406 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
ratchet mechanism.
a. Suppose the external forcef=0.Will the F-ratchet make net progress? In which direction?
[Hint: Think about the case where the temperatureTequals absolute zero.]
b. Recall Sullivan’s critique of the G-ratchet: “Couldn’t you wrap your shaft into a circle? Then
your machine would go around forever, violating the Second Law.” Figure 10.30 shows such a
device. Here the one-way mechanism on the left is a spring (the “pawl”) that jams against the
asymmetric teeth on a wheel when it tries to rotate backward. Reply to Sullivan’s remark in the
context of this circular F-ratchet. [Hint: First review Section 6.5.3.]
10.3Ion pump energetics
Textbooks quote the value ∆G′^0 =− 7. 3 kcal/mole for the hydrolysis of ATP (Figure 2.14). Chap-
ter 11 will introduce a molecular machine that uses one ATP per step and does useful work equal to
14 kBTr.Reconcile these statements, using that typical intracellular concentrations are [ATP]=0.01
(that is,cATP=10mM), [ADP]=0.001, and [Pi]=0.01.
10.4Competitive inhibition
Section 10.4.2 described competitive inhibition as one strategy to control the activity of an enzyme;
for example the protease inhibitors used to treat HIV use this strategy. In this mechanism an analog
molecule, which we will call I, binds to the active site of an enzyme E, blocking it from processing
its substrate.
a. Write down the Mass Action rule for the reaction I+EEI, with some equilibrium constant
Keq,I.
b. Now repeat the derivation of the Michaelis–Menten rule in Section 10.4.1, with the change that
now E can be in any of three states:PE+PES+PEI=1.Show that the reaction velocity can be
written as
v=vmax
cS
αKM+cS
. competitive inhbition (10.32)
Hereαis a quantity that you are to find, involving the parameters of the uninhibited enzyme (KM
andvmax),Keq,I,and the concentrationcIof inhibitor.
c. Suppose we measure the initial reaction velocity as a function of substrate concentration for two
fixed values ofcI,then plot the two datasets in Lineweaver–Burk form. Describe the two curves we
will get if I is a competitive inhibitor.
d. Ethanol and methanol are two similar, small molecules. Methanol is quite toxic: The liver enzyme
alcohol dehydrogenase converts it to formaldehyde, which can cause blindness. The kidneys will
eventually remove methanol from the blood, but not fast enough to avert this damage. Why do you
suppose a therapy for methanol poisoning involves gradual intravenous injection of ethanol over
several hours?
10.5Uncompetitive inhibition
Modify the derivation of the Michaelis-Menten rule for enzyme kinetics (Section 10.4.1) to account
foruncompetitive inhibition.^14 That is, augment the basic reaction diagram
E+ScSk 1
k- 1ES
k 2
⇀E+P(^14) T 2 Uncompetitive inhibition is a mathematical simplification of a more realistic situation called “noncompet-
itive inhibition.” For a full discussion see Nelson & Cox, 2000.