- Track 2[[Student version, January 17, 2003]] 399
Your Turn 10f
a. Check what happens when the load is close to the thermodynamic stall point,f∼/L.
b. What happens to Equation 10.26 atf→0? How can the ratchet move to the right, as implied
bySullivan’s third point? Doesn’t the formulaj(1d)=MfDA/kBT,along with Equation 10.6,
imply thatv→ 0 whenf→0?
c. Find the limit of very high drive, kBT,and compare with the result in Equation 10.7.
d. Find the limit fL kBT,and comment on Sullivan’s fourth assertion.- The physical discussion of Figure 10.12 on page 366 was subtle; an equivalent way to express
the logic may be helpful. Rather than wrap the ratchet into a circle, we can take it to be straight
and infinitely long. Then the probability distribution willnotbeperiodic, nor will it be time-
independent. Instead,P(x)will look like a broad peak (or envelope function), modulated by the
spikes of Figure 10.12. The envelope function drifts with some speedv,while the individual spkes
remain fixed at the multiples ofL.Tomake contact with the discussion given in Section 10.2.3, we
imagine sitting at the peak of the envelope function. After the system has evolved a long time, the
envelope will be very broad, and hence nearly flat at its peak. HenceP(x, t)will be approximately
periodic and time-independent. Thus our earlier analysis, leading to the Smoluchowski equation,
is sufficient to find the average speed of advance.
10.3.2′
- The ultimate origin of the energy landscape lies in quantum mechanics. For the case of simple
molecules in isolation (that is, in a gas), one can calculate this landscape explicitly. It suffices to
treat only the electrons quantum-mechanically. Thus, in the discussion of Section 10.3.2 the phrase
“positions of atoms” is interpreted as “positions of nuclei.” One imagines nailing the nuclei at
specified locations, computing the ground-state energy of the electrons, and adding in the mutual
electrostatic energy of the nuclei, to obtain the energy landscape. This procedure is known as
the “Born–Oppenheimer approximation.” For example, it could be used to generate the energy
landscape of Figure 10.14.
Formacromolecules in solution, more phenomenological approaches are widely used. Here one
attempts to replace the complicated effects of the surrounding water (hydrophobic interaction and so
on) by empirical interatomic potentials involving only the atoms of the macromolecule itself. Many
more sophisticated calculations than these have been developed. But quite generally the strategy
of understanding chemical reactions as essentially classical random walks has proved successful for
many biochemical processes. (An example of the exceptional, intrinsically quantum-mechanical
processes is the detection of single photons in the retina.) - The alert reader may notice that in passing from Section 10.2.3 to Section 10.3.2 the word
“energy” changed to “free energy.” To understand this shift, we first note that in a complex
molecule there may bemanycritical paths, each accomplishing the same reaction, not just one as
shown in Figure 10.14. In this case the reaction’s rate gets multiplied by the numberNof paths;
equivalently we can replace the barrier energyE‡byan effective barrierE‡−kBTlnN.Interpreting
the second term as the entropy of the transition state (and neglecting the difference between energy
and enthalpy), gives that the reaction is really suppressed byG‡,notE‡.Indeed, we already knew
thatequilibriumbetween two complex states is controlled by free energy differences (Section 6.6.4
on page 198).