322 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
unit is roughly ∆Sconf≈−kBln(3×3), with a corresponding contribution to the free energy change
of≈+kBTln 9.
The statement that ∆Ebond>0mayseem paradoxical. If alpha-helix formation is energetically
unfavorable, and it also reduces the conformational entropy of the chain, then why do helices ever
form atanytemperature? This paradox, like the related one involving depletion interactions (see
the end of Section 7.2.2 on page 221), goes away when we consider all the actors on the stage.
It is true that extending the helix brings a reduction in the polypeptide’s conformational entropy,
∆Sconf <0. But the formation of an intramolecular H-bond also changes the entropy of the
surrounding solvent molecules. If this entropy change ∆Sbondis positive, and big enough that the
netentropy change ∆Stot=∆Sbond+∆Sconfis positive, then increasing the temperature can indeed
drive helix formation, since then ∆Gbond=∆Ebond−T∆Stotwill become negative at high enough
temperature. We have already met a similar apparent paradox in the context of self-assembly:
Tubulin monomers can be induced to polymerize into microtubules, lowering their entropy, by an
increasein temperature (Section 7.5.2 on page 243). Again, the resolution of this paradox involved
the entropy of the small, but numerous, water molecules.
Summarizing, we have identified two helix-extension parameters ∆Ebondand ∆Stotdescribing
agiven helix-coil transition. We defineα≡(∆Ebond−T∆Stot)/(− 2 kBT), so that extending an
alpha-helical stretch of the polypeptide by one unit changes the free energy by− 2 αkBT. (Some
authors refer to the related quantity e^2 αas the “propagation parameter” of the system.) Thus:
The free energy to extend the helix,α,isafunction of the polypeptide’s tem-
perature and chemical environment. A positive value ofαmeans that extending
ahelical region is thermodynamically favorable.(9.23)
Clearly a first-principles prediction ofαwould be a very difficult problem, involving all the
physics of the H-bond network of the solvent and so on. We will not attempt this level of prediction.
But the ideas of Section 9.1.1 give us an alternative approach: We can view ∆Ebondand ∆Stotas
just two phenomenological parameters to be determined from experiment. If we get more than two
nontrivial testable predictions out of the model, then we will have learned something. In fact, the
complete shapes of all three curves in Figure 9.6 follow from these two numbers (plus one more, to
bediscussed momentarily).
It’s convenient to rearrange the above expression forαslightly. Introducing the abbreviation
Tm≡∆Ebond/∆Stotgives
α=1
2
∆Ebond
kBT−Tm
TTm. (9.24)
The formula shows thatTmis themidpoint temperature,atwhichα=0.Atthis temperature,
extending a helical section by one unit makes no change in the free energy.
The cooperativity parameter So far each monomer has been treated as an independent, two-
state system. If this were true, then we’d be done—you found〈σ〉in a two-state system in Prob-
lem 6.5. But so far we have neglected an important feature of the physics of alpha-helix formation:
Extending a helical section requires the immobilization of two flexible bonds, butcreatingahelical
section in the first place requires that we immobilizeallthe bonds between unitsiandi+4.That
is, the polymer must immobilize one full turn of its nascent helix before it gains any of the benefit
of forming its first H-bond. The quantity 2αkBTintroduced above thus exaggerates the decrease of
free energy upon initiating a helical section. We define thecooperativity parameterγbywriting