346 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
Problems....................................................
9.1Big business
DNA is a highlychargedpolymer. That is, its neutral form is a salt, with many small positive
counterions that dissociate and float away in water solution. A charged polymer of this type is
called a “polyelectrolyte.” A very big industrial application for polyelectrolytes is in the gels filling
disposable diapers. What physical properties of polyelectrolytes do you think make them especially
suitable for this vitally important technology?
9.2Geometry of bending
Verify Equation 9.4 explicitly as follows. Consider the circular arc in thexyplane defined by
r(s)=(Rcos(s/R),Rsin(s/R)) (see Figure 9.1). Show thatsreally is contour length, find the unit
tangent vectorˆt(s)and its derivative, and hence verify Equation 9.4 on page 304.
9.3Energy sleuthing
The freely jointed chain picture is a simplification of the real physics of a polymer: Really the joints
are not quite free. Each polymer molecule consists of a chain of identical individual units, which
stack best in a straight line (or in a helix with a straight axis). Thus Equation 9.2 says that bending
the chain into a tanglecostsenergy. And yet, a rubber band certainly can do work on the outside
world as it retracts. Reconcile these observations qualitatively: Where does the energy needed to
do mechanical work come from?
9.4Thermodynamics of rubber
Takeawide rubber band. Hold it to your upper lip (moustache wearers may use some other
sensitive, but public, part) and rapidly stretch it. Leave it stretched for a moment, then rapidly let
it relax while still in contact with your lip. You will feel distinct thermal phenomena during each
process.
a. Discuss what happened upon stretching, both in terms of energy and in terms of order.
b. Similarly discuss what happened upon release.
9.5Simplified helix-coil transition
In this problem you’ll work through a simplified version of the cooperative helix–coil transition,
assuming that the transition isinfinitelycooperative. That is, each polypeptide molecule is assumed
to be either allα-helix or all random-coil. The goal of the problem is to understand qualitatively a
keyfeature of the experimental data shown in Figure 9.6, namely that longer chains have a sharper
helix-coil transition. Let the chain haveNamino-acid units.
a. The observed optical rotationθof the solution varies continuously fromθmin toθmaxas we
change experimental conditions. How can an all-or-none model reproduce this observed behavior?
b. Section 9.5.1 argued that, for the conditions in Doty and Iso’s experiments,
- Theα-helix form has greater energy per monomer than the random-coil form, or
∆Ebond>0. - Forming the H-bond increases the entropy of the solvent, by an amount ∆Sbond>0.
- But forming a bond alsodecreases the molecule’s conformational entropy, by ∆Sconf<
0.
The total free energy change to extend a helical region is then ∆G=∆Ebond−T∆Sbond−T∆Sconf.
Suppose that the total free-energy change for conversion of a chain were simplyN∆G.What then