Biological Physics: Energy, Information, Life

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

2. The discussion at the end of Section 10.4.3 simplified the reaction diagram of kinesin, replacing
   Figure 10.21 by^13

E

cATPk+


k

ES 1

equil


 ES′ 1 ···

kn

⇀E.

Instead of writing ATP explicitly as a participant, we are thinking of E as spontaneously isomerizing
to ES with a rate proportional tocATP. (Some authors call the combinationcATPk+a“pesudo-
first order rate constant.”) The dots represent possible other substeps, which we ignore; as usual,
the last step (hydrolysis of ATP and release of Pi)isassumed to be effectively irreversible, as in
Section 10.4.1.
Proceeding almost as in Section 10.4.1, we first note that each kinesin head must be in one of
the three states E, ES 1 ,orES′ 1 .Weassumed near-equilibrium between the latter two states. The
appropriate equilibrium constant will reflect an intrinsic free energy change, ∆G^0 ,plus a force-
dependent termf(see Section 6.7 on page 199). Finally, while the state E is not in equilibrium
with the others, we do assume that the whole reaction is in a quasi-steady state. All together, then,
weare to solve three equations for the three unknown probabilitiesPE,PES 1 ,andPES′ 1 :

1=PE+PES 1 +PES′ 1 (normalization) (10.27)

PES 1 = PES′ 1 e(∆G

(^0) +f)/kBT
(near-equilibrium) (10.28)
0=
d
dtPE=−cATPk+PE+k-PES^1 +knPES
′ 1. (quasi-steady state) (10.29)
Solving gives
v=kn×(8nm)×

[

k-e(∆G

(^0) +f)/kBT
+kn
k+cATP
+e(∆G
(^0) +f)/kBT
+1

]− 1

.

Foranyfixed value of load forcef,this expression is of Michaelis–Menten form (Equation 10.20 on
page 381), with load-dependent parameters analogous to Equation 10.19 given by

KM=

1

k+

k-e(∆G

(^0) +f)/kBT
+kn
e(∆G^0 +f)/kBT+1

(10.30)

and
vmax=kn×(8nm)×[e(∆G

(^0) +f)/kBT
+1]−^1. (10.31)
Figure 10.24 on page 390 shows the kinetic data of Table 10.1, along with solid curves showing
the above functions with the parameter choices =3. 7 nm,∆G^0 =− 5. 1 kBTr,kn = 103 s−^1 ,
k+=1. 3 μM−^1 s−^1 ,andk-= 690 s−^1.
Schnitzer and coauthors actually compared their model to a larger dataset than the one shown
here, including additional measurements of speed versus force at fixedcATP,and again found
satisfactory agreement. Their model, however, is not the only one that gives satisfactory agreement
with experiments. Other models also account for the observed statistical properties of kinesin
stepping, stall forces, and the loss of processivity at high loads (see for example Fisher & Kolomeisky,
2001).
10.4.4′

1. C351 is not itself a naturally occurring motor, but rather a construct designed to have certain

(^13) See Problem 10.6 for another example of an enzymatic mechanism with a rapid-isomerization step.