432 Chapter 11. Machines in membranes[[Student version, January 17, 2003]]
as sketched in Figure 11.10a. They labeled the filament with a fluorescent dye and anchored the
αandβelements to a glass slide, so that relative rotary motion of the c element would crank the
entire actin filament. The resulting motion pictues showed that the motor took random (Brownian)
steps, with no net progress, until ATP was added. With ATP, it moved in one direction at speeds
up to about six revolutions per second. The motion was not uniform; slowing the F1 motor by
using low ATP levels showed discrete, 120◦steps. Such steps are just what we would expect on
structural grounds: The structure of F1 shows threeβsubunits, each one-third of a revolution from
the others. (Compare the steps taken by kinesin, Figure 10.22 on page 385.) A later experiment
used the entire F0F1 complex, not just F1, to confirm that the F0 really is rigidly connected to F1
(see Figure 11.10).
The experiments just described also allow an estimate of the torque generated by ATP hydrolysis
(or the torque required for ATP synthesis), using ideas from low Reynolds-number flow. The
experimenters found that an actin filament 1μmlong rotated at about 6 revolutions per second, or
an angular velocity of 2π× 6 radians per second, when ATP was supplied. Section 5.3.1 on page 153
claimed that the viscous drag force on a thin rod, dragged sideways through a fluid, is proportional
to its speed,v,and to the viscosity of water,η.The force should also be proportional to the rod’s
length. Detailed calculation for a rod of length 1μm,with the thickness of an actin filament, gave
Kinosita and coauthors the constant of proportionality:
f≈ 3. 0 ηLv. (11.17)Your Turn 11e
Equation 11.17 gives the force needed to drag a rod at a given speedv.But we want thetorque
needed to crank a rod pivoted at one end atangularvelocityω.
a. Work this out from Equation 11.17. Evaluate your answer for a rod of length 1μmrotating at
6 revolutions per second.
b. How much work must the F1 motor do for every one-third revolution of the actin filament?More precisely, the rotation rate just quoted was achieved when ATP was supplied at a concen-
trationcATP=2mM,along withcADP=10μMandcPi=10mM.
Your Turn 11f
a. Find ∆Gfor ATP hydrolysis under these conditions (recall Section 8.2.2 and Problem 10.3).
b. If each ATP hydrolysis cranks theγelement by one-third of a revolution, how efficiently does
F1 transduce chemical free energy to mechanical work?Thus F1 is a highly efficient transducer, when operated in its ATPase mode. Under natural condi-
tions, F1 operates in the opposite direction (converting mechanical energy supplied by F0 to ATP
production) with a similarly high efficiency, contributing to the overall high efficiency of ærobic
metabolism.
11.3.5 Vista: Cells use chemiosmotic coupling in many other contexts
Section 11.2 introduced ion pumping across membranes as a practical necessity, reconciling
- The need to segregate macromolecules inside a cellular compartment, so they can do their
jobs in a controlled chemical environment,