Biological Physics: Energy, Information, Life

392 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]

a

b2

c2

d2

U

(x

)

0

L

2 +δ

Lx

P

(x

)

P

(x

)

P 0 P 1 P^2

• L

P

(x

)

P− 1

P 0

P 1

Figure 10.25: (Schematic; sketch graphs.) Diffusing ratchet (or “D-ratchet”) model for single-headed kinesin
motility.Left panels:Bound ATP is denoted by “T”; ADP and Pimolecules are not shown. As in Figure 10.21, the
beta subunits of the microtubule are denotedβn.(b1)Initially the kinesin monomer is strongly bound to sitenon
the microtubule. (c1)Inthe weakly bound state the kinesin wanders freely along the microtubule. (d1)When the
kinesin reenters the strongly bound state, it is most likely to rebind to its original site, somewhat likely to rebind to
the next site, and least likely to bind to the previous site. Relative probabilities are represented by shading.Right
panels:(a)Aperiodic but asymmetric potential for the strongly bound (“s”) state, as a function of positionxalong
the microtubule track. The minimum of the potential is not midway between the maxima, but rather is shifted by a
distanceδ.The potential repeats every distanceL(L=8nmfor a microtubule). (b2)Quasi-equilibrium probability
distribution for a motor in its “s” state, trapped in the neighborhood of the minimum atx=0.The motor now
suddenly switches to its “w” (or weakly binding) state. (c2)(Change of vertical scale.) The probability distribution
just before the motor switches out of its “w” state. The dark gray region represents all the motors in an initial
ensemble that are about to fall back into the microtubule binding site atx=0;the area under this part of the curve
isP 0 .The light gray region represents those motors about to fall into the site atx=L;the corresponding area is
P 1. The black regions to the left and right have areasP− 1 andP 2 ,respectively. (d2)(Change of vertical scale.)
The probability distribution just before the motor switches back to the “w” state. The areasPkfrom (c2) have each
collapsed to sharp spikes. SinceP 1 >P− 1 ,the mean position has shifted slightly to the right.

there will be a net shift even without any power stroke! To see this, examine Figure 10.25 and its
caption. The dark gray part of the curve in panel (c2) of the figure represents all the motors in
the original collection that are about to rebind to the microtubule at their original position,x=0.
Thus the probability of takingnostep is the areaP 0 under this part of the curve. The two flanking
parts of the curve, black and light gray, represent respectively those motors about to rebind to the
microtubule at positions shifted by−Lor +L,respectively. But the areas under these parts of the
curve are not equal:P 1 =P− 1 .The motor is more likely to diffuse over to the basin of attraction
atx=+Lthan to the one atx=−L,simply because the latter’s boundary is farther away from
the starting position.
Thus the diffusing ratchet model predicts that a one-headed molecular motor can make net