334 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
The big picture
This chapter has worked through some case studies in which weak, nearest-neighbor couplings, be-
tween otherwise independent actors, created sharp transitions in the nanoworld of single molecules,
overturning the na ̈ıve expectation that everything would be very disorderly there. Admittedly we
have hardly scratched the surface of protein structure and dynamics—our calculations involved
only linear chains with nearest-neighbor cooperativity, whereas the allosteric couplings of greatest
interest in cell biology involvethree-dimensional protein structures. But as usual our goal was only
to address the question of “How could anything like that happen at all?” using simplified but
explicit models.
Cooperativity is a pervasive theme in both physics and cell biology, at all levels of organization.
Thus, while this chapter mentioned its role in creating well-defined allosteric transitions in single
macromolecules, Chapter 12 will turn to the cooperative behaviorbetweenthousands of proteins, the
ion channels in a single neuron. Each channel has a sharp transition between “open” and “closed”
states, but makes that transition in a noisy, random way (see Figure 12.17). Each channel also
communicates weakly with its neighbors, via its effect on the membrane’s potential. Nevertheless,
even such weak cooperativity leads to the reliable transmission of nerve impulses.
Key formulas
- Elastic rod: In the elastic rod model of a polymer, the elastic energy of a short segment
of rod is dE=^12 kBT
[
Aβ^2 +Bu^2 +Cω^2 +2Duω]
ds(Equation 9.2). HereAkBT,CkBT,
BkBT,andDkBT are the bend stiffness, twist stiffness, stretch stiffness, and twist-stretch
coupling, and dsis the length of the segment. (The quantitiesAandCare also called the
bend and twist persistence lengths.) u,β,andωare the stretch, bend, and twist density.
Assuming that the polymer is inextensible, and ignoring twist effects, led us to a simplified
elastic-rod model (Equation 9.3). This model retains only the first term of the elastic energy
above.- Stretched freely jointed chain: The fractional extension〈z〉/Ltotof a one-dimentional, freely
jointed chain is its mean end-to-end distance divided by its total unstretched length. If
westretch the chain with a forcef,the fractional extension is equal to tanh(fL(1d)seg/kBT)
(Equation 9.10), whereL(1d)seg is the effective link length. The bending stiffness of the real
molecule being represented by the FJC model determines the effective segment lengthL(1d)seg. - Alpha-helix formation: Letα(T)bethe free-energy change per monomer for the transition
from the alpha-helix to the random-coil conformation at temperatureT.Interms of the
energy difference ∆Ebondbetween the helix and coil forms, and the melting temperature
Tm,wefoundα(T)=^12 ∆EkbBondTTT−Tmm (Equation 9.24). The optical rotation of a solution of
polypeptide is then predicted to be
θ=C 1 +
C 2 sinhα
√
sinh^2 α+e−^4 γ,
whereC 1 andC 2 are constants andγdescribes the cooperativity of the transition (Equa-
tion 9.25).- Simple binding: The oxygen saturation curve of myoglobin is of the formY=[O 2 ]/([O 2 ]+
Keq−^1 )(Your Turn 9m). Hemoglobin instead follows the formula you found in Your Turn 9n.