326 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
result from Your Turn 9l(c). This calculation yields the lower two curves^9 in Figure 9.6.
Summary Earlier, we saw how fitting our model to the large-Ndata yields a satisfactory account
of the helix–coil transition for long polypeptide chains. The result was slightly unsatisfying, though,
because we adjusted several free parameters to achieve the agreement. Moreover, the data seem to
underdetermine the parameters, including the most interesting one, the cooperativity parameterγ
(see Figure 9.8a).
Nevertheless, we agreed to take seriously the value ofγobtained from the large-Ndata. We then
successfully predicted, with no further fitting, the finite-Ndata. In fact, the finite-Nbehavior of
our model does depend sensitively on the separate values of ∆Ebondandγ,aswesee in Figure 9.8b:
Both the non-cooperative and the too-cooperative models, each of which seemed to do a reasonable
job with the large-N data, fail miserably to predict the finite-N curves! It’s remarkable that
the large-N data, which seemed so indifferent to the separate values of ∆Ebondandγ,actually
determine them well enough to predict successfully the finite-Ndata.
Wecan interpret our results physically as follows.
a. A two-state transition can be sharp either because its∆Eis large, or because
of cooperativity between many similar units.
b. A modest amount of cooperativity can give as much sharpness as a very large
∆E,because it’seγthat appears in the maximum slope (as you found in Your
Turn 9j). Thus cooperativity holds the key to giving sharp transitions between
macromolecular states using only weak interactions (like H-bonds).
c. A hallmark of cooperativity is a dependence on the system’s size and dimen-
sionality.
(9.26)
Extending the last point, in the noncooperative case, each element behaves independently (top
curves of Figure 9.8a,b), and so the sharpness of the transition is independent ofN.With cooper-
ativity, the sharpness goes down for smallN(middle and bottom curves of Figure 9.8b).
T 2 Section 9.5.3′on page 344 refines the analysis of the helix–coil transition by accounting for the
sample’s polydispersity.
9.5.4 DNA also displays a cooperative “melting” transition
DNA famously consists of two strands wound around each other (Figure 2.17 on page 45); it’s
often called the “DNA duplex.” Each strand has a strong, covalently bonded backbone, but the
twostrands are only attached to each other by weak interactions, the hydrogen bonds between
complementary bases in a pair. This hierarchy of interactions is crucial for DNA’s function: Each
strand must strictly preserve the linear sequence of the bases, but the cell frequently needs to unzip
the two strands temporarily, in order to read or copy its genome. Thus the marginal stability of
the DNA duplex is essential for its function.
The weakness of the inter-strand interaction leaves us wondering, however, why DNA’s structure
is so well-defined when it isnotbeing read. We get a clue when we notice that simply heating DNA in
solution up to around 90◦Cdoes make it fall apart into two strands, or “melt.” Other environmental
changes, such as replacing the surrounding water by a nonpolar solvent, also destabilize the duplex.
The degree of melting again follows a sigmoidal (S-shaped) curve, similar to Figure 9.6 but with
(^9) T 2 Asmall correction is discussed in Section 9.5.3′on page 344 below.