine the coefficients a i using the initial condition c ( r, 0). An- other identity that will prove useful when doing the calculation of the coef- ficients is: R zJ 0 ( z ) dz = zJ 1 ( z ). 2. Average Occupancy of Receptors. In class, I did the problem of ligand-receptor binding in three different ways. Redo the derivation that I did using chemical potentials (i.e. the grand canonical ensemble) and show that the average number of ligands bound is given by h N i = 1 β ∂μ ln Z . (4) 2
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You will find it helpful to note that the average particle number can be written as h N i = 1 Z X i N i e - β ( E i - N i μ ) , (5)
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such that your answer has 5 accuracy in the region of...

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