Biological Physics: Energy, Information, Life

310 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 1 10

z/

L

tot

force, pN

3d elastic rod model

1d freely jointed chain model

Figure 9.4: (Experimental data with fits.) Log-log plot of relative extensionz/Ltotat low applied stretching
forceffor lambda phage DNA in 10mMphosphate buffer. The points show experimental data corresponding to the
regions A–B in Figure 9.3. The curves show various theoretical models discussed in the text. For comparison, in
each model the value ofLseghas been adjusted so that all the curves agree at low force.Topcurve:one-dimensional
freely jointed chain (Equation 9.10), withL(1d)seg =35nm.Dashed gray curve:one-dimensional cooperative chain (see
Your Turn 9h(b)), withL(1d)seg held fixed at 35nmandγvery large.Dotted gray curve:three-dimensional FJC (Your
Turn 9o), withLseg= 104 nm.Black curve through data points:three-dimensional elastic rod model (Section 9.4.1′
on page 341), withA=51nm.[Data kindly supplied by V. Croquette; see also Bouchiat et al., 1999.]

Your Turn 9b

If you haven’t yet worked Problem 6.5, do it now. Explain why this is mathematically the same

problem as the one we just solved.

Solving Equation 9.10 forf shows thatthe force needed to maintain a given extensionzis
proportional to the absolute temperature. This property is the hallmark of any purely entropic
force, for example ideal-gas pressure or osmotic pressure; we anticipated it in Section 9.1.3.
The function in Equation 9.10 interpolates between two important limiting behaviors:

• Athigh force,〈z〉→Ltot. This is what we expect from an inextensible chain: Once
  it’s fully straight, it can’t lengthen any more.


• Atlow force,〈z〉→f/k,wherek=kBT/(LtotL(1d)seg).

The second point means that

Atlow extension, a polymer behaves as a spring, that is, it obeys a Hooke rela-

tion,f=k〈z〉.Inthe FJC model, the effective spring constantkis proportional

to the temperature.

(9.11)

Figure 9.4 shows some experimental data obtained by stretching DNA, together with the function
in Equation 9.10 (top curve). The figure shows that takingL(1d)seg =35nmmakes the curve pass
through the first data point. Although the one-dimensional freely jointed chain correctly captures