Biological Physics: Energy, Information, Life

376 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]

free energy

nS+kP

(n−1)S + (k+1)P

(n−2)S + (k+2)P generalized reaction

coordinate

Figure 10.18:(Sketch graph.) The free energy landscape of Figure 10.17b, duplicated and shifted to show three
steps in a cyclic reaction. The reaction coordinate of Figure 10.17 has been generalized to include changes in the
number of enzyme and substrate molecules; the curve shown connects the state withnsubstrate andkproduct
molecules to the state with three fewer S (and three more P).

of the free energy landscape, slowing or halting the unbinding of product. Just as in any chemical
reaction, a large enough concentration of P can even reverse the sign of ∆G,and hence reverse the
direction of the net reaction (see Section 8.2.1).
Wecan now make a simple but crucial observation: The state of our enzyme/substrate/product
system depends on how many molecules of S have been processed into P. Although the enzyme
returns to its original state after one cycle, still the whole system’s free energy falls by ∆Gevery
time it takes one net step. We can acknowledge this fact by generalizing the reaction coordinate to
include the progress of the reaction, for example the numberNSof remaining substrate molecules.
Thenthe complete free-energy landscape consists of many copies of Figure 10.17b, each shifted
downward by∆Gso as to make a continuous curve (Figure 10.18). In fact, this curve looks
qualitatively like one we already studied, namely Figure 10.11c! We identify the barrierfLin that
figure asG‡,and the net dropfL−as ∆G.Inshort,

Many enzymes can be regarded as simple cyclic machines; they work by random-

walking down a one-dimensional free energy landscape. The net descent of this

landscape in one forward step is the value of∆Gfor the reaction S⇀P.

(10.14)

Idea 10.14 gives an immediate qualitative payoff: We see at once why so many enzymes exhibit
saturation kinetics (Section 10.1.2). Recall what this means. The rate of an enzyme-catalyzed
reaction S⇀Ptypically levels off at high concentration of S, instead of being proportional tocS
as simple collision theory might have led us to expect. Viewing enzyme catalysis as a walk on
afree energy landscape shows that saturation kinetics is a result we’ve already obtained, namely
our result for the speed of a perfect ratchet (Idea 10.9c on page 369). A large concentration of
Spulls the left side of the free energy landscape upward. This means that the step from E+S to
ES in Figure 10.17b is steeply downhill. Such a steep downward step makes the process effectively
irreversible, essentially forbidding backward steps, but it doesn’t speed up the net progress, as seen