net motility benefit as the difference between the
motility benefit and the power required for swimming (16). In intermediate turbulence (D = 1.2 ×
10−8 W kg−1), the net motility benefit is maximal
for VC ≈ 15 to 25 mm s−1 (Fig. 3B), corresponding
to swimming speeds VS of 45 to 70 mm s−1 (16).
These values are in good agreement with speeds
recorded for several species of marine bacteria
(6–8, 18, 19), suggesting that motility in marine
bacteria might be under selection for chemotactic
The effectiveness of chemotaxis as a foraging
strategy further depends on the intensity of turbulence through the stirring and mixing of nutrient patches. We quantified this dependence by
varying the turbulence intensity, while keeping
the chemotactic velocity constant (VC = 20 mm s−1).
For an initial patch size of s = 2.5 mm, chemotaxis
is optimally favored at weak turbulence intensities (D ≈ 10−9 W kg−1), characteristic of the
ocean thermocline (20), where the motility benefit is slightly larger than in the absence of turbulence (Fig. 3C). In contrast, the motility benefit is
fivefold smaller at D = 10−6 W kg−1 (Fig. 3C),
indicating that chemotaxis is less effective in
highly turbulent regions, such as the upper ocean.
For larger patches the optimum turbulence intensity shifts to intermediate values (D ≈ 10−8 to
10−7 W kg−1 for s = 7.5 mm; Fig. 3C), characteristic of the upper thermocline (20). Although
observations of motility in the ocean are insufficient to test these predictions, it will be interesting
to determine whether changes in the prevalence
of motility and chemotaxis with depth revealed
by metagenomic studies (21) are in part determined by turbulence levels.
The existence of an optimal turbulence intensity points to a second, more subtle trade-off:
that between stirring and mixing. Stirring increases the surface area between the nutrient
patch and the surrounding water (Fig. 2). Mixing
refers to homogenization of the nutrients, which
is aided by stirring but ultimately occurs by molecular diffusion. Stronger turbulence produces
thinner filaments and steeper nutrient gradients
that elicit faster chemotaxis, but also accelerates
mixing, which erases the motility benefit. This
trade-off results in an optimal turbulence intensity, whereby the maximum motility benefit depends jointly on the size and lifetime of DOM
Constraints on chemotaxis can be understood
in terms of three fundamental time scales (16).
The chemotaxis time scale, tC = lB/VC, is the time
it takes a bacterium to swim to the core of a nu-
trient filament, whose characteristic width is the
Batchelor scale, lB. Stronger turbulence creates
finer filaments (smaller lB and tC), but also de-
creases the filaments’ lifetime, which is charac-
terized by the mixing time scale, tM = lB2/kC,
where kC is the nutrient diffusivity. One thus ex-
pects that the motility benefit depends on the
relative magnitude of tC and tM. A further con-
dition for motility to be beneficial is that the con-
sumption of the patch through uptake is slower
than chemotactic migration, i.e., tC < tU. We
quantify the relative magnitude of the three time
scales by means of two Frost numbers, FrM = tC/tM
and FrU = tC/tU (16, 22). When FrM >> 1 or FrU >>
1, chemotaxis is too slow relative to mixing (the
“mixing-limited regime”) or consumption (the
“uptake-limited regime”), respectively, for motile
bacteria to gain appreciable benefit. This argu-
ment is verified by a formal scaling analysis (16),
whose prediction (Eq. S40) is in good agreement
with the DNS results.
References and Notes
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