10.4. Kinetics of real enzymes and machines[[Student version, January 17, 2003]] 389
Let us see how to make these ideas quantitative.
Kinetic predictions Wesimplify the problem by lumping all the states other than ES 1 and ES′ 1
into a single state called E, just as Equation 10.16 on page 380 lumped EP with E. The model
sketched in Figure 10.21 then amounts to splitting the bound state ES of the Michaelis–Menten
model into two substates, ES 1 and ES′ 1 .Toextract predictions from this model, Schnitzer and
coauthors proposed thatthe stepES 1 ES′ 1 is nearly in equilibrium. That is, they assumed
that the activation barrier to this transition is small enough, and hence the step is rapid enough
compared to the others, that the relative populations of the two states stay close to their equilibrium
values.^11 Wecan consider these two states together, thinking of them jointly as acomposite state.
In the language of Equation 10.16, the fraction of time spent in ES 1 effectively lowers the ratek 2
of leaving the composite state in the forward direction. Similarly, the fraction of time spent in ES′ 1
effectively lowers the ratek- 1 of leaving the composite state in the backward direction.
Wewish to understand the effect of an applied load force, that is, an external force directed
awayfrom the “+” end of the microtubule. To do this, note that the step ES 1 ⇀ES′ 1 ,besides
throwing head Kbforward, also moves the common connecting chains to a new average position,
shifted forward by some distance.All we know aboutis that it is greater than zero, but less than
afull step of 8nm.Since a spatial step does work against the external load, the applied load force
will affect the composite state: It shifts the equilibrium away from ES′ 1 and toward ES 1 .Schnitzer
and coauthors neglected other possible load dependences, focusing only on this one effect.
Wenow apply the arguments of the previous two paragraphs to the definitions of the MM
parameters (Equation 10.19 on page 380), finding that load reducesvmax,asobserved, and moreover
may increaseKM byeffectively increasingk- 1 bymore than it reducesk 2 .Thuswehavethe
possibility of explaining the data in Table 10.1 with the proposed mechanism.
Tosee if the mechanism works we must see if it can model the actual data. That is, we must see
if we can choose the free energy change ∆Gof the isomerization ES 1 ES′ 1 ,aswell as the substep
length,inawaythat explains the numbers in Table 10.1. Some mathematical details are given
in Section 10.4.3′on page 401. A reasonably good fit can indeed be found (Figure 10.24). More
important than the literal fit shown is the observation that the simplest power-stroke model does
not fit the data, but an almost equally simple model, based on structural and biochemical clues,
reproduces the qualitative facts of Michaelis–Menten kinetics, withKMrising andvmaxfalling as
the load is increased.
The fit value of the equilibrium constant for the isomerization reaction is reasonable: It corre-
sponds to a ∆G^0 of about− 5 kBTr.The fit value ofis about 4nm,which is also reasonable: It’s
half the full step length. The existence of these substeps is a key prediction of the model.
T 2 Section 10.4.3′on page 401 completes the analysis, obtaining the relation between speed, load,
and ATP availability in this model.
10.4.4 Molecular motors can move even without tight coupling or a power stroke
Section 10.4.3 argued that deep within the details of kinesin’s mechanochemical cycle, there lies a
simple mechanism: Two-headed kinesin slides down a valley in its free energy landscape. Even while
admitting that the basic idea is simple, though, we can still marvel at the elaborate mechanism
(^11) Some authors refer to this assumption as “rapid isomerization.”