360 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
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UtotθUload=(w 1 R)θUtot=(w 1 R−τ)θUmotor=−τθθUloadUmotorθ×
U, arbitrary unitsUFigure 10.7: (Sketch graphs.) Energy landscapes for the one-dimensional machine in Figure 10.6a. The vertical
scale is arbitrary. (a)Lower dotted line:The coiled spring contributesUmotor=−τθto the potential energy.Upper
dashed line:The external load contributesUload=w 1 Rθ.Solid line:the total potential energy functionUtot(θ)is
the sum of these energies; it decreases in time, reflecting the frictional dissipation of mechanical energy into thermal
form. (b)The same, but for an imperfect (“bumpy”) shaft.Solid curve:Under load, the machine will stop at the
pointθ×.Lower dotted curve:Without load, the machine will slow down, but proceed, atθ×.
with increasingθ. The slope of the total energy is downward, soτ =−dU/dθis a positive net
torque. In a viscous medium the angular speed is proportional to this torque: We can think of the
device as “sliding down” its energy landscape.
Forthe cyclic machine shown in Figure 10.6b, the graph is similar. HereUmotoris a constant,
but there is a third contributionUdrive=−w 2 Rθfrom the external driving weight, giving the same
curve forUtot(θ).
Real machines are not perfect. Irregularities in the pivot may introduce bumps in the potential
energy function, “sticky” spots where an extra push is needed to move forward. We can describe
this effect by replacing the ideal potential energy−τθbysome functionUmotor(θ)(lower dotted
curve in Figure 10.7b). As long as the resulting total potential energy (solid curve) is everywhere
sloping downward, the machine will still run. If a bump in the potential is too large, however, then a
minimum forms inUtot(pointθ×), and the machine will stop there. Note that the meaning of “too
large” depends on the load: In the example shown, the unloaded machinecanproceed beyondθ×.
Even in the unloaded case, however, the machine will slow down atθ×:The net torque−dUtot/dθ
is small at that point, as we see by examining the slope of the dotted curve in Figure 10.7b.
Tosummarize, the first two machines in Figure 10.6 operate by sliding down the potential
energy landscapes shown in Figure 10.7. These landscapes give “height” (that is, potential energy)
in terms of one coordinateθ,sowecall them “one-dimensional.”
Two-dimensional landscapes Our third machine involves gears. In the macroworld, the sort
of gears we generally encounter link the anglesαandβtogether rigidly:α=β,ormore generally
α=β+2πn/N,whereNis the number of teeth in each gear andnis any integer. But we could
also imagine “rubber gears,” which canslipover each other under high load by deforming. Then the
energy landscape for this machine will involve twoindependentcoordinates,αandβ.Figure 10.8
shows an imagined energy landscape for the internal energyUmotorof such gears withN=3.The