316 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
As always, it’s interesting to check the behavior of your solution at very low force (α→0). We
again find that〈z〉→f/k,where now the spring constant is
k=kBT/(e^2 γ Ltot). (9.21)So at least we have not spoiled the partial success we had with the FJC: The low-force limit of
the extension, where the FJC was successful, has the same form in the cooperative-chain model,
as long as we choose andγto satisfy e^2 γ=L(1d)seg.Wenowask whether the cooperative-chain
model can do a better job than the FJC of fitting the data at thehigh-force end.
The dashed curve in Figure 9.4 shows the function you found in Your Turn 9h. The cooperativity
γhas been taken very large, while holding fixedL(1d)seg.The graph shows that the cooperative one-
dimensional chain indeed does a somewhat better job of representing the data than the FJC.
Our 1d cooperative chain model is still not very realistic, though. The lowest curve on the graph
shows that the three-dimensional cooperative chain (that is, the elastic rod model, Equation 9.3)
gives a very good fit to the data. This result is a remarkable vindication of the highly reductionist
model of DNA as a uniform elastic rod. Adjusting just one phenomenological parameter (the
bend persistence lengthA)gives a quantitative account of the relative extension of DNA, a very
complex object (see Figure 2.17 on page 45). This success makes sense in the light of the discussion
in Section 9.1.1: It is a consequence of the large difference in length scales between the typical
thermal bending radius (≈ 100 nm)and the diameter of DNA (2nm).
T 2 Section 9.4.1′on page 341 works out the force-extension relation for the full, three-dimensional
elastic rod model.
9.4.2 DNA also exhibits linear stretching elasticity at moderate applied force
Wehave arrived at a reasonable understanding of the data in the low- to moderate-force regimes
Aand B shown in Figure 9.3. Turning to regime C, we see that at high force the curve doesn’t
really flatten out as predicted by the inextensible chain picture. Rather, the DNA molecule is
actuallystretching,not just straightening. In other words, an external force can induce structural
rearrangements of the atoms in a macromolecule. We might have expected such a result—we arrived
at the simple model in Equation 9.3 in part by neglecting the possibility of stretching, that is, by
discarding the second term of Equation 9.2 on page 303. Now it’s time to reinstate this term, and
in so doing formulate anextensible rodmodel, due to T. Odijk.
Toapproximate the effects of this intrinsic stretching, we note that the applied force now has
twoeffects: Each element of the chain aligns as before, but now each element also lengthens slightly.
The relative extension is a factor of 1 +u,whereuis the stretch defined in Section 9.1.2 on page
- Consider astraightsegment of rod, initially of contour length ∆s.Under an applied stretching
forcef,the segment will lengthen byu×∆s,whereutakes the value that minimizes the energy
functionkBT[^12 Bu^2 ∆s−fu∆s](see Equation 9.2). Performing the minimization givesu=kBfTB.
Wewill make the approximation that this formula holds also for the full fluctuating rod. In this
approximation each segment of the rod again lengthens by a relative factor of 1 +u,so〈z/Ltot〉
equals the inextensible elastic rod chain result, multiplied by 1 +kBfTB.
Tosee if we are on the right track, Figure 9.5 shows some experimental data on the stretching
of DNA at moderate forces. Intrinsic stretching is negligible at low force, so the low force data