404 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
Your Turn 10g
Show that for the two flanking bins, centered at±L,the mean position also shifts by〈k〉bin± 1 =
δ/L,and similarly for all the other pairs of bins,〈k〉bin±i.Wehave divided the entire range ofxinto strips, in each of which the mean position shifts by the
same amountδ/L.Hence the total mean shift per step,u,isalsoδ/L. According to Idea 10.23,
then,v≡uL/∆tis given byδ/∆t.You can also show, using Idea 10.23 on page 393, that the
increase in the variance ofxpercycle just equals the diffusive spread, 2Dtw.
The rate of ATP hydrolysis per motor under the conditions of the experiment was known to be
(∆t)−^1 ≈ 100 s−^1 .Substituting the experimental numbers then yields
140 nm s−^1 ≈δ×(100s−^1 ) and 88 000nm^2 s−^1 ≈(100s−^1 )× 2 Dtw,orδ =1. 4 nmandDtw= 440 nm^2. The first of these gives a value for the asymmetry of the
kinesin–microtubule binding that is somewhat smaller than the size of the binding sites. That’s
reasonable. The second result justifies a postieri our assumption thatDtw L^2 =64nm^2 .That’s
good. Finally, biochemical studies imply that the mean durationtwof the weak-binding state is
several milliseconds; thusD≈ 10 −^13 m^2 s−^1. This diffusion constant is consistent with measured
values for other proteins that move passively along linear polymers. Everything fits.
- It is not currently possible to apply a load force to single-headed kinesin molecules, as it is
with two-headed kinesin. Nevertheless, the velocity calculation, corresponding to the result for the
tightly coupled case (Equation 10.26), is instructive. See for example Peskin et al., 1994. The
motor will stall when its backward drift in the “w” state equals the net forward motion expected
from the asymmetry of the potential.
But shouldn’t the condition for the motor to stall depend on the chemical potential of the
“food” molecule? This question involves the first assumption made when defining the diffusing
ratchet model, that the hydrolysis cycle is unaffected by microtubule binding (page 391). This
assumption is chemically unrealistic, but it is not a bad approximation when ∆Gis very large
compared to the mechanical work done on each step. If the chemical potential of ATP is too small,
this assumption fails; the timestsandtwspent in the strong- and weak-binding states will start
to depend on the locationxalong the track. Then the probabilitiesPkto land atkLwill not be
given simply by the areas under the diffusion curve (see Figure 10.25 on page 392), and the stall
force will be smaller for smaller ∆G.For more details see J ̈ulicher et al., 1997; Astumian, 1997.
More generally, suppose a particle diffuses along an asymmetric potential energy landscape,
which is kicked by some external mechanism. The particle will make net progress only if the external
kicks correspond to a nonequilibrium process. Such disturbances will have time correlations absent
in pure Brownian motion. Some authors use the terms “correlation ratchet” or “flashing ratchet”
instead of “diffusing ratchet” to emphasize this aspect of the physics. (Still other related terms in
the literature include “Brownian ratchet,” “thermal ratchet,” and “entropic ratchet.”) For a general
argument that asymmetry and an out-of-equilibrium step are sufficient to get net directed motion
see Magnasco, 1993; Magnasco, 1994. This result is a particular case of the general result that
whenever a reaction graph contains closed loops and is coupled to an out-of-equilibrium process,
there will be circulation around one of the loops.