HW_2_2011_final

Then use the fact that we can write h r 2 i h n x i 1

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= 0 and explain why that makes sense. Then, use the fact that we can write h R 2 i = h N X i =1 N X j =1 a e i · a e j i (13) Explain both the merits and the problems with applying this kind of analysis to pictures like that of the E. coli genome given in fig. 8.6. (b) As practice with the use of the binomial distribution and to make sure you feel the N result given above “in your bones”, we will derive the end-to-end distance in one dimension by using the binomial distribution. Your goal is to compute the average end-to-end distance h x i and the standard deviation q h x 2 i . Use the fact that x = ( n r - n l ) a where n r is the number of right pointing monomers and n l is the number of left pointing monomers. Make a simple diagram of a one-dimensional random-walk configuration that shows the end-to-end distance and how it depends upon the number of monomers of each type. Explain your result for both of these averages and their signif- icance for the estimate of the genome size. HINT: Eliminate n l by using the relation N = n l + n r . To proceed, show that the probability of having n r steps pointing to the right out of a total of N steps is p ( n r , N ) = N ! n r !( N - n r )! p n r r p n l l , (14) where p r is the probability of a right step and p l is the probability of a left step. Note that later we will invoke p r = p l = 1 / 2, but for now leave this as is since it will help us with our analysis. Now, to obtain h x i and q h x 2 i all 5

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you need to do is compute h n r i and h n 2 r i . Show that these averages are of the form h n r i = N X n r =0 n r N ! n r !( N - n r )! p n r r p N - n r l , (15) and h n 2 r i = N X n r =0 n 2 r N ! n r !( N - n r )! p n r r p N - n r l . (16) To evaluate these expressions, note that h n r i = p r ∂p r N X n r =0 N ! n r !( N - n r )! p n r r p N - n r l . (17) But the binomial theorem tells us that ( p r + p l ) N = N X n r =0 N ! n r !( N - n r )! p n r r p N - n r l . (18) With these tips, you have everything you need to evaluate the averages you are asked to compute. When you get your result, make sure to connect it to the result of part (a) and to explain how this can be used to find the “size” of a one-dimensional DNA molecule. (c) Using fig. 8.6, make a best estimate at the number of basepairs in the E. coli genome, remembering that the Kuhn length a = 300 bp. Compare your estimate to the observed genome length for E. coli and give a thoughtful discussion of how the estimate might have gone wrong. 6
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