Physics 450 -- Problem Set 4 Physics 450 - Problem Set 4 - Due Monday March 26 2001

1. We showed that the random-flight polymer model had a relation between its extension and tension of
z
L
 = é
ê
ë 		coth b f
kB T
 - kB T
b f
 ù
ú
û
where L = Nb is the total contour length of the polymer.

Plot the extension (z/L) for:
(a) denatured polypeptide (b = 0.3 nm)
(b) single-stranded NA (b = 0.7 nm)
(c) double-stranded NA (b = 100 nm)

(d) For what forces are these three types of polymers 60% extended?

2. In class we saw that that the size of a polymer was slightly increased by repulsive segment-segment interactions (w > b3/N1/2), and that we could predict that size using the Flory free energy
F
kB T
 = R2
N b2
 + w N c
where c = N/R3 is the segment concentration in the polymer coil. The polymer size was determined by minimization of F with respect to R, and we found R µ N3/5.

(a) Find the size of a polymer swollen by repulsive segment-segment interactions in one, two and four dimensions (Hint: c = N/Rd in d dimensions).

(b) You should find that R µ Nn; therefore state the value of the exponent n in 1, 2, 3 and 4 dimensions.

(c) Explain in words the one-dimensional result.

(d) Can you think of a way to do an experiment to test the two-dimensional result?

(e) You could also predict n in five and higher dimensional space using the Flory theory, but the prediction can't be right. Why?

3. Suppose that the segments of a flexible polymer each carry an electric charge q, and that this polymer is immersed in solution where the Debye screening length is very long. Make a Flory-like theory, where the interaction energy is electrostatic, to find the coil size R of the polymer.

Hint: the electrostatic energy of N charges confined in a region of size R is roughly k N2 q2 / (eR)

4. Use the model of Breslauer et al discussed in lecture (Proc. Natl. Acad. Sci. USA 83, 3746 (1986)) to predict the Gibbs free energy difference as a function of temperature (i.e. the linear form DG = DH0 - T DS0) between the double-stranded and single-stranded forms of the short double-stranded DNA:

5'-aggtcgccgccc-3'
3'-tccagcggcggg-5'

Find the temperature at which DG = 0, which is the temperature at which the double helix thermally `melts'.

5. Consider a double-stranded DNA of N bases. Suppose that the base-pairing energy is e = 3 kB T per base (at room temperature).

(a) Use the Boltzmann distribution to find a formula for the probability of separation of the two strands relative to the paired, double-stranded state. Plot your results for 1 £ N £ 24 (you may ignore the effect of concentration).

(b) Estimate the time that you would have to wait to have spontaneous thermal separation (dissociation) of the two strands, assuming that one new `attempt' at dissociation is made very t0 = 10-11 sec.

6. Suppose a long double-stranded DNA is `unzipped' by pulling the two strands apart, thus progressively breaking the base-pairs. You may assume the single-stranded DNA to have a segment length b = 0.7 nm, and a base-pairing energy of e = 3 kB T per base pair.

(a) Roughly estimate the force that must be applied to start unzipping the DNA strands (Hint: the work done during unzipping should be about e per base).

(b) Use a Flory-like free energy for the partially unzipped DNA to predict the number n of bases unzipped if the two ends of the DNA are separated a given distance x

Hint 1: You may assume the free energy of an N-base single-stranded DNA stretched a distance R to have the simple harmonic form

Fstretch = 3 kB T R2
2 N b2
.

Hint 2: you will have to minimize your free energy with respect to n, not x.

You might want to look at the paper by Essevaz-Roulet et al, Proc. Natl. Acad. Sci. USA 94, 11935 (1997), to read about the actual results of this experiment.

Note that PNAS is available online via UIC computers.

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On 8 Mar 2001, 09:56.