10.4. Kinetics of real enzymes and machines[[Student version, January 17, 2003]] 391
weshould expect that solving the Smoluchowski equation onanytwo-dimensional free energy
landscape will reveal net motion, as long as the landscape is tilted in the chemical (α)direction
and asymmetric in the spatial (β)direction.
As mentioned earlier, however, it’s not easy to solve the Smoluchowski equation (Equation 10.4
on page 367) in two dimensions, nor do we even know a realistic free energy landscape for any real
motor. To show the essence of the D-ratchet mechanism, then, we will as usual construct a simplified
mathematical model. Our model motor will contain a catalytic site, which hydrolyzes ATP, and
another site, which binds to the microtubule. We will assume that an allosteric interaction couples
the ATPase cycle to the microtubule binding, in a particular way:
1.The chemical cycle is autonomous—it’s not significantly affected by the interaction with the
microtubule. The motor snaps back and forth between two states, which we will call “s”
(or “strong-binding”) and “w” (or “weak-binding”). After entering the “s” state, it waits
an average timetsbefore snapping over to “w”; after entering the “w” state it waits some
other average time,tw,before snapping back to “s.” (One of these states could be the one
with the nucleotide-binding site empty, and the other one E·ATP,as drawn in Figure 10.25.)
The assumption is thattsandtware both independent of the motor’s positionxalong the
microtubule.
2.However, the binding energy of the motor to the microtubule does depend on the state of the
chemical cycle. Specifically, we will assume that in the “s” state, the motor prefers to sit at
specific binding sites on the microtubule, separated by a distance of 8nm.Inthe “w” state,
the motor will be assumed to have no positional preference at all—it diffuses freely along the
microtubule.
3.In the strongly binding state, the motor feels an asymmetric (that is, lopsided) potential
energyU(x)asafunction of its positionx.This potential is sketched as the sawtooth curve
in Figure 10.25a; asymmetry means that this curve is not the same if we flip it end-for-end.
Indeed, we do expect the microtubule, a polar structure, to give rise to such an asymmetric
potential.In the D-ratchet model the free energy of ATP hydrolysis can be thought of as entering the motion
solely by an assumed allosteric conformational change, which alternately glues the motor onto the
nearest binding site, then pries it off. To simplify the math, we will assume that the motor spends
enough time in the “s” state to find a binding site, then binds and stays there until the next switch
to the “w” state.
Let’s see how the three assumptions listed above yield directed motion, following the left panels
of Figure 10.25. As in Section 10.2.3, imagine a collection ofmanymotor-microtubule systems, all
starting at one position,x=0(panels (b1) and (b2)). At later times we then seek the probability
distributionP(x)tofind the motor at various positionsx.Attime zero the motor snaps from “s”
to “w.” The motor then diffuses freely along the track (panel (c1)), so its probability distribution
spreads out into a Gaussian centered onx 0 (panel (c2)). After an average wait oftw,the motor
snaps back to its “s” state. Now it suddenly finds itself strongly attracted to the periodically spaced
binding sites. Accordingly, it drifts rapidly down the gradient ofU(x)tothe first such minimum,
leading to the probability distribution symbolized by Figure 10.25(d2). The cycle then repeats.
The key observation is now that the average position of the motor after one cycle is now shifted
relative to where it was originally. Some of this shift may arise from conformational changes,
“power stroke” shifts analogous to those in myosin or two-headed kinesin. But the surprise is that