by the number of patches occurring in the computational volume in a day, based on a carbon
injection rate of Ċinj ¼ 0:12 gC m−3day−1 (16),
and by the carbon content in one cell (16), the
motility benefit can be expressed in units of new
cell equivalents produced by each bacterium per
day. For the scenario shown in Fig. 2, the motility
benefit peaks 13.2 s after injection of the patch.
At this time, motile cells consume 23% more
than nonmotile cells, an equivalent benefit of
more than one new cell per day (per individual) if
the uptake difference was sustained at this level
(relative to 4.5 new cells per day produced by
each cell of either species in the absence of
chemotaxis). Instead, the motility benefit nearly
vanishes after 50 s, even though 59% of the nutrient is still available, because what remains has
been mixed, erasing any advantage of motility.
The instantaneous motility benefit, like the nu-
trient filaments, is therefore highly transient.
To determine the chemotactic velocity that
optimizes foraging, we performed competition simulations where we varied the maximum chemotactic velocity, VC, while keeping the turbulence
intensity constant at an intermediate level (D =
1.2 × 10−8 W kg−1). The advantage afforded by
chemotaxis depends strongly on VC (Fig. 3, A
and B). The motility benefit is weak throughout
the patch lifetime for slow chemotaxers. For example, motility enhances the instantaneous uptake by at most 15%, affording a time-averaged
benefit of 0.3 new cells per day, for VC = 5 mm s−1.
A chemotactic velocity of this order is typical of
the enteric bacterium Escherichia coli (VC = 0.6
to 13.8 mm s−1) (15), the traditional model organism for the study of chemotaxis. In contrast,
marine bacteria are capable of much higher swimming speeds (up to a few hundred mm s−1) and
high-performance chemotaxis (6, 7, 18). For
chemotactic velocities of VC = 20 to 60 mm s−1,
associated with swimming speeds of VS = 60 to
170 mm s−1 (16), the motility benefit can be much
larger, with motile cells instantaneously consuming
up to 58 to 133% more than nonmotile cells and
experiencing a time-averaged benefit of 1.1 to 2.3
additional new cells per day (Fig. 3, A and B).
Motility can be costly for marine bacteria.
The motility benefit grows approximately linearly with chemotactic velocity (Fig. 3B), whereas
propulsive power increases quadratically with the
swimming speed (16). This suggests a trade-off
between enhanced uptake and swimming cost,
and the existence of an optimal chemotactic velocity. To test this prediction, we computed the
Fig. 3. Trade-offs of chemotactic foraging. (A) The instantaneous motility
benefit as a function of time since release of the nutrient patch, for different
turbulence intensities D and chemotactic velocities VC. (B) The motility benefit,
shown for three carbon injection rates Ċinj (solid lines), increases with the
chemotactic velocity VC, but the cost of swimming (dashed red line) increases
more rapidly (quadratically) with VC. The trade-off between motility benefit
and swimming cost results in an optimal chemotactic velocity (dotted lines) of
VC ≈ 15 to 25 mm s−1. (C) The trade-off between stirring and mixing results in
an optimal value of turbulence (dotted lines) that depends on the initial patch
size, s. For large patches, the motility benefit is optimal in moderate tur-
bulence (orange line), whereas smaller patches lead to an optimum in weak
turbulence (blue line). Values of the motility benefit in the absence of flow
(D = 0) are connected with dashed lines (values of D < 7.7 × 10−10 W kg−1 were
not considered owing to computational restrictions).