9.5. Thermal, chemical, and mechanical switching[[Student version, January 17, 2003]] 321
9.5.2 Three phenomenological parameters describe a given helix–coil transition
Let us try to model the data in Figure 9.6 using the ideas set out in Section 9.4. Our approach is
based on ideas pioneered by J. Schellman (1955) and extended by B. Zimm and J. Bragg (1959).
Wecan think of each monomer in an alpha-helix-forming polypeptide as being in one of two
states labeled byσ=+ 1 for alpha helix, orσ=− 1 for random-coil. More precisely, we take
σi=+1ifmonomer numberiis H-bonded to monomeri+4,and− 1 otherwise. The fraction of
monomers in the helical state can be expressed as^12 (〈σav〉+1), whereσavis the average ofσiover
the chain. If we attribute specific, different contributions to the optical rotation for each state, then
the vertical axis of Figure 9.6 can also be regarded as a linear function of〈σav〉.
The three curves in Figure 9.6 show results obtained from three different samples of polymer,
differing in their average length. The polymer was synthesized under three sets of conditions; the
mean molar mass for each sample was then determined. Let us begin by studying the top curve,
which was obtained with a sample of very long polymer chains. We need a formula for〈σav〉as
afunction of temperature. To get the required result, we adapt the analysis of Section 9.4.1,
reinterpreting the parametersαandγin Equation 9.18, as follows.
The helix-extension parameters In the polymer stretching problem, we imagined an isolated
thermodynamic system consisting of the chain, its surrounding solvent, and some kind of external
spring supplying the stretching force. The parameter 2α=2 f /kBTthen described the reduction
of the spring’s potential energy when one link switched from the unfavorable (backward,σ=−1)
to favorable (forward,σ=+1) direction. The applied forcefwasknown, but the effective link
length wasanunknown parameter to be fit to the data. In the present context, on the other
hand, the link length is immaterial. When a monomer bonds to its neighbor, its link variableσi
changes from−1to+ 1 and an H-bond forms. We must remember, however, that the participating
Hand O atoms were already H-bonded to surroundingsolventmolecules; to bond to each other
they mustbreakthese preexisting bonds, with a corresponding energycost.The net energy change
of this transaction, which we will call ∆Ebond≡Ehelix−Ecoil,maytherefore be either positive
or negative, depending on solvent conditions.^8 The particular combination of polymer and solvent
shown in Figure 9.6 has ∆Ebond>0. To see this, note that raising the temperature pushes the
equilibrium toward the alpha-helix conformation. Le Chˆatelier’s Principle then says that forming
the helix must cost energy (see Section 8.2.2 on page 265).
The formation and breaking of H-bonds also involves an entropy change, which we will call
∆Sbond.
There is a third important contribution to the free energy change when a tract of alpha-helix
extends by one more monomer. As mentioned in Section 9.5.1, the formation of intramolecular
H-bonds requires the immobilization of all the intervening flexible links, so that the participating
Hand O atoms stay within the very short range of the H-bond interaction. Each amino acid
monomer contains two relevant flexible links. Even in the random-coil state, these links are not
perfectly free, due to obstructions involving the atoms on either side of them; instead, each link
flips between three preferred positions. But to get the alpha-helix state, each link must occupy just
oneparticular position. Thus the change of conformational entropy upon extending a helix by one
(^8) T 2 More precisely, we are discussing the enthalpy change, ∆H,but in this book we do not distinguish energy
from enthalpy (see Section 6.5.1).