156 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
ff⊥f‖vv⊥ v‖zxyFigure 5.8: (Schematic.) A thin rod is dragged at low Reynolds number with velocityv. The forcefneeded to
drag the rod is the resultant of two forcesf‖andf⊥coming from the components ofvparallel to, and perpendicular
to, the rod’s axis. Even ifv‖andv⊥are the same length, as shown, the resulting components offwill not be equal;
thusfwill not point parallel tov.
bases its propulsion on this fact.
Unlike cilia,E. coli’s flagella do not flex; they are rigid, helical objects, like twisted coathangers,
so they cannot solve the propulsion problem by the means shown in Figure 5.7. Since they are
only 20nmthick, it’s not easy to visualize their three-dimensional motion under the microscope.
Initially some people claimed that the bacterium waves them about, but we know this can’t work:
It’s a reciprocal motion. Others proposed that a wave of bending travels down the flagellum, but
there hardly seemed to be room for any of the required machinery inside such a thin object. In
1973, H. Berg and R. Anderson argued that instead the bacteriumcrankedthe flagellum at its base
in a rigidrotarymotion (like the twirler in Figure 5.5b). This was a heretical idea. At that time
no true rotary engine had ever been seen in any living creature (we will, however, meet another
example in Chapter 11). Nor was it easy to imagine how to prove such a theory—it’s hard to judge
the three-dimensional character of a motion seen under the microscope.
M. Silverman and M. Simon found an elegant solution to the experimental problem. They used
amutantE. colistrain that lacks most of its flagellum, having instead only a stump (called the
“hook”). They anchored the cells to a glass coverslip by their hooks. The flagellar motor, unable
to spin the anchored flagellar hook, instead spun the whole bodies of the bacteria, a process easily
visible in the microscope! Today we know that the flagellar motor is a marvel of nanotechnology, a
rotary engine just 45nmwide (Figure 5.9).
Rotary motion certainly meets our criterion of being periodic but not reciprocal. And we are
familiar with other spinning helical objects that develop thrust along their axis, namely submarine
and boat propellers. But the details are quite different in the low-Reynolds case. Figure 5.10 shows
aschematic of the situation. A rigid helical object (representing the flagellum) is cranked about
its axis (by the flagellar motor). Two short segments of the helix have been singled out for study.
The net force dfexerted on one short segment by its two neighbors must balance the viscous drag
force on that segment. Thus for the helix to undergo the desired rotational motion, dfmust be the
vector shown in Figure 5.8. Adding up all the contributions from every rod segment, we see that
the components in thexyplane all cancel (think about the corresponding segments on the far side
of the helix, whose velocity vectors point upward). But df also has a small component directed
along the−ˆzdirection, and the dfz’s donotcancel. Rather, a net leftward force must be supplied