- Problems[[Student version, January 17, 2003]] 347
would be the expected temperature dependence ofθ?[Hint: Figure out the probability to be in
anα-helix as a function of ∆Ebond,∆S,N,andTand sketch a graph as a function ofT. Don’t
forget to normalize your probability distribution properly. Make sure that the limiting behavior of
your formula at very large and very small temperature is physically reasonable.]
c. How does the sharpness of the transition depend onN?Explain that result physically.
d. The total free-energy change for conversion of a chain isnotsimplyN∆G,however,due to end
effects, as discussed in the chapter. Instead let us suppose that the last two residues at each end
are unable to benefit from the net free energy reduction of H-bonding. What is the physical origin
of this effect? Again find the expected temperature dependence ofθ.[Hint: Same hint as in (b).]
e. Continuing part (d), find the temperatureTmat whichθis halfway betweenθminandθmax,
including end effects. How doesTmdepend onN?This is an experimentally testable qualitative
prediction; compare it to Figure 9.6 on page 319.
9.6 T 2 High-force limit
The analysis of DNA stretching experiments in Sections 9.2.2–9.4.1 made a number of simplifications
out of sheer expediency. Most egregious was working in one dimension: every link pointed either
along +ˆzor−ˆz,soevery link angle was either 0 orπ.Inreal life, every link points nearly (but not
quite)parallelto the previous one. Section 9.4.1′on page 341 took this fact into account, but the
analysis was very difficult. In this problem you’ll find a shortcut applicable to the high-force end of
the stretching curve. You’ll obtain a formula which, in that limit, agrees with the full fluctuating
elastic rod model.
In this limit the curve describing the rod’s centerline is nearly straight. Thus at the point a
distancesfrom the end, the tangent to the rodˆt(s)isnearly pointing along theˆzdirection. Let
t⊥bethe projection ofˆtto thexyplane; thust⊥’s length is very small. Then
ˆt(s)=M(s)ˆz+t⊥(s), (9.44)whereM=
√
1 −(t⊥)^2 =1−^12 (t⊥)^2 +···.The ellipsis denotes terms of higher order in powers of
t⊥.
In terms of the small variablest⊥(s)=(t 1 (s),t 2 (s)), the bending termβ^2 equals (t ̇ 1 )^2 +(t ̇ 2 )^2 +···.
(t ̇idenotes dti/ds.) overdots denote d/ds.) Thus the elastic bending energy of any configuration
of the rod is (see Equation 9.3)
E=^12 kBTA∫Ltot0ds[(t ̇ 1 )^2 +(t ̇ 2 )^2 ]+···. (9.45)Just as in Section 9.2.2, we add a term−fzto Equation 9.45, to account for the external stretching
forcef.Work in the limit of very longLtot→∞.
a. RephraseEin terms of the Fourier modes oft 1 andt 2 .[Hint: Write−fzas−f
∫Ltot
0 dsˆt(s)·ˆz
and use Equation 9.44. ExpressM(s)interms oft 1 ,t 2 as above.] ThenEbecomes the sum of a
lot of decoupled quadratic terms, a little bit (not exactly!) like a vibrating string.
b. What is the mean-square magnitude of each Fourier component oft 1 andt 2 ?[Hint: Think back
to Section 6.6.1 on page 193.]
c. We want the mean end-to-end distance〈z〉/Ltot. Use the answer from (a) to write this in a
convenient form. Evaluate it using your answer to (b).
d. Find the forcefneeded to stretch the thermally dancing rod to a fraction 1−of its full length
Ltot,whereis small. How doesfdiverge as→0? Compare your result to the 3d freely jointed
chain (Your Turn 9o) and to the 1d cooperative chain (Your Turn 9h).