Here we use concepts developed in Chapter 16 and this chapter to enhance our

Chapter 19, Problem 19.30

(choose chapter or problem)

Here we use concepts developed in Chapter 16 and this chapter to enhance our understanding of closed-circular and supercoiled DNA. (a) The average end-to-end distance of a flexible polymer (such as a fully denatured polypeptide or a strand of DNA) is N1/2l, where N is the number of groups (residues or bases) and l is the length of each group. Initially, therefore, one end of the polymer can be found anywhere within a sphere of radius N1/2l centred on the other end. When the ends join to form a circle, they are confined to a volume or radius l. What is the change in molar entropy? Plot the function you derive as a function of N. (b) The energy necessary to twist ccDNA by i turns is i = ki2, with k an empirical constant and i being negative or positive depending on the sense of the twist. For example, one twist (i = 1) makes ccDNA resemble the number 8. (i) Show that the distribution of the populations pi = ni /N of ccDNA molecules with i turns at a specified temperature has the form of a Gaussian function. (ii) Plot the expression you derived in part (a) for several values of the temperature. Does the curve has a maximum? If so, at what value of i? Comment on variations of the shape of the curve with temperature. (iii) Calculate p0, p1, p5, and p10 at 298 K.