324 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
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θ,degrees
θ,degreesT,◦C T,◦Clong chains
short chainsFigure 9.8: (Experimental data with fits.) Effect of changing the degree of cooperativity. (a)Dots:The same
long-chain experimental data as Figure 9.6.Dark gray curve:best fit to the data holding the cooperativity parameter
γfixed to the value 2. 9 (too much cooperativity). The curve was obtained by setting ∆Ebond=0. 20 kBTr,with
the other three parameters the same as in the fit shown in Figure 9.6.Light gray curve:best fit to the data fixing
γ=0(no cooperativity). Here ∆Ebondwastaken to be 57kBTr.(b)Solid and open dots: The same medium-
and short-chain data as in Figure 9.6. The curves show the unsuccessful predictions of the same two alternative
models shown in panel (a).Topcurve:the model with no cooperativity gives no length dependence at all.Lower
curves:in the model with too much cooperativity, short chains are influenced too much by their ends, and so stay
overwhelmingly in the random-coil state. Solid line,N=46; dashed line,N=26.
representing zero and infinite cooperativity respectively, were quite similar once we adjusted to
give them the same slope at the origin. Similarly, if we holdγfixed to a particular value, then
adjust ∆Ebondto get the best fit to the data, we find that we can get a visually good fit usingany
value ofγ.Tosee this, compare the top curve of Figure 9.6 to the two curves in Figure 9.8a. It
is true that unrealistic values of ∆Ebondare needed to fit the data with the alternative values ofγ
shown. But the point is that the large-Ndata alone do not really test the model—there are many
ways to get a sigmoid.
Nevertheless, while our eyes would have a hard time distinguishing the curves in Figure 9.8a
from the one in Figure 9.6, still thereisaslight difference in shape, and numerical curve-fitting says
that the latter is the best fit. In order to test our model, we must now try to make some falsifiable
predictionfrom the values we have obtained for the model’s parameters. We need some situation
in which the alternative values of ∆Ebondandγshown in Figure 9.8a give wildly different results,
so that we can see if the best-fit values are really the most successful.
Polypeptides, finiteN Toget the new experimental situation we need, Zimm and Bragg noted
that one more parameter of the system is available for experimental control: Different synthesis
protocols lead to differentlengthsNof the polymer. In general, polymer synthesis leads to a mixture
of chains with many different lengths, a “polydisperse” solution. But with care, it is possible to
arrive at a rather narrow distribution of lengths. Figure 9.6 shows data on the helix–coil transition
in samples with two different, finite values ofN.
Gilbert says: The data certainly show a big qualitative effect: The midpoint temperature is much