158 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
5.3.2 To stir or not to stir?
It’s surprisingly difficult to get anything to eat when you’re tiny. We get a hint of why when we
examine the experimental photograph, Figure 5.3 on page 149. At low Reynolds number the flow
lines just part majestically as they come to the surface of the sphere; any food molecules carried in
the fluid follow the flow lines and never arrive at the surface.
Things are not as bad as they seem. The macroscopic experiment shown in Figure 5.3 doesn’t
show the effects of diffusion, whichcancarry molecules to their receptors on a cell’s surface. But
diffusion will bring food even to a lazy, motionless cell! Similarly, diffusion will carry wasteaway,
even if the cell is too lazy to move away from its waste. So why bother swimming?
Similar remarks apply tostirring.Itwas once believed that a major job of cilia was to sweep fresh
fluid to the cell, enhancing its intake compared to passively waiting. To evaluate such arguments,
imagine the cilium as moving at some characteristic speedvand swinging through a lengthd.
These determine a time scalet=d/v,the time in which the cilium can replace its surrounding fluid
with fresh, outside fluid. On the other hand, movement of molecules a distancedwill occur just by
diffusion in a characteristic timed^2 /D,according to the diffusion law (Equation 4.5 on page 104).
Stirring will only be worthwhile (more effective than diffusion) ifd/v < d^2 /D,that is, if
v>D/d. (5.15)(Some authors call the dimensionless ratiovd/Dthe “Peclet number.”) Taking a cilium to be about
d=1μmlong, the criterion for stirring to be worthwhile is then thatv> 1000 μms−^1 .This is also
the criterion for swimming to enhance food intake significantly.
But bacteria do not swim anywhere near this fast.Stirring and swimming don’t help enhance
food intake for bacteria. (The story is different for larger creatures, even protozoa, where the
Reynolds number is still small butdandvare both bigger.) There is experimental support for
this conclusion. Mutant bacteria with defective flagellar systems manage about as well as their
wild-type cousins when food is plentiful.
5.3.3 Foraging, attack, and escape
Foraging The previous subsection may have left you wondering why wild-type bacteriadoswim.
The answer is that life in the mean, real world can be more challenging than life in a nice warm flask
of broth. While bacteria don’t need to swim around systematically scooping up available food, still
it may be necessary for a cell tofindafoodsupply. The word “find” implies a degree of volition,
and mind-boggling as it may seem, supposedly primitive organisms likeE. colican indeed perform
the computations needed to hunt for food.
The strategy is elegant. E. coliswims in a burst of more or less straight-line motion, pauses,
and then takes off in a new, randomly chosen direction. While swimming, the cell continuously
samples its environment. If the concentration of food is increasing, the bacterium extends its run.
If the food concentration is decreasing, the cell terminates the run and starts off in a new direction
sooner than it would have done in an improving environment. Thus the cell executes a form of
biased random walk, with a net drift toward higher food concentrations.
But there’s no point in making a run so short that the environment won’t be appreciably different
at the end. Because diffusion constantly tries to equalize the concentration of food (and everything
else), then, it’s necessary for the bacterium tooutrun diffusionif swimming is to be of any use in
navigating food gradients. We have already found the criterion, Equation 5.15. Now, however, we