10.4. Kinetics of real enzymes and machines[[Student version, January 17, 2003]] 393
progress. Indeed, we found thatit makes net progress even if no conformational change in the
motor drives it in thexdirection. The model also makes some predictions about experiments.
Forone thing, we see that the diffusing ratchet can make backward steps;^12 P− 1 is not zero, and
can indeed be large if the motor diffuses a long way between chemical cycles. In fact, each cycle
gives an independent displacement, with the same probability distribution{Pk}for every cycle.
Section 4.1.3 on page 105 analyzed the mathematics of such a random walk. The conclusion of that
analysis, translated into the present situation, was that
The diffusing ratchet makes net progressuLperstep, whereu=〈k〉. The
variance (mean-square spread) of the total displacement increases linearly with
the number of cycles, increasing byL^2 ×variance(k)percycle.(10.23)
In our model, the steps come every ∆t=ts+twon average, so we predict a constant mean velocity
v=uL/∆tand a constant rate of increase in the variance ofxgiven by
〈(x(t)−vt)^2 〉=t×(
L^2
∆t
×variance(k))
. (10.24)
Okada and Hirokawa tested these predictions with their single-headed kinesin construct, C351.
Although the optical resolution of the measurements, 0. 2 μm,was too large to resolve individual
steps, still Figure 10.26 shows that C351 often made net backward progress (panel (a)), unlike
conventional two-headed kinesin (panel (b)). The distribution of positions at a timetafter the initial
binding,P(x, t), also showed features characteristic of the diffusing ratchet model. As predicted
byEquation 10.24, the mean position moved steadily to larger values ofx,while the variance
steadily increased. In contrast, two-headed kinesin showed uniform motion with very little increase
in variance (panel (b)).
Tomake these qualitative observations sharp, Figure 10.26c plots the observed mean-squared
displacement,〈x(t)^2 〉. According to Equation 10.24, we expect this quantity to be a quadratic
function of time, namely (vt)^2 +tL∆^2 tvariance(k). The figure shows that the data fit such a function
well. Okada and Hirokawa concluded that although monomeric kinesin cannot be tightly coupled,
it nevertheless makes net progress in the way predicted by the diffusing ratchet model.
Subtracting away the (vt)^2 term to focus attention on the diffusive part reveals a big difference
between one- and two-headed kinesin. Figure 10.26d shows that both forms obey Equation 10.24,
but with a far greater diffusion constant of proportionality for C351, reflecting the loosely coupled
character of single-headed kinesin.
Toend this section, let us return to the question that motivated it: How could molecular motors
have evolved from something simpler? We have seen how the bare minimal requirements for a motor
are simple, indeed:
- It must cyclically process some substrate like ATP, in order to generate out-of-equilibrium
fluctuations. - These fluctuations must in turn couple allosterically to the binding affinity for another protein.
- The latter protein must be an asymmetric polymer track.
It’s not so difficult to imagine how an ATPase enzyme could gain a specific protein-binding site
bygenetic reshuffling; the required allosteric coupling would arise naturally from the general fact
that all parts of a protein are tied together. Indeed a related class of enzymes is already known in
(^12) Compare Problem 4.1.