10.4. Kinetics of real enzymes and machines[[Student version, January 17, 2003]] 379
Figure 10.19: (Structure drawn from atomic coordinates.) (a)Structure of phosphoglycerate kinase, an enzyme
composed of one protein chain of 415 amino acids. The chain folds into this distinctive shape, with two large lobes
connected by a flexible hinge. The active site, where the chemical reaction occurs, is located between the two halves.
The atoms are shown in a gray scale according to their hydrophobicity, with the most hydrophobic in white, the
most hydrophilic in black. (b)Close-up of (a), showing the active site with a bound molecule of ATP (hatched
atoms). This view is looking from the right in (a), centered on the upper lobe. Amino acids from the enzyme wrap
around and hold the ATP molecule in a specific position. [From Goodsell, 1993.]
10.4.1 The Michaelis–Menten rule describes the kinetics of simple en-
zymes
The MM rule Section 10.2.3 gave us some experience calculating the net rate of a random walk
down a free-energy landscape. We saw that such calculations boil down to solving the Smoluchowski
equation (Equation 10.4 on page 367) to find the appropriate quasi-steady state. However, we
generally don’t know the free-energy landscape. Even if we did, such a detailed analysis focuses on
the specifics of one enzyme, whereas we would like to begin by finding some very broadly applicable
lessons. Let’s instead take an extremely reductionist approach.
First, focus on a situation where initially there is no product present, or hardly any. Then the
chemical potentialμPof the product is a large negative number. This in turn means that the third
step of Figure 10.17b, EP→E+P, is steeply downhill, and so we may treat this step as one-way
forward—a perfect ratchet. We also make a related simplifying assumption, that the transition
EP→E+P is so rapid that we may neglect EP altogether as a distinct step, lumping it together
with E+P. Finally, we assume that the remaining quasi-stable states, E+S, ES, and E+P, are
well separated by large barriers, so that each transition may be treated independently. Thus the
transition involving binding of substrate from solution will also be supposed to proceed at a rate
given by a first-order rate law, that is, the rate is proportional to the substrate concentrationcS
(see Section 8.2.3).