Biological Physics: Energy, Information, Life

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

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Figure 10.9:(Mathematical function.) Energy landscape for the driven, loaded, bumpy rubber gears. The land-
scape is the same as the one in Figure 10.8, but tilted. The figure shows the case where the driving torque is larger
than the load torque; in this case the tilt favors motion to the front left of the graph. Again the scale of the vertical
axis is arbitrary. The bump in the central valley (atβ=2,α=2)isnowaspotwhere “slipping” is likely to occur.
That is, the state of the machine can hop from one valley to the next lower one at such points.

viscous dissipation. In short:

The machine in Figure 10.6 stops doing useful work (that is, stops lifting the

weightw 1 )assoonaseither

a.w 1 equalsw 2 ,sothat the machine is in mechanical equilibrium (the valleys

in Figure 10.9 no longer run downhill in a direction of decreasingβ), or

b. The rate of slippage becomes large.

(10.1)

10.2.2 Microscopic machines can step past energy barriers

The machines considered in the preceding subsection were deterministic: Noise, or random fluc-
tuations, played no important role in their operation. But we wish to study molecular machines,
which occupy a nanoworld dominated by such fluctuations.
Gilbert says: Some surprising things can happen in this world. For example, a machine need
no longer stop when it encounters a bump in the energy landscape; after a while, a large enough
thermal fluctuation will always arrive to push it over the bump. In fact, I have invented a simple
waytotranslocate a protein, using thermal motion to my advantage. I’ve named my device the
“G-ratchet”in honor of myself (Figure 10.10a). It’s a shaft with a series of beveled bolts; they keep
the shaft from taking steps to the left. Occasionally a thermal fluctuation comes along and gives
the shaft a kick with energy greater than,the energy needed to compress one of the little springs
holding the bolts. Then the shaft takes a step to the right.
Sullivan: That certainlyissurprising. I notice that you could even use your machine to pull
against a load (the external forcefshown in Figure 10.10).
Gilbert: That’s right! It just slows down a bit, since now it has to wait for a thermal push with
energy greater than+fLto take a step.
Sullivan: Ihavejust one question: Where does the workfLdone against the load come from?
Gilbert: Iguess it must come from the thermal energy giving rise to the Brownian motion....