- Track 2[[Student version, January 17, 2003]] 403
- L/2 0 L/2 L 3 L/2
bin -1 bin 0 bin +1δP(x)L xFigure 10.28:(Sketch graph.) Illustrating the calculation of the diffusing ratchet’s average stepping rate. The solid
lines delimit the bins discussed in the text. The dashed lines are the same as those on the right side of Figure 10.25 on
page 392: They mark potential maxima, or “watersheds.” Thus the region between any two neighboring dashed lines
will all be attracted to whichever minimum (0,L, 2 L,.. .)lies between them when the motor snaps to its strongly
bound state. For example, the dark gray region is the part of bin 0 attracted tox=0,whereas the light gray
region is the part of bin 0 attracted tox=L.(The width of the bins has been exaggerated for clarity; actually the
calculation assumes that the distributionP(x)isroughly constant within each bin.)
experimentally convenient properties. We nevertheless take it as emblematic of a class of natural
molecular machines simpler than conventional kinesin.
- Okada and Hirokawa also interpreted the numerical values of their fit parameters, showing that
they were reasonable Okada & Hirokawa, 1999. Their data (Figure 10.26 on page 394) gave a
mean speedvof 140nm s−^1 and a variance increase rate of 88 000nm^2 s−^1 .Tointerpret these results
wemust connect them with the unknown molecular quantitiesδ,ts,tw,and the one-dimensional
diffusion constantDfor the motor as it wanders along the microtubule in its weak-binding state.
Figure 10.28 shows again the probability distribution at the end of a weak-binding period. If
wemake the approximation that the probability distribution was very sharply peaked at the start
of this period, then the curve is just given by the fundamental solution to the diffusion equation
(Equation 4.27 on page 128). To find thePk,wemust compute the areas under the various shaded
regions in Figure 10.25, and from these compute〈k〉and variance(k). This is not difficult to do
numerically, but there is a shortcut that makes it even easier.
Webegin by dividing the line into bins of widthL(solid lines on Figure 10.28). Suppose that
Dtwis much larger thanL^2 ,sothat the motor diffuses many steps in each weak-binding time. Then
P(x)will be nearly a constant in the center bin of the figure, that is, the region between±L/2. As
wemove outward from the center,P(x)will decrease. But we can still take it to be a constant in
each bin, for example the one fromL/2to3L/2. Focus first on the center bin. Those motors lying
between−L/ 2 and +L/ 2 −δ(dark gray region of Figure 10.28) will fall back to the binding site at
x=0,whereas the ones fromL/ 2 −δtoL/ 2 (light gray region) will land atx=L.For this bin,
then, the mean position will shift by
〈k〉bin 0=P(0)
(
(L−δ)×0+δ× 1)
P(0)×L
=
δ
L