Biological Physics: Energy, Information, Life

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

L 2 L

L 2 L L 2 L

U

tot

U

tot



L 1

x

x x

x

+fL 1

U=fx

fL

fL−

f′L

f′L−

U=fx

U=f′x

a

c

b

d

unloaded G-ratchet loaded G-ratchet

S-ratchet, low load S-ratchet, high load

U

tot Utot

Figure 10.11:(Sketch graphs.) (a)Energy landscape of the unloaded G-ratchet (see Figure 10.10a). Pushing the
rod to the right compresses the spring on one of the bolts, raising the stored potential energy by an amountand
giving rise to the curved part of the graph ofUtot.Once a bolt has been retracted, the potential energy is constant
until it clears the wall; then the bolt pops up, releasing its spring, and the stored energy goes back down. (b)The
loaded G-ratchet. Rightward motion now carries a net energy penalty, the work done against the load forcef.Hence
the graph ofUtotis tilted relative to (a). (c)The S-ratchet at low loadf.Asthe rod moves rightward, its potential
energy progressively decreases, as more of its bolts get released. (d)The S-ratchet at high load,f′.The downward
steps are still of fixed height,but the upward slope is greater, so that rightward progress now carries a net energy
penalty.

Sullivan: Yes, but now something is really getting used up: The S-ratchet is a one-shot motor,
releasing potential energy stored in its compressed springs as it moves. In fact, it’s a mechanical
analog of the translocation machine (Figure 10.3). There’s no longer any obvious violation of the
Second Law.
Gilbert: Won’t your criticism of my device (that it can make backward steps) apply to yours as
well?
Sullivan: Wecan design the S-ratchet’s springs to be so stiff that they rarely retract spontaneously,
so that leftward steps are rare. But thanks to the latches, rightward steps are still easy.

10.2.3 The Smoluchowski equation gives the rate of a microscopic machine

chine

Qualitative expectations Let’s supply our protagonists with the mathematical tools they need
to clear up their controversy. Figure 10.11a,b show the energy landscapes of the G-ratchet, both
without and with a load force respectively. Rightward motion of the rod compresses a spring,
increasing the potential energy. Atx=0,L, 2 L,.. .the bolt clears the wall. It then snaps up,
dissipating the spring’s potential energy into thermal form. Panels (c, d) of the figure show the
energy landscape of the S-ratchet, with small and large loads (fandf′,respectively). Again each