400 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
Further evidence that we should use the free energy landscape comes from a fact we already
know about reaction rates. Suppose the reaction involves binding a molecule that was previously
moving independently, in a simple one-step process. In this case we expect the rate of the reaction
to increase with the concentration of that molecule in solution. The same conclusion emerges from
our current picture, if we consider a walk on the free-energy landscape. To see this, note that the
bound molecule is being withdrawn from solution as it binds, so its initial entropy,Sin,makes a
positive contribution toG‡=E‡−TS‡−(Ein−TSin), decreases the Arrhenius exponential factor
e−G‡/kBT,and so slows the predicted reaction rate. For example, if the bound molecule is present
in solution at very low concentration, then its entropy loss upon binding will be large, and the
reaction will proceed slowly, as we know it must. (Reactions are also slowed by the entropy loss
implicit inorientingthe reacting molecule properly for binding.)
More quantitatively, at small concentrations the entropy per molecule is Sin = −μ/T =
−kBlnc+const (see Equations 8.1 and 8.3), so its contribution to the exponential factor is a con-
stant timesc.This is just the familiar statement that simple binding leads to first-order kinetics
(see Section 8.2.3 on page 269).
Finally, Section 10.2.3 argued that a molecular-scale device makes no net progress when its free
energy landscape has zero average slope. But we saw in Chapter 8 that a chemical reaction makes
no net progress when its ∆Gis zero, another way to see that the free energy, not ordinary energy,
is the appropriate landscape to use. For more on this important point, see Howard, 2001, appendix
5.1.
- It’s an oversimplification to say that we can simply ignore all directions in configuration space
perpendicular to the critical path between two quasi-stable states. Certainly there will be excursions
in these directions, with their own contribution to the entropy and so on. The actual elimination
procedure involves finding the free energy by doing a partition sum over these directions, following
essentially the methods of Section 7.1; the resulting free energy function is often called the “potential
of mean force.” See Grabert, 1982.
Besides modifying the free energy landscape, the mathematical step of eliminating all but one
or two of the coordinates describing the macromolecule and its surrounding bath of water has a
second important effect. The many eliminated degrees of freedom are all in thermal motion, and
are all interacting with the one reaction coordinate we have retained. Thus all contribute not only
to generating random motion along the reaction coordinate, but also to impeding directed motion.
That is, the eliminated degrees of freedom give rise tofriction,byanEinstein relation. Again
see Grabert, 1982. H. Kramers pointed out in 1940 that this friction could be large, and that for
complicated molecules in solution the calculation reaction rates via the Smoluchowski equation is
more complete than the older Eyring theory. He reproduced Eyring’s earlier results in a special
(intermediate-friction) case, then generalized it to cover low and high friction. For a modern look at
some of the issues and experimental tests of Kramers’ theory, see Frauenfelder & Wolynes, 1985.
10.3.3′
- The discussion in Section 10.3.3 focused on the possibility that the lowest free energy state of
the enzyme–substrate complex may be one in which the substrate is geometrically deformed, to a
conformation closer to its transition state. Figure 10.27 shows two other ways in which the grip of
an enzyme can alter its substrate(s), accelerating a reaction. For more biochemical details see for
example Dressler & Potter, 1991. - The simple picture developed in Idea 10.14 is also helpful in understanding a new phenomenon