340 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
By a now-familiar argument, we see that the second term equals zero: For every conformation
in whichˆtimakes a particular angle withˆtj,there is an equally weighted conformation with the
opposite value ofˆti·ˆtj,since the joints are assumed to be free. The first term is also simple, since
bydefinition (ˆti)^2 =1always, so we find
〈r^2 〉=N(Lseg)^2 =LsegLtot. (unstretched, three-dimensional FJC) (9.33)Comparing to Equation 9.31 we find that
The freely jointed chain model correctly reproduces the size of the underlying
elastic rod’s random-coil conformation, if we choose the effective link length to
beLseg=2A.(9.34)
- This chapter regards a polymer as a stack of identical units. Strictly speaking, such objects are
called “homopolymers.” Natural DNA, in contrast, is a “heteropolymer”: It contains a message
written in an alphabet of four different bases. It turns out that the effect of sequence on the large-
scale elasticity of DNA is rather weak, essentially because the AT and GC pairs are geometrically
similar to each other. Moreover it is not hard to incorporate sequence effects into the results of the
following sections. These homopolymer results apply also to heteropolymers whenAis suitably
interpreted as a combination of elastic stiffness and intrinsic disorder. - [[... ]]
9.2.2′
- One major weakness of our discussion so far has been the fact that we used aone-dimensional
random walk to describe the three-dimensional conformation of the polymer! This weakness is not
hard to fix.
The three-dimensional freely jointed chain has as its conformational variables a set of unit
tangent vectorsˆti,which need not point only along the±ˆzdirections: They can point in any
direction, as in Equation 9.32. We takerto be the end-to-end vector, as always; with an applied
stretching force in theˆzdirection we know thatrwill point alongˆz.Thusthe end-to-end extension
zequalsr·ˆz,or(
∑
iLsegˆti)·ˆz.(The parameterLsegappearing in this formula is not the same as
the parameterL1dsegin Section 9.2.2.) The Boltzmann weight factor analogous to Equation 9.9 is
then
P(ˆt 1 ,...,ˆtN)=Ce−
(
−fLseg∑iˆti·ˆz)
/kBT. (9.35)Your Turn 9o
If you haven’t done Problem 6.9 yet, do it now. Then adapt Your Turn 9b on page 310 to arrive
at an expression for the extension of a three-dimensional FJC. Again find the limiting form of
your result at very low force.The expression you just found is shown as the third curve from the top in Figure 9.4. From
your answer to Your Turn 9o you’ll see why this time we tookLseg= 104 nm. Indeed the three-
dimensional FJC gives a somewhat better fit to the experimental data than Equation 9.10.
- The effective spring constant of a real polymer won’t really be strictly proportional to the absolute
temperature, as implied by Idea 9.11. That’s because the bend persistence length, and henceLseg,
themselves depend on temperature. Nevertheless, our qualitative prediction that the effective spring
constant increases with temperature is observed experimentally (see Section 9.1.3).