396 Chapter 10. Enzymes and molecular machines[[Student version, January 17, 2003]]
Cells contain a staggering variety of molecular motors. This chapter has made no attempt
to capture Nature’s full creative range, once again focusing on the humbler question “How could
anything like that that happen at all?” Nor did we attempt even a survey of the many beautiful ex-
perimental results now available. Rather, the goal was simply to create some explicit mathematical
models, anchored in simpler, known phenomena and displaying some of the behavior experimen-
tally observed in real motors. Such conceptually simple models are the armatures upon which more
detailed understanding must rest.
Key formulas
- Perfect ratchet: Aperfect ratchet (that is, one with an irreversible step) at zero load makes
progress at the ratev=L/tstep=2D/L(Equation 10.2). - Smoluchowski: Consider a particle undergoing Brownian motion on a potential landscape
U(x). In a steady (not necessarily equilibrium) state, the probabilityP(x)offinding the
particle at locationxis a solution to (Equation 10.4)
0=
d
dx(
dP
dx+
1
kBTP
dU
dx)
,
with appropriate boundary conditions.- Michaelis–Menten: Consider the catalyzed reaction E + S
k 1
k
-1ES
k 2
⇀E+P. A steady,
nonequilibrium state can arise when the supply of substrate S is much larger than the supply
of enzyme E. The reaction velocity (rate of change of substrate concentrationcS)inthis case
isv=vmaxKMcS+cS (Equation 10.20), where the saturating reaction velocity isvmax=k 2 cE
and the Michaelis constant isKM=(k− 1 +k 2 )/k 1 (Equation 10.19).Further reading
Semipopular:
Enzymes: Dressler & Potter, 1991.
Intermediate:
Enzymes: Berg et al., 2002; Voet & Voet, 2003.
Chemical kinetics: Tinoco Jr. et al., 2001; Dill & Bromberg, 2002.
From actin/myosin up to muscle: McMahon, 1984.
Ratchets: Feynman et al., 1963a,§46.
Motors: Berg et al., 2002; Howard, 2001; Bray, 2001.
Computer modeling of muscle mechanics: Hoppensteadt & Peskin, 2002.
Technical:
Kramers theory: Kramers, 1940; Frauenfelder & Wolynes, 1985; H ̈anggi et al., 1990.
The abstract discussion of motors was largely drawn from the work of four groups around 1993.
Some representative reviews by these groups include J ̈ulicher et al., 1997; Astumian, 1997;
Mogilner et al., 2002; Magnasco, 1996; see also the original papers cited therein.
Single-molecule motility assays: Howard et al., 1989; Finer et al., 1994.