370 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
aone-kick passage may seem about as likely as a successful field goal in a football game played in
molasses! But the Smoluchowski equation showed us the right way to derive the rate law for large
molecules: Modeling the process as a random walk on an energy landscape gives qualitatively the
same result as the na ̈ıve argument.
Wecould go on to implement these ideas for more complex microscopic machines, like the gears
of Figure 10.6c. Rather than studying rolling on the potential energy surface (Figure 10.9 on page
362), we would set up atwo-dimensionalSmoluchowski equation on the surface, again arriving at
conclusions similar to Idea 10.1. The following sections will not follow this program, however,
instead seeking shortcuts to see the qualitative behavior without the difficult mathematics.
T 2 Section 10.2.3′on page 398 generalizes the above discussion to get the force-velocity relation
for an imperfect ratchet.
10.3 Molecular implementation of mechanical principles
The discussion of purely mechanical machines in Section 10.2 generated some nice formulas, but
still leaves us with many questions:
- Molecular-scale machines are presumably made of molecules, unlike the simple but rather
fanciful sketches shown so far. Can we apply our ideas to single molecules? - Westill have no candidate model for acyclicmachine that eatschemicalenergy. Won’t we
need some totally new ideas to create this? - Most important of all, how can we make contact with experimental data?
Tomake progress on the first question, it’s time to gather a number of ideas about single molecules
developed in previous chapters.
10.3.1 Three ideas
Here are some of those ideas.
First, the statistical physics of Chapter 6 was constructed to be applicable to single-molecule
subsystems. For example, Section 6.6.3 on page 197 showed that such systems drive to minimize
their free energy, just like macroscopic systems, though not necessarily in a one-way, deterministic
fashion.
Second, we saw in Chapter 8 how chemical forces are nothing but changes in free energy, in
principle interconvertible with other changes involving energy (for example, the release of the little
bolts in the S-ratchet). Chemical forces drive a reaction in a direction determined by its ∆G,a
quantity involving the stoichiometry of the reaction but otherwise reflecting only the concentrations
of molecules in the reservoir outside the reaction proper. We expressed this succinctly in Idea 8.22
on page 270, or in the slogan that “equilibrium doesn’t care what happens inside the phone booth”
(Section 8.2.2).
Third, Chapter 9 showed how even large, complex macromolecules, with tens of thousands of
atoms all in random thermal motion, can nevertheless act as though they had just a few discrete
states. Indeed macromolecules can snap crisply between those states, almost like a macroscopic
light switch. We identified the source of this “multistable” behavior in the cooperative action of
many weak physical interactions such as hydrogen bonds. Thus for example cooperativity made